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From art to food, make some space in your diary for some unmissable exhibitions and get to know the chef on everyone's lips, Skye Gyngell.
Erik Madigan Heck, the Crème de la Crème of Fashion Photography.
When he’s not busy on haute couture fashion shoots or photographing Adele for TIME Magazine, Erik Madigan Heck is finding time to write and release his fourth book, Old Future. A favourite amongst renowned names in fashion, including Erdem and Comme des Garçons, Madigan Heck is showing no signs of slowing down as he is at the top of his game in the world of contemporary fashion photography.
Recently released in March 2017, Old Future showcases striking imagery of the photographer’s best work across his blossoming career. Madigan Heck reveals that it was quite a laborious effort to compile the pictures for this book as he narrowed down the final 100 images from 1,000 photographs!
In order to get into the zone on set, it is essential that this photographic legend listens to classical music and uses natural light to keep calm and let the magic happen.
In keeping with London’s Food Month, it is only right to put a spotlight on one of London’s most applauded chefs, Skye Gyngell. This Australian-born chef extraordinaire has already had a flourishing career and continues to exceed herself. Gyngell’s successes so far include being head chef at highly-praised Petersham Nurseries in Richmond, publishing three very successful books and opening an exquisite restaurant, Spring, in Somerset House.
Brace yourself for the evocative artwork by master sculptor, Alberto Giacometti, at Tate Modern this summer. Tate Modern is proud to be showcasing the work of one of the world’s most noteworthy artists in Surrealism, presenting over 250 masterpieces, some of which have never been unveiled to the public before.
Born in 1901 in the blissful Swiss Alps, Giacometti decided to pursue his artistic passion and move to Paris at the tender age of 21. He was won over by the artistic movements of Surrealism and Cubism and eventually joined the likes of Picasso as a renowned global artist. | english |
મફત આપવાના ચક્કરમાં ક્યાંક ભારતની હાલત પણ શ્રીલંકા જેવી ન થઇ જાયઃ SC ચૂંટણી સમયે જનતાને મોટા અને આકર્ષક વચનો આપવા પર સુપ્રીમ કોર્ટે કડક શબ્દોમાં કહ્યું કે, મફતની લ્હાણી કરવાના ચક્કરમાં ભારતમાં શ્રીલંકા જેવી સ્થિતિ ન હોવી જોઈએ. સુપ્રીમ કોર્ટમાં એક જનહિતની અરજીની સુનાવણી દરમિયાન શ્રીલંકામાં મફત વસ્તુઓ આપવાનું ઉદાહરણ આપતા કહ્યું કે, ત્યાં દરેક વસ્તુ મફતમાં વહેંચવાને કારણે આવી સ્થિતિ આવી છે અને ભારત પણ તે જ રસ્તે જઈ રહ્યું છે. વરિષ્ઠ વકીલ અશ્વની ઉપાધ્યાય દ્વારા દાખલ કરવામાં આવેલી અરજી પર સુપ્રીમ કોર્ટની ટિપ્પણી આવી છે. વકીલ અશ્વની ઉપાધ્યાયે સુપ્રીમ કોર્ટમાં અરજી દાખલ કરીને માગણી કરી છે કે, રાજકીય પક્ષોએ ચૂંટણી દરમિયાન આવા વચનો આપવાનું બંધ કરવું જોઈએ જેમાં ચૂંટણી જીત્યા પછી જનતાને મફત સુવિધાઓ અથવા વસ્તુઓનું વિતરણ કરવાનું કહેવામાં આવે છે. ચીફ જસ્ટિસ એનવી રમન્નાએ કહ્યું કે આ ખૂબ જ ગંભીર મુદ્દો છે, આ મતદારને લાંચ આપવા જેવું છે. જ્યારે મુખ્ય ન્યાયાધીશે કેન્દ્ર સરકારના વકીલ કેએમ નટરાજ પાસેથી તેમનો અભિપ્રાય માંગ્યો તો તેમણે કહ્યું કે તે ચૂંટણી પંચે નક્કી કરવાનું છે, આમાં કેન્દ્ર સરકારની કોઈ દખલગીરી નથી. તેના પર CJIએ નારાજગી વ્યક્ત કરતા કહ્યું કે, કેન્દ્ર સરકાર પોતાને તેનાથી અલગ કરી શકે નહીં. કોર્ટે ફરી કેન્દ્ર સરકારને એફિડેવિટ દાખલ કરવા અને પોતાનું સ્ટેન્ડ સમજાવવા કહ્યું છે. જસ્ટિસ રમન્નાએ કોર્ટમાં હાજર વકીલ અને પૂર્વ મંત્રી કપિલ સિબ્બલને કહ્યું કે, તેઓ પણ પોતાના અનુભવથી આ મામલે પોતાનો અભિપ્રાય આપી શકે છે. સિબ્બલે કોર્ટને કહ્યું કે, આમાં કેન્દ્ર સરકારની બહુ ભૂમિકા નથી, નાણાપંચે આ કામ જોવું જોઈએ. સિબ્બલના મતે નાણાપંચ એક નિષ્પક્ષ એજન્સી છે જે રાજ્યોને ભંડોળ પૂરું પાડે છે. આવી સ્થિતિમાં, રાજ્ય સરકારોને ભંડોળ આપતા પહેલા, નાણાં પંચ કહી શકે છે કે તમને મફત સુવિધાઓ આપવા માટે ભંડોળ ફાળવવામાં આવશે નહીં. સિબ્બલે કહ્યું કે, સીધી સરકારો પર તેને નિયંત્રિત કરવાની જવાબદારી નાંખવાથી ઉકેલ નહીં આવે. એ પછી CJIએ આ કેસની સુનાવણી 3 ઓગસ્ટ પર ટાળી છે. કોર્ટે કહ્યું કે, કેન્દ્ર સરકાર આ દરમિયાન એ બતાવે કે આની પર નાણા પંચ શું કરી શકે? સુપ્રીમ કોર્ટે કેન્દ્રને નાણાપંચ પાસેથી એ જાણવા માટે કહ્યું કે, શું પહેલાથી જ દેવામાં ડૂબેલા રાજ્યમાં મફત યોજનાઓનો અમલ અટકાવી શકાય છે? અરજદાર અશ્વિની ઉપાધ્યાયે કહ્યું કે દરેક રાજ્ય પર લાખોકરોડોનું દેવું છે, પંજાબની જેમ ઉત્તર પ્રદેશ પર પણ દેવું છે. આવી સ્થિતિમાં જો સરકાર મફત સુવિધાઓ આપે તો આ દેવું વધુ વધી જશે. અશ્વિની ઉપાધ્યાયે કહ્યું કે, શ્રીલંકામાં પણ એવી જ રીતે દેશની અર્થવ્યવસ્થા કથળી છે. | gujurati |
શંખેશ્વરના ખીજડીયારીમાં લગ્નની ખુશી માતમમાં ફેરવાઈ, ભત્રીજાની જાન ઉપડવાના થોડા સમય પહેલા જ કાકાનું મોત વઢિયાર પંથકના રણના કાંધે અડીને આવેલા ખીજડીયારી ગામે પાકા મકાનની દીવાલ ધરાશાયી થઈ હતી. જેથી દંપતી દીવાલ નીચે દટાયા હતા. જેમાં પતિનું મોત થયું હતું. ભત્રીજાની જાન ઉપડવાના 2 કલાક પહેલા જ દુર્ધટનામાં કાકાનું મોત થયું હતું. પોલીસે અકસ્માતે મોતનો ગુનો નોંધી વધુ તપાસ હાથ ધરી છે. શંખેશ્વર તાલુકાના ખીજડીયારી ગામે રહેતા કાન્તિજી બાજુજી અને તેમના પત્ની ગૌરીબેન બીમાર ગાયની દેખરેખ રાખવા તબેલામાં સૂતા હતાં. શનિવારે વહેલી પરોઢે બાજુમાં આવેલા ખોડાજી વજુજીના પાકા મકાનની દીવાલ ધરાશાયી થતાં દંપતી દટાયા હતા. જોકે, સવારે વહેલા કાન્તિજીના ભત્રીજાની જાન જવાની હોવાથી લોકો જાગતા હોઈ તરત બન્નેને બહાર કાઢી શંખેશ્વર સામુહિક આરોગ્ય કેન્દ્રમાં લવાયા હતા. જ્યાં હાજર ડોક્ટરે કાન્તિજીને મૃત જાહેર કર્યા હતા અને તેમના પત્નીને વધુ સારવાર અર્થે પાટણના ધારપુર રીફર કરવામાં આવ્યા હતા. નોંધનીય છે કે, સવારે પાંચ વાગ્યે મૃતકના ભત્રીજાની જાન જવાની હતી. ત્યારે આ ઘટના બનતાં લગ્નની ખુશી માતમમાં ફેરવાઈ હતી. આ અંગે શંખેશ્વર પોલીસ મથકે અકસ્માતે ગુનો દાખલ કરવામાં આવ્યો હતો. જેની તપાસ પી. એસ. આઈ. સોલંકી ચલાવી રહ્યાં છે. | gujurati |
ਇੱਕ ਜੇ. ਐੱਸ. ਓ. ਐੱਨ. ਫੀਲਡ ਕਿਸ ਕਿਸਮ ਦੀ ਹੋ ਸਕਦੀ ਹੈ?'] ", 'ਇੱਕ ਜੇ. ਐੱਸ. ਓ. ਐੱਨ. ਫੀਲਡ ਕਿਹੋ ਜਿਹੀਆਂ ਕਿਸਮਾਂ ਦੀ ਹੋ ਸਕਦੀ ਹੈ? \n '] | punjabi |
function get_sets()
----------------------------------------------------------------------
-- Bind the keys you wish to use with GearSwap
----------------------------------------------------------------------
send_command('bind f9 gs c toggle idle set')
----------------------------------------------------------------------
-- Idle set
----------------------------------------------------------------------
sets.idle = {
ammo="Ginsen",
head="Malignance Chapeau",
body="Hiza. Haramaki +2",
hands={ name="Adhemar Wristbands", augments={'DEX+10','AGI+10','Accuracy+15',}},
legs={ name="Samnuha Tights", augments={'STR+10','DEX+10','"Dbl.Atk."+3','"Triple Atk."+3',}},
feet="Malignance Boots",
neck="Loricate Torque +1",
waist="Grunfeld Rope",
left_ear="Odnowa Earring",
right_ear="Odnowa Earring +1",
left_ring="Fortified Ring",
right_ring="Defending Ring",
back="Moonbeam Cape",
} -- end Idle
----------------------------------------------------------------------
-- Melee set
----------------------------------------------------------------------
sets.melee = {
ammo="Ginsen",
head={ name="Adhemar Bonnet", augments={'DEX+10','AGI+10','Accuracy+15',}},
body={ name="Adhemar Jacket", augments={'DEX+10','AGI+10','Accuracy+15',}},
hands={ name="Adhemar Wristbands", augments={'DEX+10','AGI+10','Accuracy+15',}},
legs={ name="Samnuha Tights", augments={'STR+10','DEX+10','"Dbl.Atk."+3','"Triple Atk."+3',}},
feet={ name="Herculean Boots", augments={'Accuracy+24 Attack+24','"Triple Atk."+2','DEX+6','Accuracy+7','Attack+14',}},
neck="Moonlight Nodowa",
waist="Windbuffet Belt +1",
left_ear="Telos Earring",
right_ear="Dedition Earring",
left_ring="Epona's Ring",
right_ring="Gere Ring",
back={ name="Andartia's Mantle", augments={'DEX+20','Accuracy+20 Attack+20','Accuracy+10','"Store TP"+10','Damage taken-5%',}},
} -- end Melee
----------------------------------------------------------------------
-- Precast sets
----------------------------------------------------------------------
-- Initialize an array to begin storing set data
sets.precast = {}
sets.precast.fastCast = {}
-- Precast : Fast Cast (61%)
sets.precast.fastCast.default = {
ammo="Sapience Orb",
head={ name="Herculean Helm", augments={'Mag. Acc.+4','"Fast Cast"+6','MND+5',}},
body={ name="Herculean Vest", augments={'Mag. Acc.+9','"Fast Cast"+5','"Mag.Atk.Bns."+7',}},
hands={ name="Leyline Gloves", augments={'Accuracy+15','Mag. Acc.+15','"Mag.Atk.Bns."+15','"Fast Cast"+3',}},
legs="Arjuna Breeches",
feet={ name="Herculean Boots", augments={'"Fast Cast"+5','INT+13','"Mag.Atk.Bns."+13',}},
neck="Orunmila's Torque",
left_ear="Loquac. Earring",
right_ear="Etiolation Earring",
left_ring="Kishar Ring",
right_ring="Prolix Ring",
back={ name="Andartia's Mantle", augments={'"Fast Cast"+10',}},
} -- end Fast Cast
-- Precast : Utsusemi Fast Cast
sets.precast.fastCast.utsusemi = set_combine(sets.precast.fastCast.default, {
neck="Magoraga Beads",
})
----------------------------------------------------------------------
-- Midcast sets
----------------------------------------------------------------------
-- Initialize an array to begin storing set data
sets.midcast = {}
sets.midcast.ws = {}
-- Midcast : Magic
sets.midcast.magic = {
ammo="Impatiens",
} -- end Magic
-- Midcast : Weapon Skill : Default
sets.midcast.ws.default = {
ammo="Expeditious Pinion",
head={ name="Dampening Tam", augments={'DEX+10','Accuracy+15','Mag. Acc.+15','Quadruple Attack +3',}},
body={ name="Adhemar Jacket", augments={'DEX+10','AGI+10','Accuracy+15',}},
hands={ name="Adhemar Wristbands", augments={'DEX+10','AGI+10','Accuracy+15',}},
legs={ name="Samnuha Tights", augments={'STR+10','DEX+10','"Dbl.Atk."+3','"Triple Atk."+3',}},
feet={ name="Herculean Boots", augments={'Accuracy+24 Attack+24','"Triple Atk."+2','DEX+6','Accuracy+7','Attack+14',}},
neck="Fotia Gorget",
waist="Fotia Belt",
left_ear="Telos Earring",
right_ear={ name="Moonshade Earring", augments={'Accuracy+4','TP Bonus +250',}},
left_ring="Epona's Ring",
right_ring="Gere Ring",
back={ name="Andartia's Mantle", augments={'AGI+20','Accuracy+20 Attack+20','AGI+10','Weapon skill damage +10%',}},
} -- end Weapon Skill Default
-- Midcast : Weapon Skill : Blade: Hi
sets.midcast.ws["Blade: Hi"] = {
ammo="Expeditious Pinion",
head="Mummu Bonnet +2",
body={ name="Adhemar Jacket", augments={'DEX+10','AGI+10','Accuracy+15',}},
hands="Mummu Wrists +2",
legs="Mummu Kecks +2",
feet="Mummu Gamash. +2",
neck="Moonlight Nodowa",
waist="Windbuffet Belt +1",
left_ear="Ishvara Earring",
right_ear="Brutal Earring",
left_ring="Mummu Ring",
right_ring="Regal Ring",
back={ name="Andartia's Mantle", augments={'AGI+20','Accuracy+20 Attack+20','AGI+10','Weapon skill damage +10%',}},
} -- end Blade: Hi
-- Midcast : Weapon Skill : Blade: Ku
sets.midcast.ws["Blade: Ku"] = {
ammo="Expeditious Pinion",
head={ name="Adhemar Bonnet", augments={'DEX+10','AGI+10','Accuracy+15',}},
body={ name="Adhemar Jacket", augments={'DEX+10','AGI+10','Accuracy+15',}},
hands={ name="Adhemar Wristbands", augments={'DEX+10','AGI+10','Accuracy+15',}},
legs={ name="Samnuha Tights", augments={'STR+10','DEX+10','"Dbl.Atk."+3','"Triple Atk."+3',}},
feet={ name="Herculean Boots", augments={'Accuracy+24 Attack+24','"Triple Atk."+2','DEX+6','Accuracy+7','Attack+14',}},
neck="Caro Necklace",
waist="Fotia Belt",
left_ear="Lugra Earring",
right_ear="Lugra Earring +1",
left_ring="Regal Ring",
right_ring="Gere Ring",
back={ name="Andartia's Mantle", augments={'DEX+20','Accuracy+20 Attack+20','Accuracy+10','"Store TP"+10',}},
} -- end Blade: Ku
-- Midcast : Weapon Skill : Blade: Shun
sets.midcast.ws["Blade: Shun"] = {
ammo="Expeditious Pinion",
head={ name="Adhemar Bonnet", augments={'DEX+10','AGI+10','Accuracy+15',}},
body={ name="Adhemar Jacket", augments={'DEX+10','AGI+10','Accuracy+15',}},
hands={ name="Adhemar Wristbands", augments={'DEX+10','AGI+10','Accuracy+15',}},
legs={ name="Samnuha Tights", augments={'STR+10','DEX+10','"Dbl.Atk."+3','"Triple Atk."+3',}},
feet={ name="Herculean Boots", augments={'Accuracy+24 Attack+24','"Triple Atk."+2','DEX+6','Accuracy+7','Attack+14',}},
neck="Caro Necklace",
waist="Fotia Belt",
left_ear="Lugra Earring",
right_ear="Lugra Earring +1",
left_ring="Regal Ring",
right_ring="Gere Ring",
back={ name="Andartia's Mantle", augments={'DEX+20','Accuracy+20 Attack+20','Accuracy+10','"Store TP"+10',}},
} -- end Blade: Shun
----------------------------------------------------------------------
-- Utility Sets (not bound to a key)
----------------------------------------------------------------------
-- Initialize an array to begin storing set data
sets.utility = {}
-- Futae set
sets.utility.futae = {
} -- end Futae
-- Innin set
sets.utility.innin = {
} -- end Innin
-- Migawari set
sets.utility.migawari = {
} -- end Migawari
-- Mikage set
sets.utility.mikage = {
back={ name="Andartia's Mantle", augments={'DEX+20','Accuracy+20 Attack+20','Accuracy+10','"Store TP"+10',}},
} -- end Mikage
-- Utsusemi set
sets.utility.utsusemi = {
feet="Iga Kyahan +2",
back={ name="Andartia's Mantle", augments={'DEX+20','Accuracy+20 Attack+20','Accuracy+10','"Store TP"+10',}},
} -- end Utsusemi Fast Cast
-- Utsusemi set
sets.utility.utsusemiFastCast = set_combine(sets.precast.fastCast, {
neck="Magoraga Beads",
})
-- Yonin set
sets.utility.yonin = {
} -- end Yonin
----------------------------------------------------------------------
-- Spell arrays
----------------------------------------------------------------------
MigawariSpells = {
["Migawari: Ichi"] = true,
}
UtsusemiSpells = {
["Utsusemi: Ichi"] = true,
["Utsusemi: Ni"] = true,
["Utsusemi: San"] = true,
}
end -- end get_sets()
----------------------------------------------------------------------
-- Callback for when casting begins
----------------------------------------------------------------------
function precast(spell)
-- Check if the action is a magic spell
if spell.action_type == 'Magic' then
if UtsusemiSpells[spell.english] then
equip(sets.precast.fastCast.utsusemi)
else
equip(sets.precast.fastCast.default)
end
end
end -- end precast()
----------------------------------------------------------------------
-- Callback for after casting begins, but before it fires
----------------------------------------------------------------------
function midcast(spell)
-- Check if the action is a weapon skill
if spell.type == 'WeaponSkill' then
if sets.midcast.ws[spell.english] then
equip(sets.midcast.ws[spell.english])
else
equip(sets.midcast.ws.default)
end
-- Check if the action is a magic spell
elseif spell.action_type == 'Magic' then
if MigawariSpells[spell.english] then
equip(sets.utility.migawari)
elseif UtsusemiSpells[spell.english] then
equip(sets.utility.utsusemi)
else
equip(sets.midcast.magic)
end
-- Check if the action is a job ability
elseif spell.type == 'JobAbility' then
if spell.english == 'Futae' then
equip(sets.utility.futae)
elseif spell.english == 'Innin' then
equip(sets.utility.innin)
elseif spell.english == 'Mikage' then
equip(sets.utility.mikage)
elseif spell.english == 'Yonin' then
equip(sets.utility.yonin)
end
end
end -- end midcast()
----------------------------------------------------------------------
-- Callback for after casting has fired
----------------------------------------------------------------------
function aftercast(spell)
if player.status =='Engaged' then
equip(sets.melee)
else
equip(sets.idle)
end
end -- end aftercast()
----------------------------------------------------------------------
-- Callback for whenever engagment status changes
----------------------------------------------------------------------
function status_change(new,old)
if new =='Engaged' then
equip(sets.melee)
else
equip(sets.idle)
end
end -- end status_change()
----------------------------------------------------------------------
-- In game alerts to gear set changes
----------------------------------------------------------------------
function self_command(command)
-- Equip the idle set
if command == 'toggle idle set' then
send_command('@input /echo <----- Idle Set Equipped ----->')
equip(sets.idle)
end -- end if
end -- end self_command()
----------------------------------------------------------------------
-- Event Listener
----------------------------------------------------------------------
-- Callback for when the job is changed
----------------------------------------------------------------------
isInitialChange = true
function job_change(mainId, mainLvl, subId, subLvl)
equip(sets.idle)
if isInitialChange then
coroutine.schedule(function() send_command('input /macro book 7;wait .5;input /macro set 1;input /lockstyleset 2') end, 10)
isInitialChange = false
end
end -- end job_change()
windower.register_event('job change', job_change)
----------------------------------------------------------------------
-- Event Listener
----------------------------------------------------------------------
-- Callback for when entering a zone
----------------------------------------------------------------------
windower.register_event('zone change', function() equip(sets.idle) end) | code |
Apply Offline For WCDD Posts डब्ल्यूसीडीडी पदों के लिए करें ऑफलाइन आवेदन Apply Offline For WCDD Posts इंडिया न्यूज। Apply Offline For WCDD Posts महिला और बाल विकास विभाग डब्ल्यूसीडीडी ने सदस्य भर्ती 2022 के 6 पदों के लिए ऑफलाइन आवेदन मांगें है । इन 6 पदों मे से 2 पद महिलाओं के लिए आरक्षित है । वहीं आवेदकों के लिए आवेदन प्रक्रिया 16 फरवरी से शुरु हो गई है जोकि 25 फरवरी तक चलेगी आमंत्रित किए हैं। वे उम्मीदवार जो डब्ल्यूसीडीडी हरियाणा रिक्ति 2022 विवरण में रुचि रखते हैं और सभी पात्रता मानदंड को पूरा करते हैं, वे अधिसूचना पढ़ कर ही आवेदन करें । महिला एवं बाल विकास विभाग का संगठनडब्ल्यूसीडीडी हरियाणा Apply Offline For WCDD Posts पद का नाम सदस्य पद कुल पद 06 पद 02 पद महिलाओं के लिए आरक्षित डब्ल्यूसीडीडी पोस्ट आयु सीमा अधिकतम: 60 वर्ष योग्यता विवरण उत्कृष्टता, ईमानदार, सक्षम और अनुभवी व्यक्ति कार्यकाल 3 वर्ष कार्यस्थल:हरियाणा आवेदन संबंधित महत्वपूर्ण तिथियां ऑफलाइन आवेदन पत्र जमा करने की प्रारंभिक तिथि: 16022022 ऑफलाइन आवेदन पत्र जमा करने की अंतिम तिथि: 25022022 हरियाणा डब्ल्यूसीडीडी सदस्य आवेदन पत्र 2022 कैसे लागू करें आधिकारिक अधिसूचना से पात्रता की जांच करें और योग्य उम्मीदवार डब्ल्यूसीडीडी हरियाणा भर्ती 2022 के लिए आवेदन कर सकते हैं। आवेदन पत्र भरें: अपना मूल विवरण जैसे नाम,संपर्क नंबर,ईमेल आईडी आदि प्रदान करके डब्ल्यूसीडीडी रिक्ति 2022 के लिए अपनी आवेदन प्रक्रिया शुरू करें। आवेदन संबंधित मेल के माध्यम से किया जाएगा । Apply Offline For WCDD Posts MORE :Know How You Will Save By Registering in CET जाने कैसे होगी सीइटी में पंजीकरण से आपकी बचत Connect With Us: Twitter Facebook The post Apply Offline For WCDD Posts डब्ल्यूसीडीडी पदों के लिए करें ऑफलाइन आवेदन appeared first on India News. | hindi |
पीएमसीएच का कर्मी बन बैंक से मांगा 2.4 करोड़ कर्ज पटना। जालसाज अब फर्जी कागजातों पर भारी भरकम कर्ज लेकर बैंकों को चपत लगाने की जुगत लगा रहे हैं। मंगलवार को प्रदेश के सबसे बड़े व प्रतिष्ठित मेडिकल कालेज पीएमसीएच में ऐसा ही मामला सामने आया। चार लोगों ने खुद को यहां का कर्मचारी बताकर एक निजी फाइनेंस कंपनी से 6060 लाख रुपये कर्ज का आवेदन दिया था। हालांकि, एक शख्स के नाम में कुमार व रंजन का अंतर होने पर बैंककर्मी उसकी जानकारी लेने प्राचार्य कार्यालय पहुंच गया। प्राचार्य डा. विद्यापति चौधरी ने सिर्फ एक व्यक्ति के नेत्र रोग विभाग में चतुर्थ श्रेणी कर्मी होने की बात कही। इसके बाद जांच को आए बैंककर्मी के होश उड़ गए। वह प्राचार्य से कार्रवाई की मांग कर रहा था। प्राचार्य ने इसे बैंक का मामला बताते हुए उन्हें थाने में जाकर प्राथमिकी कराने को कहा है। अटेंडेंस रजिस्टर में भी थे हस्ताक्षर बैंककर्मी ने प्राचार्य कार्यालय जाकर नेत्र रोग विभाग के तीन और एनाटामी विभाग के एक कर्मचारी की बाबत पूछताछ की। उसने बताया कि खुद को लैब टेक्नीशियन बताकर अपनी पैथोलाजी खोलने के लिए 6060 लाख रुपये कर्ज का आवेदन दिया था। नेत्र रोग विभाग के कर्मी के नाम और अन्य दस्तावेज में रंजन और कुमार का अंतर मिलने के बाद बैंक ने कर्मचारी को वेरिफिकेशन के लिए भेजा था। प्राचार्य ने जब तीन को कर्मचारी नहीं माना तो उसने वेतन पर्ची, आइ कार्ड की छायाप्रति दिखाई। इस पर जब कर्मचारियों की उपस्थिति का रजिस्ट्रर चेक किया गया तो उसमें भी उनके नाम और उसके आगे उपस्थिति दर्ज पाई गई। इसके बाद पीएमसीएच में हड़कंप मच गया कि जो कर्मचारी नहीं हैं, उनकी उपस्थिति कैसे दर्ज हो गई। प्राचार्य ने मामले की जांच करने की बात कही है। | hindi |
మలకపల్లి కేసులో దోషులను వదిలిపెట్టం ఎస్సి కమిషన్ సభ్యులు ఆనంద్ప్రకాష్ ప్రజాశక్తి తాళ్లపూడి మలకపల్లిలో ఈ నెల మొదటి వారంలో జరిగిన దళిత యువకుడు గెడ్డెం శ్రీను అనుమానాస్పద మృతికి సంబంధించి పూర్తి వివరాలు సేకరించి దోషులు ఎంతటి వారైనా వదిలిపెట్టబోమని ఎస్సి కమిషన్ సభ్యులు చెల్లెం ఆనంద్ప్రకాష్ అన్నారు. ఆయన మంగళవారం మలకపల్లి గ్రామానికి వచ్చారు. శ్రీను మరణించిన ప్రదేశాన్ని పరిశీలించారు. మతుని కుటుంబ సభ్యులను, స్థానికులను కలిసి వివరాలు అడిగి తెలుసుకున్నారు. ఈ కేసుకు సంబంధించి వాస్తవాలు వెలికితీసేందుకు అవసరమైతే పోలీసులను పక్కనపెట్టి వేరే మార్గాల్లో విచారణ సాగిస్తామన్నారు. తమ పరిశీలన అనంతరం పూర్తి విషయాలను జిల్లా కలెక్టర్, ఎస్పిల దృష్టిలో పెట్టి శ్రీనివాసరావుది ఆత్మహత్యా, హాత్యా అనేది తేలుస్తామన్నారు. కొవ్వూరు రూరల్: చట్టాన్ని ఎవ్వరూ చేతుల్లోకి తీసుకోరాదని, అటువంటి వారిని ఉపేక్షించబోమని రాష్ట్ర ఎస్సి కమిషన్ సభ్యులు చెల్లెం ఆనంద్ ప్రకాష్ స్పష్టం చేశారు. స్థానిక ఆర్అండ్బి అతిధిగృహంలో ఏర్పాటు చేసిన విలేకరుల సమావేశంలో ఆయన మాట్లాడారు. గెడ్డం శ్రీను మృతి ఘటన సమాజానికి మంచిది కాదన్నారు. శ్రీను మతి ఘటనలో విఆర్ఒ తప్పుడు నివేదిక ఇచ్చారని, సంబంధిత తహశీల్దార్, పోలీసులు సరిగ్గా స్పందించలేదని, దీనిపై విచారణ జరపాలని ఆర్డిఒకు, డిఎస్పికి సూచించామన్నారు. దీనిపై కొవ్వూరు ఆర్డిఒ ఎస్.మల్లిబాబు మాట్లాడుతూ శ్రీను మృతి ఘటనలో లోపాలు జరిగినట్లు గుర్తించామని, విఆర్కు ఛార్జిమెమో ఇస్తున్నామని, తహశీల్దార్పై చర్యలకు కలెక్టర్కు నివేదిక సమర్పిస్తున్నట్లు తెలిపారు. డిఎస్పి బి.శ్రీనాధ్ మాట్లాడుతూ ఎస్ఐపై ఆరోపణలు వచ్చాయని, శాఖపరమైన చర్యలకు, ఇన్ఛార్జి ఆఫీసర్గా వేరేవారిని నియమించాలని ఎస్పికి నివేదిస్తానని తెలిపారు. | telegu |
Tiger 3: नोएडा में होगी फिल्म के आखिरी चरण की शूटिंग, सलमान और कैटरीना के साथ इमरान हाशमी भी दिखेंगे यशराज फिल्म्स के बैनर तले बन रही सलमान खान और कैटरीना कैफ स्टारर फिल्म टाइगर 3 की शूटिंग अपने आखिरी दौर में है। इस फिल्म के कुछ सीन दिल्ली एनसीआर के अलगअलग लोकेशन पर फिल्माए जाने हैं। इसके साथ ही नोएडा स्थित एक निजी यूनिवर्सिटी में भी टाइगर 3 की शूटिंग होगी। फिल्म की शूटिंग के लिए नोएडा में लगभग सभी तैयारियां पूरी हो चुकी हैं। रविवार को दिनभर शूटिंग की तैयारियां की गईं। फिल्म के इस शेड्यूल में सारे प्रमुख कलाकार हिस्सा लेने वाले हैं। सूत्रों से मिली जानकारी के मुताबिक, टाइगर 3 के कुछ महत्वपूर्ण दृश्य सोमवार को फिल्माए जाएंगे। बताया जा रहा है कि फिल्म की शूटिंग से पहले सभी कलाकारों के साथ स्टाफ का कोविड टेस्ट कराया जा चुका है। वहीं जिनका टेस्ट नहीं हुआ है उनका एंटीजन टेस्ट शूटिंग लोकेशन पर ही किया जाएगा। सेट पर बहुत ज्यादा भीड़ ना हो इसके लिए शूटिंग लोकेशन के बारे में जानकारी नहीं दी गई है। आपको बता दें यशराज फिल्मस स्टूडियो में फरवरी की शुरुआत में फिल्म टाइगर 3 की शूटिंग करीब हफ्ते भर चली और इसके बाद शनिवार और इतवार यानी 12 या 13 फरवरी को सलमान खान और कैटरीना कैफ दिल्ली के लिए रवाना हो गए। खबर है कि वैलेंटाइन डे से सलमान खान दिल्ली एनसीआर में फिल्म टाइगर 3 की शूटिंग करने वाले हैं। दिल्ली में फिल्म की शूटिंग करीब 2 हफ्ते तक चलेगी और इसी के साथ ही इस फिल्म की शूटिंग पूरी हो जाएगी। मेगा बजट इस फिल्म को हिंदी के साथ अंग्रेजी के अलावा कई दूसरी भाषाओं में भी रिलीज किया जाएगा। | hindi |
ડોન કા ઇન્તેજાર... જાણો કોણ છે કુખ્યાત ગેંગસ્ટર? જેની ગુજરાત ATS એ કરી ધરપકડ ઉદય રંજન, અમદાવાદ: ભારતના ગુજરાત, રાજસ્થાન, કેરળ સહિતના અનેક રાજ્યોમાં હત્યા, લૂંટ, ધાડ, ખંડણી સહિત અનેક ગુનાઓ આચરેલ રાજસ્થાનના ગેંગસ્ટરની ગુજરાત ATS એ ધરપકડ કરી. કોણ છે આ ગેંગ અને આ ગેંગસ્ટર? શું છે ગેંગસ્ટરની ક્રાઇમ કુંડળી? આવો જોઈએ અહેવાલમાં... ગુજરાત ATS ની ટીમને માહિતી મળી હતી કે, રાજસ્થાનનો ગેંગસ્ટર અમદાવાદના હીરાવાડી વિસ્તારમાં આવનાર છે. જે બાતમીના આધારે ગુજરાત ATS ની ટીમે છટકું ગોઠવી રાજસ્થાનના કુખ્યાત ગેંગસ્ટર અરવિંદસિંહ શેતાનસિંહ બિકાને ઝડપી લીધો હતો. ગેંગસ્ટર અરવિંદસિંહ બીકા કઈક કરે એ પહેલા જ ગુજરાત ATS ની ટીમે ઘૂંટણિયે પાડી દીધો હતો. પ્રેમીના આપઘાત બાદ મૃતકના પિતાએ યુવતીને કહ્યું તું રખેલ બનીને રે, જાણો શું છે સમગ્ર મામલો રાજસ્થાનનો આ કુખ્યાત ગુનેગાર એક નહિ બે નહિ પરંતુ 35 જેટલા ગુનાઓમાં સંડોવાયેલો છે. જેમાં મુખ્ય હત્યા, હત્યાનો પ્રયાસ, લૂંટ, ધાડ, ચોરી, ખંડણી, જેલ તોડીને ભાગી જવું, પોલીસ જાપ્તામાંથી ફરાર થવું, પોલીસ જાપ્તા પર ફાયરિંગ કરવું, પોલીસ જાપ્તા પર ફાયરિંગ કરી પોતાના સાગરીતને ભગાવવા જેવા 35 ગંભીર ગુનાઓ આચરેલ છે. અમદાવાદમાં વધુ એક આયશા: સાબરમતીમાં આપઘાત કરે તે પહેલા બચાવી, જાણો શું છે સમગ્ર મામલો ગુજરાત ATS ની ટીમે આ ગેંગસ્ટરને ઝડપ્યો ત્યારે ઘાતક હથિયારો પણ મળી આવ્યા હતા. જેમાં બે પિસ્તોલ, એક દેશી તમંચો અને પાંચ જીવતા કારતૂસ મળી આવતા ATS એ અરવિંદસિંહ વિરુદ્ધ આર્મ્સ એક્ટ હેઠળ ગુનો નોંધી ધરપકડ કરી વધુ પૂછપરછ હાથ ધરી છે. વરસાદે તો ભારે કરી: અમદાવાદ સહિત આ જિલ્લામાં આગામી 3 કલાકમાં તૂટી પડશે વરસાદ જાણો ગેંગસ્ટર અરવિંદસિંહ બિકાની ક્રાઈમ કુંડળી 1 વર્ષ 2016 માં રાજસ્થાનમાં પોલીસ પર ફાયરિંગ કરી પોતાના સાગરીતને ભગાડ્યો હતો. 2 રાજસ્થાનના સિરોહીના શિવગંજ વિસ્તારમાં યુવક પર ફાયરિંગ કર્યું હતું. 3 2016 માં અમદાવાદના કાલુપુરમાં આંગડિયા પેઢીના કર્મચારી પર ફાયરિંગ કરી લુંટ ચલાવી હતી. 4 2017 માં બનાસકાંઠાના ધાનેરામાં બેંક લૂંટ કરી હતી. 5 2017 માં ડીસા જેલ તોડી ફરાર થયો હતો. 6 2018 માં પ્રાંતિજમાં તમાચો બતાવી લૂંટ ચલાવી. 8 2018 માં પાટણમાં ચાણસ્મામાં આંગડિયા પેઢીમાં કર્મીને હથિયાર બતાવી લૂંટ ચલાવી. 8 બનાસકાંઠામાં પેટ્રોલ પંપ પર લૂંટ ચલાવી હતી. 9 બનાસકાંઠામાં દૂધ મંડળીમાં કર્મી પાસેથી 18 લાખની લૂંટ ચલાવી હતી. એક ગુજરાતી ચલાવતો હતો ચાઈલ્ડ પોર્નોગ્રાફીનું વૈશ્વિક નેટવર્ક, જાણો કઈ રીતે થયો પર્દાફાશ ઝડપાયેલ ગેંગસ્ટર પોતાની ગેંગમાં ૨૦ જેટલા સક્રિય સાગરીતો રાખે છે. ત્યારે ગુજરાતના અમદાવાદમાં આ ગેંગસ્ટરનું શું કામ આવવાનું થયું? કોને મળવાનો હતો? અમદાવાદમાં કોના સંપર્કમાં છે? આ તમામ સવાલોના જવાબ ગુજરાત ATS શોધી રહી છે. | gujurati |
سیدن ووٚنُکھ آ تِکیٛازِ تَس اوس سفر کرنُک تہٕ دُنیا ہَس منٛز کینٛہہ وُچھنُک سخت شوق | kashmiri |
Dlip Ghosh: তৃণমূল থেকেও দ্রৌপদীর সমর্থনে ভোট! ক্রসভোটিংয়ের জল্পনা উস্কে দিলেন দিলীপ কলকাতা: একদিন আগেই বঙ্গ সফর সেরে দিল্লি ফিরে গিয়েছেন দ্রৌপদী মুর্মু Draupadi Murmu তার পরই আসন্ন রাষ্ট্রপতি নির্বাচনে ক্রস ভোটিংয়ের জল্পনা উস্কে দিলেন বিজেপির সর্বভারতীয় সহ সভাপতি দিলীপ ঘোষ Dilip Ghosh তাঁর দাবি, তৃণমূলের অনেকেই বিজেপি নেতৃত্বাধীন এনডিএ প্রার্থী দ্রৌপদীকেই ভোট দেবেন তৃণমূল নেত্রী মমতা বন্দ্যোপাধ্যায় Mamata Banerjee এ নিয়ে রাজনীতি করছেন বলেও অভিযোগ করেন তিনি দিলীপের মন্তব্যে জোর জল্পনা জনজাতি নেত্রী দ্রৌপদীকে রাষ্ট্রপতি পদপ্রার্থী ঘোষণা করেছে এনডিএ পক্ষ অন্য দিকে বিরোধীদের তরফে প্রার্থী করা হয়েছে যশবন্ত সিন্হাকে Yashwant Sinha তা নিয়ে সম্প্রতি মুখ খোলেন মমতা জনজাতি নেত্রীকে রাষ্ট্রপতি পদপ্রার্থী করা হচ্ছে বলে বিজেপির তরফে জানানো হলে তিনি ভেবে দেখবেন বলে মন্তব্য করেন তার পরই থেকেই তৃণমূলের TMC অবস্থান নিয়ে প্রশ্ন তুলতে শুরু করে বিজেপি তাতে আরও ইন্ধন জোগালেন দিলীপ দিলীপের কথায়, একজন আদিবাসী মহিলা এই প্রথম দেশের রাষ্ট্রপতির আসন অলঙ্কৃত করবেন বিদেশের মাটিতে ভারতের হয়ে প্রতিনিধিত্ব করবেন গোটা দুনিয়া দেখছে, কী ভাবে একেবারে শূন্য থেকে উত্থান ঘটছে ভারতের বিবেকানন্দ বলেছিলেন, শূদ্রের উত্থান ঘটলে, তবেই দেশের উত্থান সম্ভব মোদিজি সেই কাজ করছেন মুখ্যমন্ত্রী এখন বলছেন, আগে জানলে ভেবে দেখতেন অথচ আগেই নিজের দলের সহ সভাপতিকে মনোনীত করেছেন কে রাজনীতি করছেন, এ থেকেই বোঝা যায় দিদিমণির কষ্ট করার দরকার নেই ওঁর দলের অনেকেই | bengali |
ಟ್ವಿಟರ್ ನಲ್ಲಿ ಮುಂದುವರಿದ ಸುಮಲತಾ ಟಾಕ್ ವಾರ್ ಬೆಂಗಳೂರು: ಅಕ್ರಮ ಗಣಿಗಾರಿಕೆ ವಿರುದ್ಧ ಸಿಡಿದೆದ್ದಿರುವ ಮಂಡ್ಯ ಸಂಸದೆ ಸುಮಲತಾ ಅಂಬರೀಶ್ ಟ್ವಿಟರ್ ನಲ್ಲಿ ತಮ್ಮ ಟಾಕ್ ವಾರ್ ಮುಂದುವರಿಸಿದ್ದಾರೆ. ಜೆಡಿಎಸ್ ನಾಯಕರ ಜೊತೆ ಮಾತಿನ ಚಕಮಕಿಯಾದ ಬಳಿಕ ಕೊಂಚ ಮಟ್ಟಿಗೆ ಕದನ ವಿರಾಮ ಘೋಷಿಸಿದ್ದ ಎರಡೂ ಕಡೆಯ ನಾಯಕರು ಇದೀಗ ಸಾಮಾಜಿಕ ಜಾಲತಾಣದ ಮೂಲಕ ಮಾತು ಮುಂದುವರಿಸಿದ್ದಾರೆ.ಟ್ವಿಟರ್ ನಲ್ಲಿ ಸೇವ್ ಕೆಆರ್ ಎಸ್, ಸ್ಟಾಪ್ ಇಲ್ಲೀಗಲ್ ಮೈನಿಂಗ್ ಎಂದು ಹ್ಯಾಶ್ ಟ್ಯಾಗ್ ನಡಿಯಲ್ಲಿ ಕಿಡಿ ಕಾರಿರುವ ಸುಮಲತಾ ಬೇರೆಯವರಿಗೆ ಗಂಧ ಹಚ್ಚಲು ಹೊರಟರೆ ಮೊದಲು ನಮ್ಮ ಕೈಗೆ ಗಂಧವಾಗುತ್ತದೆ, ಬೇರೆಯವರಿಗೆ ಕೆಸರು ಹೆಚ್ಚಲು ಹೊರಟರೆ ನಮ್ಮ ಕೈ ಮೊದಲು ಕೆಸರಾಗುತ್ತದೆ ಎಂದು ಪರೋಕ್ಷವಾಗಿ ಎದುರಾಳಿಗಳಿಗೆ ಟಾಂಗ್ ಕೊಟ್ಟಿದ್ದಾರೆ. | kannad |
ਪਾਈਥਨ ਵਿੱਚ ਟਾਈਮ ਸਟੈਂਪ ਕਿਸਮ ਦੇ ਮੁੱਲਾਂ ਤੋਂ ਟਾਈਮ ਕੰਪੋਨੈਂਟ ਨੂੰ ਕਿਵੇਂ ਹਟਾਇਆ ਜਾਵੇ'] ", 'ਪਾਈਥਨ ਵਿੱਚ ਟਾਈਮ ਸਟੈਂਪ ਕਿਸਮ ਦੇ ਮੁੱਲਾਂ ਤੋਂ ਟਾਈਮ ਕੰਪੋਨੈਂਟ ਨੂੰ ਕਿਵੇਂ ਹਟਾਇਆ ਜਾਵੇ \n '] | punjabi |
ப்ளீஸ்.. அவரை மட்டும் கைவிட்ராதிங்க இந்திய அணியின் சீனியர் வீரருக்கு ஆதரவு கொடுத்த முன்னாள் வீரர் !! இந்திய அணியின் முன்னாள் வீரர் பிரவீன் ஆம்ரே இந்திய அணியின் முன்னாள் துணை கேப்டன் அஜிங்கியா ரஹானே குறித்து பத்திரிகையாளர்கள் சந்திப்பின்போது பேசியுள்ளார். தென் ஆப்ரிக்கா சென்றுள்ள இந்திய கிரிக்கெட் அணி, தென் ஆப்ரிக்கா அணியுடன் மூன்று டெஸ்ட் போட்டிகள் கொண்ட தொடரில் பங்கேற்க உள்ளது.இதில் முதலில் நடைபெறும் டெஸ்ட் தொடரின் முதல் போட்டி 26ம் தேதி துவங்க உள்ளது. தென் ஆப்ரிக்கா அணியை அதன் சொந்த மண்ணில் வீழ்த்துவது எளிதான காரியம் அல்ல என்பதால் இந்தியா தென் ஆப்ரிக்கா இடையேயான தொடருக்காக ஒட்டுமொத்த கிரிக்கெட் ரசிகர்களும் மிகுந்த ஆவலுடன் காத்துள்ளனர். அதே போல் ரோஹித் சர்மா, ரவீந்திர ஜடேஜா போன்ற சீனியர் வீரர்கள் இல்லாமல் இந்திய அணி தென் ஆப்ரிக்காவை எப்படி எதிர்கொள்ள போகிறது என்பதை பார்க்கவும் ரசிகர்கள் ஆவலுடன் காத்திருக்கின்றனர். ரசிகர்களை போன்றே இந்த தொடருக்காக மிகுந்த எதிர்பார்ப்புடன் காத்திருக்கும் முன்னாள் வீரர்கள் பலர், இந்த தொடர் குறித்தான தங்களது கருத்துக்களையும், கணிப்புகளையும் ஓபனாக வெளிப்படுத்தி வருகின்றனர். இந்த நிலையில் கடந்த சில போட்டிகளில் மிகவும் மோசமான பார்மை வெளிப்படுத்தி வரும் இந்திய அணியின் அஜிங்கிய ரஹானே குறித்து இந்திய அணியின் முன்னாள் வீரர் பிரவீன் ஆம்ரே தனது கருத்தை தெரிவித்துள்ளார். அதில், எப்படிப்பட்ட சிறந்த பேட்ஸ்மேனாக இருந்தாலும் அனைத்து டெஸ்ட் போட்டிகளிலும் சிறந்த முறையில் பேட்டிங் செய்ய முடியாது, குறிப்பாக இந்தியாவில் நடைபெற்ற இங்கிலாந்துக்கு எதிரான டெஸ்ட் தொடரில் மைதானம் பேட்ஸ்மேன்களுக்கு உகந்ததாக இல்லை இதனால் பெரும்பாலான பேட்ஸ்மேன்கள் அதிகமான ரன்களை அடிக்கவில்லை, ஆனால் நாம் உண்மையான காரணத்தை எல்லாம் புறந்தள்ளிவிட்டு அவர்களை கடுமையாக திட்டுவதில் மட்டும் மும்முரமாக ஈடுபடும். குறிப்பாக இந்திய அணியின் அனுபவ வீரர் அஜிங்கியா ரஹானேவை கடுமையாக விமர்சித்து வருகிறோம், ஆனால் அவர் வெளிநாட்டு மைதானங்களில் மிகச் சிறந்த முறையில் பேட்டிங் செய்து அசத்துவார், குறிப்பாக ஆஸ்திரேலியாவுக்கு எதிரான டெஸ்ட் போட்டியில் இந்திய அணியின் சீனியர் வீரர்கள் இல்லாத நிலையிலும் இந்திய அணியை திறம்பட வழிநடத்தி மமற்றும் பேட்டிங்கில் சிறப்பான ஆட்டத்தை வெளிப்படுத்தி ஆஸ்திரேலிய அணியை அதன் சொந்த மண்ணில் வீழ்த்தி சாதனை படைத்தார். தற்பொழுது தனது பார்மை வெளிப்படுத்துவதில் திணறி வரும் ரஹானே டெஸ்ட் போட்டிகளில் 4795 ரன்கள் அடித்து அசத்தியுள்ளார், இவரை இந்திய அணி புறந்தள்ளி விடக்கூடாது என்று பிரவீன் தெரிவித்திருந்தது குறிப்பிடத்தக்கது. | tamil |
આરોપી પ્રેમિકાએ તેના પ્રેમીનું ગળું દબાવી, લાશ ઢસડીને તેના ઘરની બારી પાસે રાખી દીધી એક પરિણીત પ્રેમિકા Married Girlfriend એ તેના પ્રેમીની ગળું દબાવી હત્યા Murder કરી નાખી. પ્રેમી Lover ની હત્યા કર્યા બાદ પ્રેમિકા કામ અર્થે ફેક્ટરીમાં ગઈ હતી. પોલીસને ચાર દિવસ પહેલા પ્રેમીનો મૃતદેહ મળી આવ્યો હતો. આ મામલામાં તપાસ બાદ પોલીસે એક પછી એક કડીઓ જોડીને હત્યાની આરોપી પ્રેમિકાની ધરપકડ કરી છે. પૂછપરછ દરમિયાન તેણે પોલીસની સામે સમગ્ર સત્ય ઉઘાડ્યું છે. આરોપી પ્રેમિકાના કહેવા પ્રમાણે, તે તેની સાથે બળજબરીથી શારીરિક સંબંધ બાંધવાનો પ્રયાસ Forcibly Physical relation કરી રહ્યો હતો. જેથી તેનું ગળું દબાવી હત્યા કરી નાખી હતી. આરોપી મહિલાના તેના પ્રેમી સાથે બે વર્ષથી ગેરકાયદેસર સંબંધો હતા. પોલીસને આરોપીના મોબાઈલમાંથી અશ્લીલ વીડિયો મળી આવ્યો છે. પોલીસ સમગ્ર મામલાના તળિયે જવાનો પ્રયાસ કરી રહી છે. રાજસ્થાન Rajasthan જયપુર Jaipur ના ડેપ્યુટી કમિશનર ઓફ પોલીસ પશ્ચિમ રિચા તોમરે જણાવ્યું કે હત્યાની આ ઘટના 6 માર્ચે જયપુરના કરધની પોલીસ સ્ટેશન વિસ્તારમાં બની હતી. હત્યાનો ભોગ બનનાર સુભાષ કુમાવત 27 બૈનાડ રોડ પર ફકીરા નગરમાં ભાડાના મકાન સાથે રહેતો હતો. મૂળ તે જયપુરના ગોવિંદગઢ પોલીસ સ્ટેશન વિસ્તારના ધોડસર ગામનો રહેવાસી હતો. સુભાષની હત્યા કરનાર તેની ગર્લફ્રેન્ડ વિનોદ કંવર પાસેના મકાનમાં રહેતી હતી. બંને વચ્ચે લગભગ બે વર્ષથી ગેરકાયદે સંબંધો હતા. 6 માર્ચે સવારે કરવામાં આવી હતી હત્યાપોલીસની પૂછપરછમાં જાણવા મળ્યું કે 6 માર્ચના રોજ વિનોદના પતિ સવારે 7 વાગ્યે કામ પર જવા નીકળ્યા હતા. વિનોદ ઘરે એકલો હતો. એ પછી સુભાષ ત્યાં આવ્યો. આ દરમિયાન સુભાષે તેની સાથે બળજબરીથી શારીરિક સંબંધ બાંધવાનો પ્રયાસ કર્યો હતો. વિનોદે ના પાડતાં બંને વચ્ચે મારામારી થઈ હતી. આના પર વિનોદે સુભાષના મોઢું અને ગળું દબાવી દીધું હતું, જેના કારણે તેનું મોત થયું હતું. હત્યા બાદ વિનોદે સુભાષને ખેંચીને તેના રૂમમાં બારી પાસે સુવડાવ્યો હતો. અને પોતે ફેક્ટરીમાં કામ કરવા નીકળી ગઈ હતી પતિ અને પરિવાર ને પડી ગઈ હતી ખબરપૂછપરછ દરમિયાન વિનોદના પતિ અને તેના પરિવારના સભ્યોને પણ તેના અવૈધ સંબંધોની જાણ થઈ હોવાનું સામે આવ્યું હતું. વિનોદે પોલીસને જણાવ્યું હતું કે સુભાષ તેની સાથે બે વર્ષથી સંબંધ ધરાવે છે. તે તેને પોતાની સાથે રાખવાની વાત પણ કરતો હતો. આ અંગે તેના પતિને જાણ થઈ હતી. જેના કારણે વિનોદના પતિ સાથે અણબનાવ થયો હતો. આ અણબનાવને કારણે તેણે તેના પતિ વિરુદ્ધ દેહજનો કેસ પણ નોંધાવ્યો હતો. પતિની ગેરહાજરીમાં આવતો હતો પ્રેમીવિનોદના પતિને સુભાષ પર શંકા જતાં તે તેની પત્નીને ગામડે પણ લઈ ગયો હતો. પરંતુ વિનોદે જયપુર પાછા આવવાની જીદ કરી હતી અને ફરીથી સાથે રહેવા લાગી હતી. પતિની ગેર હાજરીમાં સુભાષ તેના ઘરે આવતો હતો અને સંબંધ બાંધીને પાછો જતો હતો. આરોપી પ્રેમિકા વિનોદે જણાવ્યું કે સુભાષ તેને છેલ્લા એક મહિનાથી ખૂબ જ પરેશાન કરી રહ્યો હતો. તેણે તેનો અશ્લીલ વીડિયો બનાવ્યો હતો. જો સંબંધ રાખવાની ના પાડે તો જાનથી મારી નાખવાની ધમકી આપતો હતો. વિનોદે પોલીસને ગેરમાર્ગે દોર્યાઆરોપી પ્રેમિકા વિનોદે પોલીસને ગેરમાર્ગે દોરવામાં કોઈ કસર છોડી ન હતી. તેણે સાંજ સુધી હત્યા અંગે કોઈને જણાવ્યું ન હતું. સાંજે પતિને સુભાષને ચા માટે બોલાવવા મોકલ્યો. પરંતુ જ્યારે વિનોદના પતિ સુભાષના રૂમમાં પહોંચ્યા તો તેમને તેમની લાશ મળી. આ અંગે તેણે મકાન માલિકને જાણ કરી હતી. મકાન માલિકે પોલીસ બોલાવી. પોલીસ સામે વિનોદ સાવ અજાણી જ રહી. પરંતુ જ્યારે પોલીસને સુભાષના મોબાઈલમાં વિનોદનો અશ્લીલ વીડિયો મળી આવ્યો, આ અંગે પોલીસે વિનોદની કડક પૂછપરછ કરી હતી. આનાથી તેણી ભાંગી પડી અને તેને પોલીસને સત્ય કહ્યું. | gujurati |
ખાધાખોરાકી નહીં ચૂકવનાર દુબઇ રહેતા પતિને કસ્ટડીમાં લેવા આદેશ હુકમનું પાલન નહીં કરતા હાઇકોર્ટ આકરાં પાણીએ ગુજરાત હાઈકોર્ટ દ્વારા ભરણપોષણના કેસ મામલે મહત્વનો નિર્દેશ આપવામાં આવ્યો છે. આ કેસમાં ખરેખર પતિએ કોર્ટના આદેશનો તિરસ્કાર કર્યો હતો જેથી તેની સામે આકરા પગલા લેવામાં આવ્યા છે. પતિ દુબઈમાં રહે છે તેમ છતા કોર્ટ દ્વારા તેની સામે આકરા પગલા લેવામાં આવ્યા અને મહત્વનો નિર્દેશ આપવામાં આવ્યો છે. સાંભળીને આપને નવાઈ લાગશે કે દુબઈમાં રહેતા પતિને પણ કોર્ટ દ્વારા કસ્ટડીમાં લેવા માટે આદેશ આપવામાં આવ્યો છે. જેમા હાઈકોર્ટ દ્વારા ઈન્ડિયન એમ્બેસીને નિર્દેશ કરવામાં આવ્યો કે પતિને દુબઈથી કસ્ટડીમાં લેવામાં આવે. અત્યાર સુધીમાં કોર્ટ દ્વારા પહેલી વખત આવો નિર્દેશ આપવામાં આવ્યો છે. જેમા લોકો પણ કોર્ટના આ નિર્દેશના વખાણ કરી રહ્યા છે. પતિએ તેની પત્ની સાથે છૂટાછેડા લિધા હતા જેથી પત્નીને દર મહિને 25 હજાર ચૂકવવા આદેશ આપ્યો હતો પરંતુ પતિ તેની પત્નીને રૂૂપિયા ચૂકવી નહોતો રહ્યો જેથી આ મામલો હાઈકોર્ટ સુધી પહોચ્યો અને હાઈકોર્ટ દ્વારા હવે આ મામલે મહત્વના આદેશ આપવામાં આવ્યા છે. ઉલ્લેખનીય છે કે કોર્ટ દ્વારા વર્ષ 2015માં પતિને ભરણપોષણ આપવા માટે આદેશ આપ્યો હતો. તેમ છતા પણ પતિ ભરણપોષ આપતો ન હતો. આ મામલે કોર્ટ દ્વરા અગાઉ પતિ સામે બિજ જામીનપાજ્ઞ વોરંટ પણ ઈશ્યું કરવામાં આવ્યો હતો. જોકે હવે હાઈકોર્ટે પતિને કસ્ટડીમાં લેવા આદેશ આપ્યો છે. જેમા કોર્ટ દ્વારા ઈન્ડિયન એમ્બસીને નિર્દેશ આપવામાં આવ્યો છે પતિને દુબઈથી કસ્ટડીમાં લેવા આદેશ આપ્યો છે. | gujurati |
! { dg-do run }
!
! PR 64209: [OOP] runtime segfault with CLASS(*), INTENT(OUT) dummy argument
!
! Contributed by Miha Polajnar <[email protected]>
MODULE m
IMPLICIT NONE
TYPE :: t
CLASS(*), ALLOCATABLE :: x(:)
CONTAINS
PROCEDURE :: copy
END TYPE t
INTERFACE
SUBROUTINE copy_proc_intr(a,b)
CLASS(*), INTENT(IN) :: a
CLASS(*), INTENT(OUT) :: b
END SUBROUTINE copy_proc_intr
END INTERFACE
CONTAINS
SUBROUTINE copy(self,cp,a)
CLASS(t), INTENT(IN) :: self
PROCEDURE(copy_proc_intr) :: cp
CLASS(*), INTENT(OUT) :: a(:)
INTEGER :: i
IF( .not.same_type_as(self%x(1),a(1)) ) STOP -1
DO i = 1, size(self%x)
CALL cp(self%x(i),a(i))
END DO
END SUBROUTINE copy
END MODULE m
PROGRAM main
USE m
IMPLICIT NONE
INTEGER, PARAMETER :: n = 3, x(n) = [ 1, 2, 3 ]
INTEGER :: copy_x(n)
TYPE(t) :: test
ALLOCATE(test%x(n),SOURCE=x)
CALL test%copy(copy_int,copy_x)
! PRINT '(*(I0,:2X))', copy_x
CONTAINS
SUBROUTINE copy_int(a,b)
CLASS(*), INTENT(IN) :: a
CLASS(*), INTENT(OUT) :: b
SELECT TYPE(a); TYPE IS(integer)
SELECT TYPE(b); TYPE IS(integer)
b = a
END SELECT; END SELECT
END SUBROUTINE copy_int
END PROGRAM main
| code |
పీకేకు కరోనా పాజిటివ్.. మాస్క్ లేకుండానే పబ్లిక్ మీటింగ్! తెలంగాణలో కోవిడ్ కేసులు తగ్గినట్లు ప్రభుత్వ లెక్కల్లో కనిపిస్తున్నా క్షేత్రస్థాయిలో మాత్రం పరిస్థితి మరోలా కనిపిస్తోంది. జిల్లాల్లో కొవిడ్ కేసులు మళ్లీ పెరుగుతున్నట్లు తెలుస్తోంది. తాజాగా రెండు రోజుల క్రితమే బీఎస్పీలో చేరిన రిటైర్డ్ ఐపీఎస్ అధికారి ఆర్ఎస్ ప్రవీణ్ కుమార్ కు కొవిడ్ పాజిటివ్ వచ్చింది. కొవిడ్ నిర్దారణ కావడంతో ఆయన గాంధీ ఆసుపత్రికి వెళ్లి కాక్ టెయిల్ వ్యాక్సిన్ డోస్ తీసుకున్నారు. కొద్దిసేపు అక్కడే ఉండి మధ్యాహ్నానికి ఇంటికి వెళ్లిపోయారు. ప్రస్తుతం ఆయన ఆరోగ్య పరిస్థితి బాగానే ఉందని, మైల్డ్ లక్షణాలే కనిపిస్తున్నాయని చెబుతున్నారు. గత నెలలో వాలంటరీ రిటైర్మెంట్ తీసుకున్న ఆర్ఎస్ ప్రవీణ్ కుమార్ ఆదివారం బీఎస్పీ పార్టీలో చేరారు.అన్ని కోవిడ్ అప్డేట్స్ గురించి తెలుసుకునేందుకు ఇక్కడ చదవండి నల్గొండ ఎన్జీ కాలేజీలో నిర్వహించిన భారీ సభలో ఆయన బీఎస్పీ కండువా కప్పుకున్నారు. ఈ సభకు తెలంగాణలోని అన్ని జిల్లాల నుంచి లక్ష మంది వరకు హాజరయ్యారు. తాజాగా ఆయనకు కరోనా పాజిటివ్గా నిర్ధారణ కావడంతో ప్రజలు భయాందోళనకు గురవుతున్నారు. ఇటీవల ఆయన చాలా ప్రాంతాల్లో పర్యటించారు. ఉమ్మడి జిల్లా కేంద్రాల్లో స్వేరోస్ సభలు నిర్వహించారు. వేలాది మందితో ఇంటరాక్ట్ అయ్యారు. ఇటీవల జరిన సమావేశాల్లో ప్రవీణ్ కుమార్ ఎక్కువగా మాస్క్ ధరించలేదని చెబుతున్నారు. నల్గొండ సభలోనూ ఎక్కువ సమయం ఆయన మాస్క్ లేకుండానే కనిపించారు. వేదికపై ఉన్న అతిథులు కూడా ఎవరూ మాస్క్ పెట్టుకోలేదు. ఆయన సభలకు వచ్చిన వారు కూడా కొవిడ్ రూల్స్ పాటించలేదని అంటున్నారు. ఈ నేపథ్యంలో ఆర్ఎస్ ప్రవీణ్ కుమార్ కు కొవిడ్ సోకడంతో.. ఆయన సభలకు వచ్చిన వారంతా ఆందోళనకు గురవుతున్నారు. | telegu |
રણજી ટ્રોફી 13 ફેબ્રુઆરીથી શરૂ થશે, બીસીસીઆઈ પ્રમુખ સૌરવ ગાંગુલીએ કરી પુષ્ટિ ભારતીય ક્રિકેટ કંટ્રોલ બોર્ડે રણજી ટ્રોફી 2022 ની તારીખો જાહેર કરી છે. ગયા વર્ષે કોરોના રોગચાળાને કારણે ટૂર્નામેન્ટનું આયોજન કરવામાં આવ્યું ન હતું અને આ વખતે તે 13 જાન્યુઆરીથી શરૂ કરવાની યોજના હતી, પરંતુ દેશમાં કોરોનાની ત્રીજી લહેરને જોતા તેને અનિશ્ચિત સમય માટે સ્થગિત કરવામાં આવી હતી. જો કે હવે BCCI પ્રમુખ સૌરવ ગાંગુલીએ પોતે કહ્યું છે કે રણજી ટ્રોફી 2022 ક્યારે શરૂ થશે. બોર્ડ ચીફના જણાવ્યા અનુસાર BCCI 13 ફેબ્રુઆરીથી ટૂર્નામેન્ટ શરૂ કરવાની યોજના બનાવી રહ્યું છે. તેણે એમ પણ કહ્યું કે ટૂર્નામેન્ટના ફોર્મેટમાં કોઈ ફેરફાર કરવામાં આવશે નહીં. ગાંગુલીએ ટૂર્નામેન્ટની શરૂઆતની તારીખની પુષ્ટિ કરી હતી. તેમણે કહ્યું કે તમામ ટીમોને 5 ગ્રુપમાં વહેંચવામાં આવશે. દરેક ગ્રુપમાં 6 ટીમો હશે. જ્યારે પ્લેટ ગ્રૂપમાં 8 ટીમો હશે. ટુર્નામેન્ટ બે તબક્કામાં યોજાશે. પ્રથમ તબક્કો એક મહિનાનો હશે જે IPL 2022 પહેલા રમાશે. બોર્ડે રણજી ટ્રોફીનું બે તબક્કામાં આયોજન કરવાનો નિર્ણય લીધો છે. પ્રથમ તબક્કામાં લીગ સ્તરની મેચો હશે અને નોકઆઉટ જૂનમાં રમાશે. તેણે કહ્યું, આઈપીએલ 2022 27 માર્ચથી યોજાવાની છે અને આવી સ્થિતિમાં રણજી ટ્રોફીની નોકઆઉટ મેચ જૂન અને જુલાઈમાં આયોજિત કરવામાં આવશે. ફોર્મેટમાં કોઈ ફેરફાર કરવામાં આવશે નહીં. કોરોનાના કેસને ધ્યાનમાં રાખીને ટૂર્નામેન્ટ માટે સ્થળ શોધવામાં આવી રહ્યું છે. | gujurati |
ಮುತ್ತು ತಂದ ಆಪತ್ತು: ಕಿಸ್ ಕೊಟ್ಟು ಕೆಲ್ಸ ಕಳೆದುಕೊಂಡ ಆರೋಗ್ಯ ಸಚಿವ! ಲಂಡನ್: ಸಹೋದ್ಯೋಗಿಯೊಬ್ಬರಿಗೆ ಕಚೇರಿಯಲ್ಲಿಯೇ ಮುತ್ತು ಕೊಟ್ಟು ಆರೋಗ್ಯ ಸಚಿವರೊಬ್ಬರು ಕೆಲಸ ಕಳೆದುಕೊಂಡಿರುವ ಘಟನೆ ಬ್ರಿಟನ್ನಲ್ಲಿ ನಡೆದಿದೆ. ಮ್ಯಾಟ್ ಹಾನ್ಕಾಕ್ ಮುತ್ತು ಕೊಟ್ಟ ತಪ್ಪಿಗೆ ಕೆಲಸ ಕಳೆದುಕೊಂಡಿದ್ದಾರೆ. ಅಂದಹಾಗೆ ಇವರು ಎಲ್ಲರ ಎದುರು, ಕಚೇರಿಯ ಅವಧಿಯಲ್ಲಿ ಮುತ್ತು ಕೊಟ್ಟಿದ್ದಕ್ಕೆ ಈ ಶಿಕ್ಷೆಯಲ್ಲ, ಬದಲಿಗೆ ಕರೊನಾ ನಿಯಮ ಉಲ್ಲಂಘನೆ ಮಾಡಿದರು ಎನ್ನುವ ಕಾರಣಕ್ಕೆ! ಆರೋಗ್ಯ ಸಚಿವರಾಗಿದ್ದಕೊಂಡು ಕರೊನಾ ಮಾರ್ಗಸೂಚಿ ಉಲ್ಲಂಘನೆ ಮಾಡಿದ್ದಾರೆ ಎನ್ನುವ ಕಾರಣಕ್ಕೆ ಇವರ ರಾಜೀನಾಮೆ ಪಡೆಯಲಾಗಿದೆ. ಕೋವಿಡ್ ಮಾರ್ಗಸೂಚಿ ಕಡ್ಡಾಯವಾಗಿ ಪಾಲಿಸಬೇಕು ಎಂದು ಎಲ್ಲರಿಗೂ ಬುದ್ಧಿಹೇಳುವ ಆರೋಗ್ಯ ಸಚಿವರು ಹೀಗೆ ಮಾಡಿದ್ದು ತಪ್ಪು ಎನ್ನುವುದು ಇದಕ್ಕೆ ಕಾರಣ. ಈ ಸಚಿವರು ಚುಂಬಿಸಿದ ಫೋಟೋಗಳು ವೈರಲ್ ಆಗಿದ್ದವು. ನಂತರ ಈ ಬಗ್ಗೆ ಅವರಿಗೆ ಸಮಜಾಯಿಷಿ ಕೇಳಿದಾಗ ಕರೊನಾ ಮಾರ್ಗಸೂಚಿ ಉಲ್ಲಂಘನೆಯಾಗಿದೆ ಎಂಬುದನ್ನು ಅವರು ಒಪ್ಪಿಕೊಂಡಿದ್ದಾರೆ. ನಂತರ ರಾಜೀನಾಮೆ ನೀಡಿದ್ದಾರೆ. ಇವರ ಜಾಗಕ್ಕೆ ಪಾಕಿಸ್ತಾನದ ಸಂಸತ್ ಸದಸ್ಯ ಸಾಜಿದ್ ಜಾವಿದ್ ಬ್ರಿಟನ್ ನೂತನ ಆರೋಗ್ಯ ಸಚಿವರಾಗಿ ಪ್ರಮಾಣವಚನ ಸ್ವೀಕರಿಸುವ ಸಾಧ್ಯತೆ ಇದೆ. ಮೋದಿ ಸಂಪುಟಕ್ಕೆ ಜುಲೈನಲ್ಲಿ ಸರ್ಜರಿ? 27 ಹೊಸ ಮುಖಗಳು ರೇಸ್ನಲ್ಲಿ ಯಾರಿದ್ದಾರೆ ನೋಡಿ ಮೊಬೈಲ್ ಖರೀದಿಗೆ ಬಾಲಕಿ ಮಾರಿದಳು 12 ಮಾವಿನಹಣ್ಣು ಸಿಕ್ಕಿದ್ದು 1.20 ಲಕ್ಷ ರೂ! | kannad |
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{\mathcal B}etagin{document}
\title{On tangent cones in Wasserstein space}
\author{John Lott}
\address{Department of Mathematics\\
University of California - Berkeley\\
Berkeley, CA 94720-3840\\
USA} \email{[email protected]}
\thetaanks{Research partially supported
by NSF grant DMS-1207654 and a Simons Fellowship}
\date{August 6, 2016}
{\mathcal S}ubjclass[2000]{}
{\mathcal B}etagin{abstract}
If $M$ is a smooth compact Riemannian manifold, let $P(M)$
denote the Wasserstein
space of probability measures on $M$. If $S$ is an embedded
submanifold of $M$, and $\mu$ is an absolutely continuous measure on $S$,
then we compute the tangent cone of $P(M)$ at $\mu$.
\end{abstract}
\maketitle
{\mathcal S}ection{Introduction} \lambdabel{section1}
In optimal transport theory, a displacement interpolation is a
one-parameter family of measures that represents the most efficient
way of displacing mass between two given probability measures.
Finding a displacement interpolation between two
probability measures
is the same as finding a
minimizing geodesic in the space of probability
measures, equipped with the Wasserstein metric $W_2$
\cite[Proposition 2.10]{Lott-Villani (2009)}.
For background on optimal transport and Wasserstein space, we refer
to Villani's book \cite{Villani (2009)}.
If $M$ is a compact connected Riemannian manifold with
nonnegative sectional curvature
then $P(M)$ is a compact length space with nonnegative curvature
in the sense of Alexandrov
\cite[Theorem A.8]{Lott-Villani (2009)},
\cite[Proposition 2.10]{Sturm (2006)}.
Hence one can define the tangent cone $T_{\mu} P(M)$
of $P(M)$ at a measure $\mu \in P(M)$.
If $\mu$ is absolutely continuous with respect to the volume form
$\operatorname{dvol}_M$ then $T_{\mu} P(M)$ is a Hilbert space
\cite[Proposition A.33]{Lott-Villani (2009)}. More generally,
one can define tangent cones of $P(M)$ without any curvature assumption
on $M$,
using Ohta's $2$-uniform structure on $P(M)$ \cite{Ohta (2009)}.
Gigli showed that
$T_{\mu} P(M)$ is a Hilbert space if and only if $\mu$ is a ``regular''
measure, meaning that it gives zero measure to any hypersurface which,
locally, is the graph of the difference of two convex functions
\cite[Corollary 6.6]{Gigli (2011)}.
It is natural to ask
what the tangent cones are at other measures.
A wide class of tractable measures comes from submanifolds.
Suppose that $S$ is a smooth embedded submanifold of a
compact connected Riemannian manifold $M$. Suppose that
$\mu$ is an absolutely continuous probability measure on $S$. We can
also view $\mu$ as an element of $P(M)$. For simplicity, we assume that
${\mathcal S}upp(\mu) = S$.
{\mathcal B}etagin{theorem} \lambdabel{theorem1.1}
We have
{\mathcal B}etagin{equation} \lambdabel{1.2}
T_{\mu} P(M) = H \oplus \int_{s \in S} P_2(N_sM) \: d\mu(s),
\end{equation}
where
{\mathcal B}etagin{itemize}
\item $H$ is the Hilbert space of gradient vector fields
$\overline{\operatorname{Im}({\mathcal N}abla)} {\mathcal S}ubset L^2(TS, d\mu)$,
\item $N_sM$ is the normal space to $S {\mathcal S}ubset M$ at $s \in S$ and
\item $P_2(N_sM)$ is the metric cone of probability measures on $N_sM$
with finite second moment, equipped with the $2$-Wasserstein metric.
\end{itemize}
\end{theorem}
The homotheties in the metric cone structure on $P_2(N_sM)$ arise from
radial rescalings of $N_sM$. The direct sum and integral in (\operatorname{Re}f{1.2}) refer
to computing square distances.
The proof of Theorem \operatorname{Re}f{theorem1.1}
amounts to understanding optimal transport
starting from a measure supported on a submanifold. This seems to be a
natural question in its own right which has not been considered much.
Gangbo and McCann proved results about
optimal transport between measures supported on hypersurfaces in
Euclidean space \cite{Gangbo-McCann (2000)}.
McCann-Sosio and Kitagawa-Warren gave more refined results about optimal
transport between two measures supported on a sphere
\cite{Kitagawa-Warren (2012),McCann-Sosio (2011)}.
Castillon considered optimal transport between a measure supported on a
submanifold of Euclidean space and a measure supported on a linear subspace
\cite{Castillon (2010)}.
In the setting of Theorem \operatorname{Re}f{theorem1.1},
a Wasserstein geodesic $\{\mu_t\}_{t \in [0,
\epsilonilon]}$ starting from $\mu$ consists of a family of geodesics shooting
off from $S$ in various directions.
The geometric meaning of Theorem \operatorname{Re}f{theorem1.1}
is that the tangential component
of these directions is the gradient of a function on $S$. To motivate
this statement, in Section \operatorname{Re}f{section2} we give a Benamou-Brenier-type
variational approach to the problem of optimally tranporting a
measure supported on one hypersurface to a measure supported on a disjoint
hypersurface, through a family of measures supported on hypersurfaces.
One finds that the only constraint is the aforementioned tangentiality
constraint. The rigorous proof of Theorem \operatorname{Re}f{theorem1.1} is in
Section \operatorname{Re}f{section3}.
The structure of this paper is as follows. In Section \operatorname{Re}f{section2} we
give a formal derivation of the equation for optimal transport between
two measures supported on disjoint hypersurfaces of a Riemannian manifold.
The derivation is based on a variational method. In Section \operatorname{Re}f{section3}
we prove Theorem \operatorname{Re}f{theorem1.1}.
I thank C\'edric Villani for helpful comments, and
Robert McCann for references to the literature. I thank the
referee for his/her remarks.
{\mathcal S}ection{Variational approach} \lambdabel{section2}
Let $M$ be a smooth closed Riemannian manifold. Let
$S$ be a smooth closed manifold and let $S_0, S_1$ be
disjoint codimension-one submanifolds of $M$ diffeomorphic to $S$.
Let ${\mathcal R}ho_0 \operatorname{dvol}_{S_0}$ and ${\mathcal R}ho_1 \operatorname{dvol}_{S_1}$ be smooth
probability measures on $S_0$ and $S_1$, respectively.
We consider the problem of optimally transporting
${\mathcal R}ho_0 \operatorname{dvol}_{S_0}$ to ${\mathcal R}ho_1 \operatorname{dvol}_{S_1}$ through
a family of measures supported on codimension-one
submanifolds $\{S_t\}_{t \in [0,1]}$.
We will specify the intermediate submanifolds to be level sets of
a function $T$, which in turn will become one of the
variables in the optimization problem.
We assume that there is a
codimension-zero submanifold-with-boundary $U$ of $M$,
with $\partialartial U = S_0 \cup S_1$.
We also assume that there is a smooth submersion $T : U {\mathcal R}ightarrow [0,1]$
so that $T^{-1}(0) = S_0$ and $T^{-1}(1) = S_1$.
For $t \in [0,1]$, put $S_t = T^{-1}(t)$.
These are the intermediate hypersurfaces.
We now want to describe a family of measures
$\{\mu_t\}_{t \in [0,1]}$ that live
on the hypersurfaces
$\{S_t\}_{t \in [0,1]}$. It is convenient to think of these
measures as fitting together to form a measure on $U$.
Let $\mu$ be a
smooth measure on $U$.
In terms of the fibering $T : U {\mathcal R}ightarrow [0,1]$, decompose
$\mu$ as
$\mu = \mu_t dt$ with $\mu_t$ a measure on $S_t$.
We assume that $\mu_0 = {\mathcal R}ho_0 \operatorname{dvol}_{S_0}$ and
$\mu_1 = {\mathcal R}ho_1\operatorname{dvol}_{S_1}$.
Let $V$ be a vector field on $U$.
We want the flow $\{\operatorname{ph}i_s\}$ of $V$ to send level sets
of $T$ to level sets.
Imagining that there is an external clock, it's convenient
to think of $S_t$ as the evolving hypersurface at time $t$.
Correlating the flow of $V$ with the clock gives the constraint
{\mathcal B}etagin{equation} \lambdabel{2.1}
VT=1.
\end{equation}
Then
$\operatorname{ph}i_s$ maps $S_t$ to $S_{t+s}$.
We also want the flow to be compatible with the measures
$\{\mu_t\}_{t \in [0,1]}$ in the sense that $\operatorname{ph}i_s^* \mu_{t+s} = \mu_t$.
Now $\operatorname{ph}i_s^* dT = d \operatorname{ph}i_s^* T = d(T+s) = dT$, so it is equivalent
to require that $\operatorname{ph}i_s^*$ preserves the measure $\mu =
\mu_t dt$. This gives the constraint
{\mathcal B}etagin{equation} \lambdabel{2.2}
{\mathcal L}_V \mu = 0.
\end{equation}
In particular, each $\mu_t$ is a probability measure.
To define a functional along the lines of Benamou and Brenier
\cite{Benamou-Brenier (2000)},
put
{\mathcal B}etagin{equation} \lambdabel{2.3}
E = {\mathcal f}rac12 \int_U |V|^2 \: d\mu =
{\mathcal f}rac12 \int_0^1 \int_{S_t} |V|^2 \: d\mu_t \: dt.
\end{equation}
We want to minimize $E$ under the constraints
${\mathcal L}_V \mu = 0$, $VT = 1$,
$\mu_0 = {\mathcal R}ho_0 \operatorname{dvol}_{S_0}$ and
$\mu_1 = {\mathcal R}ho_1 \operatorname{dvol}_{S_1}$.
Let
$\operatorname{ph}i$ and $\eta$ be new functions on $U$, which
will be Lagrange multipliers for the constraints.
Then we want to extremize
{\mathcal B}etagin{equation} \lambdabel{2.4}
{\mathcal E} = \int_U \left[ {\mathcal f}rac12 |V|^2 \: d\mu
+ \operatorname{ph}i {\mathcal L}_V d\mu + \eta (VT-1) d\mu {\mathcal R}ight]
\end{equation}
with respect to $V$, $\mu$, $\operatorname{ph}i$ and $\eta$.
We will use the equations
{\mathcal B}etagin{align} \lambdabel{2.5}
\int_U \operatorname{ph}i {\mathcal L}_V d\mu
= & \int_U \left[
{\mathcal L}_V(\operatorname{ph}i d\mu)
- ({\mathcal L}_V \operatorname{ph}i) d\mu
{\mathcal R}ight] \\
= & - \int_U
(V \operatorname{ph}i) d\mu
+ \int_{S_1} \operatorname{ph}i(1) d\mu_1 -
\int_{S_0} \operatorname{ph}i(0) d\mu_0 {\mathcal N}otag
\end{align}
and
{\mathcal B}etagin{align} \lambdabel{2.6}
\int_U \eta VT d\mu = &
\int_U \left[ {\mathcal L}_V (T \eta d\mu) -
T {\mathcal L}_V (\eta d\mu) {\mathcal R}ight] \\
= &- \int_U T {\mathcal L}_V (\eta d\mu)
+ \int_{S_1} \eta(1) d\mu_1. {\mathcal N}otag
\end{align}
The Euler-Lagrange equation for $V$ is
{\mathcal B}etagin{equation} \lambdabel{2.7}
V - {\mathcal N}abla \operatorname{ph}i + \eta {\mathcal N}abla T = 0.
\end{equation}
The Euler-Lagrange equation for $\mu$ is
{\mathcal B}etagin{equation} \lambdabel{2.8}
{\mathcal f}rac12 |V|^2 - V\operatorname{ph}i = 0.
\end{equation}
Varying $T$ gives
{\mathcal B}etagin{equation} \lambdabel{2.9}
0 = {\mathcal L}_V(\eta d\mu) = (V\eta) d\mu,
\end{equation}
so the Euler-Lagrange equation for $T$ is
{\mathcal B}etagin{equation} \lambdabel{2.10}
V\eta = 0.
\end{equation}
Substituting (\operatorname{Re}f{2.7}) into (\operatorname{Re}f{2.8}) gives
$|{\mathcal N}abla \operatorname{ph}i|^2 = \eta^2 |{\mathcal N}abla T|^2$, so
$\eta = \partialm {\mathcal f}rac{|{\mathcal N}abla \operatorname{ph}i|}{|{\mathcal N}abla T|}$.
Then (\operatorname{Re}f{2.7}) becomes
{\mathcal B}etagin{equation}
V = {\mathcal N}abla \operatorname{ph}i \mp {\mathcal f}rac{|{\mathcal N}abla \operatorname{ph}i|}{|{\mathcal N}abla T|} {\mathcal N}abla T.
\end{equation}
Equation (\operatorname{Re}f{2.1}) gives
{\mathcal B}etagin{equation} \lambdabel{added}
1 = \lambdangle {\mathcal N}abla \operatorname{ph}i, {\mathcal N}abla T {\mathcal R}ightarrowngle \mp |{\mathcal N}abla \operatorname{ph}i| \operatorname{CD}ot
|{\mathcal N}abla T|.
\end{equation}
If the ``$\mp$'' is ``$-$'' then the right-hand side of
(\operatorname{Re}f{added}) is nonpositive, which is a contradiction.
Thus
{\mathcal B}etagin{equation}
1 = \lambdangle {\mathcal N}abla \operatorname{ph}i, {\mathcal N}abla T {\mathcal R}ightarrowngle+ |{\mathcal N}abla \operatorname{ph}i| \operatorname{CD}ot
|{\mathcal N}abla T|
\end{equation}
and
{\mathcal B}etagin{equation}
V = {\mathcal N}abla \operatorname{ph}i + {\mathcal f}rac{|{\mathcal N}abla \operatorname{ph}i|}{|{\mathcal N}abla T|} {\mathcal N}abla T.
\end{equation}
Equation (\operatorname{Re}f{2.10}) becomes
{\mathcal B}etagin{equation}
V {\mathcal f}rac{|{\mathcal N}abla \operatorname{ph}i|}{|{\mathcal N}abla T|} = 0,
\end{equation}
which is equivalent to
{\mathcal B}etagin{equation} \lambdabel{2.14}
{\mathcal f}rac12 V |V|^2 = 0.
\end{equation}
Equation (\operatorname{Re}f{2.14}) says that $V$ has constant
length along its flowlines.
The measure $\mu$ must still satisfy the conservation law (\operatorname{Re}f{2.2}).
From (\operatorname{Re}f{2.8}), the evolution of $\operatorname{ph}i$ between level sets
is given by
{\mathcal B}etagin{equation} \lambdabel{2.16}
V\operatorname{ph}i = {\mathcal f}rac12 |V|^2 = {\mathcal f}rac12 {\mathcal f}rac{|{\mathcal N}abla \operatorname{ph}i|}{|{\mathcal N}abla T|}.
\end{equation}
The normal line to a level set $S_t$ is spanned by ${\mathcal N}abla T$.
It follows from (\operatorname{Re}f{2.7}) that the tangential part of $V$
is the gradient of a function on $S_t$ :
{\mathcal B}etagin{equation} \lambdabel{2.17}
V_{tan} = {\mathcal N}abla_{S_t} \left( \operatorname{ph}i \operatorname{B}ig|_{S_t} {\mathcal R}ight).
\end{equation}
The normal part of $V$ is
{\mathcal B}etagin{equation} \lambdabel{2.18}
V_{norm} =
{\mathcal f}rac{\lambdangle V, {\mathcal N}abla T {\mathcal R}ightarrowngle}{|{\mathcal N}abla T|^2} {\mathcal N}abla T =
{\mathcal f}rac{1}{|{\mathcal N}abla T|^2} {\mathcal N}abla T,
\end{equation}
as must be the case from (\operatorname{Re}f{2.1}).
The conclusion is that the tangential part of $V$ on $S_t$ is a gradient
vector field on $S_t$, while the normal part of $V$ on $S_t$ is unconstrained.
{\mathcal S}ection{Tangent cones} \lambdabel{section3}
{\mathcal S}ubsection{Optimal transport from submanifolds}
Let $M$ be a smooth closed Riemannian
manifold. Let $i : S {\mathcal R}ightarrow M$ be an
embedding.
Let $\partiali : TM {\mathcal R}ightarrow M$ be the projection map.
Given $\epsilonilon > 0$, define
$E_\epsilonilon : TM {\mathcal R}ightarrow TM$ by
$E_\epsilonilon(m,v) = \left( \exp_m (\epsilonilon v),
d(\exp_m)_{\epsilonilon v} \epsilonilon v {\mathcal R}ight)$.
We define $\partiali^S$ and $E^S_\epsilonilon$ similarly, replacing $M$ by $S$.
Put $T_S M = i^* TM$, a vector bundle on $S$ with projection map
$\partiali_{T_SM} : T_S M {\mathcal R}ightarrow S$. There is an orthogonal
splitting $T_S M = TS \oplus N_S M$ into the tangential part and
the normal part. Let
$\partiali_{N_SM} : N_S M {\mathcal R}ightarrow S$
be the projection to the base of $N_SM$.
Given
$v \in TS$, let $v^T \in TS$ denote its tangential part and let
$v^\partialerp \in NS$ denote its normal part. Let
$p^T : T_SM {\mathcal R}ightarrow TS$ be the orthogonal projection.
A function $F : S {\mathcal R}ightarrow {\mathbb R} \cup \{\infty\}$ is {\it semiconvex}
if there is some $\lambdambda \in {\mathbb R}$ so that for all minimizing
constant-speed geodesics $\gammamma : [0,1] {\mathcal R}ightarrow S$, we have
{\mathcal B}etagin{equation} \lambdabel{3.1}
F(\gammamma(t)) \le t F(\gammamma(1)) + (1-t) F(\gammamma(0)) - {\mathcal f}rac12
\lambdambda t (1-t) d_S(\gammamma(0), \gammamma(1))^2
\end{equation}
for all $t \in [0,1]$.
Suppose that $F$ is a semiconvex function on $S$. Then $(s, w) \in TS$
lies in the {\em subdifferential set}
${\mathcal N}abla^- F$ if for all $w^\operatorname{pr}ime \in T_sS$,
{\mathcal B}etagin{equation} \lambdabel{3.2}
F(s) + \lambdangle w, w^\operatorname{pr}ime {\mathcal R}ightarrowngle \le F(\exp_s w^\operatorname{pr}ime) +
o(|w^\operatorname{pr}ime|).
\end{equation}
Define the cost function $c : S \times M {\mathcal R}ightarrow {\mathbb R}$ by $c(s,x) = {\mathcal f}rac12
d(s,x)^2$. Given $\eta : M {\mathcal R}ightarrow {\mathbb R} \cup \{ - \infty \}$, its
$c$-transform is the function
$\eta^c : S {\mathcal R}ightarrow {\mathbb R} \cup \{\infty\}$ given by
{\mathcal B}etagin{equation} \lambdabel{3.3}
\eta^c(s) = {\mathcal S}up_{x \in M} \left( \eta(x) - {\mathcal f}rac12 d^2(s,x) {\mathcal R}ight).
\end{equation}
Given $\partialsi : S {\mathcal R}ightarrow {\mathbb R} \cup \{ \infty \}$, its
$c$-transform is the function
$\partialsi^c : M {\mathcal R}ightarrow {\mathbb R} \cup \{ - \infty\}$ given by
{\mathcal B}etagin{equation} \lambdabel{3.4}
\partialsi^c(x) = \inf_{s \in S} \left( \partialsi(s) + {\mathcal f}rac12 d^2(s,x) {\mathcal R}ight).
\end{equation}
A function $\partialsi : S {\mathcal R}ightarrow {\mathbb R} \cup \{ \infty \}$ is
{\em $c$-convex}
if $\partialsi = \eta^c$ for some
$\eta : M {\mathcal R}ightarrow {\mathbb R} \cup \{ - \infty \}$.
A function $\eta : M {\mathcal R}ightarrow {\mathbb R} \cup \{ -\infty \}$ is
{\em $c$-concave}
if $\eta = \partialsi^c$ for some
$\partialsi : S {\mathcal R}ightarrow {\mathbb R} \cup \{ \infty \}$.
From \cite[Proposition 5.8]{Villani (2009)},
a function $F : S {\mathcal R}ightarrow {\mathbb R} \cup \{- \infty\}$ is
$c$-convex if and only if $F = (F^c)^c$, i.e. for all $s \in S$,
{\mathcal B}etagin{equation} \lambdabel{3.5}
F(s) = {\mathcal S}up_{x \in M} \inf_{s^\operatorname{pr}ime \in S}
\left( F(s^\operatorname{pr}ime) + {\mathcal f}rac12 d^2(s^\operatorname{pr}ime,x) -
{\mathcal f}rac12 d^2(s,x) {\mathcal R}ight).
\end{equation}
The next lemma appears in \cite[Lemma 2.9]{Gigli (2011)} when $S=M$.
{\mathcal B}etagin{lemma} \lambdabel{lemma3.6}
If $F : S {\mathcal R}ightarrow {\mathbb R} \cup \{\infty\}$ is a semiconvex function
then there is some $\epsilonilon > 0$ so that $\epsilonilon F$ is
$c$-convex.
\end{lemma}
{\mathcal B}etagin{proof}
Clearly
{\mathcal B}etagin{equation} \lambdabel{3.7}
\epsilonilon F(s) \ge {\mathcal S}up_{x \in M} \inf_{s^\operatorname{pr}ime \in S}
\left( \epsilonilon F(s^\operatorname{pr}ime) + {\mathcal f}rac12 d^2(s^\operatorname{pr}ime,x) -
{\mathcal f}rac12 d^2(s,x) {\mathcal R}ight),
\end{equation}
as is seen by taking $s^\operatorname{pr}ime = s$ on the right-hand side of (\operatorname{Re}f{3.7}).
Hence we must show that for suitable $\epsilonilon > 0$, for all $s \in S$
we have
{\mathcal B}etagin{equation} \lambdabel{3.8}
\epsilonilon F(s) \le {\mathcal S}up_{x \in M} \inf_{s^\operatorname{pr}ime \in S}
\left( \epsilonilon F(s^\operatorname{pr}ime) + {\mathcal f}rac12 d^2(s^\operatorname{pr}ime,x) -
{\mathcal f}rac12 d^2(s,x) {\mathcal R}ight).
\end{equation}
For this, it suffices to show that for each $s \in S$, there is
some $x \in M$ so that
{\mathcal B}etagin{equation} \lambdabel{3.9}
\epsilonilon F(s) \le
\inf_{s^\operatorname{pr}ime \in S}
\left( \epsilonilon F(s^\operatorname{pr}ime) + {\mathcal f}rac12 d^2(s^\operatorname{pr}ime,x) -
{\mathcal f}rac12 d^2(s,x) {\mathcal R}ight).
\end{equation}
That is, it suffices to show that for each $s \in S$, there is
some $x \in M$ so that for all $s^\operatorname{pr}ime \in S$, we have
{\mathcal B}etagin{equation} \lambdabel{3.10}
\epsilonilon F(s) \le
\epsilonilon F(s^\operatorname{pr}ime) + {\mathcal f}rac12 d^2(s^\operatorname{pr}ime,x) -
{\mathcal f}rac12 d^2(s,x),
\end{equation}
i.e.
{\mathcal B}etagin{equation} \lambdabel{3.11}
\epsilonilon F(s) + {\mathcal f}rac12 d^2(s,x) \le
\epsilonilon F(s^\operatorname{pr}ime) + {\mathcal f}rac12 d^2(s^\operatorname{pr}ime,x).
\end{equation}
We know that $F$ is $K$-Lipschitz for some $K < \infty$
\cite[Theorem 10.8 and Proposition 10.12]{Villani (2009)}.
Hence if $v \in {\mathcal N}abla^-_s F$ then $|v| \le K$.
Given $s$, choose $v \in {\mathcal N}abla^-_s F$ and put $x = \exp_s (
\epsilonilon v) \in M$. Then $d(s,x) \le \epsilonilon K$.
Put $G(s^\operatorname{pr}ime) =
\epsilonilon F(s^\operatorname{pr}ime) + {\mathcal f}rac12 d^2( s^\operatorname{pr}ime,x)$.
We want to show that $G(s) \le G(s^\operatorname{pr}ime)$ for all $s^\operatorname{pr}ime \in S$.
Suppose not.
Let $s^\operatorname{pr}ime$ be a minimum point for $G$; then
$G(s^\operatorname{pr}ime) < G(s)$.
We claim first that $s^\operatorname{pr}ime \in B_{4 \epsilonilon K}(s)$.
To see this, if $d(s,s^\operatorname{pr}ime) \ge 4 \epsilonilon K$ then since
{\mathcal B}etagin{equation} \lambdabel{3.12}
d(s^\operatorname{pr}ime,x) \ge d(s,s^\operatorname{pr}ime) - d(s,x) \ge
d(s,s^\operatorname{pr}ime) - \epsilonilon K,
\end{equation}
we have
{\mathcal B}etagin{align} \lambdabel{3.13}
{\mathcal f}rac12 d^2(s^\operatorname{pr}ime,x) - {\mathcal f}rac12 d^2(s,x) & \ge
{\mathcal f}rac12 \left( d(s,s^\operatorname{pr}ime) - \epsilonilon K {\mathcal R}ight)^2 -
{\mathcal f}rac12 \left( \epsilonilon K {\mathcal R}ight)^2 \\
& = {\mathcal f}rac12
(d(s,s^\operatorname{pr}ime) - 2 \epsilonilon K) \operatorname{CD}ot d(s,s^\operatorname{pr}ime) {\mathcal N}otag \\
& \ge \epsilonilon K d(s,s^\operatorname{pr}ime) \ge \epsilonilon(F(s) - F(s^\operatorname{pr}ime)), {\mathcal N}otag
\end{align}
which contradicts that $G(s^\operatorname{pr}ime) < G(s)$. This proves the claim.
If $10 \epsilonilon K$ is less than the injectivity radius of $M$ then there is a
unique minimizing geodesic from $s$ to $x$, and its tangent
vector at $s$ is $\epsilonilon v$.
It follows that $0 \in {\mathcal N}abla_s^- G$.
Finally, since $d(s,x) \le
\epsilonilon K$, we can choose an $\epsilonilon$ (depending on $K$, $S$ and $M$)
to ensure that $G$ is strictly
convex on $B_{4\epsilonilon K}(s)$, with the latter being a totally convex set.
Considering the function $G$ along a minimizing geodesic from
$s$ to $s^\operatorname{pr}ime$, we obtain a contradiction to the assumed strict
convexity of $G$, along with the facts that $0 \in {\mathcal N}abla_s^- G$ and
$0 \in {\mathcal N}abla_{s^\operatorname{pr}ime}^- G$.
Thus $G$ is minimized at $s$, which
implies (\operatorname{Re}f{3.11}).
\end{proof}
Let ${\mathcal N}u$ be a compactly-supported probability measure on $T_S M
{\mathcal S}ubset TM$.
Let $L < \infty$ be such that the support of ${\mathcal N}u$ is contained in
$\{v \in T_SM \: : \: |v| \le L \}$.
Put $\mu_\epsilonilon = \partiali_* (E_\epsilonilon)_* {\mathcal N}u$.
{\mathcal B}etagin{proposition} \lambdabel{proposition3.14}
a. Let $f$ be a semiconvex function on $S$. Suppose that
${\mathcal N}u$ is supported on $\{v \in T_SM \: : \: v^T \in {\mathcal N}abla^- f \}$.
Then there is some $\epsilonilon > 0$ so that the $1$-parameter family of
measures $\{\mu_t\}_{t \in [0, \epsilonilon]}$ is
a Wasserstein geodesic. \\
b. Given ${\mathcal N}u$, suppose that
for some $\epsilonilon > 0$, the $1$-parameter family of
measures $\{\mu_t\}_{t \in [0, \epsilonilon]}$ is
a Wasserstein geodesic. Then there is a
semiconvex function $f$ on $S$ so that
${\mathcal N}u$ is supported on $\{v \in T_SM \: : \: v^T \in {\mathcal N}abla^- f \}$.\\
\end{proposition}
{\mathcal B}etagin{proof}
a. For $t > 0$, define $\eta_t : M {\mathcal R}ightarrow {\mathbb R}$ by
$\eta_t = (tf)^c$. From Lemma \operatorname{Re}f{lemma3.6}, if $t$ is small enough then
$tf$ is $c$-convex. It follows from
\cite[Proposition 5.8]{Villani (2009)} that
$(\eta_t)^c = tf$.
From \cite[Theorem 5.10]{Villani (2009)},
if a set $\Gammamma_t {\mathcal S}ubset S \times M$
is such that $\eta_t(x) = tf(s) + {\mathcal f}rac12 d^2(s,x)$ for all
$(s,x) \in S \times M$ then any
probability measure $\Pi_t$ with support in $\Gammamma_t$ is an optimal
transport plan.
We take
{\mathcal B}etagin{equation} \lambdabel{3.15}
\Gammamma_t = \{(s,x) \in S \times M \: : \: \eta_t(x) = tf(s) + {\mathcal f}rac12 d^2(s,x) \}.
\end{equation}
Now $\eta_t(x) = tf(s) + {\mathcal f}rac12 d^2(s,x)$ if
for all $s^\operatorname{pr}ime \in S$, we have
{\mathcal B}etagin{equation} \lambdabel{3.16}
t f(s) + {\mathcal f}rac12
d^2(s,x) \le t f(s^\operatorname{pr}ime) + {\mathcal f}rac12
d^2(s^\operatorname{pr}ime,x).
\end{equation}
To prove part a. of the proposition,
it suffices to show that for all sufficiently
small $t$, equation (\operatorname{Re}f{3.16}) is satisfied for
$s,s^\operatorname{pr}ime \in S$ and $x = \exp_s (tv)$, where
$v \in T_sM$ lies in the support of ${\mathcal N}u$ and
satisfies $v^T \in {\mathcal N}abla^- f$.
Given $s$ and $v$, we know that $d(s,x) \le tL$. Put
$G(s^\operatorname{pr}ime) = t f(s^\operatorname{pr}ime) +
{\mathcal f}rac12 d^2(s^\operatorname{pr}ime, x)$. Let $s^\operatorname{pr}ime$ be a minimum point of
$G$ and suppose, to get a contradiction, that $G(s^\operatorname{pr}ime) < G(s)$.
Let $K < \infty$ be the Lipschitz constant of $f$.
We claim first that
$s^\operatorname{pr}ime \in B_{t(2K+2L)}(s)$.
To see this, if $d(s,s^\operatorname{pr}ime) \ge t(2K+2L)$ then
{\mathcal B}etagin{equation} \lambdabel{3.17}
d(s^\operatorname{pr}ime,x) \ge d(s,s^\operatorname{pr}ime) - d(s,x) \ge
d(s,s^\operatorname{pr}ime) - tL
\end{equation}
and
{\mathcal B}etagin{align} \lambdabel{3.18}
{\mathcal f}rac12 d^2(s^\operatorname{pr}ime,x) - {\mathcal f}rac12 d^2(s,x) & \ge
{\mathcal f}rac12 \left( d(s,s^\operatorname{pr}ime) - tL {\mathcal R}ight)^2 - (tL)^2 \\
& = {\mathcal f}rac12 (d(s,s^\operatorname{pr}ime) - 2tL) \operatorname{CD}ot d(s,s^\operatorname{pr}ime) {\mathcal N}otag \\
& \ge tK d(s,s^\operatorname{pr}ime) \ge t(f(s) - f(s^\operatorname{pr}ime)), {\mathcal N}otag
\end{align}
which is a contradiction and proves the claim.
There is some $\epsilonilon > 0$ (depending on $L$, $S$ and $M$)
so that if $t \in [0, \epsilonilon]$ then we are ensured that
there is a unique minimizing geodesic from $s$ to $x$, and its
tangent vector at $s$ is $tv$. It follows that $0 \in {\mathcal N}abla_s^- G$.
Finally, since $d(s,x) \le \epsilonilon L$, we can choose $\epsilonilon$
(depending on $K$, $L$, $S$ and $M$) to ensure that $G$ is strictly
convex on $B_{t(2K+2L)}(s)$, the latter being totally convex.
Considering the function $G$ along a
minimizing geodesic from $s$ to $s^\operatorname{pr}ime$, we obtain a contradiction
to the assumed strict convexity of $G$, along with the facts that
$0 \in {\mathcal N}abla_s^- G$ and $0 \in {\mathcal N}abla_{s^\operatorname{pr}ime}^- G$.
This proves part (a) of the proposition.
Now suppose that $\{\mu_t\}_{t \in [0,\epsilonilon]}$ is a Wasserstein
geodesic. From \cite[Theorem 5.10]{Villani (2009)}, there is a $c$-convex
function $\epsilonilon f$ on $S$ so that if we define
its conjugate $(\epsilonilon f)^c$ using (\operatorname{Re}f{3.4}) then
$\{(s, \exp_s(\epsilonilon v)\}_{(s,v) \in {\mathcal S}upp({\mathcal N}u)}$
is contained in
{\mathcal B}etagin{equation} \lambdabel{3.19}
\Gammamma_\epsilonilon =
\left\{(s,x) \in S \times M \: : \: (\epsilonilon f)^c(x) =
\epsilonilon f(s) + {\mathcal f}rac12 d^2(s,x) {\mathcal R}ight\}.
\end{equation}
That is, for all $s^\operatorname{pr}ime \in S$,
{\mathcal B}etagin{equation} \lambdabel{3.20}
\epsilonilon f(s) + {\mathcal f}rac12 d^2(s,\exp_s(\epsilonilon v))
\le \epsilonilon f(s^\operatorname{pr}ime) +
{\mathcal f}rac12 d^2(s^\operatorname{pr}ime,\exp_s(\epsilonilon v)).
\end{equation}
Without loss of generality, we can shrink $\epsilonilon$ as desired.
Define a curve in $S$ by $s^\operatorname{pr}ime(u) = \exp_s(- u w^\operatorname{pr}ime)$ where
$w^\operatorname{pr}ime \in T_sS$, $u$ varies over
a small interval $(-\deltalta, \deltalta)$ and $\exp_s$ denotes here the
exponential map for the submanifold $S$.
Let $\{\gammamma_u : [0, \epsilonilon] {\mathcal R}ightarrow M \}_{u \in
(- \deltalta, \deltalta)}$ be a smooth
$1$-parameter family with $\gammamma_0(t) = \exp_s(tv)$,
$\gammamma_u(0) = s^\operatorname{pr}ime(u)$ and $\gammamma_u(\epsilonilon) =
\exp_s(\epsilonilon v)$. Let $L(u)$ be the length of $\gammamma_u$.
Then
{\mathcal B}etagin{equation} \lambdabel{3.21}
\epsilonilon f(s) + {\mathcal f}rac12 d^2(s,\exp_s(\epsilonilon v))
\le \epsilonilon f(s^\operatorname{pr}ime(u)) +
{\mathcal f}rac12 L^2(u).
\end{equation}
By the first variation formula,
{\mathcal B}etagin{equation} \lambdabel{3.22}
{\mathcal f}rac{d}{du} \operatorname{B}ig|_{u=0} {\mathcal f}rac12 L^2(u) = \epsilonilon \lambdangle v^T,
w^\operatorname{pr}ime {\mathcal R}ightarrowngle.
\end{equation}
It follows that $\epsilonilon v^T \in {\mathcal N}abla_s^- (\epsilonilon f)$, so
$v^T \in {\mathcal N}abla_s^- f$.
\end{proof}
{\mathcal B}etagin{remark}
The phenomenon of possible nonuniqueness,
in the normal component of the optimal transport between two measures supported
on convex hypersurfaces in Euclidean space, was recognized in
\cite[Proposition 4.3]{Gangbo-McCann (2000)}.
\end{remark}
{\mathcal B}etagin{example} \lambdabel{example3.23} Put $M = S^1 \times {\mathbb R}$. (It is noncompact, but this will
be irrelevant for the example.) Let $F \in C^\infty(S^1)$ be a positive
function. Put $S = \{(x, F(x)) : x \in S^1\}$.
Define $p : S {\mathcal R}ightarrow S^1 \times \{0\}$ by
$p(x, F(x)) = (x,0)$.
Let $\mu_0$ be a
smooth measure on $S$. Put $\mu_1 = p_* \mu_0$. The Wasserstein
geodesic from $\mu_0$ to $\mu_1$ moves the measure down along
vertical lines. Defining $f$ on $S$ by
$f(x, F(x)) = - {\mathcal f}rac12 \left( F(x) {\mathcal R}ight)^2$, one finds that
$v^T = {\mathcal N}abla f$. Compare with \cite[Corollary 2.6]{Castillon (2010)}.
\end{example}
{\mathcal S}ubsection{Tangent cones}
If $X$ is a complete length space with Alexandrov curvature bounded below
then one can define the tangent cone $T_xX$ at $x \in X$ as follows.
Let $\Sigmagma_x^\operatorname{pr}ime$ be the space of equivalence classes of minimal geodesic
segments emanating from $x$, with the equivalence relation identifying
two segments if they form a zero angle at $x$ (which means that one
segment is contained in the other). The metric on $\Sigmagma_x^\operatorname{pr}ime$ is
the angle. By definition, the space of directions $\Sigmagma_x$ is the
metric completion of $\Sigmagma_x^\operatorname{pr}ime$. The tangent cone $T_xX$ is the
union of ${\mathbb R}^+ \times \Sigmagma_x$ and a ``vertex'' point, with the metric
described in \cite[\S 10.9]{Burago-Burago-Ivanov (2001)}.
If $X$ is finite-dimensional then one can also describe $T_x X$ as the
pointed Gromov-Hausdorff limit $\lim_{\lambdambda {\mathcal R}ightarrow \infty}
\left( \lambdambda X, x {\mathcal R}ight)$. This latter description doesn't make sense
if $X$ is infinite-dimensional, whereas the preceding definition does.
If $M$ is a smooth compact connected Riemannian manifold, and it has
nonnegative sectional curvature, then $P(M)$ has nonnegative Alexandrov
curvature and one can talk about a tangent cone $T_{\mu} P(M)$
\cite[Appendix A]{Lott-Villani (2009)}. If
$M$ does not have nonnegative sectional curvature then $P(M)$ will not
have Alexandrov curvature bounded below. Nevertheless, one can still
define $T_{\mu} P(M)$ in the same way \cite[Section 3]{Ohta (2009)}.
As a point of terminology, what is called a tangent cone here, and in
\cite{Lott-Villani (2009)},
is called the ``abstract tangent space'' in
\cite{Gigli (2011)}. The linear part of the tangent cone is called the
``tangent space'' in \cite{Ambrosio-Gigli (2008)} and the
``space of gradients'' or ``tangent vector fields'' in \cite{Gigli (2011)}.
A minimal geodesic segment emanating from $\mu \in P(M)$ is determined by
a probability measure $\Pi$ on the space of constant-speed minimizing geodesics
{\mathcal B}etagin{equation}
\Gammamma = \{ \gammamma : [0,1] {\mathcal R}ightarrow M \: : \: L(\gammamma) =
d_M(\gammamma(0), \gammamma(1)) \},
\end{equation}
which has the property that under the time-zero
evaluation $e_0 : \Gammamma {\mathcal R}ightarrow M$,
we have $(e_0)_* \Gammamma = \mu$
\cite[Section 2]{Lott-Villani (2009)}. The corresponding geodesic segment
is given by $\mu_t = (e_t)_* \Pi$, where $e_t : \Gammamma {\mathcal R}ightarrow M$
is time-$t$ evaluation.
Using this characterization of minimizing geodesic segments, one can
describe $T_{\mu} P(M)$ as follows. With $\partiali : TM {\mathcal R}ightarrow M$ being
projection to the base, put
{\mathcal B}etagin{equation}
P_2(TM)_{\mu} = \{ {\mathcal N}u \in P_2(TM) \: : \: \partiali_* {\mathcal N}u = \mu \},
\end{equation}
where $P_2$ refers to measures with finite second moment.
Given ${\mathcal N}u^1, {\mathcal N}u^2 \in P_2(TM)_\mu$, decompose them as
{\mathcal B}etagin{equation}
{\mathcal N}u^i = \int_M {\mathcal N}u^i_m \: d\mu(m),
\end{equation}
with ${\mathcal N}u^i_m \in P_2(T_mM)$. Define $W_{\mu} ({\mathcal N}u^1, {\mathcal N}u^2)$ by
{\mathcal B}etagin{equation}
W_{\mu}^2 ({\mathcal N}u^1, {\mathcal N}u^2) = \int_M W_2^2({\mathcal N}u^1_m, {\mathcal N}u^2_m) \: d\mu(m).
\end{equation}
Let $\partialartialir_\mu$ be the set of elements ${\mathcal N}u \in P_2(TM)_\mu$ with the
property that $\{\partiali_* (E_t)_* {\mathcal N}u \}_{t \in [0, \epsilonilon]}$ describes
a minimizing Wasserstein geodesic for some $\epsilonilon$. Then
$T_{\mu} P(M)$ is isometric to the metric completion of $\partialartialir_{\mu}$
with respect to $W_\mu$ \cite[Theorem 5.5]{Gigli (2011)}.
We note that since $M$ is compact, any element of $\partialartialir_\mu$ has compact
support. This is because for ${\mathcal N}u$-almost all $v \in TM$, the
geodesic $\{\exp_{\partiali(v)} tv \}_{t \in [0, \epsilonilon]}$ must be
minimizing \cite[Proposition 2.10]{Lott-Villani (2009)}, so
$|v| \le \epsilonilon^{-1} \operatorname{diam}(M)$. \\ \\
{\it Proof of Theorem \operatorname{Re}f{theorem1.1} : } From Proposition
\operatorname{Re}f{proposition3.14},
$\partialartialir_{\mu}$ is the set of compactly-supported measures
${\mathcal N}u \in P(T_SM) {\mathcal S}ubset P(TM)$ so that $\partiali_* {\mathcal N}u = \mu$ and
there is a semiconvex function $f$ on $S$ such that
${\mathcal N}u$ has support on $\{v \in T_SM \: : \: v^T \in {\mathcal N}abla^- f\}$.
Because $\mu$ has full support on $S$ by assumption, ${\mathcal N}abla^- f$ is
single-valued at $\mu$-almost all $s \in S$. Equivalently, there is a
compactly-supported ${\mathcal N}u^N \in P(N_SM)$, which decomposes under
$\partiali_{N_SM} \: : \: N_SM {\mathcal R}ightarrow S$ as
${\mathcal N}u^N = \int_S {\mathcal N}u^N_s \: d\mu(s)$ with
${\mathcal N}u^N_s \in P_2(N_sM)$, so that for all
$F \in C(T_SM) = C(TS \oplus N_SM)$,
we have
{\mathcal B}etagin{equation}
\int_{T_SM} F \: d{\mathcal N}u = \int_S \int_{N_sM} F({\mathcal N}abla^-f(s), w) \:
d{\mathcal N}u^N_s(w) \: d\mu(s).
\end{equation}
Given two such measures ${\mathcal N}u^1, {\mathcal N}u^2$, it follows that
{\mathcal B}etagin{equation} \lambdabel{w2}
W^2_{\mu}({\mathcal N}u^1, {\mathcal N}u^2) = \int_S \lambdangle {\mathcal N}abla^-f^1, {\mathcal N}abla^-f^2 {\mathcal R}ightarrowngle
\: d\mu + \int_S W_2^2({\mathcal N}u^{1,N}_s, {\mathcal N}u^{2,N}_s) \: d\mu(s).
\end{equation}
Upon taking the metric completion of $\partialartialir_{\mu}$, the tangential term
in (\operatorname{Re}f{w2}) gives
the closure of the space of gradient vector fields
in the Hilbert space $L^2(TS, d\mu)$ of
square-integrable sections of $TS$
\cite[Proposition A.33]{Lott-Villani (2009)}.
The normal term gives
$\int_{s \in S} P_2(N_sM) \: d\mu(s)$, where the metric comes from
the last term in (\operatorname{Re}f{w2}). This proves the theorem. \qed
{\mathcal B}etagin{remark}
In Section 2 we considered transports in which the intermediate
measures were supported on hypersurfaces. This corresponds to
Wasserstein geodesics starting from $\mu$ for which the initial
velocity, as an element of $T_\mu P(M)$, comes from a section of
$T_SM$. In terms of Theorem \operatorname{Re}f{theorem1.1}, this means that the data
for the initial velocity consisted of a gradient vector field
${\mathcal N}abla \operatorname{ph}i$ on $S$
and a section ${\mathcal N}$ of $N_SM$, with the element of
$P_2(N_sM)$ being the delta measure at ${\mathcal N}(s)$.
\end{remark}
{\mathcal S}ubsection{Gauss map as an optimal transport map}
In this subsection, which is an addendum to the preceding subsections,
we give an example of optimal transport coming from the Gauss map of
a convex hypersurface in ${\mathbb R}^n$.
Let $S$ be the boundary of a compact convex subset of ${\mathbb R}^n$. We assume
that near any point, $S$ is locally the graph of a $C^2$-regular function. Let
$N : S {\mathcal R}ightarrow S^{n-1}$ be the outward unit normal.
Let $\kappa \in C^0(S)$ be the Gaussian curvature function, the
product of the principal values. Then
$N_* (\kappa \operatorname{dvol}_S) = \operatorname{dvol}_{S^{n-1}}$.
The optimal transport plans in ${\mathbb R}^n$ for the cost function
${\mathcal f}rac12 |m_1-m_2|^2$ are the same as those for
the cost function $- \lambdangle m_1, m_2 {\mathcal R}ightarrowngle$.
Given $R > 0$, $s \in S$ and $x \in S^{n-1}$, the cost
function of the points $s$ and $Rx$ becomes $- \: R \lambdangle s,x {\mathcal R}ightarrowngle$.
Considering an optimal transport problem between $S$ and
$R \operatorname{CD}ot S^{n-1}$,
the optimal transport plans for the cost function
$- \: R \lambdangle s,x {\mathcal R}ightarrowngle$ are the same as those for the cost function
$- \: \lambdangle s,x {\mathcal R}ightarrowngle$.
This motivates considering the cost function
$c : S \times S^{n-1} {\mathcal R}ightarrow {\mathbb R}$
given by $c(s,x) \: = \: - \lambdangle s, x {\mathcal R}ightarrowngle$.
Here we imagine taking $R {\mathcal R}ightarrow \infty$ so that
$S^{n-1}$ is a ``sphere at infinity'', not
an embedded sphere in ${\mathbb R}^n$, although when we write $\lambdangle s, x {\mathcal R}ightarrowngle$
we are treating $x$ as a unit vector.
The analog of (\operatorname{Re}f{3.15}) is
{\mathcal B}etagin{equation}
\Gammamma_t = \{(s,x) \in S \times S^{n-1} \: : \:
\eta_t(x) = tf(s) - \lambdangle s,x {\mathcal R}ightarrowngle \}.
\end{equation}
Now $\eta_t(x) = tf(s) - \lambdangle s,x {\mathcal R}ightarrowngle$ if
for all $s^\operatorname{pr}ime \in S$, we have
{\mathcal B}etagin{equation}
t f(s) - \lambdangle s,x {\mathcal R}ightarrowngle
\le t f(s^\operatorname{pr}ime) - \lambdangle s^\operatorname{pr}ime, x {\mathcal R}ightarrowngle.
\end{equation}
Taking $f = 0$, one sees that for all $s \in S$ we have
$(s, N(s)) \in \Gammamma_1$, since the convexity of $S$ implies that
$\lambdangle s^\operatorname{pr}ime - s, N(s) {\mathcal R}ightarrowngle \le 0$ for all $s^\operatorname{pr}ime \in S$.
Hence $N$ is an optimal transport map from the measure $\kappa \operatorname{dvol}_S$
on $S$, to the measure $\operatorname{dvol}_{S^{n-1}}$ on $S^{n-1}$.
{\mathcal B}etagin{remark}
In a different direction,
Aleksandrov's problem of realizing a given curvature function
was related to optimal transport on a sphere in \cite{Oliker (2007)},
using a
certain cost function; see also
\cite{Bertrand (2015)}.
\end{remark}
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\end{document} | math |
ಇಂದು ಸಂಜೆ 4ಗಂಟೆಗೆ ವಿವಾ ಟೆಕ್ ನಲ್ಲಿ ಪ್ರಧಾನಿ ಭಾಷಣ ಇಂದು ಸಂಜೆ 4ಗಂಟೆಗೆ ವಿವಾ ಟೆಕ್ 5ನೇ ಆವೃತ್ತಿಯಲ್ಲಿ ವಿಡಿಯೋ ಕಾನ್ಫರೆನ್ಸ್ ಮೂಲಕ ಪ್ರಧಾನಿ ನರೇಂದ್ರ ಮೋದಿಯವರು ಮಾತನಾಡಲಿದ್ದಾರೆ. ಈ ಕಾರ್ಯಕ್ರಮದಲ್ಲಿ ಫ್ರಾನ್ಸ್ ಅಧ್ಯಕ್ಷ ಎಮ್ಯಾನುಯೆಲ್ ಮಾರ್ಕೋನ್, ಸ್ಪೇನ್ ಪ್ರಧಾನಿ ಪೆಡ್ರೋ ಸಾಂಚೆಜ್ ಮತ್ತು ಐರೋಪ್ಯ ದೇಶಗಳ ಸಚಿವರು, ಸಂಸದರು ಪ್ರಮುಖ ಭಾಷಣಕಾರರಾಗಿ ಪಾಲ್ಗೊಳ್ಳಲಿದ್ದಾರೆ. ಕಾರ್ಪೋರೇಟ್ ವಲಯದಲ್ಲಿ ದಿಗ್ಗಜರಾಗಿ ಗುರುತಿಸಿಕೊಂಡಿರುವ ಆಯಪಲ್ ಕಂಪನಿ ಸಿಇಒ ಟಿಮ್ ಕುಕ್, ಫೇಸ್ಬುಕ್ ಅಧ್ಯಕ್ಷ, ಸಿಇಒ ಮಾರ್ಕ್ ಜುಕನ್ ಬರ್ಗ್, ಮೈಕ್ರೋಸಾಫ್ಟ್ ಅಧ್ಯಕ್ಷ ಬ್ರಾಡ್ ಸ್ಮಿತ್ ಕೂಡ ವಿವಾ ಟೆಕ್ ನಲ್ಲಿ ಭಾಗವಹಿಸಲಿದ್ದಾರೆ.ವಿವಾಟೆಕ್ 2021ರ 5ನೇ ಆವೃತ್ತಿಯಲ್ಲಿ ಮಾತನಾಡುವಂತೆ ಪ್ರಧಾನಮಂತ್ರಿ ನರೇಂದ್ರ ಮೋದಿಯವರನ್ನು ಮುಖ್ಯಅತಿಥಿಯನ್ನಾಗಿ ಆಹ್ವಾನಿಸಲಾಗಿದೆ ಎಂದು ಪ್ರಧಾನಮಂತ್ರಿ ಕಚೇರಿ ಹೇಳಿಕೆ ಬಿಡುಗಡೆ ಮಾಡಿದೆ. ವಿಡಿಯೋ ಕಾನ್ಫರೆನ್ಸ್ ನಲ್ಲಿ ಟೆಕ್ ಮತ್ತು ಸ್ಟಾರ್ಟ್ ಆಯಪ್ ಜಗತ್ತಿನಲ್ಲಿ ಭಾರತದ ಪ್ರಗತಿಯ ಬಗ್ಗೆ ಮಾತನಾಡಲಿದ್ದೇನೆ ಎಂದು ಪ್ರಧಾನಿ ಟ್ವೀಟ್ ಮಾಡಿದ್ದಾರೆ. ವಿವಾ ಟೆಕ್ ಎಂಬುದು ಯುರೋಪ್ ನ ಅತ್ಯಂತ ದೊಡ್ಡ ಡಿಜಿಟಲ್ ಮತ್ತು ಸ್ಟಾರ್ಟ್ ಅಪ್ ಕಾರ್ಯಕ್ರಮವಾಗಿದೆ. ಈ ವರ್ಷದ ವಿವಾ ಟೆಕ್ ಜೂ.16 ರಿಂದ 19 ರವರೆಗೆ ನಡೆಯಲಿದೆ. ಕೋವಿಡ್19ನಿಂದ ಗುಣಮುಖರಾಗಿದ್ದ 34 ವರ್ಷದ ವ್ಯಕ್ತಿಯಲ್ಲಿ ಗ್ರೀನ್ ಫಂಗಸ್ ಪತ್ತೆ : ಮುಂಬೈಗೆ ರೋಗಿಯ ಏರ್ ಲಿಫ್ಟ್ | kannad |
Susquehanna University will host a screening of the documentary Someone You Love: The HPV Epidemic at 7 p.m. Wednesday, April 3, in the Degenstein Center Theater.
Narrated by Vanessa Williams, this documentary looks at the lives of five women affected by HPV, the widely misunderstood and controversial virus that causes several types of cancer, including cervical.
Each of these women has an intimate story to tell. For Susie, Tamika and Christine, it's a story of survivorship that comes with misconceptions, stigma, shame, heartbreak, pain and triumph. For the Forbes family, it's about coping with the loss of their daughter, Kristen, and trying to prevent it from happening to others like her. The cameras also follow Kelly, who at 31 years began her cancer journey, on her epic battle to save her marriage, her career, her family and ultimately, her life.
Eighty percent of all people under 50 years of age will have a strain of the virus at some point in their lives and most will not even realize they have it. Cervical cancer is almost exclusively caused by HPV and it is the second leading cancer in women. Worldwide, cervical cancer kills over 250,000 women every year.
Susquehanna's screening is cosponsored by the Student Health Center and the Panhellenic Council.
The documentary will be followed by remarks and a question-and-answer session from documentary producer Cheryl Staurulakis, whose son, Nick, is a 2016 Susquehanna graduate.
Someone You Love: The HPV Epidemic is directed by Frederic Lumiere and written by Mark Hefti. | english |
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This one-of-a-kind item has been sold. Please contact us if you are interested in commissioning a similar item. | english |
There’s nothing worse than buying a piece of furniture only to find that when it gets delivered it either doesn’t fitor likewise looks ridiculously small in the space you’ve got. When it comes to purchasing furniture, there are three simple things to remember, size, shape and style. A measuring tape is your best friend, and the golden rule is that you should always measure the space you have, taking into account the width of your doorways and any surrounding furniture before you commit to buying anything – no matter how much you love the design.
First things first, consider the proportionsof your room. Can it accommodate a large sofa and chairs, or will the sofatake up too much space on its own? If your room is small, you should probably move away from large three-piece suites and consider smaller options, or individual pieces that all work together. If you have a larger living room, then you are in a goodposition to be able to fit a three-piece suite, but still,remember the golden rule of measure, and measure again. The range of Arlo and Jacob sofascome in a variety of deep and shallow proportions designed to fit the proportionsof multiple living rooms.
Sofa and furniture depths can vary considerably. Think about the width of your space and how much room you need in front of your sofa for coffee tables, TVs and walkways. If you have the depth of space have you considered a chaise or corner units for extra comfort and to make full use of the additionalfloor area. If you’re tight on depth, there are plenty of compactsofas that have shallower seats but are still equally as comfortable as deep sofas.
If you’re lucky enough to live in a period property that has high ceilings, then high-backed furniturewill look grand and naturally fit into space. However, if you live in a more modern housewith lower ceilings, you may want to consider lower-backed piecesas the shorter proportions will help convey more head space than is actuallythere. Likewise consider the architecture such as ceiling roses, architraves and picture rails. If these are ornate and decorative, then a sofa with details like buttoning and fluting may work; however, if the spaceis full of clean, simple lines, then fuss-free modern shapes will suit better. The golden rule applies here; be sure to measure how deep the spaceis and the depth of the furniture.
Some people forget that their furniture needs to clean the door both when getting it into your home, and when it’s sat in position. A new piece of furniture will quickly get damaged if a door continuously bangs into it (this also goes for cupboard doors and drawers too).
The style of your furniture ultimately comes down to your personaltaste, but it’s also worth thinking about whether your chosen design reallygoes with the décor in your living room. Fabric can have a real impact on the overall look, for instance, a bright velvet fabricon a more traditional shape can make it look far more modern.
This post has been sponsored by Arlo & Jacob. | english |
பிபின் ராவத் பற்றிய பத்து தகவல்கள் இந்தியாவுக்கு தனியாக முப்படைகளின் தலைமை தளபதி என்ற பதவி உருவாக்கப்படும், இது நமது படைகளை சிறப்பாக செயலாற்ற வைக்கும் என பிரதமர் நரேந்திர மோடியால் அறிவிக்கப்பட்ட சில நாட்களிலேயே இந்தியாவின் முதல் முப்படை தலைமை தளபதியாக பொறுப்பேற்றுக் கொண்டவர் பிபின் ராவத் பற்றிய பத்து தகவல்கள். 1. தேசிய பாதுகாப்பு அகாடமி, இந்திய ராணுவ அகாடமியில் மாணவராக பயிற்சி பெற்றவர் பிபின் ராவத். 2. தந்தை பணியாற்றிய அதே பிரிவில் 1978ஆம் ஆண்டு பிபின் ராவத் ராணுவத்தில் இணைந்தார். 3. படைப்பிரிவின் தளபதி, கமாண்டிங் இன் சீப், தெற்கு கட்டளை அதிகாரி, உள்ளிட்ட பல்வேறு உயர் பொறுப்புகளை பிபின் ராவத் வகித்துள்ளார். 4. கர்னல் ராணுவ செயலாளர், ராணுவ இணைச் செயலாளராகவும் பிபின் ராவத் பணியாற்றியுள்ளார். 5. ஐக்கிய நாடுகளின் அமைதி காக்கும் படையிலும் பிபின் ராவத் அங்கம் வகித்துள்ளார். 6. வடகிழக்கு மாநில எல்லைப்பகுதிகள், இந்திய சீன எல்லைப்பகுதி என பல்வேறு களங்களில் பணியாற்றிய அனுபவம் பிபின் ராவத்திற்கு உண்டு. 7. இந்திய ராணுவத்தின் 27 வது தலைமை தளபதியாக டிசம்பர் 31, 2016 முதல் பொறுப்பேற்றார். 8. பிபின் ராவத்தின் பதவிக்காலம் முடிவடையும் அன்று முப்படைகளின் தலைமை தளபதி என்ற புதிய பொறுப்பு உருவாக்கப்பட்டது. 9. முப்படைகளின் தலைமை தளபதி என்ற புதிய பதவி உருவாக்கப்பட்ட பிறகு முதல் நபராக பிபின் ராவத் அந்த பதவிக்கு நியமிக்கப்பட்டார். 10. பிபின் ராவத்தின் சேவையை பாராட்டி பரம் விசிஷ்ட் சேவா விருது, யுத்தம் யுத்த சேவா விருது, சேனா விருது என பல்வேறு விருதுகள் வழங்கப்பட்டுள்ளது. Advertisement: SHARE | tamil |
<TS language="nb" version="2.1">
<context>
<name>AddressBookPage</name>
<message>
<source>Right-click to edit address or label</source>
<translation>Høyreklikk for å redigere adressen eller beskrivelsen</translation>
</message>
<message>
<source>Create a new address</source>
<translation>Opprett en ny adresse</translation>
</message>
<message>
<source>&New</source>
<translation>&Ny</translation>
</message>
<message>
<source>Copy the currently selected address to the system clipboard</source>
<translation>Kopier den valgte adressen til utklippstavlen</translation>
</message>
<message>
<source>&Copy</source>
<translation>&Kopier</translation>
</message>
<message>
<source>C&lose</source>
<translation>&Lukk</translation>
</message>
<message>
<source>Delete the currently selected address from the list</source>
<translation>Slett den valgte adressen fra listen</translation>
</message>
<message>
<source>Enter address or label to search</source>
<translation>Oppgi adresse, eller stikkord, for å søke</translation>
</message>
<message>
<source>Export the data in the current tab to a file</source>
<translation>Eksporter data i den valgte fliken til en fil</translation>
</message>
<message>
<source>&Export</source>
<translation>&Eksport</translation>
</message>
<message>
<source>&Delete</source>
<translation>&Slett</translation>
</message>
<message>
<source>Choose the address to send coins to</source>
<translation>Velg en adresse å sende mynter til</translation>
</message>
<message>
<source>Choose the address to receive coins with</source>
<translation>Velg adressen som skal motta myntene</translation>
</message>
<message>
<source>C&hoose</source>
<translation>&Velg</translation>
</message>
<message>
<source>Sending addresses</source>
<translation>Avsender adresser</translation>
</message>
<message>
<source>Receiving addresses</source>
<translation>Mottager adresser</translation>
</message>
<message>
<source>These are your Bitcoin addresses for sending payments. Always check the amount and the receiving address before sending coins.</source>
<translation>Dette er dine Bitcoin adresser for å sende å sende betalinger. Husk å sjekke beløp og mottager adresser før du sender mynter.</translation>
</message>
<message>
<source>These are your Bitcoin addresses for receiving payments. Use the 'Create new receiving address' button in the receive tab to create new addresses.
Signing is only possible with addresses of the type 'legacy'.</source>
<translation>Dette er dine Bitcoin adresser for å motta betalinger. Bruk 'Lag ny mottaksadresse' knappen i motta tabben for å lage nye adresser. Signering er bare mulig for adresser av typen 'legacy'.</translation>
</message>
<message>
<source>&Copy Address</source>
<translation>&Kopier adresse</translation>
</message>
<message>
<source>Copy &Label</source>
<translation>Kopier &beskrivelse</translation>
</message>
<message>
<source>&Edit</source>
<translation>R&ediger</translation>
</message>
<message>
<source>Export Address List</source>
<translation>Eksporter adresse listen</translation>
</message>
<message>
<source>Comma separated file (*.csv)</source>
<translation>Komma separert fil (*.csv)</translation>
</message>
<message>
<source>Exporting Failed</source>
<translation>Eksporten feilet</translation>
</message>
<message>
<source>There was an error trying to save the address list to %1. Please try again.</source>
<translation>Fet oppstod en feil ved lagring av adresselisten til %1. Vennligst prøv igjen.</translation>
</message>
</context>
<context>
<name>AddressTableModel</name>
<message>
<source>Label</source>
<translation>Beskrivelse</translation>
</message>
<message>
<source>Address</source>
<translation>Adresse</translation>
</message>
<message>
<source>(no label)</source>
<translation>(ingen beskrivelse)</translation>
</message>
</context>
<context>
<name>AskPassphraseDialog</name>
<message>
<source>Passphrase Dialog</source>
<translation>Passord dialog</translation>
</message>
<message>
<source>Enter passphrase</source>
<translation>Oppgi passord setning</translation>
</message>
<message>
<source>New passphrase</source>
<translation>Ny passord setning</translation>
</message>
<message>
<source>Repeat new passphrase</source>
<translation>Repeter passorsetningen</translation>
</message>
<message>
<source>Show passphrase</source>
<translation>Vis adgangsfrase</translation>
</message>
<message>
<source>Encrypt wallet</source>
<translation>Krypter lommeboken</translation>
</message>
<message>
<source>This operation needs your wallet passphrase to unlock the wallet.</source>
<translation>Denne operasjonen krever passordsetningen for å låse opp lommeboken.</translation>
</message>
<message>
<source>Unlock wallet</source>
<translation>Lås opp lommeboken</translation>
</message>
<message>
<source>This operation needs your wallet passphrase to decrypt the wallet.</source>
<translation>Denne operasjonen krever passordsetningen for å dekryptere lommeboken.</translation>
</message>
<message>
<source>Decrypt wallet</source>
<translation>Dekrypter lommeboken</translation>
</message>
<message>
<source>Change passphrase</source>
<translation>Endre passordsetningen</translation>
</message>
<message>
<source>Confirm wallet encryption</source>
<translation>Bekreft kryptering av lommeboken</translation>
</message>
<message>
<source>Warning: If you encrypt your wallet and lose your passphrase, you will <b>LOSE ALL OF YOUR BITCOINS</b>!</source>
<translation>Advarsel: Dersom du krypterer lommeboken og mister passordsetningen vil du <b>MISTE ALLE DINE BITCOIN</b>!</translation>
</message>
<message>
<source>Are you sure you wish to encrypt your wallet?</source>
<translation>Er du sikker på at du vil kryptere lommeboken?</translation>
</message>
<message>
<source>Wallet encrypted</source>
<translation>Lommeboken er kryptert</translation>
</message>
<message>
<source>Enter the new passphrase for the wallet.<br/>Please use a passphrase of <b>ten or more random characters</b>, or <b>eight or more words</b>.</source>
<translation>Angi den nye passordfrasen for lommeboken.<br/> Vennglist du bruker en passordfrase <b> ti eller tilfeldige tegn </b>, eller <b> åtte eller flere ord.</translation>
</message>
<message>
<source>Enter the old passphrase and new passphrase for the wallet.</source>
<translation>Svriv inn den gamle passfrasen og den nye passordfrasen for lommeboken.</translation>
</message>
<message>
<source>Remember that encrypting your wallet cannot fully protect your bitcoins from being stolen by malware infecting your computer.</source>
<translation>Husk at å kryptere lommeboken ikke vil beskytte dine bitcoins fullstendig fra å bli stjålet av skadevare som infiserer datamaskinen din.</translation>
</message>
<message>
<source>Wallet to be encrypted</source>
<translation>Lommebok som skal bli kryptert</translation>
</message>
<message>
<source>Your wallet is about to be encrypted. </source>
<translation>Din lommebok er i ferd med å bli kryptert.</translation>
</message>
<message>
<source>Your wallet is now encrypted. </source>
<translation>Din lommebok er nå kryptert.</translation>
</message>
<message>
<source>IMPORTANT: Any previous backups you have made of your wallet file should be replaced with the newly generated, encrypted wallet file. For security reasons, previous backups of the unencrypted wallet file will become useless as soon as you start using the new, encrypted wallet.</source>
<translation>VIKTIG: Alle tidligere sikkerhetskopier du har tatt av lommebokfilen bør erstattes med den nye krypterte lommebokfilen. Av sikkerhetsgrunner vil tidligere sikkerhetskopier av lommebokfilen bli ubrukelige når du begynner å bruke den ny kypterte lommeboken.</translation>
</message>
<message>
<source>Wallet encryption failed</source>
<translation>Kryptering av lommeboken feilet</translation>
</message>
<message>
<source>Wallet encryption failed due to an internal error. Your wallet was not encrypted.</source>
<translation>Lommebokkrypteringen feilet pga. en intern feil. Lommeboken din ble ikke kryptert.</translation>
</message>
<message>
<source>The supplied passphrases do not match.</source>
<translation>De oppgitte passordsetningene er forskjellige.</translation>
</message>
<message>
<source>Wallet unlock failed</source>
<translation>Opplåsing av lommeboken feilet</translation>
</message>
<message>
<source>The passphrase entered for the wallet decryption was incorrect.</source>
<translation>Passordsetningen som ble oppgitt for å dekryptere lommeboken var feil.</translation>
</message>
<message>
<source>Wallet decryption failed</source>
<translation>Dekryptering av lommeboken feilet</translation>
</message>
<message>
<source>Wallet passphrase was successfully changed.</source>
<translation>Passordsetningen for lommeboken ble endret</translation>
</message>
<message>
<source>Warning: The Caps Lock key is on!</source>
<translation>Advarsel: Caps Lock er på!</translation>
</message>
</context>
<context>
<name>BanTableModel</name>
<message>
<source>IP/Netmask</source>
<translation>IP/Nettmaske</translation>
</message>
<message>
<source>Banned Until</source>
<translation>Utestengt Til</translation>
</message>
</context>
<context>
<name>BitcoinGUI</name>
<message>
<source>Sign &message...</source>
<translation>Signer &melding</translation>
</message>
<message>
<source>Synchronizing with network...</source>
<translation>Synkroniserer med nettverket</translation>
</message>
<message>
<source>&Overview</source>
<translation>&Oversikt</translation>
</message>
<message>
<source>Show general overview of wallet</source>
<translation>Vis generell oversikt over lommeboken</translation>
</message>
<message>
<source>&Transactions</source>
<translation>&Transaksjoner</translation>
</message>
<message>
<source>Browse transaction history</source>
<translation>Bla gjennom transaksjoner</translation>
</message>
<message>
<source>E&xit</source>
<translation>&Avslutt</translation>
</message>
<message>
<source>Quit application</source>
<translation>Avslutt program</translation>
</message>
<message>
<source>&About %1</source>
<translation>&Om %1</translation>
</message>
<message>
<source>Show information about %1</source>
<translation>Vis informasjon om %1</translation>
</message>
<message>
<source>About &Qt</source>
<translation>Om &Qt</translation>
</message>
<message>
<source>Show information about Qt</source>
<translation>Vis informasjon om Qt</translation>
</message>
<message>
<source>&Options...</source>
<translation>&Alternativer</translation>
</message>
<message>
<source>Modify configuration options for %1</source>
<translation>Endre konfigurasjonsalternativer for %1</translation>
</message>
<message>
<source>&Encrypt Wallet...</source>
<translation>&Krypter lommebok...</translation>
</message>
<message>
<source>&Backup Wallet...</source>
<translation>&Sikkerhetskopier lommebok</translation>
</message>
<message>
<source>&Change Passphrase...</source>
<translation>&Endre passordsetning</translation>
</message>
<message>
<source>Open &URI...</source>
<translation>Åpne &URI</translation>
</message>
<message>
<source>Create Wallet...</source>
<translation>Lag lommebok...</translation>
</message>
<message>
<source>Create a new wallet</source>
<translation>Lag en ny lommebok</translation>
</message>
<message>
<source>Wallet:</source>
<translation>Lommebok:</translation>
</message>
<message>
<source>Click to disable network activity.</source>
<translation>Klikk for å slå av nettverksaktivitet.</translation>
</message>
<message>
<source>Network activity disabled.</source>
<translation>Nettverksaktivitet er slått av</translation>
</message>
<message>
<source>Click to enable network activity again.</source>
<translation>Klikk for å slå på nettverksaktivitet igjen.</translation>
</message>
<message>
<source>Syncing Headers (%1%)...</source>
<translation>Synkroniserer Headers (%1%)...</translation>
</message>
<message>
<source>Reindexing blocks on disk...</source>
<translation>Reindekserer blokker på disken</translation>
</message>
<message>
<source>Proxy is <b>enabled</b>: %1</source>
<translation>Proxy er <b>slått på</b>: %1</translation>
</message>
<message>
<source>Send coins to a Bitcoin address</source>
<translation>Send mynter til en Bitcoin adresse</translation>
</message>
<message>
<source>Backup wallet to another location</source>
<translation>Sikkerhetskopier lommeboken til en annen lokasjon</translation>
</message>
<message>
<source>Change the passphrase used for wallet encryption</source>
<translation>Endre passordsetningen for kryptering av lommeboken</translation>
</message>
<message>
<source>&Verify message...</source>
<translation>&Verifiser meldingen...</translation>
</message>
<message>
<source>&Send</source>
<translation>&Sende</translation>
</message>
<message>
<source>&Receive</source>
<translation>&Motta</translation>
</message>
<message>
<source>&Show / Hide</source>
<translation>Vi&s / Skjul</translation>
</message>
<message>
<source>Show or hide the main Window</source>
<translation>Vis, eller skjul, hovedvinduet</translation>
</message>
<message>
<source>Encrypt the private keys that belong to your wallet</source>
<translation>Krypter de private nøklene som tilhører lommeboken din</translation>
</message>
<message>
<source>Sign messages with your Bitcoin addresses to prove you own them</source>
<translation>Signer meldingene med Bitcoin adresse for å bevise at diu eier dem</translation>
</message>
<message>
<source>Verify messages to ensure they were signed with specified Bitcoin addresses</source>
<translation>Verifiser meldinger for å sikre at de ble signert med en angitt Bitcoin adresse</translation>
</message>
<message>
<source>&File</source>
<translation>&Fil</translation>
</message>
<message>
<source>&Settings</source>
<translation>Inn&stillinger</translation>
</message>
<message>
<source>&Help</source>
<translation>&Hjelp</translation>
</message>
<message>
<source>Tabs toolbar</source>
<translation>Hjelpelinje for fliker</translation>
</message>
<message>
<source>Request payments (generates QR codes and bitcoin: URIs)</source>
<translation>Be om betalinger (genererer QR-koder og bitcoin-URIer)</translation>
</message>
<message>
<source>Show the list of used sending addresses and labels</source>
<translation>Vis lista over brukte sendeadresser og merkelapper</translation>
</message>
<message>
<source>Show the list of used receiving addresses and labels</source>
<translation>Vis lista over brukte mottakeradresser og merkelapper</translation>
</message>
<message>
<source>&Command-line options</source>
<translation>&Kommandolinjealternativer</translation>
</message>
<message numerus="yes">
<source>%n active connection(s) to Bitcoin network</source>
<translation><numerusform>%n aktiv tilkobling til Bitcoin nettverket</numerusform><numerusform>%n aktive tilkoblinger til Bitcoin nettverket</numerusform></translation>
</message>
<message>
<source>Indexing blocks on disk...</source>
<translation>Indekserer blokker på disken...</translation>
</message>
<message>
<source>Processing blocks on disk...</source>
<translation>Behandler blokker på disken…</translation>
</message>
<message numerus="yes">
<source>Processed %n block(s) of transaction history.</source>
<translation><numerusform>Har prosessert %n blokk av transaksjonshistorien</numerusform><numerusform>Har prosessert %n blokker av transaksjonshistorien</numerusform></translation>
</message>
<message>
<source>%1 behind</source>
<translation>%1 bak</translation>
</message>
<message>
<source>Last received block was generated %1 ago.</source>
<translation>Siste mottatte blokk ble generert for %1 siden.</translation>
</message>
<message>
<source>Transactions after this will not yet be visible.</source>
<translation>Transaksjoner etter dette vil ikke være synlige ennå.</translation>
</message>
<message>
<source>Error</source>
<translation>Feilmelding</translation>
</message>
<message>
<source>Warning</source>
<translation>Advarsel</translation>
</message>
<message>
<source>Information</source>
<translation>Informasjon</translation>
</message>
<message>
<source>Up to date</source>
<translation>Oppdatert</translation>
</message>
<message>
<source>&Load PSBT from file...</source>
<translation>&Last PSBT fra fil...</translation>
</message>
<message>
<source>Load Partially Signed Bitcoin Transaction</source>
<translation>Last delvis signert Bitcoin transaksjon</translation>
</message>
<message>
<source>Load PSBT from clipboard...</source>
<translation>Last PSBT fra utklippstavlen...</translation>
</message>
<message>
<source>Load Partially Signed Bitcoin Transaction from clipboard</source>
<translation>Last Delvis Signert Bitcoin Transaksjon fra utklippstavle</translation>
</message>
<message>
<source>Node window</source>
<translation>Nodevindu</translation>
</message>
<message>
<source>Open node debugging and diagnostic console</source>
<translation>Åpne nodens konsoll for feilsøk og diagnostikk</translation>
</message>
<message>
<source>&Sending addresses</source>
<translation>&Avsender adresser</translation>
</message>
<message>
<source>&Receiving addresses</source>
<translation>&Mottaker adresser</translation>
</message>
<message>
<source>Open a bitcoin: URI</source>
<translation>Åpne en bitcoin: URI</translation>
</message>
<message>
<source>Open Wallet</source>
<translation>Åpne Lommebok</translation>
</message>
<message>
<source>Open a wallet</source>
<translation>Åpne en lommebok</translation>
</message>
<message>
<source>Close Wallet...</source>
<translation>Lukk Lommebok...</translation>
</message>
<message>
<source>Close wallet</source>
<translation>Lukk lommebok</translation>
</message>
<message>
<source>Close All Wallets...</source>
<translation>Lukk alle lommebøker...</translation>
</message>
<message>
<source>Close all wallets</source>
<translation>Lukk alle lommebøker</translation>
</message>
<message>
<source>Show the %1 help message to get a list with possible Bitcoin command-line options</source>
<translation>Vis %1-hjelpemeldingen for å få en liste over mulige Bitcoin-kommandolinjealternativer</translation>
</message>
<message>
<source>&Mask values</source>
<translation>&Masker verdier</translation>
</message>
<message>
<source>Mask the values in the Overview tab</source>
<translation>Masker verdiene i oversiktstabben</translation>
</message>
<message>
<source>default wallet</source>
<translation>standard lommebok</translation>
</message>
<message>
<source>No wallets available</source>
<translation>Ingen lommebøker tilgjengelig</translation>
</message>
<message>
<source>&Window</source>
<translation>&Vindu</translation>
</message>
<message>
<source>Minimize</source>
<translation>Minimer</translation>
</message>
<message>
<source>Zoom</source>
<translation>Zoom</translation>
</message>
<message>
<source>Main Window</source>
<translation>Hovedvindu</translation>
</message>
<message>
<source>%1 client</source>
<translation>%1-klient</translation>
</message>
<message>
<source>Connecting to peers...</source>
<translation>Kobler til peers...</translation>
</message>
<message>
<source>Catching up...</source>
<translation>Tar igjen…</translation>
</message>
<message>
<source>Error: %1</source>
<translation>Feil: %1</translation>
</message>
<message>
<source>Warning: %1</source>
<translation>Advarsel: %1</translation>
</message>
<message>
<source>Date: %1
</source>
<translation>Dato: %1
</translation>
</message>
<message>
<source>Amount: %1
</source>
<translation>Mengde: %1
</translation>
</message>
<message>
<source>Wallet: %1
</source>
<translation>Lommeboik: %1
</translation>
</message>
<message>
<source>Type: %1
</source>
<translation>Type: %1
</translation>
</message>
<message>
<source>Label: %1
</source>
<translation>Merkelapp: %1
</translation>
</message>
<message>
<source>Address: %1
</source>
<translation>Adresse: %1
</translation>
</message>
<message>
<source>Sent transaction</source>
<translation>Sendt transaksjon</translation>
</message>
<message>
<source>Incoming transaction</source>
<translation>Innkommende transaksjon</translation>
</message>
<message>
<source>HD key generation is <b>enabled</b></source>
<translation>HD nøkkel generering er <b>slått på</b></translation>
</message>
<message>
<source>HD key generation is <b>disabled</b></source>
<translation>HD nøkkel generering er <b>slått av</b></translation>
</message>
<message>
<source>Private key <b>disabled</b></source>
<translation>Privat nøkkel <b>deaktivert</b></translation>
</message>
<message>
<source>Wallet is <b>encrypted</b> and currently <b>unlocked</b></source>
<translation>Lommeboken er <b>kryptert</b> og for tiden <b>låst opp</b></translation>
</message>
<message>
<source>Wallet is <b>encrypted</b> and currently <b>locked</b></source>
<translation>Lommeboken er <b>kryptert</b> og for tiden <b>låst</b></translation>
</message>
<message>
<source>Original message:</source>
<translation>Opprinnelig melding</translation>
</message>
<message>
<source>A fatal error occurred. %1 can no longer continue safely and will quit.</source>
<translation>En fatal feil har skjedd. %1 kan ikke lenger trygt fortsette og kommer til å avslutte.</translation>
</message>
</context>
<context>
<name>CoinControlDialog</name>
<message>
<source>Coin Selection</source>
<translation>Mynt Valg</translation>
</message>
<message>
<source>Quantity:</source>
<translation>Mengde:</translation>
</message>
<message>
<source>Bytes:</source>
<translation>Bytes:</translation>
</message>
<message>
<source>Amount:</source>
<translation>Beløp:</translation>
</message>
<message>
<source>Fee:</source>
<translation>Avgift:</translation>
</message>
<message>
<source>Dust:</source>
<translation>Støv:</translation>
</message>
<message>
<source>After Fee:</source>
<translation>Totalt:</translation>
</message>
<message>
<source>Change:</source>
<translation>Veksel:</translation>
</message>
<message>
<source>(un)select all</source>
<translation>velg (fjern) alle</translation>
</message>
<message>
<source>Tree mode</source>
<translation>Trevisning</translation>
</message>
<message>
<source>List mode</source>
<translation>Listevisning</translation>
</message>
<message>
<source>Amount</source>
<translation>Beløp</translation>
</message>
<message>
<source>Received with label</source>
<translation>Mottatt med merkelapp</translation>
</message>
<message>
<source>Received with address</source>
<translation>Mottatt med adresse</translation>
</message>
<message>
<source>Date</source>
<translation>Dato</translation>
</message>
<message>
<source>Confirmations</source>
<translation>Bekreftelser</translation>
</message>
<message>
<source>Confirmed</source>
<translation>Bekreftet</translation>
</message>
<message>
<source>Copy address</source>
<translation>Kopiér adresse</translation>
</message>
<message>
<source>Copy label</source>
<translation>Kopiér merkelapp</translation>
</message>
<message>
<source>Copy amount</source>
<translation>Kopiér beløp</translation>
</message>
<message>
<source>Copy transaction ID</source>
<translation>Kopier transaksjons-ID</translation>
</message>
<message>
<source>Lock unspent</source>
<translation>Lås ubrukte</translation>
</message>
<message>
<source>Unlock unspent</source>
<translation>Lås opp ubrukte</translation>
</message>
<message>
<source>Copy quantity</source>
<translation>Kopiér mengde</translation>
</message>
<message>
<source>Copy fee</source>
<translation>Kopiér gebyr</translation>
</message>
<message>
<source>Copy after fee</source>
<translation>Kopiér totalt</translation>
</message>
<message>
<source>Copy bytes</source>
<translation>Kopiér bytes</translation>
</message>
<message>
<source>Copy dust</source>
<translation>Kopiér støv</translation>
</message>
<message>
<source>Copy change</source>
<translation>Kopier veksel</translation>
</message>
<message>
<source>(%1 locked)</source>
<translation>(%1 låst)</translation>
</message>
<message>
<source>yes</source>
<translation>ja</translation>
</message>
<message>
<source>no</source>
<translation>nei</translation>
</message>
<message>
<source>This label turns red if any recipient receives an amount smaller than the current dust threshold.</source>
<translation>Denne merkelappen blir rød hvis en mottaker får mindre enn gjeldende støvterskel.</translation>
</message>
<message>
<source>Can vary +/- %1 satoshi(s) per input.</source>
<translation>Kan variere +/- %1 satoshi(er) per input.</translation>
</message>
<message>
<source>(no label)</source>
<translation>(ingen beskrivelse)</translation>
</message>
<message>
<source>change from %1 (%2)</source>
<translation>veksel fra %1 (%2)</translation>
</message>
<message>
<source>(change)</source>
<translation>(veksel)</translation>
</message>
</context>
<context>
<name>CreateWalletActivity</name>
<message>
<source>Creating Wallet <b>%1</b>...</source>
<translation>Lager lommebok <b>%1<b>...</translation>
</message>
<message>
<source>Create wallet failed</source>
<translation>Lage lommebok feilet</translation>
</message>
<message>
<source>Create wallet warning</source>
<translation>Lag lommebokvarsel</translation>
</message>
</context>
<context>
<name>CreateWalletDialog</name>
<message>
<source>Create Wallet</source>
<translation>Lag lommebok</translation>
</message>
<message>
<source>Wallet Name</source>
<translation>Lommeboknavn</translation>
</message>
<message>
<source>Encrypt the wallet. The wallet will be encrypted with a passphrase of your choice.</source>
<translation>Krypter lommeboken. Lommeboken blir kryptert med en passordfrase du velger.</translation>
</message>
<message>
<source>Encrypt Wallet</source>
<translation>Krypter Lommebok</translation>
</message>
<message>
<source>Disable private keys for this wallet. Wallets with private keys disabled will have no private keys and cannot have an HD seed or imported private keys. This is ideal for watch-only wallets.</source>
<translation>Deaktiver private nøkler for denne lommeboken. Lommebøker med private nøkler er deaktivert vil ikke ha noen private nøkler og kan ikke ha en HD seed eller importerte private nøkler. Dette er ideelt for loomebøker som kun er klokker.</translation>
</message>
<message>
<source>Disable Private Keys</source>
<translation>Deaktiver Private Nøkler</translation>
</message>
<message>
<source>Make a blank wallet. Blank wallets do not initially have private keys or scripts. Private keys and addresses can be imported, or an HD seed can be set, at a later time.</source>
<translation>Lag en tom lommebok. Tomme lommebøker har i utgangspunktet ikke private nøkler eller skript. Private nøkler og adresser kan importeres, eller et HD- frø kan angis på et senere tidspunkt.</translation>
</message>
<message>
<source>Make Blank Wallet</source>
<translation>Lag Tom Lommebok</translation>
</message>
<message>
<source>Use descriptors for scriptPubKey management</source>
<translation>Bruk deskriptorer for scriptPubKey styring</translation>
</message>
<message>
<source>Descriptor Wallet</source>
<translation>Deskriptor lommebok</translation>
</message>
<message>
<source>Create</source>
<translation>Opprett</translation>
</message>
<message>
<source>Compiled without sqlite support (required for descriptor wallets)</source>
<translation>Kompilert uten sqlite støtte (kreves for deskriptor lommebok)</translation>
</message>
</context>
<context>
<name>EditAddressDialog</name>
<message>
<source>Edit Address</source>
<translation>Rediger adresse</translation>
</message>
<message>
<source>&Label</source>
<translation>&Merkelapp</translation>
</message>
<message>
<source>The label associated with this address list entry</source>
<translation>Merkelappen koblet til denne adresseliste oppføringen</translation>
</message>
<message>
<source>The address associated with this address list entry. This can only be modified for sending addresses.</source>
<translation>Adressen til denne oppføringen i adresseboken. Denne kan kun endres for utsendingsadresser.</translation>
</message>
<message>
<source>&Address</source>
<translation>&Adresse</translation>
</message>
<message>
<source>New sending address</source>
<translation>Ny utsendingsadresse</translation>
</message>
<message>
<source>Edit receiving address</source>
<translation>Rediger mottaksadresse</translation>
</message>
<message>
<source>Edit sending address</source>
<translation>Rediger utsendingsadresse</translation>
</message>
<message>
<source>The entered address "%1" is not a valid Bitcoin address.</source>
<translation>Den angitte adressen "%1" er ikke en gyldig Bitcoin-adresse.</translation>
</message>
<message>
<source>Address "%1" already exists as a receiving address with label "%2" and so cannot be added as a sending address.</source>
<translation>Adresse "%1" eksisterer allerede som en mottaksadresse merket "%2" og kan derfor ikke bli lagt til som en sendingsadresse.</translation>
</message>
<message>
<source>The entered address "%1" is already in the address book with label "%2".</source>
<translation>Den oppgitte adressen ''%1'' er allerede i adresseboken med etiketten ''%2''.</translation>
</message>
<message>
<source>Could not unlock wallet.</source>
<translation>Kunne ikke låse opp lommebok.</translation>
</message>
<message>
<source>New key generation failed.</source>
<translation>Generering av ny nøkkel feilet.</translation>
</message>
</context>
<context>
<name>FreespaceChecker</name>
<message>
<source>A new data directory will be created.</source>
<translation>En ny datamappe vil bli laget.</translation>
</message>
<message>
<source>name</source>
<translation>navn</translation>
</message>
<message>
<source>Directory already exists. Add %1 if you intend to create a new directory here.</source>
<translation>Mappe finnes allerede. Legg til %1 hvis du vil lage en ny mappe her.</translation>
</message>
<message>
<source>Path already exists, and is not a directory.</source>
<translation>Snarvei finnes allerede, og er ikke en mappe.</translation>
</message>
<message>
<source>Cannot create data directory here.</source>
<translation>Kan ikke lage datamappe her.</translation>
</message>
</context>
<context>
<name>HelpMessageDialog</name>
<message>
<source>version</source>
<translation>versjon</translation>
</message>
<message>
<source>About %1</source>
<translation>Om %1</translation>
</message>
<message>
<source>Command-line options</source>
<translation>Kommandolinjevalg</translation>
</message>
</context>
<context>
<name>Intro</name>
<message>
<source>Welcome</source>
<translation>Velkommen</translation>
</message>
<message>
<source>Welcome to %1.</source>
<translation>Velkommen til %1.</translation>
</message>
<message>
<source>As this is the first time the program is launched, you can choose where %1 will store its data.</source>
<translation>Siden dette er første gang programmet starter, kan du nå velge hvor %1 skal lagre sine data.</translation>
</message>
<message>
<source>When you click OK, %1 will begin to download and process the full %4 block chain (%2GB) starting with the earliest transactions in %3 when %4 initially launched.</source>
<translation>Når du klikker OK, vil %1 starte nedlasting og behandle hele den %4 blokkjeden (%2GB) fra de eldste transaksjonene i %3 når %4 først startet.</translation>
</message>
<message>
<source>Reverting this setting requires re-downloading the entire blockchain. It is faster to download the full chain first and prune it later. Disables some advanced features.</source>
<translation>Gjenoppretting av denne innstillingen krever at du laster ned hele blockchain på nytt. Det er raskere å laste ned hele kjeden først og beskjære den senere Deaktiver noen avanserte funksjoner.</translation>
</message>
<message>
<source>This initial synchronisation is very demanding, and may expose hardware problems with your computer that had previously gone unnoticed. Each time you run %1, it will continue downloading where it left off.</source>
<translation>Den initielle synkroniseringen er svært krevende, og kan forårsake problemer med maskinvaren i datamaskinen din som du tidligere ikke merket. Hver gang du kjører %1 vil den fortsette nedlastingen der den sluttet.</translation>
</message>
<message>
<source>If you have chosen to limit block chain storage (pruning), the historical data must still be downloaded and processed, but will be deleted afterward to keep your disk usage low.</source>
<translation>Hvis du har valgt å begrense blokkjedelagring (beskjæring), må historiske data fortsatt lastes ned og behandles, men de vil bli slettet etterpå for å holde bruken av lagringsplass lav.</translation>
</message>
<message>
<source>Use the default data directory</source>
<translation>Bruk standard datamappe</translation>
</message>
<message>
<source>Use a custom data directory:</source>
<translation>Bruk en egendefinert datamappe:</translation>
</message>
<message>
<source>Bitcoin</source>
<translation>Bitcoin</translation>
</message>
<message>
<source>Discard blocks after verification, except most recent %1 GB (prune)</source>
<translation>Kast blokker etter bekreftelse, bortsett fra de siste %1 GB (sviske)</translation>
</message>
<message>
<source>At least %1 GB of data will be stored in this directory, and it will grow over time.</source>
<translation>Minst %1 GB data vil bli lagret i denne mappen og den vil vokse over tid.</translation>
</message>
<message>
<source>Approximately %1 GB of data will be stored in this directory.</source>
<translation>Omtrent %1GB data vil bli lagret i denne mappen.</translation>
</message>
<message>
<source>%1 will download and store a copy of the Bitcoin block chain.</source>
<translation>%1 vil laste ned og lagre en kopi av Bitcoin blokkjeden.</translation>
</message>
<message>
<source>The wallet will also be stored in this directory.</source>
<translation>Lommeboken vil også bli lagret i denne mappen.</translation>
</message>
<message>
<source>Error: Specified data directory "%1" cannot be created.</source>
<translation>Feil: Den oppgitte datamappen "%1" kan ikke opprettes.</translation>
</message>
<message>
<source>Error</source>
<translation>Feilmelding</translation>
</message>
<message numerus="yes">
<source>%n GB of free space available</source>
<translation><numerusform>%n GB med ledig lagringsplass</numerusform><numerusform>%n GB med ledig lagringsplass</numerusform></translation>
</message>
<message numerus="yes">
<source>(of %n GB needed)</source>
<translation><numerusform>(av %n GB som trengs)</numerusform><numerusform>(av %n GB som trengs)</numerusform></translation>
</message>
</context>
<context>
<name>ModalOverlay</name>
<message>
<source>Form</source>
<translation>Skjema</translation>
</message>
<message>
<source>Recent transactions may not yet be visible, and therefore your wallet's balance might be incorrect. This information will be correct once your wallet has finished synchronizing with the bitcoin network, as detailed below.</source>
<translation>Det kan hende nylige transaksjoner ikke vises enda, og at lommeboksaldoen dermed blir uriktig. Denne informasjonen vil rette seg når synkronisering av lommeboka mot bitcoin-nettverket er fullført, som anvist nedenfor.</translation>
</message>
<message>
<source>Attempting to spend bitcoins that are affected by not-yet-displayed transactions will not be accepted by the network.</source>
<translation>Forsøk på å bruke bitcoin som er påvirket av transaksjoner som ikke er vist enda godtas ikke av nettverket.</translation>
</message>
<message>
<source>Number of blocks left</source>
<translation>Antall gjenværende blokker</translation>
</message>
<message>
<source>Unknown...</source>
<translation>Ukjent...</translation>
</message>
<message>
<source>Last block time</source>
<translation>Tidspunkt for siste blokk</translation>
</message>
<message>
<source>Progress</source>
<translation>Fremgang</translation>
</message>
<message>
<source>Progress increase per hour</source>
<translation>Fremgangen stiger hver time</translation>
</message>
<message>
<source>calculating...</source>
<translation>kalkulerer...</translation>
</message>
<message>
<source>Estimated time left until synced</source>
<translation>Estimert gjenstående tid før ferdig synkronisert</translation>
</message>
<message>
<source>Hide</source>
<translation>Skjul</translation>
</message>
<message>
<source>Esc</source>
<translation>Esc</translation>
</message>
<message>
<source>%1 is currently syncing. It will download headers and blocks from peers and validate them until reaching the tip of the block chain.</source>
<translation>%1 synkroniseres for øyeblikket. Den vil laste ned overskrifter og blokker fra jevnaldrende og validere dem til de når spissen av blokkjeden.</translation>
</message>
<message>
<source>Unknown. Syncing Headers (%1, %2%)...</source>
<translation>Ukjent.Synkroniser overskrifter (%1,%2%)...</translation>
</message>
</context>
<context>
<name>OpenURIDialog</name>
<message>
<source>Open bitcoin URI</source>
<translation>Åpne bitcoin URI</translation>
</message>
<message>
<source>URI:</source>
<translation>URI:</translation>
</message>
</context>
<context>
<name>OpenWalletActivity</name>
<message>
<source>Open wallet failed</source>
<translation>Åpne lommebok feilet</translation>
</message>
<message>
<source>Open wallet warning</source>
<translation>Advasel om åpen lommebok.</translation>
</message>
<message>
<source>default wallet</source>
<translation>standard lommebok</translation>
</message>
<message>
<source>Opening Wallet <b>%1</b>...</source>
<translation>Åpner Lommebok <b>%1</b>...</translation>
</message>
</context>
<context>
<name>OptionsDialog</name>
<message>
<source>Options</source>
<translation>Innstillinger</translation>
</message>
<message>
<source>&Main</source>
<translation>&Hoved</translation>
</message>
<message>
<source>Automatically start %1 after logging in to the system.</source>
<translation>Start %1 automatisk etter å ha logget inn på systemet.</translation>
</message>
<message>
<source>&Start %1 on system login</source>
<translation>&Start %1 ved systeminnlogging</translation>
</message>
<message>
<source>Size of &database cache</source>
<translation>Størrelse på &database hurtigbuffer</translation>
</message>
<message>
<source>Number of script &verification threads</source>
<translation>Antall script &verifikasjonstråder</translation>
</message>
<message>
<source>IP address of the proxy (e.g. IPv4: 127.0.0.1 / IPv6: ::1)</source>
<translation>IP-adressen til proxyen (f.eks. IPv4: 127.0.0.1 / IPv6: ::1)</translation>
</message>
<message>
<source>Shows if the supplied default SOCKS5 proxy is used to reach peers via this network type.</source>
<translation>Viser hvorvidt angitt SOCKS5-mellomtjener blir brukt for å nå noder via denne nettverkstypen.</translation>
</message>
<message>
<source>Hide the icon from the system tray.</source>
<translation>Skjul ikonet fra systemkurven.</translation>
</message>
<message>
<source>&Hide tray icon</source>
<translation>&Skjul systemkurvsikon</translation>
</message>
<message>
<source>Minimize instead of exit the application when the window is closed. When this option is enabled, the application will be closed only after selecting Exit in the menu.</source>
<translation>Minimer i stedet for å avslutte applikasjonen når vinduet lukkes. Når dette er valgt, vil applikasjonen avsluttes kun etter at Avslutte er valgt i menyen.</translation>
</message>
<message>
<source>Third party URLs (e.g. a block explorer) that appear in the transactions tab as context menu items. %s in the URL is replaced by transaction hash. Multiple URLs are separated by vertical bar |.</source>
<translation>Tredjepart URLer (f. eks. en blokkutforsker) som dukker opp i transaksjonsfanen som kontekst meny elementer. %s i URLen er erstattet med transaksjonen sin hash. Flere URLer er separert av en vertikal linje |.</translation>
</message>
<message>
<source>Open the %1 configuration file from the working directory.</source>
<translation>Åpne %1-oppsettsfila fra arbeidsmappen.</translation>
</message>
<message>
<source>Open Configuration File</source>
<translation>Åpne oppsettsfil</translation>
</message>
<message>
<source>Reset all client options to default.</source>
<translation>Tilbakestill alle klient valg til standard</translation>
</message>
<message>
<source>&Reset Options</source>
<translation>&Tilbakestill Instillinger</translation>
</message>
<message>
<source>&Network</source>
<translation>&Nettverk</translation>
</message>
<message>
<source>Disables some advanced features but all blocks will still be fully validated. Reverting this setting requires re-downloading the entire blockchain. Actual disk usage may be somewhat higher.</source>
<translation>Deaktiver noen avanserte funksjoner, men alle blokker vil fortsatt være fullglyldig. Gjenoppretting av denne innstillingen krever at du laster ned hele blockchain på nytt. Faktisk diskbruk kan være noe høvere.</translation>
</message>
<message>
<source>Prune &block storage to</source>
<translation>Beskjær og blokker lagring til</translation>
</message>
<message>
<source>GB</source>
<translation>GB</translation>
</message>
<message>
<source>Reverting this setting requires re-downloading the entire blockchain.</source>
<translation>Gjenoppretting av denne innstillingen krever at du laster ned hele blockchain på nytt</translation>
</message>
<message>
<source>MiB</source>
<translation>MiB</translation>
</message>
<message>
<source>(0 = auto, <0 = leave that many cores free)</source>
<translation>(0 = automatisk, <0 = la så mange kjerner være ledig)</translation>
</message>
<message>
<source>W&allet</source>
<translation>L&ommebok</translation>
</message>
<message>
<source>Expert</source>
<translation>Ekspert</translation>
</message>
<message>
<source>Enable coin &control features</source>
<translation>Aktiver &myntkontroll funksjoner</translation>
</message>
<message>
<source>If you disable the spending of unconfirmed change, the change from a transaction cannot be used until that transaction has at least one confirmation. This also affects how your balance is computed.</source>
<translation>Hvis du sperrer for bruk av ubekreftet veksel, kan ikke vekselen fra transaksjonen bli brukt før transaksjonen har minimum en bekreftelse. Dette påvirker også hvordan balansen din blir beregnet.</translation>
</message>
<message>
<source>&Spend unconfirmed change</source>
<translation>&Bruk ubekreftet veksel</translation>
</message>
<message>
<source>Automatically open the Bitcoin client port on the router. This only works when your router supports UPnP and it is enabled.</source>
<translation>Åpne automatisk Bitcoin klientporten på ruteren. Dette virker kun om din ruter støtter UPnP og dette er påslått.</translation>
</message>
<message>
<source>Map port using &UPnP</source>
<translation>Sett opp port ved hjelp av &UPnP</translation>
</message>
<message>
<source>Accept connections from outside.</source>
<translation>Tillat tilkoblinger fra utsiden</translation>
</message>
<message>
<source>Allow incomin&g connections</source>
<translation>Tillatt innkommend&e tilkoblinger</translation>
</message>
<message>
<source>Connect to the Bitcoin network through a SOCKS5 proxy.</source>
<translation>Koble til Bitcoin-nettverket gjennom en SOCKS5 proxy.</translation>
</message>
<message>
<source>&Connect through SOCKS5 proxy (default proxy):</source>
<translation>&Koble til gjennom SOCKS5 proxy (standardvalg proxy):</translation>
</message>
<message>
<source>Proxy &IP:</source>
<translation>Proxy &IP:</translation>
</message>
<message>
<source>&Port:</source>
<translation>&Port:</translation>
</message>
<message>
<source>Port of the proxy (e.g. 9050)</source>
<translation>Proxyens port (f.eks. 9050)</translation>
</message>
<message>
<source>Used for reaching peers via:</source>
<translation>Brukt for å nå noder via:</translation>
</message>
<message>
<source>IPv4</source>
<translation>IPv4</translation>
</message>
<message>
<source>IPv6</source>
<translation>IPv6</translation>
</message>
<message>
<source>Tor</source>
<translation>Tor</translation>
</message>
<message>
<source>&Window</source>
<translation>&Vindu</translation>
</message>
<message>
<source>Show only a tray icon after minimizing the window.</source>
<translation>Vis kun ikon i systemkurv etter minimering av vinduet.</translation>
</message>
<message>
<source>&Minimize to the tray instead of the taskbar</source>
<translation>&Minimer til systemkurv istedenfor oppgavelinjen</translation>
</message>
<message>
<source>M&inimize on close</source>
<translation>M&inimer ved lukking</translation>
</message>
<message>
<source>&Display</source>
<translation>&Visning</translation>
</message>
<message>
<source>User Interface &language:</source>
<translation>&Språk for brukergrensesnitt</translation>
</message>
<message>
<source>The user interface language can be set here. This setting will take effect after restarting %1.</source>
<translation>Brukergrensesnittspråket kan endres her. Denne innstillingen trer i kraft etter omstart av %1.</translation>
</message>
<message>
<source>&Unit to show amounts in:</source>
<translation>&Enhet for visning av beløper:</translation>
</message>
<message>
<source>Choose the default subdivision unit to show in the interface and when sending coins.</source>
<translation>Velg standard delt enhet for visning i grensesnittet og for sending av bitcoins.</translation>
</message>
<message>
<source>Whether to show coin control features or not.</source>
<translation>Skal myntkontroll funksjoner vises eller ikke.</translation>
</message>
<message>
<source>Connect to the Bitcoin network through a separate SOCKS5 proxy for Tor onion services.</source>
<translation>Kobl til Bitcoin nettverket gjennom en separat SOCKS5 proxy for Tor onion tjenester. </translation>
</message>
<message>
<source>Use separate SOCKS&5 proxy to reach peers via Tor onion services:</source>
<translation>Bruk separate SOCKS&5 proxy for å nå peers via Tor onion tjenester:</translation>
</message>
<message>
<source>&Third party transaction URLs</source>
<translation>Tredjepart transaksjon URLer</translation>
</message>
<message>
<source>Options set in this dialog are overridden by the command line or in the configuration file:</source>
<translation>Alternativer som er satt i denne dialogboksen overstyres av kommandolinjen eller i konfigurasjonsfilen:</translation>
</message>
<message>
<source>&OK</source>
<translation>&OK</translation>
</message>
<message>
<source>&Cancel</source>
<translation>&Avbryt</translation>
</message>
<message>
<source>default</source>
<translation>standardverdi</translation>
</message>
<message>
<source>none</source>
<translation>ingen</translation>
</message>
<message>
<source>Confirm options reset</source>
<translation>Bekreft tilbakestilling av innstillinger</translation>
</message>
<message>
<source>Client restart required to activate changes.</source>
<translation>Omstart av klienten er nødvendig for å aktivere endringene.</translation>
</message>
<message>
<source>Client will be shut down. Do you want to proceed?</source>
<translation>Klienten vil bli lukket. Ønsker du å gå videre?</translation>
</message>
<message>
<source>Configuration options</source>
<translation>Oppsettsvalg</translation>
</message>
<message>
<source>The configuration file is used to specify advanced user options which override GUI settings. Additionally, any command-line options will override this configuration file.</source>
<translation>Oppsettsfil brukt for å angi avanserte brukervalg som overstyrer innstillinger gjort i grafisk brukergrensesnitt. I tillegg vil enhver handling utført på kommandolinjen overstyre denne oppsettsfila.</translation>
</message>
<message>
<source>Error</source>
<translation>Feilmelding</translation>
</message>
<message>
<source>The configuration file could not be opened.</source>
<translation>Kunne ikke åpne oppsettsfila.</translation>
</message>
<message>
<source>This change would require a client restart.</source>
<translation>Denne endringen krever omstart av klienten.</translation>
</message>
<message>
<source>The supplied proxy address is invalid.</source>
<translation>Angitt proxyadresse er ugyldig.</translation>
</message>
</context>
<context>
<name>OverviewPage</name>
<message>
<source>Form</source>
<translation>Skjema</translation>
</message>
<message>
<source>The displayed information may be out of date. Your wallet automatically synchronizes with the Bitcoin network after a connection is established, but this process has not completed yet.</source>
<translation>Informasjonen som vises kan være foreldet. Din lommebok synkroniseres automatisk med Bitcoin-nettverket etter at tilkobling er opprettet, men denne prosessen er ikke ferdig enda.</translation>
</message>
<message>
<source>Watch-only:</source>
<translation>Kun observerbar:</translation>
</message>
<message>
<source>Available:</source>
<translation>Tilgjengelig:</translation>
</message>
<message>
<source>Your current spendable balance</source>
<translation>Din nåværende saldo</translation>
</message>
<message>
<source>Pending:</source>
<translation>Under behandling:</translation>
</message>
<message>
<source>Total of transactions that have yet to be confirmed, and do not yet count toward the spendable balance</source>
<translation>Totalt antall ubekreftede transaksjoner som ikke teller med i saldo</translation>
</message>
<message>
<source>Immature:</source>
<translation>Umoden:</translation>
</message>
<message>
<source>Mined balance that has not yet matured</source>
<translation>Minet saldo har ikke modnet enda</translation>
</message>
<message>
<source>Balances</source>
<translation>Saldoer</translation>
</message>
<message>
<source>Total:</source>
<translation>Totalt:</translation>
</message>
<message>
<source>Your current total balance</source>
<translation>Din nåværende saldo</translation>
</message>
<message>
<source>Your current balance in watch-only addresses</source>
<translation>Din nåværende balanse i kun observerbare adresser</translation>
</message>
<message>
<source>Spendable:</source>
<translation>Kan brukes:</translation>
</message>
<message>
<source>Recent transactions</source>
<translation>Nylige transaksjoner</translation>
</message>
<message>
<source>Unconfirmed transactions to watch-only addresses</source>
<translation>Ubekreftede transaksjoner til kun observerbare adresser</translation>
</message>
<message>
<source>Mined balance in watch-only addresses that has not yet matured</source>
<translation>Utvunnet balanse i kun observerbare adresser som ennå ikke har modnet</translation>
</message>
<message>
<source>Current total balance in watch-only addresses</source>
<translation>Nåværende totale balanse i kun observerbare adresser</translation>
</message>
<message>
<source>Privacy mode activated for the Overview tab. To unmask the values, uncheck Settings->Mask values.</source>
<translation>Privat mode er aktivert for oversiktstabben. For å se verdier, uncheck innstillinger->Masker verdier</translation>
</message>
</context>
<context>
<name>PSBTOperationsDialog</name>
<message>
<source>Dialog</source>
<translation>Dialog</translation>
</message>
<message>
<source>Sign Tx</source>
<translation>Signer Tx</translation>
</message>
<message>
<source>Broadcast Tx</source>
<translation>Kringkast Tx</translation>
</message>
<message>
<source>Copy to Clipboard</source>
<translation>Kopier til utklippstavle</translation>
</message>
<message>
<source>Save...</source>
<translation>Lagre...</translation>
</message>
<message>
<source>Close</source>
<translation>Lukk</translation>
</message>
<message>
<source>Failed to load transaction: %1</source>
<translation>Lasting av transaksjon: %1 feilet</translation>
</message>
<message>
<source>Failed to sign transaction: %1</source>
<translation>Signering av transaksjon: %1 feilet</translation>
</message>
<message>
<source>Could not sign any more inputs.</source>
<translation>Kunne ikke signere flere inputs.</translation>
</message>
<message>
<source>Signed %1 inputs, but more signatures are still required.</source>
<translation>Signerte %1 inputs, men flere signaturer kreves.</translation>
</message>
<message>
<source>Signed transaction successfully. Transaction is ready to broadcast.</source>
<translation>Signering av transaksjon var vellykket. Transaksjon er klar til å kringkastes.</translation>
</message>
<message>
<source>Unknown error processing transaction.</source>
<translation>Ukjent feil når den prossesserte transaksjonen.</translation>
</message>
<message>
<source>Transaction broadcast successfully! Transaction ID: %1</source>
<translation>Kringkasting av transaksjon var vellykket! Transaksjon ID: %1</translation>
</message>
<message>
<source>Transaction broadcast failed: %1</source>
<translation>Kringkasting av transaksjon feilet: %1</translation>
</message>
<message>
<source>PSBT copied to clipboard.</source>
<translation>PSBT kopiert til utklippstavle.</translation>
</message>
<message>
<source>Save Transaction Data</source>
<translation>Lagre Transaksjonsdata</translation>
</message>
<message>
<source>Partially Signed Transaction (Binary) (*.psbt)</source>
<translation>Delvis Signert Transaksjon (Binær) (*.psbt)</translation>
</message>
<message>
<source>PSBT saved to disk.</source>
<translation>PSBT lagret til disk.</translation>
</message>
<message>
<source> * Sends %1 to %2</source>
<translation>* Sender %1 til %2</translation>
</message>
<message>
<source>Unable to calculate transaction fee or total transaction amount.</source>
<translation>Klarte ikke å kalkulere transaksjonsavgift eller totalt transaksjonsbeløp.</translation>
</message>
<message>
<source>Pays transaction fee: </source>
<translation>Betaler transasjonsavgift:</translation>
</message>
<message>
<source>Total Amount</source>
<translation>Totalbeløp</translation>
</message>
<message>
<source>or</source>
<translation>eller</translation>
</message>
<message>
<source>Transaction has %1 unsigned inputs.</source>
<translation>Transaksjon har %1 usignert inputs.</translation>
</message>
<message>
<source>Transaction is missing some information about inputs.</source>
<translation>Transaksjonen mangler noe informasjon om inputs.</translation>
</message>
<message>
<source>Transaction still needs signature(s).</source>
<translation>Transaksjonen trenger signatur(er).</translation>
</message>
<message>
<source>(But this wallet cannot sign transactions.)</source>
<translation>(Men denne lommeboken kan ikke signere transaksjoner.)</translation>
</message>
<message>
<source>(But this wallet does not have the right keys.)</source>
<translation>(Men denne lommeboken har ikke de rette nøkklene.)</translation>
</message>
<message>
<source>Transaction is fully signed and ready for broadcast.</source>
<translation>Transaksjonen er signert og klar til kringkasting.</translation>
</message>
<message>
<source>Transaction status is unknown.</source>
<translation>Transaksjonsstatus er ukjent.</translation>
</message>
</context>
<context>
<name>PaymentServer</name>
<message>
<source>Payment request error</source>
<translation>Feil ved betalingsforespørsel</translation>
</message>
<message>
<source>Cannot start bitcoin: click-to-pay handler</source>
<translation>Kan ikke starte bitcoin: Klikk-og-betal håndterer</translation>
</message>
<message>
<source>URI handling</source>
<translation>URI-håndtering</translation>
</message>
<message>
<source>'bitcoin://' is not a valid URI. Use 'bitcoin:' instead.</source>
<translation>'bitcoin: //' er ikke en gyldig URI. Bruk 'bitcoin:' i stedet.</translation>
</message>
<message>
<source>Cannot process payment request because BIP70 is not supported.</source>
<translation>Kan ikke behandle betalingsforespørsel fordi BIP70 ikke støttes.</translation>
</message>
<message>
<source>Due to widespread security flaws in BIP70 it's strongly recommended that any merchant instructions to switch wallets be ignored.</source>
<translation>På grunn av utbredte sikkerhetsfeil i BIP70 anbefales det på det sterkeste at alle selgerinstruksjoner for å bytte lommebok ignoreres.</translation>
</message>
<message>
<source>If you are receiving this error you should request the merchant provide a BIP21 compatible URI.</source>
<translation>Hvis du mottar denne feilen, bør du be selgeren gi en BIP21-kompatibel URI.</translation>
</message>
<message>
<source>Invalid payment address %1</source>
<translation>Ugyldig betalingsadresse %1</translation>
</message>
<message>
<source>URI cannot be parsed! This can be caused by an invalid Bitcoin address or malformed URI parameters.</source>
<translation>URI kan ikke fortolkes! Dette kan være forårsaket av en ugyldig bitcoin-adresse eller feilformede URI-parametre.</translation>
</message>
<message>
<source>Payment request file handling</source>
<translation>Håndtering av betalingsforespørselsfil</translation>
</message>
</context>
<context>
<name>PeerTableModel</name>
<message>
<source>User Agent</source>
<translation>Brukeragent</translation>
</message>
<message>
<source>Node/Service</source>
<translation>Node/Tjeneste</translation>
</message>
<message>
<source>NodeId</source>
<translation>NodeId</translation>
</message>
<message>
<source>Ping</source>
<translation>Nettverkssvarkall</translation>
</message>
<message>
<source>Sent</source>
<translation>Sendt</translation>
</message>
<message>
<source>Received</source>
<translation>Mottatt</translation>
</message>
</context>
<context>
<name>QObject</name>
<message>
<source>Amount</source>
<translation>Beløp</translation>
</message>
<message>
<source>Enter a Bitcoin address (e.g. %1)</source>
<translation>Oppgi en Bitcoin-adresse (f.eks. %1)</translation>
</message>
<message>
<source>%1 d</source>
<translation>%1 d</translation>
</message>
<message>
<source>%1 h</source>
<translation>%1 t</translation>
</message>
<message>
<source>%1 m</source>
<translation>%1 m</translation>
</message>
<message>
<source>%1 s</source>
<translation>%1 s</translation>
</message>
<message>
<source>None</source>
<translation>Ingen</translation>
</message>
<message>
<source>N/A</source>
<translation>-</translation>
</message>
<message>
<source>%1 ms</source>
<translation>%1 ms</translation>
</message>
<message numerus="yes">
<source>%n second(s)</source>
<translation><numerusform>%n sekund</numerusform><numerusform>%n sekunder</numerusform></translation>
</message>
<message numerus="yes">
<source>%n minute(s)</source>
<translation><numerusform>%n minutt</numerusform><numerusform>%n minutter</numerusform></translation>
</message>
<message numerus="yes">
<source>%n hour(s)</source>
<translation><numerusform>%n time</numerusform><numerusform>%n timer</numerusform></translation>
</message>
<message numerus="yes">
<source>%n day(s)</source>
<translation><numerusform>%n dag</numerusform><numerusform>%n dager</numerusform></translation>
</message>
<message numerus="yes">
<source>%n week(s)</source>
<translation><numerusform>%n uke</numerusform><numerusform>%n uker</numerusform></translation>
</message>
<message>
<source>%1 and %2</source>
<translation>%1 og %2</translation>
</message>
<message numerus="yes">
<source>%n year(s)</source>
<translation><numerusform>%n år</numerusform><numerusform>%n år</numerusform></translation>
</message>
<message>
<source>%1 B</source>
<translation>%1 B</translation>
</message>
<message>
<source>%1 KB</source>
<translation>%1 KB</translation>
</message>
<message>
<source>%1 MB</source>
<translation>%1 MB</translation>
</message>
<message>
<source>%1 GB</source>
<translation>%1 GB</translation>
</message>
<message>
<source>Error: Specified data directory "%1" does not exist.</source>
<translation>Feil: Den spesifiserte datamappen "%1" finnes ikke.</translation>
</message>
<message>
<source>Error: %1</source>
<translation>Feil: %1</translation>
</message>
<message>
<source>Error initializing settings: %1</source>
<translation>Initialisering av innstillinger feilet: %1</translation>
</message>
<message>
<source>%1 didn't yet exit safely...</source>
<translation>%1 har ikke avsluttet trygt enda…</translation>
</message>
<message>
<source>unknown</source>
<translation>ukjent</translation>
</message>
</context>
<context>
<name>QRImageWidget</name>
<message>
<source>&Save Image...</source>
<translation>&Lagre bilde...</translation>
</message>
<message>
<source>&Copy Image</source>
<translation>&Kopier bilde</translation>
</message>
<message>
<source>Resulting URI too long, try to reduce the text for label / message.</source>
<translation>Resulterende URI er for lang, prøv å redusere teksten for merkelapp / melding.</translation>
</message>
<message>
<source>Error encoding URI into QR Code.</source>
<translation>Feil ved koding av URI til QR-kode.</translation>
</message>
<message>
<source>QR code support not available.</source>
<translation>Støtte for QR kode ikke tilgjengelig.</translation>
</message>
<message>
<source>Save QR Code</source>
<translation>Lagre QR-kode</translation>
</message>
<message>
<source>PNG Image (*.png)</source>
<translation>PNG-bilde (*.png)</translation>
</message>
</context>
<context>
<name>RPCConsole</name>
<message>
<source>N/A</source>
<translation>-</translation>
</message>
<message>
<source>Client version</source>
<translation>Klientversjon</translation>
</message>
<message>
<source>&Information</source>
<translation>&Informasjon</translation>
</message>
<message>
<source>General</source>
<translation>Generelt</translation>
</message>
<message>
<source>Using BerkeleyDB version</source>
<translation>Bruker BerkeleyDB versjon</translation>
</message>
<message>
<source>Datadir</source>
<translation>Datamappe</translation>
</message>
<message>
<source>Blocksdir</source>
<translation>Blocksdir</translation>
</message>
<message>
<source>Startup time</source>
<translation>Oppstartstidspunkt</translation>
</message>
<message>
<source>Network</source>
<translation>Nettverk</translation>
</message>
<message>
<source>Name</source>
<translation>Navn</translation>
</message>
<message>
<source>Number of connections</source>
<translation>Antall tilkoblinger</translation>
</message>
<message>
<source>Block chain</source>
<translation>Blokkjeden</translation>
</message>
<message>
<source>Memory Pool</source>
<translation>Hukommelsespulje</translation>
</message>
<message>
<source>Current number of transactions</source>
<translation>Nåværende antall transaksjoner</translation>
</message>
<message>
<source>Memory usage</source>
<translation>Minnebruk</translation>
</message>
<message>
<source>Wallet: </source>
<translation>Lommebok:</translation>
</message>
<message>
<source>(none)</source>
<translation>(ingen)</translation>
</message>
<message>
<source>&Reset</source>
<translation>&Tilbakestill</translation>
</message>
<message>
<source>Received</source>
<translation>Mottatt</translation>
</message>
<message>
<source>Sent</source>
<translation>Sendt</translation>
</message>
<message>
<source>&Peers</source>
<translation>&Noder</translation>
</message>
<message>
<source>Banned peers</source>
<translation>Utestengte noder</translation>
</message>
<message>
<source>Select a peer to view detailed information.</source>
<translation>Velg en node for å vise detaljert informasjon.</translation>
</message>
<message>
<source>Direction</source>
<translation>Retning</translation>
</message>
<message>
<source>Version</source>
<translation>Versjon</translation>
</message>
<message>
<source>Starting Block</source>
<translation>Startblokk</translation>
</message>
<message>
<source>Synced Headers</source>
<translation>Synkroniserte Blokkhoder</translation>
</message>
<message>
<source>Synced Blocks</source>
<translation>Synkroniserte Blokker</translation>
</message>
<message>
<source>The mapped Autonomous System used for diversifying peer selection.</source>
<translation>Det kartlagte autonome systemet som brukes til å diversifisere valg av fagfeller.</translation>
</message>
<message>
<source>Mapped AS</source>
<translation>Kartlagt AS</translation>
</message>
<message>
<source>User Agent</source>
<translation>Brukeragent</translation>
</message>
<message>
<source>Node window</source>
<translation>Nodevindu</translation>
</message>
<message>
<source>Current block height</source>
<translation>Nåværende blokkhøyde</translation>
</message>
<message>
<source>Open the %1 debug log file from the current data directory. This can take a few seconds for large log files.</source>
<translation>Åpne %1-feilrettingsloggfila fra gjeldende datamappe. Dette kan ta et par sekunder for store loggfiler.</translation>
</message>
<message>
<source>Decrease font size</source>
<translation>Forminsk font størrelsen</translation>
</message>
<message>
<source>Increase font size</source>
<translation>Forstørr font størrelse</translation>
</message>
<message>
<source>Permissions</source>
<translation>Rettigheter</translation>
</message>
<message>
<source>Services</source>
<translation>Tjenester</translation>
</message>
<message>
<source>Connection Time</source>
<translation>Tilkoblingstid</translation>
</message>
<message>
<source>Last Send</source>
<translation>Siste Sendte</translation>
</message>
<message>
<source>Last Receive</source>
<translation>Siste Mottatte</translation>
</message>
<message>
<source>Ping Time</source>
<translation>Ping-tid</translation>
</message>
<message>
<source>The duration of a currently outstanding ping.</source>
<translation>Tidsforløp for utestående ping.</translation>
</message>
<message>
<source>Ping Wait</source>
<translation>Ping Tid</translation>
</message>
<message>
<source>Min Ping</source>
<translation>Minimalt nettverkssvarkall</translation>
</message>
<message>
<source>Time Offset</source>
<translation>Tidsforskyvning</translation>
</message>
<message>
<source>Last block time</source>
<translation>Tidspunkt for siste blokk</translation>
</message>
<message>
<source>&Open</source>
<translation>&Åpne</translation>
</message>
<message>
<source>&Console</source>
<translation>&Konsoll</translation>
</message>
<message>
<source>&Network Traffic</source>
<translation>&Nettverkstrafikk</translation>
</message>
<message>
<source>Totals</source>
<translation>Totalt</translation>
</message>
<message>
<source>In:</source>
<translation>Inn:</translation>
</message>
<message>
<source>Out:</source>
<translation>Ut:</translation>
</message>
<message>
<source>Debug log file</source>
<translation>Loggfil for feilsøk</translation>
</message>
<message>
<source>Clear console</source>
<translation>Tøm konsoll</translation>
</message>
<message>
<source>1 &hour</source>
<translation>1 &time</translation>
</message>
<message>
<source>1 &day</source>
<translation>1 &dag</translation>
</message>
<message>
<source>1 &week</source>
<translation>1 &uke</translation>
</message>
<message>
<source>1 &year</source>
<translation>1 &år</translation>
</message>
<message>
<source>&Disconnect</source>
<translation>&Koble fra</translation>
</message>
<message>
<source>Ban for</source>
<translation>Bannlys i</translation>
</message>
<message>
<source>&Unban</source>
<translation>&Opphev bannlysning</translation>
</message>
<message>
<source>Welcome to the %1 RPC console.</source>
<translation>Velkommen til %1 RPC-konsoll.</translation>
</message>
<message>
<source>Use up and down arrows to navigate history, and %1 to clear screen.</source>
<translation>Bruk ↑ og ↓ til å navigere historikk, og %1 for å tømme skjermen.</translation>
</message>
<message>
<source>Type %1 for an overview of available commands.</source>
<translation>Skriv %1 for en oversikt over tilgjengelige kommandoer.</translation>
</message>
<message>
<source>For more information on using this console type %1.</source>
<translation>For mer informasjon om hvordan konsollet brukes skriv %1.</translation>
</message>
<message>
<source>WARNING: Scammers have been active, telling users to type commands here, stealing their wallet contents. Do not use this console without fully understanding the ramifications of a command.</source>
<translation>Advarsel: Svindlere har vært på ferde, i oppfordringen om å skrive kommandoer her, for å stjele lommebokinnhold. Ikke bruk konsollen uten at du forstår alle ringvirkningene av en kommando.</translation>
</message>
<message>
<source>Network activity disabled</source>
<translation>Nettverksaktivitet avskrudd</translation>
</message>
<message>
<source>Executing command without any wallet</source>
<translation>Utfør kommando uten noen lommebok</translation>
</message>
<message>
<source>Executing command using "%1" wallet</source>
<translation>Utfør kommando med lommebok "%1"</translation>
</message>
<message>
<source>(node id: %1)</source>
<translation>(node id: %1)</translation>
</message>
<message>
<source>via %1</source>
<translation>via %1</translation>
</message>
<message>
<source>never</source>
<translation>aldri</translation>
</message>
<message>
<source>Inbound</source>
<translation>Innkommende</translation>
</message>
<message>
<source>Outbound</source>
<translation>Utgående</translation>
</message>
<message>
<source>Unknown</source>
<translation>Ukjent</translation>
</message>
</context>
<context>
<name>ReceiveCoinsDialog</name>
<message>
<source>&Amount:</source>
<translation>&Beløp:</translation>
</message>
<message>
<source>&Label:</source>
<translation>&Merkelapp:</translation>
</message>
<message>
<source>&Message:</source>
<translation>&Melding:</translation>
</message>
<message>
<source>An optional message to attach to the payment request, which will be displayed when the request is opened. Note: The message will not be sent with the payment over the Bitcoin network.</source>
<translation>En valgfri melding å tilknytte betalingsetterspørringen, som vil bli vist når forespørselen er åpnet. Meldingen vil ikke bli sendt med betalingen over Bitcoin-nettverket.</translation>
</message>
<message>
<source>An optional label to associate with the new receiving address.</source>
<translation>En valgfri merkelapp å tilknytte den nye mottakeradressen.</translation>
</message>
<message>
<source>Use this form to request payments. All fields are <b>optional</b>.</source>
<translation>Bruk dette skjemaet til betalingsforespørsler. Alle felt er <b>valgfrie</b>.</translation>
</message>
<message>
<source>An optional amount to request. Leave this empty or zero to not request a specific amount.</source>
<translation>Et valgfritt beløp å etterspørre. La stå tomt eller null for ikke å etterspørre et spesifikt beløp.</translation>
</message>
<message>
<source>An optional label to associate with the new receiving address (used by you to identify an invoice). It is also attached to the payment request.</source>
<translation>En valgfri etikett for å knytte til den nye mottaksadressen (brukt av deg for å identifisere en faktura). Det er også knyttet til betalingsforespørselen.</translation>
</message>
<message>
<source>An optional message that is attached to the payment request and may be displayed to the sender.</source>
<translation>En valgfri melding som er knyttet til betalingsforespørselen og kan vises til avsenderen.</translation>
</message>
<message>
<source>&Create new receiving address</source>
<translation>&Lag ny mottakeradresse</translation>
</message>
<message>
<source>Clear all fields of the form.</source>
<translation>Fjern alle felter fra skjemaet.</translation>
</message>
<message>
<source>Clear</source>
<translation>Fjern</translation>
</message>
<message>
<source>Native segwit addresses (aka Bech32 or BIP-173) reduce your transaction fees later on and offer better protection against typos, but old wallets don't support them. When unchecked, an address compatible with older wallets will be created instead.</source>
<translation>Innfødte segwit-adresser (også kalt Bech32 eller BIP-173) reduserer transaksjonsgebyrene senere og gir bedre beskyttelse mot skrivefeil, men gamle lommebøker støtter dem ikke. Når du ikke har merket av, opprettes en adresse som er kompatibel med eldre lommebøker.</translation>
</message>
<message>
<source>Generate native segwit (Bech32) address</source>
<translation>Generer nativ segwit (Bech32) adresse</translation>
</message>
<message>
<source>Requested payments history</source>
<translation>Etterspurt betalingshistorikk</translation>
</message>
<message>
<source>Show the selected request (does the same as double clicking an entry)</source>
<translation>Vis den valgte etterspørringen (gjør det samme som å dobbelklikke på en oppføring)</translation>
</message>
<message>
<source>Show</source>
<translation>Vis</translation>
</message>
<message>
<source>Remove the selected entries from the list</source>
<translation>Fjern de valgte oppføringene fra listen</translation>
</message>
<message>
<source>Remove</source>
<translation>Fjern</translation>
</message>
<message>
<source>Copy URI</source>
<translation>Kopier URI</translation>
</message>
<message>
<source>Copy label</source>
<translation>Kopiér merkelapp</translation>
</message>
<message>
<source>Copy message</source>
<translation>Kopier melding</translation>
</message>
<message>
<source>Copy amount</source>
<translation>Kopier beløp</translation>
</message>
<message>
<source>Could not unlock wallet.</source>
<translation>Kunne ikke låse opp lommebok.</translation>
</message>
<message>
<source>Could not generate new %1 address</source>
<translation>Kunne ikke generere ny %1 adresse </translation>
</message>
</context>
<context>
<name>ReceiveRequestDialog</name>
<message>
<source>Request payment to ...</source>
<translation>Be om betaling til...</translation>
</message>
<message>
<source>Address:</source>
<translation>Adresse:</translation>
</message>
<message>
<source>Amount:</source>
<translation>Beløp:</translation>
</message>
<message>
<source>Label:</source>
<translation>Merkelapp:</translation>
</message>
<message>
<source>Message:</source>
<translation>Melding:</translation>
</message>
<message>
<source>Wallet:</source>
<translation>Lommebok:</translation>
</message>
<message>
<source>Copy &URI</source>
<translation>Kopier &URI</translation>
</message>
<message>
<source>Copy &Address</source>
<translation>Kopier &Adresse</translation>
</message>
<message>
<source>&Save Image...</source>
<translation>&Lagre Bilde...</translation>
</message>
<message>
<source>Request payment to %1</source>
<translation>Forespør betaling til %1</translation>
</message>
<message>
<source>Payment information</source>
<translation>Betalingsinformasjon</translation>
</message>
</context>
<context>
<name>RecentRequestsTableModel</name>
<message>
<source>Date</source>
<translation>Dato</translation>
</message>
<message>
<source>Label</source>
<translation>Beskrivelse</translation>
</message>
<message>
<source>Message</source>
<translation>Melding</translation>
</message>
<message>
<source>(no label)</source>
<translation>(ingen beskrivelse)</translation>
</message>
<message>
<source>(no message)</source>
<translation>(ingen melding)</translation>
</message>
<message>
<source>(no amount requested)</source>
<translation>(inget beløp forespurt)</translation>
</message>
<message>
<source>Requested</source>
<translation>Forespurt</translation>
</message>
</context>
<context>
<name>SendCoinsDialog</name>
<message>
<source>Send Coins</source>
<translation>Send Bitcoins</translation>
</message>
<message>
<source>Coin Control Features</source>
<translation>Myntkontroll Funksjoner</translation>
</message>
<message>
<source>Inputs...</source>
<translation>Inndata...</translation>
</message>
<message>
<source>automatically selected</source>
<translation>automatisk valgte</translation>
</message>
<message>
<source>Insufficient funds!</source>
<translation>Utilstrekkelige midler!</translation>
</message>
<message>
<source>Quantity:</source>
<translation>Mengde:</translation>
</message>
<message>
<source>Bytes:</source>
<translation>Bytes:</translation>
</message>
<message>
<source>Amount:</source>
<translation>Beløp:</translation>
</message>
<message>
<source>Fee:</source>
<translation>Gebyr:</translation>
</message>
<message>
<source>After Fee:</source>
<translation>Etter Gebyr:</translation>
</message>
<message>
<source>Change:</source>
<translation>Veksel:</translation>
</message>
<message>
<source>If this is activated, but the change address is empty or invalid, change will be sent to a newly generated address.</source>
<translation>Hvis dette er aktivert, men adressen for veksel er tom eller ugyldig, vil veksel bli sendt til en nylig generert adresse.</translation>
</message>
<message>
<source>Custom change address</source>
<translation>Egendefinert adresse for veksel</translation>
</message>
<message>
<source>Transaction Fee:</source>
<translation>Transaksjonsgebyr:</translation>
</message>
<message>
<source>Choose...</source>
<translation>Velg...</translation>
</message>
<message>
<source>Using the fallbackfee can result in sending a transaction that will take several hours or days (or never) to confirm. Consider choosing your fee manually or wait until you have validated the complete chain.</source>
<translation>Bruk av tilbakefallsgebyr kan medføre at en transaksjon tar flere timer eller dager, å fullføre, eller aldri gjør det. Overvei å velge et gebyr manuelt, eller vent til du har bekreftet hele kjeden.</translation>
</message>
<message>
<source>Warning: Fee estimation is currently not possible.</source>
<translation>Advarsel: Gebyroverslag er ikke tilgjengelig for tiden.</translation>
</message>
<message>
<source>Specify a custom fee per kB (1,000 bytes) of the transaction's virtual size.
Note: Since the fee is calculated on a per-byte basis, a fee of "100 satoshis per kB" for a transaction size of 500 bytes (half of 1 kB) would ultimately yield a fee of only 50 satoshis.</source>
<translation>Spesifiser en tilpasset avgift per kB (1000 byte) av transaksjonens virtuelle størrelse.
Merk: Siden avgiften er beregnet per byte-basis, vil et gebyr på "100 satoshis per kB" for en transaksjonsstørrelse på 500 byte (halvparten av 1 kB) til slutt gi et gebyr på bare 50 satoshis.</translation>
</message>
<message>
<source>per kilobyte</source>
<translation>per kilobyte</translation>
</message>
<message>
<source>Hide</source>
<translation>Skjul</translation>
</message>
<message>
<source>Recommended:</source>
<translation>Anbefalt:</translation>
</message>
<message>
<source>Custom:</source>
<translation>Egendefinert:</translation>
</message>
<message>
<source>(Smart fee not initialized yet. This usually takes a few blocks...)</source>
<translation>(Smartgebyr ikke innført ennå. Dette tar vanligvis noen blokker...)</translation>
</message>
<message>
<source>Send to multiple recipients at once</source>
<translation>Send til flere enn en mottaker</translation>
</message>
<message>
<source>Add &Recipient</source>
<translation>Legg til &Mottaker</translation>
</message>
<message>
<source>Clear all fields of the form.</source>
<translation>Fjern alle felter fra skjemaet.</translation>
</message>
<message>
<source>Dust:</source>
<translation>Støv:</translation>
</message>
<message>
<source>Hide transaction fee settings</source>
<translation>Skjul innstillinger for transaksjonsgebyr</translation>
</message>
<message>
<source>When there is less transaction volume than space in the blocks, miners as well as relaying nodes may enforce a minimum fee. Paying only this minimum fee is just fine, but be aware that this can result in a never confirming transaction once there is more demand for bitcoin transactions than the network can process.</source>
<translation>Når det er mindre transaksjonsvolum enn plass i blokkene, kan gruvearbeidere så vel som videresende noder håndheve et minimumsgebyr. Å betale bare denne minsteavgiften er helt greit, men vær klar over at dette kan resultere i en aldri bekreftende transaksjon når det er større etterspørsel etter bitcoin-transaksjoner enn nettverket kan behandle.</translation>
</message>
<message>
<source>A too low fee might result in a never confirming transaction (read the tooltip)</source>
<translation>For lavt gebyr kan føre til en transaksjon som aldri bekreftes (les verktøytips)</translation>
</message>
<message>
<source>Confirmation time target:</source>
<translation>Bekreftelsestidsmål:</translation>
</message>
<message>
<source>Enable Replace-By-Fee</source>
<translation>Aktiver Replace-By-Fee</translation>
</message>
<message>
<source>With Replace-By-Fee (BIP-125) you can increase a transaction's fee after it is sent. Without this, a higher fee may be recommended to compensate for increased transaction delay risk.</source>
<translation>Med Replace-By-Fee (BIP-125) kan du øke transaksjonens gebyr etter at den er sendt. Uten dette aktivert anbefales et høyere gebyr for å kompensere for risikoen for at transaksjonen blir forsinket.</translation>
</message>
<message>
<source>Clear &All</source>
<translation>Fjern &Alt</translation>
</message>
<message>
<source>Balance:</source>
<translation>Saldo:</translation>
</message>
<message>
<source>Confirm the send action</source>
<translation>Bekreft sending</translation>
</message>
<message>
<source>S&end</source>
<translation>S&end</translation>
</message>
<message>
<source>Copy quantity</source>
<translation>Kopier mengde</translation>
</message>
<message>
<source>Copy amount</source>
<translation>Kopier beløp</translation>
</message>
<message>
<source>Copy fee</source>
<translation>Kopier gebyr</translation>
</message>
<message>
<source>Copy after fee</source>
<translation>Kopiér totalt</translation>
</message>
<message>
<source>Copy bytes</source>
<translation>Kopiér bytes</translation>
</message>
<message>
<source>Copy dust</source>
<translation>Kopiér støv</translation>
</message>
<message>
<source>Copy change</source>
<translation>Kopier veksel</translation>
</message>
<message>
<source>%1 (%2 blocks)</source>
<translation>%1 (%2 blokker)</translation>
</message>
<message>
<source>Cr&eate Unsigned</source>
<translation>Cr & eate Usignert</translation>
</message>
<message>
<source>%1 to %2</source>
<translation>%1 til %2</translation>
</message>
<message>
<source>Do you want to draft this transaction?</source>
<translation>Vil du utarbeide denne transaksjonen?</translation>
</message>
<message>
<source>Are you sure you want to send?</source>
<translation>Er du sikker på at du vil sende?</translation>
</message>
<message>
<source>Create Unsigned</source>
<translation>Lag usignert</translation>
</message>
<message>
<source>Save Transaction Data</source>
<translation>Lagre Transaksjonsdata</translation>
</message>
<message>
<source>Partially Signed Transaction (Binary) (*.psbt)</source>
<translation>Delvis Signert Transaksjon (Binær) (*.psbt)</translation>
</message>
<message>
<source>PSBT saved</source>
<translation>PSBT lagret</translation>
</message>
<message>
<source>or</source>
<translation>eller</translation>
</message>
<message>
<source>You can increase the fee later (signals Replace-By-Fee, BIP-125).</source>
<translation>Du kan øke gebyret senere (signaliserer Replace-By-Fee, BIP-125).</translation>
</message>
<message>
<source>Please, review your transaction proposal. This will produce a Partially Signed Bitcoin Transaction (PSBT) which you can save or copy and then sign with e.g. an offline %1 wallet, or a PSBT-compatible hardware wallet.</source>
<translation>Se over ditt transaksjonsforslag. Dette kommer til å produsere en Delvis Signert Bitcoin Transaksjon (PSBT) som du kan lagre eller kopiere og så signere med f.eks. en offline %1 lommebok, eller en PSBT kompatibel hardware lommebok.</translation>
</message>
<message>
<source>Please, review your transaction.</source>
<translation>Vennligst se over transaksjonen din.</translation>
</message>
<message>
<source>Transaction fee</source>
<translation>Transaksjonsgebyr</translation>
</message>
<message>
<source>Not signalling Replace-By-Fee, BIP-125.</source>
<translation>Signaliserer ikke Replace-By-Fee, BIP-125</translation>
</message>
<message>
<source>Total Amount</source>
<translation>Totalbeløp</translation>
</message>
<message>
<source>To review recipient list click "Show Details..."</source>
<translation>For å se gjennom mottakerlisten, klikk "Vis detaljer ..."</translation>
</message>
<message>
<source>Confirm send coins</source>
<translation>Bekreft forsendelse av mynter</translation>
</message>
<message>
<source>Confirm transaction proposal</source>
<translation>Bekreft transaksjonsforslaget</translation>
</message>
<message>
<source>Send</source>
<translation>Send</translation>
</message>
<message>
<source>Watch-only balance:</source>
<translation>Kun-observer balanse:</translation>
</message>
<message>
<source>The recipient address is not valid. Please recheck.</source>
<translation>Mottakeradressen er ikke gyldig. Sjekk den igjen.</translation>
</message>
<message>
<source>The amount to pay must be larger than 0.</source>
<translation>Betalingsbeløpet må være høyere enn 0.</translation>
</message>
<message>
<source>The amount exceeds your balance.</source>
<translation>Beløper overstiger saldo.</translation>
</message>
<message>
<source>The total exceeds your balance when the %1 transaction fee is included.</source>
<translation>Totalbeløpet overstiger saldo etter at %1-transaksjonsgebyret er lagt til.</translation>
</message>
<message>
<source>Duplicate address found: addresses should only be used once each.</source>
<translation>Gjenbruk av adresse funnet: Adresser skal kun brukes én gang hver.</translation>
</message>
<message>
<source>Transaction creation failed!</source>
<translation>Opprettelse av transaksjon mislyktes!</translation>
</message>
<message>
<source>A fee higher than %1 is considered an absurdly high fee.</source>
<translation>Et gebyr høyere enn %1 anses som absurd høyt.</translation>
</message>
<message>
<source>Payment request expired.</source>
<translation>Tidsavbrudd for betalingsforespørsel</translation>
</message>
<message numerus="yes">
<source>Estimated to begin confirmation within %n block(s).</source>
<translation><numerusform>Antatt bekreftelsesbegynnelse innen %n blokk.</numerusform><numerusform>Antatt bekreftelsesbegynnelse innen %n blokker.</numerusform></translation>
</message>
<message>
<source>Warning: Invalid Bitcoin address</source>
<translation>Advarsel Ugyldig bitcoin-adresse</translation>
</message>
<message>
<source>Warning: Unknown change address</source>
<translation>Advarsel: Ukjent vekslingsadresse</translation>
</message>
<message>
<source>Confirm custom change address</source>
<translation>Bekreft egendefinert vekslingsadresse</translation>
</message>
<message>
<source>The address you selected for change is not part of this wallet. Any or all funds in your wallet may be sent to this address. Are you sure?</source>
<translation>Adressen du valgte for veksling er ikke en del av denne lommeboka. Alle verdiene i din lommebok vil bli sendt til denne adressen. Er du sikker?</translation>
</message>
<message>
<source>(no label)</source>
<translation>(ingen beskrivelse)</translation>
</message>
</context>
<context>
<name>SendCoinsEntry</name>
<message>
<source>A&mount:</source>
<translation>&Beløp:</translation>
</message>
<message>
<source>Pay &To:</source>
<translation>Betal &Til:</translation>
</message>
<message>
<source>&Label:</source>
<translation>&Merkelapp:</translation>
</message>
<message>
<source>Choose previously used address</source>
<translation>Velg tidligere brukt adresse</translation>
</message>
<message>
<source>The Bitcoin address to send the payment to</source>
<translation>Bitcoin-adressen betalingen skal sendes til</translation>
</message>
<message>
<source>Alt+A</source>
<translation>Alt+A</translation>
</message>
<message>
<source>Paste address from clipboard</source>
<translation>Lim inn adresse fra utklippstavlen</translation>
</message>
<message>
<source>Alt+P</source>
<translation>Alt+P</translation>
</message>
<message>
<source>Remove this entry</source>
<translation>Fjern denne oppføringen</translation>
</message>
<message>
<source>The amount to send in the selected unit</source>
<translation>beløpet som skal sendes inn den valgte enheten.</translation>
</message>
<message>
<source>The fee will be deducted from the amount being sent. The recipient will receive less bitcoins than you enter in the amount field. If multiple recipients are selected, the fee is split equally.</source>
<translation>Gebyret vil bli trukket fra beløpet som blir sendt. Mottakeren vil motta mindre bitcoins enn det du skriver inn i beløpsfeltet. Hvis det er valgt flere mottakere, deles gebyret likt.</translation>
</message>
<message>
<source>S&ubtract fee from amount</source>
<translation>T&rekk fra gebyr fra beløp</translation>
</message>
<message>
<source>Use available balance</source>
<translation>Bruk tilgjengelig saldo</translation>
</message>
<message>
<source>Message:</source>
<translation>Melding:</translation>
</message>
<message>
<source>This is an unauthenticated payment request.</source>
<translation>Dette er en uautorisert betalingsetterspørring.</translation>
</message>
<message>
<source>This is an authenticated payment request.</source>
<translation>Dette er en autorisert betalingsetterspørring.</translation>
</message>
<message>
<source>Enter a label for this address to add it to the list of used addresses</source>
<translation>Skriv inn en merkelapp for denne adressen for å legge den til listen av brukte adresser</translation>
</message>
<message>
<source>A message that was attached to the bitcoin: URI which will be stored with the transaction for your reference. Note: This message will not be sent over the Bitcoin network.</source>
<translation>En melding som var tilknyttet bitcoinen: URI vil bli lagret med transaksjonen for din oversikt. Denne meldingen vil ikke bli sendt over Bitcoin-nettverket.</translation>
</message>
<message>
<source>Pay To:</source>
<translation>Betal Til:</translation>
</message>
<message>
<source>Memo:</source>
<translation>Memo:</translation>
</message>
</context>
<context>
<name>ShutdownWindow</name>
<message>
<source>%1 is shutting down...</source>
<translation>%1 lukker...</translation>
</message>
<message>
<source>Do not shut down the computer until this window disappears.</source>
<translation>Slå ikke av datamaskinen før dette vinduet forsvinner.</translation>
</message>
</context>
<context>
<name>SignVerifyMessageDialog</name>
<message>
<source>Signatures - Sign / Verify a Message</source>
<translation>Signaturer - Signer / Verifiser en Melding</translation>
</message>
<message>
<source>&Sign Message</source>
<translation>&Signer Melding</translation>
</message>
<message>
<source>You can sign messages/agreements with your addresses to prove you can receive bitcoins sent to them. Be careful not to sign anything vague or random, as phishing attacks may try to trick you into signing your identity over to them. Only sign fully-detailed statements you agree to.</source>
<translation>Du kan signere meldinger/avtaler med adresser for å bevise at du kan motta bitcoins sendt til dem. Vær forsiktig med å signere noe vagt eller tilfeldig, siden phishing-angrep kan prøve å lure deg til å signere din identitet over til dem. Bare signer fullt detaljerte utsagn som du er enig i.</translation>
</message>
<message>
<source>The Bitcoin address to sign the message with</source>
<translation>Bitcoin-adressen meldingen skal signeres med</translation>
</message>
<message>
<source>Choose previously used address</source>
<translation>Velg tidligere brukt adresse</translation>
</message>
<message>
<source>Alt+A</source>
<translation>Alt+A</translation>
</message>
<message>
<source>Paste address from clipboard</source>
<translation>Lim inn adresse fra utklippstavlen</translation>
</message>
<message>
<source>Alt+P</source>
<translation>Alt+P</translation>
</message>
<message>
<source>Enter the message you want to sign here</source>
<translation>Skriv inn meldingen du vil signere her</translation>
</message>
<message>
<source>Signature</source>
<translation>Signatur</translation>
</message>
<message>
<source>Copy the current signature to the system clipboard</source>
<translation>Kopier valgt signatur til utklippstavle</translation>
</message>
<message>
<source>Sign the message to prove you own this Bitcoin address</source>
<translation>Signer meldingen for å bevise at du eier denne Bitcoin-adressen</translation>
</message>
<message>
<source>Sign &Message</source>
<translation>Signer &Melding</translation>
</message>
<message>
<source>Reset all sign message fields</source>
<translation>Tilbakestill alle felter for meldingssignering</translation>
</message>
<message>
<source>Clear &All</source>
<translation>Fjern &Alt</translation>
</message>
<message>
<source>&Verify Message</source>
<translation>&Verifiser Melding</translation>
</message>
<message>
<source>Enter the receiver's address, message (ensure you copy line breaks, spaces, tabs, etc. exactly) and signature below to verify the message. Be careful not to read more into the signature than what is in the signed message itself, to avoid being tricked by a man-in-the-middle attack. Note that this only proves the signing party receives with the address, it cannot prove sendership of any transaction!</source>
<translation>Skriv inn mottakerens adresse, melding (forsikre deg om at du kopier linjeskift, mellomrom, faner osv. nøyaktig) og underskrift nedenfor for å bekrefte meldingen. Vær forsiktig så du ikke leser mer ut av signaturen enn hva som er i den signerte meldingen i seg selv, for å unngå å bli lurt av et man-in-the-middle-angrep. Merk at dette bare beviser at den som signerer kan motta med adressen, dette beviser ikke hvem som har sendt transaksjoner!</translation>
</message>
<message>
<source>The Bitcoin address the message was signed with</source>
<translation>Bitcoin-adressen meldingen ble signert med</translation>
</message>
<message>
<source>The signed message to verify</source>
<translation>Den signerte meldingen for å bekfrefte</translation>
</message>
<message>
<source>The signature given when the message was signed</source>
<translation>signaturen som ble gitt da meldingen ble signert</translation>
</message>
<message>
<source>Verify the message to ensure it was signed with the specified Bitcoin address</source>
<translation>Verifiser meldingen for å være sikker på at den ble signert av den angitte Bitcoin-adressen</translation>
</message>
<message>
<source>Verify &Message</source>
<translation>Verifiser &Melding</translation>
</message>
<message>
<source>Reset all verify message fields</source>
<translation>Tilbakestill alle felter for meldingsverifikasjon</translation>
</message>
<message>
<source>Click "Sign Message" to generate signature</source>
<translation>Klikk "Signer melding" for å generere signatur</translation>
</message>
<message>
<source>The entered address is invalid.</source>
<translation>Innskrevet adresse er ugyldig.</translation>
</message>
<message>
<source>Please check the address and try again.</source>
<translation>Sjekk adressen og prøv igjen.</translation>
</message>
<message>
<source>The entered address does not refer to a key.</source>
<translation>Innskrevet adresse refererer ikke til noen nøkkel.</translation>
</message>
<message>
<source>Wallet unlock was cancelled.</source>
<translation>Opplåsning av lommebok ble avbrutt.</translation>
</message>
<message>
<source>No error</source>
<translation>Ingen feil</translation>
</message>
<message>
<source>Private key for the entered address is not available.</source>
<translation>Privat nøkkel for den angitte adressen er ikke tilgjengelig.</translation>
</message>
<message>
<source>Message signing failed.</source>
<translation>Signering av melding feilet.</translation>
</message>
<message>
<source>Message signed.</source>
<translation>Melding signert.</translation>
</message>
<message>
<source>The signature could not be decoded.</source>
<translation>Signaturen kunne ikke dekodes.</translation>
</message>
<message>
<source>Please check the signature and try again.</source>
<translation>Sjekk signaturen og prøv igjen.</translation>
</message>
<message>
<source>The signature did not match the message digest.</source>
<translation>Signaturen samsvarer ikke med meldingsporteføljen.</translation>
</message>
<message>
<source>Message verification failed.</source>
<translation>Meldingsverifiseringen mislyktes.</translation>
</message>
<message>
<source>Message verified.</source>
<translation>Melding bekreftet.</translation>
</message>
</context>
<context>
<name>TrafficGraphWidget</name>
<message>
<source>KB/s</source>
<translation>KB/s</translation>
</message>
</context>
<context>
<name>TransactionDesc</name>
<message numerus="yes">
<source>Open for %n more block(s)</source>
<translation><numerusform>Åpen for %n blokk til</numerusform><numerusform>Åpen for %n flere blokker</numerusform></translation>
</message>
<message>
<source>Open until %1</source>
<translation>Åpen til %1</translation>
</message>
<message>
<source>conflicted with a transaction with %1 confirmations</source>
<translation>gikk ikke overens med en transaksjon med %1 bekreftelser</translation>
</message>
<message>
<source>0/unconfirmed, %1</source>
<translation>0/ubekreftet, %1</translation>
</message>
<message>
<source>in memory pool</source>
<translation>i hukommelsespulje</translation>
</message>
<message>
<source>not in memory pool</source>
<translation>ikke i hukommelsespulje</translation>
</message>
<message>
<source>abandoned</source>
<translation>forlatt</translation>
</message>
<message>
<source>%1/unconfirmed</source>
<translation>%1/ubekreftet</translation>
</message>
<message>
<source>%1 confirmations</source>
<translation>%1 bekreftelser</translation>
</message>
<message>
<source>Status</source>
<translation>Status</translation>
</message>
<message>
<source>Date</source>
<translation>Dato</translation>
</message>
<message>
<source>Source</source>
<translation>Kilde</translation>
</message>
<message>
<source>Generated</source>
<translation>Generert</translation>
</message>
<message>
<source>From</source>
<translation>Fra</translation>
</message>
<message>
<source>unknown</source>
<translation>ukjent</translation>
</message>
<message>
<source>To</source>
<translation>Til</translation>
</message>
<message>
<source>own address</source>
<translation>egen adresse</translation>
</message>
<message>
<source>watch-only</source>
<translation>kun oppsyn</translation>
</message>
<message>
<source>label</source>
<translation>merkelapp</translation>
</message>
<message>
<source>Credit</source>
<translation>Kreditt</translation>
</message>
<message numerus="yes">
<source>matures in %n more block(s)</source>
<translation><numerusform>modner om %n blokk</numerusform><numerusform>modner om %n blokker</numerusform></translation>
</message>
<message>
<source>not accepted</source>
<translation>ikke akseptert</translation>
</message>
<message>
<source>Debit</source>
<translation>Debet</translation>
</message>
<message>
<source>Total debit</source>
<translation>Total debet</translation>
</message>
<message>
<source>Total credit</source>
<translation>Total kreditt</translation>
</message>
<message>
<source>Transaction fee</source>
<translation>Transaksjonsgebyr</translation>
</message>
<message>
<source>Net amount</source>
<translation>Nettobeløp</translation>
</message>
<message>
<source>Message</source>
<translation>Melding</translation>
</message>
<message>
<source>Comment</source>
<translation>Kommentar</translation>
</message>
<message>
<source>Transaction ID</source>
<translation>Transaksjons-ID</translation>
</message>
<message>
<source>Transaction total size</source>
<translation>Total transaksjonsstørrelse</translation>
</message>
<message>
<source>Transaction virtual size</source>
<translation>Virtuell transaksjonsstørrelse</translation>
</message>
<message>
<source>Output index</source>
<translation>Utdatainndeks</translation>
</message>
<message>
<source> (Certificate was not verified)</source>
<translation>(sertifikatet ble ikke bekreftet)</translation>
</message>
<message>
<source>Merchant</source>
<translation>Forretningsdrivende</translation>
</message>
<message>
<source>Generated coins must mature %1 blocks before they can be spent. When you generated this block, it was broadcast to the network to be added to the block chain. If it fails to get into the chain, its state will change to "not accepted" and it won't be spendable. This may occasionally happen if another node generates a block within a few seconds of yours.</source>
<translation>Genererte bitcoins må modne %1 blokker før de kan brukes. Da du genererte denne blokken ble den kringkastet på nettverket for å bli lagt til i kjeden av blokker. Hvis den ikke kommer med i kjeden vil den endre seg til "ikke akseptert", og vil ikke kunne brukes. Dette vil noen ganger skje hvis en annen node genererer en blokk innen noen sekunder av din.</translation>
</message>
<message>
<source>Debug information</source>
<translation>Feilrettingsinformasjon</translation>
</message>
<message>
<source>Transaction</source>
<translation>Transaksjon</translation>
</message>
<message>
<source>Inputs</source>
<translation>Inndata</translation>
</message>
<message>
<source>Amount</source>
<translation>Beløp</translation>
</message>
<message>
<source>true</source>
<translation>sant</translation>
</message>
<message>
<source>false</source>
<translation>usant</translation>
</message>
</context>
<context>
<name>TransactionDescDialog</name>
<message>
<source>This pane shows a detailed description of the transaction</source>
<translation>Her vises en detaljert beskrivelse av transaksjonen</translation>
</message>
<message>
<source>Details for %1</source>
<translation>Detaljer for %1</translation>
</message>
</context>
<context>
<name>TransactionTableModel</name>
<message>
<source>Date</source>
<translation>Dato</translation>
</message>
<message>
<source>Type</source>
<translation>Type</translation>
</message>
<message>
<source>Label</source>
<translation>Beskrivelse</translation>
</message>
<message numerus="yes">
<source>Open for %n more block(s)</source>
<translation><numerusform>Åpen for én blokk til</numerusform><numerusform>Åpen for %n blokker til</numerusform></translation>
</message>
<message>
<source>Open until %1</source>
<translation>Åpen til %1</translation>
</message>
<message>
<source>Unconfirmed</source>
<translation>Ubekreftet</translation>
</message>
<message>
<source>Abandoned</source>
<translation>Forlatt</translation>
</message>
<message>
<source>Confirming (%1 of %2 recommended confirmations)</source>
<translation>Bekrefter (%1 av %2 anbefalte bekreftelser)</translation>
</message>
<message>
<source>Confirmed (%1 confirmations)</source>
<translation>Bekreftet (%1 bekreftelser)</translation>
</message>
<message>
<source>Conflicted</source>
<translation>Gikk ikke overens</translation>
</message>
<message>
<source>Immature (%1 confirmations, will be available after %2)</source>
<translation>Umoden (%1 bekreftelser, vil være tilgjengelig etter %2)</translation>
</message>
<message>
<source>Generated but not accepted</source>
<translation>Generert, men ikke akseptert</translation>
</message>
<message>
<source>Received with</source>
<translation>Mottatt med</translation>
</message>
<message>
<source>Received from</source>
<translation>Mottatt fra</translation>
</message>
<message>
<source>Sent to</source>
<translation>Sendt til</translation>
</message>
<message>
<source>Payment to yourself</source>
<translation>Betaling til deg selv</translation>
</message>
<message>
<source>Mined</source>
<translation>Utvunnet</translation>
</message>
<message>
<source>watch-only</source>
<translation>kun oppsyn</translation>
</message>
<message>
<source>(n/a)</source>
<translation>(i/t)</translation>
</message>
<message>
<source>(no label)</source>
<translation>(ingen beskrivelse)</translation>
</message>
<message>
<source>Transaction status. Hover over this field to show number of confirmations.</source>
<translation>Transaksjonsstatus. Hold pekeren over dette feltet for å se antall bekreftelser.</translation>
</message>
<message>
<source>Date and time that the transaction was received.</source>
<translation>Dato og tid for mottak av transaksjonen.</translation>
</message>
<message>
<source>Type of transaction.</source>
<translation>Transaksjonstype.</translation>
</message>
<message>
<source>Whether or not a watch-only address is involved in this transaction.</source>
<translation>Hvorvidt en oppsynsadresse er involvert i denne transaksjonen.</translation>
</message>
<message>
<source>User-defined intent/purpose of the transaction.</source>
<translation>Brukerdefinert intensjon/hensikt med transaksjonen.</translation>
</message>
<message>
<source>Amount removed from or added to balance.</source>
<translation>Beløp fjernet eller lagt til saldo.</translation>
</message>
</context>
<context>
<name>TransactionView</name>
<message>
<source>All</source>
<translation>Alt</translation>
</message>
<message>
<source>Today</source>
<translation>I dag</translation>
</message>
<message>
<source>This week</source>
<translation>Denne uka</translation>
</message>
<message>
<source>This month</source>
<translation>Denne måneden</translation>
</message>
<message>
<source>Last month</source>
<translation>Forrige måned</translation>
</message>
<message>
<source>This year</source>
<translation>Dette året</translation>
</message>
<message>
<source>Range...</source>
<translation>Rekkevidde…</translation>
</message>
<message>
<source>Received with</source>
<translation>Mottatt med</translation>
</message>
<message>
<source>Sent to</source>
<translation>Sendt til</translation>
</message>
<message>
<source>To yourself</source>
<translation>Til deg selv</translation>
</message>
<message>
<source>Mined</source>
<translation>Utvunnet</translation>
</message>
<message>
<source>Other</source>
<translation>Andre</translation>
</message>
<message>
<source>Enter address, transaction id, or label to search</source>
<translation>Oppgi adresse, transaksjons-ID eller merkelapp for å søke</translation>
</message>
<message>
<source>Min amount</source>
<translation>Minimumsbeløp</translation>
</message>
<message>
<source>Abandon transaction</source>
<translation>Avbryt transaksjon</translation>
</message>
<message>
<source>Increase transaction fee</source>
<translation>Øk overføringsgebyret</translation>
</message>
<message>
<source>Copy address</source>
<translation>Kopier adresse</translation>
</message>
<message>
<source>Copy label</source>
<translation>Kopiér merkelapp</translation>
</message>
<message>
<source>Copy amount</source>
<translation>Kopier beløp</translation>
</message>
<message>
<source>Copy transaction ID</source>
<translation>Kopier transaksjons-ID</translation>
</message>
<message>
<source>Copy raw transaction</source>
<translation>Kopier råtransaksjon</translation>
</message>
<message>
<source>Copy full transaction details</source>
<translation>Kopier helhetlig transaksjonsdetaljering</translation>
</message>
<message>
<source>Edit label</source>
<translation>Rediger merkelapp</translation>
</message>
<message>
<source>Show transaction details</source>
<translation>Vis transaksjonsdetaljer</translation>
</message>
<message>
<source>Export Transaction History</source>
<translation>Eksporter transaksjonshistorikk</translation>
</message>
<message>
<source>Comma separated file (*.csv)</source>
<translation>Komma separert fil (*.csv)</translation>
</message>
<message>
<source>Confirmed</source>
<translation>Bekreftet</translation>
</message>
<message>
<source>Watch-only</source>
<translation>Kun oppsyn</translation>
</message>
<message>
<source>Date</source>
<translation>Dato</translation>
</message>
<message>
<source>Type</source>
<translation>Type</translation>
</message>
<message>
<source>Label</source>
<translation>Beskrivelse</translation>
</message>
<message>
<source>Address</source>
<translation>Adresse</translation>
</message>
<message>
<source>ID</source>
<translation>ID</translation>
</message>
<message>
<source>Exporting Failed</source>
<translation>Eksporten feilet</translation>
</message>
<message>
<source>There was an error trying to save the transaction history to %1.</source>
<translation>En feil oppstod ved lagring av transaksjonshistorikk til %1.</translation>
</message>
<message>
<source>Exporting Successful</source>
<translation>Eksportert</translation>
</message>
<message>
<source>The transaction history was successfully saved to %1.</source>
<translation>Transaksjonshistorikken ble lagret til %1.</translation>
</message>
<message>
<source>Range:</source>
<translation>Rekkevidde:</translation>
</message>
<message>
<source>to</source>
<translation>til</translation>
</message>
</context>
<context>
<name>UnitDisplayStatusBarControl</name>
<message>
<source>Unit to show amounts in. Click to select another unit.</source>
<translation>Enhet å vise beløper i. Klikk for å velge en annen enhet.</translation>
</message>
</context>
<context>
<name>WalletController</name>
<message>
<source>Close wallet</source>
<translation>Lukk lommebok</translation>
</message>
<message>
<source>Closing the wallet for too long can result in having to resync the entire chain if pruning is enabled.</source>
<translation>Å lukke lommeboken for lenge kan føre til at du må synkronisere hele kjeden hvis beskjæring er aktivert.</translation>
</message>
<message>
<source>Close all wallets</source>
<translation>Lukk alle lommebøker</translation>
</message>
<message>
<source>Are you sure you wish to close all wallets?</source>
<translation>Er du sikker på at du vil lukke alle lommebøker?</translation>
</message>
</context>
<context>
<name>WalletFrame</name>
<message>
<source>No wallet has been loaded.
Go to File > Open Wallet to load a wallet.
- OR -</source>
<translation>Ingen lommebok har blitt lastet.
Gå til Fil > Åpne lommebok for å laste en lommebok.
- ELLER -</translation>
</message>
<message>
<source>Create a new wallet</source>
<translation>Lag en ny lommebok</translation>
</message>
</context>
<context>
<name>WalletModel</name>
<message>
<source>Send Coins</source>
<translation>Send mynter</translation>
</message>
<message>
<source>Fee bump error</source>
<translation>Gebyrforhøyelsesfeil</translation>
</message>
<message>
<source>Increasing transaction fee failed</source>
<translation>Økning av transaksjonsgebyr mislyktes</translation>
</message>
<message>
<source>Do you want to increase the fee?</source>
<translation>Ønsker du å øke gebyret?</translation>
</message>
<message>
<source>Do you want to draft a transaction with fee increase?</source>
<translation>Vil du utarbeide en transaksjon med gebyrøkning?</translation>
</message>
<message>
<source>Current fee:</source>
<translation>Nåværede gebyr:</translation>
</message>
<message>
<source>Increase:</source>
<translation>Økning:</translation>
</message>
<message>
<source>New fee:</source>
<translation>Nytt gebyr:</translation>
</message>
<message>
<source>Confirm fee bump</source>
<translation>Bekreft gebyrøkning</translation>
</message>
<message>
<source>Can't draft transaction.</source>
<translation>Kan ikke utarbeide transaksjon.</translation>
</message>
<message>
<source>PSBT copied</source>
<translation>PSBT kopiert</translation>
</message>
<message>
<source>Can't sign transaction.</source>
<translation>Kan ikke signere transaksjon</translation>
</message>
<message>
<source>Could not commit transaction</source>
<translation>Kunne ikke sende inn transaksjon</translation>
</message>
<message>
<source>default wallet</source>
<translation>standard lommebok</translation>
</message>
</context>
<context>
<name>WalletView</name>
<message>
<source>&Export</source>
<translation>&Eksport</translation>
</message>
<message>
<source>Export the data in the current tab to a file</source>
<translation>Eksporter data i den valgte fliken til en fil</translation>
</message>
<message>
<source>Error</source>
<translation>Feilmelding</translation>
</message>
<message>
<source>Unable to decode PSBT from clipboard (invalid base64)</source>
<translation>Klarte ikke å dekode PSBT fra utklippstavle (ugyldig base64)</translation>
</message>
<message>
<source>Load Transaction Data</source>
<translation>Last transaksjonsdata</translation>
</message>
<message>
<source>Partially Signed Transaction (*.psbt)</source>
<translation>Delvis signert transaksjon (*.psbt)</translation>
</message>
<message>
<source>PSBT file must be smaller than 100 MiB</source>
<translation>PSBT-fil må være mindre enn 100 MiB</translation>
</message>
<message>
<source>Unable to decode PSBT</source>
<translation>Klarte ikke å dekode PSBT</translation>
</message>
<message>
<source>Backup Wallet</source>
<translation>Sikkerhetskopier lommebok</translation>
</message>
<message>
<source>Wallet Data (*.dat)</source>
<translation>Lommeboksdata (*.dat)</translation>
</message>
<message>
<source>Backup Failed</source>
<translation>Sikkerhetskopiering mislyktes</translation>
</message>
<message>
<source>There was an error trying to save the wallet data to %1.</source>
<translation>Feil under forsøk på lagring av lommebokdata til %1</translation>
</message>
<message>
<source>Backup Successful</source>
<translation>Sikkerhetskopiert</translation>
</message>
<message>
<source>The wallet data was successfully saved to %1.</source>
<translation>Lommebokdata lagret til %1.</translation>
</message>
<message>
<source>Cancel</source>
<translation>Avbryt</translation>
</message>
</context>
<context>
<name>bitcoin-core</name>
<message>
<source>Distributed under the MIT software license, see the accompanying file %s or %s</source>
<translation>Lisensiert MIT. Se tilhørende fil %s eller %s</translation>
</message>
<message>
<source>Prune configured below the minimum of %d MiB. Please use a higher number.</source>
<translation>Beskjæringsmodus er konfigurert under minimum på %d MiB. Vennligst bruk et høyere nummer.</translation>
</message>
<message>
<source>Prune: last wallet synchronisation goes beyond pruned data. You need to -reindex (download the whole blockchain again in case of pruned node)</source>
<translation>Beskjæring: siste lommeboksynkronisering går utenfor beskjærte data. Du må bruke -reindex (laster ned hele blokkjeden igjen for beskjærte noder)</translation>
</message>
<message>
<source>Pruning blockstore...</source>
<translation>Beskjærer blokklageret...</translation>
</message>
<message>
<source>Unable to start HTTP server. See debug log for details.</source>
<translation>Kunne ikke starte HTTP-tjener. Se feilrettingslogg for detaljer.</translation>
</message>
<message>
<source>The %s developers</source>
<translation>%s-utviklerne</translation>
</message>
<message>
<source>Cannot obtain a lock on data directory %s. %s is probably already running.</source>
<translation>Kan ikke låse datamappen %s. %s kjører antagelig allerede.</translation>
</message>
<message>
<source>Cannot provide specific connections and have addrman find outgoing connections at the same.</source>
<translation>Kan ikke angi spesifikke tilkoblinger og ha addrman til å finne utgående tilkoblinger samtidig.</translation>
</message>
<message>
<source>Error reading %s! All keys read correctly, but transaction data or address book entries might be missing or incorrect.</source>
<translation>Feil under lesing av %s! Alle nøkler har blitt lest rett, men transaksjonsdata eller adressebokoppføringer kan mangle eller være uriktige.</translation>
</message>
<message>
<source>More than one onion bind address is provided. Using %s for the automatically created Tor onion service.</source>
<translation>Mer enn en onion adresse har blitt gitt. Bruker %s for den automatisk lagde Tor onion tjenesten.</translation>
</message>
<message>
<source>Please check that your computer's date and time are correct! If your clock is wrong, %s will not work properly.</source>
<translation>Sjekk at din datamaskins dato og klokke er stilt rett! Hvis klokka er feil, vil ikke %s fungere ordentlig.</translation>
</message>
<message>
<source>Please contribute if you find %s useful. Visit %s for further information about the software.</source>
<translation>Bidra hvis du finner %s nyttig. Besøk %s for mer informasjon om programvaren.</translation>
</message>
<message>
<source>The block database contains a block which appears to be from the future. This may be due to your computer's date and time being set incorrectly. Only rebuild the block database if you are sure that your computer's date and time are correct</source>
<translation>Blokkdatabasen inneholder en blokk som ser ut til å være fra fremtiden. Dette kan være fordi dato og tid på din datamaskin er satt feil. Gjenopprett kun blokkdatabasen når du er sikker på at dato og tid er satt riktig.</translation>
</message>
<message>
<source>This is a pre-release test build - use at your own risk - do not use for mining or merchant applications</source>
<translation>Dette er en testversjon i påvente av utgivelse - bruk på egen risiko - ikke for bruk til blokkutvinning eller i forretningsøyemed</translation>
</message>
<message>
<source>This is the transaction fee you may discard if change is smaller than dust at this level</source>
<translation>Dette er transaksjonsgebyret du kan se bort fra hvis vekslepengene utgjør mindre enn støv på dette nivået</translation>
</message>
<message>
<source>Unable to replay blocks. You will need to rebuild the database using -reindex-chainstate.</source>
<translation>Kan ikke spille av blokker igjen. Du må bygge opp igjen databasen ved bruk av -reindex-chainstate.</translation>
</message>
<message>
<source>Unable to rewind the database to a pre-fork state. You will need to redownload the blockchain</source>
<translation>Kan ikke spole tilbake databasen til en tilstand før forgreiningen. Du må laste ned blokkjeden igjen</translation>
</message>
<message>
<source>Warning: The network does not appear to fully agree! Some miners appear to be experiencing issues.</source>
<translation>Advarsel: Nettverket ser ikke ut til å være i overenstemmelse! Noen utvinnere ser ut til å ha problemer.</translation>
</message>
<message>
<source>Warning: We do not appear to fully agree with our peers! You may need to upgrade, or other nodes may need to upgrade.</source>
<translation>Advarsel: Vi ser ikke ut til å være i full overenstemmelse med våre likemenn! Du kan trenge å oppgradere, eller andre noder kan trenge å oppgradere.</translation>
</message>
<message>
<source>-maxmempool must be at least %d MB</source>
<translation>-maxmempool må være minst %d MB</translation>
</message>
<message>
<source>Cannot resolve -%s address: '%s'</source>
<translation>Kunne ikke slå opp -%s-adresse: "%s"</translation>
</message>
<message>
<source>Change index out of range</source>
<translation>Kjedeindeks utenfor rekkevidde</translation>
</message>
<message>
<source>Copyright (C) %i-%i</source>
<translation>Kopirett © %i-%i</translation>
</message>
<message>
<source>Corrupted block database detected</source>
<translation>Oppdaget korrupt blokkdatabase</translation>
</message>
<message>
<source>Could not find asmap file %s</source>
<translation>Kunne ikke finne asmap filen %s</translation>
</message>
<message>
<source>Could not parse asmap file %s</source>
<translation>Kunne ikke analysere asmap filen %s</translation>
</message>
<message>
<source>Do you want to rebuild the block database now?</source>
<translation>Ønsker du å gjenopprette blokkdatabasen nå?</translation>
</message>
<message>
<source>Error initializing block database</source>
<translation>Feil under initialisering av blokkdatabase</translation>
</message>
<message>
<source>Error initializing wallet database environment %s!</source>
<translation>Feil under oppstart av lommeboken sitt databasemiljø %s!</translation>
</message>
<message>
<source>Error loading %s</source>
<translation>Feil ved lasting av %s</translation>
</message>
<message>
<source>Error loading %s: Wallet corrupted</source>
<translation>Feil under innlasting av %s: Skadet lommebok</translation>
</message>
<message>
<source>Error loading %s: Wallet requires newer version of %s</source>
<translation>Feil under innlasting av %s: Lommeboka krever nyere versjon av %s</translation>
</message>
<message>
<source>Error loading block database</source>
<translation>Feil ved lasting av blokkdatabase</translation>
</message>
<message>
<source>Error opening block database</source>
<translation>Feil under åpning av blokkdatabase</translation>
</message>
<message>
<source>Failed to listen on any port. Use -listen=0 if you want this.</source>
<translation>Kunne ikke lytte på noen port. Bruk -listen=0 hvis det er dette du vil.</translation>
</message>
<message>
<source>Failed to rescan the wallet during initialization</source>
<translation>Klarte ikke gå igjennom lommeboken under oppstart</translation>
</message>
<message>
<source>Failed to verify database</source>
<translation>Verifisering av database feilet</translation>
</message>
<message>
<source>Ignoring duplicate -wallet %s.</source>
<translation>Ignorerer dupliserte -wallet %s.</translation>
</message>
<message>
<source>Importing...</source>
<translation>Importerer...</translation>
</message>
<message>
<source>Incorrect or no genesis block found. Wrong datadir for network?</source>
<translation>Ugyldig eller ingen skaperblokk funnet. Feil datamappe for nettverk?</translation>
</message>
<message>
<source>Initialization sanity check failed. %s is shutting down.</source>
<translation>Sunnhetssjekk ved oppstart mislyktes. %s skrus av.</translation>
</message>
<message>
<source>Invalid amount for -%s=<amount>: '%s'</source>
<translation>Ugyldig beløp for -%s=<amount>: "%s"</translation>
</message>
<message>
<source>Invalid amount for -discardfee=<amount>: '%s'</source>
<translation>Ugyldig beløp for -discardfee=<amount>: "%s"</translation>
</message>
<message>
<source>Invalid amount for -fallbackfee=<amount>: '%s'</source>
<translation>Ugyldig beløp for -fallbackfee=<amount>: "%s"</translation>
</message>
<message>
<source>Upgrading txindex database</source>
<translation>Oppgraderer txindex databasen</translation>
</message>
<message>
<source>Loading P2P addresses...</source>
<translation>Laster maskin-til-maskin -adresser…</translation>
</message>
<message>
<source>Loading banlist...</source>
<translation>Laster inn bannlysningsliste…</translation>
</message>
<message>
<source>Not enough file descriptors available.</source>
<translation>For få fildeskriptorer tilgjengelig.</translation>
</message>
<message>
<source>Prune cannot be configured with a negative value.</source>
<translation>Beskjæringsmodus kan ikke konfigureres med en negativ verdi.</translation>
</message>
<message>
<source>Prune mode is incompatible with -txindex.</source>
<translation>Beskjæringsmodus er ikke kompatibel med -txindex.</translation>
</message>
<message>
<source>Replaying blocks...</source>
<translation>Spiller av blokker igjen…</translation>
</message>
<message>
<source>Rewinding blocks...</source>
<translation>Spoler tilbake blokker…</translation>
</message>
<message>
<source>The source code is available from %s.</source>
<translation>Kildekoden er tilgjengelig fra %s.</translation>
</message>
<message>
<source>Transaction fee and change calculation failed</source>
<translation>Transaksjonsgebyr og vekslingsutregning mislyktes</translation>
</message>
<message>
<source>Unable to bind to %s on this computer. %s is probably already running.</source>
<translation>Kan ikke binde til %s på denne datamaskinen. Sannsynligvis kjører %s allerede.</translation>
</message>
<message>
<source>Unable to generate keys</source>
<translation>Klarte ikke å lage nøkkel</translation>
</message>
<message>
<source>Unsupported logging category %s=%s.</source>
<translation>Ustøttet loggingskategori %s=%s.</translation>
</message>
<message>
<source>Upgrading UTXO database</source>
<translation>Oppgraderer UTXO-database</translation>
</message>
<message>
<source>User Agent comment (%s) contains unsafe characters.</source>
<translation>User Agent kommentar (%s) inneholder utrygge tegn.</translation>
</message>
<message>
<source>Verifying blocks...</source>
<translation>Verifiserer blokker...</translation>
</message>
<message>
<source>Wallet needed to be rewritten: restart %s to complete</source>
<translation>Lommeboka må skrives om: Start %s på nytt for å fullføre</translation>
</message>
<message>
<source>Error: Listening for incoming connections failed (listen returned error %s)</source>
<translation>Feil: Lytting etter innkommende tilkoblinger feilet (lytting returnerte feil %s)</translation>
</message>
<message>
<source>%s corrupt. Try using the wallet tool bitcoin-wallet to salvage or restoring a backup.</source>
<translation>%s korrupt. Prøv å bruk lommebokverktøyet bitcoin-wallet til å fikse det eller laste en backup.</translation>
</message>
<message>
<source>Invalid amount for -maxtxfee=<amount>: '%s' (must be at least the minrelay fee of %s to prevent stuck transactions)</source>
<translation>Ugyldig beløp for -maxtxfee=<amount>: '%s' (må være minst minimum relé gebyr på %s for å hindre fastlåste transaksjoner)</translation>
</message>
<message>
<source>The transaction amount is too small to send after the fee has been deducted</source>
<translation>Transaksjonsbeløpet er for lite til å sendes etter at gebyret er fratrukket</translation>
</message>
<message>
<source>This error could occur if this wallet was not shutdown cleanly and was last loaded using a build with a newer version of Berkeley DB. If so, please use the software that last loaded this wallet</source>
<translation>Denne feilen kan oppstå hvis denne lommeboken ikke ble avsluttet skikkelig og var sist lastet med en build som hadde en nyere versjon av Berkeley DB. Hvis det har skjedd, vær så snill å bruk softwaren som sist lastet denne lommeboken.</translation>
</message>
<message>
<source>This is the maximum transaction fee you pay (in addition to the normal fee) to prioritize partial spend avoidance over regular coin selection.</source>
<translation>Dette er maksimum transaksjonsavgift du betaler (i tillegg til den normale avgiften) for å prioritere delvis betaling unngåelse over normal mynt seleksjon.</translation>
</message>
<message>
<source>Transaction needs a change address, but we can't generate it. Please call keypoolrefill first.</source>
<translation>Transaksjon trenger en veksel adresse, men vi kan ikke generere den. Kall keypoolrefill først.</translation>
</message>
<message>
<source>You need to rebuild the database using -reindex to go back to unpruned mode. This will redownload the entire blockchain</source>
<translation>Du må gjenoppbygge databasen ved hjelp av -reindex for å gå tilbake til ubeskåret modus. Dette vil laste ned hele blokkjeden på nytt.</translation>
</message>
<message>
<source>A fatal internal error occurred, see debug.log for details</source>
<translation>En fatal intern feil oppstod, se debug.log for detaljer.</translation>
</message>
<message>
<source>Cannot set -peerblockfilters without -blockfilterindex.</source>
<translation>Kan ikke sette -peerblockfilters uten -blockfilterindex</translation>
</message>
<message>
<source>Disk space is too low!</source>
<translation>For lite diskplass!</translation>
</message>
<message>
<source>Error reading from database, shutting down.</source>
<translation>Feil ved lesing fra database, stenger ned.</translation>
</message>
<message>
<source>Error upgrading chainstate database</source>
<translation>Feil ved oppgradering av kjedetilstandsdatabase</translation>
</message>
<message>
<source>Error: Disk space is low for %s</source>
<translation>Feil: Ikke nok ledig diskplass for %s</translation>
</message>
<message>
<source>Error: Keypool ran out, please call keypoolrefill first</source>
<translation>Feil: Keypool gikk tom, kall keypoolrefill først.</translation>
</message>
<message>
<source>Fee rate (%s) is lower than the minimum fee rate setting (%s)</source>
<translation>Avgiftsrate (%s) er lavere enn den minimume avgiftsrate innstillingen (%s)</translation>
</message>
<message>
<source>Invalid -onion address or hostname: '%s'</source>
<translation>Ugyldig -onion adresse eller vertsnavn: "%s"</translation>
</message>
<message>
<source>Invalid -proxy address or hostname: '%s'</source>
<translation>Ugyldig -mellomtjeneradresse eller vertsnavn: "%s"</translation>
</message>
<message>
<source>Invalid amount for -paytxfee=<amount>: '%s' (must be at least %s)</source>
<translation>Ugyldig beløp for -paytxfee=<amount>: '%s' (må være minst %s)</translation>
</message>
<message>
<source>Invalid netmask specified in -whitelist: '%s'</source>
<translation>Ugyldig nettmaske spesifisert i -whitelist: '%s'</translation>
</message>
<message>
<source>Need to specify a port with -whitebind: '%s'</source>
<translation>Må oppgi en port med -whitebind: '%s'</translation>
</message>
<message>
<source>No proxy server specified. Use -proxy=<ip> or -proxy=<ip:port>.</source>
<translation>Ingen proxyserver er spesifisert. Bruk -proxy=<ip> eller -proxy=<ip:port>.</translation>
</message>
<message>
<source>Prune mode is incompatible with -blockfilterindex.</source>
<translation>Beskjæringsmodus er inkompatibel med -blokkfilterindex.</translation>
</message>
<message>
<source>Reducing -maxconnections from %d to %d, because of system limitations.</source>
<translation>Reduserer -maxconnections fra %d til %d, pga. systembegrensninger.</translation>
</message>
<message>
<source>Signing transaction failed</source>
<translation>Signering av transaksjon feilet</translation>
</message>
<message>
<source>Specified -walletdir "%s" does not exist</source>
<translation>Oppgitt -walletdir "%s" eksisterer ikke</translation>
</message>
<message>
<source>Specified -walletdir "%s" is a relative path</source>
<translation>Oppgitt -walletdir "%s" er en relativ sti</translation>
</message>
<message>
<source>Specified -walletdir "%s" is not a directory</source>
<translation>Oppgitt -walletdir "%s" er ikke en katalog</translation>
</message>
<message>
<source>The specified config file %s does not exist
</source>
<translation>Konfigurasjonsfilen %s eksisterer ikke
</translation>
</message>
<message>
<source>The transaction amount is too small to pay the fee</source>
<translation>Transaksjonsbeløpet er for lite til å betale gebyr</translation>
</message>
<message>
<source>This is experimental software.</source>
<translation>Dette er eksperimentell programvare.</translation>
</message>
<message>
<source>Transaction amount too small</source>
<translation>Transaksjonen er for liten</translation>
</message>
<message>
<source>Transaction too large</source>
<translation>Transaksjonen er for stor</translation>
</message>
<message>
<source>Unable to bind to %s on this computer (bind returned error %s)</source>
<translation>Kan ikke binde til %s på denne datamaskinen (binding returnerte feilen %s)</translation>
</message>
<message>
<source>Unable to generate initial keys</source>
<translation>Klarte ikke lage første nøkkel</translation>
</message>
<message>
<source>Verifying wallet(s)...</source>
<translation>Lommebokbekreftelse pågår…</translation>
</message>
<message>
<source>Warning: unknown new rules activated (versionbit %i)</source>
<translation>Advarsel: Ukjente nye regler aktivert (versionbit %i)</translation>
</message>
<message>
<source>-maxtxfee is set very high! Fees this large could be paid on a single transaction.</source>
<translation>-maxtxfee er satt veldig høyt! Så stort gebyr kan bli betalt ved en enkelt transaksjon.</translation>
</message>
<message>
<source>This is the transaction fee you may pay when fee estimates are not available.</source>
<translation>Dette er transaksjonsgebyret du kan betale når gebyranslag ikke er tilgjengelige.</translation>
</message>
<message>
<source>Total length of network version string (%i) exceeds maximum length (%i). Reduce the number or size of uacomments.</source>
<translation>Total lengde av nettverks-versionstreng (%i) er over maks lengde (%i). Reduser tallet eller størrelsen av uacomments.</translation>
</message>
<message>
<source>%s is set very high!</source>
<translation>%s er satt veldig høyt!</translation>
</message>
<message>
<source>Starting network threads...</source>
<translation>Starter nettverkstråder…</translation>
</message>
<message>
<source>The wallet will avoid paying less than the minimum relay fee.</source>
<translation>Lommeboka vil unngå å betale mindre enn minimumsstafettgebyret.</translation>
</message>
<message>
<source>This is the minimum transaction fee you pay on every transaction.</source>
<translation>Dette er minimumsgebyret du betaler for hver transaksjon.</translation>
</message>
<message>
<source>This is the transaction fee you will pay if you send a transaction.</source>
<translation>Dette er transaksjonsgebyret du betaler som forsender av transaksjon.</translation>
</message>
<message>
<source>Transaction amounts must not be negative</source>
<translation>Transaksjonsbeløpet kan ikke være negativt</translation>
</message>
<message>
<source>Transaction has too long of a mempool chain</source>
<translation>Transaksjonen har for lang hukommelsespuljekjede</translation>
</message>
<message>
<source>Transaction must have at least one recipient</source>
<translation>Transaksjonen må ha minst én mottaker</translation>
</message>
<message>
<source>Unknown network specified in -onlynet: '%s'</source>
<translation>Ukjent nettverk angitt i -onlynet '%s'</translation>
</message>
<message>
<source>Insufficient funds</source>
<translation>Utilstrekkelige midler</translation>
</message>
<message>
<source>Fee estimation failed. Fallbackfee is disabled. Wait a few blocks or enable -fallbackfee.</source>
<translation>Avgiftsberegning mislyktes. Fallbackfee er deaktivert. Vent et par blokker eller aktiver -fallbackfee.</translation>
</message>
<message>
<source>Loading block index...</source>
<translation>Laster blokkindeks...</translation>
</message>
<message>
<source>Loading wallet...</source>
<translation>Laster lommebok...</translation>
</message>
<message>
<source>Cannot downgrade wallet</source>
<translation>Kan ikke nedgradere lommebok</translation>
</message>
<message>
<source>Rescanning...</source>
<translation>Leser gjennom...</translation>
</message>
<message>
<source>Done loading</source>
<translation>Ferdig med lasting</translation>
</message>
</context>
</TS> | code |
ਇਲੈਕਟ੍ਰੀਕਲ ਇੰਜੀਨੀਅਰਿੰਗ ਦੇ 20 ਸਭ ਤੋਂ ਮੁਸ਼ਕਿਲ ਵਿਸ਼ਿਆਂ ਦੀ ਵਿਆਖਿਆ ਕਰੋ। ਸਭ ਤੋਂ ਮੁਸ਼ਕਿਲ ਨਾਲ ਸ਼ੁਰੂ ਕਰੋ'] ", 'ਇਲੈਕਟ੍ਰੀਕਲ ਇੰਜੀਨੀਅਰਿੰਗ ਦੇ 20 ਸਭ ਤੋਂ ਮੁਸ਼ਕਿਲ ਵਿਸ਼ਿਆਂ ਦੀ ਵਿਆਖਿਆ ਕਰੋ।', " ['ਇਲੈਕਟ੍ਰੀਕਲ ਇੰਜੀਨੀਅਰਿੰਗ ਵਿੱਚ 20 ਸਭ ਤੋਂ ਮੁਸ਼ਕਿਲ ਵਿਸ਼ਿਆਂ ਦੀ ਵਿਆਖਿਆ ਕਰੋ। ਸਭ ਤੋਂ ਮੁਸ਼ਕਿਲ ਨਾਲ ਸ਼ੁਰੂ ਕਰੋ', '1. ਕੁਆਂਟਮ ਇਲੈਕਟ੍ਰੋਡਾਇਨਾਮਿਕਸ-ਇਹ ਸਿਧਾਂਤਕ ਦੀ ਇੱਕ ਸ਼ਾਖਾ ਹੈ।ਭੌਤਿਕ ਵਿਗਿਆਨ ਜੋ ਕੁਆਂਟਮ ਪੱਧਰ ਉੱਤੇ ਪ੍ਰਕਾਸ਼ ਅਤੇ ਪਦਾਰਥ ਦੀ ਪਰਸਪਰ ਕ੍ਰਿਆ ਨਾਲ ਸੰਬੰਧ ਰੱਖਦਾ ਹੈ, ਜੋ ਬਹੁਤ ਗੁੰਝਲਦਾਰ ਅਤੇ ਸਮਝਣ ਵਿੱਚ ਮੁਸ਼ਕਲ ਹੋ ਸਕਦਾ ਹੈ। \\n2. ਗ਼ੈਰ-ਰੇਖਿਕਸਿਸਟਮ-ਇਹ ਉਹ ਸਿਸਟਮ ਹਨ ਜੋ ਰੇਖਿਕ ਵਿਵਹਾਰ ਨੂੰ ਪ੍ਰਦਰਸ਼ਿਤ ਨਹੀਂ ਕਰਦੇ, ਭਾਵ ਕਿ ਇੰਪੁੱਟ ਵਿੱਚ ਛੋਟੀਆਂ ਤਬਦੀਲੀਆਂ ਦੇ ਨਤੀਜੇ ਵਜੋਂ ਆਉਟਪੁੱਟ ਵਿੱਚ ਅਨੁਪਾਤਕ ਤਬਦੀਲੀਆਂ ਨਹੀਂ ਹੁੰਦੀਆਂ।ਗ਼ੈਰ-ਰੇਖਿਕ ਸਿਸਟਮਾਂ ਨੂੰ ਸਮਝਣਾ ਅਤੇ ਉਹਨਾਂ ਦਾ ਵਿਸ਼ਲੇਸ਼ਣ ਕਰਨਾ ਬਹੁਤ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦਾ ਹੈ। \\n3. ਅੰਸ਼ਕ ਭਿੰਨਤਾਸੂਚਕ ਸਮੀਕਰਨਾਂ-ਇਹ ਸਮੀਕਰਨਾਂ ਹਨ ਜੋ ਵਰਣਨ ਕਰਨ ਲਈ ਵਰਤੀਆਂ ਜਾਂਦੀਆਂ ਹਨ।ਭੌਤਿਕ ਵਰਤਾਰੇ ਜਿਨ੍ਹਾਂ ਵਿੱਚ ਅੰਸ਼ਕ ਡੈਰੀਵੇਟਿਵ ਸ਼ਾਮਲ ਹੁੰਦੇ ਹਨ, ਜਿਨ੍ਹਾਂ ਨੂੰ ਹੱਲ ਕਰਨਾ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਕਰਨਾ ਮੁਸ਼ਕਲ ਹੋ ਸਕਦਾ ਹੈ। \\n4. ਸਿਗਨਲ ਪ੍ਰੋਸੈਸਿੰਗ-ਇਸ ਵਿੱਚ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੁੰਦਾ ਹੈ,ਸੰਕੇਤਾਂ ਦੀ ਹੇਰਾਫੇਰੀ ਅਤੇ ਵਿਆਖਿਆ, ਜੋ ਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦੀ ਹੈ, ਖਾਸ ਕਰਕੇ ਰੌਲੇ-ਰੱਪੇ ਵਾਲੇ ਵਾਤਾਵਰਣ ਵਿੱਚ। \\n5. ਕੰਟਰੋਲ ਸਿਸਟਮ-ਇਹਇਸ ਵਿੱਚ ਉਹਨਾਂ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੈ ਜਿਨ੍ਹਾਂ ਨੂੰ ਨਿਯੰਤਰਿਤ ਕੀਤਾ ਜਾ ਸਕਦਾ ਹੈ, ਜਿਵੇਂ ਕਿ ਰੋਬੋਟ ਅਤੇ ਖੁਦਮੁਖਤਿਆਰ ਵਾਹਨ, ਜੋ ਬਹੁਤ ਗੁੰਝਲਦਾਰ ਹੋ ਸਕਦੇ ਹਨ।ਐਨਰਜੀ ਸਿਸਟਮ-ਇਸ ਵਿੱਚ ਉਹਨਾਂ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੈ ਜੋ ਨਵਿਆਉਣਯੋਗ ਐਨਰਜੀ ਸਰੋਤਾਂ ਦੀ ਵਰਤੋਂ ਕਰਦੇ ਹਨ, ਜਿਵੇਂ ਕਿ ਸੋਲਰ ਅਤੇ ਵਿੰਡ ਪਾਵਰ, ਜੋ ਹੋ ਸਕਦੇ ਹਨਇਹਨਾਂ ਸਰੋਤਾਂ ਦੀ ਪਰਿਵਰਤਨਸ਼ੀਲਤਾ ਕਾਰਨ ਚੁਣੌਤੀਪੂਰਨ. \\n7. ਪਾਵਰ ਸਿਸਟਮ ਸੁਰੱਖਿਆ-ਇਸ ਵਿੱਚ ਉਹਨਾਂ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਸ਼ਾਮਲ ਹੁੰਦਾ ਹੈ ਜੋ ਪਾਵਰ ਗਰਿੱਡਾਂ ਨੂੰ ਇਸ ਤੋਂ ਬਚਾਉਂਦੇ ਹਨ।ਨੁਕਸ ਅਤੇ ਅਸਫਲਤਾਵਾਂ, ਜੋ ਬਹੁਤ ਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦੀਆਂ ਹਨ। \\n8. ਆਰਟੀਫਿਸ਼ਲ ਇੰਟੈਲੀਜੈਂਸ-ਇਸ ਵਿੱਚ ਬੁੱਧੀਮਾਨ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਕਾਸ ਸ਼ਾਮਲ ਹੈ।ਸਿਸਟਮ, ਜੋ ਬੁੱਧੀਮਾਨ ਵਿਵਹਾਰ ਦੀ ਗੁੰਝਲਤਾ ਕਾਰਨ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦੇ ਹਨ। \\n9. ਡਿਜੀਟਲ ਸਿਗਨਲ ਪ੍ਰੋਸੈਸਿੰਗ-ਇਸ ਵਿੱਚ ਵਿਸ਼ਲੇਸ਼ਣ ਅਤੇਡਿਜੀਟਲ ਸੰਕੇਤਾਂ ਦੀ ਹੇਰਾਫੇਰੀ, ਜੋ ਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦੀ ਹੈ, ਖਾਸ ਕਰਕੇ ਸ਼ੋਰ-ਸ਼ਰਾਬੇ ਵਾਲੇ ਵਾਤਾਵਰਣ ਵਿੱਚ। \\n10. ਵਾਇਰਲੈੱਸ ਸੰਚਾਰ-ਇਸ ਵਿੱਚ ਸ਼ਾਮਲ ਹੈ।ਵਾਇਰਲੈੱਸ ਸੰਚਾਰ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ, ਜੋ ਕਿ ਵਾਇਰਲੈੱਸ ਮਾਧਿਅਮ ਦੀ ਪਰਿਵਰਤਨਸ਼ੀਲਤਾ ਕਾਰਨ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦਾ ਹੈ। \\n11. ਮਸ਼ੀਨਸਿੱਖਣਾ-ਇਸ ਵਿੱਚ ਮਸ਼ੀਨ ਲਰਨਿੰਗ ਐਲਗੋਰਿਦਮ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਕਾਸ ਸ਼ਾਮਲ ਹੈ, ਜੋ ਬੁੱਧੀਮਾਨ ਦੀ ਗੁੰਝਲਤਾ ਕਾਰਨ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦਾ ਹੈ।ਵਿਵਹਾਰ. \\n12. ਕੰਪਿਊਟਰ ਨੈੱਟਵਰਕ-ਇਸ ਵਿੱਚ ਕੰਪਿਊਟਰ ਨੈੱਟਵਰਕ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੈ, ਜੋ ਇਸ ਕਾਰਨ ਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦਾ ਹੈ।ਨੈੱਟਵਰਕ ਮਾਧਿਅਮ ਦੀ ਪਰਿਵਰਤਨਸ਼ੀਲਤਾ। \\n13. ਰੋਬੋਟਿਕਸ-ਇਸ ਵਿੱਚ ਰੋਬੋਟਾਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੈ, ਜੋ ਇਸ ਦੀ ਜ਼ਰੂਰਤ ਦੇ ਕਾਰਨ ਬਹੁਤ ਗੁੰਝਲਦਾਰ ਹੋ ਸਕਦਾ ਹੈ।ਸੰਤੁਲਨ ਕਾਰਜਕੁਸ਼ਲਤਾ, ਭਰੋਸੇਯੋਗਤਾ ਅਤੇ ਲਾਗਤ। \\n14. ਨੈਨੋ ਟੈਕਨੋਲੋਜੀ-ਇਸ ਵਿੱਚ ਨੈਨੋਸਕੇਲ ਉੱਤੇ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੁੰਦਾ ਹੈ, ਜੋ ਹੋ ਸਕਦਾ ਹੈ -ਇਸ ਪੈਮਾਨੇ ਉੱਤੇ ਸਮੱਗਰੀ ਦੀਆਂ ਵਿਲੱਖਣ ਵਿਸ਼ੇਸ਼ਤਾਵਾਂ ਦੇ ਕਾਰਨ ਬਹੁਤ ਚੁਣੌਤੀਪੂਰਨ ਹੈ। \\n15. ਬਾਇਓਮੈਡਿਕਲ ਇੰਜੀਨੀਅਰਿੰਗ-ਇਸ ਵਿੱਚ ਇੰਜੀਨੀਅਰਿੰਗ ਦੀ ਵਰਤੋਂ ਸ਼ਾਮਲ ਹੈ।ਜੈਵਿਕ ਪ੍ਰਣਾਲੀਆਂ ਦੇ ਸਿਧਾਂਤ, ਜੋ ਜੈਵਿਕ ਪ੍ਰਣਾਲੀਆਂ ਦੀ ਪਰਿਵਰਤਨਸ਼ੀਲਤਾ ਕਾਰਨ ਗੁੰਝਲਦਾਰ ਹੋ ਸਕਦੇ ਹਨ। \\n16. ਇਲੈਕਟ੍ਰਿਕ ਵਾਹਨ-ਇਸ ਵਿੱਚ ਡਿਜ਼ਾਈਨ ਸ਼ਾਮਲ ਹੁੰਦਾ ਹੈ।ਅਤੇ ਇਲੈਕਟ੍ਰਿਕ ਵਾਹਨਾਂ ਦਾ ਵਿਸ਼ਲੇਸ਼ਣ, ਜੋ ਕਾਰਜਕੁਸ਼ਲਤਾ, ਭਰੋਸੇਯੋਗਤਾ ਅਤੇ ਲਾਗਤ ਨੂੰ ਸੰਤੁਲਿਤ ਕਰਨ ਦੀ ਜ਼ਰੂਰਤ ਕਾਰਨ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦਾ ਹੈ। \\n17. ਸਾਈਬਰ ਸੁਰੱਖਿਆ-ਇਹਇਸ ਵਿੱਚ ਸਾਈਬਰ ਹਮਲਿਆਂ ਤੋਂ ਬਚਾਅ ਕਰਨ ਵਾਲੀਆਂ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੈ, ਜੋ ਬਹੁਤ ਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦੇ ਹਨ। \\n18. ਟਿਕਾਊ ਊਰਜਾਪ੍ਰਣਾਲੀਆਂ-ਇਸ ਵਿੱਚ ਉਹਨਾਂ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੁੰਦਾ ਹੈ ਜੋ ਹਵਾ ਅਤੇ ਸੂਰਜੀ ਸ਼ਕਤੀ ਵਰਗੇ ਟਿਕਾਊ ਊਰਜਾ ਸਰੋਤਾਂ ਦੀ ਵਰਤੋਂ ਕਰਦੇ ਹਨ, ਜੋ ਹੋ ਸਕਦੇ ਹਨਇਹਨਾਂ ਸਰੋਤਾਂ ਦੀ ਪਰਿਵਰਤਨਸ਼ੀਲਤਾ ਕਾਰਨ ਚੁਣੌਤੀਪੂਰਨ. \\n19. ਚਿੱਤਰ ਪ੍ਰਕਿਰਿਆ-ਇਸ ਵਿੱਚ ਚਿੱਤਰਾਂ ਦਾ ਵਿਸ਼ਲੇਸ਼ਣ ਅਤੇ ਹੇਰਾਫੇਰੀ ਸ਼ਾਮਲ ਹੈ, ਜੋ ਹੋ ਸਕਦੀ ਹੈਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ, ਖਾਸ ਕਰਕੇ ਰੌਲੇ-ਰੱਪੇ ਵਾਲੇ ਵਾਤਾਵਰਣ ਵਿੱਚ। \\n20. ਡਾਟਾ ਸਾਇੰਸ-ਇਸ ਵਿੱਚ ਵੱਡੇ ਡੇਟਾ ਸੈੱਟਾਂ ਦਾ ਵਿਸ਼ਲੇਸ਼ਣ ਅਤੇ ਵਿਆਖਿਆ ਸ਼ਾਮਲ ਹੈ, ਜੋਡੇਟਾ ਦੀ ਪਰਿਵਰਤਨਸ਼ੀਲਤਾ ਦੇ ਕਾਰਨ ਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦਾ ਹੈ। ',' ਕੀ ਤੁਸੀਂ ਕੁਆਂਟਮ ਇਲੈਕਟ੍ਰੋਡਾਇਨਾਮਿਕਸ ਦੀ ਵਿਆਖਿਆ ਕਰ ਸਕਦੇ ਹੋ '] ", 'ਇਲੈਕਟ੍ਰੀਕਲ ਇੰਜੀਨੀਅਰਿੰਗ ਦੇ 20 ਸਭ ਤੋਂ ਮੁਸ਼ਕਿਲ ਵਿਸ਼ਿਆਂ ਦੀ ਵਿਆਖਿਆ ਕਰੋ।', ' [\'ਇਲੈਕਟ੍ਰੀਕਲ ਇੰਜੀਨੀਅਰਿੰਗ ਵਿੱਚ 20 ਸਭ ਤੋਂ ਮੁਸ਼ਕਿਲ ਵਿਸ਼ਿਆਂ ਦੀ ਵਿਆਖਿਆ ਕਰੋ। ਸਭ ਤੋਂ ਮੁਸ਼ਕਿਲ ਨਾਲ ਸ਼ੁਰੂ ਕਰੋ\', \'1. ਕੁਆਂਟਮ ਇਲੈਕਟ੍ਰੋਡਾਇਨਾਮਿਕਸ-ਇਹ ਸਿਧਾਂਤਕ ਦੀ ਇੱਕ ਸ਼ਾਖਾ ਹੈ।ਭੌਤਿਕ ਵਿਗਿਆਨ ਜੋ ਕੁਆਂਟਮ ਪੱਧਰ ਉੱਤੇ ਪ੍ਰਕਾਸ਼ ਅਤੇ ਪਦਾਰਥ ਦੀ ਪਰਸਪਰ ਕ੍ਰਿਆ ਨਾਲ ਸੰਬੰਧ ਰੱਖਦਾ ਹੈ, ਜੋ ਬਹੁਤ ਗੁੰਝਲਦਾਰ ਅਤੇ ਸਮਝਣ ਵਿੱਚ ਮੁਸ਼ਕਲ ਹੋ ਸਕਦਾ ਹੈ। \\n2. ਗ਼ੈਰ-ਰੇਖਿਕਸਿਸਟਮ-ਇਹ ਉਹ ਸਿਸਟਮ ਹਨ ਜੋ ਰੇਖਿਕ ਵਿਵਹਾਰ ਨੂੰ ਪ੍ਰਦਰਸ਼ਿਤ ਨਹੀਂ ਕਰਦੇ, ਭਾਵ ਕਿ ਇੰਪੁੱਟ ਵਿੱਚ ਛੋਟੀਆਂ ਤਬਦੀਲੀਆਂ ਦੇ ਨਤੀਜੇ ਵਜੋਂ ਆਉਟਪੁੱਟ ਵਿੱਚ ਅਨੁਪਾਤਕ ਤਬਦੀਲੀਆਂ ਨਹੀਂ ਹੁੰਦੀਆਂ।ਗ਼ੈਰ-ਰੇਖਿਕ ਸਿਸਟਮਾਂ ਨੂੰ ਸਮਝਣਾ ਅਤੇ ਉਹਨਾਂ ਦਾ ਵਿਸ਼ਲੇਸ਼ਣ ਕਰਨਾ ਬਹੁਤ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦਾ ਹੈ। \\n3. ਅੰਸ਼ਕ ਭਿੰਨਤਾਸੂਚਕ ਸਮੀਕਰਨਾਂ-ਇਹ ਸਮੀਕਰਨਾਂ ਹਨ ਜੋ ਵਰਣਨ ਕਰਨ ਲਈ ਵਰਤੀਆਂ ਜਾਂਦੀਆਂ ਹਨ।ਭੌਤਿਕ ਵਰਤਾਰੇ ਜਿਨ੍ਹਾਂ ਵਿੱਚ ਅੰਸ਼ਕ ਡੈਰੀਵੇਟਿਵ ਸ਼ਾਮਲ ਹੁੰਦੇ ਹਨ, ਜਿਨ੍ਹਾਂ ਨੂੰ ਹੱਲ ਕਰਨਾ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਕਰਨਾ ਮੁਸ਼ਕਲ ਹੋ ਸਕਦਾ ਹੈ। \\n4. ਸਿਗਨਲ ਪ੍ਰੋਸੈਸਿੰਗ-ਇਸ ਵਿੱਚ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੁੰਦਾ ਹੈ,ਸੰਕੇਤਾਂ ਦੀ ਹੇਰਾਫੇਰੀ ਅਤੇ ਵਿਆਖਿਆ, ਜੋ ਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦੀ ਹੈ, ਖਾਸ ਕਰਕੇ ਰੌਲੇ-ਰੱਪੇ ਵਾਲੇ ਵਾਤਾਵਰਣ ਵਿੱਚ। \\n5. ਕੰਟਰੋਲ ਸਿਸਟਮ-ਇਹਇਸ ਵਿੱਚ ਉਹਨਾਂ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੈ ਜਿਨ੍ਹਾਂ ਨੂੰ ਨਿਯੰਤਰਿਤ ਕੀਤਾ ਜਾ ਸਕਦਾ ਹੈ, ਜਿਵੇਂ ਕਿ ਰੋਬੋਟ ਅਤੇ ਖੁਦਮੁਖਤਿਆਰ ਵਾਹਨ, ਜੋ ਬਹੁਤ ਗੁੰਝਲਦਾਰ ਹੋ ਸਕਦੇ ਹਨ।ਐਨਰਜੀ ਸਿਸਟਮ-ਇਸ ਵਿੱਚ ਉਹਨਾਂ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੈ ਜੋ ਨਵਿਆਉਣਯੋਗ ਐਨਰਜੀ ਸਰੋਤਾਂ ਦੀ ਵਰਤੋਂ ਕਰਦੇ ਹਨ, ਜਿਵੇਂ ਕਿ ਸੋਲਰ ਅਤੇ ਵਿੰਡ ਪਾਵਰ, ਜੋ ਹੋ ਸਕਦੇ ਹਨਇਹਨਾਂ ਸਰੋਤਾਂ ਦੀ ਪਰਿਵਰਤਨਸ਼ੀਲਤਾ ਕਾਰਨ ਚੁਣੌਤੀਪੂਰਨ. \\n7. ਪਾਵਰ ਸਿਸਟਮ ਸੁਰੱਖਿਆ-ਇਸ ਵਿੱਚ ਉਹਨਾਂ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਸ਼ਾਮਲ ਹੁੰਦਾ ਹੈ ਜੋ ਪਾਵਰ ਗਰਿੱਡਾਂ ਨੂੰ ਇਸ ਤੋਂ ਬਚਾਉਂਦੇ ਹਨ।ਨੁਕਸ ਅਤੇ ਅਸਫਲਤਾਵਾਂ, ਜੋ ਬਹੁਤ ਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦੀਆਂ ਹਨ। \\n8. ਆਰਟੀਫਿਸ਼ਲ ਇੰਟੈਲੀਜੈਂਸ-ਇਸ ਵਿੱਚ ਬੁੱਧੀਮਾਨ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਕਾਸ ਸ਼ਾਮਲ ਹੈ।ਸਿਸਟਮ, ਜੋ ਬੁੱਧੀਮਾਨ ਵਿਵਹਾਰ ਦੀ ਗੁੰਝਲਤਾ ਕਾਰਨ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦੇ ਹਨ। \\n9. ਡਿਜੀਟਲ ਸਿਗਨਲ ਪ੍ਰੋਸੈਸਿੰਗ-ਇਸ ਵਿੱਚ ਵਿਸ਼ਲੇਸ਼ਣ ਅਤੇਡਿਜੀਟਲ ਸੰਕੇਤਾਂ ਦੀ ਹੇਰਾਫੇਰੀ, ਜੋ ਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦੀ ਹੈ, ਖਾਸ ਕਰਕੇ ਸ਼ੋਰ-ਸ਼ਰਾਬੇ ਵਾਲੇ ਵਾਤਾਵਰਣ ਵਿੱਚ। \\n10. ਵਾਇਰਲੈੱਸ ਸੰਚਾਰ-ਇਸ ਵਿੱਚ ਸ਼ਾਮਲ ਹੈ।ਵਾਇਰਲੈੱਸ ਸੰਚਾਰ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ, ਜੋ ਕਿ ਵਾਇਰਲੈੱਸ ਮਾਧਿਅਮ ਦੀ ਪਰਿਵਰਤਨਸ਼ੀਲਤਾ ਕਾਰਨ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦਾ ਹੈ। \\n11. ਮਸ਼ੀਨਸਿੱਖਣਾ-ਇਸ ਵਿੱਚ ਮਸ਼ੀਨ ਲਰਨਿੰਗ ਐਲਗੋਰਿਦਮ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਕਾਸ ਸ਼ਾਮਲ ਹੈ, ਜੋ ਬੁੱਧੀਮਾਨ ਦੀ ਗੁੰਝਲਤਾ ਕਾਰਨ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦਾ ਹੈ।ਵਿਵਹਾਰ. \\n12. ਕੰਪਿਊਟਰ ਨੈੱਟਵਰਕ-ਇਸ ਵਿੱਚ ਕੰਪਿਊਟਰ ਨੈੱਟਵਰਕ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੈ, ਜੋ ਇਸ ਕਾਰਨ ਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦਾ ਹੈ।ਨੈੱਟਵਰਕ ਮਾਧਿਅਮ ਦੀ ਪਰਿਵਰਤਨਸ਼ੀਲਤਾ। \\n13. ਰੋਬੋਟਿਕਸ-ਇਸ ਵਿੱਚ ਰੋਬੋਟਾਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੈ, ਜੋ ਇਸ ਦੀ ਜ਼ਰੂਰਤ ਦੇ ਕਾਰਨ ਬਹੁਤ ਗੁੰਝਲਦਾਰ ਹੋ ਸਕਦਾ ਹੈ।ਸੰਤੁਲਨ ਕਾਰਜਕੁਸ਼ਲਤਾ, ਭਰੋਸੇਯੋਗਤਾ ਅਤੇ ਲਾਗਤ। \\n14. ਨੈਨੋ ਟੈਕਨੋਲੋਜੀ-ਇਸ ਵਿੱਚ ਨੈਨੋਸਕੇਲ ਉੱਤੇ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੁੰਦਾ ਹੈ, ਜੋ ਹੋ ਸਕਦਾ ਹੈ -ਇਸ ਪੈਮਾਨੇ ਉੱਤੇ ਸਮੱਗਰੀ ਦੀਆਂ ਵਿਲੱਖਣ ਵਿਸ਼ੇਸ਼ਤਾਵਾਂ ਦੇ ਕਾਰਨ ਬਹੁਤ ਚੁਣੌਤੀਪੂਰਨ ਹੈ। \\n15. ਬਾਇਓਮੈਡਿਕਲ ਇੰਜੀਨੀਅਰਿੰਗ-ਇਸ ਵਿੱਚ ਇੰਜੀਨੀਅਰਿੰਗ ਦੀ ਵਰਤੋਂ ਸ਼ਾਮਲ ਹੈ।ਜੈਵਿਕ ਪ੍ਰਣਾਲੀਆਂ ਦੇ ਸਿਧਾਂਤ, ਜੋ ਜੈਵਿਕ ਪ੍ਰਣਾਲੀਆਂ ਦੀ ਪਰਿਵਰਤਨਸ਼ੀਲਤਾ ਕਾਰਨ ਗੁੰਝਲਦਾਰ ਹੋ ਸਕਦੇ ਹਨ। \\n16. ਇਲੈਕਟ੍ਰਿਕ ਵਾਹਨ-ਇਸ ਵਿੱਚ ਡਿਜ਼ਾਈਨ ਸ਼ਾਮਲ ਹੁੰਦਾ ਹੈ।ਅਤੇ ਇਲੈਕਟ੍ਰਿਕ ਵਾਹਨਾਂ ਦਾ ਵਿਸ਼ਲੇਸ਼ਣ, ਜੋ ਕਾਰਜਕੁਸ਼ਲਤਾ, ਭਰੋਸੇਯੋਗਤਾ ਅਤੇ ਲਾਗਤ ਨੂੰ ਸੰਤੁਲਿਤ ਕਰਨ ਦੀ ਜ਼ਰੂਰਤ ਕਾਰਨ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦਾ ਹੈ। \\n17. ਸਾਈਬਰ ਸੁਰੱਖਿਆ-ਇਹਇਸ ਵਿੱਚ ਸਾਈਬਰ ਹਮਲਿਆਂ ਤੋਂ ਬਚਾਅ ਕਰਨ ਵਾਲੀਆਂ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੈ, ਜੋ ਬਹੁਤ ਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦੇ ਹਨ। \\n18. ਟਿਕਾਊ ਊਰਜਾਪ੍ਰਣਾਲੀਆਂ-ਇਸ ਵਿੱਚ ਉਹਨਾਂ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੁੰਦਾ ਹੈ ਜੋ ਹਵਾ ਅਤੇ ਸੂਰਜੀ ਸ਼ਕਤੀ ਵਰਗੇ ਟਿਕਾਊ ਊਰਜਾ ਸਰੋਤਾਂ ਦੀ ਵਰਤੋਂ ਕਰਦੇ ਹਨ, ਜੋ ਹੋ ਸਕਦੇ ਹਨਇਹਨਾਂ ਸਰੋਤਾਂ ਦੀ ਪਰਿਵਰਤਨਸ਼ੀਲਤਾ ਕਾਰਨ ਚੁਣੌਤੀਪੂਰਨ. \\n19. ਚਿੱਤਰ ਪ੍ਰਕਿਰਿਆ-ਇਸ ਵਿੱਚ ਚਿੱਤਰਾਂ ਦਾ ਵਿਸ਼ਲੇਸ਼ਣ ਅਤੇ ਹੇਰਾਫੇਰੀ ਸ਼ਾਮਲ ਹੈ, ਜੋ ਹੋ ਸਕਦੀ ਹੈਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ, ਖਾਸ ਕਰਕੇ ਰੌਲੇ-ਰੱਪੇ ਵਾਲੇ ਵਾਤਾਵਰਣ ਵਿੱਚ। \\n20. ਡਾਟਾ ਸਾਇੰਸ-ਇਸ ਵਿੱਚ ਵੱਡੇ ਡੇਟਾ ਸੈੱਟਾਂ ਦਾ ਵਿਸ਼ਲੇਸ਼ਣ ਅਤੇ ਵਿਆਖਿਆ ਸ਼ਾਮਲ ਹੈ, ਜੋਡੇਟਾ ਦੀ ਪਰਿਵਰਤਨਸ਼ੀਲਤਾ ਦੇ ਕਾਰਨ ਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦਾ ਹੈ। \',\' ਕੀ ਤੁਸੀਂ ਕੁਆਂਟਮ ਇਲੈਕਟ੍ਰੋਡਾਇਨਾਮਿਕਸ ਦੀ ਵਿਆਖਿਆ ਕਰ ਸਕਦੇ ਹੋ \',\' ਕੁਆਂਟਮ ਇਲੈਕਟ੍ਰੋਡਾਇਨਾਮਿਕਸ (ਕਿਊਈਡੀ) ਇੱਕ ਹੈ।ਸਿਧਾਂਤਕ ਭੌਤਿਕ ਵਿਗਿਆਨ ਦੀ ਸ਼ਾਖਾ ਜੋ ਕੁਆਂਟਮ ਪੱਧਰ ਉੱਤੇ ਪ੍ਰਕਾਸ਼ ਅਤੇ ਪਦਾਰਥ ਦੀ ਪਰਸਪਰ ਕ੍ਰਿਆ ਨਾਲ ਸੰਬੰਧਿਤ ਹੈ। ਇਹ ਇੱਕ ਸਾਪੇਖਿਕ ਕੁਆਂਟਮ ਫੀਲਡ ਥਿਊਰੀ ਹੈ ਜੋਇਲੈਕਟ੍ਰੋਮੈਗਨੈਟਿਕ ਫੀਲਡ ਨਾਲ ਪਰਸਪਰ ਕ੍ਰਿਆ ਕਰਨ ਵਾਲੇ ਇਲੈਕਟ੍ਰੌਨਾਂ ਅਤੇ ਹੋਰ ਚਾਰਜ ਕੀਤੇ ਕਣਾਂ ਦੇ ਵਿਵਹਾਰ ਦਾ ਵਰਣਨ ਕਰਦਾ ਹੈ। \\n\\n ਕਲਾਸੀਕਲ ਇਲੈਕਟ੍ਰੋਡਾਇਨਾਮਿਕਸ ਵਿੱਚ,ਇਲੈਕਟ੍ਰੋਮੈਗਨੈਟਿਕ ਫੀਲਡ ਨੂੰ ਇੱਕ ਨਿਰੰਤਰ ਫੀਲਡ ਵਜੋਂ ਦਰਸਾਇਆ ਗਿਆ ਹੈ ਜੋ ਸਾਰੀ ਸਪੇਸ ਵਿੱਚ ਫੈਲਦੀ ਹੈ। ਹਾਲਾਂਕਿ, ਕਿਊਈਡੀ ਵਿੱਚ, ਫੀਲਡ ਨੂੰ ਇਹਨਾਂ ਦਾ ਸੰਗ੍ਰਹਿ ਮੰਨਿਆ ਜਾਂਦਾ ਹੈ।ਫੋਟੌਨਾਂ ਵਜੋਂ ਜਾਣੇ ਜਾਂਦੇ ਕਣ, ਜਿਨ੍ਹਾਂ ਵਿੱਚ ਐਨਰਜੀ ਅਤੇ ਮੋਮੈਂਟਮ ਹੁੰਦੇ ਹਨ। ਇਹ ਫੋਟੌਨ ਚਾਰਜ ਕੀਤੇ ਕਣਾਂ, ਜਿਵੇਂ ਕਿ ਇਲੈਕਟ੍ਰੌਨਾਂ ਨਾਲ ਪਰਸਪਰ ਕ੍ਰਿਆ ਕਰ ਸਕਦੇ ਹਨ ਅਤੇ ਉਹਨਾਂ ਨੂੰ ਬਦਲ ਸਕਦੇ ਹਨ।ਵਿਵਹਾਰ. \\n\\nਕਿਊਈਡੀ ਦੀਆਂ ਮੁੱਖ ਭਵਿੱਖਬਾਣੀਆਂ ਵਿੱਚੋਂ ਇੱਕ "ਇਲੈਕਟ੍ਰੌਨ ਕਲਾਉਡ" ਦੀ ਹੋਂਦ ਹੈ, ਜੋ ਇੱਕ ਇਲੈਕਟ੍ਰੌਨ ਦੇ ਦੁਆਲੇ ਸਪੇਸ ਦਾ ਇੱਕ ਖੇਤਰ ਹੈ ਜਿੱਥੇਇੱਕ ਫੋਟੌਨ ਲੱਭਣ ਦੀ ਸੰਭਾਵਨਾ ਉੱਚੀ ਹੁੰਦੀ ਹੈ। ਇਹ ਇੱਕ ਨਿਸ਼ਚਿਤ ਸਥਿਤੀ ਦੇ ਨਾਲ ਇੱਕ ਬਿੰਦੂ ਕਣ ਦੇ ਰੂਪ ਵਿੱਚ ਇਲੈਕਟ੍ਰੌਨ ਦੇ ਕਲਾਸੀਕਲ ਦ੍ਰਿਸ਼ਟੀਕੋਣ ਦੇ ਉਲਟ ਹੈ ਅਤੇਗਤੀ. \\n\\nQED ਇੱਕ ਉੱਚ ਗਣਿਤਿਕ ਥਿਊਰੀ ਹੈ, ਅਤੇ ਇਸ ਵਿੱਚ ਉੱਨਤ ਗਣਿਤ ਸੰਕਲਪਾਂ ਜਿਵੇਂ ਕਿ ਟੈਂਸਰ ਕੈਲਕੁਲਸ, NAME _ 1 ਵੇਰੀਏਬਲ, ਦੀ ਵਰਤੋਂ ਸ਼ਾਮਲ ਹੈ।ਅਤੇ ਫੇਨਮੈਨ ਚਿੱਤਰ। ਇਸ ਨੂੰ ਪ੍ਰਯੋਗਾਤਮਕ ਨਤੀਜਿਆਂ ਦੁਆਰਾ ਵਿਆਪਕ ਤੌਰ ਉੱਤੇ ਪਰਖਿਆ ਅਤੇ ਤਸਦੀਕ ਕੀਤਾ ਗਿਆ ਹੈ, ਅਤੇ ਇਸ ਦੀ ਵਰਤੋਂ ਬਹੁਤ ਸਾਰੀਆਂ ਸਹੀ ਭਵਿੱਖਬਾਣੀਆਂ ਕਰਨ ਲਈ ਕੀਤੀ ਗਈ ਹੈ।ਇਲੈਕਟ੍ਰੌਨਾਂ ਅਤੇ ਫੋਟੌਨਾਂ ਦੇ ਵਿਵਹਾਰ ਬਾਰੇ। \\n\\nQED ਦੇ ਬਹੁਤ ਸਾਰੇ ਵਿਹਾਰਕ ਉਪਯੋਗ ਹਨ, ਜਿਨ੍ਹਾਂ ਵਿੱਚ ਲੇਜ਼ਰਾਂ, ਆਪਟੀਕਲ ਫਾਈਬਰਾਂ ਅਤੇ ਹੋਰਾਂ ਦੇ ਡਿਜ਼ਾਈਨ ਸ਼ਾਮਲ ਹਨ।ਦੂਰਸੰਚਾਰ ਵਿੱਚ ਵਰਤੇ ਜਾਣ ਵਾਲੇ ਯੰਤਰ। ਇਹ ਉੱਚ ਊਰਜਾਵਾਂ ਉੱਤੇ ਕਣਾਂ ਦੇ ਵਿਵਹਾਰ ਬਾਰੇ ਸਾਡੀ ਸਮਝ ਵਿੱਚ ਵੀ ਮਹੱਤਵਪੂਰਨ ਭੂਮਿਕਾ ਨਿਭਾਉਂਦਾ ਹੈ, ਜਿਵੇਂ ਕਿ ਉਹ ਜੋ ਪਾਏ ਜਾਂਦੇ ਹਨ।ਕਣ ਐਕਸਲੇਟਰਾਂ ਵਿੱਚ. \',\' ਹਾਂ ਬੱਦਲ ਆਪਸ ਵਿੱਚ ਜੁਡ਼ਿਆ ਹੋਇਆ ਹੈ, ਉਦਾਹਰਣ ਵਜੋਂ ਰੇਡੀਓ ਤਰੰਗਾਂ ਲਈ ਤਰੰਗ ਆਵਾਜਾਈ ਹੈ \'] ', 'ਇਲੈਕਟ੍ਰੀਕਲ ਇੰਜੀਨੀਅਰਿੰਗ ਦੇ 20 ਸਭ ਤੋਂ ਮੁਸ਼ਕਿਲ ਵਿਸ਼ਿਆਂ ਦੀ ਵਿਆਖਿਆ ਕਰੋ।', ' [\'ਇਲੈਕਟ੍ਰੀਕਲ ਇੰਜੀਨੀਅਰਿੰਗ ਵਿੱਚ 20 ਸਭ ਤੋਂ ਮੁਸ਼ਕਿਲ ਵਿਸ਼ਿਆਂ ਦੀ ਵਿਆਖਿਆ ਕਰੋ। ਸਭ ਤੋਂ ਮੁਸ਼ਕਿਲ ਨਾਲ ਸ਼ੁਰੂ ਕਰੋ\', \'1. ਕੁਆਂਟਮ ਇਲੈਕਟ੍ਰੋਡਾਇਨਾਮਿਕਸ-ਇਹ ਸਿਧਾਂਤਕ ਦੀ ਇੱਕ ਸ਼ਾਖਾ ਹੈ।ਭੌਤਿਕ ਵਿਗਿਆਨ ਜੋ ਕੁਆਂਟਮ ਪੱਧਰ ਉੱਤੇ ਪ੍ਰਕਾਸ਼ ਅਤੇ ਪਦਾਰਥ ਦੀ ਪਰਸਪਰ ਕ੍ਰਿਆ ਨਾਲ ਸੰਬੰਧ ਰੱਖਦਾ ਹੈ, ਜੋ ਬਹੁਤ ਗੁੰਝਲਦਾਰ ਅਤੇ ਸਮਝਣ ਵਿੱਚ ਮੁਸ਼ਕਲ ਹੋ ਸਕਦਾ ਹੈ। \\n2. ਗ਼ੈਰ-ਰੇਖਿਕਸਿਸਟਮ-ਇਹ ਉਹ ਸਿਸਟਮ ਹਨ ਜੋ ਰੇਖਿਕ ਵਿਵਹਾਰ ਨੂੰ ਪ੍ਰਦਰਸ਼ਿਤ ਨਹੀਂ ਕਰਦੇ, ਭਾਵ ਕਿ ਇੰਪੁੱਟ ਵਿੱਚ ਛੋਟੀਆਂ ਤਬਦੀਲੀਆਂ ਦੇ ਨਤੀਜੇ ਵਜੋਂ ਆਉਟਪੁੱਟ ਵਿੱਚ ਅਨੁਪਾਤਕ ਤਬਦੀਲੀਆਂ ਨਹੀਂ ਹੁੰਦੀਆਂ।ਗ਼ੈਰ-ਰੇਖਿਕ ਸਿਸਟਮਾਂ ਨੂੰ ਸਮਝਣਾ ਅਤੇ ਉਹਨਾਂ ਦਾ ਵਿਸ਼ਲੇਸ਼ਣ ਕਰਨਾ ਬਹੁਤ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦਾ ਹੈ। \\n3. ਅੰਸ਼ਕ ਭਿੰਨਤਾਸੂਚਕ ਸਮੀਕਰਨਾਂ-ਇਹ ਸਮੀਕਰਨਾਂ ਹਨ ਜੋ ਵਰਣਨ ਕਰਨ ਲਈ ਵਰਤੀਆਂ ਜਾਂਦੀਆਂ ਹਨ।ਭੌਤਿਕ ਵਰਤਾਰੇ ਜਿਨ੍ਹਾਂ ਵਿੱਚ ਅੰਸ਼ਕ ਡੈਰੀਵੇਟਿਵ ਸ਼ਾਮਲ ਹੁੰਦੇ ਹਨ, ਜਿਨ੍ਹਾਂ ਨੂੰ ਹੱਲ ਕਰਨਾ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਕਰਨਾ ਮੁਸ਼ਕਲ ਹੋ ਸਕਦਾ ਹੈ। \\n4. ਸਿਗਨਲ ਪ੍ਰੋਸੈਸਿੰਗ-ਇਸ ਵਿੱਚ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੁੰਦਾ ਹੈ,ਸੰਕੇਤਾਂ ਦੀ ਹੇਰਾਫੇਰੀ ਅਤੇ ਵਿਆਖਿਆ, ਜੋ ਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦੀ ਹੈ, ਖਾਸ ਕਰਕੇ ਰੌਲੇ-ਰੱਪੇ ਵਾਲੇ ਵਾਤਾਵਰਣ ਵਿੱਚ। \\n5. ਕੰਟਰੋਲ ਸਿਸਟਮ-ਇਹਇਸ ਵਿੱਚ ਉਹਨਾਂ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੈ ਜਿਨ੍ਹਾਂ ਨੂੰ ਨਿਯੰਤਰਿਤ ਕੀਤਾ ਜਾ ਸਕਦਾ ਹੈ, ਜਿਵੇਂ ਕਿ ਰੋਬੋਟ ਅਤੇ ਖੁਦਮੁਖਤਿਆਰ ਵਾਹਨ, ਜੋ ਬਹੁਤ ਗੁੰਝਲਦਾਰ ਹੋ ਸਕਦੇ ਹਨ।ਐਨਰਜੀ ਸਿਸਟਮ-ਇਸ ਵਿੱਚ ਉਹਨਾਂ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੈ ਜੋ ਨਵਿਆਉਣਯੋਗ ਐਨਰਜੀ ਸਰੋਤਾਂ ਦੀ ਵਰਤੋਂ ਕਰਦੇ ਹਨ, ਜਿਵੇਂ ਕਿ ਸੋਲਰ ਅਤੇ ਵਿੰਡ ਪਾਵਰ, ਜੋ ਹੋ ਸਕਦੇ ਹਨਇਹਨਾਂ ਸਰੋਤਾਂ ਦੀ ਪਰਿਵਰਤਨਸ਼ੀਲਤਾ ਕਾਰਨ ਚੁਣੌਤੀਪੂਰਨ. \\n7. ਪਾਵਰ ਸਿਸਟਮ ਸੁਰੱਖਿਆ-ਇਸ ਵਿੱਚ ਉਹਨਾਂ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਸ਼ਾਮਲ ਹੁੰਦਾ ਹੈ ਜੋ ਪਾਵਰ ਗਰਿੱਡਾਂ ਨੂੰ ਇਸ ਤੋਂ ਬਚਾਉਂਦੇ ਹਨ।ਨੁਕਸ ਅਤੇ ਅਸਫਲਤਾਵਾਂ, ਜੋ ਬਹੁਤ ਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦੀਆਂ ਹਨ। \\n8. ਆਰਟੀਫਿਸ਼ਲ ਇੰਟੈਲੀਜੈਂਸ-ਇਸ ਵਿੱਚ ਬੁੱਧੀਮਾਨ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਕਾਸ ਸ਼ਾਮਲ ਹੈ।ਸਿਸਟਮ, ਜੋ ਬੁੱਧੀਮਾਨ ਵਿਵਹਾਰ ਦੀ ਗੁੰਝਲਤਾ ਕਾਰਨ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦੇ ਹਨ। \\n9. ਡਿਜੀਟਲ ਸਿਗਨਲ ਪ੍ਰੋਸੈਸਿੰਗ-ਇਸ ਵਿੱਚ ਵਿਸ਼ਲੇਸ਼ਣ ਅਤੇਡਿਜੀਟਲ ਸੰਕੇਤਾਂ ਦੀ ਹੇਰਾਫੇਰੀ, ਜੋ ਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦੀ ਹੈ, ਖਾਸ ਕਰਕੇ ਸ਼ੋਰ-ਸ਼ਰਾਬੇ ਵਾਲੇ ਵਾਤਾਵਰਣ ਵਿੱਚ। \\n10. ਵਾਇਰਲੈੱਸ ਸੰਚਾਰ-ਇਸ ਵਿੱਚ ਸ਼ਾਮਲ ਹੈ।ਵਾਇਰਲੈੱਸ ਸੰਚਾਰ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ, ਜੋ ਕਿ ਵਾਇਰਲੈੱਸ ਮਾਧਿਅਮ ਦੀ ਪਰਿਵਰਤਨਸ਼ੀਲਤਾ ਕਾਰਨ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦਾ ਹੈ। \\n11. ਮਸ਼ੀਨਸਿੱਖਣਾ-ਇਸ ਵਿੱਚ ਮਸ਼ੀਨ ਲਰਨਿੰਗ ਐਲਗੋਰਿਦਮ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਕਾਸ ਸ਼ਾਮਲ ਹੈ, ਜੋ ਬੁੱਧੀਮਾਨ ਦੀ ਗੁੰਝਲਤਾ ਕਾਰਨ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦਾ ਹੈ।ਵਿਵਹਾਰ. \\n12. ਕੰਪਿਊਟਰ ਨੈੱਟਵਰਕ-ਇਸ ਵਿੱਚ ਕੰਪਿਊਟਰ ਨੈੱਟਵਰਕ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੈ, ਜੋ ਇਸ ਕਾਰਨ ਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦਾ ਹੈ।ਨੈੱਟਵਰਕ ਮਾਧਿਅਮ ਦੀ ਪਰਿਵਰਤਨਸ਼ੀਲਤਾ। \\n13. ਰੋਬੋਟਿਕਸ-ਇਸ ਵਿੱਚ ਰੋਬੋਟਾਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੈ, ਜੋ ਇਸ ਦੀ ਜ਼ਰੂਰਤ ਦੇ ਕਾਰਨ ਬਹੁਤ ਗੁੰਝਲਦਾਰ ਹੋ ਸਕਦਾ ਹੈ।ਸੰਤੁਲਨ ਕਾਰਜਕੁਸ਼ਲਤਾ, ਭਰੋਸੇਯੋਗਤਾ ਅਤੇ ਲਾਗਤ। \\n14. ਨੈਨੋ ਟੈਕਨੋਲੋਜੀ-ਇਸ ਵਿੱਚ ਨੈਨੋਸਕੇਲ ਉੱਤੇ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੁੰਦਾ ਹੈ, ਜੋ ਹੋ ਸਕਦਾ ਹੈ -ਇਸ ਪੈਮਾਨੇ ਉੱਤੇ ਸਮੱਗਰੀ ਦੀਆਂ ਵਿਲੱਖਣ ਵਿਸ਼ੇਸ਼ਤਾਵਾਂ ਦੇ ਕਾਰਨ ਬਹੁਤ ਚੁਣੌਤੀਪੂਰਨ ਹੈ। \\n15. ਬਾਇਓਮੈਡਿਕਲ ਇੰਜੀਨੀਅਰਿੰਗ-ਇਸ ਵਿੱਚ ਇੰਜੀਨੀਅਰਿੰਗ ਦੀ ਵਰਤੋਂ ਸ਼ਾਮਲ ਹੈ।ਜੈਵਿਕ ਪ੍ਰਣਾਲੀਆਂ ਦੇ ਸਿਧਾਂਤ, ਜੋ ਜੈਵਿਕ ਪ੍ਰਣਾਲੀਆਂ ਦੀ ਪਰਿਵਰਤਨਸ਼ੀਲਤਾ ਕਾਰਨ ਗੁੰਝਲਦਾਰ ਹੋ ਸਕਦੇ ਹਨ। \\n16. ਇਲੈਕਟ੍ਰਿਕ ਵਾਹਨ-ਇਸ ਵਿੱਚ ਡਿਜ਼ਾਈਨ ਸ਼ਾਮਲ ਹੁੰਦਾ ਹੈ।ਅਤੇ ਇਲੈਕਟ੍ਰਿਕ ਵਾਹਨਾਂ ਦਾ ਵਿਸ਼ਲੇਸ਼ਣ, ਜੋ ਕਾਰਜਕੁਸ਼ਲਤਾ, ਭਰੋਸੇਯੋਗਤਾ ਅਤੇ ਲਾਗਤ ਨੂੰ ਸੰਤੁਲਿਤ ਕਰਨ ਦੀ ਜ਼ਰੂਰਤ ਕਾਰਨ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦਾ ਹੈ। \\n17. ਸਾਈਬਰ ਸੁਰੱਖਿਆ-ਇਹਇਸ ਵਿੱਚ ਸਾਈਬਰ ਹਮਲਿਆਂ ਤੋਂ ਬਚਾਅ ਕਰਨ ਵਾਲੀਆਂ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੈ, ਜੋ ਬਹੁਤ ਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦੇ ਹਨ। \\n18. ਟਿਕਾਊ ਊਰਜਾਪ੍ਰਣਾਲੀਆਂ-ਇਸ ਵਿੱਚ ਉਹਨਾਂ ਪ੍ਰਣਾਲੀਆਂ ਦਾ ਡਿਜ਼ਾਈਨ ਅਤੇ ਵਿਸ਼ਲੇਸ਼ਣ ਸ਼ਾਮਲ ਹੁੰਦਾ ਹੈ ਜੋ ਹਵਾ ਅਤੇ ਸੂਰਜੀ ਸ਼ਕਤੀ ਵਰਗੇ ਟਿਕਾਊ ਊਰਜਾ ਸਰੋਤਾਂ ਦੀ ਵਰਤੋਂ ਕਰਦੇ ਹਨ, ਜੋ ਹੋ ਸਕਦੇ ਹਨਇਹਨਾਂ ਸਰੋਤਾਂ ਦੀ ਪਰਿਵਰਤਨਸ਼ੀਲਤਾ ਕਾਰਨ ਚੁਣੌਤੀਪੂਰਨ. \\n19. ਚਿੱਤਰ ਪ੍ਰਕਿਰਿਆ-ਇਸ ਵਿੱਚ ਚਿੱਤਰਾਂ ਦਾ ਵਿਸ਼ਲੇਸ਼ਣ ਅਤੇ ਹੇਰਾਫੇਰੀ ਸ਼ਾਮਲ ਹੈ, ਜੋ ਹੋ ਸਕਦੀ ਹੈਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ, ਖਾਸ ਕਰਕੇ ਰੌਲੇ-ਰੱਪੇ ਵਾਲੇ ਵਾਤਾਵਰਣ ਵਿੱਚ। \\n20. ਡਾਟਾ ਸਾਇੰਸ-ਇਸ ਵਿੱਚ ਵੱਡੇ ਡੇਟਾ ਸੈੱਟਾਂ ਦਾ ਵਿਸ਼ਲੇਸ਼ਣ ਅਤੇ ਵਿਆਖਿਆ ਸ਼ਾਮਲ ਹੈ, ਜੋਡੇਟਾ ਦੀ ਪਰਿਵਰਤਨਸ਼ੀਲਤਾ ਦੇ ਕਾਰਨ ਗੁੰਝਲਦਾਰ ਅਤੇ ਚੁਣੌਤੀਪੂਰਨ ਹੋ ਸਕਦਾ ਹੈ। \',\' ਕੀ ਤੁਸੀਂ ਕੁਆਂਟਮ ਇਲੈਕਟ੍ਰੋਡਾਇਨਾਮਿਕਸ ਦੀ ਵਿਆਖਿਆ ਕਰ ਸਕਦੇ ਹੋ \',\' ਕੁਆਂਟਮ ਇਲੈਕਟ੍ਰੋਡਾਇਨਾਮਿਕਸ (ਕਿਊਈਡੀ) ਇੱਕ ਹੈ।ਸਿਧਾਂਤਕ ਭੌਤਿਕ ਵਿਗਿਆਨ ਦੀ ਸ਼ਾਖਾ ਜੋ ਕੁਆਂਟਮ ਪੱਧਰ ਉੱਤੇ ਪ੍ਰਕਾਸ਼ ਅਤੇ ਪਦਾਰਥ ਦੀ ਪਰਸਪਰ ਕ੍ਰਿਆ ਨਾਲ ਸੰਬੰਧਿਤ ਹੈ। ਇਹ ਇੱਕ ਸਾਪੇਖਿਕ ਕੁਆਂਟਮ ਫੀਲਡ ਥਿਊਰੀ ਹੈ ਜੋਇਲੈਕਟ੍ਰੋਮੈਗਨੈਟਿਕ ਫੀਲਡ ਨਾਲ ਪਰਸਪਰ ਕ੍ਰਿਆ ਕਰਨ ਵਾਲੇ ਇਲੈਕਟ੍ਰੌਨਾਂ ਅਤੇ ਹੋਰ ਚਾਰਜ ਕੀਤੇ ਕਣਾਂ ਦੇ ਵਿਵਹਾਰ ਦਾ ਵਰਣਨ ਕਰਦਾ ਹੈ। \\n\\n ਕਲਾਸੀਕਲ ਇਲੈਕਟ੍ਰੋਡਾਇਨਾਮਿਕਸ ਵਿੱਚ,ਇਲੈਕਟ੍ਰੋਮੈਗਨੈਟਿਕ ਫੀਲਡ ਨੂੰ ਇੱਕ ਨਿਰੰਤਰ ਫੀਲਡ ਵਜੋਂ ਦਰਸਾਇਆ ਗਿਆ ਹੈ ਜੋ ਸਾਰੀ ਸਪੇਸ ਵਿੱਚ ਫੈਲਦੀ ਹੈ। ਹਾਲਾਂਕਿ, ਕਿਊਈਡੀ ਵਿੱਚ, ਫੀਲਡ ਨੂੰ ਇਹਨਾਂ ਦਾ ਸੰਗ੍ਰਹਿ ਮੰਨਿਆ ਜਾਂਦਾ ਹੈ।ਫੋਟੌਨਾਂ ਵਜੋਂ ਜਾਣੇ ਜਾਂਦੇ ਕਣ, ਜਿਨ੍ਹਾਂ ਵਿੱਚ ਐਨਰਜੀ ਅਤੇ ਮੋਮੈਂਟਮ ਹੁੰਦੇ ਹਨ। ਇਹ ਫੋਟੌਨ ਚਾਰਜ ਕੀਤੇ ਕਣਾਂ, ਜਿਵੇਂ ਕਿ ਇਲੈਕਟ੍ਰੌਨਾਂ ਨਾਲ ਪਰਸਪਰ ਕ੍ਰਿਆ ਕਰ ਸਕਦੇ ਹਨ ਅਤੇ ਉਹਨਾਂ ਨੂੰ ਬਦਲ ਸਕਦੇ ਹਨ।ਵਿਵਹਾਰ. \\n\\nਕਿਊਈਡੀ ਦੀਆਂ ਮੁੱਖ ਭਵਿੱਖਬਾਣੀਆਂ ਵਿੱਚੋਂ ਇੱਕ "ਇਲੈਕਟ੍ਰੌਨ ਕਲਾਉਡ" ਦੀ ਹੋਂਦ ਹੈ, ਜੋ ਇੱਕ ਇਲੈਕਟ੍ਰੌਨ ਦੇ ਦੁਆਲੇ ਸਪੇਸ ਦਾ ਇੱਕ ਖੇਤਰ ਹੈ ਜਿੱਥੇਇੱਕ ਫੋਟੌਨ ਲੱਭਣ ਦੀ ਸੰਭਾਵਨਾ ਉੱਚੀ ਹੁੰਦੀ ਹੈ। ਇਹ ਇੱਕ ਨਿਸ਼ਚਿਤ ਸਥਿਤੀ ਦੇ ਨਾਲ ਇੱਕ ਬਿੰਦੂ ਕਣ ਦੇ ਰੂਪ ਵਿੱਚ ਇਲੈਕਟ੍ਰੌਨ ਦੇ ਕਲਾਸੀਕਲ ਦ੍ਰਿਸ਼ਟੀਕੋਣ ਦੇ ਉਲਟ ਹੈ ਅਤੇਗਤੀ. \\n\\nQED ਇੱਕ ਉੱਚ ਗਣਿਤਿਕ ਥਿਊਰੀ ਹੈ, ਅਤੇ ਇਸ ਵਿੱਚ ਉੱਨਤ ਗਣਿਤ ਸੰਕਲਪਾਂ ਜਿਵੇਂ ਕਿ ਟੈਂਸਰ ਕੈਲਕੁਲਸ, NAME _ 1 ਵੇਰੀਏਬਲ, ਦੀ ਵਰਤੋਂ ਸ਼ਾਮਲ ਹੈ।ਅਤੇ ਫੇਨਮੈਨ ਚਿੱਤਰ। ਇਸ ਨੂੰ ਪ੍ਰਯੋਗਾਤਮਕ ਨਤੀਜਿਆਂ ਦੁਆਰਾ ਵਿਆਪਕ ਤੌਰ ਉੱਤੇ ਪਰਖਿਆ ਅਤੇ ਤਸਦੀਕ ਕੀਤਾ ਗਿਆ ਹੈ, ਅਤੇ ਇਸ ਦੀ ਵਰਤੋਂ ਬਹੁਤ ਸਾਰੀਆਂ ਸਹੀ ਭਵਿੱਖਬਾਣੀਆਂ ਕਰਨ ਲਈ ਕੀਤੀ ਗਈ ਹੈ।ਇਲੈਕਟ੍ਰੌਨਾਂ ਅਤੇ ਫੋਟੌਨਾਂ ਦੇ ਵਿਵਹਾਰ ਬਾਰੇ। \\n\\nQED ਦੇ ਬਹੁਤ ਸਾਰੇ ਵਿਹਾਰਕ ਉਪਯੋਗ ਹਨ, ਜਿਨ੍ਹਾਂ ਵਿੱਚ ਲੇਜ਼ਰਾਂ, ਆਪਟੀਕਲ ਫਾਈਬਰਾਂ ਅਤੇ ਹੋਰਾਂ ਦੇ ਡਿਜ਼ਾਈਨ ਸ਼ਾਮਲ ਹਨ।ਦੂਰਸੰਚਾਰ ਵਿੱਚ ਵਰਤੇ ਜਾਣ ਵਾਲੇ ਯੰਤਰ। ਇਹ ਉੱਚ ਊਰਜਾਵਾਂ ਉੱਤੇ ਕਣਾਂ ਦੇ ਵਿਵਹਾਰ ਬਾਰੇ ਸਾਡੀ ਸਮਝ ਵਿੱਚ ਵੀ ਮਹੱਤਵਪੂਰਨ ਭੂਮਿਕਾ ਨਿਭਾਉਂਦਾ ਹੈ, ਜਿਵੇਂ ਕਿ ਉਹ ਜੋ ਪਾਏ ਜਾਂਦੇ ਹਨ।ਕਣ ਐਕਸਲੇਟਰਾਂ ਵਿੱਚ. \',\' ਹਾਂ ਬੱਦਲ ਆਪਸ ਵਿੱਚ ਜੁਡ਼ਿਆ ਹੋਇਆ ਹੈ, ਉਦਾਹਰਣ ਵਜੋਂ ਰੇਡੀਓ ਤਰੰਗਾਂ ਲਈ ਲਹਿਰ ਆਵਾਜਾਈ ਹੈ \',\' ਵਾਯੂਮੰਡਲ ਵਿੱਚ, ਅਤੇ ਕਾਰਨਤਰੰਗ ਵਿੱਚ ਵਿਗਾਡ਼। \\n\\n ਇਸੇ ਲਈ ਰੇਡੀਓ ਤਰੰਗਾਂ ਵਾਯੂਮੰਡਲ ਦੀਆਂ ਸਥਿਤੀਆਂ, ਜਿਵੇਂ ਕਿ ਤਾਪਮਾਨ, ਦਬਾਅ ਅਤੇ ਨਮੀ ਦੁਆਰਾ ਪ੍ਰਭਾਵਿਤ ਹੋ ਸਕਦੀਆਂ ਹਨ।ਇਹ ਸਥਿਤੀਆਂ ਵਾਯੂਮੰਡਲ ਵਿੱਚ ਚਾਰਜ ਕੀਤੇ ਕਣਾਂ ਦੇ ਵਿਵਹਾਰ ਨੂੰ ਬਦਲ ਸਕਦੀਆਂ ਹਨ, ਅਤੇ ਬਦਲੇ ਵਿੱਚ ਰੇਡੀਓ ਲਹਿਰ ਦੇ ਪ੍ਰਸਾਰ ਨੂੰ ਪ੍ਰਭਾਵਤ ਕਰਦੀਆਂ ਹਨ।ਰੇਡੀਓ ਸਿਗਨਲ ਕਮਜ਼ੋਰ ਜਾਂ ਵਿਗਾਡ਼ ਹੋ ਸਕਦੇ ਹਨ ਕਿਉਂਕਿ ਉਹ ਵਾਯੂਮੰਡਲ ਵਿੱਚੋਂ ਲੰਘਦੇ ਹਨ, ਅਤੇ ਰੇਡੀਓ ਤਰੰਗਾਂ ਦੀਆਂ ਵੱਖ-ਵੱਖ ਫ੍ਰੀਕੁਐਂਸੀਆਂ ਕਿਉਂ ਪ੍ਰਭਾਵਿਤ ਹੋ ਸਕਦੀਆਂ ਹਨਇੱਕੋ ਜਿਹੀਆਂ ਸਥਿਤੀਆਂ ਦੁਆਰਾ ਵੱਖਰੇ ਤਰੀਕੇ ਨਾਲ। \\n\\nSo ਸੰਖੇਪ ਵਿੱਚ, ਇਲੈਕਟ੍ਰੌਨ ਕਲਾਉਡ ਅਤੇ ਇਲੈਕਟ੍ਰੋਮੈਗਨੈਟਿਕ ਫੀਲਡ ਆਪਸ ਵਿੱਚ ਜੁਡ਼ੇ ਹੋਏ ਹਨ, ਅਤੇ ਇਸਦੇ ਇੱਕ ਹਿੱਸੇ ਵਿੱਚ ਤਬਦੀਲੀਆਂ ਹੁੰਦੀਆਂ ਹਨ।ਖੇਤਰ ਦੂਜੇ ਹਿੱਸਿਆਂ ਨੂੰ ਪ੍ਰਭਾਵਿਤ ਕਰ ਸਕਦਾ ਹੈ। ਇਸ ਲਈ ਵਾਤਾਵਰਣ ਵਿੱਚ ਚਾਰਜ ਕੀਤੇ ਕਣਾਂ ਦਾ ਵਿਵਹਾਰ ਤਰੰਗਾਂ ਦੇ ਪ੍ਰਸਾਰ ਨੂੰ ਪ੍ਰਭਾਵਿਤ ਕਰ ਸਕਦਾ ਹੈ, ਜਿਵੇਂ ਕਿ ਰੇਡੀਓਤਰੰਗਾਂ, ਅਤੇ ਰੇਡੀਓ ਤਰੰਗਾਂ ਦੀਆਂ ਵੱਖ-ਵੱਖ ਫ੍ਰੀਕੁਐਂਸੀਆਂ ਇੱਕੋ ਜਿਹੀਆਂ ਸਥਿਤੀਆਂ ਦੁਆਰਾ ਵੱਖ-ਵੱਖ ਤਰੀਕੇ ਨਾਲ ਕਿਉਂ ਪ੍ਰਭਾਵਿਤ ਹੋ ਸਕਦੀਆਂ ਹਨ। \',\' ਆਹ ਹਾਂ ਇਹ ਓਵਰਟੋਨ ਵਰਤਾਰਾ ਹੈ, ਕਿਸਮਕਿਸੇ ਹੋਰ ਬਾਰੰਬਾਰਤਾ ਉੱਤੇ ਇੱਕ ਬੈਕਅੱਪ ਸਿਗਨਲ ਦਾ] ', 'ਇਲੈਕਟ੍ਰੀਕਲ ਇੰਜੀਨੀਅਰਿੰਗ ਦੇ 20 ਸਭ ਤੋਂ ਮੁਸ਼ਕਿਲ ਵਿਸ਼ਿਆਂ ਦੀ ਵਿਆਖਿਆ ਕਰੋ। \n '] | punjabi |
કોણ છે સલીમ ફ્રુટ, જેને NIAએ ઝડપી લીધો હતો? નામની કહાની રસપ્રદ છે, છોટા શકીલના સાળાના સંબંધમાં નેશનલ ઇન્વેસ્ટિગેશન એજન્સી NIA એ સોમવારે 9 મે, 2022 ગેંગસ્ટર છોટા શકીલના સહયોગી સલીમ કુરેશી ઉર્ફે સલીમ ફ્રુટની પૂછપરછ માટે અટકાયત કરી હતી. આજે ગેંગસ્ટર દાઉદ ઈબ્રાહિમના 20 થી વધુ જગ્યાઓની સર્ચ કરવામાં આવી હતી, ત્યારબાદ સલીમ ફ્રુટને કસ્ટડીમાં લેવામાં આવ્યો છે. સર્ચમાં કેન્દ્રીય એજન્સીએ અનેક ગુનાહિત દસ્તાવેજો જપ્ત કર્યા છે. ચાલો જાણીએ કોણ છે સલીમ કુરેશી અને તેની પૃષ્ઠભૂમિ શું છે? કોણ છે સલીમ ફળ? સલીમ કુરેશી ઉર્ફે સલીમ ફ્રુટ ગેંગસ્ટર છોટા શકીલનો સાળો છે, જે દાઉદ ઈબ્રાહિમનો સાગરિત છે. સલીમ કુરેશીનો દક્ષિણ મુંબઈમાં ફ્રુટ ફેમિલી બિઝનેસ છે, જેના કારણે તેઓ સલીમ ફ્રુટ તરીકે પણ ઓળખાય છે. તે દાઉદ ઈબ્રાહિમનો નજીકનો સહયોગી હોવાનું કહેવાય છે. છોટા શકીલ એક ગેંગસ્ટર અને કોન્ટ્રાક્ટ કિલર છે જે તેના સાગરિતો દ્વારા ખંડણીનું રેકેટ ચલાવતો હતો. કહેવાય છે કે છોટા શકીલ દાઉદ ઈબ્રાહિમ માટે કામ કરે છે અને તેને પાકિસ્તાનથી ઓપરેટ કરવામાં આવે છે. સૂત્રોનું કહેવું છે કે, સલીમ ફ્રુટ ત્રણથી ચાર વખત પાકિસ્તાનમાં છોટા શકીલના ઘરે પણ ગયો હતો. આ પણ વાંચો: ત્યારે કેન્દ્રીય કર્મચારીઓના મોંઘવારી ભથ્થામાં 34 ટકાનો વધારો થવા જઈ રહ્યો છે આ પણ વાંચો: ત્યારે કેન્દ્રીય કર્મચારીઓના મોંઘવારી ભથ્થામાં 34 ટકાનો વધારો થવા જઈ રહ્યો છે સલીમ ફ્રુટ સામે અન્ય કયા કેસ છે? સલીમ ફ્રુટ પર 2000ની શરૂઆતમાં શકીલ અને ઈબ્રાહીમ માટે વિદેશી ખંડણીનું રેકેટ ચલાવવાનો આરોપ હતો અને 2006માં UAE સરકાર દ્વારા તેને ભારત મોકલી દેવામાં આવ્યો હતો. છોટા શકીલને સંડોવતા ખંડણીના કેસમાં પણ તેની ધરપકડ કરવામાં આવી હતી. તે 2010 સુધી જેલમાં હતો. | gujurati |
ആന്തരികമായ ബഹുമാനത്തിന്റെ ബാഹ്യമായ പ്രകടനം സല്യൂട്ട്..! പോലീസ് ആര്ക്കൊക്കെ സല്യൂട്ട് ചെയ്യണം..? സുരേഷ് ഗോപിയുടെ ആവശ്യം ന്യായമോ. കോട്ടയം: ജീപ്പില് നിന്നിറങ്ങാതിരുന്ന എസ്ഐയെ സുരേഷ് ഗോപി എംപി വിളിച്ചുവരുത്തി സല്യൂട്ട് ചെയ്യിപ്പിച്ച സംഭവത്തിന് പിന്നാലെ പോലീസിന്റെ സല്യൂട്ട് ആര്ക്കൊക്കെ നല്കാം എന്നതില് വലിയ ചര്ച്ചകള് നടക്കുകയാണ്. പോലീസ് നിയമ പ്രകാരം എംപിമാര്ക്ക് പോലീസിന്റെ സല്യൂട്ട് ശരിക്കും ലഭിക്കുമോ? കേരള പോലീസിലെ സ്റ്റാന്ഡിംഗ് ഓര്ഡര് പ്രകാരം സല്യൂട്ട് നല്കാനുള്ള മാനദണ്ഡങ്ങള് എന്തെല്ലാമാണ്? ആര്ക്കൊക്കെയാണ് പോലീസ് ബഹുമാന സൂചകമായി സല്യൂട്ട് അടിക്കേണ്ടത്. പരിശോധിക്കാം. ആന്തരികമായ ബഹുമാനത്തിന്റെ ബാഹ്യമായ പ്രകടനം എന്നാണ് സല്യൂട്ട് പദത്തിന്റെ അര്ത്ഥം. 19ാം നൂറ്റാണ്ടിലാണ് ഇന്നത്തെ സല്യൂട്ട് ശരിക്കും പിറവി കൊള്ളുന്നത്. ബ്രിട്ടീഷ് സൈന്യത്തില് തലയിലെ തൊപ്പി അല്പ്പമൊന്നുയര്ത്തിയായിരുന്നു ആദരവ്, ഇത് പരിഷ്കരിച്ചാണ് പിന്നീട് ഇന്നത്തെ സല്യൂട്ടില് എത്തിചേര്ന്നത്. കേരള പൊലീസ് സ്റ്റാന്ഡിംഗ് ഓര്ഡറില് 18ാം അധ്യായത്തില് കേരള പോലീസ് ആര്ക്കൊക്കെ സല്യൂട്ട് നല്കണമെന്ന കാര്യം വിശദമായി തന്നെ പറയുന്നുണ്ട്. ഈ സ്റ്റാന്ഡിംഗ് ഓര്ഡര് പ്രകാരം The post ആന്തരികമായ ബഹുമാനത്തിന്റെ ബാഹ്യമായ പ്രകടനം സല്യൂട്ട്..! പോലീസ് ആര്ക്കൊക്കെ സല്യൂട്ട് ചെയ്യണം..? സുരേഷ് ഗോപിയുടെ ആവശ്യം ന്യായമോ appeared first on RashtraDeepika. | malyali |
మరో టీడీపీ నేతకు జస్టిస్ లలిత బెయిల్ మొన్న పట్టాభి.. నేడు బ్రహ్మంచౌదరికి.. 20 వేలతో రెండు పూచీకత్తుల సమర్పణకు ఆదేశం మూడు వారాల పాటు మంగళగిరి, తాడేపల్లి పోలీస్స్టేషన్ల పరిధిలోకి రావొద్దని ఆదేశం అడ్వాన్స్ ఆర్డర్ రూపంలో ఉత్తర్వులు సాక్షి, అమరావతి: టీడీపీకి చెందిన మరో నాయకుడికి హైకోర్టు న్యాయమూర్తి జస్టిస్ లలిత బెయిల్ మంజూరు చేశారు. ఇప్పటికే టీడీపీ అధికార ప్రతినిధి కొమ్మారెడ్డి పట్టాభికి బెయిల్ మంజూరు చేసిన జస్టిస్ లలిత.. తాజాగా ఆ పార్టీ కార్యదర్శి నాదెండ్ల బ్రహ్మంచౌదరికి బెయిల్ ఇచ్చారు. రూ.20 వేలతో రెండు పూచీకత్తులు సమర్పించాలని బ్రహ్మంచౌదరిని ఆదేశించారు. మూడు వారాల పాటు బ్రహ్మంచౌదరి మంగళగిరి, తాడేపల్లి పోలీస్స్టేషన్ల పరిధిలోకి ప్రవేశించరాదని న్యాయమూర్తి స్పష్టం చేశారు. ఈ మేరకు ఆమె అడ్వాన్స్ ఆర్డర్ జారీ చేశారు. పట్టాభి బెయిల్ సందర్భంగా కూడా ఆమె అడ్వాన్స్ ఆర్డర్ రూపంలో ఉత్తర్వులిచ్చిన సంగతి తెలిసిందే. టీడీపీ కార్యాలయంలో జరిగిన గొడవ సందర్భంగా అక్కడకు వెళ్లిన తనను పలువురు టీడీపీ నేతలు కులం పేరుతో దూషించి, హత్యాయత్నం చేశారంటూ రిజర్వ్ ఇన్స్పెక్టర్ సక్రూనాయక్ ఇచ్చిన ఫిర్యాదు ఆధారంగా పోలీసులు కేసు నమోదు చేశారు. ఈ కేసులో అరెస్ట్ అయిన బ్రహ్మంచౌదరి బెయిల్ కోసం హైకోర్టులో పిటిషన్ దాఖలు చేశారు. ఈ వ్యాజ్యంపై జస్టిస్ లలిత సోమవారం మరోసారి విచారణ జరిపారు. పిటిషనర్ తరఫున సీనియర్ న్యాయవాది దమ్మాలపాటి శ్రీనివాస్, న్యాయవాది కేఎం కృష్ణారెడ్డి వాదనలు వినిపించారు. బ్రహ్మంచౌదరిని అరెస్ట్ చేసిన పోలీసులు అతన్ని కొట్టారని శ్రీనివాస్ తెలిపారు. ఈ విషయాన్ని ఆయన సంబంధిత కోర్టు మేజిస్ట్రేట్కు ఫిర్యాదు చేశారని, అయితే మేజిస్ట్రేట్ మాత్రం ఎలాంటి వైద్య పరీక్షలకు ఆదేశించలేదన్నారు. అంతేకాక మంగళగిరి పోలీసులు బ్రహ్మంచౌదరిని అరెస్ట్ చేసి మేడికొండూరు పోలీసులకు అప్పగించారని, భౌతిక హాని తలపెట్టాలన్న ఉద్దేశంతోనే ఇలా చేశారని ఆయన వివరించారు. విధుల్లో ఉన్న పోలీసును కులం పేరుతో దూషించారు.. పోలీసుల తరఫున అదనపు పబ్లిక్ ప్రాసిక్యూటర్ ఎస్.దుష్యంత్రెడ్డి వాదనలు వినిపిస్తూ.. విధుల్లో ఉన్న పోలీసు అధికారిని పిటిషనర్, ఇతర టీడీపీ నేతలు కులం పేరుతో దూషించారని తెలిపారు. విధులు నిర్వర్తించకుండా అడ్డుకున్నారని చెప్పారు. ఈ కేసులోనే కాక మరో మూడు కేసుల్లో కూడా బ్రహ్మంచౌదరి నిందితుడుగా ఉన్నారని వివరించారు. పోలీసుల చిత్తశుద్ధిని పరిగణనలోకి తీసుకోవాలని, మొదట హత్యాయత్నం కేసు నమోదు చేయగా, తర్వాత దానిని తొలగించారని దుష్యంత్ కోర్టు దృష్టికి తెచ్చారు. న్యాయమూర్తి జస్టిస్ లలిత స్పందిస్తూ.. పోలీసులు కొట్టారంటూ బ్రహ్మంచౌదరి గాయాలు చూపినప్పుడు మేజిస్ట్రేట్ ఎందుకు వైద్య పరీక్షలకు ఆదేశించలేదని ప్రశ్నించారు. అలా చేయకుండా కేవలం కొట్టారన్న విషయాన్ని రికార్డ్ చేసి ఊరుకోవడం ఎంత మాత్రం సబబని ప్రశ్నించారు. మేజిస్ట్రేట్ చట్ట ప్రకారమే వ్యవహరించారు... దీనికి దుష్యంత్ స్పందిస్తూ.. కొట్టారని పిటిషనర్ చెప్పగానే, దానిపై మేజిస్ట్రేట్ పోలీసుల వివరణ కోరాని, రేపు పోలీసులిచ్చే సమాధానం సంతృప్తికరంగా లేకుంటే, షోకాజ్ నోటీసులు ఇవ్వడంతో పాటు విచారణకు సైతం మేజిస్ట్రేట్ ఆదేశించవచ్చని తెలిపారు. చట్టం నిర్దేశించిన విధి విధానాల ప్రకారమే మేజిస్ట్రేట్ వ్యవహరించారని వివరించారు. ఇరుపక్షాల వాదనలు విన్న న్యాయమూర్తి జస్టిస్ లలిత.. పిటిషనర్పై పోలీసులు నమోదు చేసిన కేసులో ఏడేళ్ల కన్నా తక్కువ శిక్ష పడే కేసులన్నారు. అందువల్ల అతనికి బెయిల్ మంజూరు చేస్తున్నట్టు చెప్పారు. ఆ సమయంలో దుష్యంత్ జోక్యం చేసుకుంటూ.. రెండు మూడు వారాల పాటు మంగళగిరి, తాడేపల్లి పోలీస్స్టేషన్ల పరిధిలో ప్రవేశించకుండా బ్ర హ్మంచౌదరిని నియంత్రిస్తూ ఉత్తర్వులివ్వాలని కోరగా.. అందుకు న్యాయమూర్తి సానుకూలం గా స్పందించారు. పిటిషనర్ గాయాలను చూపినప్పుడు వైద్య పరీక్షలకు ఎందుకు ఆదేశించలేదో ఓ నివేదికను తమ ముందుం చాలని న్యాయమూర్తి ఆదేశించారు. | telegu |
/*
@file: apps/afl/afl.js
*/
function afl() {
this.appStart = appStart;
this.appQuit = appQuit;
this.pickClub = pickClub;
this.team_list = team_list;
var team_list = {
"swans": 1,
"blues": 2,
"magpies": 3,
"bombers": 4,
"hawks": 5,
"crows": 6,
"cats": 7,
"giants": 8,
"kangaroos": 9,
"saints": 10,
"lions": 11,
"eagles": 12,
"suns": 13,
"bulldogs": 14,
"tigers": 15,
"dockers": 16,
"power": 17,
"demons": 18
};
function old_pickClub(team_name) {
team = team_name.substring(1);
console.log("team_name " + team);
console.log(team_list[team])
currentLight.afl(team_list[team]);
}
function pickClub(team_name) {
var team = team_name.substring(1);
console.log("team_name " + team);
var entry = team_list[team] -1;
// Copy the array of light values to the string
for (var j=0; j < 50; j++) {
tr = team_lights[entry][j][0];
tg = team_lights[entry][j][1];
tb = team_lights[entry][j][2];
currentLight.fastset(tr,tg,tb,j);
}
currentLight.fastlights(); // Render to the device
}
function appStart() {
console.log("afl.appStart");
$("head").append('<link rel="stylesheet" href="afl.css" />');
}
function appQuit() {
console.log("afl.appQuit");
}
// Color definitions
var WHITE = [ 0xff, 0xff, 0xff ];
var BLACK = [ 0x00, 0x00, 0x00 ];
var RED = [ 0xff, 0x00, 0x00 ];
var BLUE = [ 0x00, 0x00, 0xff ];
var PURPLE = [ 0xFF, 0x00, 0xFF ];
// Team definitions
var BLUES_BLUE = [ 0x00,0x00, 0x80 ];
var BLUES_WHITE = [ 0xff, 0xff, 0xff ];
var blues = [
BLUES_BLUE,
BLUES_BLUE,
BLUES_WHITE,
BLUES_WHITE,
BLUES_BLUE,
BLUES_BLUE,
BLUES_WHITE,
BLUES_WHITE,
BLUES_BLUE,
BLUES_BLUE,
BLUES_WHITE,
BLUES_WHITE,
BLUES_BLUE,
BLUES_BLUE,
BLUES_WHITE,
BLUES_WHITE,
BLUES_BLUE,
BLUES_BLUE,
BLUES_WHITE,
BLUES_WHITE,
BLUES_BLUE,
BLUES_BLUE,
BLUES_WHITE,
BLUES_WHITE,
BLUES_BLUE,
BLUES_BLUE,
BLUES_WHITE,
BLUES_WHITE,
BLUES_BLUE,
BLUES_BLUE,
BLUES_WHITE,
BLUES_WHITE,
BLUES_BLUE,
BLUES_BLUE,
BLUES_WHITE,
BLUES_WHITE,
BLUES_BLUE,
BLUES_BLUE,
BLUES_WHITE,
BLUES_WHITE,
BLUES_BLUE,
BLUES_BLUE,
BLUES_WHITE,
BLUES_WHITE,
BLUES_BLUE,
BLUES_BLUE,
BLUES_WHITE,
BLUES_WHITE,
BLUES_BLUE,
BLUES_BLUE
];
var BOMBERS_RED = [ 0xff, 0x00, 0x00 ];
var bombers = [
BOMBERS_RED,
BOMBERS_RED,
BLACK,
BLACK,
BOMBERS_RED,
BOMBERS_RED,
BLACK,
BLACK,
BOMBERS_RED,
BOMBERS_RED,
BLACK,
BLACK,
BOMBERS_RED,
BOMBERS_RED,
BLACK,
BLACK,
BOMBERS_RED,
BOMBERS_RED,
BLACK,
BLACK,
BOMBERS_RED,
BOMBERS_RED,
BLACK,
BLACK,
BOMBERS_RED,
BOMBERS_RED,
BLACK,
BLACK,
BOMBERS_RED,
BOMBERS_RED,
BLACK,
BLACK,
BOMBERS_RED,
BOMBERS_RED,
BLACK,
BLACK,
BOMBERS_RED,
BOMBERS_RED,
BLACK,
BLACK,
BOMBERS_RED,
BOMBERS_RED,
BLACK,
BLACK,
BOMBERS_RED,
BOMBERS_RED,
BLACK,
BLACK,
BOMBERS_RED,
BOMBERS_RED
];
var BULLDOGS_RED = [ 0xff, 0x00, 0x00 ];
var BULLDOGS_WHITE = [ 0xff, 0xff, 0xff ];
var BULLDOGS_BLUE = [ 0x00, 0x00, 0xff ];
var bulldogs = [
BULLDOGS_RED,
BULLDOGS_RED,
BULLDOGS_WHITE,
BULLDOGS_WHITE,
BULLDOGS_BLUE,
BULLDOGS_BLUE,
BULLDOGS_RED,
BULLDOGS_RED,
BULLDOGS_WHITE,
BULLDOGS_WHITE,
BULLDOGS_BLUE,
BULLDOGS_BLUE,
BULLDOGS_RED,
BULLDOGS_RED,
BULLDOGS_WHITE,
BULLDOGS_WHITE,
BULLDOGS_BLUE,
BULLDOGS_BLUE,
BULLDOGS_RED,
BULLDOGS_RED,
BULLDOGS_WHITE,
BULLDOGS_WHITE,
BULLDOGS_BLUE,
BULLDOGS_BLUE,
BULLDOGS_RED,
BULLDOGS_RED,
BULLDOGS_WHITE,
BULLDOGS_WHITE,
BULLDOGS_BLUE,
BULLDOGS_BLUE,
BULLDOGS_RED,
BULLDOGS_RED,
BULLDOGS_WHITE,
BULLDOGS_WHITE,
BULLDOGS_BLUE,
BULLDOGS_BLUE,
BULLDOGS_RED,
BULLDOGS_RED,
BULLDOGS_WHITE,
BULLDOGS_WHITE,
BULLDOGS_BLUE,
BULLDOGS_BLUE,
BULLDOGS_RED,
BULLDOGS_RED,
BULLDOGS_WHITE,
BULLDOGS_WHITE,
BULLDOGS_BLUE,
BULLDOGS_BLUE,
BULLDOGS_RED,
BULLDOGS_RED
];
var CATS_BLUE = [ 0x00, 0x00, 0x3F ];
var CATS_WHITE = [ 0xff, 0xff, 0xff ];
var cats = [
CATS_BLUE,
CATS_BLUE,
CATS_WHITE,
CATS_WHITE,
CATS_BLUE,
CATS_BLUE,
CATS_WHITE,
CATS_WHITE,
CATS_BLUE,
CATS_BLUE,
CATS_WHITE,
CATS_WHITE,
CATS_BLUE,
CATS_BLUE,
CATS_WHITE,
CATS_WHITE,
CATS_BLUE,
CATS_BLUE,
CATS_WHITE,
CATS_WHITE,
CATS_BLUE,
CATS_BLUE,
CATS_WHITE,
CATS_WHITE,
CATS_BLUE,
CATS_BLUE,
CATS_WHITE,
CATS_WHITE,
CATS_BLUE,
CATS_BLUE,
CATS_WHITE,
CATS_WHITE,
CATS_BLUE,
CATS_BLUE,
CATS_WHITE,
CATS_WHITE,
CATS_BLUE,
CATS_BLUE,
CATS_WHITE,
CATS_WHITE,
CATS_BLUE,
CATS_BLUE,
CATS_WHITE,
CATS_WHITE,
CATS_BLUE,
CATS_BLUE,
CATS_WHITE,
CATS_WHITE,
CATS_BLUE,
CATS_BLUE
];
var CROWS_BLUE = [ 0x00, 0x00, 0x7F ];
var CROWS_RED = [ 0xff, 0x00, 0x00 ];
var CROWS_YELLOW = [ 0xff, 0xd7, 0x00 ];
var crows = [
CROWS_RED,
CROWS_RED,
CROWS_YELLOW,
CROWS_YELLOW,
CROWS_BLUE,
CROWS_BLUE,
CROWS_RED,
CROWS_RED,
CROWS_YELLOW,
CROWS_YELLOW,
CROWS_BLUE,
CROWS_BLUE,
CROWS_RED,
CROWS_RED,
CROWS_YELLOW,
CROWS_YELLOW,
CROWS_BLUE,
CROWS_BLUE,
CROWS_RED,
CROWS_RED,
CROWS_YELLOW,
CROWS_YELLOW,
CROWS_BLUE,
CROWS_BLUE,
CROWS_RED,
CROWS_RED,
CROWS_YELLOW,
CROWS_YELLOW,
CROWS_BLUE,
CROWS_BLUE,
CROWS_RED,
CROWS_RED,
CROWS_YELLOW,
CROWS_YELLOW,
CROWS_BLUE,
CROWS_BLUE,
CROWS_RED,
CROWS_RED,
CROWS_YELLOW,
CROWS_YELLOW,
CROWS_BLUE,
CROWS_BLUE,
CROWS_RED,
CROWS_RED,
CROWS_YELLOW,
CROWS_YELLOW,
CROWS_BLUE,
CROWS_BLUE,
CROWS_RED,
CROWS_RED
];
var DEMON_BLUE = [ 0x00, 0x00, 0x1f ];
var DEMON_RED = [ 0xff, 0x00, 0x00 ];
var demons =[
DEMON_BLUE,
DEMON_BLUE,
DEMON_RED,
DEMON_RED,
DEMON_BLUE,
DEMON_BLUE,
DEMON_RED,
DEMON_RED,
DEMON_BLUE,
DEMON_BLUE,
DEMON_RED,
DEMON_RED,
DEMON_BLUE,
DEMON_BLUE,
DEMON_RED,
DEMON_RED,
DEMON_BLUE,
DEMON_BLUE,
DEMON_RED,
DEMON_RED,
DEMON_BLUE,
DEMON_BLUE,
DEMON_RED,
DEMON_RED,
DEMON_BLUE,
DEMON_BLUE,
DEMON_RED,
DEMON_RED,
DEMON_BLUE,
DEMON_BLUE,
DEMON_RED,
DEMON_RED,
DEMON_BLUE,
DEMON_BLUE,
DEMON_RED,
DEMON_RED,
DEMON_BLUE,
DEMON_BLUE,
DEMON_RED,
DEMON_RED,
DEMON_BLUE,
DEMON_BLUE,
DEMON_RED,
DEMON_RED,
DEMON_BLUE,
DEMON_BLUE,
DEMON_RED,
DEMON_RED,
DEMON_BLUE,
DEMON_BLUE
];
var DOCKERS_PURPLE = [ 0xFF, 0x00, 0xFF ];
var DOCKERS_WHITE = [ 0xff, 0xff, 0xff ];
var dockers = [
DOCKERS_PURPLE,
DOCKERS_PURPLE,
DOCKERS_WHITE,
DOCKERS_WHITE,
DOCKERS_PURPLE,
DOCKERS_PURPLE,
DOCKERS_WHITE,
DOCKERS_WHITE,
DOCKERS_PURPLE,
DOCKERS_PURPLE,
DOCKERS_WHITE,
DOCKERS_WHITE,
DOCKERS_PURPLE,
DOCKERS_PURPLE,
DOCKERS_WHITE,
DOCKERS_WHITE,
DOCKERS_PURPLE,
DOCKERS_PURPLE,
DOCKERS_WHITE,
DOCKERS_WHITE,
DOCKERS_PURPLE,
DOCKERS_PURPLE,
DOCKERS_WHITE,
DOCKERS_WHITE,
DOCKERS_PURPLE,
DOCKERS_PURPLE,
DOCKERS_WHITE,
DOCKERS_WHITE,
DOCKERS_PURPLE,
DOCKERS_PURPLE,
DOCKERS_WHITE,
DOCKERS_WHITE,
DOCKERS_PURPLE,
DOCKERS_PURPLE,
DOCKERS_WHITE,
DOCKERS_WHITE,
DOCKERS_PURPLE,
DOCKERS_PURPLE,
DOCKERS_WHITE,
DOCKERS_WHITE,
DOCKERS_PURPLE,
DOCKERS_PURPLE,
DOCKERS_WHITE,
DOCKERS_WHITE,
DOCKERS_PURPLE,
DOCKERS_PURPLE,
DOCKERS_WHITE,
DOCKERS_WHITE,
DOCKERS_PURPLE,
DOCKERS_PURPLE
];
var EAGLES_BLUE = [ 0x00, 0x00, 0x3F ];
var EAGLES_WHITE = [ 0xff, 0xff, 0xff ];
var EAGLES_GOLD = [ 0xff, 0xa4, 0x03 ];
var eagles = [
EAGLES_BLUE,
EAGLES_BLUE,
EAGLES_WHITE,
EAGLES_GOLD,
EAGLES_GOLD,
EAGLES_BLUE,
EAGLES_BLUE,
EAGLES_WHITE,
EAGLES_GOLD,
EAGLES_GOLD,
EAGLES_BLUE,
EAGLES_BLUE,
EAGLES_WHITE,
EAGLES_GOLD,
EAGLES_GOLD,
EAGLES_BLUE,
EAGLES_BLUE,
EAGLES_WHITE,
EAGLES_GOLD,
EAGLES_GOLD,
EAGLES_BLUE,
EAGLES_BLUE,
EAGLES_WHITE,
EAGLES_GOLD,
EAGLES_GOLD,
EAGLES_BLUE,
EAGLES_BLUE,
EAGLES_WHITE,
EAGLES_GOLD,
EAGLES_GOLD,
EAGLES_BLUE,
EAGLES_BLUE,
EAGLES_WHITE,
EAGLES_GOLD,
EAGLES_GOLD,
EAGLES_BLUE,
EAGLES_BLUE,
EAGLES_WHITE,
EAGLES_GOLD,
EAGLES_GOLD,
EAGLES_BLUE,
EAGLES_BLUE
];
var GIANTS_ORANGE = [ 0xff, 0xa5, 0x00 ];
var GIANTS_CHARCOAL = [ 0x02, 0x02, 0x02 ];
var GIANTS_WHITE = [ 0xff, 0xff, 0xff ];
var giants = [
GIANTS_ORANGE,
GIANTS_ORANGE,
GIANTS_CHARCOAL,
GIANTS_CHARCOAL,
GIANTS_WHITE,
GIANTS_WHITE,
GIANTS_ORANGE,
GIANTS_ORANGE,
GIANTS_CHARCOAL,
GIANTS_CHARCOAL,
GIANTS_WHITE,
GIANTS_WHITE,
GIANTS_ORANGE,
GIANTS_ORANGE,
GIANTS_CHARCOAL,
GIANTS_CHARCOAL,
GIANTS_WHITE,
GIANTS_WHITE,
GIANTS_ORANGE,
GIANTS_ORANGE,
GIANTS_CHARCOAL,
GIANTS_CHARCOAL,
GIANTS_WHITE,
GIANTS_WHITE,
GIANTS_ORANGE,
GIANTS_ORANGE,
GIANTS_CHARCOAL,
GIANTS_CHARCOAL,
GIANTS_WHITE,
GIANTS_WHITE,
GIANTS_ORANGE,
GIANTS_ORANGE,
GIANTS_CHARCOAL,
GIANTS_CHARCOAL,
GIANTS_WHITE,
GIANTS_WHITE,
GIANTS_ORANGE,
GIANTS_ORANGE,
GIANTS_CHARCOAL,
GIANTS_CHARCOAL,
GIANTS_WHITE,
GIANTS_WHITE,
GIANTS_ORANGE,
GIANTS_ORANGE,
GIANTS_CHARCOAL,
GIANTS_CHARCOAL,
GIANTS_WHITE,
GIANTS_WHITE,
GIANTS_ORANGE,
GIANTS_ORANGE
];
var HAWKS_GOLD = [ 0xff, 0xa4, 0x03 ]; // blue should be 0x23
var HAWKS_BROWN = [ 0x57, 0x26, 0x00 ]; // red should be 0x27 and green 0x16
var hawks = [
HAWKS_GOLD,
HAWKS_GOLD,
HAWKS_BROWN,
HAWKS_BROWN,
HAWKS_GOLD,
HAWKS_GOLD,
HAWKS_BROWN,
HAWKS_BROWN,
HAWKS_GOLD,
HAWKS_GOLD,
HAWKS_BROWN,
HAWKS_BROWN,
HAWKS_GOLD,
HAWKS_GOLD,
HAWKS_BROWN,
HAWKS_BROWN,
HAWKS_GOLD,
HAWKS_GOLD,
HAWKS_BROWN,
HAWKS_BROWN,
HAWKS_GOLD,
HAWKS_GOLD,
HAWKS_BROWN,
HAWKS_BROWN,
HAWKS_GOLD,
HAWKS_GOLD,
HAWKS_BROWN,
HAWKS_BROWN,
HAWKS_GOLD,
HAWKS_GOLD,
HAWKS_BROWN,
HAWKS_BROWN,
HAWKS_GOLD,
HAWKS_GOLD,
HAWKS_BROWN,
HAWKS_BROWN,
HAWKS_GOLD,
HAWKS_GOLD,
HAWKS_BROWN,
HAWKS_BROWN,
HAWKS_GOLD,
HAWKS_GOLD,
HAWKS_BROWN,
HAWKS_BROWN,
HAWKS_GOLD,
HAWKS_GOLD,
HAWKS_BROWN,
HAWKS_BROWN,
HAWKS_GOLD,
HAWKS_GOLD
];
var KANGAROOS_BLUE = [ 0x00, 0x00, 0x1F ];
var KANGAROOS_WHITE = [ 0xff, 0xff, 0xff ];
var kangaroos = [
KANGAROOS_BLUE,
KANGAROOS_BLUE,
KANGAROOS_WHITE,
KANGAROOS_WHITE,
KANGAROOS_BLUE,
KANGAROOS_BLUE,
KANGAROOS_WHITE,
KANGAROOS_WHITE,
KANGAROOS_BLUE,
KANGAROOS_BLUE,
KANGAROOS_WHITE,
KANGAROOS_WHITE,
KANGAROOS_BLUE,
KANGAROOS_BLUE,
KANGAROOS_WHITE,
KANGAROOS_WHITE,
KANGAROOS_BLUE,
KANGAROOS_BLUE,
KANGAROOS_WHITE,
KANGAROOS_WHITE,
KANGAROOS_BLUE,
KANGAROOS_BLUE,
KANGAROOS_WHITE,
KANGAROOS_WHITE,
KANGAROOS_BLUE,
KANGAROOS_BLUE,
KANGAROOS_WHITE,
KANGAROOS_WHITE,
KANGAROOS_BLUE,
KANGAROOS_BLUE,
KANGAROOS_WHITE,
KANGAROOS_WHITE,
KANGAROOS_BLUE,
KANGAROOS_BLUE,
KANGAROOS_WHITE,
KANGAROOS_WHITE,
KANGAROOS_BLUE,
KANGAROOS_BLUE,
KANGAROOS_WHITE,
KANGAROOS_WHITE,
KANGAROOS_BLUE,
KANGAROOS_BLUE,
KANGAROOS_WHITE,
KANGAROOS_WHITE,
KANGAROOS_BLUE,
KANGAROOS_BLUE,
KANGAROOS_WHITE,
KANGAROOS_WHITE,
KANGAROOS_BLUE,
KANGAROOS_BLUE
];
var LIONS_MAROON = [0x1f, 0x00, 0x01 ];
var LIONS_BLUE = [ 0x00, 0x00, 0xff ];
var LIONS_GOLD = [ 0xff, 0xa4, 0x03 ];
var lions = [
LIONS_MAROON,
LIONS_MAROON,
LIONS_BLUE,
LIONS_BLUE,
LIONS_GOLD,
LIONS_GOLD,
LIONS_MAROON,
LIONS_MAROON,
LIONS_BLUE,
LIONS_BLUE,
LIONS_GOLD,
LIONS_GOLD,
LIONS_MAROON,
LIONS_MAROON,
LIONS_BLUE,
LIONS_BLUE,
LIONS_GOLD,
LIONS_GOLD,
LIONS_MAROON,
LIONS_MAROON,
LIONS_BLUE,
LIONS_BLUE,
LIONS_GOLD,
LIONS_GOLD,
LIONS_MAROON,
LIONS_MAROON,
LIONS_BLUE,
LIONS_BLUE,
LIONS_GOLD,
LIONS_GOLD,
LIONS_MAROON,
LIONS_MAROON,
LIONS_BLUE,
LIONS_BLUE,
LIONS_GOLD,
LIONS_GOLD,
LIONS_MAROON,
LIONS_MAROON,
LIONS_BLUE,
LIONS_BLUE,
LIONS_GOLD,
LIONS_GOLD,
LIONS_MAROON,
LIONS_MAROON,
LIONS_BLUE,
LIONS_BLUE,
LIONS_GOLD,
LIONS_GOLD,
LIONS_MAROON,
LIONS_MAROON
];
var magpies = [
WHITE,
WHITE,
BLACK,
BLACK,
WHITE,
WHITE,
BLACK,
BLACK,
WHITE,
WHITE,
BLACK,
BLACK,
WHITE,
WHITE,
BLACK,
BLACK,
WHITE,
WHITE,
BLACK,
BLACK,
WHITE,
WHITE,
BLACK,
BLACK,
WHITE,
WHITE,
BLACK,
BLACK,
WHITE,
WHITE,
BLACK,
BLACK,
WHITE,
WHITE,
BLACK,
BLACK,
WHITE,
WHITE,
BLACK,
BLACK,
WHITE,
WHITE,
BLACK,
BLACK,
WHITE,
WHITE,
BLACK,
BLACK,
WHITE,
WHITE
];
var POWER_TEAL =[ 0x00, 0x1F, 0x1F ]; // should be 0, 128, 128 but whatever
var POWER_BLACK = [ 0x00, 0x00, 0x00 ];
var POWER_WHITE = [ 0xff, 0xff, 0xff ];
var POWER_SILVER = [ 0x10, 0x10, 0x10 ];
var power = [
POWER_TEAL,
POWER_BLACK,
POWER_WHITE,
POWER_SILVER,
POWER_TEAL,
POWER_BLACK,
POWER_WHITE,
POWER_SILVER,
POWER_TEAL,
POWER_BLACK,
POWER_WHITE,
POWER_SILVER,
POWER_TEAL,
POWER_BLACK,
POWER_WHITE,
POWER_SILVER,
POWER_TEAL,
POWER_BLACK,
POWER_WHITE,
POWER_SILVER,
POWER_TEAL,
POWER_BLACK,
POWER_WHITE,
POWER_SILVER,
POWER_TEAL,
POWER_BLACK,
POWER_WHITE,
POWER_SILVER,
POWER_TEAL,
POWER_BLACK,
POWER_WHITE,
POWER_SILVER,
POWER_TEAL,
POWER_BLACK,
POWER_WHITE,
POWER_SILVER,
POWER_TEAL,
POWER_BLACK,
POWER_WHITE,
POWER_SILVER,
POWER_TEAL,
POWER_BLACK,
POWER_WHITE,
POWER_SILVER,
POWER_TEAL,
POWER_BLACK,
POWER_WHITE,
POWER_SILVER,
POWER_TEAL,
POWER_WHITE
];
var SAINTS_RED = [ 0xff, 0x00, 0x00 ];
var SAINTS_WHITE = [ 0xff, 0xff, 0xff ];
var SAINTS_BLACK = [ 0x00, 0x00, 0x00 ];
var saints = [
SAINTS_RED,
SAINTS_RED,
SAINTS_WHITE,
SAINTS_WHITE,
SAINTS_BLACK,
SAINTS_BLACK,
SAINTS_RED,
SAINTS_RED,
SAINTS_WHITE,
SAINTS_WHITE,
SAINTS_BLACK,
SAINTS_BLACK,
SAINTS_RED,
SAINTS_RED,
SAINTS_WHITE,
SAINTS_WHITE,
SAINTS_BLACK,
SAINTS_BLACK,
SAINTS_RED,
SAINTS_RED,
SAINTS_WHITE,
SAINTS_WHITE,
SAINTS_BLACK,
SAINTS_BLACK,
SAINTS_RED,
SAINTS_RED,
SAINTS_WHITE,
SAINTS_WHITE,
SAINTS_BLACK,
SAINTS_BLACK,
SAINTS_RED,
SAINTS_RED,
SAINTS_WHITE,
SAINTS_WHITE,
SAINTS_BLACK,
SAINTS_BLACK,
SAINTS_RED,
SAINTS_RED,
SAINTS_WHITE,
SAINTS_WHITE,
SAINTS_BLACK,
SAINTS_BLACK,
SAINTS_RED,
SAINTS_RED,
SAINTS_WHITE,
SAINTS_WHITE,
SAINTS_BLACK,
SAINTS_BLACK,
SAINTS_RED,
SAINTS_RED
];
var SUNS_RED = [ 0xff, 0x00, 0x00 ];
var SUNS_GOLD = [ 0xff, 0xa4, 0x03 ];
var SUNS_BLUE = [ 0x00, 0x00, 0xff ];
var suns = [
SUNS_RED,
SUNS_RED,
SUNS_GOLD,
SUNS_GOLD,
SUNS_BLUE,
SUNS_BLUE,
SUNS_RED,
SUNS_RED,
SUNS_GOLD,
SUNS_GOLD,
SUNS_BLUE,
SUNS_BLUE,
SUNS_RED,
SUNS_RED,
SUNS_GOLD,
SUNS_GOLD,
SUNS_BLUE,
SUNS_BLUE,
SUNS_RED,
SUNS_RED,
SUNS_GOLD,
SUNS_GOLD,
SUNS_BLUE,
SUNS_BLUE,
SUNS_RED,
SUNS_RED,
SUNS_GOLD,
SUNS_GOLD,
SUNS_BLUE,
SUNS_BLUE,
SUNS_RED,
SUNS_RED,
SUNS_GOLD,
SUNS_GOLD,
SUNS_BLUE,
SUNS_BLUE,
SUNS_RED,
SUNS_RED,
SUNS_GOLD,
SUNS_GOLD,
SUNS_BLUE,
SUNS_BLUE,
SUNS_RED,
SUNS_RED,
SUNS_GOLD,
SUNS_GOLD,
SUNS_BLUE,
SUNS_BLUE,
SUNS_RED,
SUNS_RED
];
var SWANS_RED = [ 0xff, 0x00, 0x00 ];
var SWANS_WHITE = [ 0xff, 0xff, 0xff ];
var swans = [
SWANS_RED,
SWANS_RED,
SWANS_WHITE,
SWANS_WHITE,
SWANS_RED,
SWANS_RED,
SWANS_WHITE,
SWANS_WHITE,
SWANS_RED,
SWANS_RED,
SWANS_WHITE,
SWANS_WHITE,
SWANS_RED,
SWANS_RED,
SWANS_WHITE,
SWANS_WHITE,
SWANS_RED,
SWANS_RED,
SWANS_WHITE,
SWANS_WHITE,
SWANS_RED,
SWANS_RED,
SWANS_WHITE,
SWANS_WHITE,
SWANS_RED,
SWANS_RED,
SWANS_WHITE,
SWANS_WHITE,
SWANS_RED,
SWANS_RED,
SWANS_WHITE,
SWANS_WHITE,
SWANS_RED,
SWANS_RED,
SWANS_WHITE,
SWANS_WHITE,
SWANS_RED,
SWANS_RED,
SWANS_WHITE,
SWANS_WHITE,
SWANS_RED,
SWANS_RED,
SWANS_WHITE,
SWANS_WHITE,
SWANS_RED,
SWANS_RED,
SWANS_WHITE,
SWANS_WHITE,
SWANS_RED,
SWANS_RED
];
var TIGERS_YELLOW = [ 0xff, 0xd7, 0x00 ];
var TIGERS_BLACK = BLACK;
var tigers = [
TIGERS_YELLOW,
TIGERS_YELLOW,
TIGERS_BLACK,
TIGERS_BLACK,
TIGERS_YELLOW,
TIGERS_YELLOW,
TIGERS_BLACK,
TIGERS_BLACK,
TIGERS_YELLOW,
TIGERS_YELLOW,
TIGERS_BLACK,
TIGERS_BLACK,
TIGERS_YELLOW,
TIGERS_YELLOW,
TIGERS_BLACK,
TIGERS_BLACK,
TIGERS_YELLOW,
TIGERS_YELLOW,
TIGERS_BLACK,
TIGERS_BLACK,
TIGERS_YELLOW,
TIGERS_YELLOW,
TIGERS_BLACK,
TIGERS_BLACK,
TIGERS_YELLOW,
TIGERS_YELLOW,
TIGERS_BLACK,
TIGERS_BLACK,
TIGERS_YELLOW,
TIGERS_YELLOW,
TIGERS_BLACK,
TIGERS_BLACK,
TIGERS_YELLOW,
TIGERS_YELLOW,
TIGERS_BLACK,
TIGERS_BLACK,
TIGERS_YELLOW,
TIGERS_YELLOW,
TIGERS_BLACK,
TIGERS_BLACK,
TIGERS_YELLOW,
TIGERS_YELLOW,
TIGERS_BLACK,
TIGERS_BLACK,
TIGERS_YELLOW,
TIGERS_YELLOW,
TIGERS_BLACK,
TIGERS_BLACK,
TIGERS_YELLOW,
TIGERS_YELLOW
];
var team_lights = [
swans,
blues,
magpies,
bombers,
hawks,
crows,
cats,
giants,
kangaroos,
saints,
lions,
eagles,
suns,
bulldogs,
tigers,
dockers,
power,
demons
];
}
| code |
திருப்புகழ் கதைகள்: திருவிடைமருதூர்! thiruppugazh stories திருப்புகழ்க் கதைகள் 166உயிர்க்கூடு விடும் பழநி திருவிடைமருதூர் முனைவர் கு.வை. பாலசுப்பிரமணியன் அருணகிரியார் இத்திருப்புகழில் குறிப்பிடும் அடுத்த சிவத்தலம் திருவிடைமருதூர் ஆகும். இங்கே உள்ள மகாலிங்கேசுவரர் திருக்கோயில் சம்பந்தர், அப்பர், சுந்தரர் மூவரது தேவாரப் பாடல் பெற்ற தலமாகும். சோழ நாட்டின் காவிரி தென்கரைத் தலங்களில், காவிரி கரையில் அமைந்துள்ள 30ஆவது சிவத்தலமாகும். இத்தலம் திருவிசைப்பா திருப்பல்லாண்டு திருத்தலங்களில் ஒன்றாகும். கருவூர்த் தேவர், மாணிக்கவாசகர், பட்டினத்தார் ஆகியோரும் இத்தலத்தைப் பாடியுள்ளனர். இது தஞ்சை மாவட்டத்தில் அமைந்துள்ளது. தலச் சிறப்பு மருத மரத்தைத் தல மரமாகக் கொண்டு சிறப்புற விளங்குகின்ற சிவன் கோயில்கள் இந்தியாவில் மூன்று. முதலாவது ஆந்திர மாநிலம் கர்னூல் அருகே உள்ள மல்லிகார்ஜுனம் எனும் ஸ்ரீசைலம் திருக்கோயில். இரண்டாவது மத்தியார்ஜுனம் எனப்படுகின்ற தமிழ்நாடு தஞ்சாவூர் மாவட்டம் கும்பகோணம் அருகேயுள்ள திருவிடைமருதூர். மூன்றாவது புடார்ச்சுனம் எனப்படுகின்ற தமிழ்நாடு திருநெல்வேலிக்கு அருகே அம்பாசமுத்திரம் அருகில் உள்ள திருப்புடைமருதூர். இவை முறையே மல்லிகார்ஜுனம், மத்தியார்ஜுனம், புடார்சுனம் தலைமருது, இடைமருது, கடைமருது எனப் புகழப்பெறுகின்றன. வரகுணபாண்டியனின் ப்ரும்மஹத்தி தோஷம் திருவிடைமருதூர் தலம் வரகுண பாண்டியன் என்ற பாண்டிய நாட்டு அரசனின் வாழ்க்கையுடன் சம்பந்தம் உடையதாகும். ஒருமுறை வரகுண பாண்டியன் அருகிலுள்ள காட்டிற்கு வேட்டையாடச் சென்றான். மாலை நேரம் முடிந்து இரவு தொடங்கிவிட்ட நேரத்தில் அரசன் குதிரை மீதேறி திரும்பி வந்து கொண்டு இருக்கும் போது வழியில் உறங்கிக் கொண்டிருந்த ஒரு அந்தணன் குதிரையின் காலில் மிதிபட்டு இறந்துவிட்டான். இச்சம்பவம் அவனறியாமல் நடந்திருந்தாலும் ஒரு அந்தணனைக் கொன்றதால் அரசனை பிரம்மஹத்தி தோஷம் பற்றிக்கொண்டது. அந்தணின் ஆவியும் அரசனைப் பற்றிக்கொண்டது. சிறந்த சிவபக்தனான வரகுண பாண்டியன் மதுரை சோமசுந்தரரை வணங்கி இதிலிருந்து விடுவிக்க வேண்டும் என்று வேண்டிக்கொண்டான். மதுரை சோமசுந்தரக் கடவுளும் அரசனுடைய கனவில் தோன்றி திருவிடைமருதூர் சென்று அங்கு தன்னை வழிபடும்படி கூறினார். thiruvidaimaruthur எதிரி நாடான சோழ நாட்டிலுள்ள திருவிடைமருதூருக்கு எப்படிச் செல்வது என்று கவலைப்பட்டுக் கொண்டிருந்த அரசனுக்கு சோழ மன்னன் பாண்டிய நாட்டின் மேல் படையெடுத்து வந்திருக்கும் செய்தி கிடைத்தது. சோழ மன்னனுடன் போருக்குச் சென்ற வரகுண பாண்டியன் சோழ மன்னனை போரில் தோற்கடித்து சோழநாடு வரை துரத்திச் சென்றான். அப்போது திருவிடைமருதூர் சென்று இங்குள்ள இறைவனை வழிபட ஆலயத்தினுள் பிரதான கிழக்கு வாயில் வழியாக நுழைந்தான். வரகுண பாண்டியனைப் பற்றியிருந்த பிரம்மஹத்தியும் அந்தணனின் ஆவியும் அரசனைப் பின்பற்றி கோவிலினுள் செல்ல தைரியமின்றி வெளியிலேயே தங்கிவிட்டன. அரசன் திரும்பி வரும்போது மறுபடியும் அவனை பிடித்துக் கொள்ளலாம் என்று காத்திருந்தன. ஆனால் திருவிடைமருதூர் இறைவனோ வரகுண பாண்டியனை மேற்கு வாயில் வழியாக வெளியேறிச் செல்லும்படி அசரீரியாக ஆணையிட்டு அவனுக்கு அருள் புரிந்தார். அரசனும் பிரம்மஹத்தி நீங்கியவனாக பண்டியநாடு திரும்பினான். இதை நினைவுகூறும் வகையில் இன்றளவும் இவ்வாலயத்திற்கு வரும் பக்தர்கள் பிரதான கிழக்கு வாயில் வழியாக உள்ளே சென்று மேற்கிலுள்ள அம்மன் சந்நிதி கோபுரவாயில் வழியாக வெளியே செல்லும் முறையைக் கடைப்பிடித்து வருகிறார்கள். இத்தலத்தில் உள்ள இறைவன் சுயம்பு லிங்க மூர்த்தியாகும். இறைவன் மகாலிங்கேஸ்வரர் தன்னைத்தானே அர்ச்சித்துக் கொண்டு பூஜா விதிகளை சப்தரிஷிகள் மற்றுமுள்ள முனிவர்களுக்கு போதித்து அருளிய தலம் திருவிடைமருதூர். மார்க்கண்டேய முனிவருக்கு அவரின் விருப்பப்படி அர்த்தநாரீஸ்வரர் உருவத்தில் இத்தலத்து இறைவன் காட்சி கொடுத்துள்ளார். thiruvidaimaruthur1 மூகாம்பிகை சன்னதி இவ்வாலயத்தில் உள்ள மூகாம்பிகை சந்நிதி மிகவும் புகழ் பெற்றது. அம்பாள் சந்நிதிக்கு தெற்குப் பக்கம் இந்த மூகாம்பிகை சந்நிதி அமைந்துள்ளது. மூகாம்பிகைக்கு இந்தியாவில் திருவிடைமருதூரிலும், கர்நாடக மாநிலத்திலுள்ள கொல்லூரிலும் மட்டும் சந்நிதி இருப்பது குறிப்பிடத்தக்கது. வரகுண பாண்டியனுக்கு பிரம்மஹத்தி தோஷம் நீக்கிய தலமென்பதால் பிரம்மஹத்தி தோஷ நிவாரண தலம் இது. அருள்மிகு மகாலிங்கஸ்வரர் சுயம்புலிங்கமாக அருள்பாழிக்கிறார். மகாலிங்கேஸ்வரர் திருத்தலத்தை சுற்றி நான்கு வீதிகளிலும் சிவ ஸ்தலங்கள் உள்ளதால் இத்தலம் பஞ்சலிங்கத் தலம் என்றும் அழைக்கப்படுகிறது. இத்தலத்தின் தலவிநாயகர் ஆண்ட விநாயகர். இத்தலத்தில் சுவாமி, அம்பாள் இருவரது சந்நிதிகளும் கிழக்கு நோக்கியவை. இத்திருக்கோயிலில் இருபத்தியேழு நட்சத்திர லிங்கங்கள் அமைந்துள்ளன. இக்கோயிலில் அமைந்துள்ள மகாலிங்கேஸ்வரர் திருத்தேர் தமிழ்நாட்டிலே மூன்றாவது பெரியத் தேர் ஆகும். பட்டினத்தார் மற்றும் பத்திரகிரியார் ஆகியோர்க்கு கிழக்கு மற்றும் மேற்கு கோபுர வாசல்களில் சன்னதி உள்ளது. அம்முனு அம்மையார் பாவை விளக்கு தஞ்சாவூரில் அரசுக்கட்டிலை இழந்த மராத்திய அரசன், அமரசிம்மன் திருவிடைமருதூர் அரண்மனையில் தங்கி வாழ்ந்தார். அவரது மகன் பிரதாபசிம்மனை அவருடைய அம்மான் பெண்ணான அம்முனு அம்மணி விரும்பி, தன் திருமணம் நிறைவேற இக்கோயிலுக்கு லட்ச தீபம் ஏற்றுவதாக வேண்டிக்கொண்டாள். அவருடைய பிரார்த்தனை நிறைவேறிய பின்னர் இலட்சத் தீபம் ஏற்றி, அவற்றுள் ஒரு விளக்காக தன்னுடைய உருவத்தையே பாவை விளக்காக்கித் தன்னுடைய சிற்பமே தீபம் ஏந்தும் வகை செய்தாள். 120 செ.மீ. உயரமுள்ள, பித்தளையால் ஆன இந்த பாவை விளக்கு அழகிய பீடத்தின் மீது உள்ளது. thiruvidaimaruthur2 நின்ற நிலையில் அம்முனு அம்மணி தன் இரு கரங்களாலும் விளக்கினை ஏந்தியுள்ளார். அவருடைய தோளில் கிளி ஒன்று உள்ளது. இதன் பீடத்தில் அம்முனு அம்மணியின் காதல் காவியம் தமிழ்க் கல்வெட்டாக இடம் பெற்றுள்ளது. ஒரு மன்னரின் மனைவியே தீபம் ஏந்திய பாவை விளக்காக இன்றும் நிற்பது வரலாற்றுச் சிறப்புடைய நிகழ்வாகும். திருவிடைமருதூர் ஸ்ரீமகாலிங்கசுவாமி திருக்கோயில் என்ற நூலில் மருதவன வரலாறு, கோயில் செய்திகள், ஆண்டவிநாயகர், அகத்தியர் தரிசனம், தவக்கோலம், மகாலிங்கப்பெருமான், பிரணவப்பிரகாரம், ஆயர்பாடி கிருஷ்ணன், நட்சத்திரலிங்க வழிபாடு, பட்டினத்தார், பத்ரகிரியார், பிரம்மஹத்தி தோஷ பரிகாரம், கங்கையும் காருண்யாமிர்தமும், சக்ர மகாமேரு வழிபாடு, அசுவமேதப் பிரகாரம் உள்ளிட்ட பல தலைப்புகளில் திருவிடைமருதூர் மகாலிங்கசுவாமி கோயிலைப் பற்றி விவாதிக்கப்பட்டுள்ளன. | tamil |
શું હોય છે અખાડા, કેવી રીતે બને છે, જાણો તેની પરંપરા વીશે.. શૈવ, વૈષ્ણવ અને ઉદાસીન પંથના સન્યાસીઓની માન્યતા પ્રાપ્ત કુલ 13 અખાડા છે. પહેલા આશ્રમોના અખાડાને બેડા એટલે કે સાધુઓનો સમૂહ કહેવામાં આવતો હતો. પહેલા અખાડા શબ્દનું ચલણ ન હતું. સાધુઓના સમૂહમાં પીર અને તદ્દવીર હોતા હતા. અખાડા શબ્દનું ચલણ મુગલકાળથી શરુ થયું. અખાડા સાધુઓનો તે સમૂહ છે જે સત્રવિદ્યામાં પણ પારંગત રહે છે. કેટલાક વિદ્વાનોનું માનવું છે કે, અલખ શબ્દ માંથી જ અખાડા શબ્દ બન્યો છે. કેટલાક માને છે કે અક્ખડ ઉપરથી કે આશ્રમ ઉપરથી આ શબ્દ બન્યો છે. અખાડા પરિષદના પ્રમુખની પસંદગી કુંભમેળા જેવા વિશાળ ધાર્મિક કાર્યક્રમોના અવસર પર સંતો સંતો વચ્ચે સંઘર્ષની વધતી ઘટનાઓને રોકવા માટે અખાડા પરિષદની સ્થાપના કરવામાં આવી હતી. તેમાં સરકાર દ્વારા માન્યતા પ્રાપ્ત કુલ 13 અખાડાઓ હોય છે. આ તમામ અખાડા લોકશાહી રીતે ચૂંટાયેલા પ્રમુખ અને સચિવ દ્વારા ચલાવવામાં આવે છે. અખાડા પરિષદની બેઠકમાં સર્વસંમતિથી અધ્યક્ષની પસંદગી કરવામાં આવે છે. મુખ્ય પારંપરીક 13 અખાડા કુંભ કે અર્ધકુંભમાં સાધુસંતોના કુલ 13 અખાડાઓ દ્વારા ભાગ લેવામાં આવે છે. આ અખાડાઓની પ્રાચીન કાળથી જ સ્નાન પર્વની પરંપરા ચાલતી આવી રહી છે. જેમાં શૈવ સન્યાસી સંપ્રદાયના 7 અખાડા છે જેમાં શ્રી પંચાયતી અખાડા મહાનિર્વાણી દારાગંજ પ્રયાગ ઉત્તર પ્રદેશ, શ્રી પંચ અટલ અખાડા ચૈક હનુમાન, વારાણસી ઉત્તર પ્રદેશ. શ્રી પંચાયતી અખાડા નીરંજની દારાગંજ, પ્રયાગ ઉત્તર પ્રદેશ, શ્રી તપોનિધિ આનંદ અખાડા પંચાયતી ત્ર્યંબકેશ્વર, નાસિક મહારાષ્ટ્ર, શ્રી પંચદશનામ જુના અખાડા બાબા હનુમાન ઘાટ, વારાણસી ઉત્તર પ્રદેશ, શ્રી પંચદશનામ આવાહન અખાડા દશાશ્વમેઘ ઘાટ, વારાણસી ઉત્તર પ્રદેશ, શ્રી પંચદશનામ પંચ અગ્નિ અખાડા ગીરીનગર, ભવનાથ, જુનાગઢ ગુજરાત છે. જયારે વૈરાગી વૈષ્ણવ સંપ્રદાયના 3 અખાડા છે જેમાં શ્રી દિગમ્બર અણી અખાડા શામળાજી ખાકચોક મંદિર, સાંબરકાંઠા ગુજરાત, શ્રી નિર્વાની આની અખાડા હનુમાન ગાદી, અયોધ્યા ઉત્તર પ્રદેશ, શ્રી પંચ નિર્મોહી અણી અખાડા ધીર સમીર મંદિર બંસીવટ, વૃંદાવન, મથુરા ઉત્તર પ્રદેશનો સમાવે શ થાય છે. જ્યારે ઉદાસીન સંપ્રદાયના 3 અખાડા છે. શ્રી પંચાયતી બડા ઉદાસીન અખાડા ક્રષ્ણનગર, કીટગંજ, પ્રયાગરાજ ઉત્તર પ્રદેશ, શ્રી પંચાયતી અખાડા નયા ઉદાસીન કનખલ, હરિદ્વાર ઉત્તરાખંડ, છે. આઠમી સદીમાં બન્યા હતા આ અખાડા કહેવામાં આવે છે કે આદિ શંકરાચાર્યએ આઠમી સદીમાં 13 અખાડા બનાવ્યા હતા. આજ સુધી તે અખાડા કાયમ છે. અન્ય કુંભ મેળામાં બધા અખાડા એક સાથે સ્નાન કરે છે, પણ નાસિકના કુંભમાં વૈષ્ણવ અખાડા નાસિકમાં અને શૈવ અખાડા ત્ર્યંબકેશ્વરમાં સ્નાન કરે છે. આ વ્યવસ્થા પેશવાના સમયમાં શરુ કરવામાં આવી જે ઈ.સ. 1772 થી ચાલી રહી છે. અખાડા સાથે જોડાયેલી મહત્વની વાતો અટલ અખાડાઃ આ અખાડામાં માત્ર બ્રામણ, ક્ષત્રીય અને વૈશ્ય દીક્ષા લઇ શકે છે અને બીજા કોઈ આ અખાડામાં આવી શકતા નથી. અવાહન અખાડાઃ બીજા અખાડાઓમાં મહિલા સાધ્વીઓને પણ દીક્ષા આપવામાં આવે છે પણ આ અખાડામાં એવી કોઈ પરંપરા નથી. નિરંજની અખાડાઃ આ અખાડો સૌથી વધુ શિક્ષિત અખાડો છે. આ અખાડામાં લગભગ 50 મહામંડલેશ્ચર છે. અગ્નિ અખાડાઃ આ અખાડામાં માત્ર બ્રહ્મચારી બ્રાહ્મણ જ દિશા લઇ શકે છે. કોઈ બીજા દીક્ષા નથી લઇ શકતા. મહાનિર્વાણી અખાડાઃ મહાકાલેશ્વર જ્યોતિર્લીંગની પૂજાની જવાબદારી આ અખાડા પાસે છે. આ પરંપરા વર્ષોથી ચાલી રહી છે. આનંદ અખાડાઃ આ શૈવ અખાડો છે જેને આજ સુધી એક પણ મહામંડલેશ્વર નથી બનાવ્યા. આ અખાડાના આચાર્યનું પદ જ મુખ્ય હોય છે. દિગંબર અણી અખાડાઃ આ અખાડાને વૈષ્ણવ સંપ્રદાયમાં રાજા કહેવામાં આવે છે. આ અખાડામાં સૌથી વધુ 431 ખાલસા છે. નિર્મોહી અણી અખાડાઃ વૈષ્ણવ સંપ્રદાયના ત્રણ અણી અખાડાઓ માંથી આમાં સૌથી વધુ 9 અખાડાનો સમાવેશ થાય છે. નિર્વાણી અણી અખાડાઃ આ અખાડામાં કુશ્તી મુખ્ય હોય છે જે તેમના જીવનનો એક ભાગ છે. આ કારણથી અખાડાના ઘણા સંત પ્રોફેશનલ પહેલવાન રહી ચુક્યા છે. બડા ઉદાસીન અખાડાઃ આ અખાડાનો ઉદેશ્ય સેવા કરવાનો છે. આ અખાડામાં માત્ર 4 મહંત હોય છે જે ક્યારેય કામ માંથી નિવૃત્ત થતા નથી. નયા ઉદાસીન અખાડાઃ આ અખાડામાં 8 થી 12 વર્ષ સુધીના બાળકો જેમને દાઢી મુછ ન નીકળી હોય તેમને નાગાસાધુ બનાવવામાં આવે છે. નિર્મલ અખાડાઃ આ અખાડામાં બીજા અખાડાની જેમ ધુમ્રપાનની મંજુરી નથી. તેના વિષે અખાડામાં બધા કેન્દ્રોના ગેટ ઉપર તેની સુચના લખેલી હોય છે. mantavyanews.com Copyright 2021 Mantavya News | gujurati |
स्व. डा. शुभदर्शन को नम आंखों से दी श्रद्धांजलि संवाद सहयोगी, अमृतसर: एक सांध्य दैनिक समाचार पत्र के मुख्य संपादक स्वर्गीय डा. शुभदर्शन के शनिवार को अंतिम रस्म चौथे पर शहर के राजनीतिज्ञ, पत्रकार समूह और समाजसेवी संस्थाओं से जुड़े सदस्यों ने उन्हें श्रद्धांजलि अर्पित की। सांसद गुरजीत औजला, विधायक सुनील दत्ती, डेवलपमेंट प्लानिग बोर्ड के चेयरमैन राजकंवल सिंह लकी, होली हार्ट स्कूल की डायरेक्टर अंजना सेठ, समाज सेविका डॉक्टर स्वराज ग्रोवर, मजीठा से नगर कौंसिल के पूर्व अध्यक्ष तरुण अबरोल, एडवोकेट एमके शर्मा, गुरशरण सिंह बब्बर, डा. विनोद तनेजा, डा. किरण खन्ना, डा. अतुला भास्कर सहित कई अन्यों ने डॉ. शुभदर्शन को निर्भीक, ईमानदार पत्रकार बताते हुए उनके द्वारा समाज में दी गई सेवाओं की सराहना की। स्वर्गीय डा. शुभ दर्शन पत्रकारिता से पिछले 40 साल से जुड़े थे। वह मजीठा से पांच वर्ष तक नगर कौंसिल के सदस्य भी रहे। वह हिदी जगत में भी प्रख्यात कवि के रूप में जाने जाते थे। उनके द्वारा लिखी गई कविताओं की कई पुस्तकों ने हिदी जगत पर छाप छोड़ी। कविता के क्षेत्र में भारत सरकार से कई पुरस्कार प्राप्त कर चुके हैं। इसके अलावा उन्होंने हिदी प्रचार प्रसार सोसायटी के अध्यक्ष के रूप में हिदी जगत के कई प्रमुख हस्तियों को जोड़कर एक परिवार की तरह एक माला में पिरो रखा था। | hindi |
आखिर 17 साल बाद समर्पित हुआ सामुदायिक भवन जागरण संवाददाता, शिमला : नगर निगम शिमला के भराड़ी वार्ड के कलस्टन में 1.02 करोड़ रुपये से बने सामुदायिक भवन का वीरवार को शहरी विकास मंत्री सुरेश भारद्वाज और पूर्व केंद्रीय मंत्री एवं राज्यसभा सदस्य आनंद शर्मा ने उद्घाटन किया। वर्ष 2005 में इस भवन का निर्माण कार्य शुरू हुआ था, लेकिन इसे तैयार होने में 17 साल लग गए।Click here to get the latest updates on State Elections 2022 सांसद निधि और निगम कोष से इस पर बजट खर्च किया गया है। इस भवन का निर्माण कार्य राज्य लोक निर्माण विभाग ने किया है। इसके लिए 54.25 लाख सांसद निधि और 48.50 लाख निगम कोष से जारी हुआ है। पूर्व केंद्रीय मंत्री आनंद शर्मा ने कहा कि वर्ष 2005 में सांसद निधि से उन्होंने इसके लिए बजट जारी किया था। निर्माण कार्य में देरी की वजह से इसकी लागत बढ़ी है, उन्होंने कहा कि इस भवन के बनने से लोगों को सुविधा मिलेगी। शादीसमारोह के अलावा अन्य कार्यक्रम के लिए लोगों को अब होटल व अन्य हाल बुक नहीं करने पड़ेंगे। उन्होंने कहा कि जनता को अपने वार्ड में ही सुविधा मिल सकेगी। इस मौके पर कांग्रेस प्रदेश अध्यक्ष कुलदीप राठौर, विधायक अनिरुद्ध सिंह, रोहित ठाकुर, कांग्रेस प्रवक्ता नरेश चौहान, जिला कांग्रेस कमेटी शिमला शहरी अध्यक्ष जितेंद्र चौधरी, जिला कांग्रेस कमेटी अध्यक्ष शिमला ग्रामीण यशवंत छाजटा, भराड़ी वार्ड की पार्षद तनुजा चौधरी, नगर निगम की महापौर सत्या कौंडल, आयुक्त नगर निगम आशीष कोहली, सुरेंद्र चौहान, रितेश कपरेट, हरिकृष्ण हिमराल मौजूद रहे। देरी से पहुंचे शहरी विकास मंत्री, आनंद शर्मा को करना पड़ा इंतजार सामुदायिक भवन का उद्घाटन कार्यक्रम साढ़े 11 बजे प्रस्तावित था। शहरी विकास मंत्री सुरेश भारद्वाज विधानसभा में थे। दोपहर 12 बजे विधानसभा में प्रश्नकाल समाप्त होने के बाद शहरी विकास मंत्री कार्यक्रम के लिए रवाना हुए। आनंद शर्मा का घर कलस्टन में ही है ऐसे में वह 11:45 पर कार्यक्रम स्थल पर पहुंच गए थे। शहरी विकास मंत्री के न आने के चलते उन्हें इंतजार करना पड़ा। उद्घाटन 12:20 पर हुआ। अधिकारियों ने किए फोन सुबह के समय मौसम का मिजाज सही नहीं था। आनंद शर्मा पौने 12 बजे कार्यक्रम स्थल पर पहुंच गए। वहां पर राज्य लोक निर्माण विभाग, नगर निगम के अधिकारी मौजूद थे। इसी दौरान बारिश शुरू हो गई और लोगों की भीड़ ज्यादा हो गई थी। आनंद शर्मा के आने के बाद निगम व लोक निर्माण विभाग के अधिकारी शहरी विकास मंत्री के पर्सनल स्टाफ को फोन करते रहे। उन्हें इसकी चिता थी कि देरी के कारण कोई विवाद न हो जाए। गैस में भड़की आग, काबू पाने से बड़ा हादसा टला कार्यक्रम में आने वाले लोगों के लिए चायपानी की व्यवस्था की गई थी। सामुदायिक भवन के बेसमेंट में रिफ्रैशमेंट तैयार की जा रही थी। इस दौरान गैस में आग भड़क गई, लेकिन इस पर तुरंत काबू पा लिया गया, जिससे बड़ा हादसा होने से टल गया। उद्घाटन पर इसलिए था विरोध पूर्व मनोनीत भाजपा पार्षद संजीव सूद का आरोप है कि इस भवन का नक्शा बनकर तैयार नहीं हुआ है। इसके अलावा भवन बनकर तैयार नहीं हुआ था। उन्होंने इसके निर्माण कार्य पर सवाल उठाए थे। इस पर यहां पर उद्घाटन पर विरोध हो गया था। सांसद निधि से कब कितना बजट आया वर्ष,बजट आया 2005,10 लाख 2008,15 लाख 2012,10 लाख 2015,10 लाख 2015,9.25 लाख कुल,54.25 लाख निगम कोष से कब कितना बजट 2018,25 लाख 2021,9.50 लाख 2021,9 लाख 2022,5 लाख कुल,48.50 लाख | hindi |
વિન્ડિઝ સામે કાલે ટી20 મુકાબલો નહીં રમે કોહલીપંત: બોર્ડે આપ્યો આરામ કોલકત્તા, તા.19કોલકત્તાના ઈડન ગાર્ડન ઉપર ભારતવિન્ડિઝ વચ્ચે કાલે શ્રેણીની અંતિમ અને ત્રીજી ટી20 રમાય તે પહેલાં ક્રિકેટ બોર્ડે એક મહત્ત્વનો નિર્ણય લેતાં વિરાટ કોહલી અને ઋષભ પંતને આરામ આપ્યો છે. કોહલી શ્રીલંકા સામે ટેસ્ટ શ્રેણી પહેલાં બાયોબબલમાંથી આરામ ઈચ્છતો હોવાથી તેને બ્રેક આપવામાં આવ્યો છે. બ્રેક મળતાં જ કોહલી ઘેર જવા માટે રવાના થઈ ગયો છે. કોહલીએ વિન્ડિઝ વિરુદ્ધ અમદાવાદમાં ત્રણ વનડે અને કોલકત્તામાં બે ટી20 મુકાબલા રમ્યા છે.વિન્ડિઝ વિરુદ્ધ ટી20 શ્રેણી બાદ ટીમ ઈન્ડિયા શ્રીલંકા વિરુદ્ધ ત્રણ ટી20 અને બે ટેસ્ટ મુકાબલાની શ્રેણી રમશે. કોહલીને શ્રીલંકા સામેની ટેસ્ટ શ્રેણીને ધ્યાનમાં રાખીને બ્રેક આપવામાં આવ્યો છે. શ્રીલંકા વિરુદ્ધ શ્રેણીની શરૂઆત 24 ફેબ્રુઆરીથી લખનૌમાં પહેલી ટી20 સાથે થશે. પહેલી ટી20 લખનૌ અને બાકીની બન્ને મેચ ધર્મશાલામાં રમાશે.ટી20 શ્રેણી બાદ શ્રીલંકા સામે બે ટેસ્ટ મુકાબલા રમાશે જેની શરૂઆત ચાર માર્ચથી થશે. વિરાટ કોહલીની સાથે વિકેટકિપરબેટર ઋષભ પંતને પણ બાયોબબલમાંથી આરામ આપવા માટે ત્રીજા ટી20માંથી બ્રેક આપવામાં આવ્યો છે. બન્નેએ ગઈકાલે રમાયેલી મેચમાં શાનદાર પ્રદર્શન કર્યું હતું. કોહલીએ આ મેચમાં 52 રનની ઈનિંગ રમી તો પંતે પણ 52 રન ફટકારી મેન ઑફ ધ મેચનો ખિતાબ મેળવ્યો હતો. | gujurati |
B2 Places to Visit Email is practice for the Cambridge Assessment English B2 First (FCE) examination. For this example, it is an informal email replying to a friend asking about the best places to visit during their stay.
You have received an email from your English friend, Sam.
Read this part of the letter and then write your letter to Sam.
I’m so excited about coming to visit you this summer for a few weeks. It’s going to be great, isn’t it? First, I’d like to visit some places where you like to go. Then, I’d like to do some shopping. Can you tell me a bit about where we might do these things together? | english |
## FINDINGS:
Scattered calcific focal plaque involving both carotid systems.
That on the left involving the ICA is moderate. Peak systolic velocities on
the left are 112, 174, 121, 83 and 101 cm/sec for the proximal, mid and distal
ICA, CCA and ECA respectively. Similar values on the right are 81, 94, 110,
81 and 116 cm/sec. The ICA to CCA ratio is 1.4 on the right and 2.1 on the
left. There is antegrade flow involving both vertebral arteries.
## IMPRESSION:
Findings as stated above which indicate:
1. Approximately 60-69% left mid ICA stenosis.
2. No significant right ICA stenosis (graded as less than 40% ).
| medical |
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DIN AISI / SAE / ASTM AFNOR B.S. UNI JIS Fe 330 Fe 330 B FU 1.0034 RSt 34 - 2 (SG2T) - A 34 - 2 NE 1449 34/20 HIR.HS.CR.CS Fe 330 B FN - (Fe 590 2) Fe 590 - 2 FN Fe 60 - 2 St 60 - 2 4360 - 55 E / 55 C Fe 590 E360 (Fe 690 2) Fe 70 - 2 The standards that correspond with DIN Material Numbers can only be compared approximately.
UNI 5867 Fe E 420 KGN steel equivalent material: Fe E 420 KGN steel plate/sheet is in UNI 5867 standard, the grade material is 1.8902. The equivalent material of Fe E 420 KGN steel are UNE36081 AE420KG/AE420KW, EN 10111 S420N, DIN17102 StE420, UN FeE420KGN.
Material Equivalents; prev. next. out of 25. Post on 04-Oct-2015. 300 views. Category: Documents. 7 download. Report. Download; 2 CARBON STEEL BOLTS A 307 GRADE A&B B.S. 970 En 3 X C 18 S NF A35-552 MSZ 229-63 Fe 37 UNI 5334 SIS 14 13 11 11 423. GRADE 2H B.S. 1750 GR. 2H MSZ 1745-68 GR. MC C 60 UNI 5332. 2 ALLOY STEEL.
Mechanical Properties. The mechanical properties of AISI 8630 alloy steel are displayed in the following table.
In the meantime, if the format of any material on our web pages interferes with your ability to access the information, please contact us to request assistance or if you have questions or comments about our website’s accessibility. Please call 800-942-5334 or submit an inquiry.
Mortonagrion megabinluyog spec. nov. from Brunei (Odonata: Zygoptera: Coenagrionidae) Mortonagrion megabinluyog spec. nov. from Brunei Type material examined.
Students learn the basic 2D tools of drafting while producing complex 3D models that can be used for perspective renderings, animations, material specifications and construction drawings. ArchiCAD is an integral design tool in interior design.
bacco, 60 acres corn, 15 acres chufas and 19 hogs. His 1950 program consists of: 3oo N. H. red chicks, 40 acres of Blue Lupine, 12 acres of Hairy Indigo, 5 acres of Pensacola-Bahia grass and put up 120 rods new fence which is in addition to his regular crops of: 2 acres tobacco, 60 acres of corn, 25 acres of peanuts and 30 hogs.
Writing as Material Technology: Orientation within landscapes of the Classic Maya world Sarah E. Jackson University of Cincinnati Introduction Writing as Material Technology Our shared charge in this volume is to consider writing as material practice.
Milias Liu, Thomas Leichtweiß, Jürgen Janek, Manfred Martin, In-situ structural investigation of non-stoichiometric HfO 2-x films using quick-scanning extended X-ray absorption fine structure, Thin Solid Films 2013, 539, 60-64, M. Crumbach, M. Martin, Non-Soichiometric HfO 2-x Thin Films, Frontiers in Electronic Materials 2012, 691.
60 The Tamil language is characterised by diglossia, according to which two varieties of the language are functionally differentiated. The high variety is used in the formal contexts and in the written form, while the low variety is used in informal contexts and in orality.
Course Descriptions and Symbols. Course Numbering. 3310 Printmaking. Studio course in printmaking techniques with focus on stenciling and relief printing. FE Integration of didactic and clinical material in the supervised practice of individual, group, marital, and family therapy.
Full text of "Italy : handbook for travellers"
Depois de clicar no material da lista resultante, uma lista de subgrupos aparecerá. Na Total Materia, o termo "subgrupos" refere-se a especificações que definem as propriedades do tipo de aço, no presente caso, a especificação DIN 1654-5 está selecionado.
Filter Lookup For Industrial Filters Find the right NAPA Industrial Filter for your needs Search by NAPA Filter part number or cross reference a competing manufacturer's part number.
All Recipient's rights under this Agreement shall terminate if it fails to comply with any of the material terms or conditions of this Agreement and does not cure such failure in a reasonable period of time after becoming aware of such noncompliance.
Z.D. Zheng, X.C. Wang and W.B. Mi, Tunable electronic structure and spin splitting in single and multiple Fe-adsorbed g-C 2 N with different layers: A first-principles study, Journal of Physics and Chemistry of Solids, 115, (221), (2018).
The resulting bulk material is a powder termed dry water, one macroscopic element of which is a liquid water marble. (72) Alongside the many applications of liquid marbles, (11) we have recently investigated different aspects including freezing, (73) levitation, coalescence, and self-propulsion.
Tabernaculo De La Fe Mundial 413 Wyandotte St. Talian Frank A 618 Hayes St. Leon Eleazar PO Box 60. Meyerhoffer Ethel M 6 Oakwood Dr. Nicholas Andrew PO Box 543. Stence William 5334 RT 333. Switzer Robert C Sr RR 4 Box 232. Ulsh Marion, Harry RD 2 Box 47.
Bai, C., K. M. Eskridge, and Y. Li, Analysis of the fate and transport of nC60 nanoparticles in the subsurface using response surface methodology, Journal of Contaminant Hydrology, 152, 60-69, 2013. | english |
ڈھاکہ سپورٹس ڈیسک بنگلہ دیش کرکٹ بورڈ نے کہا کہ وہ نے والے دنوں میں قومی معاہدہ کرنے والی خواتین کرکٹرز کی تنخواہ بڑھانے کا فیصلہ کیا ہے بی سی بی کی ویمن ونگ کے چیئرمین نڈیل چودھری نے بتایا کہ انہوں نے بورڈ کو خواتین کرکٹرز کی تنخواہوں میں اضافے کے ساتھ بین الاقوامی میچ فیس میں اضافے کی تجویز پیش کی ہے | urdu |
\begin{document}
\author[Robert Laterveer]
{Robert Laterveer}
\address{Institut de Recherche Math\'ematique Avanc\'ee,
CNRS -- Universit\'e
de Strasbourg,\
7 Rue Ren\'e Des\-car\-tes, 67084 Strasbourg CEDEX,
FRANCE.}
\email{[email protected]}
\title[Zero--cycles on self--products of surfaces]{Zero--cycles on self--products of surfaces: some new examples verifying Voisin's conjecture}
\begin{abstract} An old conjecture of Voisin describes how $0$--cycles of a surface $S$ should behave when pulled--back to the self--product $S^m$ for $m>p_g(S)$.
We exhibit some surfaces with large $p_g$ that verify Voisin's conjecture.
\end{abstract}
\keywords{Algebraic cycles, Chow groups, motives, Voisin conjecture, Kimura finite--dimensionality conjecture}
\subjclass[2010]{Primary 14C15, 14C25, 14C30.}
\maketitle
\section{Introduction}
Let $X$ be a smooth projective variety over $\mathbb{C}$, and let $A^i(X)_{\mathbb{Z}}:=CH^i(X)_{}$ denote the Chow groups of $X$ (i.e. the groups of codimension $i$ algebraic cycles on $X$ with $\mathbb{Z}$--coefficients, modulo rational equivalence \cite{F}). Let $A^i_{hom}(X)_{\mathbb{Z}}$ (and $A^i_{AJ}(X)_{\mathbb{Z}}$) denote the subgroup of homologically trivial (resp. Abel--Jacobi trivial) cycles.
The Bloch--Beilinson--Murre conjectures present a beautiful and coherent dream--world in which Chow groups are determined by cohomology and the coniveau filtration \cite{J2}, \cite{J4}, \cite{Mur}, \cite{Kim}, \cite{MNP}, \cite{Vo}. The following particular instance of this dream--world was first formulated by Voisin:
\begin{conjecture}[Voisin 1993 \cite{V9}]\label{conj} Let $S$ be a smooth projective surface. Let $m$ be an integer larger than the geometric genus $p_g(S)$. Then for any $0$--cycles $a_1,\ldots,a_m\in A^2_{AJ}(S)_{\mathbb{Z}}$, one has
\[ \sum_{\sigma\in\mathfrak S_m} \hbox{sgn}(\sigma) a_{\sigma(1)}\times\cdots\times a_{\sigma(m)}=0\ \ \ \hbox{in}\ A^{2m}(S^m)_{\mathbb{Z}}\ .\]
(Here $\mathfrak S_m$ is the symmetric group on $m$ elements, and $ \hbox{sgn}(\sigma)$ is the sign of the permutation $\sigma$.)
\end{conjecture}
For surfaces of geometric genus $0$, Conjecture \ref{conj} reduces to Bloch's conjecture \cite{B}. For surfaces $S$ of geometric genus $1$, Conjecture \ref{conj} takes on a particularly simple form: in this case, the conjecture stipulates that any $a_1, a_2\in A^2_{AJ}(S)_{\mathbb{Z}}$ should verify the equality
\[ a_1\times a_2 =a_2\times a_1\ \ \ \hbox{in}\ A^4(S\times S)_{\mathbb{Z}}\ .\]
This conjecture is still open for a general $K3$ surface;
examples of surfaces of geometric genus $1$ verifying this conjecture are given in \cite{V9}, \cite{16.5}, \cite{19}, \cite{21}. One can also formulate versions of Conjecture \ref{conj} for higher--dimensional varieties; this is studied in \cite{V9}, \cite{17}, \cite{24.4}, \cite{24.5}, \cite{BLP}, \cite{LV}, \cite{Ch}.
On a historical note, it is interesting to observe that Voisin's Conjecture \ref{conj} antedates Kimura's conjecture ``all varieties have finite--dimensional motive'' \cite{Kim}. Both conjectures have a similar flavour: Chow groups of a surface $S$ should have controlled behaviour when pulled--back to the self--product $S^m$, for large $m$.
The difference between Voisin's conjecture and Kimura's conjecture lies in the index $m$ which is much lower in Voisin's conjecture. In fact (as explained in \cite{BLP}), Voisin's conjecture follows from a combination of Kimura's conjecture with a strong form of the generalized Hodge conjecture.
The goal of the present note is to collect some (easy) examples of surfaces with geometric genus larger than $1$ verifying Voisin's conjecture.
\begin{nonumbering}[=Corollaries \ref{main1}, \ref{cor2}, \ref{cor4} and \ref{last}] The following surfaces verify Conjecture \ref{conj}:
\item
{(\romannumeral1)} generalized Burniat type surfaces in the family $\mathcal S_{16}$ of \cite{BCF} ($p_g(S)=3$);
\item
{(\romannumeral2)} the hypersurfaces $S\subset A/\iota$ considered in \cite{LNP}, where $A$ is an abelian threefold and $\iota$ is the $-1$-involution ($p_g(S)=3$);
\item
{(\romannumeral3)} minimal surfaces $S$ of general type with $p_g(S)=q(S)=3$ and $K^2_S=6$;
\item{(\romannumeral4)} the double cover of certain cubic surfaces (among which the Fermat cubic)
branched along the Hessian ($p_g(S)=4$);
\item
{(\romannumeral5)} the Fano surface of lines in a smooth cubic threefold ($p_g(S)=10$);
\item{(\romannumeral6)} the quotient $S=F/\iota$, where $F$ is the Fano surface of conics in a Verra threefold and $\iota$ is a certain involution ($p_g(S)=36$);
\item{(\romannumeral7)} the surface of bitangents $S$ of a general quartic in $\mathbb{P}^3$ ($p_g(S)=45$);
\item{(\romannumeral8)} the singular locus $S$ of a general EPW sextic ($p_g(S)=45$).
\end{nonumbering}
A by--product of the proof is that these surfaces all have finite--dimensional motive, in the sense of Kimura \cite{Kim} (this appears to be a new observation for cases (\romannumeral6)--(\romannumeral8)). Also,
certain instances of the generalized Hodge conjecture are verified:
\begin{nonumberingc}[=Corollary \ref{ghc}] Let $S$ be any of the above surfaces, and let $m>p_g(S)$. Then the sub--Hodge structure
\[ \wedge^m H^2(S,\mathbb{Q})\ \subset\ H^{2m}(S^m,\mathbb{Q}) \]
is supported on a divisor.
\end{nonumberingc}
The surfaces considered in this note have an interesting feature in common (which makes it easy to prove Conjecture \ref{conj} for them): for many of them, intersection product induces a surjection
\[ A^1_{hom}(S)\otimes A^1_{hom}(S)\ \twoheadrightarrow\ A^2_{AJ}(S)\ .\]
In the other cases (cases (\romannumeral2), (\romannumeral4), (\romannumeral6)--(\romannumeral8), which have $q(S)=0$), the surface $S$ is dominated by a surface $T$ with the property that the intersection product map
\[ A^1_{hom}(T)\otimes A^1_{hom}(T)\ \to\ A^2_{AJ}(T)\ \]
surjects onto $\operatorname{i}a \bigl( A^2_{AJ}(S)\to A^2_{AJ}(T)\bigr)$.
Using this feature, to prove Conjecture \ref{conj} for these surfaces one is reduced to a problem concerning $0$--cycles on abelian varieties. This last problem has recently been solved by Vial \cite{Ch}, using a strong version of the generalized Hodge conjecture for generic abelian varieties.
\vskip0.6cm
\begin{convention} In this note, the word {\sl variety\/} will refer to a reduced irreducible scheme of finite type over $\mathbb{C}$. A {\sl subvariety\/} is a (possibly reducible) reduced subscheme which is equidimensional.
{\bf Unless indicated otherwise, all Chow groups will be with rational coefficients}: we will denote by $A_j(X)$ the Chow group of $j$--dimensional cycles on $X$ with $\mathbb{Q}$--coefficients (and by $A_j(X)_{\mathbb{Z}}$ the Chow groups with $\mathbb{Z}$--coefficients); for $X$ smooth of dimension $n$ the notations $A_j(X)$ and $A^{n-j}(X)$ are used interchangeably.
The notations $A^j_{hom}(X)$, $A^j_{AJ}(X)$ will be used to indicate the subgroups of homologically trivial, resp. Abel--Jacobi trivial cycles.
The contravariant category of Chow motives (i.e., pure motives with respect to rational equivalence as in \cite{Sc}, \cite{MNP}) will be denoted $\mathcal M_{\rm rat}$.
We will write $H^j(X)$
to indicate singular cohomology $H^j(X,\mathbb{Q})$.
\end{convention}
\section{Generalized Burniat type surfaces with $p_g=3$}
\begin{definition}[\cite{BCF}]\label{gbt} Let $A=E_1\times E_2\times E_3$ be a product of elliptic curves. A {\em generalized Burniat type surface\/} (or ``GBT surface'')
is a quotient $S=Y/G$, where $Y\subset A$ is a smooth hypersurface corresponding to the square of a principal polarization, and $G\cong \mathbb{Z}_2^3$ acts freely.
\end{definition}
\begin{remark} GBT surfaces are minimal surfaces of general type with $p_g(S)=q(S)$ ranging from $0$ to $3$. There are $16$ irreducible families of GBT surfaces, labelled $\mathcal S_1,\ldots \mathcal S_{16}$ in \cite{BCF}. The families $\mathcal S_1, \mathcal S_2$ have moduli--dimension $4$, the other families are $3$--dimensional.
\end{remark}
\begin{theorem}[Peters \cite{Chris}]\label{Gbt} Let $S$ be a GBT surface with $p_g(S)=3$ (i.e., $S$ is in the family labelled $\mathcal S_{16}$ in \cite{BCF}), and let
$A$ be the abelian threefold as in definition \ref{gbt}.
\noindent
(\romannumeral1)
$S$ has finite--dimensional motive, and there are natural isomorphisms
\[ A^2_{(2)}(A)\ \xrightarrow{\cong}\ A^2_{AJ}(S)\ \xrightarrow{\cong}\ A^3_{(2)}(A)\ .\]
(Here $A^\ast_{(\ast)}(A)$ refers to Beauville's decomposition \cite{Beau}.)
\noindent
(\romannumeral2) Intersection product induces a surjection
\[ A^1_{hom}(S)\otimes A^1_{hom}(S)\ \twoheadrightarrow\ A^2_{AJ}(S)\ .\]
\end{theorem}
\begin{proof} Part (\romannumeral1) is \cite[Theorem 4.2]{Chris}.
Part (\romannumeral2) follows from (\romannumeral1), in view of the fact that intersection product induces a surjection
\[ A^1_{hom}(A)\otimes A^1_{hom}(A)\ \twoheadrightarrow\ A^2_{(2)}(A) \ \]
\cite[Proposition 4]{Beau}.
\end{proof}
Property (\romannumeral2) of Theorem \ref{Gbt} is relevant to Conjecture \ref{conj}:
\begin{proposition}\label{handy0} Let $S$ be a smooth projective surface, and assume that intersection product induces a surjection
\[ A^1_{hom}(S)\otimes A^1_{hom}(S)\ \twoheadrightarrow\ A^2_{AJ}(S)\ .\]
Then $S$ has finite--dimensional motive.
Also, Conjecture \ref{conj} is true for $S$ with $m>{q(S)\choose 2}$.
(In particular, in case of equality $p_g(S)= {q(S)\choose 2}$ the full Conjecture \ref{conj} is true for $S$.)
\end{proposition}
\begin{proof} Let $\alpha\colon S\to A:=\hbox{Alb}(S)$ be the Albanese map. There is a commutative diagram
\[ \begin{array}[c]{ccc}
A^1_{hom}(S)\otimes A^1_{hom}(S) &\to& A^2_{AJ}(S)\\
&&\\
\ \ \ \ \uparrow{\scriptstyle (\alpha^\ast,\alpha^\ast)}&& \ \ \ \ \uparrow{\scriptstyle \alpha^\ast}\\
&&\\
A^1_{hom}(A)\otimes A^1_{hom}(A) &\to& A^2_{(2)}(A)\\
\end{array} \]
(where horizontal maps are induced by intersection product, and $A^\ast_{(\ast)}(A)$ refers to the Beauville decomposition \cite{Beau} of the Chow ring of any abelian variety). As the left vertical map is an isomorphism, the assumption implies that the right vertical map is surjective. In view of \cite[Theorem 3.11]{V3}, this implies $S$ has finite--dimensional motive. (For an alternative proof of \cite[Theorem 3.11]{V3} in terms of birational motives, cf. \cite[Theorem B.7]{LNP}. For a similar result, cf. \cite[Proposition 2.1]{Diaz}.)
Next, let us consider Conjecture \ref{conj} for $S$. Thanks to Rojtman's result \cite{Ro}, it suffices to establish Conjecture \ref{conj} for $0$--cycles with $\mathbb{Q}$--coefficients.
Because $\alpha^\ast\colon A^2_{(2)}(A)\to A^2_{AJ}(S)$ is surjective, to prove Conjecture \ref{conj} for $S$ it suffices to prove (a version of) Conjecture \ref{conj} for elements $b_1,\ldots,b_m\in A^2_{(2)}(A)$. We now reduce to $0$--cycles on $A$: given $b_j\in A^2_{(2)}(A)$, let
\[ c_j:= b_j\cdot h^{q-2}\ \ \in\ A^q_{(2)}(A)\ ,\ \ \ j=1,\ldots,m\ ,\]
be $0$--cycles, where $q:=q(S)$ is the dimension of $A$ and $h\in A^1(A)$ is a symmetric ample divisor.
Let us consider the $\mathfrak S_m$--invariant ample divisor
\[ H:= \sum_{j=1}^m (pr_j)^\ast(h)\ \ \ \in\ A^1(A^m)\ .\]
From K\"unnemann's hard Lefschetz result \cite{Kun}, we know that the map
\[ \cdot H^{m(q-2)}\colon\ \ A^{2m}_{(2m)}(A^m)\ \to\ A^{qm}_{(2m)}(A^m) \]
is an isomorphism. On the other hand,
\[ \begin{split}
c_{\sigma(1)}\times\cdots\times c_{\sigma(m)}&= \bigl(b_{\sigma(1)}\times\cdots\times b_{\sigma(m)} \bigr)\cdot \bigl( h^{q-2}\times\cdots\times h^{q-2}\bigr)\\
&= \bigl(b_{\sigma(1)}\times\cdots\times b_{\sigma(m)} \bigr)\cdot H^{m(q-2)}\ \ \ \hbox{in}\ A^{qm}_{(2m)}(A^m)\\
\end{split}\]
(since intersecting $A^2(A)$ with a power $h^r, r>q-2$ gives $0$).
We are thus reduced to proving that for any $c_1,\ldots,c_m\in A^q_{(2)}(A)$, where $m>{q\choose 2}$, there is equality
\[ \sum_{\sigma\in\mathfrak S_m} \hbox{sgn}(\sigma) \, c_{\sigma(1)}\times\cdots\times c_{\sigma(m)}=0\ \ \ \hbox{in}\ A^{gm}(A^m)_{}\ .\]
At this point, we can invoke the following general result on $0$--cycles on abelian varieties to conclude:
\begin{theorem}[Vial \cite{Ch}] Let $A$ be an abelian variety of dimension $g$, and let $c_1,\ldots,c_m\in A^g_{(k)}(A)$.
If $k$ is even and $m>{g\choose k}$, there is vanishing
\[ \sum_{\sigma\in\mathfrak S_m} \hbox{sgn}(\sigma) \, c_{\sigma(1)}\times\cdots\times c_{\sigma(m)}=0\ \ \ \hbox{in}\ A^{mg}(A^m)_{}\ .\]
If $k$ is odd and $m>{g\choose k}$, there is vanishing
\[ \sum_{\sigma\in\mathfrak S_m} c_{\sigma(1)}\times\cdots\times c_{\sigma(m)}=0\ \ \ \hbox{in}\ A^{mg}(A^m)_{}\ .\]
\end{theorem}
\begin{proof} This is \cite[Theorem 4.1]{Ch}, whose proof uses the concept of ``generically defined cycles on abelian varieties'', and a strong form of the generalized Hodge conjecture for powers of generic abelian varieties, due to Hazama \cite[Theorem 2.12]{Ch}. The case $k=g$ was proven earlier (and differently) in
\cite[Example 4.40]{Vo}.
\end{proof}
This ends the proof of Proposition \ref{handy0}.
\end{proof}
We can now prove that surfaces in the family $\mathcal S_{16}$ verify Voisin's conjecture:
\begin{corollary}\label{main1} Let $S$ be a GBT surface with $p_g(S)=3$ (i.e., $S$ is in the family labelled $\mathcal S_{16}$ in \cite{BCF}).
Then $S$ verifies Conjecture \ref{conj}: for any $m>3$ and $a_1,\ldots,a_m\in A^2_{AJ}(S)$, there is equality
\[ \sum_{\sigma\in\mathfrak S_m} \hbox{sgn}(\sigma) a_{\sigma(1)}\times\cdots\times a_{\sigma(m)}=0\ \ \ \hbox{in}\ A^{2m}(S^m)\ .\]
\end{corollary}
\begin{proof} This follows from Proposition \ref{handy0}, in view of Theorem \ref{Gbt} plus the fact that $q(S)=p_g(S)=3$.
\end{proof}
We recall that the truth of Conjecture \ref{conj} implies a certain instance of the generalized Hodge conjecture:
\begin{corollary}\label{ghc} Let $S$ be a surface verifying Conjecture \ref{conj}, and let $m>p_g(S)$. Then the sub--Hodge structure
\[ \wedge^m H^2(S,\mathbb{Q})\ \subset\ H^{2m}(S^m,\mathbb{Q}) \]
is supported on a divisor.
\end{corollary}
\begin{proof} This is already observed in \cite{V9}. Consider the Chow motive $\wedge^m h^2(S)$ defined by the idempotent
\[ \Gamma:= \bigl(\sum_{\sigma\in\mathfrak S_m} \hbox{sgn}(\sigma) \Gamma_\sigma\bigr)\circ \bigl(\pi^2_S\times\cdots\times \pi^2_S\bigr)\ \ \ \in\ A^{2m}(S^m\times S^m)\ .\]
Conjecture \ref{conj} is equivalent to saying that $A_0(\wedge^m h^2(S))=0$.
Applying the Bloch--Srinivas argument \cite{BS} to $\Gamma$, one obtains a rational equivalence
\[ \Gamma=\gamma\ \ \ \hbox{in}\ A^{2m}(S^m\times S^m)\ ,\]
where $\gamma$ is a cycle supported on $S^m\times D$ for some divisor $D\subset S^m$.
On the other hand, $\Gamma$ acts on $H^{2m}(S^m,\mathbb{Q})$ as projector on $\wedge^m H^2(S,\mathbb{Q})$. It follows that $ \wedge^m H^2(S,\mathbb{Q})$ is supported on $D$.
\end{proof}
\section{A criterion}
The approach of the last section can be conveniently rephrased as follows:
\begin{proposition}\label{handy} Let $S$ be a smooth projective surface. Assume that $S$ has finite--dimensional motive, and that cup product induces an isomorphism
\[ C\colon\ \ \wedge^2 H^1(S,\mathcal O_S) \ \xrightarrow{\cong}\ H^2(S,\mathcal O_S)\ .\]
Then Conjecture \ref{conj} is true for $S$.
\end{proposition}
\begin{proof} Surjectivity of $C$, combined with finite--dimensionality of the motive of $S$, ensures that intersection product induces a surjection
\[ A^1_{hom}(S)\otimes A^1_{hom}(S)\ \twoheadrightarrow\ A^2_{AJ}(S)\ \]
\cite{moib}. The assumption that $C$ is an isomorphism implies that $p_g(S)={{q(S)}\choose{2}}$. The result now follows from Proposition \ref{handy0}.
\end{proof}
This takes care of two more cases announced in the introduction:
\begin{corollary}\label{cor2} Conjecture \ref{conj} is true for the following surfaces:
\item
{(\romannumeral1)} minimal surfaces of general type with $p_g(S)=q(S)=3$ and $K^2=6$;
\item
{(\romannumeral2)} the Fano surface of lines in a cubic threefold ($p_g(S)=10$).
\end{corollary}
\begin{proof}
In case (\romannumeral1), it is known that $S$ is the symmetric square $S=C^{(2)}$ where $C$ is a genus $3$ curve \cite{CCML} (cf. also \cite[Theorem 9]{BCP}). Thus, the assumptions of Proposition \ref{handy} are clearly satisfied.
As for case (\romannumeral2), it is well--known this satisfies the assumptions of Proposition \ref{handy} (finite--dimensionality is proven in \cite{Diaz} and \cite{22}). Alternatively, one could apply Proposition \ref{handy0} directly (the assumption of Proposition \ref{handy0} is satisfied by the Fano surface thanks to \cite{B}; an alternative proof is sketched in \cite[Remark 20.8]{SV}).
\end{proof}
\section{A variant criterion}
Let us now state a variant version of Proposition \ref{handy0}:
\begin{proposition}\label{handy1} Let $S$ be a smooth projective surface. Assume that $S=S^\prime/<\iota>$, where $\iota$ is an automorphism of a surface $S^\prime$
such that intersection product induces a surjection
\[ A^1_{hom}(S^\prime)\otimes A^1_{hom}(S^\prime) \ \twoheadrightarrow\ A^2_{AJ}(S^\prime)^\iota\ .\]
Then $S$ has finite--dimensional motive.
Also, Conjecture \ref{conj} is true for $S$ with $m>{q(S^\prime)\choose 2}$.
(In particular, if $p_g(S)={q(S^\prime)\choose 2}$ the full Conjecture \ref{conj} is true for $S$.)
\end{proposition}
\begin{proof} This is proven just as Proposition \ref{handy0}.
\end{proof}
This takes care of several more cases announced in the introduction:
\begin{corollary}\label{cor4} Conjecture \ref{conj} is true for the following surfaces:
\item
{(\romannumeral1)} surfaces $S=T/<\iota>$, where $T$ is a smooth divisor in the linear system $\vert 2\Theta\vert$ on a principally polarized abelian threefold, and $\iota$ is the $(-1)$--involution ($p_g(S)=3$);
\item{(\romannumeral2)} the quotient $S=F/\iota$, where $F$ is the Fano surface of conics in a general Verra threefold and $\iota$ is a certain involution ($p_g(S)=36$);
\item{(\romannumeral3)} the surface of bitangents $S$ of a general quartic in $\mathbb{P}^3$ ($p_g(S)=45$);
\item{(\romannumeral4)} the surface $S$ that is the singular locus of a general EPW sextic ($p_g(S)=45$).
\end{corollary}
\begin{proof}
\noindent
\item{(\romannumeral1)} The surface $S$ verifies the assumptions of Proposition \ref{handy1} with $S^\prime=T$, according to \cite[Subsection 7.2]{LNP}.
\noindent
\item{(\romannumeral3)} More generally, one may consider the surface $S$ studied by Welters \cite{Wel} and defined as follows. Let $Y$ be a {\em quartic double solid\/}, i.e.
$Y\to\mathbb{P}^3$ is a double cover branched along a smooth quartic $Q$. Let $T$ be the surface of conics contained in $Y$, and let $\iota\in\aut(T)$ be the involution induced by the covering involution of $Y$.
Then the surface $S:=T/<\iota>$ is a smooth surface of general type with $p_g(S)=45$.
(The generic quartic $K3$ surface $Q$ does not contain a line. In this case, as explained in \cite{Fer} (cf. also \cite[Example 3.5]{Beau1} and \cite[Remark 8.5]{Huy1}), the surface $S$ is (isomorphic to) the so--called ``surface of bitangents'', which is the fixed locus of Beauville's anti--symplectic involution
\[ Q^{[2]}\ \to\ Q^{[2]} \]
first considered in \cite{Beau0}. As noted in \cite[Example 3.5]{Beau1}, this gives another proof of the fact that $p_g(S)=45$.)
Voisin has proven \cite[Corollaire 3.2(b)]{V8} (cf. also \cite[Remarque 3.4]{V8}) that intersection product induces a surjection
\[ A^1_{hom}(T)\otimes A^1_{hom}(T)\ \twoheadrightarrow\ A^2_{AJ}(T)^\iota=A^2_{AJ}(S)\ .\]
Since $p_g(S)=45$ and $q(T)=10$ \cite{Wel}, the assumptions of Proposition \ref{handy1} are met with.
\noindent
\item
{(\romannumeral2)} A {\em Verra threefold\/} $Y$ is a divisor of bidegree $(2,2)$ in $\mathbb{P}^2\times\mathbb{P}^2$ (these varieties were introduced in \cite{Ver}). Let $F$ be the Fano surface of conics of bidegree $(1,1)$ contained in $Y$. As observed in \cite[Section 5]{IKKR}, $F$ admits an involution $\iota$ such that $(F,\iota)$ enters into the set--up of Voisin's work \cite{V8}. Thus, \cite[Corollaire 3.2(b)]{V8} implies that intersection product induces a surjection
\[ A^1_{hom}(F)\otimes A^1_{hom}(F)\ \twoheadrightarrow\ A^2_{AJ}(F)^\iota=A^2_{AJ}(S)\ .\]
Since $q(F)=9$ and $p_g(S)=36$ \cite[Proposition 5.1]{IKKR}, the assumptions of Proposition \ref{handy1} are again met with.
\noindent
\item
{(\romannumeral4)} Let $Y$ be a transverse intersection of the Grassmannian $Gr(2,5)\subset\mathbb{P}^9$ with a codimension $2$ linear subspace and a quadric (i.e., $Y$ is an {\em ordinary Gushel--Mukai threefold\/}, in the language of \cite{DK}, \cite{DK1}). For generic $Y$, the surface $F$ of conics contained in $Y$ is smooth and irreducible.
There exists a birational involution $\iota\in\hbox{Bir}(F)$, such that intersection product induces a surjection
\[ A^1_{hom}(F)\otimes A^1_{hom}(F)\ \twoheadrightarrow\ A^2_{AJ}(F)^\iota\ \]
\cite[Corollaire 3.2(b)]{V8}. The surface $F$ and the birational involution $\iota$ are also studied in \cite{Lo} and \cite{DIM}. There exists a (geometrically meaningful) birational morphism $F\to F_m$, where $F_m$ is smooth and such that $\iota$ extends to a morphism $\iota_m$ on $F_m$ \cite{Lo}, \cite[Section 6]{DIM}, \cite[Section 5.1]{IM}. For $Y$ generic, the quotient $S:=F_m/<\iota_m>$ is smooth, and it is isomorphic to the singular locus of the EPW sextic associated to $Y$.
(This is contained in \cite{Lo}, \cite{DIM}. The double cover $F_m\to S$ is also described in \cite[Theorem 5.2(2)]{DK3}.)
Since $A^1_{hom}(), A^2_{AJ}()$ are birational invariants among smooth varieties, Voisin's result implies there is also a surjection
\[ A^1_{hom}(F_m)\otimes A^1_{hom}(F_m)\ \twoheadrightarrow\ A^2_{AJ}(F_m)^{\iota_m}=A^2_{AJ}(S)\ .\]
It is known that $q(F_m)=10$ \cite{Lo} and $p_g(S)=45$ \cite{OG0} (this can also be deduced from \cite{Beau1}), and so Proposition \ref{handy1} applies.
\end{proof}
\begin{remark} In cases (\romannumeral2), (\romannumeral3) and (\romannumeral4) of Corollary \ref{cor4}, the surface $S$ is the fixed locus of an anti--symplectic involution of a hyperk\"ahler fourfold. For the surface of bitangents, this is Beauville's involution on the Hilbert square $Q^{[2]}$.
For the singular locus $S$ of a general EPW sextic, this is (isomorphic to) the fixed locus of the anti--symplectic involution of the associated double EPW sextic.
For the surface $S$ of (\romannumeral2), this is the anti--symplectic involution of the ``double EPW quartic'' (double EPW quartics form a $19$--dimensional family of hyperk\"ahler fourfolds, introduced in \cite{IKKR}).
Is this merely a coincidence, or is there something fundamental going on ? Do other two--dimensional fixed loci of anti--symplectic involutions of hyperk\"ahler fourfolds
also enter in the set--up of Proposition \ref{handy1} ?
\end{remark}
\begin{remark} Inspired by the famous results concerning the Fano surface of the cubic threefold, Voisin \cite{V8} systematically studies the Fano surface $F$ of conics contained in Fano threefolds $Y$. Under certain conditions, she is able to prove \cite[Corollaire 3.2]{V8} that there is a birational involution $\iota$ on $F$, with the property that
\[ A^1_{hom}(F)\otimes A^1_{hom}(F)\ \to\ A^2_{AJ}(F)^{<\iota>} \]
is surjective (and so one could hope to apply Proposition \ref{handy1} to find more examples of surfaces verifying Conjecture \ref{conj}).
Examples given in \cite{V8} (other than those mentioned in Corollary \ref{cor4} above) include:
\noindent
\item{(1)}
Fano threefolds $Y$ of index $1$, Picard number $1$ and genus $g\in[7,10]\cup\{12\}$ \cite[Section 2.4]{V8};
\noindent
\item{(2)}
a general complete intersection of two quadrics in $\mathbb{P}^5$ \cite[Section 2.7]{V8};
\noindent
\item{(3)}
the intersection of the Grassmannian $Gr(2,5)\subset\mathbb{P}^9$ with a general codimension $3$ linear subspace \cite[Section 2.7]{V8}.
(In all these cases, $\iota$ is actually the identity.)
In case (1), the surface of conics $F$ is not very interesting. (for $g=12$, $F\cong\mathbb{P}^2$ \cite[Proposition B.4.1]{KPS}; for $g=10$, $F$ is an abelian surface \cite[Proposition B.5.5]{KPS}; ; for $g=9$, $F$ is a $\mathbb{P}^1$--bundle over a curve \cite[Proposition 2.3.6]{KPS}; for $g=8$, $F$ is isomorphic to the Fano surface of a cubic threefold \cite[Proposition B.6.1]{KPS}; for $g=7$, $F$ is the symmetric product of a curve of genus $7$ \cite{Kuz05}. These results are also discussed in \cite[Section 3.1]{IM0}.)
The other two cases also turn out to reduce to known cases: Indeed, for case (2) the Fano surface of lines is isomorphic to the Jacobian of a genus $2$ curve \cite[Theorem 2]{DR}. For case (3), the Fano threefold $Y$ is birational to a cubic threefold $Y^\prime$, and the Fano surface of conics on $Y$ is birational to the Fano surface of lines on
$Y^\prime$ \cite[Theorem B and Section 6]{Puts}. Since Conjecture \ref{conj} is obviously a birationally invariant statement, Conjecture \ref{conj} for the Fano surface of case (3) thus reduces to Corollary \ref{cor2}(\romannumeral2).
\end{remark}
\begin{remark} There are interesting relations between the surfaces of Corollary \ref{cor4} and other Fano surfaces:
In case (\romannumeral2), the general Verra threefold $Y$ is birational to a one--nodal ordinary Gushel--Mukai threefold $\bar{X}$, and there is an induced birational map between the Fano surface of lines $F(Y)$ and the Fano surface of conics $F(\bar{X})$ \cite[Section 5.4 and Proposition 6.6]{DIM2}.
In case (\romannumeral3), the general quartic double solid $Y$ is known to be birational to a one--nodal ordinary degree $10$ Fano threefold $\bar{X}$, and there is an induced birational map between the Fano surface of lines $F(Y)$ and the Fano surface of conics $F(\bar{X})$ \cite[Proposition 5.2]{DIM}.
\end{remark}
\section{Double covers of cubic surfaces}
\begin{theorem}[Ikeda \cite{Ike}]\label{ike} Let $Y\subset\mathbb{P}^3$ be a smooth cubic surface, and let $\bar{S}\to Y$ be the double cover of $Y$ branched along its Hessian. Let $S\to\bar{S}$ be a minimal resolution of singularities. The surface $S$ is a minimal surface of general type with $p_g(S)=4$ and $K^2=6$.
\end{theorem}
\begin{remark} The intersection of $Y$ with its Hessian is smooth (and so $S=\bar{S}$) precisely when $Y$ has no Eckardt points. In this case, the Picard rank of $S$ is $28$ \cite[Theorem 6.1]{Ike}. At the other extreme, if $Y$ is the Fermat cubic (which is the only cubic surface attaining the maximal number of Eckardt points) the Picard rank of $S$ is $44$ \cite[Theorem 6.6]{Ike}, and so in this case $S$ is a $\rho$--maximal surface (in the sense of \cite{Beau3}). For more on Eckardt points of cubic surfaces, cf. \cite[Chapter 2 Section 3.6]{Huy}.
\end{remark}
Let us now prove Voisin's conjecture for some of Ikeda's double covers:
\begin{corollary}\label{last} Let $Y\subset\mathbb{P}^3$ be a smooth cubic surface, and let $S$ be a double cover as in theorem \ref{ike}.
Assume that $Y$ is in the pencil
\[ x_0^3 + x_1^3 +x_2^3 -3\lambda x_0 x_1 x_2 + x_3^3 =0 \ .\]
Then $S$ verifies Conjecture \ref{conj}: for any $m>4$ and $a_1,\ldots,a_m\in A^2_{hom}(S)_{\mathbb{Z}}$, there is equality
\[ \sum_{\sigma\in\mathfrak S_m} \hbox{sgn}(\sigma) a_{\sigma(1)}\times\cdots\times a_{\sigma(m)}=0\ \ \ \hbox{in}\ A^{2m}(S^m)_{\mathbb{Z}}\ .\]
\end{corollary}
\begin{proof} A first part of the argument works for arbitrary smooth cubic surfaces $Y$; only in the last step will we use that $Y$ is of a specific type.
Let us assume $Y\subset\mathbb{P}^3$ is any smooth cubic, defined by a cubic polynomial $f(x_0,\ldots,x_3)$. Let $Z\subset\mathbb{P}^4$ be the smooth cubic threefold defined by
\[ f(x_0,\ldots,x_3)+x_4^3=0\ ,\]
so $Z$ has the structure of a triple cover
\[ \rho\colon\ \ Z\ \to\ \mathbb{P}^3 \]
branched along $Y$.
Let $F(Z)$ denote the Fano surface of lines contained in $Z$. Ikeda \cite{Ike} shows that there is a dominant rational map of degree $3$
\[ f\colon\ \ F(Z)\ \dashrightarrow\ S \ ,\]
and an isomorphism
\[ f^\ast\colon\ \ H^2_{tr}(S,\mathbb{Q})\ \xrightarrow{\cong}\ H^2_{tr}(F(Z),\mathbb{Q})^{Gal(\rho)}\ .\]
This implies that there is an isomorphism of homological motives
\begin{equation}\label{homiso} {}^t \Gamma_f\colon\ \ \ t(S)\ \xrightarrow{\cong}\ t(F(Z))^{Gal(\rho)}:=(F(Z),{1\over 3}\sum_{g\in Gal(\rho)} \Gamma_g\circ \pi^2_{tr},0)\ \ \ \hbox{in}\ \mathcal M_{\rm hom}\ .\end{equation}
(Here for any surface $T$, the motive $t(T):=(T,\pi^2_{tr},0)\in\mathcal M_{\rm rat}$ denotes the {\em transcendental part of the motive\/} as in \cite{KMP}.)
According to \cite{Diaz} and \cite{22}, the Fano surface $F(Z)$ has finite--dimensional motive (in the sense of Kimura \cite{Kim}, \cite{An}, \cite{J4}). The surface $S$, being rationally dominated by $F(Z)$, also has finite--dimensional motive. Thus, one may upgrade (\ref{homiso}) to an isomorphism of Chow motives
\[ {}^t \Gamma_f\colon\ \ \ t(S)\ \xrightarrow{\cong}\ t(F(Z))^{Gal(\rho)}\ \ \ \hbox{in}\ \mathcal M_{\rm rat}\ .\]
In particular, this implies that there is an isomorphism of Chow groups
\[ f^\ast\colon A^2_{hom}(S)=A^2_{AJ}(S)\ \xrightarrow{\cong}\ A^2_{AJ}(F(Z))^{Gal(\rho)}\ .\]
Let $A$ be the 5--dimensional Albanese variety of $F(Z)$ (which is isomorphic to the intermediate Jacobian of $Z$). As observed in \cite{Diaz}, the inclusion $F(Z)\hookrightarrow A$ induces an isomorphism
\[ A^2_{(2)}(A)\cong A^2_{AJ}(F(Z))\ .\]
In particular, there is a restriction--induced isomorphism
\[ A^2_{(2)}(A)^{Gal(\rho)}\cong A^2_{AJ}(F(Z))^{Gal(\rho)}\ ,\]
where we simply use the same letter $\rho$ for the action induced by the triple cover $\rho\colon Z\to\mathbb{P}^3$.
Consequently, it suffices to prove a version of Conjecture \ref{conj} for cycles in $ A^2_{(2)}(A)^{Gal(\rho)}$. Also, using K\"unnemann's hard Lefschetz theorem (for some $Gal(\rho)$--invariant ample divisor), one reduces to a statement for cycles in $ A^5_{(2)}(A)^{Gal(\rho)}$ (i.e., $0$--cycles). This last statement can be proven, subject to some restrictions on the cubic surface $Y$, thanks to the following result:
\begin{proposition}[Vial \cite{Ch}]\label{factors} Let $B$ be an abelian variety of dimension $g$, and assume $B$ is isogenous to $ E_1^{r_1}\times E_2^{r_2}\times E_3^{r_3}$, where the $E_j$ are elliptic curves. Let $\Gamma\in A^g(B\times B)$ be an idempotent which lies in the sub--algebra generated by symmetric divisors. Assume that $\Gamma^\ast H^{j,0}(B)=0$ for all $j$. Then also
\[ \Gamma_\ast A^g(B)=0\ .\]
\end{proposition}
\begin{proof} This is a special case of \cite[Theorem 3.15]{Ch}, whose hypotheses are more general.
\end{proof}
It remains to verify that Proposition \ref{factors} applies to our set--up. If the cubic threefold $Z=Z_\lambda$ is in the pencil
\[ x_0^3 + x_1^3 +x_2^3 -3\lambda x_0 x_1 x_2 + x_3^3 +x_4^4=0 \ ,\]
its intermediate Jacobian $A$ is isogenous to $E_0^3\times E_\lambda^2$, where $E_\lambda$ is the elliptic curve
\[ x_0^3 + x_1^3 +x_2^3 -3\lambda x_0 x_1 x_2=0\ \]
\cite{Rou}.
We can apply Proposition \ref{factors} with $B:=A^m$ and
\[ \Gamma:= \bigl(\sum_{g\in Gal(\rho)} \Gamma_g\times \cdots \times\Gamma_g\bigr) \circ \bigl(\sum_{\sigma\in \mathfrak S_m} \hbox{sgn}(\sigma)\,\Gamma_\sigma\bigr) \circ \bigl( \pi^8_A\times \cdots \times \pi^8_A\bigr) \ \ \ \in A^{5m}(A^m\times A^m)\ .\]
Here $\pi^8_A$ is part of the Chow--K\"unneth decomposition of \cite{DM}, with the property that
\[ A^5_{(2)}(A)=(\pi^8_A)_\ast A^5(A)\ .\]
Since $g\in Gal(\rho)$ and $\sigma\in \mathfrak S_m$ are homomorphisms of abelian varieties, and the $\pi^8_A$ are symmetrically distinguished (in the sense of O'Sullivan \cite{OS}) and generically defined (in the sense of Vial \cite{Ch}), the correspondence $\Gamma$ is in the sub--algebra generated by symmetric divisors \cite[Proposition 3.11]{Ch}. In particular, the correspondence $\Gamma$ is symmetrically distinguished, and so (since it is idempotent in cohomology) idempotent.
The correspondence ${}^t \Gamma$ acts on cohomology as projector on
\[ \wedge^m \bigl( H^2(A)^{Gal(\rho)}\bigr)\ .\]
Since
\[ \dim \hbox{Gr}^0_F H^2(A)^{Gal(\rho)}=p_g(S)=4\ ,\]
we have that $\Gamma^\ast=({}^t \Gamma)_\ast$ is zero on $H^{j,0}(B)$ as soon as $m>4$. Applying Proposition \ref{factors}, we can prove Conjecture \ref{conj} for
$A^5_{(2)}(A)^{Gal(\rho)}$ (and hence, as explained above, also for $A^2_{AJ}(S)$): let $b_1,\ldots,b_m\in A^5_{(2)}(A)^{Gal(\rho)}$, where $m>4$. Then
\[ \sum_{\sigma\in\mathfrak S_m} \hbox{sgn}(\sigma)\, b_{\sigma(1)}\times b_{\sigma(2)}\times\cdots\times b_{\sigma(m)}=\Gamma_\ast (b_1\times b_2\times\cdots\times b_m)=0\ \ \ \hbox{in}\ A^{5m}(A^m)\ .\]
\end{proof}
\begin{remark} The argument of Corollary \ref{last} also applies to double covers of some other cubic surfaces. For instance, let $Y$ be a cubic surface, let $S$ be the double cover as in theorem \ref{ike}, and let $J(Z)$ be the intermediate Jacobian of the associated cubic threefold. If $J(Z)$ is $\rho$--maximal, then $S$ verifies conjecture \ref{conj}. Indeed, $\rho$--maximality implies that $J(Z)$ is isogenous to $E^5$ for some elliptic curve $E$ \cite[Proposition 3]{Beau3}, and so Proposition \ref{factors} applies.
\end{remark}
\vskip1cm
\begin{nonumberingt} Thanks to the wonderful staff of the Executive Lounge at the Schilik Math Research Institute.
\end{nonumberingt}
\vskip1cm
\end{document} | math |
# == Schema Information
#
# Table name: merge_requests
#
# id :integer not null, primary key
# target_branch :string(255) not null
# source_branch :string(255) not null
# source_project_id :integer not null
# author_id :integer
# assignee_id :integer
# title :string(255)
# created_at :datetime
# updated_at :datetime
# milestone_id :integer
# state :string(255)
# merge_status :string(255)
# target_project_id :integer not null
# iid :integer
# description :text
# position :integer default(0)
#
require Rails.root.join("app/models/commit")
require Rails.root.join("lib/static_model")
class MergeRequest < ActiveRecord::Base
include Issuable
include Taskable
include InternalId
belongs_to :target_project, foreign_key: :target_project_id, class_name: "Project"
belongs_to :source_project, foreign_key: :source_project_id, class_name: "Project"
has_one :merge_request_diff, dependent: :destroy
after_create :create_merge_request_diff
after_update :update_merge_request_diff
delegate :commits, :diffs, :last_commit, :last_commit_short_sha, to: :merge_request_diff, prefix: nil
attr_accessor :should_remove_source_branch
# When this attribute is true some MR validation is ignored
# It allows us to close or modify broken merge requests
attr_accessor :allow_broken
# Temporary fields to store compare vars
# when creating new merge request
attr_accessor :can_be_created, :compare_failed,
:compare_commits, :compare_diffs
state_machine :state, initial: :opened do
event :close do
transition [:reopened, :opened] => :closed
end
event :merge do
transition [:reopened, :opened, :locked] => :merged
end
event :reopen do
transition closed: :reopened
end
event :lock_mr do
transition [:reopened, :opened] => :locked
end
event :unlock_mr do
transition locked: :reopened
end
after_transition any => :locked do |merge_request, transition|
merge_request.locked_at = Time.now
merge_request.save
end
after_transition :locked => (any - :locked) do |merge_request, transition|
merge_request.locked_at = nil
merge_request.save
end
state :opened
state :reopened
state :closed
state :merged
state :locked
end
state_machine :merge_status, initial: :unchecked do
event :mark_as_unchecked do
transition [:can_be_merged, :cannot_be_merged] => :unchecked
end
event :mark_as_mergeable do
transition unchecked: :can_be_merged
end
event :mark_as_unmergeable do
transition unchecked: :cannot_be_merged
end
state :unchecked
state :can_be_merged
state :cannot_be_merged
end
validates :source_project, presence: true, unless: :allow_broken
validates :source_branch, presence: true
validates :target_project, presence: true
validates :target_branch, presence: true
validate :validate_branches
validate :validate_fork
scope :of_group, ->(group) { where("source_project_id in (:group_project_ids) OR target_project_id in (:group_project_ids)", group_project_ids: group.project_ids) }
scope :of_user_team, ->(team) { where("(source_project_id in (:team_project_ids) OR target_project_id in (:team_project_ids) AND assignee_id in (:team_member_ids))", team_project_ids: team.project_ids, team_member_ids: team.member_ids) }
scope :merged, -> { with_state(:merged) }
scope :by_branch, ->(branch_name) { where("(source_branch LIKE :branch) OR (target_branch LIKE :branch)", branch: branch_name) }
scope :cared, ->(user) { where('assignee_id = :user OR author_id = :user', user: user.id) }
scope :by_milestone, ->(milestone) { where(milestone_id: milestone) }
scope :in_projects, ->(project_ids) { where("source_project_id in (:project_ids) OR target_project_id in (:project_ids)", project_ids: project_ids) }
scope :of_projects, ->(ids) { where(target_project_id: ids) }
# Closed scope for merge request should return
# both merged and closed mr's
scope :closed, -> { with_states(:closed, :merged) }
scope :declined, -> { with_states(:closed) }
def validate_branches
if target_project == source_project && target_branch == source_branch
errors.add :branch_conflict, "You can not use same project/branch for source and target"
end
if opened? || reopened?
similar_mrs = self.target_project.merge_requests.where(source_branch: source_branch, target_branch: target_branch, source_project_id: source_project.id).opened
similar_mrs = similar_mrs.where('id not in (?)', self.id) if self.id
if similar_mrs.any?
errors.add :validate_branches,
"Cannot Create: This merge request already exists: #{
similar_mrs.pluck(:title)
}"
end
end
end
def validate_fork
return true unless target_project && source_project
if target_project == source_project
true
else
# If source and target projects are different
# we should check if source project is actually a fork of target project
if source_project.forked_from?(target_project)
true
else
errors.add :validate_fork,
'Source project is not a fork of target project'
end
end
end
def update_merge_request_diff
if source_branch_changed? || target_branch_changed?
reload_code
mark_as_unchecked
end
end
def reload_code
if merge_request_diff && open?
merge_request_diff.reload_content
end
end
def check_if_can_be_merged
if Gitlab::Satellite::MergeAction.new(self.author, self).can_be_merged?
mark_as_mergeable
else
mark_as_unmergeable
end
end
def merge_event
self.target_project.events.where(target_id: self.id, target_type: "MergeRequest", action: Event::MERGED).last
end
def closed_event
self.target_project.events.where(target_id: self.id, target_type: "MergeRequest", action: Event::CLOSED).last
end
def automerge!(current_user, commit_message = nil)
MergeRequests::AutoMergeService.
new(target_project, current_user).
execute(self, commit_message)
end
def open?
opened? || reopened?
end
def mr_and_commit_notes
# Fetch comments only from last 100 commits
commits_for_notes_limit = 100
commit_ids = commits.last(commits_for_notes_limit).map(&:id)
project.notes.where(
"(noteable_type = 'MergeRequest' AND noteable_id = :mr_id) OR (noteable_type = 'Commit' AND commit_id IN (:commit_ids))",
mr_id: id,
commit_ids: commit_ids
)
end
# Returns the raw diff for this merge request
#
# see "git diff"
def to_diff(current_user)
Gitlab::Satellite::MergeAction.new(current_user, self).diff_in_satellite
end
# Returns the commit as a series of email patches.
#
# see "git format-patch"
def to_patch(current_user)
Gitlab::Satellite::MergeAction.new(current_user, self).format_patch
end
def hook_attrs
attrs = {
source: source_project.hook_attrs,
target: target_project.hook_attrs,
last_commit: nil
}
unless last_commit.nil?
attrs.merge!(last_commit: last_commit.hook_attrs(source_project))
end
attributes.merge!(attrs)
end
def for_fork?
target_project != source_project
end
def project
target_project
end
# Return the set of issues that will be closed if this merge request is accepted.
def closes_issues
if target_branch == project.default_branch
issues = commits.flat_map { |c| c.closes_issues(project) }
issues += Gitlab::ClosingIssueExtractor.closed_by_message_in_project(description, project)
issues.uniq.sort_by(&:id)
else
[]
end
end
# Mentionable override.
def gfm_reference
"merge request !#{iid}"
end
def target_project_path
if target_project
target_project.path_with_namespace
else
"(removed)"
end
end
def source_project_path
if source_project
source_project.path_with_namespace
else
"(removed)"
end
end
def source_project_namespace
if source_project && source_project.namespace
source_project.namespace.path
else
"(removed)"
end
end
def target_project_namespace
if target_project && target_project.namespace
target_project.namespace.path
else
"(removed)"
end
end
def source_branch_exists?
return false unless self.source_project
self.source_project.repository.branch_names.include?(self.source_branch)
end
def target_branch_exists?
return false unless self.target_project
self.target_project.repository.branch_names.include?(self.target_branch)
end
# Reset merge request events cache
#
# Since we do cache @event we need to reset cache in special cases:
# * when a merge request is updated
# Events cache stored like events/23-20130109142513.
# The cache key includes updated_at timestamp.
# Thus it will automatically generate a new fragment
# when the event is updated because the key changes.
def reset_events_cache
Event.reset_event_cache_for(self)
end
def merge_commit_message
message = "Merge branch '#{source_branch}' into '#{target_branch}'"
message << "\n\n"
message << title.to_s
message << "\n\n"
message << description.to_s
message << "\n\n"
message << "See merge request !#{iid}"
message
end
# Return array of possible target branches
# dependes on target project of MR
def target_branches
if target_project.nil?
[]
else
target_project.repository.branch_names
end
end
# Return array of possible source branches
# dependes on source project of MR
def source_branches
if source_project.nil?
[]
else
source_project.repository.branch_names
end
end
def locked_long_ago?
locked_at && locked_at < (Time.now - 1.day)
end
end
| code |
जिंदगी का आखिरी सफरः रवीना टंडन के पिता का निधन, सामने आई अंतिम संस्कार की तस्वीरें बॉलीवुड इंडस्ट्री के मशहूर डायरेक्टर और एक्ट्रेस रवीना टंडन के पिता रवि टंडन का निधन हो गया है। उन्होंने 11 फरवरी यानि आज ही दोपहर को अपने जीवन की अंतिम सांस ली। रवि के निधन से उनके परिवार पर दुखों का पहाड़ टूट पड़ा है। वहीं फिल्म इंडस्ट्री में भी शोक की लहर दौड गई है। रवि टंडन का निधन आज ही मुंबई में किया जाएगा। हाल ही में उनके अंतिम संस्कार की तस्वीरें भी सामने आई हैं।सामने आई तस्वीरों में देखा जा सकता है कि 85 वर्षीय रवि टंडन का पार्थीव शरीर फूलों से सजाया गया है।उनके करीबी उनकी अर्थी को कंधा देते नजर आ रहे हैं। वहीं एक्ट्रेस और उनकी बेटी रवीना टंडन टूटे दिल से मटका और नारियल लिए रस्मों को निभा रही है।इस दौरान उनके चेहरे पर पापा को खोने का गम साफ देखा जा सकता है। | hindi |
তৃণমূলে যোগ দিতে চান আরও ৭ থেকে ৮ জন বিজেপি বিধায়ক, বিস্ফোরক দাবি মমতার তৃণমূল কংগ্রেসে যোগ দিতে চান আরও সাত থেকে আট জন বিজেপি বিধায়ক বুধবার এমনই দাবি করেছেন তৃণমূল কংগ্রেস সুপ্রিমো মমতা বন্দ্যোপাধ্যায় বুধবার পার্টির নির্বাচন ছিল তাতে বিনা প্রতিদ্বন্দ্বিতায় দলের সভানেত্রী নির্বাচিত হয়েছেন মমতা বন্দ্যোপাধ্যায় পার্টির নির্বাচনের সাক্ষী থাকার জন্য অন্যান্য রাজনৈতিক দলগুলিকেও উপস্থিত থাকার আমন্ত্রণ জানিয়েছিল তৃণমূল কংগ্রেস ফের দলের সভানেত্রী নির্বাচিত হওয়ার পর দলকে আরও শক্তিশালী করে গড়ে তোলার আহ্বান জানিয়েছেন তিনি source: oneindia.com | bengali |
राष्ट्रीयअंतरराष्ट्रीय खेल के लिए प्रशिक्षित होगी नाथनगर की बेटी, स्टेट एथलेटिक्स चैंपियनशिप में हासिल किया था गोल्ड मेडल जागरण संवाददाता, भागलपुर : गांव की खेतों में प्रैक्टिस कर स्टेट चैंपियनशिप जीतने वाली नाथनगर की बेटी खुशी कुमारी ने जिले का नाम रौशन किया है। 600 मीटर, अंडर14 वर्ग में खुशी का चयन हैदराबाद स्थित साईं सेंटर में राष्ट्रीय और अंतरराष्ट्रीय प्रतियोगिताओं के लिए प्रशिक्षण दिया जाएगा। उनके पिता दिलीप मंडल किसान हैं, जबकि मां बंदना देवी सिलाई का काम करती हैं। शनिवार को इसकी जानकारी उन्हें दी गई। खुशी गनौरा बादरपुर की रहने वाली है। वह आशादीप एथलेटिक्स क्लब में जितेंद्र मणि राकेश के सानिध्य में प्रशिक्षण लेती हैं।प्रशिक्षक जितेंद्र ने बताया कि यह भागलपुर के लिए बड़ी उपलब्धि वाली बात है। खुशी ने 2021 में मुजफ्फरपुर में आयोजित बिहार स्टेट एथलेटिक्स चैंपियनशिप के 600 और 60 मीटर की प्रतियोगिता में स्वर्ण पदक हासिल किया था। इसी आधार पर दिसंबर 2021 में खुशी को साईं सेंटर में प्रशिक्षण के लिए होने वाली प्रतियोगिता के ट्रायल के लिए बुलाया गया। उनके प्रशिक्षक से कई वीडियो भी मंगाए गए। उसकी प्रतिभा देख, ट्रायल में शामिल कराया गया। पूरे देश से इसमें चुनिंदा खिलाडिय़ों ने शिरकत की। जिसमें से केवल बिहार से खुशी और तेलांगना की एक एथलीट का चयन हुआ।अर्जन और द्रोणाचार्य अवार्डी खिलाड़ी करेंगे प्रशिक्षित बकौल जितेंद्र अब खुशी को राष्ट्रीय और अंतरराष्ट्रीय स्तर पर रिकार्ड बनाने वाले खिलाडिय़ों के साथ प्रशिक्षण का मौका मिलेगा। उसे द्रोणाचार्य और अर्जुन अवार्डी एथलीटों का सानिध्य मिलेगा। प्रशिक्षक के रूप में श्रीनिवास, महेश सूरी, नंदनी जैसे खिलाड़ी मिलेंगे। खुशी ने अपने चयन पर कहा है कि यह सफलता मातापिता के साथ उनके कोच को समर्पित हैं। जिनकी वजह से गांव की मिट्टी से सिंथेटिक ट्रैक पर प्रैक्टिस करने का मौका मिलेगा।बाल भारती ने पूर्ववर्ती छात्र के सम्मान में समारोह आयोजित कर सम्मानित कियाबाल भारती ने पूर्ववर्ती छात्र के सम्मान में समारोह आयोजित कर सम्मानित कियासंवाद सहयोगी, नवगछिया:बाल भारती ने पूर्ववर्ती छात्र के सम्मान में समारोह आयोजित कर सम्मानित किया। बाल भारती पोस्टआफिस रोड नवगछिया के पूर्ववर्ती छात्र अभिषेक कुमार को सम्मानित किया। विद्यालय के प्राचार्य मुरारी लाल पंसारी, कौशल किशोर जायसवाल व केसी मिश्रा व अन्य शिक्षकों के द्वारा श्री रामचरितमानस व घड़ी देकर सम्मानित किया।अभिषेक ने वर्ष 201718 सत्र में बोर्ड की सीबीएसई की परीक्षा 90 प्रतिशित अंको के साथ पास की थी। वे अपने पले ही प्रयास में नीट 2020 की परीक्षा पास की थी। नीट की परीक्षा में इन्होंने 536 वां रैंक हासिल कर अपने क्षेत्र व बाल भारती का नाम रौशन किया था। अभिषेक नयाटोला निवासी गोपाल चौरसियाव नीलम देवी का सपुत्र हैं। वर्तमान में वह एनएमसी के द्वूतीय वर्ष के छात्र हैं। उन्होंने अपने सफलता का श्रेय अपने माता पिता तथा विद्यालय के शिक्षकों को दिया। | hindi |
जींद : वायदाखिलाफी के विरोध में सडकों पर उतरे किसान जींद, 31 जनवरी हि.स.। किसान संयुक्त मोर्चा के आह्वान पर कृषि कानून, किसान समस्याओं, शिक्षा समेत अन्य मुद्दों को लेकर किसानों, मजदूरों, खाप प्रतिनिधियों व अन्य संगठनों ने शहर में प्रदर्शन किया। किसानों ने लघु सचिवालय के सामने प्रदर्शन किया और प्रधानमंत्री को संबोधित ज्ञापन डीसी नरेश नरवाल को सौंपा।संयुक्त किसान मोर्चा के आह्वान पर किसान संगठन, मजदूर, खाप प्रतिनिधि व अन्य संगठन नेहरू पार्क में एकत्रित हुए। वक्ताओं ने कहा कि नौ दिसम्बर को सरकार द्वारा तीन कृषि कानून को लेकर लिखित में दिए जाने पर आंदोलन स्थगित हुआ था। एमएसपी के लिए कमेटी का गठन करने जिसमें किसान संयुक्त मोर्चा प्रतिनिधियों को शामिल किया जाना था। किसान आंदोलन के दौरान देशभर में किसानों के खिलाफ दर्ज मुकद्दमों को वापस लेने पर सहमति बनी थी। अगर सरकार ने किसानों की मांगे नहीं मानी तो किसानों के पास सिवाए आंदोलन के कोई विकल्प नहीं बचता। वक्ताओं ने कहा कि कोरोना संक्रमण को लेकर दोहरे मापदंड अपनाए जा रहे हैं। पिछले दो वर्षो से शिक्षा प्रणाली ठप पडी हुई है। अन्य उपक्रम खुले हुए हैं, स्कूलों पर ताले पडे हुए हैं। बेरोजगारी में बहुत ज्यादा इजाफा हुआ है। नौकरियों को छीना जा रहा है, सरकार की गलत नीतियों के खिलाफ आवाज उठाने पर मुकद्दमें दर्ज किए जा रहे हैं। इस मौके पर भाकियू के रामफल कंडेला, टेकराम कंडेला, फूल सिंह श्योकंद, सतबीर पहलवान, ओमप्रकाश कंडेला, रामराजी ढूल पोकरीखेडी, कामरेड रमेश, भुल्ला देवी, समुंद्र फोर समेत काफी संख्या में किसान संगठन, मजदूर, खाप प्रतिनिधि मौजूद थे। भाकियू के राष्ट्रीय उपाध्यक्ष रामफल कंडेला ने कहा कि जिन बिंदूओं पर सहमति के बाद आंदोलन स्थगित हुआ था अब सरकार उन्हीं बिंदूओं पर वायदाखिलाफी कर रही है। भारतीय किसान सभा के उपाध्यक्ष फूलसिंह श्योकंद ने कहा कि किसान मुद्दों पर सरकार की सहमति के बाद आंदोलन स्थगित किया गया था। अब सरकार वायदाखिलाफी कर रही है और किसानों के साथ विश्वासघात भी। बजट सत्र शुरु हो रहा है सरकार को जिन मुद्दों पर सहमति बनी थी उन्हें लागू करना होगा। कंडेला खाप के अध्यक्ष ओमप्रकाश कंडेला ने कहा कि सरकार सहमति के बाद भी उन पर खरा नहीं उतर रही है। हिन्दुस्थान समाचार विजेंद्रसंजीव | hindi |
Our Discover Burnt Hickory Class and Reception is held several times a year in the Connections Room (2nd floor) from 11am-1:15pm.
Check-In begins at the Connection Cafe' at 10:45am. From there, everyone will move to room 286 for our 11am informational meeting. Your children should remain in BHBC Kids and Radiate at this time. At 12pm, we will break and move to the Connection Room for the reception. Children will stay in the BHBC Kids area and Students should meet their parents for the reception. During the reception, you will meet the pastors and staff, their families and learn about their ministries. We will conclude at 1:15pm, at which point you can make your way to the BHBC Kids area to pick up your children.
Please call the BHBC front office at 770.590.0334 if you have questions or wish to register over the phone. Please note that online registration will close at noon on Thursday, April 25. If you would like to attend after registration has closed, you may sign up the day of the event at 10am upstairs outside the Connections Room in the cafe. | english |
The A1 premium supermini is proving to be an astounding success for Audi. Just after one year since its launch and without any assistance from the United States or China, the 100,000th A1 rolled off the production line at the marque's Brussels facility in Belgium. The ice white milestone model was bearing the signatures of 2,400 workers in colours of the Belgian flag: yellow, red and black.
The milestone was so important that Audi invited King Albert II of Belgium to see off the 100,000th car and was treated to a factory tour with the factory's Rupert Stadler.
Audi obtained the Brussels complex from Volkswagen in 2007 and has since invested around €300 million to bring it up to date. It's understood that Audi will be expanding the A1 line-up with a 5dr, convertible, all-wheel-drive Quattro, and performance S1 variants. | english |
బెయిల్ ఇచ్చేటప్పుడు జాగ్రత్త నేర తీవ్రతను గమనించాలి: సుప్రీం కోర్టు న్యూఢిల్లీ, ఆగస్టు 24: నిందితులకు బెయిల్ మం జూరు చేసేటప్పుడు జాగ్రత్తలు పాటించాలని సర్వోన్న త న్యాయస్థానం హైకోర్టులను ఆదేశించింది. ఇందుకు కొన్ని కొలమానాలను కూడా నిర్దేశించింది. నిందితుడు నేరం చేశాడని విశ్వసించే ప్రాథమిక సాక్ష్యాధారాలు ఉన్నాయో లేదో పరిశీలించాలని సుప్రీంకోర్టు న్యాయమూర్తులు జస్టిస్ డీవై చంద్రచూడ్, జస్టిస్ ఎంఆర్ షాల ధర్మాసనం ఉత్తర్వులిచ్చింది. హత్యాభియోగం ఎదుర్కొంటున్న ఓ నిందితుడికి హైకోర్టు ఇచ్చిన బెయిల్ను రద్దు చేసింది. అతడు బెయిల్పై విడుదలైతే పిటిషనర్ భద్రత ప్రమాదంలో పడుతుందని అభిప్రాయపడింది. గతంలో కొన్ని కేసుల్లో అతడు బెయిల్పై బయటికొచ్చి మళ్లీ నేరాలకు పాల్పడ్డాడని, హైకోర్టు దీన్ని పరిగణనలోకి తీసుకోలేదని ఆక్షేపించింది. మరో కేసులో ఇదే హైకోర్టు వేరే నిందితుడికి బెయిల్ రద్దు చేసిన విషయాన్ని గుర్తు చేసింది. సుప్రీం కోర్టు కొలమానాలివీ.. నిందితుడు నేరం చేశాడని విశ్వసించేందుకు సహేతుక కారణాలు ఉన్నాయా.. ప్రాథమిక సాక్ష్యాధారాలు ఉన్నాయా? అభియోగం తీవ్రత, స్వభావం దోషిగా తేలితే ఎంత తీవ్ర శిక్ష పడొచ్చు? బెయిల్పై విడుదలైతే నిందితుడు పరారయ్యే ప్రమాదం ఉందా? నిందితుడి వ్యక్తిత్వం, ప్రవర్తన, నడత మళ్లీ నేరానికి పాల్పడే అవకాశముందా? సాక్షులను ప్రభావితం చేసే వీలుందా? బెయిల్ ఇస్తే న్యాయానికి ప్రమాదమా? | telegu |
देश के सीमावर्ती गांवों में पर्यटन के लिये जायें लोग: Modi नयी दिल्ली: प्रधानमंत्री नरेंद्र मोदी ने बुधवार को लोगों से अपील की है कि देश के सीमावर्ती गांवों के विकास के लिये जरूरी वाइब्रेंट विलेज प्रोग्राम के तहत वे इन गांवों में पर्यटन के लिये जायें। वाइब्रेट विलेज प्रोग्राम की घोषणा एक फरवरी को संसद में पेश आम बजट में की गयी है। सरकार का मकसद इस योजना के तहत चीन से लगे देश के सीमावर्ती गांवों में आधारभूत ढांचे का विकास करना है। ग्रामीण विकास और जल पर आधारित वेबीनार सत्र में लीविंग नो सिटीजन बिहाइंड थीम पर दिये अपने संबोधन में प्रधानमंत्री ने कहा कि वाइब्रेंट विलेज प्रोग्राम सीमावर्ती गांवों के विकास के लिए बहुत जरूरी है। उन्होंने कहा, यह कितना अच्छा होगा कि तहसील के लोग सीमावर्ती गांवों में जायें। वे खुद ही उसका अनुभव प्राप्त करें कि किस तरह का माहौल वहां हैं और लोग वहां किस तरह रहते हैं। यह सिर्फ शिक्षा से संबंधित गतिविधि नहीं होगी बल्कि इससे हमारी वाइब्रेंट विलेज योजना को भी मदद मिलेगी। उन्होंने साथ ही कहा कि इन सीमावर्ती गांवों के जन्मदिन का उत्सव मनाया जाना चाहिए। प्रधानमंत्री ने कहा कि महिलायें ग्रामीण अर्थव्यवस्था की धुरी हैं और वित्तीय समावेश ने यह सुनिश्चित किया है कि वे परिवार की आर्थिक गतिविधियों में हिस्सा लें। उन्होंने कहा कि स्वयं सहायता समूह के जरिये इसका दायरा बढ़ाया जाना चाहिए। प्रधानमंत्री मोदी ने कहा कि जल जीवन अभियान के जरिये उनकी सरकार का लक्ष्य चार करोड़ कनेक्शन प्रदान करने का है और इस दिशा में और भी कुछ करने की जरूरत है। उन्होंने कहा, मैँ यह जोर देना चाहता हूं कि हर राज्य पाइपलाइन के जरिये शुद्ध और अच्छी गुणवत्ता का पेयजल मुहैया करायें। उन्होंने कहा कि अब समय आ गया है कि देश के सभी लोगों को मूलभूत सुविधायें जैसे पानी, बिजली आदि मिले। हमें अपनी पूरी ताकत के साथ इस दिशा में काम करना होगा। बजट में इन योजनाओं को लागू करने का स्पष्ट खाका तैयार किया हुआ है। उन्होंने बताया कि प्रधानमंत्री आवास योजना, प्रधानमंत्री ग्रामीण सड़क योजना, जल जीवन अभियान, पूर्वोत्तर भारत संपर्क योजना, ग्रामीण भारत के लिए ब्रॉडबैंड कनेक्टिविटी योजना आदि के प्रावधान बजट में किये गये हैं। प्रधानमंत्री ने पूर्वोत्तर क्षेत्र के लिए प्रधानमंत्री विकास पहल और स्वामित्व योजना का भी उल्लेख किया और बताया कि स्वामित्व योजना के तहत 40 लाख प्रापर्टी कार्ड जारी किये गये हैं। उन्होंने प्रत्येक रुपये के उचित इस्तेमाल पर जोर दिया। देश के 100 जिलों, 1,144 प्रखंडों, 66,647 ग्राम पंचायतों और 1,37,642 गांवों में हर घर जल योजना के तहत पेयजल की आपूर्ति की जा रही है। गोवा, तेलंगाना और हरियाणा के अलावा दादर नगर हवेली, दमन दीव और पुड्डुचेरी जैसे केंद्र शासित प्रदेशों ने अपने शत प्रतिशत निवासियों को नल का पानी मुहैया कराया है। स्वामित्व योजना की शुरुआत राष्ट्रीय पंचायती राज दिवस के अवसर पर 20 अप्रैल 2020 को की गयी थी। इस योजना के पायलट चरण के तहत 2020 में ही देश के नौ राज्यों में इसे शुरु किया गया था। मौजूदा समय में देश के 29 राज्यों और केंद्र शासित प्रदेशों में यह योजना लागू है। वर्ष 2025 तक इस योजना को देश के सभी गांवों में लागू करने की लक्ष्य है। | hindi |
FRED KRUPP is President of the Environmental Defense Fund. Follow him on Twitter @FredKrupp.
When U.S. President Donald Trump and Chinese President Xi Jinping met in April at Trump’s Mar-a-Lago resort, one topic was not on the agenda: the environment. Perhaps they couldn’t find enough common ground. Xi, a chemical engineer by training, has often spoken publicly about his concerns over the effects of climate change on China, where almost 20 percent of the land is desert, an area expanding at a rate of more than 1,300 square miles per year. Analysts believe Xi is also determined to help China dominate the clean energy industry. In 2015, China installed more than one wind turbine every hour, on average, and enough solar panels to cover over two dozen soccer fields every day, according to Greenpeace. As part of its drive to clean up dangerous air pollution in Chinese cities, Beijing has canceled the construction of more than 100 coal-fired power plants this year alone. Such measures, coupled with Xi’s commitment to the 2015 Paris agreement on climate change, have turned Xi into a global leader on energy and the environment, filling a void created by the man who sat across the table from him at Mar-a-Lago.
Trump’s position on environmental protection has been consistent: he wants far less of it. Unlike Xi, Trump and many of his cabinet secretaries question the scientific consensus that human activities are the main driver of climate change. In the name of regulatory reform and job creation, they want to increase domestic fossil fuel production and roll back limits on both greenhouse gas emissions and the release of conventional pollutants. During his campaign, Trump promised to “get rid of” the Environmental Protection Agency (EPA). “We’re going to have little tidbits left, but we’re going to take a tremendous amount out,” he said in March 2016. And his administration is considering withdrawing from the Paris agreement, a move that would undermine the United States’ standing in the world, cede clean energy jobs and investment to China and Europe, and expose U. | english |
دانش فاروق بھٹ چھُ اَکھ فٹ بال پیلیر.
== حَوالہٕ == | kashmiri |
JURGEN KLOPP is not surprised in the slightest Unai Emery has stamped his authority on Arsenal so quickly.
Liverpool’s manager goes head-to-head with the Spaniard today at Anfield.
Arsenal travel to Merseyside and face a daunting task to stop Klopp’s Premier League leaders.
While the Gunners are short of defensive options due to injuries, they have been equally potent in attack.
Arsenal go into the game (5.30pm) sitting fifth and two points off the Champions League spots.
And strike duo Pierre-Emerick Aubameyang and Alexandre Lacazette are expected to start.
Klopp is wary of the threat Arsenal pose, especially as they are one of only three teams to have not lost to Liverpool in the league this term.
Arsenal held the Reds to a 1-1 draw at the Emirates in the reverse fixture.
And ahead of today’s clash, Klopp has hailed his opposite number.
“Unai is one of the Europe’s top coaching talents,” Klopp wrote for Liverpool’s official matchday programme.
“And it is zero surprise to anyone who knows his quality that he has imposed himself so quickly on the club.
“The team already plays in a manner that reflects what he looks for: organised and adventurous, with quick, clever and skilful players all over the pitch.
“This is a team that can and will punish you if you drop even 0.01 per cent of your focus.
“Our game against them in London was exhausting to watch because of the pace of it and the intensity.
“They have proved they can strike back in big matches when they fall behind and they have proved they can dominate an opponent also.
“I know a number of their players, some better than others, and I know they are winners. | english |
बिकिनी पहन समंदर में उतरीं शेफाली, पानी में जमकर की मस्ती, देखे वीडियो शेफाली जरीवाला Shefali Jariwala को कौन नहीं जानता हैं. उन्होंने कांटा लगा गाने से जमकर सुर्खियां बटोरी थीं. हालांकि, वह सिल्वर स्क्रीन पर कम नजर आती हैं लेकिन सोशल मीडिया पर अपने हुस्न से आग लगाती रहती हैं. अब उनका एक वीडियो सामने आया है, जिस देखकर फैंस आहें भरने लगे हैं. समंदर में जमकर की मस्ती वीडियो में शेफाली जरीवाला Shefali Jariwala बिकिनी पहने हुए आ नजर रही हैं. उन्होंने सनग्लासेस पहन रखे हैं. वह समंदर में उतरकर बहुत खुश लग रही हैं और जमकर मस्ती कर रही हैं. वीडियो के बैकग्राउंड में गहराइयां फिल्म का गाना डूबे सुनाई दे रहे है. उन्होंने इस वीडियो को इंस्टाग्राम अकाउंट पर शेयर किया है जिसे खूब लाइक और शेयर किया जा रहा है. वीडियो देख फैंस के उड़े होश शेफाली Shefali Jariwala के वीडियो पर एक यूजर ने कमेंट करते हुए लिखा, आपकी तो हर अदा कातिलाना है. दूसरे ने लिखा, आपने तो पानी में आग लगा दी. किसी ने कमेंट किया, सेक्सी क्वीन. इसके अलावा कुछ यूजर्स ने तो कमेंट सेक्शन में फायर इमोजी की बरसात कर दी है. शेफाली को पड़ते थे मिर्गी के दौरे काफी समय पहले शेफाली जरीवाला Shefali Jariwala ने एक वेबसाइट से बात करते हुए अपनी जिंदगी के उस राज के बारे में बताया जिसके बारे में किसी को भी नहीं पता था. शेफाली ने कहा मुझे 15 साल की उम्र से मिर्गी के दौरे पड़ने लगे थे. उस वक्त मेरे ऊपर पढ़ाई में अच्छा करने का प्रेशर था. तनाव और चिंता के कारण मेरे साथ ऐसा हुआ. मुझे कई बार क्लासरूम, बैकस्टेज और कभीकभी सड़क पर भी दौरे पड़ जाते थे. | hindi |
ఆలేరు టీఆర్ఎస్లో వేటు రాజకీయాలు..! పార్టీలలో వర్గపోరు సహజం. సమయం వచ్చినప్పుడు అది ఏ రూపంలో.. ఏ విధంగా బయట పడుతుందో చెప్పలేం. సందర్భాన్ని బట్టి అసంతృప్తి తీవ్రత ఉంటుంది. సమయం కోసం వేచి చూసేవాళ్లు ఛాన్స్ చిక్కితే అస్సలు వదలరు. ప్రస్తుతం ఆలేరు టీఆర్ఎస్లో అదే జరుగుతోందట.ఆలేరు టీఆర్ఎస్లో రచ్చ! యాదాద్రి జిల్లా ఆలేరులో సంస్థాగత ఎన్నికలు టీఆర్ఎస్ అంతర్గత విభేదాలను బయటపెట్టింది. తుర్కపల్లి మండల పార్టీ అధ్యక్ష ఎంపిక అగ్గి రాజేసింది. ఎమ్మెల్యే గొంగిడి సునీత.. ఆమె భర్త, డీసీసీబీ ఛైర్మన్ మహేందర్రెడ్డి తీరుపై పార్టీలోని ఓ వర్గం భగ్గుమంది. వారిపై ఇప్పుడు వేటు వేయడంతో టీఆర్ఎస్ వర్గాల్లో ఆలేరు చర్చగా మారింది.శ్రీనివాస్ సహా ఆరుగురిపై ఎమ్మెల్యే వేటు! తుర్కపల్లి మండల టీఆర్ఎస్ అధ్యక్షుడిగా ఆలేరు మార్కెట్ కమిటీ మాజీ ఛైర్మన్ పడల శ్రీనివాస్ ఉన్నారు. ఇప్పుడు ఆయన్ని కాకుండా పిన్నపురెడ్డి నరేందర్రెడ్డిని నియమించడంతో శ్రీనివాస్ వర్గం ఆగ్రహం వ్యక్తం చేసింది. మహేందర్రెడ్డి కారుపై రాళ్లు పడ్డాయి. దాంతో పార్టీలో గుంభనంగా ఉన్న వర్గపోరు రోడ్డుకెక్కింది. భర్త కారుపై దాడిని ఎమ్మెల్యే గొంగిడి సునీత సీరియస్గా తీసుకున్నారు. రాళ్ల దాడికి శ్రీనివాస్ అండ్ కోనే కారణమని ఆరోపిస్తూ మొత్తం ఐదుగురిని ఆరేళ్లపాటు టీఆర్ఎస్ నుంచి బహిష్కరిస్తున్నట్టు ఎమ్మెల్యే ప్రకటించారు. ఈ చర్యే ఆలేరు అధికార పార్టీలో ప్రకంపనలు సృష్టిస్తోంది. పైగా దాడి చేసినవారిపై కేసు కూడా పెట్టారు.రెండు వర్గాలకు ఎప్పటి నుంచో పడటం లేదా? టీఆర్ఎస్లో ఉంటూ పార్టీ వ్యతిరేక కార్యకలాపాలకు పాల్పడేవారు ఎంతటి వారైనా ఉపేక్షించేది లేదని ఎమ్మెల్యే తన చర్యలతో హెచ్చరించినా.. దీని వెనక ఉన్న కారణాలపై ఆరా తీస్తున్నాయట పార్టీ వర్గాలు. వాస్తవానికి శ్రీనివాస్కు, ఎమ్మెల్యే సునీత వర్గాలకు ఎప్పటి నుంచో పడటం లేదట.అవి ఇప్పుడు పార్టీ కమిటీ ఏర్పాటు రూపంలో బయటపడ్డాయట. ఇదే అవకాశం అనుకున్నారో.. ఇలాంటి ఛాన్స్ మళ్లీ రాదని భావించారో ఏమో.. పార్టీ నుంచి వేటు వేయడం.. కేసు పెట్టడం చకచకా జరిగిపోయింది.పార్టీ పెద్దలకు ఫిర్యాదు చేసే పనిలో శ్రీనివాస్ వర్గం! పది రోజుల క్రితం వంగపల్లిలో జరిగిన టీఆర్ఎస్ మీటింగ్లోనూ ఎమ్మెల్యే సునీత చేసిన కామెంట్స్ను ఈ సందర్భంగా కొందరు ప్రస్తావిస్తున్నారు. టీఆర్ఎస్లో కోవర్టులు ఉన్నారని ఆరోపించారు ఎమ్మెల్యే. ఆమె ఎవరిని ఉద్దేశించి అన్నారో అని నాడు పార్టీ వర్గాలు ఆరా తీశాయి. ఇప్పుడు గొడవలు ఇలా టర్న్ తీసుకోవడంతో.. ఆ కామెంట్స్కు.. తాజా రగడకు సంబంధం ఉందని అనుకుంటున్నాయట పార్టీ శ్రేణులు. ప్రస్తుతం రెండు వర్గాలు పోటాపోటీ సమావేశాలు నిర్వహించి ఎత్తుకు పైఎత్తు వేస్తున్నాయి. ఈ అంశాన్ని ఇంతటితో వదిలిపెట్టబోమని.. టీఆర్ఎస్ పెద్దల దృష్టి తీసుకెళ్తామని శ్రీనివాస్ వర్గం హెచ్చరిస్తోంది. తుర్కపల్లి ఘటనపై టీఆర్ఎస్ అధిష్ఠానం కూడా సీరియస్గా ఉన్నట్టు సమాచారం. క్షేత్రస్థాయి నుంచి సమాచారం తెప్పించుకున్నట్టు తెలుస్తోంది. ఇప్పుడు రెండు వర్గాలు ఫిర్యాదు చేసుకున్న తర్వాత ఈ వర్గపోరు ఎలాంటి టర్న్ తీసుకుంటుందన్నది ఆలేరు అధికారపార్టీలో ఆసక్తిగా మారింది. | telegu |
ಚಿತ್ರರಂಗ ಎಂಟ್ರಿಗೆ ಪ್ರೇಮ್ ಪುತ್ರಿ ಸಿದ್ಧತೆ ಸಿನಿಮಾರಂಗದಲ್ಲಿ ಮಿಂಚುತ್ತಿರುವ ನಾಯಕ ನಟ ನಟಿಯರ ಮಕ್ಕಳು ಕೂಡ ಚಿತ್ರರಂಗದ ಕಡೆಗೆ ಮುಖ ಮಾಡುವುದು ಹೊಸದೇನಲ್ಲ.ಕನ್ನಡ ಚಿತ್ರರಂಗದಲ್ಲೂ ಈಗಾಗಲೇ ಸಾಕಷ್ಟು ನಾಯಕ ನಟ ನಟಿಯರ ಮಕ್ಕಳು ಕಲಾವಿದರಾಗಿ, ನಿರ್ಮಾಪಕರಾಗಿ, ನಿರ್ದೇಶಕರಾಗಿ ಚಿತ್ರರಂಗಕ್ಕೆ ಅಡಿಯಿಟ್ಟಿದ್ದಾರೆ. ಅದರಲ್ಲಿ ಕೆಲವರು ಭದ್ರ ನೆಲೆಯನ್ನೂ ಕಂಡು ಕೊಂಡಿದ್ದಾರೆ. ಈಗ ಸ್ಯಾಂಡಲ್ವುಡ್ನ ರೊಮ್ಯಾಂಟಿಕ್ ಹೀರೋ ನೆನಪಿರಲಿ ಪ್ರೇಮ್ ಅವರ ಪುತ್ರಿ ಅಮೃತಾ ಪ್ರೇಮ್ಕೂಡ ಇದೇ ಸಾಲಿನಲ್ಲಿ ಚಿತ್ರರಂಗ ಪ್ರವೇಶ ಪಡೆಯುವ ತಯಾರಿಯಲ್ಲಿದ್ದಾರೆ. ಹೌದು, ಇತ್ತೀಚೆಗಷ್ಟೇ ನಟ ನೆನಪಿರಲಿ ಪ್ರೇಮ್ ಪುತ್ರಿ ಅಮೃತಾ ಪ್ರೇಮ್, ಸಾಂಪ್ರದಾಯಿಕ ಉಡುಪಿನಲ್ಲಿ ಸ್ಪೆಷಲ್ ಫೋಟೋಶೂಟ್ ಮಾಡಿಸಿಕೊಂಡಿದ್ದಾರೆ. ಸದ್ಯ ಈ ಫೋಟೋಗಳು ಸೋಶಿಯಲ್ ಮೀಡಿಯಾಗಳಲ್ಲಿ ಜೋರಾಗಿ ಹರಿದಾಡುತ್ತಿದ್ದು, ಪ್ರೇಮ್ ಪುತ್ರಿ ಚಿತ್ರರಂಗಕ್ಕೆಕಾಲಿಡಲು ತೆರೆಮರೆಯಲ್ಲಿ ಸಜ್ಜಾಗಿದ್ದಾರೆ ಎನ್ನಲಾಗುತ್ತಿದೆ. ಇನ್ನು ಸದ್ಯ ದ್ವಿತೀಯ ವರ್ಷದ ಎಂಜಿನಿಯರಿಂಗ್ ಶಿಕ್ಷಣ ಅಧ್ಯಯನ ಮಾಡುತ್ತಿರುವ ಅಮೃತಾ, ಓದಿನಲ್ಲೂ ಸಾಕಷ್ಟು ಮುಂದಿರುವ ಹುಡುಗಿ. ಬಾಲ್ಯದಿಂದಲೂ ನೃತ್ಯ ಮತ್ತುಕಲೆಯ ಕಡೆಗೆ ಆಸಕ್ತಿ ಬೆಳೆಸಿಕೊಂಡಿರುವ ಅಮೃತಾ, ಈಗಾಗಲೇ ಸೋಶಿಯಲ್ ಮೀಡಿಯಾಗಳಲ್ಲಿ ಸಾಕಷ್ಟು ಡಬ್ಸ್ಮ್ಯಾಶ್ ಮತ್ತು ಟಿಕ್ಟಾಕ್ ವಿಡಿಯೋಗಳ ಮೂಲಕ ನೆಟ್ಟಿಗರ ಗಮನ ಸೆಳೆದಿದ್ದಾರೆ. ಪ್ರೇಮ್ ತಮ್ಮ ಪುತ್ರಿಯ ಫೋಟೋವನ್ನು ಸೋಶಿಯಲ್ ಮೀಡಿಯಾದಲ್ಲಿ ಪೋಸ್ಟ್ ಮಾಡಿ, ಮಗಳೆಂದರೆ ತಂದೆಗೆ ದೇವತೆಯಂತೆ.ದೇವತೆಯನ್ನು ಪಡೆಯುವ ಅದೃಷ್ಟ ಎಲ್ಲ ತಂದೆಯರಿಗೂ ಸಿಗುವುದಿಲ್ಲ. ಅಭಿನಂದನೆಗಳು ನನ್ನ ಅದೃಷ್ಟದ ದೇವತೆಗೆ ಎಂದು ಬರೆದುಕೊಂಡಿದ್ದಾರೆ. ತಮ್ಮ ಪುತ್ರಿಯ ಚಿತ್ರರಂಗ ಎಂಟ್ರಿ ಬಗ್ಗೆ ಮಾತನಾಡುವ ಪ್ರೇಮ್, ಅವಳಿನ್ನು ಚಿಕ್ಕವಳು, ಓದುತ್ತಿದ್ದಾಳೆ. ಅವಳನ್ನು ಚಿತ್ರರಂಗಕ್ಕೆ ತರುವ ಆಸೆ ಇದೆ. ಅವಳಿಗೂ ಸಿನಿಮಾರಂಗದ ಮೇಲೆ ಆಸಕ್ತಿ ಇದೆ. ತುಂಬಾ ದಿನಗಳಿಂದ ಒಂದು ಒಳ್ಳೆಯ ಫೋಟೋಶೂಟ್ ಮಾಡಿಸಬೇಕೆಂದು ಅಂದುಕೊಂಡಿದ್ದೆವು. ಈಗ ಅದು ನಡೆದಿದೆ ಎನ್ನುತ್ತಾರೆ. | kannad |
Kuppam : కుప్పంలో టీడీపీ అభ్యర్థి నామినేషన్ పత్రాలు లాక్కుపోయిన గుర్తు తెలియని వ్యక్తులు Tension in Kuppam Municipality Election Nominations చిత్తూరు జిల్లా కుప్పం మున్సిపాలిటీ ఎన్నికల నామినేషన్ల వేస్తున్న క్రమంలో తీవ్ర ఉద్రిక్తత నెలకొంది. టీడీపీ నుంచి పోటీ చేయటానికి ఓ వ్యక్తి నామినేషన్ వేయటానికి వెళుతుండగా అతని చేతిలోంచి నామినేషన్ల పేపర్లకు కొంతమంది వ్యక్తులు లాక్కుపోయారు. అతనిపై దాడి చేసిన మరీ పేపర్లను లాక్కుపోయారని వాపోయాడు బాధితుడు. కాగా కుప్పంలో మున్సిపల్ ఎన్నికల జరుగనున్న క్రమంలో నామినేషన్లు వేస్తున్నారు పలువురు అభ్యర్ధులు. నామినేషన్లకు ఈరోజే చివరి రోజు కావటంతో ఆయా పార్టీల తరపున పోటీ చేయాలనుకునే అభ్యర్థులు నామినేషన్లు వేస్తున్నారు. ఈ క్రమంలో 14వ వార్డుకు చెందిన వెంకటేశ్ అనే వ్యక్తి నామినేషన్ వేసేందుకు వెళ్లగా.. అతడి వద్ద నుంచి కొంతమంది వ్యక్తులు నామినేషన్ పత్రాలను గుంజుకుని పోయారు. ఈక్రమంలో పెనుగులాట జరగగా సదరు వ్యక్తులు వెకటేశ్ పై దాడి చేసి మరీ పత్రాలను లాక్కుపోయారు. ఈ ఘర్షణలో వెంకటేశ్ చేతికి గాయమైంది. కాగా బాధితుడు వెంకటేశ్ గతంలో కుప్పం సర్పంచ్ గా, ఎంపిపిగా పనిచేశారు. ఈ ఘటనపై సమాచారం అందుకున్న మాజీ మంత్రి అమర్నాధ్ రెడ్డి ఘటనా స్థలానికి చేరుకుని పరిస్థితిని సమీక్షిస్తున్నారు.మరోవైపు కుప్పం మున్సిపాలిటీని కైవసం చేసుకునేందుకు టీడీపీ, వైపీసీ నేతలు పోటా పోటీగా ఉన్నారు. జిల్లావ్యాప్తంగా ఈ పురపాలిక ఎన్నికలపై ఆసక్తి నెలకొంది. పట్టు నిలుపుకోవాలని తెదేపా, పాగా వేయాలని వైకాపా ప్రయత్నాలు చేస్తున్నాయి. The post Kuppam : కుప్పంలో టీడీపీ అభ్యర్థి నామినేషన్ పత్రాలు లాక్కుపోయిన గుర్తు తెలియని వ్యక్తులు appeared first on 10TV. | telegu |
Editor’s Note: You may or may not have noticed my blog header has changed to “Beyond Words.” You may or may not have also noticed I haven’t written in a while. I’ve been in a bit of a lull regarding just the right words and just the right topics to write about. You could say the “write” words just haven’t come and, at the same time, I’ve been thinking of changing the focus and title of my blog. Sitting in mass this weekend I heard just what I needed to hear and voila: I’m back! My new “Beyond Words” blog will continue to be more of the same with maybe a few tweaks here and there. You need do nothing different to receive it and I can still be found at carlawordsmith.com. I hope you continue to read my words and as always, let me know when I get off the “write” track.
Happy Super Bowl Sunday…or as Father Larry said in his sermon, “the superest of all Sundays.” Well, kinda.
We all know it’s merely “super” because either the NFC team (New England Patriots) or the AFC team (Atlanta Falcons) will go home tonight World Champions. Again, kinda. Not really “world” champions, but Super Bowl Champions. It’s a day to be with family and friends and to eat, drink, and be merry. It’s also a day when, as Father explained, the 7 Deadly Sins rear their ugly heads.
Of course there is Gluttony. We eat foods we know we shouldn’t and we eat way too much. Many of us will also drink too much.
There is Envy. We are all envious of those pesky New England Patriots, their famed quarterback and coach, and all their past Super Bowl heroics. It’s said the only people rooting for the Patriots today are Patriot fans. The rest, are envious.
We will show Anger when the team we’re rooting for calls a bad play, turns over the ball, or when the other team scores.
Patriots fans will be full of Greed today, as they hope for their fifth Super Bowl victory. Quarterback Tom Brady will also be vying for his fifth Super Bowl title, which will make him the most prolific NFL QB in history…Super Bowl ring-wise at least.
Fans on both sides of the field will most likely display Pride as they cheer on their hometown boys.
The rest of us are sure to exhibit Sloth; sitting in front of televisions for virtually the whole day. Work? We’ll get to it tomorrow. Laundy? It can wait. Today is a day for America to take the day off!
As for the last of the deadly sins, Lust, let’s hope we don’t partake in it, if only that we lust for our team to win and not that Brady gets hurt!
All of this is, of course, fun and the game of football, but look ahead and as soon as tomorrow, and we are likely to be guilty of those very sins in our daily lives.
Do we work so hard to make more and more money that we neglect our family?
Do we eat even when we’re not hungry and serve ourselves whopping portions of what we do eat?
Do we get angry driving in traffic and envy those driving nicer cars than ours?
Are we so full of pride that we have lost our sense of humility?
Do we lie around watching mindless TV rather than going for a walk or volunteering somewhere?
And, do we lust after things that are immoral or just plain wrong and have impure thoughts or actions?
If you answered yes to any of those, no fear; you are not alone. At the same time you might tell yourself “no harm, no foul” if I cursed at the driver next to me or if I am proud of my accomplishments. The problem is, all these sins are the roots of greater sins such as murder, adultery, theft, and others.
So how can we avoid being sinfully proud, envious, and the like? By praying. It’s that simple. Simply pray every day. Pray for the gifts of the Capital Virtues.
Chastity will help you overcome lust and the infected acts it encourages.
Generosity will make you less greedy. Start by detaching from things of this world.
Temperance will overcome gluttony by helping you live in moderation.
If you have the gift of Brotherly Love, you will be less envious, which will lead to less badmouthing and a genuine happiness for others.
Anger can be weakened by meekness, which will help you control resentment while cultivating patience.
Humility will topple your pride and help you rely less on your will and more on God’s.
If you feel you have the sin of sloth, pray for Diligence and the ability to fulfill your duties in life, even when they are tiresome.
Whatever you do, don’t get overwhelmed. Instead, start with just one or two and watch how the others will be none too happy to creep into your life.
As I read my Facebook and Instagram pages these last few weeks, I see an incredible amount of envy, anger, pride, greed, and the need to insult and degrade. It is both alarming and disturbing how much hate and bitterness prevails. I have been guilty of them myself, but learned a good lesson on Friday.
As I was getting my hair done, my stylist and I were talking about the current state of our country, and after a lengthy discussion, I mentioned how I think it’s unfair that the “winners” of this election are being inundated with insults and abuse by those not victorious and are not allowed to be openly happy and celebratory. I was expecting a “Yeah, that’s just not right,” but instead my very wise and astute stylist replied, “maybe it’s an opportunity for us to be humble.” Bingo.
Thank you Mary. It’s good to be back! | english |
किशोरी से छेड़छाड़ का विरोध करने पर मनचलों ने भाई के दोस्त को पीटा नई दिल्ली, जागरण संवाददाता। कालिंदी कुंज थाना क्षेत्र में मनचलों ने किशोरी से छेड़छाड़ की। विरोध करने पर पीड़िता के भाई के दोस्त को आरोपितों ने लातघूंसों से बुरी तरह पीटा। पीड़ित की शिकायत पर पुलिस ने मामला दर्ज कर लिया है। एक आरोपित की पहचान मुन्ना के रूप में हुई है। 25 जनवरी की है घटना पीड़िता के दोस्त के भाई और पीड़ित मदनपुर खादर के प्रसन्नजीत ने बताया कि वह यहां किराए पर रहते हैं। उनके पिता कोलकाता में और माता गोविंदपुरी में रहती हैं। माता गोविंदपुरी में ही खाना बनाने का कार्य करती हैं। उन्होंने बताया कि 25 जनवरी की रात वे अपने दोस्त के घर के बाहर बैठकर आग सेंक रहे थे। इसी दौरान उनके दोस्त की बहन परेशान हालत में घर आई। पूछने पर उसने बताया कि पास के ही कुछ लड़कों ने उनसे छेड़छाड़ की है। इस पर प्रसन्नजीत ने आराेपितों के पास जाकर इसका विरोध किया तो आरोपितों ने उनकी पिटाई कर दी। प्रसन्नजीत पीड़ित किशोरी को लेकर उसके घर पहुंचा तो किशोरी के पिता ने पुलिस को फोन कर मामले की जानकारी दी। इस पर पुलिस ने मामला दर्ज कर लिया है।इधर, दिल्ली के पहाड़गंज स्थित झंडेवालान से करोड़ों की ठगी करने वाले सगे भाइयों संदीप और अमित को गिरफ्तार किया है। आरोपित फर्जी काल सेंटर के जरिये नौकरी डाट काम व मांस्टर डाट काम पर आवेदन करने वाले का डाटा निकालकर उन्हें फोन करते थे। इसके बाद नौकरी दिलाने का झांसा देकर मोटी रकम वसूल लेते थे। आरोपितों के पास से एक लाख पांच हजार रुपये, नौ डेस्क फोन, कंप्यूटर व मोबाइल फोन बरामद किए गए हैं। आरोपित दिल्ली के 90 डी बीसी ब्लाक शालीमार बाग के रहने वाले हैं। चौक निवासी सेवानिवृत्त दारोगा सैयद इरशाद हुसैन रिजवी की बेटी ने नौकरी डाट काम पर बायोडाटा अपलोड किया था। वेबसाइट से युवती का डिटेल हासिल कर आरोपितों ने उसे फोन किया। इसके बाद नौकरी दिलाने का झांसा दिया। | hindi |
\begin{document}
\title{Resolving Knudsen Layer by High Order Moment Expansion}
\begin{abstract}
We model the Knudsen layer in Kramers' problem by linearized high
order hyperbolic moment system. Due to the hyperbolicity, the
boundary conditions of the moment system is properly reduced from
the kinetic boundary condition. For Kramers' problem, we give the
analytical solutions of moment systems. With the order increasing of
the moment model, the solutions are approaching to the solution of
the linearized BGK kinetic equation. The velocity profile in the
Knudsen layer is captured with improved accuracy for a wide range of
accommodation coefficients.
\vspace*{4mm}
\end{abstract}
\section{Introduction}
In the area of kinetic theory, Kramers' problem \cite{Kramers1949} is
generally considered as the most basic way to understand the
fundamental flow physics of the wall, which defining the Knudsen layer
\cite{Lilley2007}, without some of the additional complications in
other more realistic problems, such as flow in a plane channel
\cite{Garcia2009} or cylindrical tube \cite{Higuera1989,
Grucelski2013}. It is well known \cite{Karniadakis2002, Zhang2012}
that the classical Navier-Stokes-Fourier(NSF) equations with
appropriate boundary conditions can be used to describe the flow with
satisfactory accuracy when the gas is close to a statistical
equilibrium state. However, more accurate model is needed to depict
the nonequilibrium effects near the wall, where the continuum
assumption is essentially broken down and NSF equations themselves
become inappropriate \cite{Lilley2007, Dongari2009}. This is exactly
the case in Knudsen layers.
During the past decades, various methods have been developed to
investigate the Kramers' problem based on the Boltzmann equation. Highly
accurate results on the dependence of slip coefficient for the
unmodeled Boltzmann equation and general boundary condition have been
reported \cite{Loyalka1967, Loyalka1971, Klinc1972}. Variable
collision frequency models of the Boltzmann equation
\cite{Cercignani1969, Williams2001, Loyalka1967, Loyalka1975,
Loyalka1990, Siewert2001} are extensively discussed. We note that
the direct simulation Monte Carlo (DSMC) method \cite{Bird} is widely
used to solve the Boltzmann equation numerically. Unfortunately, DSMC
calculations impose prohibitive computational demands for many
applications of current interests. The intensive computational
demands of DSMC method have motivated recent interests in the
application of higher-order hydrodynamic models to simulate rarefied
flows \cite{Reese2003, Guo2006, Gu2010, Mizzi2007}. There are many
competing sets of higher-order constitutive relations, which are
derived from the fundamental Boltzmann equation using differing
approaches. The classical approaches are the Chapman-Enskog technique
and Grad's moment method. Among these alternative macroscopic modeling
and simulation strategies \cite{Grad, Levermore}, the moment method is
quite attractive due to its numerous advantages \cite{Muller,
Struchtrup2002, TorrilhonEditorial}. It is regarded as a useful
tool to extend classical fluid dynamics, and achieves highly accurate
approximations with great efficiency.
The moment method for gas kinetic theory \cite{Grad} has been applied
on wall-bounded geometries which supplemented by slip and jump
boundary conditions \cite{Marques2001}, while its application is
seriously limited due to the lack of hyperbolicity \cite{Muller,
Grad13toR13}. Particularly for the $3$D case, the moment system is
not hyperbolic in any neighborhood of the Maxwellian. Only recently
this fatal defect has been remedied \cite{Fan, Fan_new, ANRxx} that
globally hyperbolic models can be deduced. The global hyperbolicity of
the new models provides us the information propagation directions,
and thus a proper boundary condition of the moment model may be
proposed. This motivates us to study the Kramers' problem using the
new moment models.
Starting from the globally hyperbolic moment system (HME), we first
derive a linearized hyperbolic moment model to depict the Kramers'
problem. We found that the linearized model is even simpler than one's
expectation, since the equations for velocity are decoupled from
other equations in the system involving high order moments. The number
of equations in the decoupled part related with velocity is the same
as the moment expansion order only. Then we establish the boundary
conditions for the linearized moment model according to physical and
mathematical requirements for the system. Following Grad's approach in
\cite{Grad} for the kinetic accommodation model by Maxwell
\cite{Maxwell}, we propose the general boundary conditions for shear
flows. After that, by linearizing the velocity jump and high order
terms in the expression of the general boundary conditions, it is then
adapted to the boundary condition for the linearized model. This makes us
able to give the expression of velocity by solving the decoupled
system related with velocity together with the corresponding boundary
condition. It is extensively believed that the linearized system is
accurate enough for low-speed flows, which encourages us to apply the
solution of the velocity obtained to study Kramers' problem.
To obtain the full velocity profile and the velocity slip coefficient
in Kramers' problem, one may adopt a certain direct numerical method to
solve the linearized Boltzmann equation. However, the linearized moment
system can depict the velocity profile in the Knudsen layer with
analytical expressions. This can be used to provide a convenient
correction near the wall \cite{Lockerby2008} for the lower order
macroscopic system, such as NSF equations. In the moment method, the
Knudsen layer appears as superpositions of exponential layers
\cite{Struchtrup2008.1}. For the result we give based on HME, the
number of exponential layers is increasing. Comparing with the results
given by direct numerical simulation, our solutions illustrate a
significant improvement in accuracy than the results in references
when more and more high order moments are considered. Particularly,
our results can capture the velocity profile in the Knusen layer
accurately for a wide range of accommodation coefficients. We note that
our linearized model is of the same computational cost as the lower
order moment system.
This paper is organized as follows. In Section \ref{sec:hme} we
reviewed HME for Boltzmann equations and derived the linearized HME.
The boundary conditions for HME and linearized HME are established in
Section \ref{sec:bc}. The solutions of linearized equations are solved in
detail for Kramers' problem in Section \ref{sec:kramers}. With the
solutions of the velocity profile in Knudsen layer, some important
coefficients, such as defect velocity, are compared with the other
model of kinetic solution in the same section. We then draw some
conclusions to end this paper.
\section{Linearized HME for Boltzmann Equation}
\label{sec:hme}
\subsection{Boltzmann equation}
In gas kinetic theory, the motion of particles of gas can be depicted
by the Boltzmann equation \cite{Boltzmann}
\begin{equation}\label{eq:boltzmann}
\pd{f}{t} + \bxi\cdot\nabla_{\bx}f = Q(f,f),
\end{equation}
where $f(t,\bx,\bxi)$ is the number density distribution function
which depends on the time $t\in\bbR^+$, the spatial position
$\bx\in\bbR^3$ and the microscopic particle velocity $\bxi\in\bbR^3$,
and $Q(f,f)$ is the collision term. In this paper, we limit the
discussion on the BGK collision model \cite{BGK}, which reads:
\begin{equation}\label{eq:collision}
Q(f,f) = \frac{\rho\theta}{\mu}(\mathcal{M} - f) ,
\end{equation}
where $\mu$ is the viscosity and $\mathcal{M}$ is the local
thermodynamic equilibrium, usually called the local Maxwellian,
defined by
\begin{displaymath}
\mathcal{M}=\frac{\rho}{(2\pi\theta)^{3/2}}
\exp\left( -\frac{|\bxi-\bu|^2}{2\theta} \right).
\end{displaymath}
Here the density $\rho$, the macroscopic velocity $\bu$ and the
temperature $\theta$ are related to the distribution function as
\begin{equation}
\rho =\int_{\bbR^3}f\dd\bxi,\qquad
\rho\bu =\int_{\bbR^3}\bxi f\dd\bxi,\qquad
\rho|\bu|^2+3\rho\theta =\int_{\bbR^3}|\bxi|^2f\dd\bxi.
\end{equation}
Multiplying the Boltzmann equation by $(1,\bxi,|\bxi|^2)$ and
integrating both sides over $\bbR^3$ with respect to $\bxi$, we obtain
the conservation laws of mass, momentum and energy as
\begin{equation}\label{eq:conservationlaws}
\begin{aligned}
\odd{\rho}{t}&+\rho\sum_{d=1}^3\pd{u_d}{x_d}=0,\\
\rho\odd{u_i}{t}&+\sum_{d=1}^3\pd{p_{id}}{x_d}=0,\\
\frac{3}{2}\rho\odd{\theta}{t}&+\sum_{k,d=1}^3p_{kd}\pd{u_k}{x_d}+\sum_{d=1}^3\pd{q_d}{x_d}=0,
\end{aligned}
\end{equation}
where
$\odd{\cdot}{t} := \pd{\cdot}{t} + \displaystyle \sum_{d=1}^3
u_d \pd{\cdot}{x_d}$
is the material derivative, and the pressure tensor $p_{ij}$ and the
heat flux $q_i$ are defined by
\begin{equation}
p_{ij}=\int_{\bbR^3}(\xi_i-u_i)(\xi_j-u_j)f\dd\bxi,\quad
q_i=\frac{1}{2}\int_{\bbR^3}|\bxi-\bu|^2(\xi_i-u_i)f\dd\bxi, \quad
i,j=1,2,3.
\end{equation}
For convenience, we define the pressure $p$ and the stress tensor
$\sigma_{ij}$ by
\[
p=\sum_{d=1}^3\frac{p_{dd}}{3}=\rho\theta,\quad
\sigma_{ij}=p_{ij}-p\delta_{ij},\quad i,j=1,2,3.
\]
\subsection{HME and its linearization}
The moment method in kinetic theory is first proposed by Grad in 1949
\cite{Grad}. The primary idea is to expand the distribution function
around the Maxwellian into Hermite series
\begin{equation}\label{eq:expansion}
f(t,\bx,\bxi) =
\frac{\mathcal{M}}{\rho}\sum_{\alpha\in\bbN^3}f_{\alpha}(t,\bx)\He_{\alpha}^{[\bu,\theta]}(\bxi)=
\sum_{\alpha\in\bbN^3}f_{\alpha}(t,\bx)\mH_{\alpha}^{[\bu,\theta]}(\bxi),
\end{equation}
where $\alpha = (\alpha_1, \alpha_2, \alpha_3) \in \bbN^3$ is a 3D
multi-index, and $\He_{\alpha}^{[\bu,\theta]}(\bxi)$ are generalized
Hermite polynomials defined by
\begin{equation}\label{eq:hermite-poly}
\He_{\alpha}^{[\bu,\theta]}(\bxi) =
\frac{(-1)^{|\alpha|}}{\mathcal{M}}
\dfrac{\partial^{|\alpha|} \mathcal{M}}{\partial
\xi_1^{\alpha_1} \partial \xi_2^{\alpha_2} \partial
\xi_3^{\alpha_3}}, \qquad
|\alpha|=\sum_{d=1}^3\alpha_d,
\end{equation}
and $\mH_{\alpha}^{[\bu,\theta]}(\bxi)$ is the basis function defined
by
\begin{equation}\label{eq:basis-fun}
\mH_{\alpha}^{[\bu,\theta]}(\bxi) = \frac{\mathcal{M}}{\rho}
\He_{\alpha}^{[\bu,\theta]}(\bxi).
\end{equation}
Directly calculations yield, for
$i,j=1,2,3$,
\begin{equation}
\begin{aligned}
&f_{0}=\rho,\quad
f_{e_i}=0,\quad
\sum_{d=1}^3f_{2e_d}=0,\\
p_{ij}=p\delta_{ij} &+ (1+\delta_{ij})f_{e_i+e_j},\quad
q_i = 2 f_{3e_i}+\sum_{d=1}^3 f_{e_i+2e_d}.
\end{aligned}
\end{equation}
Substituting Grad's expansion \eqref{eq:expansion} into the
Boltzmann equation, and matching the coefficient of the basis function
$\mH^{[\bu,\theta]}_{\alpha}(\bxi)$, one can obtain the governing equations of
$\bu$, $\theta$ and $f_{\alpha}$, $\alpha\in\bbN^3$. However, the
resulting system contains infinite number of equations. Choosing a
positive integer $3\leq M\in\bbN$, and discarding all the equations
including $\pd{f_{\alpha}}{t}$, $|\alpha|>M$, and setting
$f_{\alpha}=0$, $|\alpha|>M$ to closure the residual system, we obtain
the $M$-th order Grad's moment system. Since
\[
Q(f,f) = - \frac{p}{\mu} \sum_{|\alpha| \geq 2} f_{\alpha}
\mH_{\alpha}^{[\bu,\theta]}(\bxi)
=-\frac{p}{\mu}\mathrm{H}(|\alpha|-2)f_{\alpha},
\]
where $\mathrm{H}(n)$ is the Heaviside step function
\[
\mathrm{H}(n) = \left\{ \begin{array}{ll}
0, & n<0,\\
1, & n\geq0,
\end{array} \right.
\]
the $M$-th order Grad's moment system can be written as
\begin{equation}\label{eq:arbit-system}
\begin{aligned}
\odd{f_{\alpha}}{t} &+ \sum_{d=1}^3 \left( \theta \pd{f_{\alpha-e_d}}{x_d} +
(1-\delta_{|\alpha|,M})(\alpha_d + 1)\pd{f_{\alpha+e_d}}{x_d} \right) \\
+ \sum_{k=1}^3 f_{\alpha-e_k} \odd{u_k}{t} &+ \sum_{k,d=1}^3 \pd{u_k}{x_d}
\left(\theta f_{\alpha-e_k-e_d} + (\alpha_d + 1) f_{\alpha-e_k+e_d}
\right) \\
+ \frac{1}{2} \sum_{k=1}^3 f_{\alpha-2e_k} \odd{\theta}{t} &+ \sum_{k,d=1}^3
\frac{1}{2} \pd{\theta}{x_d} \left(
\theta f_{\alpha-2e_k-e_d} + (\alpha_d + 1) f_{\alpha-2e_k+e_d}
\right)\\
&= -\frac{p}{\mu}f_{\alpha}\mathrm{H}(|\alpha|-2) , \quad |\alpha| \leq M,
\end{aligned}
\end{equation}
where $\delta_{i,j}$ is Kronecker delta. Here and hereafter we agree
that $(\cdot)_{\alpha}$ is taken as zero if any component of $\alpha$
is negative.
However, as is pointed in \cite{Muller,Grad13toR13}, Grad's moment
system lacks global hyperbolicity and is not hyperbolic even in any
neighborhood of local Maxwellian. The globally hyperbolic
regularization proposed in \cite{Fan,Fan_new} figures the drawback out
essentially, and results in globally Hyperbolic Moment Equations (HME)
as
\begin{equation}\label{eq:moment-system}
\begin{aligned}
\odd{f_{\alpha}}{t} &+ \sum_{d=1}^3 \left( \theta \pd{f_{\alpha-e_d}}{x_d} +
(1 - \delta_{|\alpha|,M})(\alpha_d + 1)\pd{f_{\alpha+e_d}}{x_d} \right) \\
+ \sum_{k=1}^3 f_{\alpha-e_k} \odd{u_k}{t} &+ \sum_{k,d=1}^3 \pd{u_k}{x_d}
\left(\theta f_{\alpha-e_k-e_d} + (1 - \delta_{|\alpha|,M})(\alpha_d + 1) f_{\alpha-e_k+e_d}
\right) \\
+ \frac{1}{2} \sum_{k=1}^3 f_{\alpha-2e_k} \odd{\theta}{t} &+ \sum_{k,d=1}^3
\frac{1}{2} \pd{\theta}{x_d} \left(
\theta f_{\alpha-2e_k-e_d} + (1 - \delta_{|\alpha|,M})(\alpha_d + 1) f_{\alpha-2e_k+e_d}
\right)\\
&= -\frac{p}{\mu}f_{\alpha}\mathrm{H}(|\alpha|-2) , \quad |\alpha| \leq M.
\end{aligned}
\end{equation}
Next, we try to derive the linearized system of
\eqref{eq:moment-system}. This requires us to examine the case that
the distribution function is in a small neighborhood of an equilibrium
state
\[
f_0(\bxi) = \frac{\rho_0}{(2\pi\theta_0)^{\frac{3}{2}}} \mathrm{exp}
\left( - \frac{|\bxi|^2}{2\theta_0}\right) ,
\]
given by $\rho_0, \theta_0, \bu = 0$. We introduce the dimensionless
variables $\bar{\rho}$, $\bar{\theta}$, $\bar{\bu}$, $\bar{p}$,
$\bar{p}_{ij}$ and $\bar{f}_{\alpha}$ as
\begin{equation}\label{eq:dimensionless}
\begin{aligned}
&\rho = \rho_0 (1 + \bar{\rho}),\quad \bu = \sqrt{\theta_0} \bar{\bu},
\quad \theta = \theta_0 (1 + \bar{\theta}),
\quad p = p_0 (1 + \bar{p}),\\
&p_{ij}=p_0(\delta_{ij}+\bar{p}_{ij}),
\quad f_{\alpha}=\rho_0\theta_0^{\frac{|\alpha|}{2}} \cdot \bar{f}_{\alpha},
\quad \bx = L\cdot
\bar{\bx},\quad t = \frac{L}{\sqrt{\theta_0}}\bar{t},
\end{aligned}
\end{equation}
where $L$ is a characteristic length, $\bar{\bx}$ and $\bar{t}$ are
the dimensionless coordinates and time, respectively. Assume all the
dimensionless variables $\bar{\rho}$, $\bar{\theta}$, $\bar{\bu}$,
$\bar{p}$, $\bar{p}_{ij}$ and $\bar{f}_{\alpha}$ are small quantities.
Substituting \eqref{eq:dimensionless} into the globally hyperbolic
moment system \eqref{eq:moment-system}, and discarding all the
high-order small quantities, and noticing that $u_d \pd{\cdot}{x_d}$
is high-order small quantity, $\odd{\cdot}{t}\approx\pd{\cdot}{t}$, we
obtain the linearized HME as
\begin{equation}\label{eq:linear-system}
\begin{aligned}
&\pd{\bar{\rho}}{\bar{t}} + \sum_{d=1}^3\pd{\bar{u}_d}{\bar{x}_d} = 0,\\
&\pd{\bar{u}_k}{\bar{t}} + \pd{\bar{p}}{\bar{x}_k} +
\sum_{d=1}^3\pd{\bar{\sigma}_{kd}}{\bar{x}_d} = 0,\\
&\pd{\bar{p}_{ij}}{\bar{t}} + \sum_{d=1}^3\delta_{ij}\pd{\bar{u}_d}{\bar{x}_d} +
\pd{\bar{u}_j}{\bar{x}_i} +
\pd{\bar{u}_i}{\bar{x}_j} + \sum_{d=1}^3(e_i + e_j + e_d)!
\pd{\bar{f}_{e_i + e_j + e_d}}{\bar{x}_d} =
-\frac{\bar{\sigma}_{ij}}{{\Kn}},\\
&\begin{split}
&\pd{\bar{f}_{\alpha}}{\bar{t}} + \sum_{d=1}^3\pd{\bar{f}_{\alpha - e_d}}{\bar{x}_d}
+ \sum_{d=1}^3(\alpha_d + 1)(1 - \delta_M)
\pd{\bar{f}_{\alpha + e_d}}{\bar{x}_d} \\
&\qquad\qquad\qquad\qquad+ \sum_{d=1}^3\frac{1}{2}\delta_{\alpha,e_d+2e_k}
\pd{\bar{\theta}}{\bar{x}_d}
=-\frac{\bar{f}_{\alpha}}{{\Kn}},\quad 3 \leq |\alpha|\leq M,
\end{split}
\end{aligned}
\end{equation}
where $\bar{\sigma}_{ij}=\bar{p}_{ij}-\bar{p}\delta_{ij}$,
$i,j=1,2,3$, and $\delta_{\alpha, e_d+2e_k}$ is $1$ iff
$\alpha=e_d+2e_k$, otherwise is $0$.
The Knudsen number $\Kn$ is defined by
\begin{displaymath}
\Kn = \frac{\lambda}{L},
\end{displaymath}
where $\lambda = \frac{\mu}{p_0}\sqrt{\theta_0}$ is the mean free
path.
\section{Boundary Condition}\label{sec:bc}
In this paper, we adopt Maxwell's accommodation boundary condition
\cite{Maxwell}, which is the most commonly used boundary condition in
gas kinetic theory. It is formulated as a linear combination of the
specular
reflection and the diffuse reflection. Wall boundary only requires the
incoming half of the distribution function when $\bxi \cdot \bn > 0$,
where $\bn$ is the unit normal vector pointing into the gas. With the
given velocity $\bu^W(t,\bx)$ and temperature $\theta^W(t,\bx)$ of the
wall, the boundary condition at the wall is
\begin{equation}\label{eq:Maxwell}
f^W(t, \bx, \bxi) =
\left \{
\begin{array}{ll}
\chi f^W_M(t, \bx, \bxi) +(1 - \chi)f(t, \bx, \bxi^{\ast}),
& \bC^W \cdot \bn > 0, \\
f(t, \bx, \bxi), & \bC^W \cdot \bn \leq 0,
\end{array}
\right.
\end{equation}
where
\begin{equation}\label{eq:equilibrium}
\begin{aligned}
\bxi^{\ast} = \bxi - 2(\bC^W \cdot \bn)\bn,
\quad \bC^W = \bxi - \bu^W(t,\bx) ,\\
f^W_M(t, \bx, \bxi) =
\frac{\rho^W(t,\bx)}{(2\pi\theta^W(t,\bx))^{3/2}}
\exp\left(-\frac{|\bxi -
\bu^W(t,\bx)|^2}{2\theta^W(t,\bx)}\right),
\end{aligned}
\end{equation}
and $\chi \in [0,1]$ is the accommodation coefficient.
A boundary condition for general hyperbolic moment system was proposed
in \cite{Li}, which is derived from the Maxwell boundary condition by
calculating the expression of the moments at the wall. Here we are
purposely considering only steady shear flow, thus we adopt an
alternative approach to derive our boundary conditions. Let the unit
normal vector of the wall $\bn = (0,1,0)^T$. The velocity of the wall
$\bu^W = (u^W, 0, 0)$, and velocity for steady shear flow is
$\bu = (u_1, 0, 0)$. For $\bxi^{\ast} = (\xi_1,-\xi_2,\xi_3)$,
\eqref{eq:Maxwell} is precisely as
\begin{equation}\label{eq:wall-function}
f^W(\bx,\bxi) = \left \{
\begin{array}{ll}
\chi f^W_M(\bx, \bxi) +(1 - \chi)f(\bx, \bxi^{\ast}),
& \xi_2 > 0, \\
f(\bx, \bxi), & \xi_2 \leq 0.
\end{array}
\right .
\end{equation}
Denote $\Omega = \{\bxi \in \bbR^3\}$,
$\Omega^+ = \{\xi_1 \in \bbR, \xi_2 \in \bbR^+, \xi_3 \in \bbR \}$ and
$\Omega^- = \{\xi_1 \in \bbR, \xi_2 \in \bbR^-, \xi_3 \in \bbR \}$.
The integral of the wall distribution function
\eqref{eq:wall-function} with any function $\psi(\bC)$ gives us an
equation
\begin{equation}\label{eq:integral-equation}
\begin{aligned}
\int_{\Omega}\psi(\bC) f^W(\bx, \bxi) & \dd\bxi =
\int_{\Omega^-} \psi(\bC) f(\bx, \bxi) \dd\bxi \\
&+\int_{\Omega^+}\psi(\bC)\left(\chi f_M^W(\bx, \bxi-\bu^W) + (1 -
\chi) f(\bx, \bxi^{\ast})\right) \dd\bxi,
\end{aligned}
\end{equation}
where $\bC = (\xi_1-u_1, \xi_2, \xi_3)$.
Definitely, for HME one has to restrict the form of function
$\psi(\bC)$, otherwise \eqref{eq:integral-equation} will produce too
many boundary conditions. It is clear that we should restrict
ourselves to those $\psi$'s that the moments in the equation can be
retrieved. Thus those $\psi$'s are polynomials as $\bC^{\beta}$,
$|\beta\leq M$, where $\beta = (\beta_1, \beta_2, \beta_3) \in
\bbN^3$ is a 3D multi-index. Moreover, the distribution function of
shear flow is an
even function in the $\xi_3$ direction, which leads to
$f_{\beta} = 0$, for $\beta_3$ is odd. Following Grad's theory
\cite{Grad} to limit the number of boundary condition in order to
ensure the continuity of boundary conditions when $\chi \to 0$, only a
subset of all the moments corresponding to
\begin{equation}\label{set:bbI}
\{\bC^{\beta} \big| \beta \in \bbI \}, \qquad \text{where}\qquad
\bbI = \{|\beta| \leq M~\big|~\beta_2~\text{is odd and}~\beta_3
~\text{is even}\}
\end{equation}
can be used to construct the wall boundary conditions. Then
we reformulate the equation \eqref{eq:integral-equation} as
\begin{equation}\label{eq:bc-xi2}
\int_{\Omega^+}\bC^{\beta}f^W_M(\bx, \bxi-\bu^W) \dd\bxi =
\frac{1}{\chi}\left( \int_{\Omega^+}\bC^{\beta}\left(f(\bx, \bxi) -
(1-\chi)f(\bx, \bxi^{\ast})\right)\dd\bxi\right),
\quad \beta\in\bbI.
\end{equation}
Notice that the basis function defined in \eqref{eq:basis-fun} is
decoupled in compoents of $\bxi$. We then substitute
\eqref{eq:expansion} into \eqref{eq:bc-xi2} to calculate the integral
on both left and right hand side in \eqref{eq:bc-xi2},
respectively. To give the results, we first make some simplification
and define the following notations. Let
\begin{equation}
J_0(u,\theta) = 1,\quad
J_1(u,\theta) = u,\quad
J_{k+1}(u,\theta)= u J_k(u,\theta)+k\theta J_{k-1},
k\geq 1,
\end{equation}
then
\[
\frac{1}{\sqrt{2\pi\theta^W}}\int_{-\infty}^{\infty}(\xi_1 - u_1)^k
\exp\left(-\frac{|\xi_1-u_1^W|^2}{2\theta^W}\right)\dd\xi_1
=J_k(u_1^W-u_1,\theta^W).
\]
Let
\[
K(k,m) := \int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi}}\exp\left(
-\frac{|\xi|^2}{2} \right)\xi^k \He_m(\xi)\dd\xi,
\]
where $\He_m(\xi)$ is $m$-th Hermite polynomial, then using the
orthogonal relation of the Hermite polynomials, one can find
$K(0,m) = \delta_{0,m}$.
Denote the half space integral by
\begin{equation}
S^\star (k,m) := \int_0^{\infty}\xi^k
\He_m(\xi)\exp\left(-\frac{\xi^2}{2}\right)\dd\xi,
\end{equation}
and we have the following properties for $S^\star (k,m)$.
\begin{itemize}
\item Recursion relation:
\begin{equation}\label{eq:rec}
S^{\star}(k, m) = (k - 1) S^{\star}(k-2, m) + m S^{\star}(k-1, m-1).
\end{equation}
\item The value of $S^\star (k,m)$ is:
\begin{enumerate}
\item If $m \leq k$:
\begin{enumerate}
\item If $k - m$ is even, $S^{\star}(k, m) = \sqrt{2 \pi}\cdot A$;
\item If $k - m$ is odd, $S^{\star}(k, m) = B$;
\end{enumerate}
here $A$ and $B$ are two algebraic numbers.
\item If $m > k$ and $k - m$ is even, $S^{\star}(k, m) = 0$.
\end{enumerate}
\end{itemize}
Let
\begin{equation}\label{eq:integralS}
\begin{aligned}
S(k,m) &:= \frac{\hat{\chi}}{\sqrt{2\pi}} S^\star(k,m) \\
&\ = \frac{\theta^{(m-k)/2}}{\chi}
\int_0^{\infty}\xi^k\left(\He_m(\xi)-
(1-\chi)\He_m(-\xi)\right)\exp\left( -\frac{|\xi|^2}{2} \right)\dd\xi,
\end{aligned}
\end{equation}
where
\[
\hat{\chi} = \left\{
\begin{array}{ll}
1, & m ~\text{is even},\\
\frac{2-\chi}{\chi}, & m ~\text{is odd},
\end{array} \right.
\]
then for each $\beta \in \bbI$ in \eqref{set:bbI}, the left and right
hand side of \eqref{eq:bc-xi2} can be represented by
\begin{equation}\label{eq:bc}
\begin{aligned}
\text{lhs of \eqref{eq:bc-xi2}}
&=\frac{\rho^W \left(\theta^W\right)^{(\beta_2+\beta_3)/2}}{\sqrt{2\pi}}
J_{\beta_1}\left(u_1^W-u_1,\theta^W\right)
(\beta_2-1)!!(\beta_3-1)!!,\\
\text{rhs of \eqref{eq:bc-xi2}} &=
\sum_{\alpha\in\bbN^3} \left(
K(\beta_1,\alpha_1)
S(\beta_2,\alpha_2)
K(\beta_3,\alpha_3)
\theta^{(\beta_2-\alpha_2)/2}
\right) f_{\alpha}.
\end{aligned}
\end{equation}
Noticing $K(0,m)=\delta_{0,m}$, by setting $\beta=e_2$ in
\eqref{eq:bc}, we have
\begin{equation}\label{eq:e2bc}
\rho^W\sqrt{\frac{\theta^W}{2\pi}} = \sum_{m=0}^{\infty}
S(1,m)\frac{f_{me_2}}{{\theta}^{(m-1)/2}}.
\end{equation}
Let $p_{w} = \rho^W\sqrt{\theta^W \theta}$, then we have
\begin{equation}
p_{w} =
\sqrt{2\pi\theta}\sum_{m=0}^{\infty}S(1,m)\frac{f_{me_2}}{\theta^{(m-1)/2}}
=p + f_{2e_2} - \frac{f_{4e_2}}{\theta} + \frac{3}{\theta^2}
f_{6e_2} - \frac{15}{\theta^3}f_{8e_2} + \cdots.
\end{equation}
The boundary condition for the case $\beta = e_1+\beta_2e_2 \in \bbI$
in \eqref{set:bbI} is
\begin{equation}\label{eq:beta2oddbc}
\frac{\rho^W}{\sqrt{2\pi}}(\theta^W)^{\frac{\beta_2}{2}}(u_1^W - u_1)
(\beta_2 -1)!! = \sum_{\alpha_2} S(\beta_2,
\alpha_2) f_{e_1+ \alpha_2e_2} \theta^{(\beta_2-\alpha_2)/2}.
\end{equation}
Particularly, for the case $\beta = e_1+e_2$, one has
\[
p_{w}\sqrt{\frac{\theta^W}{2\pi\theta}}(u_1^W - u_1) =
S(1,1) \sigma_{12} + \sum_{\alpha_2 > 1}
S(1,\alpha_2) f_{e_1+\alpha_2e_2}\theta^{(1-\alpha_2)/2}.
\]
Here we only consider the boundary condition for the specific case
that $\beta = e_1+\beta_2e_2 \in \bbI$ in \eqref{eq:beta2oddbc},
which is
\begin{equation}\label{eq:bc-palpha}
p_{w}
\frac{\left(\theta^W\right)^{\frac{\beta_2-1}{2}}}{\sqrt{2\pi}}(\beta_2 -1)!!
(u_1^W - u_1) =
\sum_{\alpha_2} \theta^{\frac{1+\beta_2-\alpha_2}{2}}
S(\beta_2, \alpha_2) f_{e_1+\alpha_2e_2}.
\end{equation}
We linearize this condition at $\theta_0$ as that in
\eqref{eq:dimensionless} for our purpose, and assume
$\theta^W-\theta_0$ is a small quantity. By substituting
\eqref{eq:dimensionless} into \eqref{eq:bc-palpha}, and applying the
closure of HME, i.e $f_{\alpha}=0$, $|\alpha|>M$, the linearized
boundary condition is arrived at as
\begin{equation}\label{bc:linear}
\frac{(\beta_2 -1)!!}{\sqrt{2\pi}}(\bar{u}_1^W - \bar{u}_1) = \sum_{\alpha_2\leq M}
S(\beta_2,\alpha_2) \bar{f}_{e_1+\alpha_2e_2},
\end{equation}
where $\bar{u}_1^W$ is defined as dimensionless variable $u_1^W =
\sqrt{\theta_0}\bar{u}_1^W$, and $\beta_2$ is odd and $|\beta_2|\leq
M$.
\section{Kramers' Problem}
\label{sec:kramers}
Our setup for Kramers' problem is standard. The gas flow in a
half-space over a flat wall is considered, and the coordinates are
chosen such that $x$ direction is parallel to the wall, and $y$
direction is perpendicular to the wall. The solid wall is fixed on
$\bar{y} = 0~(\bar{u}_1^W = 0)$. The temperature and density of the
gas far from the wall are constant. Gas velocity is $\bar{\bu} =
(\bar{u}_1, 0, 0)$ and all derivatives in equations
\eqref{eq:linear-system} in $x$ and $z$ direction are zero.
\subsection{Formal solution of linearized HME}
We give the formal solution of the linearized HME at first. The setup
of Kramers' problem makes the equations of linearized moment system
\eqref{eq:linear-system} related to velocity decoupled from the whole
linearized moment system, which enables us to investigate the velocity
by studying a small system as
\begin{equation}\label{eq:velocity}
\begin{aligned}
&\od{\bar{\sigma}_{12}}{\bar{y}} = 0, \\
&\od{\bar{u}_1}{\bar{y}} + 2\od{\bar{f}_{e_1+2e_2}}{\bar{y}}
= -\frac{1}{\Kn}\bar{\sigma}_{12}, \\
&\od{\bar{\sigma}_{12}}{\bar{y}} +
3\od{\bar{f}_{e_1+3e_2}}{\bar{y}} =
-\frac{1}{\Kn}\bar{f}_{e_1+2e_2}, \\
&\cdots \\
&\od{\bar{f}_{e_1+ (M-2)e_2}}{\bar{y}}
= -\frac{1}{\Kn}\bar{f}_{e_1+(M-1)e_2}.
\end{aligned}
\end{equation}
We collect the variables involved in \eqref{eq:velocity} into a vector
\[
V = \left(\bar{u}_1, \bar{\sigma}_{12}, \bar{f}_{e_1+2e_2},
\bar{f}_{e_1+3e_2},\cdots, \bar{f}_{e_1+(M-1)e_2}\right)^T,
\]
and then \eqref{eq:velocity} is formulated as
\begin{equation}\label{eq:simple-velocity}
\bM \od{V}{\bar{y}} = -\frac{1}{\Kn}\bQ V,
\end{equation}
where
\begin{equation}\label{eq:def_MQ}
\bM = \left(
\begin{array}{cccccc}
0 & 1 & & & & \\
1 & 0 & 2 & & & \\
& 1 & 0 & 3 & & \\
& & \ddots & \ddots & \ddots & \\
& & & 1 & 0 & M - 1\\
& & & & 1 & 0
\end{array}
\right), \quad \bQ = \left(
\begin{array}{cccc}
0 & & & \\
& 1 & & \\
& & \ddots & \\
& & & 1
\end{array}
\right).
\end{equation}
Easy to check that the matrix $\bM$ is real diagonalizable.
Actually, we have the eigen-decomposition of $\bM$
as $\bM = \bR \bLambda \bR^{-1}$, where $\bR$ is the Hermite transformation matrix
\begin{equation}\label{mat:eigen-vec}
\bR = (r_{ij})_{M\times M},\quad
r_{ij}=\frac{\He_{i-1}(\lambda_j)}{(i-1)!},
\quad i,j=1,\cdots,M,
\end{equation}
and $\bLambda = \diag\{\lambda_i; i = 1, \cdots, M\}$,
where the eigenvalues $\lambda_i$, $i=1,\cdots,M$ are zeros of the
$M$-th order Hermite polynomial $\He_M(x)$. We sort the eigenvalues
$\lambda_i$ in decending order, saying $\lambda_i >
\lambda_{i+1}$. The diagonal matrix $\bLambda$ can then be written as
\begin{equation}\label{mat:Lamb}
\bLambda = \left(
\begin{array}{cc}
\bLambda_+ & \\
& \bLambda_{\leq 0}
\end{array}
\right),
\end{equation}
\begin{equation}\label{mat:pos-neg}
\begin{aligned}
& \bLambda_+ = \diag \left\{\lambda_i;~ i =
1, \cdots, \lfloor \frac{M}{2} \rfloor \right\}, \\
& \bLambda_{\leq 0} = \diag \left\{\lambda_i;~ i =
\lfloor \frac{M}{2} \rfloor + 1, \cdots, M \right\}.
\end{aligned}
\end{equation}
The first equation of \eqref{eq:simple-velocity} indicates
$\bar{\sigma}_{12}$ is a constant and the second equation of
\eqref{eq:simple-velocity} gives that
\[
\bar{u}_1(\bar{y}) = -\dfrac{\bar{y}}{\Kn} \bar{\sigma}_{12} -
2\bar{f}_{e_1+2e_2}(\bar{y}) + c_0,
\]
where $c_0$ is a constant to be determined. We denote
\[
\hat{V} = (\bar{f}_{e_1+2e_2}, \bar{f}_{e_1+3e_2}, \cdots,
\bar{f}_{e_1+(M-1)e_2})^T,
\]
which is the remaining part of $V$ excluded the first two variables
$\bar{u}_1$ and $\bar{\sigma}_{12}$. Then the system with higher order
moments is separated from \eqref{eq:simple-velocity}, which reads as
\begin{equation}\label{eq:matrix-hatM}
\hat{\bM} \od{\hat{V}}{\bar{y}} = -\frac{1}{\Kn} \hat{V},
\end{equation}
where
\[
\hat{\bM} = \left(
\begin{array}{cccccc}
0 & 3 & & & & \\
1 & 0 & 4 & & & \\
& 1 & 0 & 5 & & \\
& & \ddots & \ddots & \ddots & \\
& & & 1 & 0 & M - 1\\
& & & & 1 & 0
\end{array}
\right).
\]
Correspondingly to the matrix $\bM$, the matrix $\hat{\bM}$ is real
diagonalizable, too. Precisely, let
\[
\hat{\He}_0(x) = 1,~\hat{\He}_1(x)=x,~
\hat{\He}_{k+1}(x)=x\hat{\He}_k(x)-(k+2)\hat{\He}_{k-1}(x),
~k\geq1,
\]
then the characteristic polynomial of $\hat{\bM}$ is
$\hat{\He}_{M-2}(\lambda)$. The recursion relation implies that
$\hat{\He}_k(x)$ has $k$ real and simple zeros, and thus $\hat{\bM}$
is real diagonalizable and the eigenvalues $\hat{\lambda}_i$,
$i = 1, \cdots, M-2$, are the zeros of $\hat{\He}_{M-2}
(\lambda)$.
Furthermore, if $\hat{\lambda}_i$ is an eigenvalue of $\hat{\bM}$,
then $-\hat{\lambda}_i$ has to be an eigenvalue of $\hat{\bM}$, since
$\hat{\He}_{M-2}(x)$ is an odd function if $M$ is odd and is an even
function if $M$ is even. As for the matrix $\bM$, we sort the
eigenvalues $\hat{\lambda}_i$ in decending order, too, to make the
diagonal matrix
\[
\hat{\bLambda} = \left(
\begin{array}{cc}
\hat{\bLambda}_+ & \\
& \hat{\bLambda}_{\leq 0}
\end{array}
\right),
\]
\[
\begin{aligned}
& \hat{\bLambda}_+ = \diag \left\{ \hat{\lambda}_i;~ i = 1, \cdots,
\lfloor \frac{M}{2} \rfloor - 1 \right\}, \\
& \hat{\bLambda}_{\leq 0} = \diag \left\{ \hat{\lambda}_i;~ i =
\lfloor \frac{M}{2} \rfloor, \cdots, M - 2 \right\}.
\end{aligned}
\]
Then eigen-decomposition of $\hat{\bM}$ is
$\hat{\bM} = \hat{\bR} \hat{\bLambda} \hat{\bR}^{-1}$, where
\begin{equation}\label{eq:def_hatbR}
\hat{\bR}=(\hat{r}_{ij})_{(M-2)\times(M-2)},\quad
\hat{r}_{ij}=\frac{\hat{\He}_{i-1}(\hat{\lambda}_j)}{(i+1)!},
\quad i,j=1,\cdots,M-2.
\end{equation}
Let us define the matrix $\hat{\bR}_+$ as the left
$\lfloor \dfrac{M}{2} \rfloor - 1$ colomns of $\hat{\bR}$,
$\hat{\bR}_-$ as the right $\lceil \dfrac{M}{2} \rceil - 1$ colomns of
$\hat{\bR}$ for latter usage, and precisely, we have
\[
\hat{\bR}_+ = \left(
\begin{array}{ccc}
\frac{\hat{\He}_0(\hat{\lambda}_1)}{2!} & \cdots
& \frac{\hat{\He}_0(\hat{\lambda}_{\lfloor \frac{M}{2} \rfloor - 1})}{2!} \\
\vdots & \ddots & \vdots \\
\frac{\hat{\He}_{M-3}(\hat{\lambda}_1)}{(M-1)!} & \cdots
& \frac{\hat{\He}_{M-3}(\hat{\lambda}_{\lfloor \frac{M}{2} \rfloor - 1})}{(M-1)!}
\end{array}\right), \quad
\hat{\bR}_- = \left(
\begin{array}{ccc}
\frac{\hat{\He}_0(\hat{\lambda}_{\lfloor \frac{M}{2} \rfloor})}{2!} & \cdots
& \frac{\hat{\He}_0(\hat{\lambda}_{M-2})}{2!} \\
\vdots & \ddots & \vdots \\
\frac{\hat{\He}_{M-3}(\hat{\lambda}_{\lfloor \frac{M}{2} \rfloor})}{(M-1)!} & \cdots
& \frac{\hat{\He}_{M-3}(\hat{\lambda}_{M-2})}{(M-1)!}
\end{array}\right).
\]
Let $\hat{\bR}_{+,\text{even}}$, which is made with the even rows of
$\hat{\bR}_+$ as
\[
\begin{aligned}
\hat{\bR}_{+,\text{even}} &\triangleq
(\hat{r}_{ij}), \text{~where~} i \text{~is
even,~} j = 1,\cdots,\lfloor \frac{M}{2} \rfloor - 1, \\
&=\left(
\begin{array}{cccc}
\dfrac{\hat{\He}_1(\hat{\lambda}_1)}{3!}
& \dfrac{\hat{\He}_1(\hat{\lambda}_2)}{3!}
& \cdots
& \dfrac{\hat{\He}_1(\hat{\lambda}_{\lfloor \frac{M}{2} \rfloor - 1})}{3!} \\
\dfrac{\hat{\He}_3(\hat{\lambda}_1)}{5!}
& \dfrac{\hat{\He}_3(\hat{\lambda}_2)}{5!}
& \cdots
& \dfrac{\hat{\He}_3(\hat{\lambda}_{\lfloor \frac{M}{2} \rfloor - 1})}{5!} \\
\vdots & \vdots & \cdots & \vdots \\
\end{array}
\right),
\end{aligned}
\]
be a $\lfloor \frac{M}{2} \rfloor - 1 \times \lfloor \frac{M}{2}
\rfloor - 1$ square matrix. And corresponding to
$\hat{\bR}_{+,\text{even}}$, define $\hat{\bR}_{+,\text{odd}},
\hat{\bR}_{-,\text{odd}}, \hat{\bR}_{-,\text{odd}}$ as
\[
\begin{aligned}
\hat{\bR}_{+,\text{odd}} &\triangleq
(\hat{r}_{ij}), \text{~where~} i \text{~is
odd,~} j = 1,\cdots,\lfloor \frac{M}{2} \rfloor - 1, \\
\hat{\bR}_{-,\text{even}} &\triangleq
(\hat{r}_{ij}), \text{~where~} i \text{~is
even,~} j = \lfloor \frac{M}{2} \rfloor,\cdots,M-2, \\
\hat{\bR}_{-,\text{odd}} &\triangleq
(\hat{r}_{ij}), \text{~where~} i \text{~is
odd,~} j = \lfloor \frac{M}{2} \rfloor,\cdots,M-2.
\end{aligned}
\]
We declare that
\begin{lemma}\label{lem:R_peven_invertible}
$\hat{\bR}_{+,\text{even}}$ is invertible.
\end{lemma}
\begin{proof}
Let $\boldsymbol{P}_\sigma$ to be the permutation matrix of the
permutation
\begin{equation}\label{mat:P_sigma}
\sigma: \left\{ 1, 2, \cdots , M-2 \right\} \rightarrow \left\{ 1,
2, \cdots , M-2 \right\},
\end{equation}
that
\[
\sigma(i) = \mod(i,2) \times (\lfloor \frac{M}{2} \rfloor - 1) +
\lfloor i/2 \rfloor.
\]
The permutation maps the list of numbers $1, 2, \cdots, M -2 $ to
\[
2, 4, 6, \cdots, 1, 3, 5, \cdots
\]
that the even numbers are ahead of the odd numbers.
Then matrix $\hat{\bR}$ is re-organized by the
permutation matrix as
\[
\boldsymbol{P}^{-1}_\sigma \hat{\bR} = \left(
\begin{array}{c|c}
\hat{\bR}_{+,\text{even}} & \hat{\bR}_{-,\text{even}}\\
\hat{\bR}_{+,\text{odd}} & \hat{\bR}_{-,\text{odd}}
\end{array}
\right).
\]
Notice that each eigenvalue $\hat{\lambda}_i \in \hat{\bLambda}_+$,
$-\hat{\lambda}_i \in \hat{\bLambda}_{\leq 0}$. Then for any
eigenvector
\[
\hat{\br}_i = (\hat{\br}_{i,\text{even}} |
\hat{\br}_{i,\text{odd}})^T \in (\hat{\bR}_{+,\text{even}} |
\hat{\bR}_{+,\text{odd}})^T,
\]
there exists a column vector
\[
\hat{\br}_j = (-\hat{\br}_{i,\text{even}} |
\hat{\br}_{i,\text{odd}})^T \in (\hat{\bR}_{-,\text{even}} |
\hat{\bR}_{-,\text{odd}})^T.
\]
Then
\[
\hat{\br}_i - \hat{\br}_j = 2(\hat{\br}_{i,\text{even}} |
\boldsymbol{0})^T.
\]
The set of vectors $\hat{\br}_i - \hat{\br}_j$ are linearly
independent since $\hat{\br}_i, \hat{\br}_j$ are eigenvectors of
$\hat{\bR}$, then
columns of $\hat{\bR}_{+,\text{even}}$ are linearly independent.
Thus $\hat{\bR}_{+,\text{even}}$ is invertible.
\end{proof}
\subsubsection{Illustrative examples: $M \leq 5$}
We examine the cases for small $M$ to find out the formal solution for
generic $M$. The simplest system is the case for $M = 3$. The
variables are $V = (\bar{u}_1, \bar{\sigma}_{12},
\bar{f}_{e_1+2e_2})^T$, and matrices $\bM$ and $\bQ$ in
\eqref{eq:simple-velocity} are
\[
\bM =\left(
\begin{array}{ccc}
0 & 1 & 0 \\
1 & 0 & 2\\
0 & 1 & 0
\end{array} \right),
\quad
\bQ = \left(
\begin{array}{ccc}
0 & & \\
& 1 & \\
& & 1
\end{array} \right).
\]
Since $\bar{\sigma}_{12}$ is constant and the velocity is
\[
\bar{u}_1 = -\bar{\sigma}_{12}\frac{\bar{y}}{\Kn} + c_0,
\]
it is clear that the solution of $\bar{u}_1$ is not able to capture
the boundary layer since here $\bar{u}_1$ is a linear function of
$\bar{y}$. To capture the boundary layer of velocity, we need more
moments thus we turn to the case $M = 4$. For $M = 4$, the equations
\eqref{eq:velocity} are
\begin{equation}\label{eq:vel-M4}
\begin{aligned}
&\od{\bar{\sigma}_{12}}{\bar{y}} = 0, \\
&\od{\bar{u}_1}{\bar{y}} + 2\od{\bar{f}_{e_1+2e_2}}{\bar{y}}
= -\frac{1}{\Kn}\bar{\sigma}_{12}, \\
&\od{\bar{\sigma}_{12}}{\bar{y}} +
3\od{\bar{f}_{e_1+3e_2}}{\bar{y}} = -\frac{1}{\Kn}\bar{f}_{e_1+2e_2}, \\
&\od{\bar{f}_{e_1+2e_2}}{\bar{y}} = -\frac{1}{\Kn}\bar{f}_{e_1+3e_2}. \\
\end{aligned}
\end{equation}
The solution gives us the expression of velocity as
\begin{equation} \label{eq:velo-ori}
\bar{u}_1 = - \bar{\sigma}_{12}\frac{\bar{y}}{\Kn} - 2 \bar{f}_{e_1+2e_2} + c_0
\end{equation}
from the second equation in \eqref{eq:vel-M4}. Here we need to solve the equations
\eqref{eq:matrix-hatM} for $\hat{V} = (\bar{f}_{e_1+2e_2}, \bar{f}_{e_1+3e_2})^T$,
where
\begin{equation}\label{mat:tridiagonal}
\hat{\bM} = \left(
\begin{array}{cc}
0 & 3\\
1 & 0
\end{array}
\right ).
\end{equation}
The matrix $\hat{\bM}$ can be decomposited as
$\hat{\bM} = \hat{\bR} \hat{\bLambda} \hat{\bR}^{-1}$,
\[
\hat{\bLambda} = \left(
\begin{array}{cc}
\sqrt{3} & \\
& -\sqrt{3}
\end{array} \right), \qquad
\hat{\bR} = \left(
\begin{array}{cc}
1 & 1 \\
\frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{3}}
\end{array} \right).
\]
Hence, the solution of system \eqref{eq:matrix-hatM} is
\begin{equation*}
\hat{V} = \hat{\bR} \exp \left(-\frac{\bar{y}}{\Kn} \hat{\bLambda}^{-1} \right)
\hat{\bR}^{-1} \hat{V}^{(0)}.
\end{equation*}
By setting
$\hat{\bc}= (\hat{c}_1, \hat{c}_2)^T = \hat{\bR}^{-1}\hat{V}^{(0)}$,
the equations above result in
\begin{equation*}
\hat{V}=\left(\begin{array}{l}
\bar{f}_{e_1+2e_2}\\
\bar{f}_{e_1+3e_2}
\end{array}\right)
=\hat{\bR} \exp \left(-\frac{\bar{y}}{\Kn} \hat{\bLambda}^{-1}
\right)\hat{\bc}
=
\left( \begin{array}{l}
\hat{c}_1 \exp(-\frac{\bar{y}}{\sqrt{3}\Kn}) +
\hat{c}_2\exp(\frac{\bar{y}}{\sqrt{3}\Kn})\\
\frac{\sqrt{3}}{3} \hat{c}_1
\exp(-\frac{\bar{y}}{\sqrt{3}\Kn}) - \frac{\sqrt{3}}{3} \hat{c}_2
\exp(\frac{\bar{y}}{\sqrt{3}\Kn})
\end{array} \right).
\end{equation*}
The exponential terms provide us the boundary layer. Since all the
variables have to remain finite as $\bar{y} \to \infty$, the term
$\exp(\frac{1}{\sqrt{3}\Kn}\bar{y})$ has to be dropped. Therefore,
\[
\left(\begin{array}{l}
\bar{f}_{e_1+2e_2}\\
\bar{f}_{e_1+3e_2}
\end{array}\right)
=\hat{\bR} \left(
\begin{array}{cc}
\exp \left(-\frac{\bar{y}}{\Kn} \hat{\bLambda}_+^{-1} \right)
& \\ & \bzero
\end{array}
\right) \hat{\bc}
= \hat{c}_1 \exp(-\frac{\bar{y}}{\sqrt{3}\Kn}) \left(
\begin{array}{c}
1 \\
\frac{\sqrt{3}}{3}
\end{array}
\right).
\]
Here $\hat{\bLambda}_{+}$ is a $1 \times 1$ matrix with its entry as
$\sqrt{3}$. Applying the linearized boundary condition
\eqref{bc:linear}, i.e.
\begin{equation*}
\begin{aligned}
&- \frac{1}{\sqrt{2\pi}} \bar{u}_1 = S(1,1) \bar{\sigma}_{12} + S(1,2)
\bar{f}_{e_1+2e_2} + S(1,3)\bar{f}_{e_1+3e_2}, \\
&- \frac{2}{\sqrt{2\pi}} \bar{u}_1 = S(3,1) \bar{\sigma}_{12} + S(3,2)
\bar{f}_{e_1+2e_2} + S(3,3)\bar{f}_{e_1+3e_2},
\end{aligned}
\end{equation*}
we can obtain
\begin{equation*}
\hat{c}_1 = -\frac{\sqrt{\pi}(\chi - 2)}{2(\sqrt{3\pi}(2 -
\chi) + 2\sqrt{2}\chi)} \bar{\sigma}_{12}, \quad c_0 =
\sqrt{\frac{\pi}{2}} \frac{\chi - 2}{\chi}\left( 1 +
\frac{\sqrt{2}\chi}{4\sqrt{2}\chi + 2\sqrt{3\pi}(2 - \chi)}\right)
\bar{\sigma}_{12}.
\end{equation*}
Then the solution of velocity is
\begin{equation*}
\bar{u}_1 = - \bar{\sigma}_{12}\frac{\bar{y}}{\Kn}
-2\hat{c}_1\exp(-\frac{\bar{y}}{\sqrt{3}\Kn}) + c_0.
\end{equation*}
Similar procedure can be carried out for greater $M$. For example, if
we set $M = 5$, then
$V = (\bar{u}_1, \bar{\sigma}_{12}, \bar{f}_{e_1+2e_2},
\bar{f}_{e_1+3e_2}, \bar{f}_{e_1+4e_2})^T$
and $\hat{V} = (\bar{f}_{e_1+2e_2}, \bar{f}_{e_1+3e_2}, \bar{f}_{e_1+4e_2})^T$. The
matrix $\hat{\bM} = \hat{\bR} \hat{\bLambda}\hat{\bR}^{-1}$ in
\eqref{eq:matrix-hatM} is
\begin{equation*}
\hat{\bM} = \left(
\begin{array}{ccc}
0 & 3 & 0 \\
1 & 0 & 4 \\
0 & 1 & 0
\end{array}
\right ) \text{ with }
\hat{\bLambda} = \left(
\begin{array}{ccc}
\sqrt{7} & & \\
& 0 & \\
& & -\sqrt{7}
\end{array} \right),
~~
\hat{\bR} = \left(
\begin{array}{ccc}
1 & 1 & 1 \\
\sqrt{7}/3 & 0 & -\sqrt{7}/3 \\
1/3 & -1/4 & 1/3
\end{array} \right).
\end{equation*}
Notice that $M = 5$ is odd, so zero is a simple eigenvalue of
$\hat{\bM}$ . This vanished eigenvalue provides a constant factor in
the exponential terms in the boundary layer, while the eigenvalue
$\sqrt{7}$ of matrix $\hat{\bM}$ provides the only stable term which
survives in the solution. The solution of \eqref{eq:matrix-hatM} is
\begin{equation}
\hat{V}=
\left(\begin{array}{l}
\bar{f}_{e_1+2e_2}\\
\bar{f}_{e_1+3e_2}\\
\bar{f}_{e_1+4e_2}\\
\end{array}\right)
=\hat{\bR} \left(
\begin{array}{cc}
\exp \left(-\frac{\bar{y}}{\Kn} \hat{\bLambda}_+^{-1} \right)
& \\ & \bzero
\end{array}
\right) \hat{\bc}
= \hat{c}_1 \exp(-\frac{\bar{y}}{\sqrt{7}\Kn})
\left( \begin{array}{c}
1 \\
\frac{\sqrt{7}}{3} \\
\frac{1}{3}
\end{array} \right),
\end{equation}
where
$\hat{\bc}=(\hat{c}_1, \hat{c}_2, \hat{c}_3)^T =
\hat{\bR}^{-1}\hat{V}^{(0)}$
and the entry of the $1 \times 1$ matrix $\hat{\bLambda}_+$ is
$\sqrt{7}$. Similar as the case $M = 4$, there are $2$ coefficients
$c_0$ and $\hat{c}_1$ to be determined. To fix the coefficients, we
utilize two boundary conditions by setting $\beta_2 = 1,3$ in
\eqref{bc:linear}
\begin{equation*}
\begin{aligned}
&- \frac{1}{\sqrt{2\pi}} \bar{u}_1 = S(1,1) \bar{\sigma}_{12} + S(1,2)
\bar{f}_{e_1+2e_2} + S(1,3)\bar{f}_{e_1+3e_2} + S(1,4)\bar{f}_{e_1+4e_2} , \\
&- \frac{2}{\sqrt{2\pi}} \bar{u}_1 = S(3,1) \bar{\sigma}_{12} + S(3,2)
\bar{f}_{e_1+2e_2} + S(3,3)\bar{f}_{e_1+3e_2} + S(3,4)\bar{f}_{e_1+4e_2}.
\end{aligned}
\end{equation*}
Direct calculations yield
\begin{equation*}
\hat{c}_1 = -\frac{3\sqrt{\pi}(\chi - 2)}{2(3\sqrt{7\pi}(\chi -
2) - 10\sqrt{2}\chi)} \bar{\sigma}_{12}, \quad c_0 = \sqrt{\frac{\pi}{2}}
\frac{\chi - 2}{\chi}\left(1 -
\frac{2\sqrt{2}\chi}{3\sqrt{7\pi}(\chi - 2) - 10\sqrt{2}\chi}\right)
\bar{\sigma}_{12}.
\end{equation*}
Then the solution of velocity is given by
\begin{equation*}
\bar{u}_1 = -\bar{\sigma}_{12}\frac{\bar{y}}{\Kn} - 2\hat{c}_1
\exp\left(-\frac{\bar{y}}{\sqrt{7}\Kn}\right) + c_0.
\end{equation*}
\subsubsection{General case: arbitrary $M$}
Now we are ready to present the formal solution for arbitrary
$M$. Following the examples above, we have to drop those unbounded
factors to attain a stable solution that only the terms contributed
from the positive eigenvalues of $\hat{\bM}$ are kept. Thus the stable
solution of \eqref{eq:matrix-hatM} is
\begin{equation}\label{sol:general}
\hat{V}(\bar{y}) = \hat{\bR} \left(
\begin{array}{cc}
\exp \left(-\frac{\bar{y}}{\Kn} \hat{\bLambda}_{+}^{-1}\right)
& \\ & \bzero
\end{array}
\right)
\hat{\bc},
\end{equation}
where
$\hat{\bc}=(\hat{c}_1, \cdots, \hat{c}_{M-2})^T = \hat{\bR}^{-1}
\hat{V}^{(0)}$.
Clearly, there are only the beginning $\lfloor \dfrac{M}{2} \rfloor - 1$
entries in $\hat{\bc}$ appears in $\hat{V}(\bar{y})$. With the expression of
$\bar{f}_{e_1+2e_2}(\bar{y})$ provided as the first entry of
$\hat{V}(\bar{y})$, the velocity is again given by the second equation
in \eqref{eq:velocity} as
\begin{equation}\label{sol:velocity}
\bar{u}_1(\bar{y}) = - \bar{\sigma}_{12}\frac{\bar{y}}{\Kn} - 2
\be_1^T\hat{V}(\bar{y}) + c_0,
\end{equation}
where $\be_1 = (1, 0, \cdots, 0)^T$. Since $c_0$ in the expression of
$\bar{u}_1(\bar{y})$ is also to be determined, there are in total
$\lfloor \dfrac{M}{2} \rfloor$ indeterminate coefficients in $V(\bar{y})$.
Combining \eqref{sol:general} and \eqref{sol:velocity} with linearized
boundary condition \eqref{bc:linear}, we can obtain the boundary
condition for \eqref{eq:matrix-hatM} as
\begin{equation}\label{eq:simple-bc}
-\frac{(\beta_2 -1)!!}{\sqrt{2\pi}} \bar{u}_1 = S(\beta_2,1)\bar{\sigma}_{12} +
\sum_{\alpha_2=2}^{M-1} S(\beta_2,\alpha_2) \bar{f}_{e_1+\alpha_2e_2},
\quad \beta_2=1,3,\cdots,2\lfloor\frac{M}{2}\rfloor-1.
\end{equation}
The total number of boundary condition is $\lfloor\frac{M}{2}\rfloor$,
which may fix all coefficients in the solution of $V(\bar{y})$. Once these
coefficients are fixed by the boundary conditions, we eventually
attain $\bar{u}_1$ formated as
\begin{equation}\label{eq:sol_u1_formal}
\begin{aligned}
\bar{u}_1(\bar{y}) &= - \bar{\sigma}_{12}\frac{\bar{y}}{\Kn} - 2
\be_1^T\hat{V}(\bar{y}) + c_0 \\
&= - \bar{\sigma}_{12}\frac{\bar{y}}{\Kn} - 2 \be_1^T \hat{\bR} \left(
\begin{array}{cc}
\exp \left(-\frac{\bar{y}}{\Kn} \hat{\bLambda}_{+}^{-1}\right)
& \\ & \bzero
\end{array}
\right)
\hat{\bc} + c_0\\
&= - \bar{\sigma}_{12}\frac{\bar{y}}{\Kn} - 2\sum_{i = 1}^{\lfloor
\frac{M-2}{2} \rfloor}\hat{c_i}
\exp\left(-\frac{\bar{y}}{\Kn\hat{\lambda}_i}\right) +c_0.
\end{aligned}
\end{equation}
We let $\bar{y} = 0$ in \eqref{eq:sol_u1_formal} to have
$\bar{u}_1 = -2 \sum_{i = 1}^{\lfloor \frac{M}{2} \rfloor -1}\hat{c_i} +
c_0$
and substitute it into \eqref{eq:simple-bc} to obtain the following
linear system
\begin{equation}\label{sol:linear}
\begin{aligned}
& \quad \qquad - \frac{-2 \displaystyle \sum_{i = 1}^{\lfloor
\frac{M}{2} \rfloor-1}\hat{c_i} + c_0}{\sqrt{2\pi}}\left(
\begin{array}{c}
1 \\
(3 - 1)!!\\
\vdots\\
(2\lfloor \frac{M}{2} \rfloor - 2)!!
\end{array}\right) = \bar{\sigma}_{12} \left(
\begin{array}{c}
S(1,1) \\ (S(3,1) \\ \vdots \\
S(2\lfloor \frac{M}{2} \rfloor -1,1)
\end{array}
\right) \\
& + \left( \begin{array}{cccc}
S(1,2) & S(1,3) &\cdots & S(1,M-1) \\
S(3,2) & S(3,3) &\cdots & S(3,M-1) \\
\vdots & \vdots &\ddots & \vdots \\
S(2\lfloor \frac{M}{2} \rfloor -1,2)
& S(2\lfloor \frac{M}{2} \rfloor -1,3)
& \cdots & S(2\lfloor \frac{M}{2} \rfloor -1,M-1)
\end{array} \right)
\hat{\bR} \left(
\begin{array}{c}
\hat{c}_1 \\ \vdots \\ \hat{c}_{\lfloor \frac{M}{2} \rfloor - 1}
\\ 0 \\ \vdots \\ 0
\end{array}
\right)
\end{aligned}
\end{equation}
Clearly, this is a linear system of for
$\bc = (c_0, \hat{c}_1, \cdots, \hat{c}_{\lfloor \frac{M}{2}
\rfloor-1})^T$. Precisely, we let
\[
\bh = \frac{1}{\sqrt{2\pi}} \left(
\begin{array}{c}
1 \\
(3 - 1)!!\\
\vdots\\
(2\lfloor \frac{M}{2} \rfloor - 2)!!
\end{array}\right),
\qquad \bb = \left(
\begin{array}{c}
S(1,1)\\
S(3,1)\\
\vdots\\
S(2\lfloor \frac{M}{2} \rfloor-1,1)
\end{array} \right),
\]
\[
\bS = \left(
\begin{array}{cccc}
S(1,2) & S(1,3) & \cdots & S(1,M-1) \\
S(3,2) & S(3,3) & \cdots & S(3,M-1) \\
\vdots & \vdots & \ddots & \vdots \\
S(2\lfloor \frac{M}{2} \rfloor-1 ,2)
& S(2\lfloor \frac{M}{2} \rfloor-1, 3)
& \cdots & S(2\lfloor \frac{M}{2} \rfloor-1, M-1)
\end{array} \right),
\]
then the system \eqref{sol:linear} is formulated as
\begin{equation}\label{sol:linear1}
\bA \bc = -\sigma_{12} \bb,
\end{equation}
where $\bA = \left( \bh, (\bS - 2 \bh \be_1^T)\hat{\bR}_+ \right)$.
To fix the parameters in $\bc$, the unique solvability of
\eqref{sol:linear1} is required. Currently, we can only claim the
system \eqref{sol:linear1} is uniquely solvable when $\chi$ is an
algebraic number. Precisely, we have the following theorem:
\begin{theorem}
$\left| \bA \right| \neq 0$ if $\chi$ is an algebraic number.
\end{theorem}
\begin{proof}
We times $\bA$ by
$\left(
\begin{array}{cc}
1 & 2 \be_1^T \hat{\bR}_+ \\
\bzero & \bI \end{array}
\right)$ to obtain $(\bh, \bS \hat{\bR}_+)$. Thus $|\bA| = |(\bh,\bS
\hat{\bR}_+)|$.
We retrieve the coefficient $h_m(\chi)$ in $S(k,m)$ to have
\[
\bS = \bS^\star_0 \boldsymbol{H}
\]
where
$\boldsymbol{H} = \dfrac{1}{\sqrt{2\pi}} \diag \{ h_2(\chi),
h_3(\chi), \cdots, h_{M-1}(\chi) \}$ and
\[
\bS^\star_0 = \left(
\begin{array}{cccc}
S^\star(1,2) & S^\star(1,3) & \cdots & S^\star(1,M-1) \\
S^\star(3,2) & S^\star(3,3) & \cdots & S^\star(3,M-1) \\
\vdots & \vdots & \ddots & \vdots \\
S^\star(2\lfloor \frac{M}{2} \rfloor-1 ,2)
& S^\star(2\lfloor \frac{M}{2} \rfloor-1, 3)
& \cdots & S^\star(2\lfloor
\frac{M}{2} \rfloor-1, M-1)
\end{array} \right).
\]
By the recursion relation \eqref{eq:rec} of $S^\star(k,m)$, we have
that
\[
\boldsymbol{L} \bS^\star_0 = \left(
\begin{array}{c}
S^\star(1,2), ~ S^\star(1,3), ~ \cdots, ~ S^\star(1,M-1) \\
\bS^\star_1
\end{array}
\right),
\]
where
\[
\boldsymbol{L} = \left(
\begin{array}{ccccc}
1 &&&& \\
-2 & 1 &&& \\
& -4 & 1 && \\
& & \ddots & \ddots & \\
& & & -(2\lfloor \frac{M}{2} \rfloor-2) & 1
\end{array}
\right),
\]
and
\[
\bS^\star_1 = \left(
\begin{array}{cccc}
2S^\star(2,1) & 3S^\star(2,2) & \cdots & (M-1)S^\star(2,M-2) \\
2S^\star(4,1) & 3S^\star(4,2) & \cdots & (M-1)S^\star(4,M-2) \\
\vdots & \vdots & \ddots & \vdots \\
2S^\star(2\lfloor \frac{M}{2} \rfloor-2,1)
& 3S^\star(2\lfloor \frac{M}{2} \rfloor-2, 2)
& \cdots & (M-1)S^\star(2\lfloor
\frac{M}{2} \rfloor-2, M-2)
\end{array}
\right).
\]
Noticing that $\boldsymbol{L} \bh = (1/\sqrt{2\pi}, 0, \cdots, 0)^T$, we have
that
\[
(\bh, \bS \hat{\bR}_+) = \boldsymbol{L}^{-1} \left(
\begin{array}{cc}
1/\sqrt{2\pi} & (S^\star (1, 2), \cdots, S^\star (1, M-1)) \\
\bzero & \bS^\star_1
\end{array}
\right) \left(
\begin{array}{cc} 1 & \\ & \boldsymbol{H} \hat{\bR}_+ \end{array}
\right).
\]
Thus we need only to verify the determinant of
$\bS^\star_1 \boldsymbol{H} \hat{\bR}_+$ is not vanished.
Consider the permutation matrix in \eqref{mat:P_sigma},
we then see that
\[
\bS^\star_1 \boldsymbol{P}_\sigma = (\bS^\star_{\text{even}},
\bS^\star_{\text{odd}}),
\]
where $\bS^\star_{\text{even}}$ is made with the even columns of
$\bS^\star_1$ as
\[
\bS^\star_{\text{even}} = \left(
\begin{array}{cccc}
3S^\star(2,2) & 5S^\star(2,4) & 7S^\star(2,6) & \hdots \\
3S^\star(4,2) & 5S^\star(4,4) & 7S^\star(4,6) & \hdots \\
\vdots & \vdots & \vdots & \vdots \\
3S^\star(2\lfloor \frac{M}{2} \rfloor-2,2) & 5S^\star(2\lfloor
\frac{M}{2} \rfloor-2,4) & 7S^\star(2\lfloor \frac{M}{2}
\rfloor-2,6)& \hdots
\end{array}
\right),
\]
and $\bS^\star_{\text{odd}}$ is made with the odd columns of
$\bS^\star_1$ as
\[
\bS^\star_{\text{odd}} = \left(
\begin{array}{cccc}
2S^\star(2,1) & 4S^\star(2,3) & 6S^\star(2,5) &\hdots \\
2S^\star(4,1) & 4S^\star(4,3) & 6S^\star(2,5) &\hdots \\
\vdots & \vdots & \vdots & \vdots \\
2S^\star(2\lfloor \frac{M}{2} \rfloor-2,1) & 4S^\star(2\lfloor
\frac{M}{2} \rfloor-2,3) & 6S^\star(2\lfloor \frac{M}{2}
\rfloor-2,5) & \hdots
\end{array}
\right).
\]
With the integral properties of $S^{\star}(k,m)$ in ~\ref{sec:bc},
$\bS^{\star}_{\text{even}}$ is a lower triangular matrix and each
entry in its lower triangular part is an
algebraic number$\times \sqrt{2\pi}$.
The diagonal matrix $\boldsymbol{H}$ is turned into
\[
\boldsymbol{P}_\sigma^{-1} \boldsymbol{H} \boldsymbol{P}_\sigma =
\dfrac{1}{\sqrt{2\pi}}\left(
\begin{array}{cc}
\bI & \\
& \dfrac{2-\chi}{\chi} \bI
\end{array}
\right).
\]
We then have that
\[
\begin{aligned}
\bS^\star_1 \boldsymbol{H} \hat{\bR}_+ = &
\bS^\star_1 \boldsymbol{P}_\sigma ~ \boldsymbol{P}_\sigma^{-1}
\boldsymbol{H} \boldsymbol{P}_\sigma ~ \boldsymbol{P}_\sigma^{-1}
\hat{\bR}_+ \\
= & \dfrac{1}{\sqrt{2\pi}}
(\bS^\star_{\text{even}}, \bS^\star_{\text{odd}})
\left(
\begin{array}{cc}
\bI & \\
& \dfrac{2-\chi}{\chi} \bI
\end{array}
\right) \left(
\begin{array}{c}
\hat{\bR}_{+,\text{even}} \\
\hat{\bR}_{+,\text{odd}}
\end{array}
\right) \\
= & \dfrac{1}{\sqrt{2\pi}} \left( \bS^\star_{\text{even}}
\hat{\bR}_{+,\text{even}} + \dfrac{2-\chi}{\chi}
\bS^\star_{\text{odd}} \hat{\bR}_{+,\text{odd}} \right).
\end{aligned}
\]
Since $\hat{\bR}_{+,\text{even}}$ is invertible by Lemma
\ref{lem:R_peven_invertible},
\begin{equation}\label{eq:SHR}
\bS^\star_1 \boldsymbol{H} \hat{\bR}_+ = \dfrac{1}{\sqrt{2\pi}}
\left( \bS^\star_{\text{even}} + \dfrac{2-\chi}{\chi}
\bS^\star_{\text{odd}} \hat{\bR}_{+,\text{odd}}
\hat{\bR}_{+,\text{even}}^{-1}\right)\hat{\bR}_{+,\text{even}},
\end{equation}
and we only need to verify the matrix
$\bS^\star_{\text{even}} + \dfrac{2-\chi}{\chi}
\bS^\star_{\text{odd}} \hat{\bR}_{+,\text{odd}}
\hat{\bR}_{+,\text{even}}^{-1}$
in \eqref{eq:SHR} is not singular. Considering the polynomial of
$\lambda$ defined by
\[
p(\lambda) \triangleq
\left|\dfrac{\lambda}{\sqrt{2\pi}}\bS^\star_{\text{even}} +
\dfrac{2-\chi}{\chi} \bS^\star_{\text{odd}}
\hat{\bR}_{+,\text{odd}} \hat{\bR}_{+,\text{even}}^{-1}\right|,
\]
we point out that $p(\lambda)$ is a polynomial with all coefficients
to be algebraic numbers, since $\chi$ is assumed to be algebraic,
and entries of matrices
$\dfrac{1}{\sqrt{2\pi}}\bS^\star_{\text{even}}$,
$\hat{\bR}_{+,\text{even}}^{-1}$, $\bS^\star_{\text{odd}}$, and
$ \hat{\bR}_{+,\text{odd}}$ are all algebraic
numbers. Particularly, the coefficient of the leading term of
$p(\lambda)$ is the product of all diagonal entries in matrix
$\dfrac{1}{\sqrt{2\pi}}\bS^{\star}_{\text{even}}$ and thus is not
vanished. Therefore, $p(\lambda) = 0$ can only valid for $\lambda$
to be an algebraic number, too. Thus $p(\sqrt{2\pi}) \neq 0$ and
consequently $\left| \bA \right| \neq 0$. We conclude that the
linear system \eqref{sol:linear1} is uniquely solvable.
\end{proof}
\begin{remark}
Definitely, we speculate that $\left| \bA \right| \neq 0$ for all
$\chi$, while currently we can not prove it unfortunately. If we
take $\left| \bA \right|$ as a function of $\chi$, it is clearly a
continuous function. The theorem above declares that the value of
the function is not zero on all algebraic numbers, and by the
continuity of the function, the roots for $\left| \bA \right| = 0$
can not be dense on $\mathbb{R}$ at least.
\end{remark}
\subsection{Convergence in moment order}\label{sec:convergence}
Let us reveal below the connection of the linearized HME and the
linearized Boltzmann equation in \cite{Williams2001}. For Kramers'
problem, the boundary condition we proposed is related with that in
\cite{Williams2001}, either. Roughly speaking, our system is
illustrated to be a particular discretization of the equation in
\cite{Williams2001}. This allows to examine the convergence of the
solution of our systems to the numerical results of the equation in
\cite{Williams2001}. It is demonstrated that the solution converges to
that of the linearized Boltzmann equation in \cite{Williams2001} with
the increasing of moment order. Let us start from a brief review of
the main result on the Kramers' problem in \cite{Williams2001}.
For the time independent Boltzmann equation
\begin{equation}\label{eq:space-Boltz}
\bxi\cdot\nabla_{\bx} f = Q(f,f),
\end{equation}
considering here only Kramers' problem is studied, we linearize the
distribution function $f$ as
\begin{equation}\label{eq:linear-func}
f(\bx,\bxi) = \mathcal{M}(\bx,\bxi) [ 1 + h(\bx,\bxi) ],
\end{equation}
where $h(\bx,\bxi)$ is a disturbance term caused by the small
perturbation near the local equilibrium Maxwellian
$\mathcal{M}(\bx,\bxi)$, which has the form
\[
\mathcal{M}(\bx,\bxi) =
\frac{\rho_{0}(x,y)}{(2\pi\theta_{0}(x,y))^{3/2}}
\exp\left( -\frac{(\xi_x - u(y))^2 + \xi_y^2 + \xi_z^2}
{2\theta_{0}(x,y)} \right).
\]
Here $\rho_0$ and $\theta_0$ are same as that in
\eqref{eq:dimensionless}, and
\[
u(y) = K y.
\]
Inserting \eqref{eq:linear-func} into the time independent Boltzmann
equation \eqref{eq:space-Boltz}, and discarding the high-order small
quantities, we can obtain
\begin{equation}\label{eq:insert-Boltz}
\bxi \cdot \nabla_{\bx} \mathcal{M} + \mathcal{M} \bxi \cdot \nabla_{\bx} h =
J(\mathcal{M},h),
\end{equation}
where $J(\mathcal{M},h)$ is the linearized BGK collision
\[
J(\mathcal{M},h) = - \mathcal{M}\left\{ \nu h - \frac{\nu}{\sqrt{2\pi\theta_0}^3}
\int h(y, \bxi') \left[1 + \frac{1}{\theta_0} \bxi \cdot \bxi' +
\frac{2}{3}\left(\frac{|\bxi|^2}{2\theta_0}- \frac{3}{2}\right)
\left(\frac{|\bxi'|^2}{2\theta_0}-\frac{3}{2}\right)\right]
\exp\left(-\frac{|\bxi|^2}{2\theta_0}\right) \dd \bxi' \right\},
\]
where $\nu$ is the collision frequency of BGK model.
For convenience, we introduce the dimensionless variables
\[
\xi_i = \sqrt{\theta_0}\bar{\xi}_i, \qquad K = \sqrt{\theta_0}K_0,
\qquad \bx=L\bar{\bx},\qquad
\Kn = \frac{\sqrt{\theta_0}}{L\nu}.
\]
Then direct calculations and some simplification yield
\[
\begin{aligned}
&\bar{\xi}_x
\bar{\xi}_y K_0 + \bar{\xi}_y \pd{h(\bar{y},\bar{\bxi})}{\bar{y}} \\
&= - \frac{1}{\Kn} \left\{ h(\bar{y}, \bar{\bxi}) - \frac{1}{\sqrt{2\pi}^3} \int
h(\bar{y}, \bar{\bxi'}) \left[1 + \bar{\bxi} \cdot \bar{\bxi'} +
\frac{2}{3}\left(\frac{|\bar{\bxi}|^2-3}{2}\right)
\left(\frac{|\bar{\bxi'}|^2-3}{2}\right)\right]
\exp(-\frac{|\bar{\bxi}|^2}{2}) \dd \bar{\bxi}'\right\},
\end{aligned}
\]
and
\begin{equation}\label{eq:integralu}
\bar{u}_1(\bar{y}) = K_0 \bar{y} + \frac{1}{\sqrt{2\pi}}
\int_{-\infty}^{\infty} Z(\bar{y},\bar{\xi}) \exp\left(
-\frac{\bar{\xi}^2}{2} \right) \dd \bar{\xi},
\end{equation}
where
\[
Z(\bar{y}, \bar{\xi}_y) = \frac{1}{2\pi} \int_{-\infty}^{\infty}
\int_{-\infty}^{\infty} \bar{\xi}_x h(\bar{y}, \bar{\bxi})
\exp\left(-\frac{\bar{\xi}_x^2+\bar{\xi}_z^2}{2}\right) \dd
\bar{\xi}_x \dd \bar{\xi}_z .
\]
Then we have
\begin{equation}\label{eq:William}
K_0 \bar{\xi} + \bar{\xi} \pd{Z(\bar{y},\bar{\xi})}{\bar{y}}
=\frac{1}{\Kn} \left(
- Z(\bar{y},\bar{\xi}) + \frac{1}{\sqrt{2\pi}} \int_{\bbR}
Z(\bar{y}, \bar{\xi'}) \exp\left (-\frac{\bar{\xi'}^2}{2}\right)
\dd \bar{\xi'} \right).
\end{equation}
From \eqref{eq:sol_u1_formal} and \eqref{eq:integralu} we notice that
\[
K_0 = - \frac{\bar{\sigma}_{12}}{\Kn},
\]
and
\begin{equation}\label{eq:integralZ}
\begin{aligned}
\bar{f}_{e_1+ie_2} &= \frac{1}{i!}\frac{1}{\sqrt{2\pi}^3}
\int_{\bbR^3}\He_1(\bar{\xi}_x) \He_i(\bar{\xi}_y)
h(\bar{y},\bar{\bxi}) \exp\left(-\frac{\bar{\bxi}^2}{2}\right)
\dd\bar{\bxi}, \quad i=1, \dots,M-1,\\
&= \frac{1}{i!}\frac{1}{\sqrt{2\pi}}
\int_{\bbR} \He_i(\bar{\xi}) Z(\bar{y},\bar{\xi})
\exp\left(-\frac{\bar{\xi}^2}{2}\right)
\dd\bar{\xi}, \quad i= 1,\dots,M-1.
\end{aligned}
\end{equation}
Following \cite{Williams2001}, we use the model which is
also based on the diffuse-specular process, and boundary condition can
be written as
\[
f(0, \bxi) = \chi N f^W_M(\bx, \bxi) + (1-\chi) f(0, \bxi^{\ast}),
\]
where $f^W_M, \bxi^{\ast}$ is defined in \eqref{eq:equilibrium}, $N$
is a normalizing factor to be determined
\cite{Williams2001}. Using the zero mass flux condition
\[
\int_{\xi_y<0} \xi_y f(0,\bxi) \dd\bxi + \int_{\xi_y>0} \xi_y
f(0,\bxi) \dd\bxi = 0
\]
at $y=0$ to calculate the $N$.
After the linearization and nondimensionalization, the boundary
condition in Cartesian velocity coordinates as follows
\[
\begin{aligned}
h(0, \bar{\xi}_x, \bar{\xi}_y, \bar{\xi}_z) &= \chi[ \bar{\xi}_x
\bar{u}_1^W + \delta(\frac{\bar{\xi}^2}{2} - 2)] + (1- \chi) h(0, \bar{\xi}_x,
-\bar{\xi}_y, \bar{\xi}_z) \\
&- \frac{\chi}{2\pi} \int^0_{-\infty} \bar{\xi}'_y \dd \bar{\xi}'_y
\int_{\bbR^2} h(0, \bar{\xi}'_x, \bar{\xi}'_y, \bar{\xi}'_z)
\exp(-\frac{\bar{\bxi}'^2}{2}) \dd \bar{\xi}'_x \bar{\xi}'_z; \quad
\bar{\xi}_y > 0.
\end{aligned}
\]
Consider the Kramers' problem with boundary condition $\bar{u}_1^W =
0$ and $\delta = (\theta_W - \theta_0)/ \theta_W = 0$, then we have
\begin{equation}\label{eq:William-bc}
Z(0, \bar{\xi}) = (1 - \chi) Z(0, -\bar{\xi}) ; \quad \bar{\xi} > 0.
\end{equation}
The equation \eqref{eq:William} is an integral equation on $\bar{\xi}$
and differential equation on $y$. Here we discretize it on
$\bar{\xi}$. Consider the Gauss-Hermite quadrature with $M\in\bbN$
points, and denote the weights and integral points by $\omega_i$ and
$\bar{\xi}_i$, $i=1,\cdots,M$. If we sort the $\bar{\xi}_i$ in
decending order, then $\bar{\xi}_i = \lambda_i$ in
\eqref{mat:pos-neg}. Let $Z(\bar{y})^k =
Z(\bar{y},\bar{\xi}_k)$ and
$\bZ(\bar{y})=(Z(\bar{y})^1,\cdots,Z(\bar{y})^M)^T$ and
$\bomega=(\omega_1,\cdots,\omega_M)^T$, then we have
\begin{equation}
K_0 \bLambda \boldsymbol{1} +
\bLambda\od{\bZ(\bar{y})}{\bar{y}}=\frac{1}{\Kn}\left(
\boldsymbol{1}\bomega^T-\bI \right)\bZ(\bar{y}),
\end{equation}
where $\bLambda$ is same as the \eqref{mat:Lamb}
and $\bI$ is
the $M\times M$ identity matrix, and
$\boldsymbol{1}=(1,\cdots,1)^T\in\bbR^M$.
Let $\bW = \diag\{\omega_i;~ i=1, \dots, M\}$, since $\bW$ is
independent of $\bar{y}$ and $\bW$ and $\bLambda$ are both diagonal
matrices, the upper formulation can be rewritten as
\begin{equation}\label{eq:discrete-matrix}
K_0 \bLambda \bW \boldsymbol{1} + \bLambda
\od{\bW\bZ(\bar{y})}{\bar{y}} = \frac{1}{\Kn} \left(
\bW\boldsymbol{1}\bomega^T\bW^{-1} - \bI \right) \bW\bZ(\bar{y}).
\end{equation}
Noticing $\bar{\xi}_i$, $i=1,\cdots,M$ are Gauss-Hermite integral
points, we have $\He_M(\bar{\xi}_i)=0$, which indicates
$\bLambda = \bR^{-1} \bM \bR$, where $\bM$ and $\bR$ are defined in
\eqref{eq:def_MQ} and \eqref{mat:eigen-vec}, respectively.
The originality of the Hermite polynomial indicates
$\sum_{i=1}^Mw_i\He_j(\bar{y}_i)=\delta_{j,0}$, thus we have
\begin{equation*}
\bR\bW\boldsymbol{1} = \be_1, \quad \bomega^T = \be_1^T
\bR \bW.
\end{equation*}
Now \eqref{eq:discrete-matrix} can be rewritten as
\begin{equation}\label{eq:discrete}
\begin{aligned}
K_0 \be_2 + \bM \od{[\bR\bW \bZ(\bar{y})]}{\bar{y}} &=
\bM \od{V}{\bar{y}}\\
&= \frac{1}{\Kn} \bR \left(\bW\boldsymbol{1}\bomega^T\bW^{-1}
- \bI \right) \bW\bZ(\bar{y})\\
&= \frac{1}{\Kn}
\left(\bR\bW\boldsymbol{1}\bomega^T\bW^{-1}\bR^{-1}
- \bI \right) [\bR\bW\bZ(\bar{y})]\\
&=\frac{1}{\Kn}\left( \be_1\be_1^T-\bI \right)
[\bR\bW\bZ(\bar{y})]\\
&=-\frac{1}{\Kn}\bQ [\bR\bW\bZ(\bar{y})]
= -\frac{1}{\Kn} \bQ V,
\end{aligned}
\end{equation}
where $\bQ$ is defined in \eqref{eq:def_MQ} and
it is readily shown that $V$ can be written as
\[
V = \left\{
\begin{array}{l}
\bar{u}_1(\bar{y}) = K_0\bar{y} + \displaystyle\sum_{j=1}^M \omega_j
Z^j(\bar{y}), \\
\bar{f}_{e_1+ie_2} = (\bR\bW \bZ(\bar{y}))_{i+1} = \dfrac{1}{i!}
\displaystyle\sum_{j=1}^M
\omega_j\He_i(\bar{\xi}^j)Z^j(\bar{y}), \quad i=1,\dots,M-1,
\end{array} \right.
\]
which is the discrete form of \eqref{eq:integralu} and
\eqref{eq:integralZ}.
Similar discretization can be carried out for boundary condition
\eqref{eq:William-bc}. It can be written as
\[
Z(0, \bar{\xi}_i) = (1 - \chi) Z(0, -\bar{\xi}_i),
\quad (i=1, \dots, \lfloor \frac{M}{2} \rfloor).
\]
Since the zeros of Hermite polynomials are symmetric, the equation
\[
\omega_i Z(0, \bar{\xi}_i) = (1- \chi) \omega_j Z(0, \bar{\xi}_j),
\quad (i = 1, \dots, \lfloor \frac{M}{2} \rfloor)
\]
has to be satisfied for $j = M+1-i$. Thus we have
\begin{equation}\label{bc:discrete}
\bH_{\chi} \bW \bZ(0) = 0,
\end{equation}
where $\bZ(0) = (Z(0, \bar{\xi}_1), \dots, Z(0, \bar{\xi}_M))^T$ and
\[
\begin{aligned}
&\text{when $M$ is even:}~
\bH_{\chi} = \left(
\begin{array}{cccccc}
1 & & & & & \chi-1 \\
& \ddots & & & \iddots & \\
& & 1 & \chi-1 & &
\end{array} \right )_{\frac{M}{2} \times M} ,\\
&\text{when $M$ is odd:}~
\bH_{\chi} = \left(
\begin{array}{ccccccc}
1 & & & 0 & & & \chi-1 \\
& \ddots & & \vdots & & \iddots & \\
& & 1 & 0 &\chi-1 & &
\end{array} \right )_{\lfloor \frac{M}{2} \rfloor \times M}.
\end{aligned}
\]
Let
\[
\bK_v = \frac{1}{\chi} \left(
\begin{array}{cccc}
\bar{\xi}_1 & \bar{\xi}_2 & \dots & \bar{\xi}_{\lfloor \frac{M}{2} \rfloor}\\
\bar{\xi}^3_1 & \bar{\xi}^3_2 & \dots & \bar{\xi}^3_{\lfloor \frac{M}{2} \rfloor}\\
\vdots & \vdots & \ddots & \vdots\\
\bar{\xi}^{2\lfloor \frac{M}{2} \rfloor-1}_1 & \bar{\xi}^{2\lfloor
\frac{M}{2} \rfloor-1}_2 & \dots & \bar{\xi}^{2\lfloor \frac{M}{2}
\rfloor-1}_{\lfloor \frac{M}{2} \rfloor}
\end{array} \right)_{\lfloor \frac{M}{2} \rfloor \times \lfloor
\frac{M}{2} \rfloor}.
\]
Since $\bar{\xi}_1,\cdots,\bar{\xi}_{\lfloor\frac{M}{2}\rfloor}$ are
distinct, $\bK_v$ is invertible due to the invertibility of
Vandermonde matrix.
Let $\tilde{\bR} = (\tilde{r}_{ij})_{M\times M}$ with $\tilde{r}_{ij}
= \He_{i-1}(\lambda_j)$, $i, j = 1,\dots,M,$ then from
\eqref{mat:eigen-vec} we have
$\bR = \diag \{1,1,\frac{1}{2!},\dots,\frac{1}{(M-1)!}\} \cdot
\tilde{\bR}$.
Using the orthogonality of Hermite polynomials
\[
\frac{1}{\sqrt{2\pi}}\int_{\bbR} \He_j(x) \He_k(x)
\exp\left(-\frac{x^2}{2}\right) \dd x = j!\delta_{jk},
\]
we have
\[
\bW \tilde{\bR}^T\bR = \bI.
\]
Then we multiply matrix \eqref{bc:discrete} by $\bK_v$, and the matrix
form of boundary condition becomes
\begin{equation}\label{bc:mat-form}
\bK_v \bH_{\chi} \bW \bZ(0) = [\bK_v \bH_{\chi} \bW \tilde{\bR}^T]
\cdot [\bR\bW \bZ(0)] = 0.
\end{equation}
The discretization of $S(l,m)$ in \eqref{eq:integralS} is
\[
\begin{aligned}
S(l,m) &= \frac{1}{\chi} \sum_{j=1}^{\lfloor \frac{M}{2} \rfloor}
\{\bar{\xi}_j^l \omega_j
[\He_m(\bar{\xi}_j) - (1 - \chi)\He_m(-\bar{\xi}_j)]\} \\
&=\frac{1}{\chi} (\bar{\xi}_1^l, \bar{\xi}_2^l, \dots,
\bar{\xi}_{\lfloor \frac{M}{2} \rfloor}^l)
\cdot \bH_{\chi} \bW \cdot (\He_m(\bar{\xi}_1),
\He_m(\bar{\xi}_2),\dots, \He_m(\bar{\xi}_M))^T,
\end{aligned}
\]
then
\[
\bK_v \bH_{\chi} \bW \tilde{\bR}^T = \left(
\begin{array}{cccc}
S(1,0) & S(1,1) & \cdots & S(1,M-1) \\
S(3,0) & S(3,1) & \cdots & S(3,M-1) \\
\vdots & \vdots & \ddots & \vdots \\
S(2\lfloor \frac{M}{2} \rfloor-1,0) & S(2\lfloor \frac{M}{2} \rfloor-1,1)
& \cdots & S(2\lfloor \frac{M}{2} \rfloor-1,M-1)
\end{array} \right).
\]
And \eqref{bc:mat-form} is then turned into
\[
\left(
\begin{array}{cccc}
S(1,0) & S(1,1) & \cdots & S(1,M-1)\\
S(3,0) & S(3,1) & \cdots & S(3,M-1)\\
\vdots & \vdots & \ddots & \vdots\\
S(2\lfloor \frac{M}{2} \rfloor-1,0) & S(2\lfloor \frac{M}{2}
\rfloor-1,1) & \cdots & S(2\lfloor \frac{M}{2} \rfloor-1,M-1)
\end{array} \right) \cdot V^{(0)} = 0,
\]
which is same as the boundary condition in \eqref{bc:linear}.
\section{Quantity Validification}
In this section, we numerically study the convergence of the solutions
of the linearized HME to that of the linearized Boltzmann equation,
and Knudsen layer effect of the velocity and effective viscosity, and
compare them with the existing results. In all the tests, high
precision computation in Maple\footnote{Maple is a trademark of
Waterloo Maple Inc.} is used to reduce the numerical error.
\subsection{Convergence in moment order}\label{sec:convergenceNum}
In order to compare the results with linearized Boltzmann equation
\cite{Williams2001}, we normalized the velocity in
\eqref{eq:sol_u1_formal} as
\begin{equation}\label{eq:refer_velo}
\tilde{u}(\bar{y}) = -\Kn\dfrac{\bar{u}}{\bar{\sigma}_{12}}
= \bar{y} + \frac{\Kn}{\bar{\sigma}_{12}} \left(2\sum_{i = 1}^{\lfloor
\frac{M-2}{2} \rfloor} \hat{c_i}
\exp\left(-\frac{\bar{y}}{\Kn\hat{\lambda}_i}\right) - c_0\right).
\end{equation}
The normalized velocity can be split into three parts
\cite{Williams2001,Siewert2001}
\begin{equation}\label{eq:defe_velo}
\tilde{u}(\bar{y})
= \bar{y} + \zeta - \tilde{u}_d(\bar{y}),
\end{equation}
where $\tilde{u}_d(\bar{y})$ is the velocity defect, satisfying
$\lim\limits_{\bar{y} \to +\infty} \tilde{u}_d(\bar{y}) = 0$, and
$\zeta$ is the slip coefficient, which is
\begin{equation}\label{eq:slip_coefficient_approx}
\zeta =
\lim_{\bar{y}\to+\infty}(\tilde{u}(\bar{y})+\tilde{u}_d(\bar{y})-\bar{y})
= -\Kn \cdot \frac{c_0}{\bar{\sigma}_{12}}.
\end{equation}
Then the velocity defect is
\begin{equation}
\tilde{u}_d(\bar{y}) =
2\Kn \sum_{i = 1}^{\lfloor
\frac{M-2}{2} \rfloor} \frac{\hat{c_i}}{\bar{\sigma}_{12}}
\exp\left(-\frac{\bar{y}}{\Kn\hat{\lambda}_i}\right).
\end{equation}
Here we notice that there is always a factor $\bar{\sigma}_{12}$ in
the expression of $\hat{c}_i$ and $c_0$ in \eqref{eq:sol_u1_formal}.
In this subsection, we fix $\Kn = 1/\sqrt{2}$ as a constant
for convenience. Next we study the convergence of the velocity defect
and slip coefficient, respectively.
\begin{figure}
\caption{Profile of the defect velocity $\tilde{u}
\label{fig:diff_M_1}
\end{figure}
\begin{figure}
\caption{Profile of the defect velocity $\tilde{u}
\label{fig:diff_M_9}
\end{figure}
\begin{figure}
\caption{\label{fig:hatlambdai}
\label{fig:hatlambdai}
\end{figure}
For the velocity defect $\tilde{u}_d(\bar{y})$, the analytical results
with $M$ ranging from $5$ to $80$ are presented in Fig.
\ref{fig:diff_M_1} for $\chi = 0.1$ and Fig. \ref{fig:diff_M_9} for
$\chi = 0.9$, which are compared with the Siewert's numerical results
in \cite{Siewert2001} for the linearized BGK model. It is clear that
the results of the linearized HME converge to Siewert's result as $M$
increasing, which is consistent with the theoretical analysis in Sec.
\ref{sec:convergence}.
Meanwhile, one can find that the defect velocity of even order
converges faster to the reference solution than that of odd order.
This can be understood based on the smallest width of the boundary
layer, which is represented by $w_M:=\min\{\hat{\lambda}_i:
i=1,\cdots,\lfloor \frac{M}{2}-1\rfloor\}$. The smaller $w_M$, the
closer of the defect velocity of the linearized HME to the reference
solution. Fig. \ref{fig:hatlambdai} gives all the
$\hat{\lambda}_i$ for $M$ ranging from $3$ to $40$. One can observe
that $w_M$ for even $M$ is quite smaller than that for the adjacent
odd $M$.
Moreover, comparing with Fig. \ref{fig:diff_M_1} and
\ref{fig:diff_M_9}, one can find that for a given $M$, the relative
error in Fig. \ref{fig:diff_M_1} is a little larger than that of Fig.
\ref{fig:diff_M_9}. Actually, for smaller $\chi$, the diffusion
interaction between gas and the wall turns weak, then the distribution
function is expected to be more far from the equilibrium, which
indicates more moment is needed.
For the slip coefficient $\zeta$, the analytical results for different
$M$ are plotted in Fig. \ref{fig:zeta}. Similar convergence can be
readily observed in Fig. \ref{fig:zeta}.
All the phenomena observed in Fig. \ref{fig:diff_M_1} and
\ref{fig:diff_M_9} are also valid in Fig. \ref{fig:zeta}.
\begin{figure}
\caption{Values of the slip coefficient $\zeta$ for different $M$
and $\chi$. The reference solution is Siewert's result in
\cite{Siewert2001}
\label{fig:zeta}
\end{figure}
\subsection{Knudsen layer}
In this subsection, we study the Knudsen layer of Kramers' problem in
three aspects. The first one is the profile of the normalized velocity
$\tilde{u}(\bar{y})$ \eqref{eq:refer_velo}. For convergence, here we
also fix $\Kn$ as a constant $1 /\sqrt{2}$. Fig.
\ref{fig:diff_chi} gives the profile of $\tilde{u}(\bar{y})$ in
\eqref{eq:refer_velo} of linearized HME with $M=8$ and $M=9$.
Compared with numerical results of linearized Boltzmann equation in
\cite{Loyalka1975}, the good agreement of the solutions of the
linearized HME in Fig. \ref{fig:diff_chi} indicates the moment system
with a small $M$ is good enough to describe the velocity profile in
the Knudsen layer.
Moreover, the value of $\tilde{u}(\bar{y})$ increases, as $\chi$
decreasing. This is because the coefficients $\hat{c}_i$ and $c_0$ are
dependent on $\frac{2-\chi}{\chi}$.
As discussed in the Section \ref{sec:convergenceNum}, the diffusion
interaction between gas and the wall is weaker for smaller $\chi$.
\begin{figure}
\caption{ Profile of $\tilde{u}
\label{fig:diff_chi}
\end{figure}
\begin{figure}
\caption{Profile of $\tilde{u}
\label{fig:diff_Kn}
\end{figure}
The second one is the profile of the velocity defect
$\tilde{u}_d(\bar{y})$ in \eqref{eq:defe_velo}.
Fig. \ref{fig:diff_Kn} shows the profile of $\tilde{u}_d(\bar{y})$
with $M=20$ for different Knudsen number.
The thickness of the Knudsen layer largens as $\Kn$ increasing and the
strength of of the Knudsen layer enhances. In practical application,
more moments are needed for large $\Kn$.
The third one is the effective viscosity. The Navier-Stokes law
indicates $\sigma_{12}=-\mu\pd{u}{y}$. However, in the Knudsen layer,
the Navier-Stokes does not hold anymore. To describe
the non-Newtonian behavior inherent in the Knudsen layer, we formally
write the Navier-Stokes law on the shear stress $\sigma_{12}$ as
\begin{equation}\label{eq:effective}
\sigma_{12} = -\mu_{\mathrm{eff}}\pd{u}{y},
\end{equation}
where $\mu_{\mathrm{eff}}$ is called the ``effective viscosity''.
Since the shear stress $\sigma_{12}$ is constant in the Kramers'
problem, we have
\begin{equation}
\frac{\mu_{\mathrm{eff}}}{\mu}
= -\left(\frac{\sigma_{12}}{\partial u / \partial y}\right)
\Big{/} \left(\frac{\lambda p_0}{\sqrt{\theta_0}}\right)
= - \frac{1}{\Kn} \frac{\bar{\sigma}_{12}}{\partial \bar{u} / \partial\bar{y}}
= \frac{1}{\partial \tilde{u} / \partial\bar{y}}.
\end{equation}
Noticing the definition of the normalized velocity
\eqref{eq:refer_velo}, one can directly calculate
\begin{equation}\label{eq:eff_HME}
\mu_{\mathrm{eff}} = \frac{\mu}{1 + \sum_{i = 1}^{\lfloor
\frac{M-2}{2} \rfloor} c_i
\exp\left(-\frac{\bar{y}}{\hat{\lambda}_i\Kn}\right)},
\quad c_i =
- \frac{2\hat{c}_i}{\hat{\lambda}_i \bar{\sigma}_{12}}.
\end{equation}
\begin{figure}
\caption{Effective viscosity $\mu_{\mathrm{eff}
\label{fig:eff_vis}
\end{figure}
\begin{figure}
\caption{Comparison between effective viscosity $\mu_{\mathrm{eff}
\label{fig:eff_vis_err}
\end{figure}
In the past, the effective viscosity is well studied. For example, in
\cite{Gu_R26}, Gu investigated the R26 moment equations and predicted
the effective viscosity as
\begin{equation}\label{eq:eff_Gu}
\mu_{\mathrm{eff}} = \left[1
-\left(1.3042C_1 \exp\left(-\frac{1.265\bar{y}}{\Kn}\right)
+ 1.6751C_2 \exp\left(-\frac{0.5102\bar{y}}{\Kn}\right) \right)
\right]^{-1} \mu,
\end{equation}
where
\[
\begin{aligned}
C_1 = \frac{\chi -2}{\chi} \frac{0.81265 \times 10^{-1}\chi^2 +
1.2824\chi}{0.48517 \times 10^{-2} \chi^2 + 0.64884 \chi +
8.0995},\\
C_2 = \frac{\chi -2}{\chi} \frac{0.8565 \times 10^{-3}\chi^2 +
0.362 \chi}{0.48517 \times 10^{-2} \chi^2 + 0.64884 \chi +
8.0995}.
\end{aligned}
\]
This model is similar as the linearized HME. Actually, since R26
moment system can be derived from HME, Gu's result can be treated as a
special case of the linearized HME.
In \cite{Lockerby2008}, Lockerby et al. studied the effective
viscosity based on the two low-$\Kn$ BGK results, and proposed an
empirical expression as
\begin{equation}\label{eq:vis_Lockerby}
\mu_{\mathrm{eff}} = \left(1
+ 0.1859\bar{y}^{-0.464} \exp\left(-0.7902\bar{y}\right)
\right)^{-1} \mu.
\end{equation}
For Lockerby's model, we have $\mu_{\mathrm{eff}}\to0$ as $y\to 0+$,
which indicates the velocity gradient to approach infinity at the
wall.
For convenience, here we let $\chi=1$ and $\Kn=1/\sqrt{2}$. Fig.
\ref{fig:eff_vis} shows the profile of the effective viscosity of
these models. Due to the convergence of the linearized HME, we take
the solution the linearized HME with $M=200$ as the reference
solution. One can observe that Gu gives a relative larger effective
viscosity $\mu_{\mathrm{eff}}$, while Lockerby gives a relative smaller
one. If one want to obtain a good approximation of the effective
viscosity close to the wall, a lot of moments are needed.
We also take the solution of the linearized HME with $M=200$ as the
reference solution, and define the error as
\[
\mathrm{err} = \mu_{\mathrm{eff}}^{\mathrm{reference}} - \mu_{\mathrm{eff}}^{\mathrm{model}}.
\]
Fig. \ref{fig:eff_vis_err} shows the error of Gu's and Lockerby's
models and the linearized HME with $M=30$. Gu's model agrees with the
reference very well away from the Knudsen layer and gives too large
effective viscosity, while Lockerby's model gives too small effective
viscosity. For the linearized HME, by choosing a proper $M$, the
effective viscosity can be well captured.
\section{Conclusion}
In this paper, the globally hyperbolic moment equations (HME) is
employed to study Kramers'
problem. Firstly, the set of linearized globally hyperbolic moment
equations and their boundary conditions are built. The analytical
solutions for the defect velocity and slip coefficient have been
obtained for arbitrary order moment equations. In comparison with
data from kinetic theory, it has been shown that they can accurately
capture the Knudsen layer velocity profile over a wide range of
accommodation coefficients, especially for the small accommodation
coefficients case. The results indicate that the physics of
non-equilibrium gas flow can be captured by high-order HME system.
\end{document} | math |
KUIDFC Recruitment 2022 : 11 ಕಾರ್ಯನಿರ್ವಾಹಕ ಅಭಿಯಂತರರು ಮತ್ತು ವಿವಿಧ ಹುದ್ದೆಗಳಿಗೆ ಅರ್ಜಿ ಆಹ್ವಾನ ಕರ್ನಾಟಕ ನಗರ ಮೂಲಸೌಕರ್ಯ ಅಭಿವೃದ್ಧಿ ಮತ್ತು ಹಣಕಾಸು ನಿಗಮ ನಿಯಮಿತದಿಂದ ಅನುಷ್ಠಾನಗೊಳ್ಳುತ್ತಿರುವ ಕರ್ನಾಟಕ ನಗರ ನೀರು ಸರಬರಾಜು ಆಧುನೀಕರಣ ಯೋಜನೆ ನೇಮಕಾತಿಯ 11 ಕಾರ್ಯನಿರ್ವಾಹಕ ಅಭಿಯಂತರರು, ಸಹಾಯಕ ಅಭಿಯಂತರರು, ಕಾರ್ಯನಿರ್ವಾಹಕ, ಎಂ.ಐ.ಎಸ್ ತಜ್ಞರು, ವ್ಯವಸ್ಥಾಪಕರು ಮತ್ತು ಸಾರ್ವಜನಿಕ ಸಂಪರ್ಕ ಅಧಿಕಾರಿ ಹುದ್ದೆಗಳನ್ನು ಭರ್ತಿ ಮಾಡಲು ಅರ್ಜಿ ಆಹ್ವಾನಿಸಲಾಗಿದೆ. ಆಸಕ್ತರು ಅಧಿಕೃತ ವೆಬ್ಸೈಟ್ಗೆ ಭೇಟಿ ನೀಡಿ ಹುದ್ದೆಗಳ ಬಗೆಗಿನ ಅಧಿಸೂಚನೆಯನ್ನು ಓದಬಹುದು. ಅರ್ಹ ಅಭ್ಯರ್ಥಿಗಳು ಆನ್ಲೈನ್ ಮೂಲಕ ಜುಲೈ 25,2022 ರಿಂದ ಆಗಸ್ಟ್ 18,2022ರೊಳಗೆ ಅರ್ಜಿಯನ್ನು ಹಾಕಬಹುದು. ಈ ಹುದ್ದೆಗಳಿಗೆ ಕೇಳಲಾಗಿರುವ ಅರ್ಹತೆ, ವಯೋಮಿತಿ, ನೀಡಲಾಗುವ ವೇತನ, ಆಯ್ಕೆ ಪ್ರಕ್ರಿಯೆ ಮತ್ತು ಅರ್ಜಿ ಸಲ್ಲಿಕೆ ಬಗ್ಗೆ ಮಾಹಿತಿಯನ್ನು ಪಡೆಯಲು ಮುಂದೆ ಓದಿ. KUIDFC ನೇಮಕಾತಿ 2022 ವಿದ್ಯಾರ್ಹತೆ : ಕರ್ನಾಟಕ ನಗರ ಮೂಲಸೌಕರ್ಯ ಅಭಿವೃದ್ಧಿ ಮತ್ತು ಹಣಕಾಸು ನಿಗಮ ನಿಯಮಿತ ನೇಮಕಾತಿಯ ಕಾರ್ಯನಿರ್ವಾಹಕ ಅಭಿಯಂತರರು, ಸಹಾಯಕ ಅಭಿಯಂತರರು, ಕಾರ್ಯನಿರ್ವಾಹಕ, ಎಂ.ಐ.ಎಸ್ ತಜ್ಞರು, ವ್ಯವಸ್ಥಾಪಕರು ಮತ್ತು ಸಾರ್ವಜನಿಕ ಸಂಪರ್ಕ ಅಧಿಕಾರಿ ಹುದ್ದೆಗಳಿಗೆ ಬಿ.ಇ, ಸಿಎ, ಎಂಬಿಎ, ಸ್ನಾತಕೋತ್ತರ ಪದವಿ ವಿದ್ಯಾರ್ಹತೆಯನ್ನು ಮಾನ್ಯತೆ ಪಡೆದ ಬೋರ್ಡ್ವಿಶ್ವವಿದ್ಯಾಲಯಸಂಸ್ಥೆಯಿಂದ ಹೊಂದಿರುವ ಅಭ್ಯರ್ಥಿಗಳು ಅರ್ಜಿಯನ್ನು ಹಾಕಬಹುದು ಎಂದು ಅಧಿಸೂಚನೆಯಲ್ಲಿ ತಿಳಿಸಲಾಗಿದೆ. KUIDFC ನೇಮಕಾತಿ 2022 ವಯೋಮಿತಿ : ಕರ್ನಾಟಕ ನಗರ ಮೂಲಸೌಕರ್ಯ ಅಭಿವೃದ್ಧಿ ಮತ್ತು ಹಣಕಾಸು ನಿಗಮ ನಿಯಮಿತ ನೇಮಕಾತಿಯ ಕಾರ್ಯನಿರ್ವಾಹಕ ಅಭಿಯಂತರರು, ಸಹಾಯಕ ಅಭಿಯಂತರರು, ಕಾರ್ಯನಿರ್ವಾಹಕ, ಎಂ.ಐ.ಎಸ್ ತಜ್ಞರು, ವ್ಯವಸ್ಥಾಪಕರು ಮತ್ತು ಸಾರ್ವಜನಿಕ ಸಂಪರ್ಕ ಅಧಿಕಾರಿ ಹುದ್ದೆಗಳಿಗೆ ಗರಿಷ್ಟ 45 ಮತ್ತು 65 ವರ್ಷ ವಯೋಮಿತಿಯೊಳಗಿನ ಅಭ್ಯರ್ಥಿಗಳು ಅರ್ಜಿಯನ್ನು ಸಲ್ಲಿಸಬಹುದು. ಅಭ್ಯರ್ಥಿಗಳಿಗೆ ನೇಮಕಾತಿ ನಿಯಮಾನುಸಾರ ವಯೋಮಿತಿ ಸಡಿಲಿಕೆಯನ್ನು ನೀಡಲಾಗಿರುತ್ತದೆ. KUIDFC ನೇಮಕಾತಿ 2022 ಆಯ್ಕೆ ಪ್ರಕ್ರಿಯೆ : ಕರ್ನಾಟಕ ನಗರ ಮೂಲಸೌಕರ್ಯ ಅಭಿವೃದ್ಧಿ ಮತ್ತು ಹಣಕಾಸು ನಿಗಮ ನಿಯಮಿತ ನೇಮಕಾತಿಯ ಕಾರ್ಯನಿರ್ವಾಹಕ ಅಭಿಯಂತರರು, ಸಹಾಯಕ ಅಭಿಯಂತರರು, ಕಾರ್ಯನಿರ್ವಾಹಕ, ಎಂ.ಐ.ಎಸ್ ತಜ್ಞರು, ವ್ಯವಸ್ಥಾಪಕರು ಮತ್ತು ಸಾರ್ವಜನಿಕ ಸಂಪರ್ಕ ಅಧಿಕಾರಿ ಹುದ್ದೆಗಳಿಗೆ ಅರ್ಜಿ ಸಲ್ಲಿಸುವ ಅಭ್ಯರ್ಥಿಗಳನ್ನು ಅರ್ಹತೆ, ಅನುಭವ ಮತ್ತು ಸಂದರ್ಶನದ ಮೂಲಕ ಆಯ್ಕೆ ಮಾಡಲಾಗುತ್ತದೆ ಎಂದು ನೇಮಕಾತಿ ಅಧಿಸೂಚನೆಯಲ್ಲಿ ತಿಳಿಸಲಾಗಿದೆ. KUIDFC ನೇಮಕಾತಿ 2022 ವೇತನ : ಕರ್ನಾಟಕ ನಗರ ಮೂಲಸೌಕರ್ಯ ಅಭಿವೃದ್ಧಿ ಮತ್ತು ಹಣಕಾಸು ನಿಗಮ ನಿಯಮಿತ ನೇಮಕಾತಿಯ ಕಾರ್ಯನಿರ್ವಾಹಕ ಅಭಿಯಂತರರು, ಸಹಾಯಕ ಅಭಿಯಂತರರು, ಕಾರ್ಯನಿರ್ವಾಹಕ, ಎಂ.ಐ.ಎಸ್ ತಜ್ಞರು, ವ್ಯವಸ್ಥಾಪಕರು ಮತ್ತು ಸಾರ್ವಜನಿಕ ಸಂಪರ್ಕ ಅಧಿಕಾರಿ ಹುದ್ದೆಗಳಿಗೆ ಆಯ್ಕೆಯಾದ ಅಭ್ಯರ್ಥಿಗಳಿಗೆ ತಿಂಗಳಿಗೆ 69,116 ರಿಂದ 86,510ರೂಗಳ ವರೆಗೆ ವೇತನ ಜೊತೆಗೆ ಇತರೆ ಭತ್ಯೆಗಳನ್ನು ನೀಡಲಾಗುವುದು ಎಂದು ಅಧಿಸೂಚನೆಯಲ್ಲಿ ಹೇಳಲಾಗಿದೆ. KUIDFC ನೇಮಕಾತಿ 2022 ಅರ್ಜಿ ಶುಲ್ಕ : ಕರ್ನಾಟಕ ನಗರ ಮೂಲಸೌಕರ್ಯ ಅಭಿವೃದ್ಧಿ ಮತ್ತು ಹಣಕಾಸು ನಿಗಮ ನಿಯಮಿತ ನೇಮಕಾತಿಯ ಕಾರ್ಯನಿರ್ವಾಹಕ ಅಭಿಯಂತರರು, ಸಹಾಯಕ ಅಭಿಯಂತರರು, ಕಾರ್ಯನಿರ್ವಾಹಕ, ಎಂ.ಐ.ಎಸ್ ತಜ್ಞರು, ವ್ಯವಸ್ಥಾಪಕರು ಮತ್ತು ಸಾರ್ವಜನಿಕ ಸಂಪರ್ಕ ಅಧಿಕಾರಿ ಹುದ್ದೆಗಳಿಗೆ ಅರ್ಜಿ ಸಲ್ಲಿಸುವ ಅಭ್ಯರ್ಥಿಗಳು ಅರ್ಜಿ ಶುಲ್ಕದ ವಿವರ ಪಡೆಯಲು ನೇಮಕಾತಿ ಅಧಿಸೂಚನೆಯನ್ನು ಓದಬಹುದು. KUIDFC ನೇಮಕಾತಿ 2022 ಅರ್ಜಿ ಸಲ್ಲಿಕೆ : ಕರ್ನಾಟಕ ನಗರ ಮೂಲಸೌಕರ್ಯ ಅಭಿವೃದ್ಧಿ ಮತ್ತು ಹಣಕಾಸು ನಿಗಮ ನಿಯಮಿತ ನೇಮಕಾತಿಯ ಕಾರ್ಯನಿರ್ವಾಹಕ ಅಭಿಯಂತರರು, ಸಹಾಯಕ ಅಭಿಯಂತರರು, ಕಾರ್ಯನಿರ್ವಾಹಕ, ಎಂ.ಐ.ಎಸ್ ತಜ್ಞರು, ವ್ಯವಸ್ಥಾಪಕರು ಮತ್ತು ಸಾರ್ವಜನಿಕ ಸಂಪರ್ಕ ಅಧಿಕಾರಿ ಹುದ್ದೆಗಳಿಗೆ ಅರ್ಜಿ ಸಲ್ಲಿಸುವ ಅಭ್ಯರ್ಥಿಗಳು ಆನ್ಲೈನ್ ನಲ್ಲಿ ಅಧಿಕೃತ ವೆಬ್ಸೈಟ್ http:kuidfc.com ಗೆ ಭೇಟಿ ನೀಡಿ. ಅಲ್ಲಿ ಕೇಳಲಾಗಿರುವ ಮಾಹಿತಿಯನ್ನು ಭರ್ತಿ ಮಾಡಿ ಜೊತೆಗೆ ಅಗತ್ಯ ದಾಖಲೆಗಳನ್ನು ಲಗತ್ತಿಸುವ ಮೂಲಕ ಜುಲೈ 25,2022 ರಿಂದ ಆಗಸ್ಟ್ 18,2022ರೊಳಗೆ ಅರ್ಜಿಯನ್ನು ಹಾಕಬಹುದು. ಅಭ್ಯರ್ಥಿಗಳು ನೇಮಕಾತಿ ಬಗೆಗಿನ ಅಧಿಸೂಚನೆಯನ್ನು ಓದಲು ಇಲ್ಲಿ .By Kavya source: kannada.careerindia.com | kannad |
শেষমেশ জয়ের সরণিতে এসসি ইস্টবেঙ্গল, প্রথম ম্যাচেই বাজিমাত কোচ মারিওর Photo Google এক্সট্রা টাইম ওয়েব ডেস্ক এ যেন এলেন, দেখলেন, জয় করলেন নতুন কোচ মারিও নিজের দ্বিতীয় ইনিংসের শুরুতেই তিনি বলেছিলেন দলকে তিনি আক্রমণাত্মক খেলাবেন দলের যা অবস্থা তাতে তাঁর কথার ওপর কেউই তেমন আস্থা রাখেননি কিন্তু আজকের ম্যাচে ঝলক পাওয়া গেল সেই চেনা ইস্টবেঙ্গলের প্রতি আক্রমণে ক্রমাগত চাপ সৃষ্টি যেমন করল লাল হলুদ দল দল তেমনি নিচে নেমে ডিফেন্সও করল খেলার শুরু থেকে শেষ অবধি পরিসংখ্যান দেখলে মনে হবে পুরো আধিপত্য ছিল এফ সি গোয়ার স্কোরবোর্ডের সাথে তার কোনও মিলই নেই কিন্তু ম্যাচে দুর্দান্ত লড়াই এর মাধ্যমে তাঁদের প্রথম তিন পয়েন্ট তুলে নিল লাল হলুদ দল প্রথম থেকেই ডিফেন্স আঁটোসাঁটো রেখে প্রতি আক্রমণে খেলতে শুরু করে ইস্টবেঙ্গল গোয়া চাপ সৃষ্টি করলেও তাদের সব জারিজুরি ইস্টবেঙ্গল পেনাল্টি বক্সের বাইরে নিষ্প্রভ হয়ে যাচ্ছিল, লাল হলুদ ডিফেন্সের বদান্যতায় আজ টিমে পর্সে প্রত্যাবর্তন করেন, খেলেন আর এক বিদেশি সিডয়েলও আজ কোভিড পজিটিভ থাকায় খেলেননি হীরা মন্ডল শুরুতেই কোচ মারিও মাস্টার স্ট্রোক খেলেন অঙ্কিতের জায়গায় অমরজিত ও হীরার অনুপস্থিতিতে সেই জায়গায় খেলালেন অঙ্কিতকে সিডয়েল ও সৌরভ কে ব্লকার হিসেবে রেখে দিলেন ডিফেন্সের সামনে ব্যাস বাজিমাত এখানেই গোয়া দলের মূল দুই বিদেশি অর্টিজ ও এডু বেদিয়া জায়গা পেলেন না কোনও মাঝখান থেকে গোয়ান ডিফেন্সের ভুলে গোল করে যান নাওরেম মহেশ ম্যাচের জোড়া গোলই তার তারপর গোটা দ্বিতীয়ার্ধ দুরন্ত লড়াই করে নিজেদের এ মরশুমের প্রথম তিন পয়েন্ট তুলে নিল লাল হলুদ ব্রিগেড এবার দেখার সামনের ম্যাচগুলিতে তাদের এই জয়ের ধারা অব্যাহাত থাকে কি না View all posts | bengali |
എംഐ ഇനി ഇല്ല എല്ലാം ഷവോമിയിലേക്ക് റീബ്രാന്ഡ് ചെയ്യപ്പെടും ഇനി എംഐ ഉണ്ടാവില്ല. ഷവോമി അവരുടെ പ്രീമിയം ഉല്പ്പന്നങ്ങളെല്ലാം എംഐയില് നിന്നും ഷവോമി യിലേക്ക് റീബ്രാന്ഡ് ചെയ്യാന് പോകുന്നു. ഷവോമി, റെഡ്മി എന്നിങ്ങനെ രണ്ട് ഉപ ബ്രാന്ഡുകളായിരിക്കും ഉണ്ടായിരിക്കുക. സ്മാര്ട്ട്ഫോണുകള് ലാപ്ടോപ്പുകള് സ്മാര്ട്ട്ടിവികള് ഉള്പ്പടെ എല്ലാം റീബ്രാന്ഡ് ചെയ്യപ്പെടും. എംഐ റീബ്രാന്ഡ് ചെയ്യപ്പെടുന്നുണ്ടെങ്കിലും ലോഗോ എംഐ എന്നത് തന്നെ തുടരുമെന്നും എം ഐ അറയിച്ചു. വരാനിരിക്കുന്ന സ്മാര്ട്ട്ഫോണുകള് ടിവികള് ഫിറ്റ്നസ് ഉപകരണങ്ങള് ഉള്പ്പടെ മുന്പ് എംഐ ഉല്പ്പന്നങ്ങളായിരുന്ന എല്ലാത്തിനും പുതിയ ഷവോമി ലോഗോ ആയിരിക്കും നല്കുക. ഇപ്പോള് എംഐ ഫോണുകള് അല്ലെങ്കില് ഷവോമി ഫോണുകള് എന്ന് വിളിക്കുന്നത് ബ്രാന്ഡിന്റെ പ്രീമിയം വിഭാഗത്തില് ഉള്പെടുന്നവയെയാണ്. ഇന്ത്യന് വിപണിയില് ലഭ്യമായ എംഐ 11 അള്ട്രാ, എംഐ 11എക്സ് സീരീസ് എന്നിവ ഇതില് ഉള്പ്പെടുന്നു. എംഐ ബ്രാന്ഡ് ആയിരുന്നവ ഷവോമിയിലേക്ക് പുനര്നാമകരണം ചെയ്യപ്പെടുമ്ബോഴും, അത് നല്കിയിരുന്ന ഉന്നത സാങ്കേതികവിദ്യയും എല്ലാ വിഭാഗങ്ങളിലും പ്രീമിയം അനുഭവം വാഗ്ദാനം ചെയ്തിരുന്നതും തുടരുമെന്ന് ഷവോമി പ്രസ്താവനയില് പറഞ്ഞിട്ടുണ്ട്. ലോകമെമ്ബാടും ശക്തമായ സാന്നിധ്യമുള്ള മുന്നിര ടെക്നോളജി ബ്രാന്ഡ് ആയതിനാല്, ഞങ്ങളുടെ ലക്ഷ്യം ഒരു ഏകീകൃത സാന്നിധ്യമാണ്. ഈ പുതിയ ലോഗോ മാറ്റത്തിലൂടെ, ഞങ്ങളുടെ ബ്രാന്ഡും ഉല്പ്പന്നങ്ങളും തമ്മിലുള്ള ധാരണ വിടവ് നികത്താന് ഞങ്ങള് ഉദ്ദേശിക്കുന്നു. പുതിയ ഷവോമി ലോഗോ ഞങ്ങളുടെ പ്രീമിയം ഉല്പ്പന്നങ്ങള്ക്കായി ഉപയോഗിക്കും. ഷവോമി ഇന്ത്യയുടെ മാര്ക്കറ്റിംഗ് ഹെഡ് ജസ്കരന് സിംഗ് കപാനി പത്രക്കുറിപ്പില് പറഞ്ഞു. Tags | malyali |
The Spark menus are not working after I have upgraded my project to Primefaces 6.2. The topmenu is rendered, but no dropdown elements are triggered on mouse click.
I guess this error occurred because PF 6.2 upgraded to the latest JQuery implementation. In the latest JQuery ".size()" has removed in favor for ".length"
If possible this should probably be put into a Spark patch?
Could you please send an email to contact at primetek.com.tr?
This had been stumping me for a bit.
Hi @aragorn, I am having the same problem after upgrading to PrimeFaces 6.2.
I am currently using Spark Layout v2.1.1. I see on PrimeStore that v2.1.2 is available and it appears that its layout.js is using the new function you mentioned. Should I try to use v2.1.2 instead? Is it fully compatible with PF 6.2?
If so, then what I don't understand is why the Migration Update from v.2.1.1 to 2.1.2 doesn't mention anything about updating the layout.js file.
It appears I was wrong. The 2.1.2 version is still using the old jQuery size() method. For the moment, I manually patched the layout.js file but I would like to make sure that this is the only compatibility issue.
Can someone confirm this? Why isn't there a PrimeFaces 6.2 compatible version of the Spark layout released yet? | english |
BIGG BREAKING : ಆರ್ ಆರ್ ನಗರದಲ್ಲಿ ಬಿಜೆಪಿ ಅಭ್ಯರ್ಥಿ ಮುನಿರತ್ನ ಭರ್ಜರಿ ಗೆಲುವು : ಅಧಿಕೃತ ಘೋಷಣೆ ಬಾಕಿ ಬೆಂಗಳೂರು : ನವೆಂಬರ್ 3ರಂದು ನಡೆದಂತ ಶಿರಾ ಹಾಗೂ ಆರ್ ಆರ್ ನಗರ ಕ್ಷೇತ್ರಗಳ ಉಪಚುನಾವಣೆಯ ಮತಏಣಿಕ ಕಾರ್ಯ ಇಂದು ನಡೆಯುತ್ತಿದೆ. ಇದೀಗ ಸಿಕ್ಕಿರುವ ಮಾಹಿತಿಯಂತೆ ರಾಜರಾಜೇಶ್ವರಿನಗರ ಕ್ಷೇತ್ರದಲ್ಲಿ 44,548 ಮತಗಳ ಅಂತರದಿಂದ ಕಾಂಗ್ರೆಸ್ ಅಭ್ಯರ್ಥಿ ಕುಸುಮಾ ವಿರುದ್ಧ ಭರ್ಜರಿ ಗೆಲುವನ್ನು ಸಾಧಿಸಿದ್ದಾರೆ. ಈ ಬಗ್ಗೆ ಚುನಾವಣಾ ಆಯೋಗ ಅಧಿಕೃತವಾಗಿ ಘೋಷಿಸಬೇಕಿದೆ. ರಾಜರಾಜೇಶ್ವರಿನಗರ ಉಪ ಚುನಾವಣೆಯಲ್ಲಿ ಬಿಜೆಪಿ ಅಭ್ಯರ್ಥಿ ಮುನಿರತ್ನ ನಾಯ್ಡು ಹ್ಯಾಟ್ರಿಕ್ ಗೆಲುವನ್ನು ಸಾಧಿಸಿದ್ದಾರೆ. ಮುನಿರತ್ನ 1,03,291 ಮತಗಳನ್ನು ಗಳಿಸಿದ್ದರೇ, ಕಾಂಗ್ರೆಸ್ ನ ಹೆಚ್ ಕುಸುಮಾ 58,743 ಮತಗಳನ್ನು ಪಡೆದಿದ್ದಾರೆ. ಜೆಡಿಎಸ್ ನ ಕೃಷ್ಣ ಮೂರ್ತಿ 6,381 ಗಳಿಸಿದ್ದಾರೆ. ಬಿಜೆಪಿಯ ಅಭ್ಯರ್ಥಿ ಮುನಿರತ್ನ ಕಾಂಗ್ರೆಸ್ ಅಭ್ಯರ್ಥಿ ವಿರುದ್ಧ 44,548 ಮತಗಳ ಅಂತರದಿಂದ ಭರ್ಜರಿ ಗೆಲುವನ್ನು ಸಾಧಿಸಿದ್ದಾರೆ. ಆದ್ರೇ ಅಧಿಕೃತವಾಗಿ ಚುನಾವಣಾ ಆಯೋಗದಿಂದ ಗೆಲವನ್ನು ಘೋಷಣೆ ಮಾಡುವುದು ಬಾಕಿ ಇದೆ. | kannad |
سَنَتان سَنستھایہ ہندِس أکِس مقالس منٛز چھِ تمام ناو تفصٟل سان دِتھ | kashmiri |
టీటీడీ నిర్ణయంపై హైకోర్టు సీరియస్ విజయవాడ: టీటీటీ పాలకమండలి సభ్యుల నియామకంపై ఏపీ హైకోర్టు సీరియస్ అయింది. ప్రత్యేక ఆహ్వానితుల కోసం జారీ చేసిన జీవోను సస్పెండ్ చేసింది. టీటీటీ బోర్డు సభ్యుల నియామకాన్ని సవాల్ చేస్తూ హైకోర్టులో పలు పిటిషన్లు దాఖలయ్యాయి. నిబంధనలకు విరుద్ధంగా ప్రత్యేక ఆహ్వానితులను నియమించారని వాదనలు జరిగాయి. టీటీడీ నిర్ణయం సామాన్య భక్తులపై ప్రభావం చూపుతుందని పిటిషనర్ల తరపు న్యాయవాది యలమంజుల బాలాజీ.. కోర్టులో వాదనలు వినిపించారు. వాదనలు విన్న హైకోర్టు.. కేసు విచారణను నాలుగు వారాలకు వాయిదా వేసింది. అలాగే దీనిపై టీటీడీ, వైసీపీ ప్రభుత్వానికి కోర్టు.. నోటీసులు జారీ చేసింది. వైసీపీ ప్రభుత్వం పంచాయితీరాజ్ వ్యవస్థను నీరుగారుస్తోంది: అయ్యన్న | telegu |
Glacial blue eyes closed tightly, it did no good. Mathis? legs were no longer able to support the crushing soul pain, which drove him to his knees. The force behind his fall cracked the ice beneath him. The part of his blood tying him to Tierreglacia and his family burned, as his world did. He?d gathered himself off the grass that now lay frigid and frozen around him. Summoning his power of ice he straightened upright and created an ice bridge over the ground to ferry him home. But no matter the speed he slid across the ice, he still returned too late to do anything other than stand on the ridge and witness his world burn. | english |
Jammu Kashmir: जम्मू कश्मीर में भारी बर्फबारी, जनजीवन अस्तव्यस्त, हवाई उड़ानें भी प्रभावित Jammu Kashmir Snowfall: जम्मूकश्मीर में बीते दिन शुरू हुई हल्की बर्फबारी अब व्यापक रूप ले चुकी है। भारी बर्फबारी के चलते स्थानीय लोगों को कई समस्याओं का सामना करना पड़ रहा है। आमजन की समस्याओं के साथ ही हवाई अड्डे पर भी भारी बर्फबारी के चलते उड़ानें देरी से संचालित करनी पड़ गई हैं। बर्फबारी के चलते दृश्यता की समस्या आन खड़ी हुई है। अचानक से मौसम के करवट बदलने के चलते हुई शुरू हुई भारी बर्फबारी से लोगों की मुश्किलें बढ़ गई है। बर्फबारी के चलते उड़ानों में देरी होने की जानकारी देते हुए श्रीनगर हवाई अड्डा प्रबंधन ने बताया किहमारे हवाई अड्डे पर लगातार भारी बर्फबारी हो रही है, जिसके चलते रनवे पर बर्फ इकट्ठा हो गयी है। हालांकि बर्फ को हटाने के लिए हमारा ऑपरेशन निरंतर प्रगति पर है। इस बीच दृश्यता महज 400 मीटर है, जिसके चलते सभी एयरलाइंस की उड़ानें देरी से चल रही हैं। हम उड़ानों की स्थिति को लगातार अपडेट करते रहेंगे। मौसम विज्ञान विभाग में जारी की थी चेतावनी जम्मूकश्मीर और लद्दाख के ऊंचाई वाले इलाकों में बीते दिन मंगलवार को हल्की बर्फबारी हुई थी तथा इसी दौरान जम्मूकश्मीर और लद्दाख के मैदानी इलाकों में बारिश भी देखी गयी थी। इसी के चलते मौसम विभाग ने अगले 24 घंटों के दौरान भारी बारिश और बर्फबारी का अनुमान लगाया था, जो कि आज सुबह की शुरुआत के साथ ही शुरू हो गया है। मौसम विज्ञान विभाग के एक अधिकारी ने बीते दिन इस विषय में जानकरी देते हुए कहा था कि अगले 24 घंटों के दौरान जम्मूकश्मीर और लद्दाख में हल्की से मध्यम बारिश व हिमपात होने की संभावना है। बीते दिन का दर्ज तापमान श्रीनगर में बीते दिन न्यूनतम तापमान 3.8 डिग्री सेल्सियस, पहलगाम में माइनस 2.4 डिग्री और गुलमर्ग में माइनस 5 डिग्री सेल्सियस दर्ज किया गया। इसके अतीरिक्त जम्मू शहर में न्यूनतम तापमान 12.8 डिग्री सेल्सियस, कटरा में 12.2 डिग्री, बनिहाल में 4.6 डिग्री और भद्रवाह में 5.1 डिग्री सेल्सियस रहा। | hindi |
இவர் முன் நடிக்கும் போது எனக்கும் ஆப்பிள் பாக்ஸ் போட்டார்கள் சிவகார்த்திகேயன் சொன்ன அந்த நடிகர் யார் சென்னை: தமிழக சினிமாவின் மிக முக்கிய நடிகராக இருக்கும் சிவகார்த்திகேயன் நடிப்பில், இயக்குநர் நெல்சன் இயக்கியுள்ள டாக்டர் திரைப்படம், ரசிகர்களிடையேயும் வர்த்தக வட்டாரங்களிடையேயும் இந்த வருடத்தின் மிகவும் எதிர்பார்க்கபடும் படங்களில் ஒன்றாக உள்ளது. Nelson பேசுறது Comedyஆனு seriousஆ தெரியாது Sivakarthikeyan, Vinay, Priyanka Mohan இப்படத்தை Sivakarthikeyan Productions உடன் இணைந்து, KJR Studios சார்பில் கோட்டபாடி J ராஜேஷ் தயாரித்துள்ளார். உலகமெங்கும் அக்டோபர் 9 ஆம் தேதி, இப்படம் வெளியாகவுள்ள நிலையில், பட வெளியீட்டை ஒட்டி, படக்குழுவினர் பத்திரிகையாளர்களை சந்தித்து சமீபத்தில் பேசினர் . மாநாடு படத்தின் வெற்றிக்கு வாழ்த்திய டாக்டர் ஹீரோ இந்த படத்தில் நடித்த நடிகர் ரெடின் பேசியதாவது... இந்தப்படம் மிக கலகலப்பாக இருந்தது. நான் நடித்த காட்சிகள் நிர்மல் கட் பண்ணிவிடுவாரோ என்ற பயம் இருந்தது. நெல்சன் என்னை வித்தியாசமாக காட்டியிருக்கிறார். படம் முழுக்க பிரமாண்ட செட் போட்டு தான் படம் எடுத்திருக்கிறோம்.குறிப்பாக ஷாப்பிங் செய்யும் பெரிய பில்டிங்ஸ் பிரமிக்க வைக்கும் பாருங்கள் என்று சொல்லி அனைவருக்கும் நன்றி தெரிவித்தார் . டி வி புகழ் அர்ச்சனா இந்த மகிழ்ச்சியை தெரிவிக்க வார்த்தைகளே இல்லை. எனக்கும் என் மகளுக்கும் ஒரு சேஃபான உணர்வை தந்தார்கள். இந்தப்படத்தில் ஒரு குடும்பமாக தான் வேலை பார்த்தோம் மிக அழகான படமாக, அனைவருக்கும் பிடிக்கும் படமாக இப்படம் இருக்கும். கண்டிப்பாக அனைவரும் தியேட்டரில் போய் படம் பாருங்கள் நன்றி என்று கூறினார் . வினய் கனவு நிறைவேறியது நெல்சன், சிவா உங்கள் இருவருக்கும் நன்றி. இந்த 15 வருடத்தில் 15 படங்கள் பன்ணியிருக்கிறேன். அனைவருடனும் இன்றும் நல்ல உறவு இருக்கிறது. இந்தப்படத்தில் வாய்ப்பு கிடைத்ததே ஒரு கனவு நிறைவேறியது போலவே இருந்தது. இந்தப்படம் முழுதுமே ஒரு இனிமையான பயணமாக இருந்தது. அனைவருக்கும் இப்படம் பிடிக்கும். உங்களுடன் மீண்டும் இணைந்து பணிபுரிய காத்திருக்கிறேன் என்று சொன்னது மட்டும் அல்லாமல் சிவகார்த்திகேயனுடன் தனது அம்மாவை வீடியோ கால் மூலம் பேச வைத்து மகிழ்ந்தார் நடிகர் வினய் உண்மையிலும் அண்ணாவாக நாயகி பிரியங்கா அருள் மோகன் பேசியதாவது ....இப்படம் எனக்கு கிடைத்ததை ஆசிர்வாதமாகத்தான் பார்க்கிறேன். என்னுடைய அறிமுக படமே பெரிய படமாக கிடைத்தது மகிழ்ச்சி. SK நடிகர், தயாரிப்பாளர், பாடலாசிரியர், பாடகர் என வளர்ந்துகொண்டே போகிறார் அவருக்கு நன்றி. என்னை அறிமுகப்படுத்திய இயக்குநருக்கு நன்றி. அனிருத் இசை அட்டாகாசமாக இருகிறது. அருண் எனக்கு அண்ணாவாக நடித்தார் உண்மையிலும் அண்ணாவாக இருந்தார். இப்படம் மிகப்பெரிய புகழை பெற்று தந்திருக்கிறது. மிகப்பெரிய சந்தோஷத்தை தந்துள்ளது. படமும் மிகப்பெரிய வெற்றி பெறும் என்று நம்புவுதாக சொல்லி மகிழ்ச்சி தெரிவித்தார் . மிகப்பெரும் வருத்தம் இயக்குநர் நெல்சன் பேசியதாவது....முதலில் சிவகார்த்திகேயன் வழக்கமாக அவரது படங்கள் போல் இல்லாமல் இருக்க வேண்டும் என்று அவரிடம் பேசி தான் இப்படம் எடுக்கலாம் என்று முடிவு செய்தோம். அவரிடம் இரண்டு ஐடியா சொன்னேன் இரண்டுமே நல்லாருக்கு நீங்களே முடிவு பண்ணுங்கள் என்றார். என் கடமை அதிகமாகிவிட்டது. படம் எடுக்க ஆரம்பித்த இரண்டு வாரத்தில் இது நன்றாக வந்துவிடும் என்ற நம்பிக்கை வந்துவிட்டது. படம் நினைத்தது போலவே ஒரு நல்ல படமாக வந்துள்ளது. சிவகார்த்திகேயனே படத்தயாரிப்பாளர் என்பதால், அது எனக்கு உதவியாக இருந்தது. என்னை கேள்வி கேட்காமல் எனக்கு முழு சுதந்திரம் அளிக்கப்பட்டது. விஜய் கார்த்திக் ஒவ்வொரு ஷாட்டுக்கும் கேள்வி கேட்டுக்கொண்டே இருப்பார். ஆனால் நினைத்ததை கொண்டு வந்து விடுவார். முழுப்படத்தையும் எடிட் பண்ணிதற்குப்புறம், எனக்கே தெரியாமல் எடிட் செய்துவிட்டார் நிர்மல். அந்தளவு படத்துடன் ஒன்றியிருப்பார். ப்ரியங்கா அவரது முழுத்திறமை இந்தப்படத்தில் வெளிப்படவில்லை, அவருடன் மீண்டும் படங்கள் செய்வேன். வினய் பார்த்து பழகும் போது அப்பாவியாக இருந்தார் ஆனால் படத்தில் வில்லனாக அசத்தியுள்ளார். அருணை ரொம்ப மிஸ் பண்ணுகிறேன். அவரை எல்லாப்படத்திலும் வைத்து கொள்ள வேண்டும் என்று நினைத்திருந்தேன் அவர் இல்லாதது மிகப்பெரும் வருத்தம். அனிருத்தை வைத்து தான் திரைக்கதையே எழுதுவேன் அவர் இப்படத்திற்கு மிகப்பெரிய பலம். படமும் நினைத்தது போல அழகாக வந்திருக்கிறது. படம் பாருங்கள் உங்கள் அனைவருக்கும் பிடிக்கும் என்று சொல்லி எதிர்பார்ப்பை கூட்டினார் நெல்சன் . சிவகார்த்திகேயன் நின்ற ஆப்பிள் பாக்ஸ் இரண்டு வருடங்கள் கழித்து அனைவரையும் சந்தித்தது மகிழ்ச்சி. எனக்கு பாட்டு எழுதும் நம்பிக்கை எல்லாம் இருந்தது இல்லை. நெல்சன் தான் அவரது முதல் படத்தில் ஆரம்பித்து வைத்தார். இப்படத்தில் செல்லம்மா பாடல் எளிதாக இருந்தது. ஆனால் ஓ பேபி பாடல் கொஞ்சம் கஷ்டமாக பயமாக இருந்தது. அதிலும் நல்ல பெயர் எடுக்க வேண்டும் என்ற எண்ணம் இருந்தது. மக்களுக்கு பிடித்தது மகிழ்ச்சியாக இருக்கிறது. இந்தப்படத்தில் எனக்கு டயாலாக்கே இல்லை. மொத்தமாகவே ஒரு பத்து டயலாக் தான். எல்லோரும் பேசிக்கொண்டிருக்கும் போது நான் படத்தில் பேசாமல் இருந்தது கஷ்டமாக இருந்தது. ஆனால் நெல்சன் எப்படி என்னை இப்படி யோசித்தார் என்று தோன்றியது. வினய் உன்னாலே உன்னாலே படம் பார்த்ததில் இருந்து பிடிக்கும். நான் உயரமாக இருக்கிறேன் என்று நினைத்து கொண்டிருந்தேன் ஆனால் வினய் முன்னால் நடிக்கும் போது எனக்கே ஆப்பிள் பாக்ஸ் போட்டு தான் நின்றேன். மனுஷன் மிகப்பெரிய உயரமாக இருந்தார். அவரது குரலும் பாடியும் படத்திற்கு மிகப்பெரிய பலம். ப்ரியங்காவிற்கு தமிழ் தெரிந்தது மிகப்பெரிய உதவியாக இருந்தது. தமிழ் தெரிந்த நடிகையுடன் நடிக்கும் போது, படப்பிடிப்பிலேயே காட்சி எப்படி வரும் என்ற தெளிவு இருக்கும். ரெடின், யோகிபாபு இப்படத்தில் கலக்கியுள்ளனர். அருண் ப்ரோ அவர் இப்படத்தில் செய்தது காலாகாலத்திற்கும் பேசப்படும், அவரை ரொம்ப மிஸ் பண்ணுகிறேன். விஜய் கார்த்திக் ஒளிப்பதிவு படத்தில் அட்டகாசமாக இருக்கும், தியேட்டரில் பார்த்தால் உங்களுக்கு புரியும். அனிருத் தான் இந்தப்படத்தை அறிவித்ததிலிருந்தே, இதற்கு அடையாளமாக இருந்தவரே அவர்தான். இந்தப்படம் நடித்த அனைவருக்குமே முக்கியமான படமாக இருக்கும். இந்தப்படம் எல்லாருக்கும் பிடிக்கும் ஒரு படமாக இருக்கும் என்று ரசிகர்களை நம்பி சிரித்த முகத்துடன் பேசி முடித்தார் . அக்டோபர் 9ஆம் தேதி நெல்சன் திலீப்குமார் எழுதி இயக்கியுள்ள டாக்டர் திரைப்படத்தை , சிவகார்த்திகேயனின் Sivakarthikeyan Productions உடன் இணைந்து, KJR Studios சார்பில் கோட்டபாடி J ராஜேஷ் தயாரிக்கிறார். சிவகார்த்திகேயன் முதன்மை கதாபாத்திரத்தில் நடிக்க, பிரியங்கா அருள் மோகன் கதாநாயகியாகவும், வினய் ராய் வில்லனாகவும் நடிக்கின்றனர். அனிருத் இசையமைத்துள்ளார். ஒளிப்பதிவை விஜய் கார்த்திக் செய்துள்ளார். R. நிர்மல் எடிட்டிங் செய்துள்ளார். அக்டோபர் 9 ந்தேதி இப்படம் உலகமெங்கும் திரையரங்குகளில் வெளியாகிறது. source: filmibeat.com | tamil |
ટીવી રિયાલિટી શો લોકઅપમાં કંગના રનૌતનો ખુલાસો બાળપણમાં થઇ હતી સેક્યુઅલ હેરેસમેન્ટનો શિકાર ટીવી રિયાલિટી શો લોકઅપના રવિવારે પ્રસારિત થયેલા એપિસોડમાં શોની હોસ્ટ કંગના રનૌતે ખૂબ જ ચોંકાવનારો ખુલાસો કર્યો છે. કંગનાએ કહ્યું કે બાળપણમાં પણ તેને જાતીય શોષણનો સામનો કરવો પડ્યો હતો. વાસ્તવમાં શોના એક સ્પર્ધક મુનવ્વર ફારૂકીએ એક ટાસ્ક દરમિયાન કહ્યું હતું કે તે બાળપણમાં જાતીય શોષણનો શિકાર બન્યો હતો. મુનવ્વરની વાત સાંભળ્યા બાદ કંગનાને પણ તેની સાથે બનેલી આવી જ ઘટના યાદ કરી હતી. વાસ્તવમાં શોના અન્ય એક સ્પર્ધક સાયશા શિંદેએ શોની એક સ્પર્ધકને તેના જીવનનું એક મોટું રહસ્ય બધાને જણાવવા માટે કહ્યું હતું. આ સ્થિતિમાં મુનવ્વર ફારૂકીએ આગળ આવીને કહ્યું કે તે માત્ર 11 વર્ષનો હતો જ્યારે તેના જ નજીકના સંબંધી દ્વારા તેનું જાતીય શોષણ કરવામાં આવ્યું હતું. મુનવ્વરે જણાવ્યું હતું કે મારી જાતીય સતામણી થઈ ત્યારે હું માત્ર છ વર્ષનો હતો અને હું 11 વર્ષનો થયો ત્યાં સુધી આ ચાલુ રહ્યુ હતું. જાતીય શોષણ કરનાર બીજુ કોઇ નહી પરંતુ મારા બે સગા હતા. એ વખતે હું નાનો હતો અને આ બધી વાતો સમજી શકતો નહોતો. પછી એક વાર બહુ થઈ ગયું પછી પેલા બંને સગાંઓને લાગ્યું કે હવે અમારે છોડી દેવું જોઇએ. મુનવ્વર એ વધુમાં જણાવ્યું કે જ્યારે તેણે તેના પિતાને આ વિશે કહ્યું તો તેઓ ખૂબ ગુસ્સે થયા અને કહ્યું કે આવી વાતો બહાર ન આવવી જોઈએ. આ બધું સાંભળીને કંગનાએ કહ્યું કે દર વર્ષે ઘણા બાળકો જાતીય શોષણનો શિકાર બને છે. તેઓ ક્યારેય આ અંગે ખુલીને વાત નથી કરતા. અભિનેત્રીએ કહ્યું કે, મેં પણ આવી પરિસ્થિતિનો સામનો કર્યો છે, હું ત્યારે બાળક હતી અને અમારા ટાઉનશીપનો છોકરો મને અયોગ્ય રીતે સ્પર્શ કરતો હતો, ત્યારે હું પણ નાની હતી અને મને આ વાતનો અર્થ સમજાતો નહોતો. LSG vs MI: કેએલ રાહુલે મુંબઈ સામે સીઝનની બીજી સદી ફટકારી સાથે આ અનોખો રેકોર્ડ પણ બનાવ્યો... બિકીની પર હિંદુ દેવીદેવતાઓની તસવીરો છાપવા બદલ સોશિયલ મીડિયા યુઝર્સે કપડાની બ્રાન્ડની ઝાટકણી કાઢી Nia Sharma Video: રેડ સાડીમાં ટિપ ટિપ બરસા પાની ગીત પર નિયા શર્માનો હોટ ડાન્સ, જુઓ વીડિયો WHOનો મોટો દાવોઃ બાળકોમાં જોવા મળી રહ્યા છે અજ્ઞાત મૂળના હેપેટાઈટિસના કેસ, આ મોટા દેશ ઝપેટમાં આવ્યા | gujurati |
\begin{document}
\title[Self-Similar Solutions to the Mullins' Equation]{Self-Similar Grooving Solutions to the Mullins' Equation}
\author{Habiba V.~Kalantarova$^\ast$$^1$}
\email{$^[email protected]}
\author{Amy Novick-Cohen$^\ast$$^2$}
\email{$^[email protected]}
\address{$^\ast$Department of Mathematics, Technion-IIT, Haifa 32000, Israel}
\date{\today}
\begin{abstract}
In 1957, Mullins proposed surface diffusion motion as a model for
thermal grooving. By adopting a small slope approximation,
he reduced the model to the \textit{Mullins' linear surface diffusion equation,}
\begin{equation} \nonumber
({\rm{ME}})\quad\quad y_t + B y_{xxxx}=0,
\end{equation}
known also more simply as the \textit{Mullins' equation}. Mullins sought self-similar solutions to (ME) for planar initial conditions, prescribing boundary conditions at the thermal groove, as well as far field decay. He found explicit series solutions which are routinely used in analyzing thermal grooving to this day.
While (ME) and the small slope approximation are physically reasonable, Mullins' choice of boundary conditions is not always appropriate. Here we present an in depth study of self-similar solutions to the Mullins' equation for general self-similar boundary conditions, explicitly identifying four linearly independent solutions defined on $\mathbb{R}\setminus\{0\}$; among these four solutions,
two exhibit unbounded growth and two exhibit asymptotic decay, far from the origin. We indicate how the full set of solutions can be used in analyzing the effective boundary conditions from experimental profiles and in evaluating the governing physical parameters.
\end{abstract}
\maketitle
\section{Introduction}\label{sec:intr}
Motion by surface diffusion
\begin{equation} \label{msd}
V_n = - B \triangle_s \kappa,
\end{equation}
describes a geometric motion for an evolving surface. In (\ref{msd}), $V_n$ and $\kappa$ denote, respectively, the normal velocity and the mean curvature of the evolving surface, $\triangle_s$ denotes the Laplace-Beltrami operator known also as the surface Laplacian, and $B$ is the \textit{Mullins' coefficient}. Motion by surface diffusion, as well as motion by mean curvature, were first proposed by Mullins \cite{Mullins1956,Mullins1957} in modeling the evolution of microstructure in polycrystalline materials. Polycrystalline materials contain numerous crystals or \textit{grains}, separated by grain boundaries, and there is a tendency for thermal grooves to form where interior grain boundaries intersect the exterior surface of the polycrystalline specimen. The
evolution of the microstructure, including the phenomenon of thermal grooving, are of quite general interest, since the microstructure and grooving in particular are highly
influential in determining the strength, the stability, as well as many other properties of polycrystalline materials.
In using (\ref{msd}) to model the development of thermal grooves, various possible effects have been neglected, such as bulk diffusion \cite{Hardy1991}, surface energy anisotropy \cite{Davi1990}, \cite{Klinger2001}, \cite{Xin2003}, as well as evaporation and condensation \cite{Mullins1957}.
The Mullins' coefficient is frequently prescribed as $B= D_s \gamma_{ext}\, \Omega^2\nu/(kT)$, where
$D_{s}$ is the surface diffusion coefficient, $\gamma_{ext}$ is the surface-free energy per unit area of the exterior surface, $\Omega$ is the atomic volume, $\nu$ is the number of mobile atoms per unit area, $k$ is the Boltzmann constant and $T$ is the temperature.
In studying the formation of thermal grooves, it is constructive to focus on the normal cross-section to some particular thermal
groove. Under the assumption that the height of the exterior surface can be described in the normal cross-section as the graph of function, $y=y(t,x)$,
relative to an initially planar exterior surface, $y(0,x)\equiv 0,$ and
that there is little out of plane variation in the shape of the thermal groove relative to the cross-sectional plane, then (\ref{msd}) implies that
\begin{equation}\label{mllnsoe}
y_{t}=-B[\kappa_{x}(1+y_{x}^{2})^{-1/2}]_{x}, \kappa=y_{xx}(1+y_{x}^{2})^{-3/2},\ x\in \mathbb{R}\setminus \{0\},\ t>0.
\end{equation}
In writing (\ref{mllnsoe}), it has been implicitly assumed that the thermal groove is initially located at $x=0$ and maintains its location there, and that there are no
additional effects influencing the shape of the exterior surface.
To obtain a complete problem formulation for (\ref{mllnsoe}), it is reasonable to impose conditions at $x=0$ as well as far field conditions, in addition to the initial planarity condition $y(0,x)\equiv 0$. Mullins \cite{Mullins1957} effectively imposed symmetry with respect to $x=0$, implying that the grain boundary attached below the thermal groove is constrained to lie along the $y-$axis and remain
orthogonal to the planar surface $y\equiv 0$ for $t \ge 0$.
In accordance with balance of mechanical forces (Herring's law), he required that
$y_x(t,0^+):=\lim_{x\rightarrow0^{+}}y_{x}(t,x)=m/\sqrt{4-m^2}$, where $m=\gamma_{gb}/\gamma_{ext}$ and $\gamma_{gb}$, $\gamma_{ext}$ denote, respectively, the surface energies of the grain boundary and of the exterior surface. Zero mass flux along the thermal groove was assumed. Noting that the resultant problem was non-trivial, Mullins observed that typically $0< m=\gamma_{gb}/\gamma_{ext}< 1/3.$ This allowed him to treat $m$ as a small dimensionless parameter and to make the
physically reasonable assumption that the slope of the exterior surface remained small at all times.
Based on the {small slope assumption}, Mullins \cite{Mullins1957} obtained a simpler linear problem formulation, namely, Mullins' linear surface diffusion equation
\begin{equation}\label{b2}
y_{t}+B y_{xxxx}=0, \quad x\in \mathbb{R}\setminus \{0\},\quad t>0,
\end{equation}
often referred to more simply as the \textit{Mullins' equation}, (ME), together with the initial condition
\begin{equation}\label{icb2}
y(0,x)=0, \quad x\in \mathbb{R},
\end{equation}
the boundary conditions at $x=0$,
\begin{equation} \label{bcb2}
\lim_{x\rightarrow0^{\pm}}y_x(t,x)=\pm m/2,\ \lim_{x\rightarrow0^{\pm}}y_{xxx}(t,x)=0, \quad t>0,
\end{equation}
as well as far field decay.\footnote{In \cite{Mullins1957} the assumption is made that the "solution $\ldots$ behaves properly at infinity."}
Mullins sought symmetric self-similar solutions of the form
\begin{equation} \label{ssform}
y(t,x) = (Bt)^{1/4} Z(x/(Bt)^{1/4}),
\end{equation}
where $Z=Z(u)$ satisfies
\begin{equation}\label{rm34}
Z^{(4)}(u)-\frac{1}{4}uZ'(u)+\frac{1}{4}Z(u)=0,\quad u\in\mathbb{R},
\end{equation}
for the problem prescribed in (\ref{b2})--(\ref{bcb2}), guided by the form of the Laplace transform of (\ref{b2}). He obtained a power series solution with recursively defined coefficients, and this solution implied the now classical formula for the depth
of the thermal groove as a function of time, $d(t):=y(t,0),$ namely
\begin{equation} \label{depth}
d(t)= -\frac{m (Bt)^{1/4}}{2\sqrt{2}\Gamma(5/4)},\quad t\geq 0.
\end{equation}
Studies of this problem in the physical literature typically rely strongly on the linear solution derived by Mullins. In \cite{Martin2009}, Martin obtained an integral representation for Mullins' solution by using Fourier cosine transforms, which led him to conclude that Mullins' solution exhibited far field decay to planarity.
Often Mullins' assumptions regarding the accompanying boundary conditions are not overly accurate.
Possible concerns in this direction include the following:
The underlying grain boundary may not remain vertical due to internal motion of the grain boundaries, hence the symmetry assumption may not be valid. Often there
is some amount of mass flux along the grain boundary which reaches and interacts with the thermal groove, so the vanishing mass flux assumption may not be realistic, \cite{Amram2014}.
Since all specimens are necessarily of finite extent,
far field planarity is not obvious, and it often is of interest to analyze the development of thermal grooves which are not well isolated from their
surroundings. Accordingly with these issues in mind, we return in this paper to consider self-similar solutions to (ME) on $(t,x)\in (0, T)\times \mathbb{R}\setminus\{0\}$ with more general boundary conditions, without explicitly imposing far field decay or initial planarity.
We begin by treating the resultant more general problem by making use of the theory of generalized hypergeometric differential equations (GHDE), \cite{NIST}, to demonstrate that all self-similar solutions to \eqref{b2} of the form \eqref{ssform} may be expressed as
\begin{equation}\label{d41}
y(t,x)=(Bt)^{1/4} \sum_{i=0}^{3} C_i\, z_i(u), \quad\quad u=x/(Bt)^{1/4}, \quad C_i \in\mathbb{R},
\end{equation}
where
\begin{eqnarray}
&z_{1}(u)=\prescript{}{1}{F}_{3}^{}(-\frac{1}{4};\frac{1}{4},\frac{1}{2},\frac{3}{4};\frac{u^{4}}{256}),\quad &z_{2}(u)=u, \nonumber\\[2ex]
&z_{3}(u)=u^{2}\prescript{}{1}{F}_{3}^{}(\frac{1}{4};\frac{3}{4},\frac{5}{4},\frac{3}{2};\frac{u^{4}}{256}),\quad & z_{4}(u)=u^{3}\prescript{}{1}{F}_{3}^{}(\frac{1}{2};\frac{5}{4},\frac{3}{2},\frac{7}{4};\frac{u^{4}}{256}),\nonumber
\end{eqnarray}
and the functions $\prescript{}{1}{F}_{3}^{}(a_1;b_1,b_2,b_3;\cdot)$ with $a_1,\, b_1,\, b_2,\, b_3 \in \mathbb{R}$ denote generalized hypergeometric functions.
The functions $\{z_i(u)\}_{i=1}^{i=4}$ defined above are linearly independent entire functions which satisfy \eqref{rm34}; moreover, $z_1(u)$, $z_3(u)$ are even and
$z_2(u)$, $z_4(u)$ are odd. It can also be readily shown that
\begin{equation} \label{zij}
z_i^{(j-1)}(0)=\delta_{i\,j} (j-1)!, \quad i,j=1,2,3,4.
\end{equation}
From (\ref{ssform}), (\ref{d41}), (\ref{zij}), it follows that
\begin{equation} \label{bcz}
C_{i}=\frac{1}{(Bt)^{(2-i)/4} (i-1)!}\frac{\partial^{(i-1)} y(t,0)}{\partial x^{(i-1)}} , \quad i=1,2,3,4.
\end{equation}
Recalling that by assumption, in our geometry a thermal groove is forming at $x=0$, which effectively reflects the development of a singularity,
it is reasonable to consider the behavior of solutions on either side of the thermal groove separately. This leads us to define for $t>0$
\begin{equation}\label{solution_lin}
y(t,x) =
\begin{cases}
(Bt)^{1/4}\sum_{i=1}^{4} C_{i}^{+} z_i(u), & \quad C_{i}^{+}\in\mathbb{R},\ x>0,\\[1ex]
(Bt)^{1/4}\sum_{i=1}^{4} C_{i}^{-} z_i(u), & \quad C_{i}^{-}\in\mathbb{R},\ x<0,
\end{cases}
\end{equation}
where $u=\frac{x}{(Bt)^{1/4}}$ and
\begin{equation} \label{bc_lin}
C_{i}^{\pm}=\frac{1}{(Bt)^{(2-i)/4} (i-1)!} \lim_{x\rightarrow 0^{\pm}}\frac{\partial^{(i-1)} y(t,x)}{\partial x^{(i-1)}},\quad i=1, 2, 3, 4.
\end{equation}
Note in particular that it follows from (\ref{bc_lin}) that the coefficients in (\ref{solution_lin}) are directly proportional to the derivatives of the surface profile at
the thermal groove. This feature makes the solution representation (\ref{solution_lin}) useful for data fitting.
An alternative approach to solving the Mullins' equation is via Laplace transform methods under the assumption of initial planarity and general boundary conditions at zero in accordance with the self-similar form \eqref{ssform}. Proceeding in this fashion yields
four linearly independent self-similar solutions of the form \eqref{ssform}, which we denote by $\{y_i\}_{i=1}^{i=4}$. Recalling \eqref{rm34} and \eqref{d41}, it follows that the set of self-similar solutions of the form \eqref{ssform} to (ME) is spanned by four linearly independent functions. Hence each of the functions $y_{i}(t,x)$, $i=1,2,3,4,$ may be expressed as a linear combination of the functions $\{(Bt)^{1/4}z_{i}(u)\}_{i=1}^{4}$, where $u=\frac{x}{(Bt)^{1/4}}$.
Accordingly by evaluating
\begin{equation}
\frac{\partial^{(j-1)}y_{i}(t,0)}{\partial x^{(i-1)}},\quad i, j=1, 2, 3, 4\nonumber
\end{equation}
\noindent from their Laplace transforms and taking \eqref{zij} into consideration, we find for $t>0$ and $x\in\mathbb{R}\setminus\{0\}$ that
\begin{equation}
\begin{bmatrix}
y_{1}(t,x)\\[0.3em]
y_{2}(t,x)\\[0.3em]
y_{3}(t,x)\\[0.3em]
y_{4}(t,x)
\end{bmatrix}
=(Bt)^{1/4}\begin{bmatrix}
0 & \frac{1}{\sqrt{2}} & -1 &\frac{1}{\sqrt{2}} \\[0.5em]
1 & -\frac{1}{\sqrt{2}} & 0 &\frac{1}{\sqrt{2}} \\[0.5em]
0 & \frac{1}{\sqrt{2}} & 1 &\frac{1}{\sqrt{2}} \\[0.5em]
1 &\frac{1}{\sqrt{2}} & 0 &-\frac{1}{\sqrt{2}}
\end{bmatrix}
\begin{bmatrix}
\frac{1}{\Gamma\left(\frac{5}{4}\right)}z_{1}(u)\\[0.3em]
\frac{1}{\Gamma(1)} z_{2}(u)\\[0.3em]
\frac{1}{2\Gamma\left(\frac{3}{4}\right)}z_{3}(u)\\[0.3em]
\frac{1}{6\Gamma\left(\frac{1}{2}\right)}z_{4}(u)
\end{bmatrix},\nonumber
\end{equation}
\noindent where $u=\frac{x}{(Bt)^{1/4}}$. Here as in \eqref{solution_lin}-\eqref{bc_lin} we may define solutions separately on either side of the thermal groove.
It is easy to show that
\begin{equation}
y_{1}(t,x)=-y_{3}(t,-x)\quad\mbox{ and }\quad y_{2}(t,x)=y_{4}(t,-x)\nonumber
\end{equation}
for $t>0$, $x\in\mathbb{R}\setminus\{0\}$. Moreover for $t>0$, $\{y_{i}(t,x)\}_{i=1}^{i=2}$ and $\{y_{i}(t,-x)\}_{i=3}^{i=4}$ are asymptotically flat as $x\rightarrow\infty$, and for $x>0$, $\{y_{i}(t,x)\}_{i=1}^{i=2}$ and $\{y_{i}(t,-x)\}_{i=3}^{i=4}$ satisfy the initial planarity condition. By examining series solution expressions for $\{y_i(t,x)\}_{i=3}^{i=4}$ and $\{y_{i}(t,-x)\}_{i=1}^{i=2}$ for $t>0$, we show that they exhibit unbounded far field growth as $x\rightarrow\infty$, and that they do not satisfy initial planarity.
Martin \cite{Martin2009} demonstrated that it is possible to obtain an integral representation for Mullins' solution by using Fourier cosine transforms;
we demonstrate that it is possible to obtain two linearly independent solutions by Fourier cosine transform method, and the integral representations obtained by this approach for these solutions allow us to ascertain that both solutions tend to zero as $x\rightarrow\infty$, for fixed $t>0$.
Earlier we mentioned that the solution representation given in (\ref{solution_lin}) is useful for data fitting.
By undertaking a direct statistical least squares comparison with experimental data from Amram et al., \cite{Amram2014}, we show in Section \ref{sec:df} that our results can be successfully used to fit data and to distinguish between experiments in which Mullins' boundary conditions are accurate from experiments in which other boundary conditions such as the boundary conditions proposed in Amram et al.~\cite{Amram2014}, namely
\begin{equation}\nonumber
y_x(t,0)=m/2, \quad y_{xx}(t,0)=0, \quad \lim_{x \rightarrow \infty} y(t,x)=0,\quad t>0,
\end{equation}
are more accurate. It is not difficult to verify that Mullins' solution \cite{Mullins1957} may be expressed in terms of the functions $\{y_{i}(t,x)\}_{i=1}^{4}$ as
\begin{equation}\label{msol}
\frac{m}{2\sqrt{2}}(y_{1}(t,x)-y_{2}(t,x)),
\end{equation}
and that the solution discussed by Amram et al. \cite{Amram2014} corresponds to
\begin{equation}\label{asol}
-\frac{m}{\sqrt{2}}y_{2}(t,x),
\end{equation}
for details see Section \ref{asymp_dec}. Essentially, our results yield a method for "reading off" the effective boundary conditions from the measurements, see \cite{Kalantarova2019data}. To the authors' knowledge, this is the first study presenting an analytical solution to the Mullins' equation that makes such data fitting possible, which is perhaps the main advantage of our approach.
The paper is organized as follows. In Section 2, we state and prove our main results regarding the existence of a four dimensional self-similar solution to (ME), which can be expressed in terms of generalized hypergeometric solutions
as well as via Laplace transforms, and we describe the far field behavior of these solutions. In Section 3, we briefly demonstrate how these results can be used for data fitting. In Appendix A, we discuss the generalized hypergeometric solutions, indicating in detail how a specific GHDE may be identified whose solutions yield $\{z_{i}(u)\}_{i=1}^{4}$. In Appendix B, we prove in detail the initial and far field behavior of the solutions, as well as demonstrating \eqref{msol}, \eqref{asol}.
\section{Self-similar Solutions to the Mullins' Equation}\label{sec:lt}
Mullins' linear surface diffusion equation (ME)
\begin{equation}
\label{m22}y_{t}+B y_{xxxx}=0,\quad x\in\mathbb{R}\setminus\{0\}, \quad t>0,
\end{equation}
along with the initial and boundary conditions
\begin{eqnarray}
&&\label{lic}y(0,x)=0\quad x\in\mathbb{R}\setminus\{0\},\\
&&\label{slope}\lim_{x\rightarrow0^{\pm}}y_x(t,x)=\pm\frac{m}{2},\quad t>0,
\end{eqnarray}
has the following scaling symmetry, namely, given any solution $y(t,x)$ to \eqref{m22}-\eqref{slope},
\begin{equation}
y_{\lambda}(t,x)=\lambda^{-1}y(\lambda^{4}t, \lambda x)\nonumber
\end{equation}
\noindent is also a solution of \eqref{m22}, for any $\lambda>0$. This scaling property leads one to seek similarity solutions of the form,
\begin{equation}\label{lt4}
y(t,x)=(Bt)^{1/4}Z\left(\frac{x}{(Bt)^{1/4}}\right),
\end{equation}
\noindent see \cite{Bluman2010}, \cite{Mullins1957}. The nonlinear problem \eqref{mllnsoe}, \eqref{lic} also has this scaling property, \cite{Derkach2018, Mullins1957}, but our focus here is on similarity solutions for the linear problem.
Substituting \eqref{lt4} into \eqref{m22} and making the change of variable $u=\frac{x}{(Bt)^{1/4}}$, yields that
\begin{equation}
\label{m34}Z^{(4)}(u)-\frac{1}{4}uZ'(u)+\frac{1}{4}Z(u)=0,\quad u\in\mathbb{R}.
\end{equation}
Having obtained \eqref{m34}, Mullins \cite{Mullins1957} went on to look for a power series solution assuming zero flux at the groove root and far field decay and calculated its coefficients. Here we consider \eqref{m34} without imposing further restrictions on the solutions, $Z$, of \eqref{m34}, such as no flux or decay, and this allows us to gain a more complete understanding of \eqref{m34} and its solutions.
\begin{theorem}
The fourth order linear ordinary differential equation \eqref{m34}
\begin{equation}
Z^{(4)}(u)-\frac{1}{4}uZ'(u)+\frac{1}{4}Z(u)=0,\quad u\in\mathbb{R},\nonumber
\end{equation}
has the following fundamental set of solutions
\begin{eqnarray}
\label{d9a}&&z_{1}(u)=\prescript{}{1}{F}_{3}^{}(-\frac{1}{4};\frac{1}{4},\frac{1}{2},\frac{3}{4};\frac{u^{4}}{256}),\\
\label{d9b}&&z_{2}(u)=u,\\
\label{d9c}&&z_{3}(u)=u^{2}\prescript{}{1}{F}_{3}^{}(\frac{1}{4};\frac{3}{4},\frac{5}{4},\frac{3}{2};\frac{u^{4}}{256}),\\
\label{d9d}&&z_{4}(u)=u^{3}\prescript{}{1}{F}_{3}^{}(\frac{1}{2};\frac{5}{4},\frac{3}{2},\frac{7}{4};\frac{u^{4}}{256}),
\end{eqnarray}
where $\prescript{}{p}{F}_{q}^{}(a_{1}\ldots,a_{p};b_{1},\ldots,b_{q};u)$ for $\{a_{i}\}_{i=1}^{p}$, $\{b_{i}\}_{i=1}^{q}\in\mathbb{R}$ denotes the generalized hypergeometric function (or the generalized hypergeometric series) defined as
\begin{multline}\label{def:hf}
\prescript{}{p}{F}_{q}^{}(a_{1}\ldots,a_{p};b_{1},\ldots,b_{q};\nu)=\sum_{k=0}^{\infty}\frac{(a_{1})_{k}\ldots(a_{p})_{k}}{(b_{1})_{k}\ldots(b_{q})_{k}}\frac{\nu^{k}}{k!}\\
=1+\frac{a_{1}\ldots a_{p}}{b_{1}\ldots b_{q}}\nu+\frac{a_{1}(a_{1}+1)\ldots a_{p}(a_{p}+1)}{b_{1}(b_{1}+1)\ldots b_{q}(b_{q}+1)2!}\nu^{2}+\ldots,
\end{multline}
in which
\begin{equation}
(\lambda)_{k}=\frac{\Gamma(\lambda+k)}{\Gamma(\lambda)}=\lambda(\lambda+1)\ldots(\lambda+k-1)\nonumber
\end{equation}
is the Pochhammer symbol.
\end{theorem}
\begin{proof}
Employing the change of variable
\begin{equation}\label{d33}
u(v)=4v^{1/4}
\end{equation}
in equation \eqref{m34}, yields the equation
\begin{equation}\label{d43}
[v\frac{d}{dv}(v\frac{d}{dv}+b_{1}-1)(v\frac{d}{dv}+b_{2}-1)(v\frac{d}{dv}+b_{3}-1)-v(v\frac{d}{dv}+a_{1})]V=0,
\end{equation}
for $V(v)=Z(u)\big|_{u=u(v)}$, where
\begin{equation}\label{d21}
a_{1}=-\frac{1}{4},\quad b_{1}=\frac{1}{4},\quad b_{2}=\frac{1}{2},\quad b_{3}=\frac{3}{4}.
\end{equation}
Equation \eqref{d43} constitutes a generalized hypergeometric equation.
Generalized hypergeometric differential equations (\textbf{GHDE})
\begin{equation}\label{d36}
v\frac{d}{dv}\left(\prod_{i=1}^{q}(v\frac{d}{dv}+b_{i}-1)\right)V-v\left(\prod_{j=1}^{p} (v\frac{d}{dv}+a_{j})\right)V=0,
\end{equation}
for $V=V(v)$, where $p,\,q \in \mathbb{Z}_+$, $p, \, q>2,$ and $\{a_{j}\}_{j=1}^{j=p}, \; \{b_{i}\}_{i=1}^{i=q}, v \in \mathbb{C}$, were first studied by Thomae \cite{Thomae1870}. In particular, Thomae showed that there
exists a solution to equation \eqref{d36}, which he denoted as
\begin{equation}
\prescript{}{p}{F}_{q}^{}(a_{1},\ldots,a_{p};b_{1},\ldots,b_{q};v).\nonumber
\end{equation}
Accordingly, it follows from \eqref{d43}-\eqref{d21} that
\begin{equation}\label{d34}
\prescript{}{1}{F}_{3}^{}(-\frac{1}{4};\frac{1}{4},\frac{1}{2},\frac{3}{4};\frac{u^{4}}{256})
\end{equation}
is a solution to \eqref{m34}.
From the theory of generalized hypergeometric equations, see e.g. \cite[Chapter 16]{NIST}, since $p<q$ in (\ref{d21}) and since none of the differences between the numbers $0, b_{1}, b_{2}, b_{3}$ is an integer, it follows that
\begin{equation}
V_0(v)=\prescript{}{1}{F}_{3}^{}(a_{1};b_{1},b_{2},b_{3};v)\nonumber
\end{equation}
together with
\begin{equation}\label{d35}
\quad V_j(v)= v^{1-b_{j}}\prescript{}{1}{F}_{3}^{}(1+a_{1}-b_{j}; 1+b_{1}-b_{j},\ldots,\ast,\ldots,1+b_{q}-b_{j};v),
\end{equation}
where $j=1,2,3,$ and where $\ast$ indicates that the $j$th entry is replaced by $2-b_{j},$ form a fundamental set of linearly independent solutions to \eqref{d43}, \eqref{d21}, for $v\in\mathbb{C}$.
Recalling the change of variables (\ref{d33}), it now follows that
\begin{eqnarray}
&&\label{d37}z_{1}(u)=\prescript{}{1}{F}_{3}^{}(-\frac{1}{4};\frac{1}{4},\frac{1}{2},\frac{3}{4};\frac{u^{4}}{256})=\sum_{k=0}^{\infty}\frac{\left(-\frac{1}{4}\right)_{k}}{\left(\frac{1}{4}\right)_{k}\left(\frac{1}{2}\right)_{k}\left(\frac{3}{4}\right)_{k}256^{k}k!}u^{4k},\\
&&\label{d38}z_{2}(u)=u\prescript{}{1}{F}_{3}^{}(0;\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{u^{4}}{256})= u,\\
&&\label{d39}z_{3}(u)=u^{2}\prescript{}{1}{F}_{3}^{}(\frac{1}{4};\frac{3}{4},\frac{5}{4},\frac{3}{2};\frac{u^{4}}{256})=\sum_{k=0}^{\infty}\frac{\left(\frac{1}{4}\right)_{k}}{\left(\frac{3}{4}\right)_{k}\left(\frac{5}{4}\right)_{k}\left(\frac{3}{2}\right)_{k}256^{k}k!}u^{4k+2},\\
&&\label{d40}z_{4}(u)=u^{3}\prescript{}{1}{F}_{3}^{}(\frac{1}{2};\frac{5}{4},\frac{3}{2},\frac{7}{4};\frac{u^{4}}{256})=\sum_{k=0}^{\infty}\frac{\left(\frac{1}{2}\right)_{k}}{\left(\frac{5}{4}\right)_{k}\left(\frac{3}{2}\right)_{k}\left(\frac{7}{4}\right)_{k}256^{k}k!}u^{4k+3},
\end{eqnarray}
form a fundamental set of solutions to \eqref{m34} for $u \in \mathbb{R}$.
\end{proof}
The elements of the fundamental set of solutions $\{z_{i}(u)\}_{i=1}^{4}$ of \eqref{m34} are portrayed in Fig.\ref{fig:fss}. They converge for all finite values of $u\in\mathbb{R}$ and define entire functions, \cite{NIST}. Moreover they exhibit the following asymptotic behavior
\begin{figure}
\caption{\footnotesize{The elements of the fundamental set of solutions.}
\label{fig:fss}
\end{figure}
\begin{flushleft}
\begin{eqnarray}
&&\lim_{u\rightarrow\infty}z_{1}(u)=-\infty,\quad\quad \lim_{u\rightarrow\infty}z_{i}(u)=\infty,\quad i=2,3,4,\nonumber\\
&&\lim_{u\rightarrow-\infty}z_{i}(u)=-\infty,\quad i=1,2,4,\quad\quad \lim_{u\rightarrow-\infty}z_{3}(u)=\infty.\nonumber
\end{eqnarray}
\end{flushleft}
From the definitions \eqref{d37}-\eqref{d40}, it is easy to verify that $z_{i}^{(j-1)}(0)$, $i, j= 1,2,3,4,$ satisfies \eqref{zij}. Returning to \eqref{lt4}, \eqref{m34}, we obtain
\begin{theorem}
If $y(t,x)$ is a self-similar solution to \eqref{m22} for $x>0$ (or $x<0$), $t>0$, which is of the form \eqref{lt4}, then $y(t,x)$ may be expressed as
\begin{equation}\label{d10}
y(t,x)=(Bt)^{1/4}\sum_{i=1}^{4}C_{i}z_{i}\left(\frac{x}{(Bt)^{1/4}}\right),\quad x>0,\ (\mbox{or }x<0),\ t>0,
\end{equation}
where $C_{i}$, $i=1,\ldots, 4$ are arbitrary constants, and the functions $\{z_{i}\}_{i=1}^{4}$ are prescribed in \eqref{d37}-\eqref{d40}.
\end{theorem}
The following theorem allows us to distinguish between decaying and growing solutions to \eqref{m22} of the form \eqref{lt4}.
\begin{theorem}
The functions $\{y_{i}(t,x)\}_{i=1}^{4}$ defined by
\begin{equation}\label{d20}
\begin{bmatrix}
y_{1}(t,x)\\[0.3em]
y_{2}(t,x)\\[0.3em]
y_{3}(t,x)\\[0.3em]
y_{4}(t,x)
\end{bmatrix}
=(Bt)^{1/4}\begin{bmatrix}
0 & \frac{1}{\sqrt{2}} & -1 &\frac{1}{\sqrt{2}} \\[0.5em]
1 & -\frac{1}{\sqrt{2}} & 0 &\frac{1}{\sqrt{2}} \\[0.5em]
0 & \frac{1}{\sqrt{2}} & 1 &\frac{1}{\sqrt{2}} \\[0.5em]
1 &\frac{1}{\sqrt{2}} & 0 &-\frac{1}{\sqrt{2}}
\end{bmatrix}
\begin{bmatrix}
\frac{1}{\Gamma\left(\frac{5}{4}\right)}z_{1}(u)\\[0.3em]
\frac{1}{\Gamma(1)} z_{2}(u)\\[0.3em]
\frac{1}{2\Gamma\left(\frac{3}{4}\right)}z_{3}(u)\\[0.3em]
\frac{1}{6\Gamma\left(\frac{1}{2}\right)}z_{4}(u)
\end{bmatrix},
\end{equation}
\noindent where $u=\frac{x}{(Bt)^{1/4}}$, form a fundamental set of self-similar solutions to \eqref{m22}, namely
\begin{equation}
y_{t}+B y_{xxxx}=0,\quad x\in\mathbb{R}\setminus\{0\}, \quad t>0,\nonumber
\end{equation}
satisfying \eqref{lt4}, which are linearly independent over the domain of definition. Moreover for $t>0$,
\begin{eqnarray}
&&\label{d24a}
\lim_{x\rightarrow\infty}y_{1}(t,x)=\lim_{x\rightarrow\infty}y_{2}(t,x)=0,\\
&&\label{d24b}
\lim_{x\rightarrow-\infty}y_{3}(t,x)=\lim_{x\rightarrow-\infty}y_{4}(t,x)=0,\\
&&\label{d25a}
\lim_{x\rightarrow-\infty} y_{1}(t, x)=\lim_{x\rightarrow-\infty} y_{2}(t, x)=-\infty,\\
&&\label{d25b}
\lim_{x\rightarrow\infty} y_{3}(t,x)=\infty,\quad \lim_{x\rightarrow\infty}y_{4}(t,x)=-\infty,
\end{eqnarray}
and furthermore
\begin{eqnarray}
\label{d26a}&&\lim_{t\rightarrow0}y_{1}(t,x)=\lim_{t\rightarrow0}y_{2}(t,x)=0,\quad x>0,\\
\label{d26b}&&\lim_{t\rightarrow0}y_{3}(t,x)=\lim_{t\rightarrow0}y_{4}(t,x)=0, \quad x<0.
\end{eqnarray}
\end{theorem}
\begin{proof}
Let $y(t,x)$ be a self-similar solution of the form \eqref{lt4} to \eqref{m22}. Then formally taking the Laplace transform of \eqref{m22} with respect to the time variable $t$, we get
\begin{equation}
\label{m26}p\overline{y}+B\overline{y}_{xxxx}=0,\quad x\in\mathbb{R}\setminus\{0\},\quad p>0,
\end{equation}
\noindent where
\begin{equation}
\overline{y}(p,x)=\int_{0}^{\infty}e^{-pt}y(t,x)dt.\nonumber
\end{equation}
From the assumed self-similarity of $y(t,x)$, it follows from \eqref{d10} and \eqref{zij} that
\begin{equation}
\label{d44}\frac{\partial^{(i-1)}y}{\partial {x}^{(i-1)}}(t,0)=(Bt)^{(2-i)/4}C_{i}(i-1)!,\quad i=1,2,3,4.
\end{equation}
By taking the Laplace transform of \eqref{d44}, we get
\begin{equation}\label{lic1}
\frac{\partial^{(i-1)}\overline{y}}{\partial x^{(i-1)}}(p,0)=C_{i}(i-1)!B^{(2-i)/4}\ \Gamma\left(\frac{6-i}{4}\right)p^{(i-6)/4},\quad i=1,2,3,4.
\end{equation}
Let us now note that the ODE given in \eqref{m26} has a set of four fundamental solutions, $\{\overline{y}_{j}(p,x)\}_{j=1}^{4}$,
\begin{eqnarray}\label{d28}
&&\overline{y}_{1}(p,x)=B^{1/4}p^{-5/4}\exp\left(-\frac{p^{1/4}}{B^{1/4}\sqrt{2}}x\right)\sin\left(\frac{p^{1/4}}{B^{1/4}\sqrt{2}}x\right),\\
&&\label{d29}\overline{y}_{2}(p,x)=B^{1/4}p^{-5/4}\exp\left(-\frac{p^{1/4}}{B^{1/4}\sqrt{2}}x\right)\cos\left(\frac{p^{1/4}}{B^{1/4}\sqrt{2}}x\right),\\
&&\label{d30}\overline{y}_{3}(p,x)=B^{1/4}p^{-5/4}\exp\left(\frac{p^{1/4}}{B^{1/4}\sqrt{2}}x\right)\sin\left(\frac{p^{1/4}}{B^{1/4}\sqrt{2}}x\right),\\
&&\label{d31}\overline{y}_{4}(p,x)=B^{1/4}p^{-5/4}\exp\left(\frac{p^{1/4}}{B^{1/4}\sqrt{2}}x\right)\cos\left(\frac{p^{1/4}}{B^{1/4}\sqrt{2}}x\right).
\end{eqnarray}
It follows from \eqref{d28}-\eqref{d31} that
\begin{equation}
\frac{\partial^{(i-1)}\overline{y}_{j}(p,0)}{\partial x^{(i-1)}}=d^{\ast}_{ij}B^{(2-i)/4}p^{(i-6)/4},\quad i,j\in\{1,2,3,4\},\nonumber
\end{equation}
where $d^{\ast}_{ij}=[D^{T}]_{ij}=[D]_{ji}$,
\begin{equation}
D:=\begin{bmatrix}
0 & \frac{1}{\sqrt{2}} & -1 &\frac{1}{\sqrt{2}} \\[0.5em]
1 & -\frac{1}{\sqrt{2}} & 0 &\frac{1}{\sqrt{2}} \\[0.5em]
0 & \frac{1}{\sqrt{2}} & 1 &\frac{1}{\sqrt{2}} \\[0.5em]
1 &\frac{1}{\sqrt{2}} & 0 &-\frac{1}{\sqrt{2}}
\end{bmatrix}.\nonumber
\end{equation}
These solutions may be linearly combined to yield
\begin{equation}\label{lt}
\overline{y}(p,x)=\sum_{i=1}^{4}c_{i}\overline{y}_{i}(p,x),
\end{equation}
which satisfies both \eqref{m26} and \eqref{lic1} if we set
\begin{equation}\label{d32}
\begin{bmatrix}
c_{1}\\[0.3em]
c_{2}\\[0.3em]
c_{3}\\[0.3em]
c_{4}
\end{bmatrix}
=\frac{1}{2}D
\begin{bmatrix}
0!\Gamma\left(\frac{5}{4}\right) C_{1}\\[0.3em]
1!\Gamma(1)C_{2}\\[0.3em]
2!\Gamma\left(\frac{3}{4}\right)C_{3}\\[0.3em]
3!\Gamma\left(\frac{1}{2}\right)C_{4}
\end{bmatrix}.
\end{equation}
Recalling that $\overline{y}(p,x)$ is Laplace transform of $y(t,x)$, we denote by $y_{i}(t,x)$ the inverse Laplace transform of $\overline{y}_{i}(p,x)$, for $i=1,2,3,4,$ and from \eqref{lt} we obtain that
\begin{equation}\label{nh0}
y(t,x)=\sum_{i=1}^{4}c_{i}y_{i}(t,x).
\end{equation}
Since $y(t,x)$ is a self-similar solution to \eqref{m22}, by \eqref{d10} it may be expressed equivalently as
\begin{equation}\label{d27}
y(t,x)=(Bt)^{1/4}\sum_{i=1}^{4}C_{i}z_{i}\left(\frac{x}{(Bt)^{1/4}}\right),\ x>0\ (\mbox{or }x<0),\ t>0,
\end{equation}
where the coefficients in \eqref{d27} can easily be obtained from \eqref{d32} upon noting that $\frac{1}{2}DD^{T}=I$.
The proofs of the asymptotic properties \eqref{d24a}-\eqref{d25b} and \eqref{d26a}-\eqref{d26b} are rather technical and are given in Appendix \ref{de}.
\end{proof}
\section{Data Fitting}\label{sec:df}
A major advantage of our solution representations over previous solutions such as the solution given by Mullins \cite{Mullins1957} and the solution given in Amram et al.~\cite{Amram2014} is that it can be used effectively to do data fitting, enabling the identification of the effective boundary conditions and relevant physical parameters during thermal grooving. The solutions given in \cite{Mullins1957} and \cite{Amram2014} were prescribed via power series with recursively defined coefficients. By relying on the power series representation, these solutions can be plotted as truncated series (or polynomials) with unbounded growth as $x\rightarrow\pm\infty$; accordingly, the resultant plots of these solutions are primarily helpful for analyzing to the surface profiles in close proximity to the thermal groove, as in Fig.~8a \cite{Amram2014}. The solutions \eqref{d41} and \eqref{nh0} which were derived here are more general and in particular, \eqref{d41} is prescribed in terms of known functions which can be readily and accurately evaluated using common software in an arbitrarily wide neighborhood of the thermal groove.
Below, we illustrate data fitting of our solution to experimental data by Amram et al.~\cite{Amram2014}, from atomic microscopy measurements of thermal groove formation in a nickel (Ni) film, Fig.~\ref{fig:film20}, and in bulk Ni, Fig.~\ref{fig:bulk20}, after annealing of the specimens at $700^{\circ}$C for 20 minutes.
\begin{figure}
\caption{\scriptsize{Experimental data from Amram et al., \cite{Amram2014}
\label{fig:film20}
\end{figure}
\footnote{\label{acta}Reprinted from Acta Materialia, Vol. 69, D.~Amram, L.~Klinger, N.~Gazit, H.~Gluska, E.~Rabkin, Grain boundary grooving in thin films revisited: The role of interface diffusion, pp.~386-396, Copyright (2014), with permission from Elsevier.}
\begin{figure}
\caption{\scriptsize{Experimental data from Amram et al., \cite{Amram2014}
\label{fig:bulk20}
\end{figure}
\appendix
\setcounter{theorem}{0}
\section{Derivation of a Set of Fundamental Solutions}\label{dgs}
\begin{lemma} The equation \eqref{m34}
\begin{equation}
Z^{(4)}(u)-\frac{1}{4}uZ'(u)+\frac{1}{4}Z(u)=0\quad u>0\quad (\mbox{or }u<0)\nonumber
\end{equation}
can be transformed into a GHDE for $V=V(v)$,
\begin{equation}\label{a1}
v \frac{d}{dv}\Bigl( \prod_{i=1}^{q} (v\frac{d}{dv} + b_i -1) \Bigr)V -v \Bigl( \prod_{j=1}^{p} (v\frac{d}{dv} + a_j) \Bigr)V=0,
\end{equation}
where $p,\,q \in \mathbb{Z}_+$, $p, \, q>2,$ and $\{a_{j}\}_{j=1}^{j=p}, \; \{b_{i}\}_{i=1}^{i=q}, v \in \mathbb{C}$, by setting
\begin{equation}\label{a3}
p=1,\quad q=3,\quad \hbox{\, and \,}\quad a_{1}=-\frac{1}{4}, b_{1}=\frac{1}{4}, b_{2}=\frac{1}{2}, b_{3}=\frac{3}{4},
\end{equation}
and making the change of variable
\begin{equation}\label{a2}
v(u)=\frac{u^{4}}{256}.
\end{equation}
\end{lemma}
\begin{proof}
Observe that \eqref{a1} may be expanded and written as
\begin{equation}\label{a9}
v^{q}\frac{d^{q+1}V}{d v^{q+1}}+\sum_{j=1}^{q}v^{j-1}(\alpha_{j}v+\beta_{j})\frac{d^{j}V}{d v^{j}}+\alpha_{0}V=0\quad\mbox{ if }p\leq q,
\end{equation}
\noindent where $\alpha_{j}$, $\beta_{j}\in \mathbb{C}$, or as
\begin{equation}\label{a10}
v^{q}(1-v)\frac{d^{q+1}V}{d v^{q+1}}+\sum_{j=1}^{q}v^{j-1}(\tilde{\alpha}_{j}v+\tilde{\beta}_{j})\frac{d^{j}V}{d v^{j}}+\tilde{\alpha}_{0}V=0\quad\mbox{ if }p=q+1,
\end{equation}
\noindent where $\tilde{\alpha}_{j}$, $\tilde{\beta}_{j}\in\mathbb{C}$, or as
\begin{equation}\label{a10a}
v^{p}\frac{d^{p}V}{d v^{p}}+\sum_{j=1}^{p-1}v^{j-1}(\tilde{\tilde{\alpha}}_{j}v+\tilde{\tilde{\beta}}_{j})\frac{d^{j}V}{d v^{j}}+\tilde{\tilde{\alpha}}_{0}V=0\quad\mbox{ if }p> q+1,
\end{equation}
where $\tilde{\tilde{\alpha}}_{j}$, $\tilde{\tilde{\beta}}_{j}$ $\in \mathbb{C}$, \cite{NIST}. Note that if we set $q=3$ in (\ref{a9}), or (\ref{a10}), or we set $p=4$ in \eqref{a10a}, then the resultant equations are fourth order linear homogeneous ODEs in which the coefficients of $\frac{d^{j}}{d v^{j}}V$ for $j=0,\ldots, 4,$ are polynomials in $v$ of degree $d_j$ with $d_j \le j$. This allows us to postulate that via a suitable change of variables, $u=u(v)$, equation \eqref{m34} can be transformed into either \eqref{a9}, \eqref{a10} or \eqref{a10a} for some suitable choice of the parameters $p$, $q\in\mathbb{Z}_{+}$ and $\alpha_{j}$, $\beta_{j}$, $\tilde{\alpha}_{j}$, $\tilde{\beta}_{j}$, $\tilde{\tilde{\alpha}}_{j}$, $\tilde{\tilde{\beta}}_{j}\in\mathbb{R}$, since \eqref{m34} is an ordinary differential equation for $Z(u), u\in\mathbb{R}$ with real valued coefficients.
Let us first consider the case $q=3$ and $p=q+1=4$, which corresponds to (\ref{a10}), and let us attempt to find a change of variables which can transform \eqref{m34} into \eqref{a10}. Setting $u= u(v)$ in \eqref{m34} yields for $V(v)=Z(u)|_{u=u(v)}$ the equation
\begin{multline}\label{a11}
\frac{1}{(u')^{4}}\frac{d^{4}V}{dv^{4}}-6\frac{u''}{(u')^{5}}\frac{d^{3}V}{dv^{3}}+\left(15\frac{(u'')^{2}}{(u')^{6}}
-4\frac{u'''}{(u')^{5}}\right)\frac{d^{2}V}{dv^{2}}\\
+\left(10\frac{u''u'''}{(u')^{6}}-15\frac{(u'')^{3}}{(u')^{7}}-\frac{u^{(4)}}{(u')^{5}}-\frac{1}{4}\frac{u}{u'}\right)\frac{dV}{dv}+\frac{1}{4}V=0.
\end{multline}
\noindent Equating the coefficients of $\frac{d^{4}V}{dv^{4}}$ and $V$ in \eqref{a10} and \eqref{a11} implies that
\begin{equation}\label{a22}
(u')^{-4}=v^{3}-v^{4}.
\end{equation}
\noindent However by using \eqref{a22} to evaluate the coefficient of $\frac{d^{2}V}{dv^{2}}$ in \eqref{a11}, we find that
\begin{equation}
15\frac{(u'')^{2}}{(u')^{6}}-4\frac{u'''}{(u')^{5}}=-\frac{v(51-168v+112v^{2})}{16(v-1)},\nonumber
\end{equation}
\noindent which is not of the form $\tilde{\alpha}_{2}v^{2}+\tilde{\beta}_{2}v$ for $\tilde{\alpha}_{2}$, $\tilde{\beta}_{2}\in\mathbb{R}$. Hence we conclude that there does not exist a change of variables which transforms \eqref{m34} into \eqref{a10}.
A similar argument allows us to conclude that also in the case $p > q + 1$, which corresponds to \eqref{a10a}, there is no change of variables which can transform \eqref{m34} into \eqref{a10a}. This can be seen as follows. In accordance with the form of \eqref{a10a} we equate the coefficients of $\frac{d^{4}V}{dv^{4}}$ and $V$ in \eqref{a10a} and \eqref{a11}, which implies
\begin{equation}\label{a10b}
(u')^{-4}=v^{4}.
\end{equation}
\noindent Using \eqref{a10b} to calculate the coefficient of $\frac{dV}{dv}$ in \eqref{a11}, we get
\begin{equation}
10\frac{u''u'''}{(u')^{6}}-15\frac{(u'')^{3}}{(u')^{7}}-\frac{u^{(4)}}{(u')^{5}}-\frac{1}{4}\frac{u}{u'}=v-\frac{v}{4}(\ln(v)+C), \quad C\in\mathbb{R},\nonumber
\end{equation}
\noindent which is not of the form $\tilde{\tilde{\alpha}}_{1}v+\tilde{\tilde{\beta}}_{1}$ for $\tilde{\tilde{\alpha}}_{1},\tilde{\tilde{\beta}}_{1}\in\mathbb{R}$. This implies that there is no change of variables that can transform \eqref{m34} into \eqref{a10a}.
So let us now focus on the case $p\leq q =3$ which corresponds to (\ref{a9}), and let us look for a change of variable which can transform \eqref{m34} into \eqref{a9}. Equating the coefficients of $\frac{d^{4}}{dv^{4}}V$ in \eqref{a11} and in \eqref{a9}, we get that
$(u')^{-4}=v^{3},$
which implies that
\begin{equation} \label{cv0}
u(v)=4v^{1/4}+C,\quad C \in \mathbb{R}.
\end{equation}
Taking (\ref{cv0}) into account and matching the coefficients of $\frac{d^{j}}{dv^{j}}V,$ $j=1,2,3,$ in \eqref{a11} and \eqref{a9}, we get
\begin{eqnarray}
&&10\frac{u''u'''}{(u')^{6}}-15\frac{(u'')^{3}}{(u')^{7}}-\frac{u^{(4)}}{(u')^{5}}-\frac{1}{4}\frac{u}{u'}=\frac{3}{32}-\frac{C}{4}v^{3/4}-v=\alpha_{1}v+\beta_{1},\nonumber\\
&&15\frac{(u'')^{2}}{(u')^{6}}-4\frac{u'''}{(u')^{5}}=\frac{51}{16}v=\alpha_{2}v^{2}+\beta_{2}v,\quad -6\frac{u''}{(u')^{5}}=\frac{9}{2}v^{2}=\alpha_{3}v^{3}+\beta_{3}v^{2},\nonumber
\end{eqnarray}
which imply that
\begin{equation}
\alpha_{0}\!=\frac{1}{4},\quad \alpha_{1}\!=-1,\quad \alpha_{2}\!=\alpha_{3}\!=0,\quad \beta_{1}\!=\frac{3}{32},\quad \beta_{2}\!=\frac{51}{16},\quad\beta_{3}=\frac{9}{2},
\end{equation}
and that $C=0$ in (\ref{cv0}), which implies that
\begin{equation}\label{a12}
u(v)=4v^{1/4}.
\end{equation}
Next, we want to write \eqref{m34} in the form \eqref{a1}, since specific knowledge of the values of $a_{i}$ and $b_{i}$ in \eqref{a1} will provide us with a set of fundamental solutions to \eqref{m34}. Since \eqref{a1} is equivalent to \eqref{a9}, \cite[Section 16.8(ii)]{NIST}, in order to identify the coefficients in \eqref{a1}, we may proceed by using the inverse of the function $u=u(v)$ defined in \eqref{a12}, namely,
\begin{equation}\label{a13}
v(u)=\frac{1}{256}u^{4},
\end{equation}
\noindent as a change of variables in \eqref{a1}.
Recalling that $q=3$ and $p\leq q$, it follows that $p\in\{0, 1, 2, 3\}$. We demonstrate that $p=1$ by eliminating the other cases. The case $p=0$ is easily eliminated, since when $p=0$ equation \eqref{a1} yields
\begin{equation}\label{a13a}
\frac{d^{4}Z}{du^{4}}+B_{3}\frac{4}{u}\frac{d^{3}Z}{du^{3}}+B_{2}\frac{16}{u^{2}}\frac{d^{2}Z}{du^{2}}+B_{1}\frac{64}{u^{3}}\frac{dZ}{du}=0,
\end{equation}
which is not equivalent to \eqref{m34}, as the coefficients of $Z$ in \eqref{a13a} and \eqref{m34} do not match.
Next, let us suppose that $p=3$. Then \eqref{a1} yields
\begin{multline}\label{a14}
\frac{d^{4}Z}{du^{4}}+(B_{3}\frac{4}{u}+A_{3}\frac{u^{3}}{64})\frac{d^{3}Z}{du^{3}}+(B_{2}\frac{16}{u^{2}}+A_{2}\frac{u^{2}}{16})\frac{d^{2}Z}{du^{2}}\\
+(B_{1}\frac{64}{u^{3}}+A_{1}\frac{u}{4})\frac{dZ}{du}+A_{0}Z=0,
\end{multline}
\noindent where
\begin{eqnarray}
\label{a15}&&B_{1}=\frac{9}{16}(b_{1}+b_{2}+b_{3}-\frac{3}{4})-\frac{3}{4}(b_{1}b_{2}+b_{1}b_{3}+b_{2}b_{3})+b_{1}b_{2}b_{3},\\
\label{a16}&&B_{2}=b_{1}b_{2}+b_{1}b_{3}+b_{2}b_{3}-\frac{5}{4}(b_{1}+b_{2}+b_{3})+\frac{19}{16},\\
\label{a17}&&B_{3}=b_{1}+b_{2}+b_{3}-\frac{3}{2},\\
&&A_{0}=-a_{1}a_{2}a_{3},\nonumber\\
&&A_{1}=-\frac{1}{4}(a_{1}+a_{2}+a_{3}+\frac{1}{4})-(a_{1}a_{2}+a_{1}a_{3}+a_{2}a_{3}),\nonumber\\
&&A_{2}=-(a_{1}+a_{2}+a_{3}+\frac{3}{4}),\nonumber\\
\label{a18}&&A_{3}=-1.
\end{eqnarray}
\noindent However, comparing the coefficients of $\frac{d^{3}Z}{du^{3}}$ in equations \eqref{a14} and \eqref{m34} implies that
\begin{equation}
B_{3}\frac{4}{u}+A_{3}\frac{u^{3}}{64}=0,\nonumber
\end{equation}
\noindent which yields a contradiction, since $A_{3}=-1$, $B_{3}$ is a constant, and $u$ is a variable.
Suppose now that $p=2$. Then \eqref{a1} yields
\begin{equation}
\frac{d^{4}Z}{du^{4}}+B_{3}\frac{4}{u}\frac{d^{3}Z}{du^{3}}+(B_{2}\frac{16}{u^{2}}+\overline{A}_{2}\frac{u^{2}}{16})\frac{d^{2}Z}{du^{2}}+(B_{1}\frac{64}{u^{3}}+\overline{A}_{1}\frac{u}{4})\frac{dZ}{du}+\overline{A}_{0}Z=0,\nonumber
\end{equation}
\noindent where $B_{3}$, $B_{2}$, $B_{1}$ are as in \eqref{a15}-\eqref{a17}, and
\begin{equation}\label{a18a}
\overline{A}_{2}=-1,\quad \overline{A}_{1}=-(a_{1}+a_{2}+\frac{1}{4}),\quad \overline{A}_{0}=-a_{1}a_{2}.
\end{equation}
Matching the coefficient of $\frac{d^{2}Z}{du^{2}}$ gives
\begin{equation}
B_{2}\frac{16}{u^{2}}+\overline{A}_{2}\frac{u^{2}}{16}=0,\nonumber
\end{equation}
\noindent which again yields a contradiction.
Finally let us suppose that $p=1$. Then \eqref{a1} yields
\begin{equation}
\frac{d^{4}Z}{du^{4}}+B_{3}\frac{4}{u}\frac{d^{3}Z}{du^{3}}+B_{2}\frac{16}{u^{2}}\frac{d^{2}Z}{du^{2}}+\left(B_{1}\frac{64}{u^{3}}-\frac{u}{4}\right)\frac{dZ}{du}-{a}_{1}Z=0.\nonumber
\end{equation}
\noindent Matching the coefficients of $\frac{d^{i}Z}{du^{i}}$, $i=0, 1, 2, 3$ in the equation above and in \eqref{m34}, we get that $a_{1}=-\frac{1}{4}$ and that
\begin{eqnarray}
&&\label{a19}b_{1}+b_{2}+b_{3}-\frac{3}{2}=0,\\
&&\label{a20}b_{1}b_{2}+b_{1}b_{3}+b_{2}b_{3}-\frac{5}{4}(b_{1}+b_{2}+b_{3})+\frac{19}{16}=0,\\
&&\label{a21}\frac{9}{16}(b_{1}+b_{2}+b_{3}-\frac{3}{4})-\frac{3}{4}(b_{1}b_{2}+b_{1}b_{3}+b_{2}b_{3})+b_{1}b_{2}b_{3}=0.
\end{eqnarray}
\noindent Noting that equations \eqref{a19}-\eqref{a21} are invariant with respect to permutations of $\{b_{1}, b_{2}, b_{3}\}$, which reflects the fact that \eqref{a1} is similarly invariant, we find, modulo permutations, that
\begin{equation}
b_{1}=\frac{1}{4},\quad b_{2}=\frac{1}{2},\quad b_{3}=\frac{3}{4}.\nonumber
\end{equation}
\end{proof}
\section{Asymptotic Behavior of the Fundamental Solutions}\label{de}
In this appendix we give detailed proofs of the asymptotic properties \eqref{d24a}-\eqref{d25b} and \eqref{d26a}-\eqref{d26b} of the solutions $\{y_{i}(t,x)\}_{i=1}^{4}$; proof of the growth properties is given in Appendix \ref{asymp_grow} and proof of the decay properties is given in Appendix \ref{asymp_dec}. In Appendix \ref{asymp_dec}, the solution representations \eqref{msol}, \eqref{asol} are also derived.
\subsection{Asymptotically Growing Solutions}\label{asymp_grow}
\begin{lemma}\label{lem:h4} For $t> 0$,
\begin{equation}
\lim_{ x\rightarrow\infty} y_{3}(t,x)=\infty,\quad \lim_{ x\rightarrow\infty} y_{4}(t,x)=-\infty,\nonumber
\end{equation}
and for $x>0$,
\begin{equation}
\lim_{ t\rightarrow 0} y_{3}(t,x)=\infty,\quad \lim_{ t\rightarrow 0} y_{4}(t,x)=-\infty.\nonumber
\end{equation}
Similarly, for $t>0$,
\begin{eqnarray}
&&\lim_{ x\rightarrow-\infty} y_{1}(t,x)=\lim_{ x\rightarrow-\infty} y_{2}(t,x)=-\infty,\nonumber
\end{eqnarray}
and for $x>0$,
\begin{eqnarray}
&&\lim_{ t\rightarrow 0} y_{1}(t,-x)=\lim_{ t\rightarrow 0} y_{2}(t,-x)=-\infty.\nonumber
\end{eqnarray}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:h4}]
Since $z_{1}$, $z_{3}$ are even functions and $z_{2}$, $z_{4}$ are odd functions, it follows from the definitions of $y_{i}(t,x)$, $i=1,2,3,4,$ given in \eqref{d20}, that
\begin{eqnarray}
\label{j23}&&\quad\quad y_{3}(t,x)=(Bt)^{1/4}\left[\frac{z_{2}\big(\frac{x}{(Bt)^{1/4}}\big)}{\sqrt{2}}+\frac{z_{3}\big(\frac{x}{(Bt)^{1/4}}\big)}{2\Gamma\left(\frac{3}{4}\right)}+\frac{z_{4}\big(\frac{x}{(Bt)^{1/4}}\big)}{6\sqrt{2}\Gamma\left(\frac{1}{2}\right)}\right]\\
&&\quad\quad\quad=(Bt)^{1/4}\left[-\frac{z_{2}\big(-\frac{x}{(Bt)^{1/4}}\big)}{\sqrt{2}}+\frac{z_{3}\big(-\frac{x}{(Bt)^{1/4}}\big)}{2\Gamma\left(\frac{3}{4}\right)}-\frac{z_{4}\big(-\frac{x}{(Bt)^{1/4}}\big)}{6\sqrt{2}\Gamma\left(\frac{1}{2}\right)}\right]\nonumber\\
&&\quad\quad\quad=-y_{1}(t,-x),\quad t>0,\ x\in\mathbb{R}\setminus\{0\},\nonumber
\end{eqnarray}
and
\begin{multline}
\label{j24}y_{4}(t,x)=(Bt)^{1/4}\left[\frac{z_{1}\big(\frac{x}{(Bt)^{1/4}}\big)}{\Gamma\left(\frac{5}{4}\right)}+\frac{z_{2}\big(\frac{x}{(Bt)^{1/4}}\big)}{\sqrt{2}}-\frac{z_{4}\big(\frac{x}{(Bt)^{1/4}}\big)}{6\sqrt{2}\Gamma\left(\frac{1}{2}\right)}\right]\\
=(Bt)^{1/4}\left[\frac{z_{1}\big(-\frac{x}{(Bt)^{1/4}}\big)}{\Gamma\left(\frac{5}{4}\right)}-\frac{z_{2}\big(-\frac{x}{(Bt)^{1/4}}\big)}{\sqrt{2}}+\frac{z_{4}\big(-\frac{x}{(Bt)^{1/4}}\big)}{6\sqrt{2}\Gamma\left(\frac{1}{2}\right)}\right]\\
=y_{2}(t,-x),\quad t>0,\ x\in\mathbb{R}\setminus\{0\}.
\end{multline}
Hence it suffices to demonstrate the indicated asymptotic behavior for $y_{3}(t,x)$ and $y_{4}(t,x)$.
From \eqref{j23}, \eqref{d37}-\eqref{d40}, and the expansion (\ref{def:hf}), it follows that
\begin{eqnarray}
y_{3}(t,x)&=&(Bt)^{1/4}\left[\frac{1}{\sqrt{2}} z_{2}(u)+\frac{1}{2\Gamma\left(\frac{3}{4}\right)}z_{3}(u)+\frac{1}{6\sqrt{2}\Gamma\left(\frac{1}{2}\right)}z_{4}(u)\right]\nonumber\\
&=&(Bt)^{1/4}\left[\frac{1}{\sqrt{2}} u+\frac{1}{2\Gamma\left(\frac{3}{4}\right)}\sum_{k=0}^{\infty}\frac{\left(\frac{1}{4}\right)_{k}}{\left(\frac{3}{4}\right)_{k}\left(\frac{5}{4}\right)_{k}\left(\frac{3}{2}\right)_{k}k!256^{k}}u^{4k+2}\right.\nonumber\\
&&\left.+\frac{1}{6\sqrt{2}\Gamma\left(\frac{1}{2}\right)}\sum_{k=0}^{\infty}\frac{\left(\frac{1}{2}\right)_{k}}{\left(\frac{5}{4}\right)_{k}\left(\frac{3}{2}\right)_{k}\left(\frac{7}{4}\right)_{k}k!256^{k}}u^{4k+3}\right],\nonumber
\end{eqnarray}
\noindent where $u=\frac{x}{(Bt)^{1/4}}$. Note that for $t, x>0$, $y_{3}(t,x)$ is a sum of positive terms of the form
\begin{equation}
c_{n}(Bt)^{1/4}u^{n},n\in\{1,4k+2,4k+3\ |\ k\in\mathbb{Z}_{+}\},\quad 0<c_{n}\in\mathbb{R}.\nonumber
\end{equation}
Thus $y_{3}(t,x)\rightarrow\infty$ as $x\rightarrow\infty$ for fixed $t>0$, and $y_{3}(t,x)\rightarrow\infty$ as $t\rightarrow 0$ for fixed $x>0$.
Next, note that
\begin{eqnarray}
y_{4}(t,x)&=&(Bt)^{1/4}\left[\frac{1}{\Gamma\left(\frac{5}{4}\right)}z_{1}(u)+\frac{1}{\sqrt{2}}z_{2}(u)-\frac{1}{6\sqrt{2}\Gamma\left(\frac{1}{2}\right)}z_{4}(u)\right]\nonumber\\
&=&(Bt)^{1/4}\left[\frac{1}{\Gamma\left(\frac{5}{4}\right)}+\frac{1}{\sqrt{2}}u-\frac{1}{6\sqrt{2}\Gamma\left(\frac{1}{2}\right)}u^{3}\right.\nonumber
\end{eqnarray}
\begin{eqnarray}
&&-\frac{1}{4\Gamma\left(\frac{5}{4}\right)}\sum_{k=1}^{\infty}\frac{\left(\frac{3}{4}\right)_{k-1}}{\left(\frac{1}{4}\right)_{k}\left(\frac{1}{2}\right)_{k}\left(\frac{3}{4}\right)_{k}k!256^{k}}u^{4k}\nonumber\\
&&\left.-\frac{1}{6\sqrt{2}\Gamma\left(\frac{1}{2}\right)}\sum_{k=1}^{\infty}\frac{\left(\frac{1}{2}\right)_{k}}{\left(\frac{5}{4}\right)_{k}\left(\frac{3}{2}\right)_{k}\left(\frac{7}{4}\right)_{k}k!256^{k}}u^{4k+3}\right].\nonumber
\end{eqnarray}
It follows from the expression above that for $t, x>0$, except for the first two terms, $y_{4}(t,x)$ can be expressed as a sum of negative terms of the form
\begin{equation}
c_{n}(Bt)^{1/4}u^{n},n\in\{3, 4k, 4k+3\ |\ k\in\mathbb{Z}_{+}\setminus\{0\}\},\quad 0>c_{n}\in\mathbb{R}.\nonumber
\end{equation}
The sum of the first three terms is negative for $u>4$. Hence $y_{4}(t,x)\rightarrow -\infty$ as $x\rightarrow\infty$ for fixed $t>0$, and $y_{4}(t,x)\rightarrow-\infty$ as $t\rightarrow 0$ for fixed $x>0$.
\end{proof}
\subsection{Asymptotically Decaying Solutions}\label{asymp_dec}
First we obtain integral representations for two linearly independent solutions of \eqref{m22}-\eqref{lic}, which we denote by $\tilde{y}_{1}(t,x)$ and $\tilde{y}_{2}(t,x)$, by using Fourier cosine transform and symmetry considerations. The solution $\tilde{y}_{1}(t,x)$ is proportional to Martin's integral representation for Mullins' series solution \cite{Martin2009,Mullins1957}. We prove in detail that for any $t>0$, both of these solutions tend to zero as $x \rightarrow \infty$, and that for any $x>0$, both of these solutions tend to zero as $t\rightarrow 0$. Then, by considering the boundary conditions satisfied by these solutions at $x=0$, we demonstrate that $y_1(t,x)$, $y_2(t,x)$ can both be expressed as linear combinations of $\tilde{y}_{1}(t,x)$, $\tilde{y}_{2}(t,x)$. This yields a closed form representation for the series solution obtained in Amram et al., and justifies \eqref{msol}, \eqref{asol}. Moreover, it allows us to conclude that $y_1(t,x)$ and $y_2(t,x)$ both tend to zero as $x \rightarrow \infty$ for $t>0$, and as $t\rightarrow 0$ for $x>0$.
In parallel with Lemma \ref{lem:h4}, we may summarize the asymptotic results obtained in this section as follows.
\begin{lemma}\label{lem:h3} Each solution $\{y_{i}(t,x)\}_{i=1}^{4}$ defined in \eqref{d20} exhibit the following asymptotic decay,
\begin{equation}
\label{j26}\lim_{x\rightarrow\infty}y_{1}(t,x)=\lim_{x\rightarrow\infty}y_{2}(t,x)=0,\mbox{ for }t>0,
\end{equation}
and
\begin{equation}
\label{j26n}\lim_{t\rightarrow 0}y_{1}(t,x)=\lim_{t\rightarrow 0}y_{2}(t,x)=0,\mbox{ for }x>0.
\end{equation}
Similarly
\begin{equation}
\label{j27}\lim_{x\rightarrow-\infty}y_{3}(t,x)=\lim_{x\rightarrow-\infty}y_{4}(t,x)=0,\mbox{ for }t>0,
\end{equation}
and
\begin{equation}
\label{j27n}\lim_{t\rightarrow 0}y_{3}(t,x)=\lim_{t\rightarrow 0}y_{4}(t,x)=0,\mbox{ for }x<0.
\end{equation}
\end{lemma}
Recalling \eqref{j23}, \eqref{j24}, we note that it suffices to demonstrate the indicated asymptotic decay for $y_{1}(t,x)$ and $y_{2}(t,x)$.
Self-similar solutions of the form \eqref{lt4} to \eqref{m22}-\eqref{lic} can be found by utilizing the Fourier cosine transform under the assumption that
\begin{equation}\label{j2}
y_{xxx}(t,x), y_{xx}(t,x), y_{x}(t,x), y(t,x)\rightarrow0\mbox{ as }x\rightarrow\infty.
\end{equation}
The Fourier cosine transform was first used in this context by Martin \cite{Martin2009}, in conjunction with the condition
\begin{equation}\label{h0}
y'''(0,t)=0,\quad t>0.
\end{equation}
Here we proceed without imposing \eqref{h0}. The resultant solution can be expressed as a linear combination of two linearly independent solutions, denoted below as $\tilde{y}_{1}(t,x)$ and $\tilde{y}_{2}(t,x)$.
Taking the Fourier cosine transform of \eqref{m22}-\eqref{lic} with respect to the variable $x$, we get for solutions having the similarity form \eqref{lt4} that
\begin{equation}\label{h1}
\frac{\partial Y_{c}}{\partial t}-B(Bt)^{-1/2}Z'''(0)+Bk^{2}Z'(0)+Bk^{4}Y_{c}=0,\ Y_{c}(0,k)=0,
\end{equation}
where
\begin{equation}
Y_{c}(t,k)=\int_{0}^{\infty}y(t,x)\cos(kx)dx=(Bt)^{1/4}\int_{0}^{\infty}Z\left(\frac{x}{(Bt)^{1/4}}\right)\cos(kx)dx.\nonumber
\end{equation}
\noindent Solving \eqref{h1} as an initial value problem in $t$, we obtain
\begin{equation}\label{h2}
Y_{c}(t,k)=Z'(0)\frac{(e^{-Btk^{4}}-1)}{k^{2}}+Z'''(0)\frac{e^{-Btk^{4}}}{k^{2}}\int_{0}^{Btk^{4}}e^{s}s^{-1/2}ds.
\end{equation}
By taking the inverse Fourier cosine transform of \eqref{h2}, we get
\begin{equation}\label{ycos}
y_{c}(t,x)=Z'(0)\tilde{y}_{1}(t,x)+Z'''(0)\tilde{y}_{2}(t,x)
\end{equation}
\noindent where
\begin{equation}
\label{h3a}\tilde{y}_{1}(t,x)=-\frac{2(Bt)^{1/4}}{\pi}\int_{0}^{\infty}\frac{(1-e^{-w^{4}})}{w^{2}}\cos\left(\frac{x}{(Bt)^{1/4}}w\right)dw,
\end{equation}
\begin{multline}
\quad\quad\label{h3b}\tilde{y}_{2}(t,x)=\\
\frac{2(Bt)^{1/4}}{\pi}\int_{0}^{\infty}\frac{e^{-w^{4}}}{w^{2}}\left(\int_{0}^{w^{4}}e^{s}s^{-1/2}ds\right)\cos\left(\frac{x}{(Bt)^{1/4}}w\right)dw.
\end{multline}
We now prove that both of these solutions tend to zero as $x\rightarrow\infty$ for fixed $t>0$ as well as when $t\rightarrow 0$ for fixed $x>0$. For the proof of these properties for $\tilde{y}_{1}$ we make use of the following auxiliary proposition.
\begin{proposition}\label{prop:h1}The following hold
\begin{equation}\label{h13}
\lim_{w\rightarrow 0}\frac{1-e^{-w^{4}}}{w^{2}}=0,
\end{equation}
\begin{equation}\label{h14}
\lim_{w\rightarrow \infty}\frac{1-e^{-w^{4}}}{w^{2}}=0,
\end{equation}
\begin{equation}\label{h15}
\lim_{w\rightarrow 0}\left(\frac{1-e^{-w^{4}}}{w^{2}}\right)^{(k)}=\begin{cases}
\frac{(-1)^{n}(4n+2)!}{(n+1)!}, &\text{if } k=4n+2,\ n\in\mathbb{Z}_{+},\\[1ex]
0, &\text{otherwise.}
\end{cases}
\end{equation}
\begin{equation}\label{h16}
\lim_{w\rightarrow\infty}\frac{\left(\frac{1-e^{-w^{4}}}{w^{2}}\right)^{(k)}}{\frac{1}{w^{2}}}=0,\quad k\in\mathbb{Z}_{+}.
\end{equation}
\end{proposition}
\begin{proof}
The limits \eqref{h13}, \eqref{h14} follow from L'Hopital's rule.
We obtain \eqref{h15} by substituting the Maclaurin series for $e^{-w^{4}}$ into $\frac{1-e^{-w^{4}}}{w^{2}}$ and then calculating the derivatives at $w=0$.
The limit \eqref{h16} follows from the identity
\begin{equation}\label{h18}
\left(\frac{1-e^{-w^{4}}}{w^{2}}\right)^{(k)}=\frac{(-1)^{k}(k+1)!}{w^{k+2}}+e^{-w^{4}}\sum_{i=0}^{k}a_{i}(k)w^{-k-2+4i},
\end{equation}
where $a_{i}(k)$ are constants which depend only on $k$, which can be proved by mathematical induction on $k$. Substituting \eqref{h18} into the expression in \eqref{h16} and using the basic properties of the exponential function and power functions, we get
\begin{multline}
\lim_{w\rightarrow\infty}\frac{\left(\frac{1-e^{-w^{4}}}{w^{2}}\right)^{(k)}}{\frac{1}{w^{2}}}=\\
\lim_{w\rightarrow\infty}\left[
\frac{(-1)^{k}(k+1)!}{w^{k}}+e^{-w^{4}}\sum_{i=0}^{k}a_{i}(k)w^{-k+4(i-1)}\right]=0,\quad\nonumber
\end{multline}
for any $k\in\mathbb{Z}_{+}$.
\end{proof}
\begin{lemma}\label{lem:h1}
Let $\tilde{y}_{1}(t,x)$ be as defined in \eqref{h3a}. Then $\tilde{y}_{1}(t,x)$ tends to zero as $x\rightarrow\infty$ for fixed $t>0$ and as $t\rightarrow 0$ for fixed $x>0$.
\end{lemma}
\begin{proof}
We set $u=\frac{x}{(Bt)^{1/4}}$ as in Section \ref{sec:lt}, and integrate $\tilde{y}_{1}$ by parts $2k$ times
\begin{eqnarray}
&&-\frac{\pi}{2(Bt)^{1/4}}\tilde{y}_{1}(t,x)=\int_{0}^{\infty}\frac{1-e^{-w^{4}}}{w^{2}}\cos(uw)dw=\nonumber\\
&&=\frac{1}{u}\sin(uw)\left.\frac{1-e^{-w^{4}}}{w^{2}}\right\vert_{0}^{\infty}-\frac{1}{u}\int_{0}^{\infty}\sin(uw)\left(\frac{1-e^{-w^{4}}}{w^{2}}\right)'dw\nonumber\\
&&\quad(\mbox{by }\eqref{h13}\mbox{ and }\eqref{h14})\nonumber\\
&=&-\frac{1}{u}\int_{0}^{\infty}\sin(uw)\left(\frac{1-e^{-w^{4}}}{w^{2}}\right)'dw\nonumber
\end{eqnarray}
\begin{eqnarray}
&=&\ldots=\frac{(-1)^{k+1}}{u^{2k-1}}\sin(uw)\left.\left(\frac{1-e^{-w^{4}}}{w^{2}}\right)^{(2k-2)}\right\vert_{0}^{\infty}\nonumber\\
&&+(-1)^{k}\frac{1}{u^{2k-1}}\int_{0}^{\infty}\sin(uw)\left(\frac{1-e^{-w^{4}}}{w^{2}}\right)^{(2k-1)}dw\nonumber\\
&&\quad(\mbox{by }\eqref{h15}\mbox{ and }\eqref{h16})\nonumber
\end{eqnarray}
\begin{eqnarray}
&=&(-1)^{k}\frac{1}{u^{2k-1}}\int_{0}^{\infty}\sin(uw)\left(\frac{1-e^{-w^{4}}}{w^{2}}\right)^{(2k-1)}dw\nonumber\\
&=&\frac{(-1)^{k+1}}{u^{2k}}\cos(uw)\left.\left(\frac{1-e^{-w^{4}}}{w^{2}}\right)^{(2k-1)}\right\vert_{0}^{\infty}\nonumber\\
&&+(-1)^{k}\frac{1}{u^{2k}}\int_{0}^{\infty}\cos(uw)\left(\frac{1-e^{-w^{4}}}{w^{2}}\right)^{(2k)}dw\nonumber\\
&&(\mbox{by }\eqref{h15}\mbox{ and }\eqref{h16})\nonumber\\
\label{h4}&=&(-1)^{k}\frac{1}{u^{2k}}\int_{0}^{\infty}\cos(uw)\left(\frac{1-e^{-w^{4}}}{w^{2}}\right)^{(2k)}dw.
\end{eqnarray}
It follows from \eqref{h15}, \eqref{h16} that the integrand in \eqref{h4} is bounded in $L^{1}((0,\infty))$ uniformly with respect to $u$, and therefore \eqref{h4} decays to zero faster than $u^{-n}$ for every positive integer $n$ as $u\rightarrow\infty$. Hence for fixed $t>0$, $\tilde{y}_{1}(t,x)\rightarrow0$ as $x\rightarrow\infty$ and for fixed $x>0$, $\tilde{y}_{1}(t,x)\rightarrow0$ as $t\rightarrow 0$.
\end{proof}
\begin{lemma}\label{lem:h2}
Let $\tilde{y}_{2}(t,x)$ be as defined in \eqref{h3b}. Then $\tilde{y}_{2}(t,x)$ tends to zero as $x\rightarrow\infty$ for fixed $t>0$ as well as when $t\rightarrow 0$ for fixed $x>0$.
\end{lemma}
To prove Lemma \ref{lem:h2} we make use of the following proposition.
\begin{proposition}\label{prop:h2}Let $\erfi(z)$ denote the modified error function, \cite[Section 6.2.11]{Luke1969},
\begin{equation}
\erfi(z)=-i\erf(iz),\nonumber
\end{equation}
where
\begin{equation}
\erf(z)=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}dt,\quad z\in\mathbb{C}.\nonumber
\end{equation}
Then the following hold
\begin{equation}\label{h6}
\int_{0}^{w^{4}}e^{s}s^{-1/2}ds=\sqrt{\pi}\erfi(w^{2}),
\end{equation}
\begin{equation}\label{h7}
\lim_{w\rightarrow0}\left(\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}\right)=\frac{2}{\sqrt{\pi}},
\end{equation}
\begin{equation}\label{h9}
\lim_{w\rightarrow 0}\left(\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}\right)^{(k)}=\begin{cases}
\frac{2^{n+1}(-1)^{n}(4n)!}{\sqrt{\pi}(2n+1)!},&\text{if } k=4n,\ n\in\mathbb{N},\\[1ex]
0, &\text{otherwise,}
\end{cases}
\end{equation}
\begin{equation}\label{h8}
\lim_{w\rightarrow\infty}\left(\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}\right)=0,
\end{equation}
\begin{equation}\label{h11}
\lim_{w\rightarrow \infty}\frac{\left(\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}\right)^{(k)}}{\frac{1}{w^{2}}}=0,\quad k\in\mathbb{Z}_{+}.
\end{equation}
\end{proposition}
\begin{proof}
The identity \eqref{h6} is obtained by applying the change of variable $-t^{2}=s$ in the definition of the $\erfi$ function.
The limits in \eqref{h7} and \eqref{h9} result from substituting the following Taylor series expansion
\begin{equation}\label{erfi:small}
\erfi(z)=\frac{2}{\sqrt{\pi}}\sum_{k=0}^{\infty}\frac{z^{2k+1}}{k!(2k+1)},
\end{equation}
which is valid for all $z\in\mathbb{C}$, see \cite[7.6.1]{NIST}, into $\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}$.
The limits in \eqref{h8}, \eqref{h11} result from following technical claim.
\textbf{Claim:}
\begin{equation}\label{h19}
\lim_{w\rightarrow\infty}\frac{\left(\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}\right)^{(n)}}{\frac{1}{w^{4+n}}}=\frac{(-1)^{n}(4)_{n}}{\sqrt{\pi}},\quad n=0,1,2,3,\ldots,
\end{equation}
where $(4)_{n}=\frac{(3+n)!}{3!}$ in accordance with the definition of the Pochhammer symbol.
\begin{proof}[Proof of the Claim:]
Both $\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}$ and $\frac{1}{w^{4+n}}$ are smooth on $(0,\infty)$. First, we prove \eqref{h19} when $n=0$. Using L'Hopital's rule
\begin{equation}
\lim_{w\rightarrow\infty}\frac{\erfi(w^{2})}{\frac{e^{w^{4}}}{w^{2}}}=\lim_{w\rightarrow\infty}\frac{\frac{2}{\sqrt{\pi}}e^{w^{4}}2w}{\frac{4w^{3}e^{w^{4}}w^{2}-2we^{w^{4}}}{w^{4}}}=\frac{1}{\sqrt{\pi}}.\nonumber
\end{equation}
Hence
\begin{equation}\label{h20}
\lim_{w\rightarrow\infty}\frac{\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}}{\frac{1}{w^{4}}}=\frac{1}{\sqrt{\pi}}.
\end{equation}
We apply L'Hopital's rule to \eqref{h20} and obtain
\begin{equation}
\frac{1}{\sqrt{\pi}}=\lim_{w\rightarrow\infty}\frac{\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}}{\frac{1}{w^{4}}}=\lim_{w\rightarrow\infty}\frac{\left(\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}\right)'}{\left(\frac{1}{w^{4}}\right)'}=\lim_{w\rightarrow\infty}\frac{\left(\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}\right)'}{-\frac{4}{w^{5}}}\nonumber
\end{equation}
which gives us
\begin{equation}
\lim_{w\rightarrow\infty}\frac{\left(\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}\right)'}{\frac{1}{w^{5}}}=-\frac{4}{\sqrt{\pi}}\nonumber
\end{equation}
namely \eqref{h19} when $n=1$.
We obtain \eqref{h19} by applying L'Hopital's rule to \eqref{h20} $n$ times.
\end{proof}
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:h2}]
Setting $u=\frac{x}{(Bt)^{1/4}}$ and integrating $\frac{\sqrt{\pi}}{2(Bt)^{1/4}}\tilde{y}_{2}(t,x)$ by parts $2k+1$ times, we get for $x>0$, $t>0$,
\begin{eqnarray}
&&\frac{\sqrt{\pi}}{2(Bt)^{1/4}}\tilde{y}_{2}(t,x)=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-w^{4}}}{w^{2}}\left(\int_{0}^{w^{4}}e^{s}s^{-1/2}ds\right)\cos(uw)dw,\nonumber\\
&&=\int_{0}^{\infty}\frac{e^{-w^{4}}}{w^{2}}\erfi(w^{2})\cos(uw)dw\quad(\mbox{by }\eqref{h6}),\nonumber
\end{eqnarray}
\begin{eqnarray}
&&=\left.\frac{\sin(uw)}{u}\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}\right\vert_{0}^{\infty}-\int_{0}^{\infty}\frac{\sin(uw)}{u}\left(\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}\right)'dw\nonumber\\
&&(\mbox{by }\eqref{h7}\mbox{ and }\eqref{h8}),\nonumber\\
&&=-\int_{0}^{\infty}\frac{\sin(uw)}{u}\left(\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}\right)'dw,\nonumber\\
&&=\ldots=\frac{(-1)^{k+1}}{u^{2k}}\cos(uw)\left.\left(\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}\right)^{(2k-1)}\right\vert_{0}^{\infty}\nonumber\\
&&+\frac{(-1)^{k}}{u^{2k}}\int_{0}^{\infty}\cos(uw)\left(\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}\right)^{(2k)}dw\nonumber\\
&&(\mbox{by }\eqref{h9}\mbox{ and }\eqref{h11}),\nonumber\\
&&=\frac{(-1)^{k}}{u^{2k}}\int_{0}^{\infty}\cos(uw)\left(\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}\right)^{(2k)}dw,\nonumber\\
&&=\frac{(-1)^{k+2}}{u^{2k+1}}\sin(uw)\left.\left(\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}\right)^{(2k)}\right\vert_{0}^{\infty}\nonumber\\
&&+\frac{(-1)^{k+1}}{u^{2k+1}}\int_{0}^{\infty}\sin(uw)\left(\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}\right)^{(2k+1)}dw\nonumber\\
&&(\mbox{by }\eqref{h9}\mbox{ and }\eqref{h11}),\nonumber
\end{eqnarray}
\begin{eqnarray}
\label{h10}&&=\frac{(-1)^{k+1}}{u^{2k+1}}\int_{0}^{\infty}\sin(uw)\left(\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}\right)^{(2k+1)}dw.
\end{eqnarray}
Clearly, the function $\frac{e^{-w^{4}}\erfi(w^{2})}{w^{2}}$ is smooth on $(0, \infty)$, since it is the quotient of smooth functions and the denominator does not vanish. This together with \eqref{h9} and \eqref{h11} imply that the integrand in \eqref{h10} is bounded in $L^{1}((0,\infty))$ uniformly with respect to $u$. Hence \eqref{h10} tends to zero as $u\rightarrow\infty$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:h3}]
As noted earlier, it suffices to prove \eqref{j26} and \eqref{j26n}, namely
\begin{equation}
\lim_{x\rightarrow\infty}y_{1}(t,x)=\lim_{x\rightarrow\infty}y_{2}(t,x)=0,\quad \mbox{ for fixed }\ t>0,\nonumber
\end{equation}
and that
\begin{equation}
\lim_{t\rightarrow 0}y_{1}(t,x)=\lim_{t\rightarrow 0}y_{2}(t,x)=0,\quad \mbox{ for fixed }\ x>0.\nonumber
\end{equation}
In \cite{Martin2009} it is proved that $\tilde{y}_{1}(t,x)$, which is defined in \eqref{h3a}, may be expressed as
\begin{equation}\label{h12}
(Bt)^{1/4}\sum_{n=0}^{\infty}a_{n}u^{n},\quad u=\frac{x}{(Bt)^{1/4}},
\end{equation}
where
\begin{equation}
a_{0}=-\frac{2}{\pi}\Gamma\left(\frac{3}{4}\right),\quad a_{1}=1,\quad a_{2}=-\frac{1}{4\pi}\Gamma\left(\frac{1}{4}\right),\quad a_{3}=0.\nonumber
\end{equation}
Since $\frac{\tilde{y}_{1}}{(Bt)^{1/4}}$ is a solution of the fourth order linear ordinary differential equation \eqref{m34}, the recursion relation
\begin{equation}\label{h103}
a_{n+4}=\frac{n-1}{4(n+1)(n+2)(n+3)(n+4)}a_{n},
\end{equation}
then determines the coefficients $a_{n}$ for $n\geq 4$.
Note that $y_{1}$ and $y_{2}$, which are defined in \eqref{d20}, are also of the form \eqref{h12} and have coefficients that satisfy \eqref{h103}. Hence, we find
\begin{equation}\label{msol2}
\tilde{y}_{1}(t,x)=\frac{1}{\sqrt{2}}y_{1}(t,x)-\frac{1}{\sqrt{2}}y_{2}(t,x)
\end{equation}
by comparing the first four coefficients. Since $\tilde{y}_{1}(t,x)$ corresponds to Mullins' solution \cite{Mullins1957}, \eqref{msol2} implies \eqref{msol}. Similarly \eqref{asol} can be demonstrated directly by comparing the first four coefficients in the solution given in \cite{Amram2014} and in $-\frac{m}{\sqrt{2}}y_{2}(t,x)$, as implied by \eqref{d20}.
Next, we want to express $\tilde{y}_{2}(t,x)$ as
\begin{equation}\label{j19}
\tilde{y}_{2}(t,x)=\sum_{i=1}^{4}\alpha_{i}y_{i}(t,x),
\end{equation}
for $\alpha_{i}\in\mathbb{R}$, $i=1,2,3,4$. Let us recall that we obtained $\tilde{y}_{2}(t,x)$ by solving \eqref{m22}-\eqref{lic}, under the assumption that $\tilde{y}_{2}(t,x)$ is of the form \eqref{lt4} with
\begin{equation}
\label{ic1}Z'(0)=0,\quad Z'''(0)=1.
\end{equation}
The conditions in \eqref{ic1} imply the following equalities
\begin{equation}\label{j5}
\frac{\alpha_{1}-\alpha_{2}+\alpha_{3}+\alpha_{4}}{\sqrt{2}}=Z'(0)=0,
\end{equation}
\begin{equation}\label{j30}
\frac{\alpha_{1}}{\sqrt{2}\Gamma\left(\frac{1}{2}\right)}+\frac{\alpha_{2}}{\sqrt{2}\Gamma\left(\frac{1}{2}\right)}+\frac{\alpha_{3}}{\sqrt{2}\Gamma\left(\frac{1}{2}\right)}-\frac{\alpha_{4}}{\sqrt{2}\Gamma\left(\frac{1}{2}\right)}=Z'''(0)=1.
\end{equation}
We can calculate $\tilde{y}_{2}(t,0)$ and $\frac{\partial^{2}\tilde{y}_{2}}{\partial x^{2}}(t,0)$ directly from \eqref{h3b}. Using \eqref{h6}, \eqref{erfi:small}, as well as the series expansions for $e^{-w^{4}}$ and $\prescript{}{2}{F}_{2}^{}$, we obtain that
\begin{eqnarray}
\tilde{y}_{2}(t,0)&=&\frac{2(Bt)^{1/4}}{\sqrt{\pi}}\int_{0}^{\infty}e^{-w^{4}}w^{-2}\erfi(w^{2})dw\nonumber\\
&=&\frac{4(Bt)^{1/4}w}{\pi}\prescript{}{2}{F}_{2}^{}(\frac{1}{4},1;\frac{5}{4},\frac{3}{2};-w^{4})\bigg|_{0}^{\infty}\nonumber\\
\label{nh1}&=&\frac{2}{\sqrt{\pi}}(Bt)^{1/4}\Gamma\left(\frac{3}{4}\right),
\end{eqnarray}
and
\begin{eqnarray}
\frac{\partial^{2}\tilde{y}_{2}}{\partial x^{2}}(t,0)&=&-\frac{2(Bt)^{-1/4}}{\sqrt{\pi}}\int_{0}^{\infty}e^{-w^{4}}\erfi(w^{2})dw\nonumber\\
&=&-\frac{4(Bt)^{-1/4}w^{3}}{3\pi}\prescript{}{2}{F}_{2}^{}(\frac{3}{4},1;\frac{3}{2},\frac{7}{4};-w^{4})\bigg|_{0}^{\infty}\nonumber\\
\label{nh3}&=&-\frac{2}{\sqrt{\pi}}(Bt)^{-1/4}\Gamma\left(\frac{5}{4}\right),
\end{eqnarray}
where the asymptotic evaluations of $\prescript{}{2}{F}_{2}^{}$ can be found in \cite[16.11(ii)]{NIST}. Both \eqref{nh1} and \eqref{nh3} can be verified by Mathematica. Combining \eqref{nh1} and \eqref{nh3} with \eqref{j19}, we get
\begin{equation}\label{j6}
\frac{\alpha_{2}(Bt)^{1/4}}{\Gamma\left(\frac{5}{4}\right)}+\frac{\alpha_{4}(Bt)^{1/4}}{\Gamma\left(\frac{5}{4}\right)}=\tilde{y}_{2}(t,0)=\frac{2}{\sqrt{\pi}}(Bt)^{1/4}\Gamma\left(\frac{3}{4}\right),
\end{equation}
\begin{equation}\label{j31}
-\frac{\alpha_{1}(Bt)^{-1/4}}{\Gamma\left(\frac{3}{4}\right)}+\frac{\alpha_{3}(Bt)^{-1/4}}{\Gamma\left(\frac{3}{4}\right)}=\frac{\partial^{2}\tilde{y}_{2}}{\partial x^{2}}(t,0)=-\frac{2}{\sqrt{\pi}}(Bt)^{-1/4}\Gamma\left(\frac{5}{4}\right).
\end{equation}
Solving \eqref{j5}, \eqref{j30}, \eqref{j6} and \eqref{j31}, we get
\begin{equation}
\alpha_{1}=\alpha_{2}=\frac{\sqrt{\pi}}{\sqrt{2}},\quad \alpha_{3}=\alpha_{4}=0.\nonumber
\end{equation}
Thus for $x>0$
\begin{equation}\label{fcsol2}
\tilde{y}_{2}(t,x)=\sqrt{\pi}\left(\frac{1}{\sqrt{2}}y_{1}(t,x)+\frac{1}{\sqrt{2}}y_{2}(t,x)\right).
\end{equation}
From \eqref{msol2}, \eqref{fcsol2}, it follows that
\begin{eqnarray}
&&y_{1}(t,x)=\frac{1}{\sqrt{2}}\tilde{y}_{1}(t,x)+\frac{1}{\sqrt{2\pi}}\tilde{y}_{2}(t,x),\nonumber\\
&&y_{2}(t,x)=-\frac{1}{\sqrt{2}}\tilde{y}_{1}(t,x)+\frac{1}{\sqrt{2\pi}}\tilde{y}_{2}(t,x),\nonumber
\end{eqnarray}
and the initial and far field properties of $y_{1}(t,x)$, $y_{2}(t,x)$ are implied by the results in Lemma \ref{lem:h1} and \ref{lem:h2}.
\end{proof}
\section*{Acknowledgments} The authors would like to thank Prof.~Eugen Rabkin for fruitful discussions regarding the experimental data, \cite{Amram2014}, that he generously shared. The authors would also like to thank Dr.~Orestis Vantzos for developing the code for the data fitting.
The authors would like to acknowledge support from the Israel Science Foundation (Grant $\#1200/16$).
\end{document} | math |
के चंद्रशेखर राव क्यों कर रहे हैं पीएम मोदी की इतनी आलोचना, इन 5 वजहों से समझिये हाल के दिनों में प्रधानमंत्री नरेंद्र मोदी और केंद्र सरकार के बड़े आलोचक बन के उभरे तेलंगाना के मुख्यमंत्री के चंद्रशेखर राव केसीआर 2024 के चुनाव को लेकर अभी से मोर्चाबंदी में जुट गए हैं। रविवार को एक दिवसीय दौरे पर मुंबई पहुंचे केसीआर ने महाराष्ट्र के मुख्यमंत्री और शिवसेना प्रमुख उद्धव ठाकरे और एनसीपी के मुखिया शरद पवार से मुलाकात की। मुलाकात के बाद दोनों नेताओं ने एक ज्वाइंट प्रेस कॉन्फ्रेंस की। केसीआर ने कहा आज देश की राजनीति और विकास का आजादी के 75 साल बाद जो हाल है। उस पर चर्चा करने के लिए मैं महाराष्ट्र आया हूं। लेकिन राष्ट्रीय राजनीति में केसीआर की अचानक सक्रियता की वजह क्या है? केसीआर भारतीय प्रधानमंत्री की आलोचना करते हुए राज्य सरकार के अधिकारों में दखल, जीएसटी, प्रशासनिक सेवाओं में नियुक्ति जैसे मुद्दे उठा रहे हैं। लेकिन क्या केसीआर प्रधानमंत्री मोदी की आलोचना बस इन्हीं वजहों से कर रहे हैं या प्रधानमंत्री मोदी की आलोचना के पीछे वास्तविक कारण कुछ और है। प्रधानमंत्री मोदी की आलोचना के पीछे वास्तविक कारणों का अंदाजा इन बातों से लगाया जा सकता है।तीसरी पारीराजनीति में कुछ भी संभव है। केसीआर ने कभी नहीं सोचा होगा कि उन्हें तीसरी पारी खेलने का मौका मिल सकता है। हाल फिलहाल के प्रेस बैठकों में वह कहते हैं जब मेरा जन्म हुआ होगा तब क्या मेरे पिता ने सोचा होगा कि 1 दिन में मुख्यमंत्री बनूंगा। राजनीति में कुछ भी संभव है। केसीआर ने अपनी सियासत तेलुगू देशम पार्टी के साथ शुरू की थी। वह मिडल लेवल के नेता थे और कुछ वक्त तक उन्होंने इस स्थिति में काम किया। यह उनके राजनीतिक जीवन की पहली पारी थी। केसीआर स्वर्गीय एनटी रामा राव के बड़े प्रशंसक हैं। उन्होंने अपने बेटे का नाम उन पर ही रखा है।सियासी जीवन की दूसरी पारी में उन्होंने अपनी पार्टी का गठन किया और बीते दो दशकों के दौरान तमाम बाधाओं को पार करते हुए अपनी पार्टी स्थापित की। तेलंगाना राज्य के गठन के साथ वो न केवल मुख्यमंत्री बने बल्कि निर्विवादित रूप से राज्य के सबसे बड़े नेता बन गए। मौजूदा समय में उन्होंने अपने लिए बड़ा लक्ष्य रखा है। उनका ध्यान दिल्ली की ओर है।प्रधानमंत्री मोदी को बनाया नया दुश्मनराजनीति में कहा जाता है कि कोई दुश्मन नहीं होता, केवल विरोधी होते हैं। लेकिन केसीआर की सियासत में हमेशा एक दुश्मन रहा है। बीजेपी की सियासत की तरह ही तेलंगाना राष्ट्र समिति की सियासत में ध्रुवीकरण एक पहलू रहा है। आंध्र प्रदेश के विभाजन के समय संसदीय प्रक्रिया का सही से पालन नहीं हुआ और तेलंगाना के गठन का प्रस्ताव संसद में बंद दरवाजे से अंधेरे में लिया गया जैसे मोदी का बयान इस्तेमाल करते हुए केसीआर उन्हें राज्य के दुश्मन के तौर पर पेश करते हैं। वो हर चुनाव में मतदाताओं को भावनात्मक रूप से जोड़ने के लिए किसी ना किसी को दुश्मन बनाते रहे हैं। 2009 में वाईएस राजशेखर रेड्डी की अगुवाई वाली कांग्रेस दुश्मन थी। 2014 में आंध्र प्रदेश के नेता दुश्मन थे। 2019 में चंद्रबाबू नायडू दुश्मन थे। आज की तारीख में मोदी उनके नए दुश्मन हैं।बीजेपी का नया लक्ष्य और बदलते समीकरणहाल फिलहाल तक बीजेपी के टीआरएस के सथ उनके रिश्ते मधुर थे। बीजेपी संसद में जो प्रस्ताव रखती टीआरएस संसद में उसका समर्थन करती थी। माना जाता है जब तक तेलंगाना बीजेपी की कमान किशन रेड्डी के हाथों में थी, तब तक दोनों पार्टी एक सीमा तक ही एक दूसरे की आलोचना करती थीं।लेकिन अब हालात बदल गए हैं। बीजेपी के लिए उत्तर भारत में अपनी ताकत बढ़ाने के लिए अब ज्यादा जगह नहीं है। ऐसे में बीजेपी उन राज्यों की ओर देख रही है जहां वह अपनी ताकत बढ़ा सकती है। भगवा पार्टी दक्षिण भारत में भी अपना विस्तार चाहती है और तेलंगाना उसी रणनीति का हिस्सा है। भाजपा ने तेलंगाना को पश्चिम बंगाल के बाद अपना नया लक्ष्य बनाया है।भाजपा का युवा नेतृत्व तेलंगाना में केसीआर पर रोज हमले कर रहा है। दरअसल हैदराबाद नगर निगम में उम्मीद से बेहतर प्रदर्शन और 2 विधानसभा सीटों के उपचुनाव में जीत ने बीजेपी कार्यकर्ताओं का उत्साह बढ़ा दिया है। जातिगत समीकरण भी कुछ बदले हुए नजर आते हैं। अतीत में राज्य बीजेपी की कमान वेलम्मा और रेड्डी जाति के नेताओं के पास थी। अब टीआरएस की कमान भी वेलम्मा नेतृत्व के हाथों में है, ऐसे में बीजेपी पिछड़ी जातियों पर भरोसा कर रही है। यहां बीजेपी की मजबूती की सबसे बड़ी वजह है कि पिछड़ी जातियां उस पर भरोसा कर रही हैं। और यह टीआरएस के लिए खतरा हो सकता है। केसीआर जिस तरह से मोदी की लगातार आलोचना कर रहे हैं उससे यह भी संकेत मिलता है कि टीआरएस के लिए मुख्य विपक्षी दल तेलंगाना में अब बीजेपी है। ऐसे में आने वाले दिनों में इन दोनों राजनीतिक दलों में टकराहट और बढ़ सकती है। रणनीति एक निशाने दो केसीआर की चुनावी राजनीति में मोदी को चुनौती देने का फैसला बेहद महत्वपूर्ण है। वह एक रणनीति से दो निशाने लगाना चाहते हैं। जब कांग्रेस नेता राहुल गांधी ने केंद्र सरकार से सर्जिकल स्ट्राइक का सबूत मांगा तो असम के मुख्यमंत्री हिमंत बिस्वा सरमा ने राहुल गांधी पर आपत्तिजनक भाषा का इस्तेमाल करते हुए हमला बोला। केसीआर ने असम के मुख्यमंत्री के बयान को आधार बनाकर मोदी पर निशाना साधते हुए कहा, ऐसी अपमानजनक भाषा से मेरी आंखों में आंसू आ गए। आप ऐसी भाषा का इस्तेमाल करते हैं? यह आप की संस्कृति है? क्या आप उन्हें पार्टी से हटाएंगे?केसीआर की यही रणनीति मानी जाती है कि वह अपना दुश्मन जिसे घोषित कर देते हैं, उस पर हमला शुरू कर देते हैं। दूसरी ओर राज्य में कांग्रेस विपक्षी पार्टी है लेकिन उसके प्रति सहानुभूति दिखाकर उन्होंने यह संकेत दिया कि राज्य में उनको कांग्रेस से कोई खतरा नहीं है। राजनीतिक विश्लेषकों के मुताबिक कांग्रेस पार्टी के बिना किसी तीसरे मोर्चे का आगे बढ़ पाना संभव नहीं होगा। केसीआर ने इस पहल के साथ आने वाले दिनों में जरूरत पड़ने पर कांग्रेस के साथ गठबंधन की संभावनाओं का रास्ता भी खोल दिया है। कांग्रेस के साथ गठबंधन टीआरएस के लिए नई बात नहीं है। दोनों पार्टियां अतीत में भी एक साथ आ चुकी हैं अतीत मेरे देखें तो केसीआर ने यहां तक घोषणा की थी कि अगर तेलंगाना को राज्य का दर्जा मिल जाता है तो वो टीआरएस इस का विलय है कांग्रेस में कर देंगे।लक्ष्य नहीं बदला गति बदली हैइन दिनों राष्ट्रीय मीडिया में किसी और केसीआर को लेकर जो खबरें और विश्लेषण लिखे जा रहे हैं आने वाले दिनों में वह और बढ़ सकते हैं। केसी र का केंद्र में दिखना भले नया लग रहा हो लेकिन उनकी महत्वकांक्षी पुरानी है। तेलंगाना में दूसरी बार सरकार बनने के बाद 2019 के आम चुनावों से ठीक पहले उन्होंने एक मोर्चा बनाने की कोशिश की थी। तब उन्होंने मौका लपकने की कोशिश की थी। वो ममता बनर्जी से मिलने बंगाल भी गए। उन्होंने एम के स्टालिन, नवीन पटनायक और पीनारायी विजयन से भी बातचीत की। केसीआर देवेगौड़ा से भी मिले।बीजेपी 2019 में बहुमत के साथ सरकार बनाने में सफल रही। अब दो साल बाद लोकसभा चुनाव होने हैं लेकिन केसीआर एक बार फिर अपनी कोशिशें शुरू कर चुके हैं। इसी कड़ी में वो तीसरा मोर्चा बनाने के लिए क्षेत्रीय दलों के नेताओं से मिल रहे हैं। उन्होंने चुनावी जंग का बिगुल फूंक दिया है और संकेत दे रहे हैं कि मोदी को चुनौती देने वाले संभावित नेताओं में सबसे आगे हैं।मौजूदा समय में तेलुगु भाषी क्षेत्र में केसीआर जैसी वाकपटुता किसी दूसरे नेता में नहीं है। धाराप्रवाह हिंदी बोलना उनकी खासियत है। केसीआर को राजनीतिक तौर पर ऐसा लग रहा है कि भारतीय राजनीति अभी उस दौर में है जहां आने वाले दिनों में कुछ भी संभव है। उन्होंने यह सुनिश्चित कर लिया है कि लोग उनकी आंख मक्का पर ध्यान दे रहे हैं और वह चर्चा में बने हुए हैं। | hindi |
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4 हजार पदों पर निकली भर्ती, 14 फरवरी तक कर सकते है अप्लाई NHM UP CHO Recruitment 2022: उत्तर प्रदेश में मेडिकल फील्ड में नौकरी पाने का इंतजार कर रहे उम्मीदवारों के लिए आवेदन करने का शानदार मौका है. यूपी नेशनल हेल्थ मिशन UP NHM की ओर से कम्युनिटी हेल्थ ऑफिसर CHO के पदों पर भर्ती के लिए नोटिफिकेशन जारी किया गया है. इस वैकेंसी UP NHM CHO Recruitment 2022 के तहत कुल 4000 पदों पर भर्तियां की जाएंगी. NHM यूपी भर्ती के लिए आवेदन का लिंक एक्टिव है, इच्छुक एवं योग्य उम्मीदवार UP NHM की ऑफिशियल वेबसाइट upnrhm.gov.in पर जाकर ऑनलाइन आवेदन कर सकते हैं.UP NHM CHO Recruitment 2022 के तहत आवेदन करने के लिए उम्मीदवारों के पास किसी भी मान्यता प्राप्त यूनिवर्सिटी या संस्थान से बीएससी नर्सिंग या पोस्ट बेसिक बीएससी नर्सिंग की डिग्री होनी चाहिए. वहीं, अधिकतर आयु सीमा 35 वर्ष निर्धारित है. बता दें कि यह भर्तियां कांट्रेक्ट बेसिस पर होंगी और उम्मीदवारों का चयन मेरिट लिस्ट के आधार पर होगा.कम्युनिटी हेल्थ ऑफिसर CHO के पदों पर चयनित उम्मीदवारों को 20,500 रुपये प्रति माह वेतन मिलेगा. बता दें कि कम्युनिटी हेल्थ ऑफिसर के इन पदों पर आवेदन की प्रक्रिया 04 फरवरी 2022 से शुरू हो गई है. आधिकारिक नोटिफिकेशन के अनुसार, आवेदन करने की अंतिम तिथि 13 फरवरी है. | hindi |
In the afternoon we decided to split up. Brian, Mia, and I decided to take a hike up to the top of the foot hill that protected Cusco. At the top of the hill was a statue of Jesus Christ (we assumed) and several crosses.
Climbing a few hundred feet gave us a good view of the rooftops of Cusco.
After taking a turn, the stairs were more ominous. This was the view back down after hiking about half way up.
We were rewarded with a resting/vista point that was spectacular.
There were also two children with their llama and alpaca. For a small donation we took the obligatory tourist picture.
We continued up the hill. About 1/2 mile from the top we reached the entrance to Sacsayhuaman, a natural park that included the peak of the mountain. We decided not to pay the somewhat pricey $70 solas admission fee and instead turned around.
When we got back to the edge of the city we took a different route, but it still had plenty of stairs. | english |
పూరీ జగన్నాథ్ భార్య ప్రవర్తన పై ప్రభాస్ షాకింగ్ కామెంట్స్..? టాలీవుడ్ ఫిల్మ్ ఇండస్ట్రీలో యంగ్ రెబల్ స్టార్ ప్రభాస్, డాషింగ్ డైరెక్టర్ పూరి జగన్నాథ్ ల కాంబినేషన్ కి ఆడియన్స్ లో ఎలాంటి క్రేజ్ ఉంటుందో ప్రత్యేకంగా చెప్పాల్సిన అవసరం లేదు. వీరిద్దరి కాంబినేషన్లో ఇప్పటికే బుజ్జిగాడు, ఏక్ నిరంజన్ వంటి సినిమాలు వచ్చాయి. ఇక వీటిలో బుజ్జిగాడు సినిమా యావరేజ్ ఫలితాన్ని అందుకోగా.. ఏక్ నిరంజన్ ప్లాప్ గా నిలిచింది. అయినా కూడా పూరి జగన్నాథ్ అంటే ప్రభాస్ కి ఎంతో ఇష్టం. ఇక ఇదిలా ఉంటే పూరి జగన్నాథ్ భార్య అయిన లావణ్య గురించి ప్రేక్షకులకు పెద్దగా తెలియదు. అయితే తాజాగా ఒక ఇంటర్వ్యూలో ప్రభాస్.. పూరి జగన్నాథ్ భార్య గురించి పలు ఆసక్తికర విషయాలను వెల్లడించారు. ఇక పూరి జగన్నాథ్ కొడుకు ఆకాష్ పూరి తాజాగా నటించిన రొమాంటిక్ సినిమా ప్రమోషన్స్ లో భాగంగా ప్రభాస్ కూడా ఇంటర్వ్యూ లో పాల్గొన్నాడు. అంతేకాదు రొమాంటిక్ సినిమా ప్రమోషన్స్ లో తాను పాల్గొనడానికి గల కారణం కూడా వివరించాడు.ఈ క్రమంలోనే బుజ్జిగాడు సినిమా షూటింగ్ సమయంలో జరిగిన ఓ సంఘటనను ఈ సందర్భంగా చెప్పుకొచ్చాడు ప్రభాస్. ఇక బుజ్జిగాడు షూటింగ్ సమయంలో పూరి జగన్నాథ్ భోజనం చేస్తుండగా మరో మహిళ కూడా అక్కడ భోజనం చేస్తున్నా రని.. అయితే తాను ఆమె ఎవరిని కనుక్కో గా పూరి జగన్నాథ్ ఇంట్లో పని చేసే పని మనిషి అనే సమాధానం తనకు వినిపించిందని తెలిపాడు. తమతో పాటు తమ పని మనిషిని కూడా భోజనానికి పక్కనే కూర్చోబెట్టుకునే మంచి గుణం లావణ్య గారిదని ప్రభాస్ వెల్లడించాడు. ఇక ఈ కారణం వల్లే తాను రొమాంటిక్ సినిమాకి సహాయం చేసానని పేర్కొన్నాడు డార్లింగ్. అంతే కాదు అమ్మను జాగ్రత్తగా చూసుకోమని ఆకాష్ పూరికి సలహా కూడా ఇచ్చాడు. ఇక ఈ ఇంటర్వ్యూలో ఆకాష్ పూరి, కేతికశర్మ లను ప్రభాస్ తన సరదా మాటలతో ఓ ఆట ఆడుకున్నాడు. అందుకు సంబంధించిన వీడియో కూడా ఇప్పుడు సోషల్ మీడియాలో వైరల్ గా మారుతుంది.ఇక ఆకాష్ పూరి, కేతికశర్మ జంటగా నటించిన రొమాంటిక్ సినిమా అక్టోబర్ 29న థియేటర్స్ లో విడుదల కానుంది...!!గంజాయిపై పవన్ ట్వీట్స్.. ఆసక్తికర విషయాలు వెల్లడి ఆ విషయంలో తండ్రిని మించిన ఆకాష్ పూరీ..!! ధీమాతో ఉన్న అంబటి...? వివేకా హత్య కేసులో సీబీఐ ఛార్జీషీట్ దాఖలు ఇన్ని విమర్శలు వచ్చినా సమంత మారలేదుగా? బిగ్ బాస్ షణ్ముఖ్ కు సినిమా ఛాన్స్... ఏంట్రా ఇది షన్నూ ? బిగ్ బాస్ 5 : ఈ వారం ఎలిమినేట్ అయ్యేది అతనే..? మోహన్ బాబు ట్రైన్ లో టీసీని చూసి బాత్రూం లో దాక్కున్నాడట? ఆఫ్ఘన్ లోనే.. అమెరికావాళ్లు.. ! సోర్స్: ఇండియాహెరాల్డ్.కామ్ Anilkumar | telegu |
Corona Update Jammu Kashmir : प्रदेश में 151 नए संक्रमित, रियासी समेत चार जिलों में कोई केस नहीं विस्तार प्रदेश में पिछले 24 घंटों में कोरोना संक्रमण के 151 मामलों की पुष्टि हुई है। चार जिलों में कोई नया मामला नहीं आया है। वहीं किसी मरीज की मौत भी नहीं हुई है। 496 मरीज स्वस्थ हुए हैं और सक्रिय मामले कम होकर 1949 ही रह गए है।यहां कोविड से संबधित सभी नए अपडेट पढ़ें रियासी, पुंछ, शोपियां और गांदरबल जिलों में कोई नया मामला पिछले चौबीस घंटों के दौरान नहीं आया है। अन्य जिलो में जम्मू में 43, उधमपुर में सात, राजोरी में दो, डोडा में 25, कठुआ में चार, किश्तवाड़ में चार और रामबन में पांच नए मामलों की पुष्टि हुई है। श्रीनगर जिले में 29, बारामुला में पांच, बडगाम में चार, पुलवामा में सात, कुपवाड़ा में आठ, अनंतनाग में तीन, बांदीपोरा में दो व कुलगाम में तीन नए मामले आए हैं। ऐसे में कुल मिलाकर प्रदेश में सक्रिय मामलों की संख्या 1949 रह गई है। जम्मू संभाग में 1030 व कश्मीर संभाग में 919 सक्रिय मरीज ही रह गए हैं। | hindi |
குற்றால அருவியில் குளிக்க 9 மாதங்களுக்கு பிறகு அனுமதி குற்றாலம் அருவியில் குளிக்க இன்று முதல் அனுமதி அளிக்கப்பட்டுள்ளது. இதற்காக தென்காசி சட்டமன்ற உறுப்பினர் பழனி தலைமையில் பட்டாசுகள் வெடித்து கொண்டாடப்பட்டது. 9 மாதங்களுக்கு பிறகு குற்றால அருவி திறக்கப்பட்டதால், சுற்றுலா பயணிகள், வியாபாரிகள் மிகுந்த மகிழ்ச்சி அடைந்துள்ளனர். அங்கே சுற்றுலா பயணிகள் கூட்டம் அலைமோதுகிறது.தென்காசி மாவட்டம் குற்றாலத்தில் கொரோனா தடுப்பு நடவடிக்கையாக கடந்த 2 ஆண்டுகளாக குற்றால அருவிகளில் குளிப்பதற்கு சுற்றுலா பயணிகளுக்கு தடை விதிக்கப்பட்டது. ஊரடங்கு தளர்வில் அவ்வப்போது சுற்றுலா பயணிகளை குளிக்க அனுமதித்தாலும், தொற்று பரவல் காரணமாக மீண்டும் தடை விதிக்கப்பட்டது.இந்நிலையில், 9 மாதங்களுக்கு பிறகு குற்றாலம் அருவிகளில் குளிக்க சுற்றுலா பயணிகளுக்கு இன்று திங்கட்கிழமை முதல் மாவட்ட நிர்வாகம் சார்பில் அனுமதி வழங்கப்பட்டுள்ளது.பிரதான அருவியான குற்றால மெயின் அருவியில் ஆண்கள் பகுதியில் ஒரே நேரத்தில் 10 பேர், பெண்கள் பகுதியில் ஒரு நேரத்தில் 6 பேரை அனுமதிக்கப்படுகின்றனர். ஐந்தருவியில் ஆண்கள் மற்றும் பெண்கள் பகுதியில் தலா 10 பேர், பழைய குற்றாலம் அருவியில் ஆண்கள் பகுதியில் 5 பேர், பெண்கள் பகுதியில் 10 பேரை அனுமதிக்கப்படுகின்றனர். : மது விலை அதிரடி குறைப்பு : தமிழகத்திற்கு வந்து வாங்குவதை தடுக்க ஆந்திர அரசு நடவடிக்கைமேலும், இங்கு வரும் சுற்றுலாப் பயணிகள் தனிமனித இடைவெளியை கடைபிடித்து நிற்க வரையப்பட்ட வட்டங்களில் நின்று செல்லவேண்டும் மேலும் இங்கு வருவோர் அனைவரும் முக கவசம் அணிந்திருக்க வேண்டும், அருவிப் பகுதிகளில் குளிக்க வரும் சுற்றுலா பயணிகள் இரண்டு டோஸ் தடுப்பூசி செலுத்தி கொண்டதற்கான சான்றிதழ் இருத்தல் வேண்டும் எனவும் மாவட்ட நிர்வாகம் சார்பில் அறிவுறுத்தப்பட்டுள்ளது. | tamil |
ടി20 ക്രിക്കറ്റില് സുവര്ണ നേട്ടം സ്വന്തമാക്കി രോഹിത് ശര്മ ദുബായ്: ടി20 ക്രിക്കറ്റില് സുവര്ണ നേട്ടം സ്വന്തമാക്കി ഇന്ത്യന് താരം രോഹിത് ശര്മ. രാജ്യാന്തര മത്സരങ്ങളില് 3000 റണ്സ് തികയ്ക്കുന്ന മൂന്നാമത്തെ താരമെന്ന നേട്ടമാണ് രോഹിത് സ്വന്തമാക്കിയത്. ടി20 ലോകകപ്പ് സൂപ്പര് 12 റൗണ്ടില് നമീബിയയ്ക്കെതിരേ മൂന്നാം ഓവറിന്റെ ആദ്യ പന്തില് ബൗണ്ടറിയിലൂടെയാണ് രോഹിത് ഈ നേട്ടം സ്വന്തമാക്കിയത്. മത്സരത്തില് തകര്പ്പന് ബാറ്റിങ് കാഴ്ചവച്ച രോഹിത് ടൂര്ണമെന്റിലെ രണ്ടാമത്തെ അര്ധസെഞ്ചുറിയും സ്വന്തമാക്കി. 37 പന്തുകളില് നിന്ന് ഏഴു ബൗണ്ടറികളും രണ്ടു സിക്സറുകളും സഹിതം 56 റണ്സാണ് രോഹിത് നേടിയത്. ഇതോടെ ട്വന്റി 20 യില് 116 മത്സരങ്ങളില് നിന്ന് 3038 റണ്സായി രോഹിതിന്റെ സമ്ബാദ്യം. 24 അര്ധസെഞ്ചുറികളും നാലു സെഞ്ചുറികളും ഹിറ്റ്മാന്റെ പേരിലുണ്ട്. 54.64 ആണ് ശരാശരി, സ്ട്രൈക്ക് റേറ്റ് 139.41. ടി20യിലെ ഉയര്ന്ന സ്കോര് 118. രാജ്യാന്തര തലത്തില് രോഹിതിനു മുമ്ബ് രണ്ടുപേര് മാത്രമാണ് 3000 കടന്നിട്ടുള്ളത്. ഇന്ത്യന് നായകന് വിരാട് കോഹ്ലിയാണ് ആദ്യം ഈ നേട്ടത്തിലെത്തിയത്. രണ്ടാമത് ന്യൂസിലന്ഡ് ഓപ്പണര് മാര്ട്ടിന് ഗുപ്റ്റിലും. 3227 റണ്സുമായി ഈ ക്ലബില് കോഹ്ലിയാണ് ഒന്നാമന്. ഗുപ്റ്റിലിന്റെ സമ്ബാദ്യം 3115 റണ്സാണ്. ന്യൂസിലന്ഡ് ലോകകപ്പ് സെമിയില് കടന്നതിനാല് ഗുപ്റ്റിലിന് കോഹ്ലിയെ മറികടക്കാന് കഴിയും. Tags rohit sharma t20 cricket world cup shortlink | malyali |
ಸಚಿವರಿಗೆ ಸ್ವಾತಂತ್ರ ದಿನಾಚರಣೆಯಂದು ಧ್ವಜಾರೋಹಣಕ್ಕೆ ಹೊಣೆಗಾರಿಕೆ ಹಂಚಿಕೆ : ಕಲಬುರ್ಗಿ, ಉಡುಪಿ, ಯಾದಗಿರಿಯಲ್ಲಿ ಡಿಸಿಗಳಿಗೆ ಹೊಣೆ ಬೆಂಗಳೂರು : ರಾಜ್ಯ ಸರ್ಕಾರದಿಂದ ಪ್ರತಿವರ್ಷದಂತೆ ಸಚಿವರಿಗೆ ಸ್ವಾತಂತ್ರ ದಿನಾಚರಣೆಯಂದು ಯಾವ ಸಚಿವರು ಯಾವ ಜಿಲ್ಲೆಯಲ್ಲಿ ಧ್ವಜಾರೋಹಣ ನೆರವೇರಿಸಬೇಕು ಎಂಬ ಪಟ್ಟಿಯನ್ನು ಬಿಡುಗಡೆ ಮಾಡಿದೆ. ಬೆಂಗಳೂರು ನಗರ ಜಿಲ್ಲೆಯನ್ನು ಹೊರತು ಪಡಿಸಿ, ವಿವಿಧ ಜಿಲ್ಲೆಗಳಿಗೆ ಸಚಿವರು ಸ್ವಾತಂತ್ರ್ಯ ದಿನಾಚರಣೆಯಂದು ಧ್ವಜಾರೋಹಣ ನೆರವೇರಿಸಲು ಸೂಚಿಸಿದೆ. ಆದ್ರೇ ಕಲಬುರ್ಗಿ, ಯಾದಗಿರಿ ಮತ್ತು ಉಡುಪಿ ಜಿಲ್ಲೆಗಳಲ್ಲಿ ಜಿಲ್ಲಾಧಿಕಾರಿಗಳೇ ಸ್ವಾತಂತ್ರ ದಿನಾಚರಣೆಯಂದು ಧ್ವಜಾರೋಹಣ ನೆರವೇರಿಸಲು ಹೊಣೆಗಾರಿಕೆ ನೀಡಿದೆ. 2020ನೇ ಸಾಲಿನ ಆಗಸ್ಟ್ 15ರ ಸ್ವಾತಂತ್ರ್ಯ ದಿನಾಚಣೆಯಂದು ಜಿಲ್ಲಾ ಕೇಂದ್ರಗಳಲ್ಲಿ ಸ್ವಾತಂತ್ರ್ಯ ದಿನಾಚರಣೆ ಪ್ರಯುಕ್ತ ಬೆಂಗಳೂರು ನಗರ ಜಿಲ್ಲೆಯನ್ನು ಹೊರತುಪಡಿಸಿ, ಈ ಕೆಳಗೆ ಸೂಚಿಸಿದಂತ ಉಪ ಮುಖ್ಯಮಂತ್ರಿಗಳು, ಸಚಿವರು ಹಾಗೂ ಜಿಲ್ಲಾಧಿಕಾರಿಗಳನ್ನು ಧ್ವಜಾರೋಹಣ ಮಾಡಲು ನೇಮಿಸಲಾಗಿದೆ. ಜಿಲ್ಲಾವಾರು ಧ್ವಜಾರೋಹಣ ನೆರೆವೇರಿಸುವ ಸಚಿವರ ಪಟ್ಟಿ ಬಾಗಲಕೋಟೆ ಉಪ ಮುಖ್ಯಮಂತ್ರಿ ಹಾಗೂ ಲೋಕೋಪಯೋಗಿ, ಸಮಾಜಕಲ್ಯಾಣ ಸಚಿವ ಗೋವಿಂದ ಎಂ ಕಾರಜೋಳ ರಾಮನಗರ ಉಪ ಮುಖ್ಯಮಂತ್ರಿ ಡಾ.ಸಿಎಸ್ ಅಶ್ವತ್ಥ್ ನಾರಾಯಣ ರಾಯಚೂರು ಉಪ ಮುಖ್ಯಮಂತ್ರಿ ಲಕ್ಷ್ಮಣ್ ಸವದಿ ಶಿವಮೊಗ್ಗ ಗ್ರಾಮೀಣಾಭಿವೃದ್ಧಿ ಸಚಿವ ಕೆ ಎಸ್ ಈಶ್ವರಪ್ಪ ಬೆಂಗಳೂರು ಗ್ರಾಮಾಂತರ ಕಂದಾಯ ಸಚಿವ ಆರ್ ಅಶೋಕ್ ಧಾರವಾಡ ಸಚಿವ ಜಗದೀಶ್ ಶೆಟ್ಟರ್ ಚಿತ್ರದುರ್ಗ ಸಚಿವ ಬಿ ಶ್ರೀರಾಮುಲು ಚಾಮರಾಜನಗರ ಸಚಿವ ಸುರೇಶ್ ಕುಮಾರ್ ಕೊಡಗು ಸಚಿವ ವಿ ಸೋಮಣ್ಣ ಚಿಕ್ಕಮಗಳೂರು ಸಚಿವ ಸಿಟಿ ರವಿ ಹಾವೇರಿ ಗೃಹ ಸಚಿವ ಬಸವರಾಜ ಬೊಮ್ಮಾಯಿ ದಕ್ಷಿಣ ಕನ್ನಡ ಸಚಿವ ಕೋಟಾ ಶ್ರೀನಿವಾಸ ಪೂಜಾರಿ ತುಮಕೂರು ಸಚಿವ ಜೆಸಿ ಮಾಧುಸ್ವಾಮಿ ಗದಗ ಸಚಿವ ಸಿಸಿ ಪಾಟೀಲ್ ಕೋಲಾರ ಸಚಿವ ಹೆಚ್ ನಾಗೇಶ್ ಬೀದರ್ ಸಚಿವ ಪ್ರಭ ಚೌವ್ಹಾಣ್ ವಿಜಯಪುರ ಸಚಿವ ಶಶಿಕಲಾ ಜೊಲ್ಲೆ ಬಳ್ಳಾರಿ ಸಚಿವ ಆನಂದ್ ಸಿಂಗ್ ದಾವಣಗೆರೆ ಸಚಿವ ಭೈರತಿ ಬಸವರಾಜ್ ಮೈಸೂರು ಸಚಿವ ಎಸ್ ಟಿ ಸೋಮಶೇಖರ್ ಕೊಪ್ಪಳ ಸಚಿವ ಬಿಸಿ ಪಾಟೀಲ್ ಚಿಕ್ಕಬಳ್ಳಾಪುರ ಸಚಿವ ಡಾ.ಕೆ.ಸುಧಾಕರ್ ಮಂಡ್ಯ ಸಚಿವ ಕೆಸಿ ನಾರಾಯಣಗೌಡ ಉತ್ತರ ಕನ್ನಡ ಸಚಿವ ಶಿವರಾಂ ಹೆಬ್ಬಾರ್ ಬೆಳಗಾವಿ ಸಚಿವ ರಮೇಶ್ ಜಾರಕಿಹೊಳಿ ಹಾಸನ ಸಚಿವ ಕೆ.ಗೋಪಾಲಯ್ಯ ಕಲಬುರ್ಗಿ ಜಿಲ್ಲಾಧಿಕಾರಿಗಳು ಉಡುಪಿ ಜಿಲ್ಲಾಧಿಕಾರಿಗಳು ಯಾದಗಿರಿ ಜಿಲ್ಲಾಧಿಕಾರಿಗಳು ಈ ಮೇಲ್ಕಂಡವರು ಅನಾರೋಗ್ಯ ಅಥವಾ ಇತರೆ ಕಾರಣಗಳಿಂದಾಗಿ ಸಚಿವರು ಧ್ವಜಾರೋಹಣ ಕಾರ್ಯಕ್ರಮಕ್ಕೆ ಅನುಪಸ್ಥಿತರಿದ್ದಲ್ಲಿ, ಆಯಾ ಜಿಲ್ಲೆಯ ಜಿಲ್ಲಾಧಿಕಾರಿಗಳು ಧ್ವಜಾರೋಹಣ ಮಾಡತಕ್ಕದ್ದು ಎಂಬುದಾಗಿ ತಿಳಿಸಿದೆ. ವಸಂತ ಬಿ ಈಶ್ವರಗೆರೆ | kannad |
ખીચડી બનાવવી થઇ હાનિકારક સાબિત ઘરમાં ચોરી કરવા ઘુસ્યો અને રસોડામાં ખિચડી બનાવતો હતો ત્યારે પોલીસે ઝડપી લીધો આસામ માં ચોરીની એક વિચિત્ર અને રમૂજી ઘટના સામે આવી છે. એક ઘરમાં ચોરી કરવા દરમિયાન ચોરને પોતાના માટે ખિચડી બનાવવી ભારે પડી ગઈ અને તે આસામ પોલીસના હાથે ઝડપાઈ ગયો. એ માણસ ઘરમાં કિંમતી વસ્તુઓ ચોરવાના ઈરાદે ઘૂસ્યો હતો, પણ એ દરમ્યાન તે રસોડામાં ગયો અને પોતાના માટે ખિચડી બનાવવા લાગ્યો. રિપોર્ટ મુજબ, ગુવાહાટી પોલીસે કહ્યું છે કે એ માણસની ધરપકડ કરી લેવામાં આવી છે. આસામ પોલીસે ટ્વિટર પર આ ઘટનાને રમૂજી અંદાજમાં જણાવી હતી. આસામ પોલીસે લખ્યું કે, નખિચડી ચોર નો વિચિત્ર કેસ! સ્વાસ્થ્યના અઢળક લાભ હોવા છતાં, એવું સામે આવ્યું છે કે ચોરી દરમ્યાન ખિચડી બનાવવી એ તમારા આરોગ્ય માટે હાનિકારક બની શકે છે. આ ચોર પકડાઈ ગયો છે અને ગુવાહાટી પોલીસ તેને ગરમાગરમ ભોજન પીરસી રહી છે. ચોરી કરવાના ઈરાદે આવેલા એ માણસે જ્યારે કિચનમાં ખિચડી બનાવવાનું શરુ કર્યું, તો આજુબાજુના લોકો સાવધ થઈ ગયા અને પોલીસને બોલાવી લીધી હતી. એ ચોર ખિચડી બનાવતો હતો એ દરમ્યાન જ પકડાઈ ગયો! આસામ પોલીસની ટ્વિટ બાદ અન્ય ટ્વિટર યુઝર્સ પણ આ ઘટનાની મજા લઈ રહ્યા છે. તો અમુક યુઝરે લખ્યું કે, એ માણસે ચોરી દરમ્યાન ખિચડી બનાવી એટલે તેને ખરેખર ભૂખ લાગી હશે. જો કે, ભારતમાં આ પ્રકારની ચોરીની ઘટના પહેલી નથી કે જેમાં ચોર ચોરી દરમ્યાન થોડો વિચલિત થઈ ગયો હોય. ગયા વર્ષે મહારાષ્ટ્રના થાણેની નૌપાડા પોલીસે એક માણસની હનુમાનના મંદિરમાંથી દાનપેટી ચોરવા બદલ ધરપકડ કરી હતી. જ્યારે પોલીસે સીસીટીવી ફૂટેજ ચેક કર્યા તો તેમને જોઈને નવાઈ લાગી કે એ સિદ્ધાંતવાદી ચોરે દાનપેટી લઈને જતાં પહેલા હનુમાનજીના ચરણ સ્પર્શ કર્યા હતા! | gujurati |
ಶ್ರೀರಾಮನ ಆಶೀರ್ವಾದದಿಂದ ಕೋವಿಡ್19 ಮಾಯವಾಗಬಹುದು: ಶಿವಸೇನಾ ಮುಂಬೈ: ದೇಶದಲ್ಲಿರುವ ಕೋವಿಡ್19 ಸಂಕಷ್ಟ ಶ್ರೀರಾಮನ ಆಶೀರ್ವಾದದಿಂದ ಮಾಯವಾಗಬಹುದು ಎಂದು ಶಿವಸೇನಾ ಹೇಳಿದೆ. ಪಕ್ಷದ ಮುಖವಾಣಿ ಸಾಮ್ನಾದಲ್ಲಿ ಆಗಸ್ಟ್ 5ರಂದು ಅಯೋಧ್ಯೆಯಲ್ಲಿ ನಡೆಯಲಿರುವ ರಾಮಮಂದಿದ ಭೂಮಿಪೂಜೆ ಬಗ್ಗೆ ಉಲ್ಲೇಖಿಸಿದ್ದು ನರೇಂದ್ರ ಮೋದಿಯವರು ಭೂಮಿ ಪೂಜೆ ಮಾಡುವುದು ಎಂಬುದು ಸುವರ್ಣ ಗಳಿಗೆ. ಇಲ್ಲಿ ಕೊರೊನಾ ವೈರಸ್ ಇದೆ. ಆದರೆ ಅದು ಶ್ರೀರಾಮನ ಆಶೀರ್ವಾದದಿಂದ ಮಾಯವಾಗಲಿದೆ ಎಂದಿದೆ. ರಾಮಮಂದಿರ ನಿರ್ಮಾಣಕ್ಕಾಗಿ ಅಭಿಯಾನ ಕೈಗೊಂಡ ನಾಯಕರಾಗಿದ್ದ ಅಡ್ವಾಣಿ ಮತ್ತು ಜೋಷಿ ವಿಡಿಯೊ ಕಾನ್ಫರೆನ್ಸ್ ಮೂಲಕ ಕಾರ್ಯಕ್ರಮ ವೀಕ್ಷಿಸಲಿದ್ದಾರೆ. ಕೋವಿಡ್ ಹರಡುತ್ತಿರುವ ಈ ಹೊತ್ತಿನಲ್ಲಿ ಈ ನಾಯಕರಿಬ್ಬರ ವಯಸ್ಸು ಪರಿಗಣಿಸಿ ಕಾರ್ಯಕ್ರಮದಲ್ಲಿ ಭಾಗವಹಿಸದಿರುವುದು ಒಳ್ಳೆಯದು ಎಂದು ಸಲಹೆ ನೀಡಲಾಗಿತ್ತು ಎಂದು ಸಾಮ್ನಾದಲ್ಲಿ ಬರೆಯಲಾಗಿದೆ. ಬಿಜೆಪಿ ನಾಯಕಿ ಉಮಾ ಭಾರತಿ ಅವರು ಸರಯೂ ನದೀತಟದಲ್ಲಿ ಅವರ ಮನಸ್ಸಿನಿಂದಲೇ ಕಾರ್ಯಕ್ರಮವನ್ನು ವೀಕ್ಷಿಸಲಿದ್ದಾರೆ ಭೂಮಿಪೂಜೆ ಕಾರ್ಯಕ್ರಮದ ಬಗ್ಗೆ ದೇಶ ಉತ್ಸುಕಗೊಂಡಿದೆ. ಕೊರೊನಾವೈರಸ್ ಅಯೋಧ್ಯೆ, ಉತ್ತರ ಪ್ರದೇಶ ಮತ್ತು ಇಡೀ ದೇಶದಲ್ಲಿ ವ್ಯಾಪಿಸಿಕೊಂಡಿದೆ. ಈ ಸಂಕಷ್ಟವು ರಾಮನ ಆಶೀರ್ವಾದದಿಂದ ಮಾಯವಾಗಿ ಹೋಗಲಿದೆ ಎಂದು ಶಿವಸೇನಾ ಹೇಳಿದೆ. | kannad |
ಕಲ್ಲಾಪು ಡೆವಲಪ್ಮೆಂಟ್ ಗ್ರೂಪ್ನಿಂದ ಉಚಿತ ಆಯುಷ್ಮಾನ್ ಕಾರ್ಡ್ ನೋಂದಣಿ ಕಾರ್ಯಕ್ರಮ ಮಂಗಳೂರು, ಆ.21: ಕಲ್ಲಾಪು ಡೆವಲಪ್ಮೆಂಟ್ ಗ್ರೂಪ್ ವತಿಯಿಂದ ಉಚಿತ ಆಯುಷ್ಮಾನ್ ಕಾರ್ಡ್ ನೋಂದಣಿ ಮತ್ತು ವಿತರಣಾ ಕಾರ್ಯಕ್ರಮ ಕಲ್ಲಾಪು ನಗರಸಭಾ ಸದಸ್ಯ ಮುಸ್ತಾಕ್ ಪಟ್ಲ ಮನೆ ವಠಾರದಲ್ಲಿ ಶುಕ್ರವಾರ ಬೆಳಗ್ಗೆ ನಡೆಯಿತು. ಉಳ್ಳಾಲ ನಗರಸಭಾ ಪೌರಾಯುಕ್ತ ರಾಯಪ್ಪಅವರು ಕಾರ್ಡ್ ನೋಂದಣಿ ಮತ್ತು ವಿತರಣೆ ನಡೆಸಿ ಕಾರ್ಯಕ್ರಮಕ್ಕೆ ಚಾಲನೆ ನೀಡಿದರು. ನಗರಸಭಾ ಜೆ.ಇ. ತುಳಸಿದಾಸ್, ಸೇವಂತಿಗುಡ್ಡೆ ಶ್ರೀ ಕೊರಗ ತನಿಯ ಸೇವಾ ಟ್ರಸ್ಟ್ ಅಧ್ಯಕ್ಷ ಮೋಹನ್ ಸಾಲ್ಯಾನ್, ಮುಹಮ್ಮದ್ ಶರೀಫ್, ನವಾಝ್, ಸಾಯಿ ಪರಿವಾರ್ ಸಂಚಾಲಕ ಪುರುಷೋತ್ತಮ ಕಲ್ಲಾಪು ಮುಂತಾದವರು ಉಪಸ್ಥಿತರಿದ್ದರು. ಸ್ಥಳೀಯ ನಗರಸಭಾ ಸದಸ್ಯ ಮುಸ್ತಾಕ್ ಪಟ್ಲ ಸ್ವಾಗತಿಸಿ, ಪ್ರಾಸ್ತಾವಿಕವಾಗಿ ಮಾತನಾಡಿದರು. | kannad |
ಸಿಬಿಎಸ್ಇ, ಸಿಐಸಿಎಸ್ಇ 12ನೇ ಪರೀಕ್ಷೆ ಶೀಘ್ರ ಅಂತಿಮ ನಿರ್ಧಾರ ಪ್ರಕಟ : ಶಿಕ್ಷಣ ಇಲಾಖೆ ಹೊಸದಿಲ್ಲಿ, ಮೇ 30: ದೇಶದಲ್ಲಿ ಕೊರೋನ ಸೋಂಕಿನ ಪರಿಸ್ಥಿತಿಯನ್ನು ಗಮನದಲ್ಲಿಟ್ಟುಕೊಂಡು ಸಿಬಿಎಸ್ಇ ಮತ್ತು ಸಿಐಎಸ್ಸಿಇಯ ಬಾಕಿ ಉಳಿದಿರುವ 12ನೇ ಪರೀಕ್ಷೆ ನಡೆಸಬೇಕೇ ಅಥವಾ ಇತರ ಆಯ್ಕೆಯತ್ತ ಆದ್ಯತೆ ನೀಡಬೇಕೇ ಎಂಬುದನ್ನು ಶೀಘ್ರ ನಿರ್ಧರಿಸಲಾಗುವುದು ಎಂದು ಮೂಲಗಳು ಹೇಳಿವೆ. ಪರೀಕ್ಷೆಯನ್ನು ರದ್ದುಗೊಳಿಸಿ, ಈ ಹಿಂದಿನ ಪರೀಕ್ಷೆಯಲ್ಲಿ ಪಡೆದ ಅಂಕಗಳ ಆಧಾರದಲ್ಲಿ ತೇರ್ಗಡೆಗೊಳಿಸುವುದು ಅಥವಾ ಕಡಿಮೆ ಅವಧಿಯ ಪರೀಕ್ಷೆಯನ್ನು ಆಗಸ್ಟ್ನಲ್ಲಿ ನಡೆಸುವುದು ಸೇರಿದಂತೆ ಹಲವು ಆಯ್ಕೆಗಳು ನಮ್ಮೆದುರಿಗಿವೆ. ಹಲವು ರಾಜ್ಯಗಳು ಆಗಸ್ಟ್ ನಲ್ಲಿ ಕಿರು ಅವಧಿಯ ಪರೀಕ್ಷೆ ನಡೆಸಲು ಒಲವು ತೋರಿಸಿವೆ ಎಂದು ಕೇಂದ್ರ ಶಿಕ್ಷಣ ಇಲಾಖೆಯ ಮೂಲಗಳು ಹೇಳಿವೆ. ಈ ಮಧ್ಯೆ, ವಿದ್ಯಾರ್ಥಿಗಳು 11ನೇ ತರಗತಿಯ ಅಂತಿಮ ಪರೀಕ್ಷೆ ಹಾಗೂ 12ನೇ ತರಗತಿಯ ಕ್ಲಾಸ್ ಪರೀಕ್ಷೆಯಲ್ಲಿ ಪಡೆದಿರುವ ಸರಾಸರಿ ಅಂಕಗಳ ಪಟ್ಟಿಯನ್ನು ಜೂನ್ 7ರೊಳಗೆ ಒದಗಿಸುವಂತೆ ಸಿಐಸಿಎಸ್ಇಗೆ ಸಂಯೋಜನೆಗೊಂಡಿರುವ ಶಾಲೆಗಳಿಗೆ ಸೂಚಿಸಲಾಗಿದೆ ಎಂದು ವರದಿಯಾಗಿದೆ. ಇದುವರೆಗೆ ಯಾವುದೇ ನಿರ್ಧಾರ ಅಂತಿಮಗೊಂಡಿಲ್ಲ. ಜೂನ್ 1ರಂದು ಅಂತಿಮ ನಿರ್ಧಾರ ಘೋಷಿಸಲಾಗುವುದು . ವಿದ್ಯಾರ್ಥಿಗಳ ಸುರಕ್ಷತೆಗೆ ಮೊದಲ ಆದ್ಯತೆಯಾಗಿದೆ. ಆದರೆ ಪರೀಕ್ಷೆಗಳೂ ಅತ್ಯಂತ ನಿರ್ಣಾಯಕ ಎಂದು ಕೇಂದ್ರ ಶಿಕ್ಷಣ ಇಲಾಖೆ ಹೇಳಿದೆ. ಸಿಬಿಎಸ್ಇಸೆಂಟ್ರಲ್ ಬೋರ್ಡ್ ಆಫ್ ಸೆಕೆಂಡರಿ ಎಜುಕೇಶನ್ ಮತ್ತು ಸಿಐಎಸ್ಸಿಇಕೌನ್ಸಿಲ್ ಫಾರ್ ದಿ ಇಂಡಿಯನ್ ಸ್ಕೂಲ್ ಸರ್ಟಿಫಿಕೇಟ್ ಎಕ್ಸಾಮಿನೇಷನ್ 12ನೇ ತರಗತಿ ಪರೀಕ್ಷೆಗಳನ್ನು ಕೊರೋನ ಸೋಂಕು ಉಲ್ಬಣಿಸಿದ್ದರಿಂದ ರದ್ದುಗೊಳಿಸಬೇಕೆಂದು ಕೋರಿ ಸಲ್ಲಿಸಿರುವ ಅರ್ಜಿಯ ವಿಚಾರಣೆ ಸುಪ್ರೀಂ ಕೋರ್ಟ್ ನಲ್ಲಿ ಮೇ 31ರಂದು ನಡೆಯಲಿದೆ. ಈ ಮಧ್ಯೆ ರವಿವಾರ ಕೇಂದ್ರ ಸರಕಾರ ನಡೆಸಿದ್ದ ಉನ್ನತ ಮಟ್ಟದ ಸಭೆಯಲ್ಲಿ ಸಿಬಿಎಸ್ಸಿ ಎರಡು ಆಯ್ಕೆಗಳನ್ನು ಪ್ರಸ್ತಾಪಿಸಿದೆ ಎಂದು ವರದಿಯಾಗಿದೆ. ಜುಲೈ 15 ಆಗಸ್ಟ್ 26ರ ಅವಧಿಯಲ್ಲಿ ಪ್ರಮುಖ ವಿಷಯಗಳಿಗೆ , ಅಧಿಸೂಚಿತ ಕೇಂದ್ರಗಳಲ್ಲಿ ಪರೀಕ್ಷೆ ನಡೆಸಿ ಸೆಪ್ಟಂಬರನಲ್ಲಿ ಫಲಿತಾಂಶ ಘೋಷಿಸುವುದು ಅಥವಾ ವಿದ್ಯಾರ್ಥಿ ದಾಖಲಾತಿ ಪಡೆದಿರುವ ಶಾಲೆಯಲ್ಲೇ ಕಿರು ಅವಧಿಯ ಪರೀಕ್ಷೆ ನಡೆಸುವುದು. ಈ ಎರಡು ಆಯ್ಯೆಯಲ್ಲಿ ಎರಡನೇ ಆಯ್ಕೆಗೆ ಬಹುಮತ ದೊರಕಿದ್ದು ಪ್ರಮುಖ ವಿಷಯಗಳಿಗೆ 90 ನಿಮಿಷದ ಪರೀಕ್ಷೆ ನಡೆಸಲು ಹಲವು ರಾಜ್ಯಗಳು ಒಲವು ಹೊಂದಿವೆ ಎಂದು ವರದಿಯಾಗಿದೆ. ಕೊರೋನ ಸೋಂಕಿನ ಹಿನ್ನೆಲೆಯಲ್ಲಿ ಎರಡೂ ಶಿಕ್ಷಣ ಮಂಡಳಿಗಳ 12ನೇ ತರಗತಿ ಪರೀಕ್ಷೆಯನ್ನು ಮೇಜೂನ್ ನಲ್ಲಿ ನಡೆಯಬೇಕಿತ್ತು ಮುಂದೂಡಲಾಗಿದೆ. ಎರಡೂ ಮಂಡಳಿಗಳು 10ನೇ ತರಗತಿ ಪರೀಕ್ಷೆಯನ್ನು ರದ್ದುಗೊಳಿಸಿವೆ. | kannad |
ਆਪਣੇ ਆਪ ਨੂੰ ਸਮਝਾਓ।'] ", 'ਆਪਣੇ ਆਪ ਨੂੰ ਸਮਝਾਓ। \n '] | punjabi |
Clarity Counseling Agency | How Do I Control My Anger?
response to situations and people that make us angry.
Let's begin with identifying the ABC's of anger.
ABC stands for Antecedent, Behavior, Consequence.
The antecedent is what led up to the situation that caused the anger.
The behavior is what you did in your response to the situation (in this case anger).
The consequence is the result of your behavior.
A negative reaction to an antecedent (situation), leads to a behavior (yelling, blowing up, property destruction, aggression) that results in a consequence (ruined relationships, poor health, legal troubles).
Anger is usually a second emotion; this means there is usually something behind our feeling of anger that has very little to do with the situation we are angry about.
In looking at a situation of anger I like to use the analogy of an iceberg. Eighty percent of an iceberg is under the water. The twenty percent of the iceberg we do see represents the outward anger we express through yelling, outbursts, conflict, etc. The eighty percent under the surface makes up the bulk of our emotional response, such as pain, depression, and anxiety.
When you begin to talk about the things underneath the surface of your anger, your negative emotions can begin to heal and eventually you can work through these feelings. This ultimately results in better outcomes and feeling better. This is especially true when working with children. If children are unable to adequately express their emotions they usually will react with increased anger and frustration.
Anger is a very quick response and likely stems from our primal "fight or flight" instinct of survival. By stopping and examining the situation when we begin to feel anger, we can start to develop a more thoughtful response.
To start developing a more thoughtful response, examine your options and make a more thoughtful behavior choice. This positive behavior choice will then lead to a better result (consequence). This can be done in a couple of different ways.
First, you can examine different situations that have created an anger response in the past and use a technique called "cognitive rehearsal." To practice cognitive rehearsal, imagine a situation that makes you angry and go through a step-by-step process of facing that anger and successfully dealing with it. Then practice these steps mentally over and over again. By reviewing these situations, you can begin to retrain your brain to have a more positive response.
When dealing with anger, the bottom line is to have plan in place to effectively cope with and effectively communicate your anger. If you experience extreme anger often, it is likely that there are unresolved emotions/ feelings underneath that are currently being expressed with escalation and outbursts. If you can begin to communicate these feelings more effectively while utilizing skills to remain in control, you will have much better outcomes/relationships with family, friends, and coworkers while feeling happier with your life. | english |
పాండురంగ స్వామికి మంత్రి పేర్ని పట్టువస్త్రాలు సమర్పణ పాండురంగ స్వామికి మంత్రి పేర్ని పట్టువస్త్రాలు సమర్పణ ప్రజాశక్తిమచిలీపట్నం పాండురంగ స్వామి దంపతులకు పట్టు వస్త్రాలను సమర్పించిన పేర్ని దంపతులు పాండురంగ రుక్మిణి స్వామి అమ్మవార్లకు మంత్రి పేర్ని వెంకట్రామయ్యనాని దంపతులు సతీసమేతంగా పట్టువస్త్రాలు సమర్పించారు. ఆదివారం ఉదయం చిలకలపూడి, కీరపండరీపురంలోని పాండురంగ ఆలయంలో కార్తీకశుద్ధ ఏకాదశి మహోత్సవం సందర్భంగా పాండురంగ స్వామి దంపతులకు రాష్ట్ర రవాణా, సమాచార పౌర సంబంధాలశాఖా మంత్రి పేర్ని వెంకట్రామయ్యనాని సతీసమేతంగా పట్టు వస్త్రాలు సమర్పించారు. ఆలయ ప్రవేశ ద్వారం వద్ద నుండి గర్భాలయం వరకు పేర్ని నాని దంపతులు నడుచుకుంటూ వెళ్లి స్వామివార్లకు పట్టు వస్త్రాలు సమర్పించారు. ఈ సందర్భంగా ఆలయ నిర్వాహకులు పేర్ని దంపతులకు మంగళవాయిద్యాలతో స్వాగతం పలికారు. అర్చకులు ప్రత్యేక పూజలు నిర్వహించి దంపతులకు ఆశీస్సులు అందజేశారు. ఉత్సవ ఏర్పాట్లపై మంత్రి ఆలయ నిర్వాహకులతో చర్చించారు. ఆయన మాట్లాడుతూ ఎప్పటిలాగే ఈ సంవత్సరం కూడా పాండురంగ స్వామి వారి ఉత్సవాలు నిర్వహించడానికి ప్రభుత్వం, మున్సిపాలిటీ తరపున ఏర్పాట్లు పూర్తి చేశామన్నారు. కరోనా కారణంగా గత సంవత్సరం పూర్తిస్థాయిలో పాండురంగ స్వామి ఉత్సవాలు నిర్వహించలేకపోయామన్నారు. ఈ సంవత్సరం పరిస్థితులు అనుకూలించడం వల్ల ఉత్సవాలకు అంతా సిద్ధం చేసినట్లు తెలిపారు. సోమవారం ఏకాదశి రోజున సాయంత్రం 7 గంటలకు కళ్యాణం, మంగళవారం మధ్యాహ్నం 3 గం.లకు రథయాత్ర పాండురంగ ఆలయం దగ్గర నుంచి మచిలీపట్నం పురవీధుల గుండా రథయాత్ర జరుగుతుందన్నారు. పౌర్ణమి రోజు గోపాలకాల మహోత్సవంతో ఉత్సవాలు ముగుస్తాయన్నారు. ఈ ఉత్సవాల్లో ప్రతి ఒక్కరూ పాల్గొని పాండురంగ స్వామి ఆశీస్సులు పొందాలని ఆయన ఆశించారు. ఆర్యవైశ్యులకు అండ | telegu |
Mamata Banerjee Calls Party Meeting : 5 মে দলের গুরুত্বপূর্ণ বৈঠক ডাকলেন মমতা মমতার ডাকা এই বৈঠকে TMC Supremo Mamata Banerjee calls party meeting আলোচনা হতে পারে পঞ্চায়েত ভোট নিয়ে দলের নেতাকর্মীদের জন্য নতুন কর্মসূচি ঘোষণা করতে পারেন তৃণমূল নেত্রী কলকাতা, 1 মে : 5 মে অর্থাত্ বৃহস্পতিবার দলের গুরুত্বপূর্ণ বৈঠক ডাকলেন তৃণমূল সুপ্রিমো মমতা বন্দ্যোপাধ্যায় Mamata Banerjee Calls Party Meeting ওই দিন আবার মমতা বন্দ্যোপাধ্যায়ের নেতৃত্বাধীন রাজ্য সরকারেরও বর্ষপূর্তি 2021 সালে তৃতীয়বার রাজ্যে ক্ষমতায় আসার পর ওই দিনই মুখ্যমন্ত্রী হিসেবে শপথ নিয়েছিলেন মমতা তৃণমূল কংগ্রেস সূত্রে খবর, গত এক বছরে তৃণমূল সরকার রাজ্যের উন্নয়নে কী কী কাজ করেছে ওই দিন থেকেই দলীয়স্তরে তার প্রচার শুরু হবে এই বৈঠকে মন্ত্রী, বিধায়ক, সাংসদদের নিয়ে আলোচনায় বসতে চলেছেন তৃণমূল নেত্রী কিছু জেলার দলীয় সভাপতি ও দলের রাজ্য কমিটির সদস্যদেরও এই বৈঠকে ডাকা হয়েছে বলে খবর 5 মে মমতার ডাকা বৈঠক মূলত দলকে দিশা দেখানোর জন্যই তাঁর নির্দেশে আগামী দিনে জনসংযোগ কর্মসূচি কীভাবে হবে, তা বাতলে দিতে পারেন তৃণমূল সুপ্রিমো সাম্প্রতিক অতীতে একাধিক বিষয়ে দলের অন্দরের দ্বন্দ্ব প্রকাশ্যে চলে এসেছে, সে বিষয়েও দলীয় নেতৃত্বকে বার্তা দিতে পারেন মমতা বন্দ্যোপাধ্যায় সেই দিক থেকে এই বৈঠক অত্যন্ত গুরুত্বপূর্ণ দলের সমস্ত শীর্ষ নেতাদের ওইদিন তৃণমূল ভবনে উপস্থিত থাকার কথা থাকতে পারেন দলের ভোটকুশলী প্রশান্ত কিশোরও সাম্প্রতিক অতীতে পিকের সঙ্গে তৃণমূলের কিছুটা দূরত্ব তৈরি হয়েছিল, সেই অতীত দূরে সরিয়ে দলকে আবার ভোটমুখী করার জন্য দলনেত্রী স্বয়ং কৌশল নির্ধারণ করতে পারেন এই বৈঠক থেকেই আরও পড়ুন : আপনাদের জন্য গর্বিত, শ্রমদিবসে বিশ্বব্যাপী কর্মীদের কুর্নিশ মমতার5 মে থেকেই টানা একমাস বিভিন্ন ইস্যুতে দলীয় কর্মসূচির ঘোষণা করতে পারেন তৃণমূল নেত্রী রাজ্যের বিধানসভা নির্বাচনে জেতার পর সম্প্রতি পৌরভোটেও বিপুল জয় পেয়েছে তৃণমূল এই অবস্থায় দলকে শৃঙ্খলার বন্ধনে বাঁধা একটা গুরুত্বপূর্ণ কাজ বলে মনে করা হচ্ছে এই বৈঠক থেকেই সে বিষয়ে পদক্ষেপ করতে পারেন মমতা এছাড়াও আগামী বছর রাজ্যে পঞ্চায়েত ভোটের বিষয়টিও ওঠে আসতে পারে এই বৈঠকে এই নির্বাচনের জন্য দলকে এখন থেকেই প্রস্তুতির নির্দেশ দিতে পারেন তৃণমূল সুপ্রিমো 2018 সালের পঞ্চায়েত নির্বাচনের পর 2019 লোকসভা ভোটের ফল ও রাজ্যে বিজেপির উত্থানের কথা ভুলে যাননি মমতা বন্দ্যোপাধ্যায় তিনি জানেন জাতীয় ক্ষেত্রে তাঁকে প্রভাব বিস্তার করতে গেলে রাজ্যের লোকসভা ভোটে আরও ভাল ফল করতে হবে তার জন্য সেমিফাইনাল হতে চলেছে পঞ্চায়েত ভোট মূলত এই বৈঠক থেকে গ্রামীণ তথা বুথস্তরের সংগঠনকে শক্তিশালী করে গোটা দেশের পরিপ্রেক্ষিতে নিজেকে আরও পোক্ত করতে চান মমতা বন্দ্যোপাধ্যায় যাতে তার সুফল মেলে 2024এর ভোটে | bengali |
मतदान दिवस को व मतदान दिवस से एक दिन पूर्व किसी भी प्रकार का राजनैतिक विज्ञापन प्रकाशित कराने की अनुमति लेना होगा अनिवार्य उन्नाव। अपर जिलाधिकारी वि.रा.उप जिला निर्वाचन अधिकरी नरेन्द्र सिंह ने बताया कि विधान सभा सामान्य निर्वाचन 2022 मतदान दिवससे एक दिन पूर्व एवं मतदान दिवस को प्रिन्ट मीडिया से संबंधित प्रीसार्टिफिकेशन राजनैतिक विज्ञापन से संबंधित मुख्य निर्वाचन अधिकारीउ.प्र. द्वारा विस्तृत दिशा निर्देश जारी किये गये हैं। उन्होंने बताया कि जनपद उन्नाव में विधानसभा सामान्य निर्वाचन 2022 चैथे चरण में होने वाले मतदान दिवस 23 फरवरी 2022 को होना हैऐसी स्थिति में दिनांक 22 एवं 23 फरवरी 2022 की तिथियों में कोई भी राजनैतिक विज्ञापन प्रकाशित कराया जाता है तो मीडिया प्रमाणन एवंमाॅनीटरिंग समिति एम.सी.एम.सी की पूर्व में अनुमति लेना अनिवार्य होगा अन्यथा की दशा में सख्त कार्यवाही की जायेगी। | hindi |
jsonp({"cep":"40349220","logradouro":"Avenida Teixeira","bairro":"Alto do Peru","cidade":"Salvador","uf":"BA","estado":"Bahia"});
| code |
// Copyright(c) 2007 Andreas Gullberg Larsen
// https://github.com/angularsen/UnitsNet
//
// Permission is hereby granted, free of charge, to any person obtaining a copy
// of this software and associated documentation files (the "Software"), to deal
// in the Software without restriction, including without limitation the rights
// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
// copies of the Software, and to permit persons to whom the Software is
// furnished to do so, subject to the following conditions:
//
// The above copyright notice and this permission notice shall be included in
// all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
// THE SOFTWARE.
using System;
using Xunit;
using UnitsNet.CustomCode.Extensions;
namespace UnitsNet.Tests.CustomCode
{
public class AmplitudeRatioTests : AmplitudeRatioTestsBase
{
protected override double DecibelMicrovoltsInOneDecibelVolt => 121;
protected override double DecibelMillivoltsInOneDecibelVolt => 61;
protected override double DecibelsUnloadedInOneDecibelVolt => 3.218487499;
protected override double DecibelVoltsInOneDecibelVolt => 1;
protected override void AssertLogarithmicAddition()
{
AmplitudeRatio v = AmplitudeRatio.FromDecibelVolts(40);
AssertEx.EqualTolerance(46.0205999133, (v + v).DecibelVolts, DecibelVoltsTolerance);
}
protected override void AssertLogarithmicSubtraction()
{
AmplitudeRatio v = AmplitudeRatio.FromDecibelVolts(40);
AssertEx.EqualTolerance(46.6982292275, (AmplitudeRatio.FromDecibelVolts(50) - v).DecibelVolts, DecibelVoltsTolerance);
}
[Theory]
[InlineData(0)]
[InlineData(-1)]
[InlineData(-10)]
public void InvalidVoltage_ExpectArgumentOutOfRangeException(double voltage)
{
ElectricPotential invalidVoltage = ElectricPotential.FromVolts(voltage);
// ReSharper disable once ObjectCreationAsStatement
Assert.Throws<ArgumentOutOfRangeException>(() => new AmplitudeRatio(invalidVoltage));
}
[Theory]
[InlineData(1, 0)]
[InlineData(10, 20)]
[InlineData(100, 40)]
[InlineData(1000, 60)]
public void ExpectVoltageConvertedToAmplitudeRatioCorrectly(double voltage, double expected)
{
// Amplitude ratio increases linearly by 20 dBV with power-of-10 increases of voltage.
ElectricPotential v = ElectricPotential.FromVolts(voltage);
double actual = AmplitudeRatio.FromElectricPotential(v).DecibelVolts;
Assert.Equal(expected, actual);
}
[Theory]
[InlineData(-40, 0.01)]
[InlineData(-20, 0.1)]
[InlineData(0, 1)]
[InlineData(20, 10)]
[InlineData(40, 100)]
public void ExpectAmplitudeRatioConvertedToVoltageCorrectly(double amplitudeRatio, double expected)
{
// Voltage increases by powers of 10 for every 20 dBV increase in amplitude ratio.
AmplitudeRatio ar = AmplitudeRatio.FromDecibelVolts(amplitudeRatio);
double actual = AmplitudeRatio.ToElectricPotential(ar).Volts;
Assert.Equal(expected, actual);
}
// http://www.maximintegrated.com/en/app-notes/index.mvp/id/808
[Theory]
[InlineData(8, -38.99)]
[InlineData(20, -26.99)]
[InlineData(40, -6.99)]
[InlineData(60, 13.01)]
public void AmplitudeRatioToPowerRatio_50OhmImpedance(double dBmV, double expected)
{
AmplitudeRatio ampRatio = AmplitudeRatio.FromDecibelMillivolts(dBmV);
double actual = Math.Round(ampRatio.ToPowerRatio(ElectricResistance.FromOhms(50)).DecibelMilliwatts, 2);
Assert.Equal(expected, actual);
}
[Theory]
[InlineData(8, -40.75)]
[InlineData(20, -28.75)]
[InlineData(40, -8.75)]
[InlineData(60, 11.25)]
public void AmplitudeRatioToPowerRatio_75OhmImpedance(double dBmV, double expected)
{
AmplitudeRatio ampRatio = AmplitudeRatio.FromDecibelMillivolts(dBmV);
double actual = Math.Round(ampRatio.ToPowerRatio(ElectricResistance.FromOhms(75)).DecibelMilliwatts, 2);
Assert.Equal(expected, actual);
}
}
} | code |
এই আয়নাতেই আঁটকে আছে অন্তত ১০ জনের অতৃপ্ত আত্মা নিউজ ডেস্কঃ ভূতে বিশ্বাস থাকুক বা না থাকুক ভুতুড়ে রহস্যময় ঘটনা সম্পর্কে আগ্রহ আমাদের সকলেরই আছে মধ্যাহ্নভোজনের পরে দুপুরে আরাম করে শুয়ে ভূতের গল্প শুনতেও নেহাত মন্দ লাগে না কারোরই আর ছোটবেলা থেকে পড়ে আসা অসংখ্য ভূতের গল্পে আমরা প্রায়ই নানা ভূতুড়ে স্থান, বাড়ি, শ্মশান প্রভৃতির কথা শুনে এসেছি অতৃপ্ত ভয়ঙ্কর এই সমস্ত আত্মাদের নিয়ে সিনেমাও নেহাত কম তৈরি হয়নি ভুতুড়ে বাড়ি, আয়না প্রভৃতি কতকিছুই না সেই সমস্ত সিনেমায় উঠে এসেছে তবে যদি বলা হয় এই পৃথিবীতে বাস্তবেই রয়েছে এমন এক আয়না ভুতুড়ে নামে কুখ্যাতি লাভ করেছে বিশ্বাস করছেন না তাইতো? ভাবছেন তাই আবার সম্ভব নাকি!ভুতুড়ে আয়নার কথা আবার সত্যি হয় কখনো ? তবে অবিশ্বাস্য লাগলেও এই তূতুড়ে আয়নার গল্প কিন্তু মিথ্যা নয় সেন্ট ফ্রান্সভিলের মির্টলেস প্ল্যান্টেসনএ রয়েছে এরকমই একটি ভয়ঙ্কর আয়না আর এই আয়না এমনই ভয়ঙ্কর যে যে কোন মুভি বা গল্পের ভুতুড়ে আয়না কেউ তা হার মানাবে স্থানীয়দের মতে এই আয়নায় একটি বা দুটি নয় প্রায় 10 জন অতৃপ্ত আত্মা আটকে রয়েছে তবে শুধু যে আয়নাটি ই ভুতুড়ে তা নয় এই বাড়িটিও ভুতুড়ে বলে কুখ্যাতি অর্জন করেছে এখানে প্রায়ই নানা রকমের ভুতুড়ে কান্ড কারখানা ঘটার কথা শুনতে পাওয়া যায় অনেকেই ওই বাড়িতে অতৃপ্ত আত্মার দেখা পর্যন্ত পেয়েছেন স্থানীয়দের মতে এখানে এরকম ভৌতিক কর্মকাণ্ড ঘটার মূল কারণ লুকিয়ে আছে এই বাড়ির ইতিহাসেই মার্কিন গৃহযুদ্ধের বহু পূর্বে এই খামার বাড়িটি তৈরি করেন জেনারেল ডেভিজ ব্র্যাডফোর্ড তবে যে সে জমির ওপর কিন্তু তৈরি হয়নি এটি এক রেড ইন্ডিয়ান গোরস্থানের ওপর তৈরি এই খামারবাড়ি প্রথম থেকেই অভিশপ্ত ছিল বলে অনেকে মনে করেন তার ওপর এই বাড়িতে নেহাত কম রক্ত ঝরেনি ইতিহাস ঘাটলে যদিও কেবল একটিমাত্র হত্যাকান্ডের কথা জানা যায় তবে,কথিত আছে এই বাড়িতে নাকি প্রায় দশটি হত্যাকাণ্ড সংঘটিত হয়েছে উইলিয়াম উইন্টার নামক এক ব্যাক্তি এই বাড়ির সিঁড়ির 17 নাম্বার ভাবে খুন হন বলে জানা যায় তারপর থেকে অনেকেই জানিয়েছেন যে সিড়ির ওই এক জায়গায় অনেকবার তারা দেখতে পেয়েছেন স্যার উইলিয়াম উইন্টারের আত্মাকে এ তো গেল কেবল বাড়ির কথা তবে এই খামারবাড়ির বদনাম মূলত এখানে থাকা এক আয়নার কারণে শোনা যায় আজ অবদি একদিনের জন্যেও আয়নাটিকে ঢেকে রাখা সম্ভব হয়নি অদ্ভুত শোনালেও এ কথাই সত্যি বারবার ঢেকে রাখা হলেও কোনো না কোনোভাবে এই আয়নার ঢাকনা বারবার সরে গেছে আর এর জন্য প্রত্যক্ষদর্শীরা অতৃপ্ত আত্মাদের দায়ী করেন তারা মনে করেন এই আয়নার ভেতরে থাকা প্রেতাত্মারা সময় বুঝে বেরিয়ে আসেন আর তখনই সরে যায় আয়নার ঢাকনা এমনকি এই বাড়িতে ঘুরতে যাওয়া অনেকেই নাকি এই আয়নার ভূতের খপ্পরে পড়েছেন কথিত আছে সারা উড্রফ নামক এক মহিলা ও তার দুই শিশু সন্তানের মৃত্যু হয় এই বাড়িতে আর তাদের মৃত্যুর পর ঢাকা না দেওয়া আয়নায় কোন রকম ভাবে তাদের আত্মা আটকা পড়ে যায় আর সেই থেকেই এই বাড়িতে দেখা যায় তাদের শোনা যায় প্রায়ই নাকি আয়নার কাচের গায়ে শিশুর হাতের ছাপ দেখা দেয় আবার কখনও কখনও এক অজানা বৃদ্ধ মহিলার চেহারাও ভেসে ওঠে এই আয়নায় বর্তমানে আমেরিকায় এই ভুতুড়ে আয়না টি এতোটাই জনপ্রিয়তা লাভ করেছে যে 2013 সালে এই আয়না কে কেন্দ্র করে এক ছবিও তৈরি হয় হলিউডে অকুলাস নামের তবে ,যতই ভুতুড়ে হিসেবে কুখ্যাতি লাভ করুক এই বাড়ি ! ভূত নিয়ে আগ্রহী বা অ্যাডভেঞ্চারপ্রিয় বহু মানুষ প্রত্যেক বছর এই বাড়ি ভ্রমণ করতে আসেন | bengali |
ভারতেও খুঁজে পাওয়া গিয়েছে দশটি মনমুগ্ধকর লেকের ঠিকানা নিউজ ডেস্ক প্রাকৃতিক সৌন্দর্যের টানে বিদেশে পাড়ি দেয় দেশবাসীরা তবে ভারতেও এমন কিছু দর্শনীয় স্থান রয়েছে যে জায়গা গুলির মুখে পড়ে বারবার ছুটে আসে পর্যটকরা পাহাড় বেষ্টিত লেক গুলির মধ্যে বিখ্যাত হলো ১ প্যাঙ্গং সো, লাদাখ লেহ থেকে ৫ ঘন্টা রাস্তা অতিক্রম করলেই বোঝা যাবে লাদাখে নীলরঙা নোনা জলের হ্রদ প্যাঙ্গংয়ে এই পদের জল লবণাক্ত হলেও চারিদিকের পাহাড় থাকায় প্রতিমুহূর্তে সৌন্দর্যে ভরপুর থাকে গোটা লেক যদিও এই জায়গা একেবারে উপযুক্ত চিত্রশিল্পীদের জন্য ২পরাশর লেক, মান্ডী সন্ন্যাসী পরাশর মুনিকে উত্সর্গ করে হিমাচল স্টাইল মন্দিরের পাশে অবস্থিত লেকের নাম দেওয়া হয়েছে পরাশর লেক মান্ডী থেকে মাত্র ৪৯ কিলোমিটার গেলেই সবুজে ঘেরা এই লেকের দর্শন পাওয়া যাবে এখানকার বিশেষত্ব হলো ছোট কিংবা বড় সকলেই ট্রেকিং করতে পারবে ৩ লেক পিচোলা, উদয়পুর উদয়পুরের সিটি প্যালেসের উপর নির্মিত প্রায় চার কিমি দীর্ঘ ও তিন কিমি প্রস্থ কৃত্রিম লেক তবে সামনা সামনি দেখলে কোনমতে এদিকে কৃত্রিম বলে মনে হবে না এই লেকে রামেশ্বর হাট থেকে দীর্ঘ খুনের জন্য নৌকাবিহার করা যায় এমনকি সূর্যদয় ও সূর্যাস্তের সময় নৌকা বিহারের সুব্যবস্থা ব্যবস্থা রয়েছে ৪ চন্দ্রতাল লেক, স্পীত চন্দ্রের ন্যায় অর্ধাকৃতির এই লেকটি লাহোল এবং স্পীতির কুঞ্জুম লা পাস থেকে প্রায় ৭ কিলোমিটার দূরে অবস্থিত ৪৩০০ মিটার উচ্চতায় অবস্থিত এই লেকের জল স্যাফারারের নীল রঙের মতো এমনকি এর চারিপাশ সবুজাভ দিয়েঘিরে থাকায় আদর্শ ও অপূর্ব পরিবেশ তৈরি করেছে ৫ লোকতাক লেক , মণিপুর উত্তরপূর্ব ভারতের সবচেয়ে বড় লেক লোকতাক লেক এছাড়া এই লেকের অন্য মাধুর্য হলো ফ্লোরা এবং ফণার আকৃতি এমনকি এখানে ভাসমান ফুমডিস রয়েছে মাটি, গাছপালা ও নানান অর্গানিক ম্যাটারের পুঞ্জিভূত অংশগুলি লেকের উপর দ্বীপপুঞ্জের আকার তৈরি করেছে যেখানে পর্যটকরা থাকতে পারে আর এই সৌন্দর্যের জন্য এই লেক বিখ্যাত ৬ সোমগো লেক, গ্যাংটক নাথু লা পাস যাওয়ার পথে গ্যাংটক থেকে মাত্র ৩৫ কিলোমিটার দূরে অবস্থিত এই লেকটি স্থানীয় ভাষা ছাঙ্গু নামে বেশি পরিচিত তুষারবৃত সর্বোচ্চ শৃঙ্গ বেষ্টিত শোভায় অভিভূত পর্যটকরা তবে এখানকার বিশেষ আকর্ষণীয় বিষয় হল লেকের চারিপাশে রংবেরঙের প্রার্থনা পতাকা ৭ ডাল লেক, শ্রীনগর কাশ্মীরের সৌন্দর্যকে বুঝতে গেলে ডাল লেকে অবশ্যই যেতে হবে কারণ এখানে শীত হোক কিংবা গ্রীষ্ম একেক ঋতুতে এক এক রকম মনোরম সৌন্দর্য দেখা যায় ১৫ কিলোমিটার জুড়ে বিস্তৃত এই লেকে রাজাদের যুগ থেকে বহু হাউসবোট উপস্থিত রয়েছে হিমশীতল আমেজের এই মনোরম লেকে বছরের প্রত্যেকটা দিনই ভিড় জমায় পর্যটকরা ৮চেম্ব্রা লেক, ওয়ানাদ চেম্ব্রা শৃঙ্গ যাওয়ার পথে পর্যটকদের নজর কাড়ে এই লেক স্থানীয় ভাষায় এই লেককে প্রেমীদের স্বর্গ বলা হয় এই লেকের মহিমা ছাড়াও এখানকার পারিপার্শ্বিক প্রকৃতির সৌন্দর্য অপার ৯ রূপকুন্ডু লেক, উত্তরাখন্ড ৫৬ দিন টানা ট্রেকিং করে রুপকুন্ডু লেকে পৌঁছাতে হয় তবে এই লোকটিকে অনেকে রহস্যময় বা মিস্ট্রি লেক বলেও ডেকে থাকেন কারণ এখানকার জল বছরের অর্ধেক দিন বরফ জমা থাকে কিন্তু যখন বরফ গলে জলে পরিণত হয় তখন লেকের ভেতর দৃশ্যমান হয় মানুষের হাড় ও কঙ্কাল যদিও এই রহস্যের পেছনে সাইন্টিফিক ব্যাখ্যা দিয়েছে বৈজ্ঞানিকরা তবুও রহস্য লেগেই রয়েছে লেককে ঘিরে ১০ ভেম্বানাদ লেক, কুমারাকম ভারতের তথা কেরালার সবচেয়ে লম্বা লেক হল কুমারাকম আশ্চর্যজনকভাবে এই লেগে বিশুদ্ধ আর নোনা জলের সংমিশ্রণে তৈরি যা নোনা জলের বেড় দিয়ে আলাদা করা ক্ষেত্রেই এখানে এক মনোরম ও শান্ত শ্রেষ্ঠ পরিবেশ তৈরি হয়েছে | bengali |
Effective today, the St. Patrick Federal Credit Union in Carleton is closed and its operations will be merged with the Monroe County Community Credit Union (MCCCU).
Mike Newman, president and chief executive officer for MCCCU, confirmed this morning that the longtime small credit union across from St. Patrick Catholic Church and Grade School is closing under a joint operating agreement. The accord has been in the works for the past month with St. Patrick and the National Credit Union Administration.
"Our people are merging the operations under this management agreement" with the board of directors, Mr. Newman told The Evening News Friday. "Due to economic conditions beyond their control, they found it very difficult to provide services to their members and decided they would be better off merging with another partner."
St. Patrick Credit Union held its last day of operations Friday. The credit union has been in business for more than 60 years and has about 700 members, some of whom are members of St. Patrick's parish, Mr. Newman said. MCCCU has operated for 58 years and has a membership of more than 31,000 people, he said.
"We are a much larger community-based credit union and can offer more personal services than they can," the president said. "Starting Monday, they will become full-fledged members and be able to use any of the five locations we have."
One Carleton woman who wished to remain anonymous said she withdrew her money from the credit union Thursday after hearing rumors about the business folding. She said she was second in line and about 30 people were waiting behind her to withdraw from their accounts. She said many of the people were angry about not being notified that the credit union was shutting its doors for good.
Mr. Newman said the reason members were not notified is because MCCCU was waiting for state approval of the joint operating agreement. He said that approval came Friday.
"We were waiting for approval from the state regulator of the pending merger," he explained. "Had we gotten that, they would have been notified much sooner."
He stressed that all deposits of members are insured by the federal government up to $250,000. He said the members' savings were insured and "never in jeopardy of losing their accounts." He called the closing a merger, not a liquidation or takeover.
"We went into this agreement voluntarily to serve the members of St. Patrick," he said. "We're much larger and able to serve them."
The Carleton woman said she was told by someone at the credit union that there was a "criminal investigation" under way. Mr. Newman declined to comment on the matter.
• 715 N. Telegraph Rd. (main office).
• 5044 N. Dixie Hwy. at Brest Rd. near Newport.
• 120 E. First St. (Monroe City Hall).
• 7408 Lewis Ave. at Sterns Rd., Temperance. | english |
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