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ఈ నెల 24న అమెరికా అధ్యక్షుడు జో బైడెన్ తో ప్రధాని మోడీ భేటీ PM Modi Biden meet On September 24th At White House అమెరికా అధ్యక్షుడు జో బైడెన్ తో ప్రధాని మోడీ ఈ నెల 24న భేటీ కానున్నారు. ఇరు దేశాలకు సంబంధించిన పలు ద్వైపాక్షిక అంశాల గురించి ఈ భేటీలో చర్చ జరుగనుంది.అన్ని కోవిడ్ అప్డేట్స్ గురించి తెలుసుకునేందుకు ఇక్కడ చదవండి ఈ భేటీకి సంబంధించిన విషయాన్ని వైట్ హౌజ్ అధికారులు కొద్దిసేపటి క్రితమే ప్రకటించారు. అయితే.. బైడెన్ కంటే ముందుగా.. ప్రధాని మోడీ వైట్ హౌజ్ లోని క్వాడ్ నేతలతో సమావేశం అవుతారు. వారితో ఆఫ్ఘనిస్తాన్ పరిణామాలు, ఇండో పసిఫిక్ సంబంధాలు, కరోనా వ్యాప్తి, పర్యావరణ మార్పులు వంటి తదితర అంశాలపై మాట్లాడనున్నారు. బైడెన్ కంటే ఒకరోజు ముందు అంటే సెప్టెంబర్ 23న మోడీ జపాన్, ఆస్ట్రేలియా ప్రధానులతో విడివిడిగా సమావేశం కానున్నారు.The post ఈ నెల 24న అమెరికా అధ్యక్షుడు జో బైడెన్ తో ప్రధాని మోడీ భేటీ first appeared on TNews Telugu. | telegu |
Digha high tide: নিম্নচাপকটালের যুগলবন্দি, জোয়ারের জলে বানভাসি দিঘা, ভাসল রাস্তা, হোটেল! নিম্নচাপ ও কটাল দুইয়ের জেরে বানভাসি দিঘা বঙ্গোপসাগরে ঝোড়ো হাওয়ার দাপটে দিঘার সমুদ্রের ঢেউ আছড়ে পড়ল তটসংলগ্ন রাস্তায় সৈকত সরণি পেরিয়ে সমুদ্রের নোনা জলে ডুবে গেল দিঘার রাস্তাঘাট জল ঢুকেছে অন্তত ৩৫টি হোটেলে প্রশাসন সূত্রে জানানো হয়েছে, বুধবার গুরুপূর্ণিমার দিন থেকেই সমুদ্রের চেহারা বদলেছে শুক্রবার তা আরও ভয়াবহ আকার নিয়েছে বৃহস্পতিবার থেকেই দিঘার সমুদ্রে নামার ক্ষেত্রে নিষেধাজ্ঞা জারি হয়েছে নজরদারির জন্য মোতায়েন রয়েছে নুলিয়া ও পুলিশ জোয়ারের সময় কেউ যাতে সমুদ্রে নেমে পড়তে না পারেন সে জন্য প্রতিটি ঘাট দড়ি দিয়ে ঘিরে রাখা হয়েছে জোয়ারের জলে এ ভাবে সৈকত শহর ভেসে যাওয়ার ঘটনা নজিরবিহীন বলেই মনে করছেন স্থানীয়েরা এর আগে ইয়াসের সময় একই ভাবে সমুদ্রের জল ঢুকে দিঘা শহর ভেসে গিয়েছিল তবে এই মুহূর্তে দিঘায় আসা পর্যটকরা সমুদ্রের এই বিশালাকায় ঢেউ দেখে নিজেদের উচ্ছ্বাস চেপে রাখতে পারেননি গত কয়েকটা দিন সমুদ্রের ভয়ানক চেহারায় জলে নামা না গেলেও সৈকত সরণিতে দাঁড়িয়ে বিশাল বিশাল ঢেউয়ে স্নান করতে দেখা গিয়েছে বহু পর্যটককেই অনেককেই সমুদ্রের ভয়ানক চেহারা ফ্রেমবন্দি করতে দেখা গিয়েছে সমুদ্রের জল রাস্তায় চলে আসায় অনেককে আবার সেই জলে নেমেই সাঁতার কেটেছেন দিঘা থানা জানিয়েছে, শুক্রবার পর্যন্ত পর্যটকদের সমুদ্রে নামায় নিষেধাজ্ঞা রয়েছে শনিবার থেকে পরিস্থিতি কিছুটা স্বাভাবিক হবে বলে মনে করা হচ্ছে তবে জোয়ারের সময় জলস্ফীতি হলে সতর্কতামূলক ব্যবস্থা নেওয়া হবে বলেও দিঘা থানা জানিয়েছে শুধু দিঘাই নয়, বঙ্গোপসাগরের জল ঢুকেছে তাজপুর, শঙ্করপুরসহ পূর্ব মেদিনীপুরের সমুদ্র তীরবর্তী একাধিক গ্রামে নষ্ট হয়েছে ফসল বিভিন্ন জায়গায় বাড়ি ভাঙারও খবর পাওয়া যাচ্ছে যে কোনও পরিস্থিতির জন্য প্রশাসন প্রস্তুত বলে রামনগর ব্লক প্রশাসন জানিয়েছে | bengali |
બ્રહ્માકુમારીઝ દ્વારા પીસ માર્ચ રેલી યોજાઇ આઝાદીના અમૃત મહોત્સવથી સ્વર્ણિમ ભારત કી ઓર અંતર્ગત બ્રહ્માકુમારીઝ દ્વારા આજે સવારે 6:1પ કલાકે 7પ શાંતિદૂતો દ્વારા સફેદ વસ્ત્ર ધારણ કરીને પીસ માર્ચ રેલીનું આયોજન કરવામાં આવ્યુ હતું. આ રેલી રેસકોર્ષ રોડ ખાતેથી શરૂ કરવામાં આવી હતી. રેલીનુ મુખ્ય લક્ષ્ય શાંતિદૂત બની શાંતિના પ્રકંપનો વાયબ્રેશન પ્રવાહિત કરવાનો, એકતાનો સંદેશ આપવાનો, પ્રકૃતિને શુધ્ધશાંતિના સંકલ્પોનુ દાન કરવાનું હતું. 7પ શાંતિદૂતોએ મળીને મૌનની તાકાતનો સંદેશ ફેલાવ્યો હતો. દરેક શાંતિદૂતોના હાથમાં હું શાંતિ દૂત છું, શાંતિની ધરોહર છું, મનમાં શાંતિ તો વિશ્ર્વમાં શાંતિ વગેરે સુવિચારોના બોર્ડ હતાં. બ્ર. કુ. ભગવતીબેન તથા બ્ર. કુ. અંજુબેને ઝંડી દ્વારા રેલીનો પ્રારંભ કરાવ્યો હતો ત્યારબાદ કિશાનપરા ચોક, એરપોર્ટ રોડ, બહુમાળી ભવન, જીલ્લા પંચાયત થઇ જાગનાથ સેવા કેન્દ્ર ખાતે રેલીનું સમાપન કરવામાં આવ્યુ હતું. આ રેલી દરમિયાન પ્રત્યેક શાંતિદૂતનાં મનમાં આ એક જ સંકલ્પ હતો કે નિગાહેં રખ આસમાન પે, ઘૂમાદે તુ રૂખ દરિયાકા... અર્થાત અંતર મનમાં પરમાત્માની યાદ હશે તો પરમાત્માનાં વાયબ્રેશન મેળવી પ્રત્યેક અસંભવ કાર્યને સંભવ બનાવવાની શકિત અને શાંતિ ફેલાવવાની શકિત પરમાત્મા પિતા પાસેથી મળશે. ઉપરોકત તસ્વીરો રેલીની છે. | gujurati |
કોરોનાનો રિકવરી રેટ ઘટીને ૯૮.૭૧ ટકા થઈ ગયો રાજ્યમાં કોરોના વાયરસના નવા કુલ ૮૯૪ કેસ નોંધાયા રાજ્યમાં આજે કોરોના કેસમાં આજે ફરી કેસમાં વધારો થયો છે. હાલ પણ રાજ્યમાં કોરોનાના ૮૯૪ કેસ નોંધાયા છે. જ્યારે કોરોના સંક્રમણથી ૬૯૧ દર્દીઓ સાજા થયા છે. ત્યારે રાહતની વાત એ છે કે રાજ્યમાં કોરોનાથી આજે એકપણ દર્દીનું મોત થયું નથી. આ સાથે કોરોનાનો રિકવરી રેટ ઘટીને ૯૮.૭૧ ટકા થઈ ગયો છે. રાજ્યમાં હાલ કુલ દર્દીઓની વિગતો જાેઈએ તો રાજ્યમાં હાલ ૫૦૯૯ એક્ટિવ કેસ છે. જેમાંથી ૮ દર્દી વેન્ટિલેટર પર છે. જ્યારે ૫૦૯૧ દર્દીઓ સ્ટેબલ છે. રાજ્યમાં કોરોનાને અત્યાર સુધીમાં કુલ ૧૨,૨૮,૯૫૫ દર્દીઓ મ્હાત આપી ચુક્યાં છે. જ્યારે રાજ્યમાં કોરોના વાયરસને કારણે કુલ ૧૦,૯૫૪ લોકોએ પોતાનો જીવ ગુમાવ્યો છે. જાેકે, જિલ્લા અને કોર્પોરેશન મુજબ કોરોના કેસની વાત કરવામાં આવે તો અમદાવાદ કોર્પોરેશનમાં સૌથી વધુ ૨૯૫ કેસ નોંધાયા છે, જ્યારે વડોદરા કોર્પોરેશનમાં ૬૫, સુરત કોર્પોરેશનમાં ૫૪, મહેસાણા ૪૫, પાટણ ૩૮, ગાંધીનગર ૩૧, રાજકોટ કોર્પોરેશન ૩૧, કચ્છ ૩૦, ગાંધીનગર કોર્પોરેશન ૨૭, સુરત ૨૪, દેવભૂમિ દ્વારકા ૨૩, ભાવનગર કોર્પોરેશન ૨૨, બનાસકાંઠા ૨૧, વલસાડ ૧૯, ભરૂચ ૧૮, રાજકોટ ૧૮, વડોદરા ૧૬, આણંદ ૧૫, મોરબી ૧૫, સાબરકાંઠા ૧૫, પોરબંદર ૧૦, અમદાવાદ ૮, જામનગર કોર્પોરેશન ૭, ખેડા ૭, નવસારી ૭, અમરેલી ૬, પંચમહાલ ૫, અરવલ્લી ૪, સુરેન્દ્રનગર ૪, તાપી ૪, ભાવનગર ૩, ગીર સોમનાથ ૩, દાહોદ ૨, જામનગર ૨ એમ કુલ ૮૯૪ કેસ નોધાયા છે. જાે રસીકરણની વાત કરવામાં આવે તો રાજ્યમાં આજે સાંજે ૫ વાગ્યા સુધીમાં કુલ ૧,૯૩,૦૭૪ લોકોને રસીના ડોઝ આપવામાં આવ્યા છે. રસીકરણના મોરચે પણ સરકાર મજબુતીથી લડી રહી છે. રાજ્યમાં ૧૮ વર્ષથી વધારેની ઉંમરના ૧૬૩૪ ને રસીનો પ્રથમ અને ૩૪૭૯ લોકોને રસીનો બીજાે ડોઝ આપવામાં આવ્યો છે. ૧૫૧૭ વર્ષના લોકો પૈકી ૧૧૭ ને રસીનો પ્રથમ અને ૭૯૪ ને રસીનો બીજાે ડોઝ આપવામાં આવ્યો છે. જ્યારે ૨૨૨૪૩ લોકોને પ્રીકોર્શન ડોઝ આપવામાં આવ્યો છે. ૧૨૧૪ વર્ષના લોકો પૈકી ૧૭૦૫ ને રસીનો પ્રથમ અને ૧૪૬૪ ને રસીનો બીજાે ડોઝ અપાયો હતો. ૧૮૫૯ વર્ષના લોકોને ૧૬૧૬૩૮ પ્રીકોશન ડોઝ આપવામાં આવ્યા છે. અત્યાર સુધીમાં કુલ ૧૧,૨૮,૩૨,૧૩૨ રસીના ડોઝ આપવામાં આવ્યો છે. | gujurati |
హీరో హీరోయిన్స్ మాత్రమే కాదు.. వీరు కూడా సినిమా లో హైలైట్!! అజయ్ భూపతి దర్శకత్వంలో తెరకెక్కిన మహా సముద్రం సినిమా ఈ రోజు ప్రేక్షకుల ముందుకు వచ్చింది. తొలి ఆటతోనే ఈ సినిమా సూపర్ హిట్ టాక్ ను తెచ్చుకోగా ప్రేక్షకులను ఎంతగానో అలరించే చేసిన సినిమాగా ఇది నిలిచిపోతుంది అని చెప్పడంలో ఎలాంటి సందేహం లేదు. సినిమాలో కథ చాలా బాగుంది దాని కంటే ఎక్కువగా నటీనటులు ఎమోషనల్ గా నటించి ప్రతి ఒక్క ప్రేక్షకుడుతో విజిల్స్ కొట్టించుకుంటున్నారు. ఏ సినిమాలో అయినా హీరో హీరోయిన్లకు మాత్రమే ఎక్కువగా పేరు వస్తూ ఉంటుంది. కానీ ఈ సినిమాలో చేసిన ప్రతి పాత్ర కూడా హీరో హీరోయిన్లకు సమానంగా పేరు దక్కించుకున్నారు. ఆ విధంగా ఈ సినిమాలో హీరో లు మరియు హీరోయిన్ ల తర్వాత రెండు పాత్రలకు ప్రేక్షకులు బాగా కనెక్ట్ అయ్యారు . అవే చుంచు మామ మరియు గూని బాబ్జీ పాత్రలు. ఈ రెండు పాత్రల్లో మహామహులు నటించడం వల్లే ఈ పాత్రలకు ఎంతో గుర్తింపు దక్కింది అని చెప్పవచ్చు. అజయ్ భూపతి కూడా ఈ పాత్రను డిజైన్ చేసే విషయంలో ఎంతో జాగ్రత్తగా తీసుకుని తనదైన మార్కును ఈ పాత్రల ద్వారా కూడా చూపించాడు. గూని బాబ్జి గా రావు రమేష్ జగపతిబాబు చుంచు మామ గా నటించగా వీరిద్దరూ తెలుగు పరిశ్రమ ఎప్పటికీ గుర్తుంచుకునే నటన ను ఈ సినిమా లో ప్రదర్శించారు. జగపతి బాబు విలన్ గా మారిన తర్వాత ఆయన రేంజ్ ఏ విధంగా మారిపోయిందో అందరికీ తెలిసిందే. సౌత్ లోనే కాదు నార్త్ లో కూడా ముఖ్యమైన పాత్ర చేయాలంటే తానే కేరాఫ్ అడ్రస్ అన్నట్లుగా ఆయన తన ఇమేజ్ మార్చుకుని ఇప్పటివరకు చాలా పాత్రల ద్వారా ప్రేక్షకులను అలరించారు. చుంచు మామ పాత్ర ఆయన కెరీర్లోనే ఒక బెస్ట్ పాత్ర అని చెప్పుకోవచ్చు. అలాగే గూని బాబ్జి గా రావు రమేష్ నటన అసాధారణంగా ఉందని చెప్పవచ్చు. ఈ రెండు పాత్రలలో వీరు తప్ప వేరే ఎవరు లేరు అన్న రీతిలో నటించి సినిమా హిట్టు ప్రధాన కారణం అయ్యారు. కళా మా తల్లి : మా లో ఆ సత్తా ఆయనకే ఉంది ..శ్రీ రెడ్డి !! ఏపీలో సకల శాఖల మంత్రి.. సర్వాంతర్యామి...! మహా సముద్రం కి అదే పెద్ద మైనస్ గా మారిందా?? మహాసముద్రం తో అజయ్ భూపతి మస్కా కొట్టాడే? తెలంగాణలో మరో ప్రయోగానికి అమీషా పూనుకున్నారా.. తర్వాత..? కృష్ణా జల వివాదంపై కేసీఆర్ మెలిక... గెజిట్ అమలవుతుందా...? 1,259 అకౌంట్స్ డిలీట్ చేసిన ఫేస్ బుక్... ఎందుకంటే ? వరల్డ్ సైట్ డే : కళ్లు ఆరోగ్యంగా ఉండాలంటే... దేవత సీరియల్ లో రుక్మిణిని చివరకు ఏంచేయనున్నారు? సోర్స్: ఇండియాహెరాల్డ్.కామ్ P.Nishanth Kumar | telegu |
This gently foaming shower gel and shampoo pampers sensitive skin of your little princesses with organic Aloe Vera, mallow extracts and hydrating castor oil. The enchanting scent of organic raspberry fascinates little ladies and makes bathtime especially pleasing.
Use Sante KIDS Shampoo and Shower Gel to cleanse your child’s hair and skin in the bath or the shower. Use on moistened skin and hair and gently massage it into the scalp or skin and then rinse off with plenty of warm water.
This sweet smelling shampoo&shower gel can also be used as a bubble bath - pour some shampoo under the running water and enjoy colourful bubbles. | english |
ಎಸಿಬಿಗೆ ಕಳಂಕಿತರನ್ನ ನಿಯೋಜನೆ ಮಾಡಬೇಡಿ: ಸರ್ಕಾರಕ್ಕೆ ಹೈಕೋರ್ಟ್ ನಿರ್ದೇಶನ ವರದಿ: ರಮೇಶ್.ಕೆಎಚ್, ಏಷ್ಯಾನೆಟ್ ಸುವರ್ಣನ್ಯೂಸ್ ಬೆಂಗಳೂರು ಜುಲೈ 11: ಎಸಿಬಿಗೆ ಕಳಂಕಿತ ಅಧಿಕಾರಿಗಳನ್ನ ನಿಯೋಜನೆ ಮಾಡಬೇಡಿ ಎಂದು ಡಿಸಿ ಕಚೇರಿಯಲ್ಲಿ ಭ್ರಷ್ಟಾಚಾರ ಪ್ರಕರಣದ ವಿಚಾರಣೆ ವೇಳೆ ಹೈಕೋರ್ಟ್ ಸರ್ಕಾರಕ್ಕೆ ಖಡಕ್ ನಿರ್ದೇಶನ ಮಾಡಿದೆ. ಎಸಿಬಿ ಭ್ರಷ್ಟಾಚಾರ ನಿಯಂತ್ರಣ ಮಾಡುವ ಸಂಸ್ಥೆಯಾಗಿದ್ದು, ಕಳಂಕ ಇರುವ ಅಧಿಕಾರಿಗಳನ್ನ ಅಲ್ಲಿಗೆ ವರ್ಗಾವಣೆ ಮಾಡಬೇಡಿ ಎಂದು ಮುಖ್ಯ ಕಾರ್ಯದರ್ಶಿ, ಡಿಪಿಎಆರ್ ಎಸಿಎಸ್ಗೆ ಹೈಕೋರ್ಟ್ ನಿರ್ದೇಶಿಸಿದೆ. ಎಸಿಬಿ ಎಡಿಜಿಪಿ ಸೀಮಂತ್ ಕುಮಾರ್ ಅವರು ತಮ್ಮ ವಿರುದ್ಧ ಯಾವುದೇ ಟೀಕೆ ಮಾಡದಂತೆ ಸುಪ್ರೀಂ ಕೋರ್ಟ್ಗೆ ಸಲ್ಲಿಸಿದ ಅರ್ಜಿ ಮಾಹಿತಿಯನ್ನ ಎಸಿಬಿ ಪರ ವಕೀಲರು ಮಾಹಿತಿ ನೀಡಿದರು.. ಅಧಿಕಾರಿಯ ವಿರುದ್ಧ ಯಾವುದೇ ವೈಯಕ್ತಿಯ ದ್ವೇಷ ಇಲ್ಲ ಎಂದು ಸ್ಪಷ್ಟಪಡಿಸಿದ ನ್ಯಾಯಮೂರ್ತಿಗಳು, ತಮಗೆ ಬಂದ ಬೆದರಿಕೆಯ ಬಗ್ಗೆಯೂ ಪ್ರಸ್ತಾಪಿಸಿದರು. ಮುಖ್ಯ ನ್ಯಾಯಾಮೂರ್ತಿ ಆಗಿದ್ದ ರಿತುರಾಜ್ ಆವಸ್ಥಿ ಅವರ ಬೀಳ್ಕೋಡುಗೆ ಸಮಾರಂಭದಲ್ಲಿ ನಾಯಾಮೂರ್ತಿಯೊಬ್ಬರು ಹೇಳಿದ ವಿಷಯ ಪ್ರಸ್ತಾಪಿಸಿದರು. ಜುಲೈ 1 ರಂದು ರಾತ್ರಿ ಭೋಜನದ ವೇಳೆ ಹಾಲಿ ನ್ಯಾಯಮೂರ್ತಿಯೊಬ್ಬರು ಬಂದು ನನ್ನ ಪಕ್ಕದಲ್ಲಿ ಕುಳಿತರು. ದೆಹಲಿಯಿಂದ ನನಗೆ ಒಂದು ಕರೆ ಬಂದಿದೆ, ಆ ವೇಳೆ ನಿಮ್ಮ ವಿಚಾರಿಸಿದ್ದರು ಎಂದು ಹೇಳಿದರು. ನಾನು ಯಾವ ಪಕ್ಷಕ್ಕೂ ಸೇರಿದವನನ್ನ ಎಂದು ಹೇಳಿದೆ. ಅಲ್ಲದೆ ಆ ಎಡಿಜಿಪಿ ಉತ್ತರ ಭಾರತದವರು ತುಂಬಾ ಪ್ರಭಾವಿ ಎಂದು ಹೇಳಿದ ಅವರು, ನ್ಯಾಯಮೂರ್ತಿಯೊಬ್ಬರ ವರ್ಗಾವಣೆ ವಿಚಾರವನ್ನೂ ಹೇಳಿದ್ರು ಎಂದು ನ್ಯಾ.ಹೆಚ್.ಪಿ ಸಂದೇಶ್ ಉಲ್ಲೇಖಿಸಿದರು. ಎಸಿಬಿ ಎಡಿಜಿಪಿ ACB ADGP ಸೀಮಂತ್ ಕುಮಾರ್ ವಿರುದ್ಧ ವಾಗ್ದಾಳಿ ನಡೆಸಿದ್ದ ಹೈಕೋರ್ಟ್ Karnataka High Court, ಅವರ ವಿರುದ್ಧದ ಸಿಬಿಐ ಪ್ರಕರಣದ cbi investigation ತನಿಖಾ ವರದಿಯನ್ನ ತರಿಸಿಕೊಂಡಿದೆ. ಇಂದು ಸಿಬಿಐ ಪರ ವಕೀಲ ಪ್ರಸನ್ನ ಕುಮಾರ್ ವರದಿ ಸಲ್ಲಿಸಿ ತನಿಖೆಯ ಪ್ರಗತಿ ಬಗ್ಗೆ ನ್ಯಾಯಪೀಠಕ್ಕೆ ವಿವರಣೆ ನೀಡಿದರು. ಇದನ್ನೂ ಓದಿ: Exclusive: ಏನಾಗ್ತಿದೆ ಎಸಿಬಿ ದಾಳಿಗಳು, ಎತ್ತ ಸಾಗುತ್ತಿದೆ ತನಿಖೆಗಳು..? ಅಸಲಿ ಕಹಾನಿ ಇದುಸೀಮಂತ್ ಕುಮಾರ್ ಪರವಾಗಿ ಯಾವುದೇ ಅರ್ಜಿ ಸಲ್ಲಿಸದೇ ವಾದಕ್ಕೆ ಮುಂದಾದ ಮಾಜಿ ಅಡ್ವೋಕೇಟ್ ಜನರಲ್ ಅಶೋಕ್ ಹಾರನಹಳ್ಳಿ ಅವರ ವಾದಕ್ಕೆ ನ್ಯಾಯಪೀಠ ಅವಕಾಶ ತಳ್ಳಿ ಹಾಕಿತು. ಅಧಿಕೃತವಾಗಿ ಅರ್ಜಿ ಸಲ್ಲಿಸುವುದು ಸರಿಯಲ್ಲ. ಅಲ್ಲದೆ ಎಸಿಬಿ ರದ್ದು ಮಾಡಿ ಲೋಕಾಯುಕ್ತಕ್ಕೆ ಅಧಿಕಾರ ನೀಡುವಂತೆ ವಾದ ಮಂಡಿಸುತ್ತಿರುವ ಅಶೋಕ್ ಹಾರನಹಳ್ಳಿ ಇಲ್ಲಿ ಎಸಿಬಿ ಎಡಿಜಿಪಿ ಪರ ವಾದ ಮಂಡನೆ ಮುಂದಾಗಿರುವುದು ವಿರೋಧಾಭಾಸ ಎಂದು ನ್ಯಾಯಮೂರ್ತಿ ಹೆ.ಪಿ.ಸಂದೇಶ್ ಕಾಲೆಳೆದರು. ಇದನ್ನೂ ಓದಿ: Bengaluru: ಭೂ ವಿವಾದ ಮುಕ್ತಾಯಗೊಳಿಸಲು ಲಂಚ: ಬಿಎಂಟಿಎಫ್ ಎಸ್ಐ ಬಲೆಗೆಸದ್ಯ ಎಸಿಬಿ ಆರಂಭವಾದಗಿನಿಂದಲೂ ದಾಖಲಾಗಿರುವ ಎಲ್ಲಾ ಪ್ರಕರಣಗಳ ಮಾಹಿತಿಯನ್ನ ಹೈಕೋರ್ಟ್ ಕೇಳಿದ್ದು ಮುಚ್ಚಿದ ಲಕೋಟೆಯಲ್ಲಿ ಮಾಹಿತಿ ಸಲ್ಲಿಸಲಾಯ್ತು. ಪ್ರಗತಿ ವರದಿಯನ್ನ ಪಡೆದು ಮುಂದಿನ ವಿಚಾರಣೆ ವೇಳೆ ಸಲ್ಲಿಸುವಂತೆ ಸೂಚಿಸಿ ವಿಚಾರಣೆಯನ್ನ ಜು.13ಕ್ಕೆ ಮುಂದೂಡಿದ್ದಾರೆ. | kannad |
Odisha Coronavirus Update: ओडिशा में कोरोना संक्रमण के 886 नए मामले, 8664 मरीज सक्रिय भुवनेश्वर, जागरण संवाददाता। ओडिशा में कोरोना संक्रमण के आज 886 नए मामले सामने आए हैं। इसमें 018 वर्षों 169 बच्चे शामिल हैं। 886 नए मामले में से 516 मरीज संगरोध से हैं जबकि 370 स्थानीय संपर्क में आने के बाद संक्रमित हुए हैं।यहां कोविड से संबधित सभी नए अपडेट पढ़ें प्रदेश में 8664 सक्रिय मामले हैं।राज्य सूचना एवं जनसंपर्क विभाग से मिली जानकारी के मुताबिक नए संक्रमित मरीजों में अनुगुल से 14, बालेश्वर जिले से 36 बरगढ़ से 10, भद्रक से 11, बलांगीर से 13, बौद्ध से 5, कटक से 40, देवगढ़ से 15, ढेंकनाल से 10, गजपति से 23, गंजम से 16, जगतसिंहपुर से 14, जाजपुर से 32, झारसुगुड़ा से 20, कालाहांडी से 12, कंधमाल से 25, केंद्रपाड़ा से 18, केन्दुझर से 10, खुर्दा से 86, कोरापुट से 66, मलकानगिरी से 3, मयूरभंज से 23, नवरंगपुर से 13, नयागढ़ से 20, नुआपाड़ा से 19, पुरी से 8, रायगड़ा से 22, संबलपुर से 179, सोनपुर से 25, सुंदरगढ़ से 79 तथा राज्य पूल से 19 नए मरीज सामने आए हैं।गौरतलब है कि राज्य के स्वास्थ्य और कल्याण विभाग ने मंगलवार को राज्य में कोरोना से और 22 लोगों की मृत्यु होने की जानकारी दी थी। इसी के साथ राज्य में अब कोरोना से मरने वालों की कुल संख्या 8,926 हो गई थी। जिन 22 कोविड मरीजों की मृत्यु की सूचना मिली है, उसमें छह बालेश्वर जिले से, पांच खुर्दा जिले से और दो भुवनेश्वर से थे।बालेश्वर जिले से 65 एवं 32 वर्षीय पुरुष तथा 75, 93, 64 तथा 73 वर्षीय महिला की मौत कोरोना से हुई। उसी तरह से भद्रक जिले की एक 60 वर्षीय महिला की मौत हुई। भुवनेश्वर से एक 76 वर्षीय महिला एवं एक 92 वर्षीय पुरुष की मृत्यु हुई। उसी तरह से कटक जिले की एक 60 वर्षीय महिला, 76 वर्षीय दो पुरुष, ढेंकनाल जिले से एक 73 वर्षीय एवं एक 66 वर्षीय पुरुष तथा एक 70 वर्षीय महिला की मौत हुई। इसके अलावा जाजपुर जिले का एक 38 वर्षीय पुरुष, खुर्दा जिले से एक 78, 82, वर्षीय पुरुष तथा एक 43 वर्षीय महिला की मौत हो गई है। मयूरभंज जिले से एक 28 तथा एक 70 वर्षीय पुरुष एवं नयागढ़ जिले का एक 45 वर्षीय पुरुष की मृत्यु हो गई है। | hindi |
چٹاگانگ سپورٹس ڈیسک تیجل اسلام کی تباہ کن بائولنگ کی بدولت بنگلہ دیش نے کالی ندھی کو پہلے کرکٹ ٹیسٹ میچ میں 64 رنز سے شکست دے کر دو میچوں کی سیریز میں کی برتری حاصل کر لی بنگلہ دیشی ٹیم میچ کے تیسرے روز اپنی دوسری اننگز میں 125 رنز بنا کر پویلین لوٹ گئی محمودا 31 رنز بنا کر نمایاں رہے دیوندرا بیشو نے اور روسٹن چیس نے وکٹیں لیں جواب میں ویسٹ انڈین ہدف کے تعاقب میں دوسری اننگز میں 139 رنز بنا کر ئوٹ ہو گئی مومن الحق کو عمدہ بلے بازی کا مظاہرہ کرنے پر میچ کے بہترین کھلاڑی کا ایوارڈ دیا گیا تیجل اسلام کی گھومتی ہوئی گیندوں کے سامنے کالی ندھی کا کوئی بھی کھلاڑی سنبھل کر نہ کھیل سکا کھلاڑی دوہرا ہندسہ بھی عبور نہ کر سکے تیجل اسلام نے وکٹیں لیں شکیب الحسن اور مہدی حسن میراز نے کھلاڑیوں کو ئوٹ کیا ہفتے کو چٹاگانگ میں کھیلے گئے پہلے ٹیسٹ میچ کے تیسرے روز بنگلہ دیشی ٹیم نے 55 رنز کھلاڑی ئوٹ پر دوسری ادھوری اننگز دوبارہ شروع کی تو پوری ٹیم 125 رنز بنا کر ئوٹ ہو گئی اور اس نے مہمان ویسٹ انڈین ٹیم کو فتح کیلئے مجموعی طور پر 204 رنز کا ہدف دیا مشفق الرحیم 19 مہدی حسن میراز 18 محمودا 31 نعیم حسن اور تیجل اسلام ایک رن بنا کر ئوٹ ہوئے بیشو نے چیس نے ویریکن نے اور گبرائل نے ایک وکٹ لی جواب میں ویسٹ انڈین ٹیم ہدف کے تعاقب میں دوسری اننگز میں 139 رنز بنا کر پویلین لوٹ گئی تیجل اسلام کی جادوئی سپن بائولنگ کے سامنے ویسٹ انڈین ٹیم کی بیٹنگ لائن اپ ریت کی دیوار ثابت ہوئی کوئی بھی کھلاڑی جم کر نہ کھیل سکا کھلاڑی دوہرا ہندسہ بھی عبور نہ کر سکے کپتان کریگ بریتھویٹ کیرن پاویل صفر شائی ہوپ ایمبرس 43 روسٹن چیس صفر شمرون ہٹمائر 27 شین ڈورچ بیشو کیمر روچ ایک اور ویریکن 41 رنز بنا کر ئوٹ ہو گئے تیجل اسلام نے | urdu |
परवन अकावद पेयजल स्कीम से जल्द मिलेगा शुद्ध पेयजल: सिंघवी जयपुर, 28 जनवरी हि.स.। परवन अकावद वृहद पेयजल परियोजना से छीपाबड़ौद तहसील के दस गावों में पेयजल स्कीम को स्वीकृति मिली है। छबड़ा विधायक व पूर्व मंत्री प्रताप सिंह सिंघवी ने बताया कि इस परियोजना से श्रीपुरा, देवरीमुंड, झनझनी, मंडोला, मातीपुराकलां, अमरपुरा, पीथपुर, पीपलहेड़ा, देवरीजोध, कुंभाखेड़ी, इत्यादी गांवों को लाभान्वित किया जाएगा। इसके अतिरिक्त ग्राम श्रीपुरा, तहसील छीपाबड़ौद को नियमित रूप से खण्ड़ छबड़ा द्वारा जल जीवन मिशन के अंतर्गत प्रस्तावित योजना से भी पीने के लिए पानी मिलता रहेगा, जिसकी प्रशासनिक एवं वित्तीय स्वीकृति एसएलएसएससी की 25 वीं बैठक में जारी की जा चुकी है।सिंघवी ने कहा कि बारां, कोटा और झालावाड़ जिले के 1402 गांवों, बारां शहर, बारां जिले के 907 गांव, कोटा जिले के 184 गांव और झालावाड़ जिले के 311 गांवों के परिवारों को सतही स्त्रोत परवन नदी निर्माणाधीन बांध से जल जीवन मिशन के अंतर्गत घरघर पेयजल नल कनेक्शन जारी कर लाभान्वित किया जाएगा। इस प्रस्तावित योजना के लिए एसएलएसएससी की 27वीं बैठक में राशि रु 3523.16 करोड़ जारी की जा चुकी है। इस परियोजना के अंतर्गत उक्त गांवों के निवासियों को शुद्ध पेयजल उपलब्ध कराया जाएगा। उन्होंने कहा कि क्षेत्र के लोग काफी लम्बे समय से पेयजल समस्या के समाधान की मांग कर रहे है। परवन अकावद वृहद पेयजल परियोजना स्वीकृत होने से निकट भविष्य में जल्द क्षेत्र लोगों को शुद्ध पेयजल पानी उपलब्ध कराया जाएगा।हिन्दुस्थान समाचारसंदीप ईश्वर | hindi |
चेर्नोबिल न्यूक्लियर प्लांट पर रूस का कब्जा, हादसे पर बन चुकी है सीरीज चेर्नोबिल न्यूक्लियर प्लांट, रूस का कब्जा, हादसे ,सीरीज,रूस ,Chernobyl Nuclear Plant, Russia Captured, Accident, Series, Russia, यूक्रेन Ukraine पर रूस Russia का हमला जारी है.Click here to get the latest updates on Ukraine Russia conflict इस बीच खबर आई है कि रूस की सेना Russian Army ने चेर्नोबिल न्यूक्लियर प्लांट Chernobyl Nuclear Plant पर कब्जा कर लिया है. इसकी जानकारी यूक्रेन के प्रधानमंत्री Denys Shmygal ने दी. ये वही न्यूक्लियर प्लांट है जहां आज से 36 साल पहले परमाणु रिसाव की वजह से भयंकर तबाही हुई थी.रूसी सेना को नहीं रोक पाया यूक्रेनयूक्रेन के प्रधानमंत्री Denys Shmygal ने कहा कि चेर्नोबिल न्यूक्लियर प्लांट Chernobyl Nuclear Plant और उसके आसपास के एक्सक्लूशन जोन पर रूसी सेना ने कब्जा कर लिया है.चेर्नोबिल न्यूक्लियर प्लांट में हादसे पर बन चुकी है सीरीजबता दें कि चेर्नोबिल न्यूक्लियर प्लांट में 26 अप्रैल 1986 को भयंकर हादसा हुआ था. चेर्नोबिल न्यूक्लियर प्लांट में रिसाव हुआ था. जान लें कि चेर्नोबिल न्यूक्लियर प्लांट की इस भयंकर घटना पर एक सीरीज भी बन चुकी है. ये सीरीज साल 2019 में बनी थी. जान लें कि चेर्नोबिल न्यूक्लियर प्लांट में रिसाव का प्रभाव रूस, बेलारूस और यूरोप के कई देशों पर हुआ था.कैसे देख सकते हैं चेर्नोबिल सीरीज?जान लें कि चेर्नोबिल न्यूक्लियर प्लांट में हुए हादसे पर बनी सीरीज को आप हॉटस्टार पर देख सकते हैं. इस सीरीज का नाम चेर्नोबिल है. इस सीरीज में 5 एपिसोड हैं. इस सीरीज को Craig Mazin बनाया है. चेर्नोबिल सीरीज में Jessie Buckley, Stellan Skarsgård, Jared Harris, Adam Nagaiti और Emily Watson ने काम किया है.न्यूक्लियर प्लांट में कैसे हुआ था हादसा?गौरतलब है कि चेर्नोबिल न्यूक्लियर प्लांट में हादसा परमाणु रिएक्टर के स्टीम टर्बाइन में सेफ्टी टेस्ट के दौरान हुआ था. ऑपरेटर की लापरवाही की वजह से रिएक्टर में विस्फोट होना शुरू हो गया था. इसके बाद मंजर तबाही में बदल गया था. इस हादसे में हजारों लोगों की जान चली गई थी. | hindi |
New research reveals that high-intensity interval training (HIIT) increases glucose metabolism in muscles as well as insulin sensitivity in type 2 diabetes. Already after a two-week training period, the glucose uptake in thigh muscles returned to a normal level.
The discovery was made in a research project led by Senior Research Fellow Kari Kalliokoski and Project Manager Jarna Hannukainen at the University of Turku, Finland. The project studied the health impacts of high-intensity interval training on healthy people and diabetics, and the results are encouraging.
- HIIT has a rapid impact on metabolism. However, no great differences have been demonstrated between the impact of HIIT and moderate intensity continuous training over a longer period of time. The main benefit of high-intensity interval training is mostly that it takes less time, says Doctoral Candidate Tanja Sjöros.
First in the study, healthy men in their forties and fifties did either high-intensity interval training or traditional, moderate intensity training. Later, a group of people with insulin resistance carried out a similar two-week training routine. Some of them had type 2 diabetes and some prediabetes, i.e. their blood sugar levels were elevated but not yet high enough to indicate type 2 diabetes.
- Before the training started, the glucose metabolism and insulin sensitivity of the insulin resistant persons were significantly reduced compared to the group of healthy individuals. However, already after two weeks of high intensity training, which amounted to six training sessions, the glucose metabolism in the thigh muscles achieved the starting level of the healthy control group, tells Sjöros.
In HIIT, the training sessions are highly intensive but short and followed by recovery period. For example, HIIT can be carried out in 30-second training sessions of maximum intensity and with a recovery sessions of a couple of minutes.
Glucose metabolism and insulin sensitivity improved after both the high-intensity training and the moderate intensity continuous training, so the study suggests that people can choose the type of training based on their own preferences.
- However, the group that did moderate intensity training achieved only half of the improvement experienced by the HIIT group during the two-week period. Therefore, this type of training requires a longer period of time. If you have only little time to spare, high-interval training could be a great alternative to traditional training that requires more time but is lower in intensity, says Sjöros.
HIIT also improves endurance. In the study, the endurance of type 2 diabetics increased only in the HIIT group, but earlier studies have shown that, when the training routine continues for over two weeks, endurance increases with the traditional, moderate intensity training just as much as it does with high-interval training.
The research results published in the Scandinavian Journal of Medicine & Science in Sports highlight the beneficial effects of exercise on glucose metabolism especially in diabetics and in those who suffer from disturbances in the glucose metabolism. According to previous research, exercise lowers blood sugar as much as diabetes medication. Therefore, exercise is an essential part of treating and preventing diabetes.
- It's particularly good news that when it comes to the glucose metabolism and endurance it does not seem to matter in whether the exercise takes place over a longer period of time as moderate training or over a short period as high-interval training. Everyone can choose the type of training that suits them best. In general, you can achieve the best results for you body by using both training methods, encourages Sjöros.
However, the researchers advise that diabetics should consult their doctor before starting a new exercise routine. For example, if the amount of exercise increases significantly, it might be necessary to check the diabetes medication. Also other possible illnesses have to be kept in mind when planning a new exercise routine. | english |
বড় খবর : মুখ্যমন্ত্রী নীতীশ কুমারের সুরক্ষায় ফাঁক ! মাত্র ১৫১৮ ফুট দূরে ফাটলো বোম প্রথম কলকাতা অনুষ্ঠানস্থলে উপস্থিত ছিলেন স্বয়ং বিহারের মুখ্যমন্ত্রী মাত্র ১৫ থেকে ১৮ ফুট দূরে ঘটল বিস্ফোরণ এই মুহূর্তে সব থেকে বড় খবর হলো এটি চারিদিকে কড়া পুলিশি পাহারা তার উপর সমস্ত রকম যথাযথ ব্যবস্থাপনার মাঝেও এই ধরনের ঘটনা কেমন ভাবে ঘটল ? বিষয়টি নিয়ে জোর তদন্ত চলছে এই ঘটনায় গ্রেপ্তার হয়েছেন একজন বিহারের নালন্দার সিলাবে গান্ধী হাইস্কুলে মুখ্যমন্ত্রী নীতীশ কুমারের গণসংলাপ অনুষ্ঠানের সময় বিস্ফোরণটি ঘটে অনুষ্ঠানস্থল থেকে মাত্র ১৫১৮ ফুট দূরে বিস্ফোরণটি ঘটে বিস্ফোরণের পর ঘটনাস্থলে বিশৃঙ্খলা দেখা দেয় বর্তমানে কোন হতাহতের খবর নেই এ ঘটনায় একজনকে আটক করেছে পুলিশ আরও পড়ুন : দুদিন পর তৃণমূল মানুষের বেঁচে থাকার অধিকার কেড়ে নেবে, দাবি শমীকের মঙ্গলবারই প্রথম পাওয়াপুরিতে যান মুখ্যমন্ত্রী নীতীশ কুমার সেখান থেকে সিলাব হয়ে রাজগীর যাওয়ার কথা ছিল বোমা বিস্ফোরণের ঘটনাটি ঘটেছে সিলাব গান্ধী হাইস্কুলে তিনি প্যান্ডেলে বসে প্রায় ২৫০ মানুষের সঙ্গে দেখা করে তাদের আবেদন শুনছিলেন সে সময় হঠাত্ করেই মঞ্চের প্যান্ডেলের পিছন থেকে বিস্ফোরণের জোর আওয়াজ আসে বিস্ফোরণের শব্দে দ্রুত মঞ্চে থাকা ব্যক্তিরা ওখান থেকে সরে আসেন মুখ্যমন্ত্রীর জনসভার সময় এই ধরনের বিস্ফোরণ ঘটায় অবাক হয়েছেন বহু মানুষ পরে জানা যায় মঞ্চের পেছনে এক যুবক বাজি ফাটিয়েছেন ওই যুবককে পুলিশ গ্রেফতার করেছে ওই যুবক ইসলামপুর ব্লকের বাসিন্দা মুখ্যমন্ত্রী নীতীশ কুমারের নালন্দায় মোট চারটি গণ সংলাপের কর্মসূচি ছিল যার কারণে তিনি সিলাবে পৌঁছান সেখানেই জোর শব্দের মঞ্চের পেছনে পটকা ফাটান এক যুবক, অথচ আওয়াজ হয়েছে প্রায় ছোটখাটো বিস্ফোরণের মত বিষয়টি নিয়ে খতিয়ে দেখছে পুলিশ | bengali |
આ 3 કંપનીઓના શેરે 3 સેશનમાં આપ્યું મજબૂત વળતર, ખરીદો, વેચો કે પકડી રાખો, જાણો શું કહે છે નિષ્ણાતો છેલ્લા 3 સેશનમાં શેરબજારમાં ઘણી વોલેટિલિટી જોવા મળી છે. આ સમયગાળા દરમિયાન, મિડ કેપ અને લાર્જ કેપ શેરોમાં ત્રણ કંપનીઓના શેરોએ તેમના રોકાણકારોને મજબૂત વળતર આપ્યું છે. જેકે લક્ષ્મી સિમેન્ટ, અદાણી વિલ્મર અને સ્ટાર હેલ્થે ત્રણ સત્રોમાં તેમના રોકાણકારોને 13.17 થી 19.41 ટકા વળતર આપ્યું છે. જોકે સોમવારે NSE પર JK લક્ષ્મી સિમેન્ટ 4.94 ટકાના ઘટાડા સાથે બંધ થયું હતું, પરંતુ છેલ્લા 3 સત્રોમાં મજબૂત વળતર આપવાના સંદર્ભમાં તે નંબર વન છે. આ સમયગાળામાં જેકે લક્ષ્મી સિમેન્ટના શેરમાં 19.41 ટકાનો ઉછાળો આવ્યો છે. જેકે લક્ષ્મી સિમેન્ટનું માર્કેટ કેપ 5522.84 કરોડ છે અને તેના શેર સોમવારે નુકસાન સાથે રૂ. 469.35 પર બંધ થયા છે. છેલ્લા એક સપ્તાહના તેના પ્રદર્શન પર નજર કરીએ તો તેણે 19.22 ટકાની વૃદ્ધિ હાંસલ કરી છે. જોકે, છેલ્લા એક વર્ષમાં તેણે તેના રોકાણકારોને લગભગ 5 ટકાનું નુકસાન કર્યું છે. તેની 52 સપ્તાહની ઊંચી સપાટી 815 અને નીચી 366.25 રૂપિયા છે. આ પણ વાંચો: વ્હાઇટ હાઉસના રાષ્ટ્રીય સુરક્ષા સલાહકારે પુષ્ટિ કરી કે તાઇવાન નવા ઇન્ડોપેસિફિક ડીલમાં નથી આ પણ વાંચો: વ્હાઇટ હાઉસના રાષ્ટ્રીય સુરક્ષા સલાહકારે પુષ્ટિ કરી કે તાઇવાન નવા ઇન્ડોપેસિફિક ડીલમાં નથી અદાણી વિલ્મર તેના લિસ્ટિંગથી તેના રોકાણકારોને સમૃદ્ધ બનાવી રહી છે, પરંતુ આ સત્રોમાં તેણે 15.72 ટકાનો ઉછાળો આપ્યો છે. સોમવારે અદાણી વિલ્મરમાં અપર સર્કિટ હતી અને તે 736.60 રૂપિયા પર પહોંચી ગયો છે. અદાણી વિલ્મરનું માર્કેટ કેપ રૂ. 95734.33 કરોડ છે અને છેલ્લા એક સપ્તાહમાં શેરમાં 27.57 ટકાનો ઉછાળો આવ્યો છે. જો કે, છેલ્લા કેટલાય સપ્તાહોથી બજારમાં આવેલા ઘટાડાથી આ શેરે એક મહિનામાં માત્ર 1.1 ટકાનું જ વળતર આપ્યું છે. જ્યારે 3 મહિનામાં તેમાં 113.48 ટકાનો વધારો થયો છે. અદાણી વિલ્મરનો 52 સપ્તાહનો હાઈ રૂ. 878 અને નીચો રૂ. 227 છે. બજારના મોટાભાગના નિષ્ણાતો તેમાં રહેવાની સલાહ આપી રહ્યા છે. | gujurati |
Apple is donating $2 million to two human rights groups as part of CEO Tim Cook’s pledge to help lead the fight against the hate that fueled the violence in Virginia during a white-nationalist rally last weekend.
Cook made the commitment late Wednesday in an internal memo obtained by The Associated Press.
Cook also told Apple employees in the memo that he strongly disagrees with President Donald Trump’s attempts to draw comparisons between the actions of the white nationalists and protesters opposing them. Cook believes equating the two “runs counter to our ideals as Americans,” making him the latest prominent CEO to distance himself from Trump’s remarks in the wake of the violence in Charlottesville, which left a woman dead and more than a dozen injured.
Apple is giving $1 million apiece to Southern Poverty Law Center and the Anti-Defamation League. It will also match employee donations to those two groups and other human rights organizations on a two-for-one basis.
Meanwhile, American Airlines will donate $150,000 to Habitat for Humanity of Greater Charlottesville, airline spokesman Matt Miller said. | english |
રણબીર કપુરને સાસુજી તરફથી મળી મોંઘીદાટ ૨.૫૦ કરોડની વોચઃ સાળીઓએ બુટ ચોરવાના માગ્યા ૧૧.૫ કરોડ મુંબઇ, તા.૧૬: રણબીર કપૂર અને આલિયા ભટ્ટના લગ્ન થઈ ગયા છે, પરંતુ લગ્ન પછી લોકો આ ખાસ દિવસે બંનેની તસવીરો અને લગ્નમાં ભજવવામાં આવેલા રિવાજો વિશે જાણવા માગે છે. રણબીરઆલિયા બંનેએ કોઈ પણ ઢોંગ વગર, માયાનગરી મુંબઈમાં જ પરિવાર અને અમુક પસંદગીના મિત્રો સાથે ખૂબ જ સાદગી સાથે તેમના લગ્નને ખાસ બનાવ્યા. લગ્ન બાદ ફેન્સ લગ્નના વીડિયો અને તસવીરો જોઈને ખુશ થઈ ગયા છે. આલિયાના મંગલસૂત્ર અને વીંટી આલિયા ભટ્ટ મંગલસૂત્ર અને લગ્નની વીંટી પછી હવે રણબીરને તેના સાસરિયાઓ તરફથી ખાસ ભેટો મળી છે અને પુત્રવધૂના જૂતાની ભારે માંગ કરવામાં આવી છે. જયારે આલિયાને કપૂર પરિવાર તરફથી અનોખી હીરાની વીંટી મળી હતી, ત્યારે આલિયાએ રણબીર કપૂરને બેન્ડ પહેરાવ્યું હતું. આ સાથે વરરાજાને સાસુ સોની રાઝદાન તરફથી એક ખાસ ભેટ મળી છે, જેની કિંમત ૨.૫ કરોડ રૂપિયા જણાવવામાં આવી રહી છે. સોની રાઝદાને તેના જમાઈ રણબીર કપૂરને ખૂબ જ મોંઘી ઘડિયાળ ભેટમાં આપી છે જે સરળતાથી ઉપલબ્ધ નથી. લગ્ન હોવા જોઈએ અને ચંપલ ચોરવાની વિધિ ન હોવી જોઈએ, આવું કેવી રીતે થઈ શકે. બોલિવૂડના આ બહુપ્રતિક્ષિત લગ્નમાં ચંપલ ચોરવાની વિધિ કરવામાં આવી અને જૂતાના બદલામાં રણબીરની ભાભીએ કરી એવી માંગ, જેને સાંભળીને તમે પણ હોંશમાં આવી જશો. આલિયા ભટ્ટની ગર્લ ગેંગે તેના સાળા પાસેથી જૂતાની ચોરી કરવા બદલ ૧૧.૫ કરોડ રૂપિયાની માંગણી કરી હતી. પૈસાને લઈને લાંબી લડાઈ અને ઝઘડા પછી આખરે તેને ૧ લાખ રૂપિયાનું પરબિડીયું આપવામાં આવ્યું. રિવાજ મુજબ લગ્નમાં આવનાર મહેમાનોને રિટર્ન ગિફટ આપવાની હતી. આ માટે ભટ્ટ પરિવાર દ્વારા પણ ખાસ તૈયારીઓ કરવામાં આવી હતી, જે આલિયાએ પોતે કરી હતી. રિટર્ન ગિફટ તરીકે, મહેમાનોને ખૂબ જ ખાસ કાશ્મીરી શાલ આપવામાં આવી હતી, જે કન્યા આલિયા ભટ્ટે પોતે પસંદ કરી હતી. આ શાલોનું ફેબ્રિક ખૂબ જ ખાસ હતું અને દરેક લોકો તેનાથી ખૂબ જ પ્રભાવિત થયા હતા. | gujurati |
ക്രയോജനിക് ഘട്ടം പാളി ഇ.ഒ.എസ്.03 വിക്ഷേപണം പരാജയം ബംഗളുരു: ഇന്ത്യയുടെ ഭൗമനിരീക്ഷണ ഉപഗ്രഹം ഇ.ഒ.എസ്.03 വിക്ഷേപണം പരാജയപ്പെട്ടു. ആകാശത്തേക്കു കുതിച്ചുയര്ന്നു 340 സെക്കന്ഡുകള്ക്കുശേഷമായിരുന്നു പരാജയം. പലതവണ മാറ്റിവച്ച വിക്ഷേപണം ആദ്യം നിശ്ചയിച്ചതില്നിന്നു ആറു മാസത്തിനു ശേഷമാണു നടത്തിയത്. ഇന്നലെ പുലര്ച്ചെ 5.43 നാണു ശ്രീഹരിക്കോട്ടയില്നിന്നു ജി.എസ്.എല്.വി. എഫ്10 റോക്കറ്റില് ഉപഗ്രഹം ബഹിരാകാശത്തേക്കു കുതിച്ചത്. 52 മീറ്റര് നീളമുള്ള റോക്കറ്റിന്റെ ആദ്യ രണ്ട് ഘട്ടങ്ങള് വിജയമായിരുന്നു. അവസാന ഘട്ടത്തില് ക്രയോജനിക് എന്ജിന്ലിക്വിഡ് ഹൈഡ്രജനം ഓക്സിജനും ഇന്ധമായി ഉപയോഗിക്കുന്ന പ്രവര്ത്തിപ്പിച്ചപ്പോഴാണു പാളിച്ച സംഭവിച്ചത്. മുന് നിശ്ചയിച്ച ദിശയില്നിന്നു റോക്കറ്റ് മാറിപ്പോകുകയായിരുന്നു. ഉപഗ്രഹത്തിന്റെ അവശിഷ്ടങ്ങള് ബംഗാള് ഉള്ക്കടലില് പതിച്ചു. ദിവസവും നാലോ അഞ്ചോ തവണ രാജ്യത്തിന്റെ സമഗ്രവും വ്യക്തവുമായ ചിത്രങ്ങള് പകര്ത്തുകയായിരുന്നുഇ.ഒ.എസ്.03യുടെ പ്രധാന ദൗത്യം. പ്രകൃതി ദുരന്തം, കാലാവസ്ഥാ വ്യതിയാനം എന്നിവയെക്കുറിച്ച് കൃത്യമായ മുന്നറിയിപ്പ് നല്കുക എന്നിവയും ദൗത്യങ്ങളായിരുന്നു. | malyali |
justinandkent, private room! Enjoy watching justinandkent video as is totally FREE! Anyhow, to live with justinandkent, view justinandkent private. | english |
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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<title>All Classes</title>
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<h1 class="bar">All Classes</h1>
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<li><a href="toTest/Main.html" title="class in toTest">Main</a></li>
<li><a href="toTest/MathLib.html" title="class in toTest">MathLib</a></li>
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| code |
जर्नलिस्ट नहीं सिर्फ चमचागिरी के है लायक ऋद्धिमान साहा को धमकी मिलने पर भड़के वीरेंद्र सहवाग श्रीलंका के खिलाफ टेस्ट टीम से दिग्गज विकेटकीपर बल्लेबाज ऋद्धिमान साहा Wridhiman Saha समेत 4 सीनियर प्लेयर को बाहर कर दिया गया है. भारतीय टीम से बाहर होने वाले विकेटकीपरबल्लेबाज ऋद्धिमान साहा को अब धमकियां भी मिलनी शुरू हो गई है. ये धमकी उन्हें एक जर्नलिस्ट से वॉट्सऐप पर मिली है, जिसका स्क्रीनशॉट उन्होंने सोशल मीडिया Social Media पर शेयर किया है. इस मामले में पूर्व भारतीय सलामी बल्लेबाज वीरेंद्र सहवाग Virendra Sehwag ने हैरानी जताते हुए साहा का समर्थन कर जर्नलिस्ट की क्लास लगाई है. साहा के समर्थन में सहवाग का ट्वीट दरअसल, सीनियर विकेटकीपर बल्लेबाज साहा Wridhiman Saha पर पत्रकार इंटरव्यू के लिए दबाव डाल रहा था. उनके रिप्लाई ना करने पर अभद्रता से पेश आते हुए धमकी बी दी. साहा ने ऐसा कर बस अपनी बात रखी है और साथ ही पत्रकारिता पर भी सवाल उठाए और उस पत्रकार के साथ हुई बातचीत का स्क्रीनशॉट सोशल मीडिया पर शेयर किया. इसके बाद साहा के समर्थन में उतरते हुए पूर्व विस्फोटक बल्लेबाज वीरेंद्र सहवाग ने साहा के ट्वीट को शेयर करते हुए ट्विटर पर रिपोर्टर की जमकर फटकार लगाई है. उन्होंने ट्वीट कर कहा ये बेहद दुख की बात है. इस तरह से किसी की भावनाओं को आहत करना. ना तो वह सम्मान के लायक है और ना ही वह जर्नलिस्ट है. सिर्फ चमचागिरी। हम आपके साथ हैं ऋद्धि. जर्नलिस्ट ने दिया था ऐसे धमकी दरअसल, विकेटकीपर बल्लेबाज ऋद्धिमान साहा ने जो स्क्रीनशॉट शेयर किया है, उसमें जर्नलिस्ट उनसे कहता है, मेरे साथ एक इंटरव्यू करोगे. यह अच्छा होगा. उन्होंने सेलेक्टर्स केवल एक ही विकेटकीपर चुना. कौन बेस्ट है. तुमने 11 जर्नलिस्ट को चुनने की कोशिश की, जोकि मेरे हिसाब से सही नहीं है. उसे चुनो जो ज्यादा मदद कर सके. तुमने कॉल नहीं किया. मैं अब तुम्हारा कभी इंटरव्यू नहीं लूंगा और मैं इसे याद रखूंगा. बता दें कि दिग्गज विकेटकीपर बल्लेबाज ऋद्धिमान साहा ने ट्विटर पर लिखा, भारतीय क्रिकेट में मेरे सभी योगदानों के बाद..एक तथाकथित सम्मानित जर्नलिस्ट से मुझे इस तरह की चीजों का सामना करना पड़ रहा है! यहीं से जर्नलिज्म खत्म. द्रविड़गांगुली पर साहा ने लगाए थे गंभीर आरोप गौरतलब है कि चयनकर्ताओं ने श्रीलंका के खिलाफ टेस्ट सीरीज के लिए भारतीय टीम का चयन किया जिसमें ऋद्धिमान साहा समेत 4 सीनियर किलाड़ियों को टीम से बाहर का रास्ता दिखा दिया गया. इसके बाद साहा ने कोच राहुल द्रविड़ पर गंभीर आरोप लगाए साथ ही बीसीसीआई अध्यक्ष सौरव गांगुली पर भी निशाना साधा है. गर्दन की दर्द से जूझते हुए 61 रन की पारी खेली थी. इसके बाद बीसीसीआई अध्यक्ष सौरव गांगुली ने भी सराहना की थी. उन्होंने बताया कि, न्यूजीलैंड के खिलाफ 61 रन वाली नाबाद पारी के बाद दादी गांगुली ने मुझे व्हाट्स एप मैसेज कर मुबारकबाद दी. और लिखा कि जब तक मैं BCCI में हूं, तुम टीम में हो. BCCI अध्यक्ष से इतनी बड़ी बात सुनने के बाद मेरा आत्मविश्वास और बढ़ गया था. लेकिन अब मैं ये समझ नहीं पा रहा हूं कि सबकुछ अचानक कैसे बदल गया? | hindi |
Hyd : భర్త ఆగడాలపై భార్య ఫిర్యాదు... జూబ్లీహిల్స్ హైదరాబాద్ : కులం పేరుతో దూషణ.. అర్థనగ్నంగా ఉండాలని.. మూత్రం తాగాలని బలవంతపెడుతున్నాడంటూ.. భర్తపై భార్య జూబ్లీహిల్స్ పోలీసులకు ఫిర్యాదు చేసింది. పోలీసుల కథనం మేరకు... నారాయణపేట మక్తల్కు చెందిన మహిళ రహమత్నగర్లో నివసిస్తున్నారు. 2016 లో ఆమెకు ఓ యువకుడితో ప్రేమ వివాహమయ్యింది. గర్భం దాల్చినా బలవంతంగా గర్భస్రావం చేయించారు. 2020 లో భర్త సోదరుడు, సోదరి, బావ ఆమెను కులం పేరుతో దూషించేవారు. చాలాసార్లు పెట్రోల్ పోసి చంపేస్తామంటూ బెదిరించేవారు. భర్తకు మహిళ రూ.1.50 లక్షలు ఇచ్చింది. ఆ తర్వాత కూడా వేధింపులు ఆపలేదు. అర్ధనగంగా కూర్చోవాలని, మూత్రం తాగాలని భర్త బలవంతం చేసేవాడు. భర్త, అతడి కుటుంబ సభ్యుల వేధింపులు తట్టుకోలేక మహిళ జూబ్లీహిల్స్ పోలీసులను ఆశ్రయించింది. నిందితులపై పోలీసులు ఎస్సీ, ఎస్టీ అట్రాసిటీ కేసులు నమోదు చేశారు. | telegu |
# -*- coding: utf-8 -*-
import pygame
__author__ = 'fyabc'
class MySprite(pygame.sprite.Sprite):
def __init__(self, *groups):
super(MySprite, self).__init__(*groups)
def draw(self, surface):
surface.blit(self.image, self.rect)
class MyGroup(pygame.sprite.Group):
def __init__(self, *sprites):
super(MyGroup, self).__init__(*sprites)
def draw(self, surface):
sprites = self.sprites()
for sprite in sprites:
sprite.draw(surface)
self.lostsprites = []
| code |
\begin{document}
\title{Comments on: "Impossibility of the existence of the universal density functional" V.B.Vobrov, S.A. Trigger. arXiv: 1012.3241 v1. Dec 2010}
\begin{small}
$\S$ email: [email protected] "Higher Institute of Technology and Applied Sciences, Havana, Cuba"
\end{small}
The paper is based in the original formulation of DFT presented by Hohenberg and Kohn\cite{Hohenberg}. In that paper the authors dealt only with systems which ground states are non-degenerated, using this condition in order to obtain most of the outcomes presented in the cited work. As a consequence, some results are either not presented or derived in a general form. For example, they defined the functional $F_{HK}[\rho]$ as $F_{HK}[\rho]=\langle\psi_{o}[\rho]|T+V|\psi_{o}[\rho]\rangle$, as it is also done in the first part of the expression (4), been this an expression that is only valid for non-degenerated ground states. The general form of the $F_{HK}[\rho]$ functional is the one presented in the second part of expression (4):
$$F_{HK}[\rho]=E_{o}(\{\rho_{o}\})-\int\varphi^{ext}(r)\rho(r)dr$$
been this the form that can be used for any ground state.
It is true that Hohemberg and Kohn assumed, without any demonstration, that the above mentioned functional is universal, in the sense that it is independent of the external potential \cite{Hohenberg}. It is also certain that in many derivations of the variational principle based on the electronic density, that can be found in the literature, the universality of the $F_{HK}[\rho]$ is used without no previous demonstration \cite{Koch,Parr}. But, the essential point here, is that the universality of the functional $F_{HK}[\rho]$ is not necessary in order to construct such variational principle. In the book "The Fundamentals of Density Functional Theory" \cite{Eschrig}, the author demonstrated that, no matter the functional $F_{HK}[\rho]$ is universal or not, a variational principle based in the electronic density can be rigorously stated.
Nevertheless, we have the fundamental question about the universality of the $F_{HK}[\rho]$ . This is a very important question, not only form the theoretical point of view, but also for practical reasons, because, in the hypothetical case that $F_{HK}[\rho]$ was not universal, then we are obliged to find a different functional for each concrete problem (i.e. external potential). Fortunately, the answer to this crucial question was given by Levy and Lieb \cite{Levy,Lieb}. In order to solve the v-representability problem of $F_{HK}[\rho]$, these authors defined a new functional:$F_{LL}[\rho]=inf_{\psi\rightarrow\rho}\langle\psi|T+V|\psi\rangle$, here the infimum search is over all N-particle wave functions (not only ground states) yielding a given density $\rho$. In this way, besides having a functional whose domain of definition is the set of all the N-representable densities, it is obtained a universal functional because in its definition it wasn't made any reference to the external potential. So, the question of the universality of the functional $F_{HK}[\rho]$ is answered through its extension, $F_{LL}[\rho]$ , over all N-representable densities.
\addcontentsline{toc}{section}{References}
\end{document} | math |
ಕೋವಿಡ್ 19: ಪತ್ರಕರ್ತ ಪವನ್ ಹೆತ್ತೂರು ಬಲಿ ಬೆಂಗಳೂರು: ಪ್ರಜಾವಾಣಿ ಪತ್ರಿಕೆಯಲ್ಲಿ ಪತ್ರಕರ್ತರಾಗಿದ್ದ ಪವನ್ ಹೆತ್ತೂರು37 ಭಾನುವಾರ ಕೋವಿಡ್ ಸೋಂಕಿಗೆ ಬಲಿಯಾಗಿದ್ದಾರೆ. ಮೈಸೂರು ಪ್ರಜಾವಾಣಿ ಬ್ಯೂರೋ ದಲ್ಲಿ ಕೆಲಸ ಮಾಡುತ್ತಿದ್ದ ಪವನ್, ಪತ್ನಿ, ಇಬ್ಬರು ಚಿಕ್ಕ ಮಕ್ಕಳನ್ನು ಅಗಲಿದ್ದಾರೆ. ಪ್ರಜಾವಾಣಿ, ವಿಜಯವಾಣಿ, ಕಸ್ತೂರಿ ಟಿವಿಯಲ್ಲಿ ಕೆಲಸ ಮಾಡಿದ್ದ ಉತ್ಸಾಹಿ ಪತ್ರಕರ್ತ ಪವನ್ ಹೆತ್ತೂರು 37 ಕೋವಿಡ್ ಸೋಂಕಿಗೆ ಬಲಿಯಾಗಿದ್ದಾರೆ. ಕೋವಿಡ್ ದೃಢವಾಗಿ ಆಸ್ಪತ್ರೆಗೆ ದಾಖಲಾಗಿದ್ದ ಪವನ್ಗೆ ಆರೋಗ್ಯ ಗಂಭೀರವಾಗಿತ್ತು. ಇಂದು ಚಿಕಿತ್ಸೆಗೆ ಸ್ಪಂಧಿಸದೆ ರಾತ್ರಿ 1.30ರ ಸುಮಾರಿಗೆ ಕೊನೆಯುಸಿರೆಳೆದಿದ್ದಾರೆ. ಮೃತರ ಅಂತ್ಯಕ್ರಿಯೆ ಹಾಸನ ಜಿಲ್ಲೆ ಸಕಲೇಶಪುರ ತಾಲೂಕಿನ ಹೆತ್ತೂರಿನಲ್ಲಿ ನಡೆಯಲಿದೆ. | kannad |
#Requirements:
Here is a mechanical descriptions of how the UML procedure works:
##Goal:
to estimate the psychometric function for speech recognition, which is a function relating the signal-to-noise ratio (SNR, x-axis) to proportion correct (y-axis).
##Procedures:
###Initialization phase (what happens before an experiment):
1. create empty vectors of potential alpha and beta values. The parameter alpha could take one of 61 values, linearly spaced from -15 to 15 (dB). The parameter beta could take one of 23 values, linearly spaced between 0.1 and 1.2.
2. create a "31x23 matrix", which will be referred to as "L". L stores the log-likelihood for each combination of the potential alpha and beta values. The initial value for each element in the matrix should be
```
L = log(normpdf(alpha,10,20)*normpdf(beta,0.4,0.4))
```
, where
```
normpdf(x,mu,sigma)=(1/(sigma*sqrt(2*pi))*exp(-(x-mu)^2/(2*sigma^2))
```
stands for the probability density function of a normal distribution with mu and sigma being the mean and standard deviation of the normal distribution. Also log(x) stands for natural log. It might be easier to store the matrix L as a long array instead of a matrix, it makes sort and finding the maximum easier to program.
3. create an array x, which stores the SNR on each trials. The total number of trials could be variable from one run to another run, so it is not clear how much space we should pre-allocate for the array. I would say initial x to be an array of 100 values.
4. The first x would be the SNR used for the first trial. For our purpose, set it to a high value, such as 10. Print this value to the screen and wait for the experimenter to return the scores.
5. The scores are in the form of "n of m correct". For example, "1 of 5 correct" means one out of five words are recognized correctly on that trial. Two arrays n and m, each with equal length to x are created.
6. Initialize the "sweet-point number" swpt to 3. Set the total number of reversals nrev to 0.
###Iteration phase:
1. Following the ith response is collected, the response obtained from the experiment is used to update the log-likelihood L in the following way.
```
L = L + n[i]*Log(pf(x[i],alpha,beta))+(m[i]-n[i])*Log(1-pf(x[i],alpha,beta))
```
, where
```
pf(x) = 1/(1+exp(-beta*(x-alpha)))
```
, x[i] is the SNR used on the ith trial. This calculate is repeated for each combination of alpha and beta, hence updates all values stored in L.
2. Find the combination of alpha and beta that corresponds to the maximum value in L, obtained from step 1. This pair of alpha and beta values forms the interim estimate of the psychometric function and shall be referred to as alpha' and beta'.
3. Find the sweet points. The three sweet points (#1-3) are on the 15.356%, 50%, and 92.344% positions on the interim estimate of the psychometric function. They can be found using the inverse function of p=pf(x):
```
x = alpha-(1/beta)*Log((1-p)/p)
```
For example, the sweet point #1 corresponds to an SNR of
```
alpha'-(1/beta')*Log((1-.15356)/.15356)
```
Note that alpha' and beta' are obtained from step 2.
4. If n/m > 0.5 for the ith trial, shift the current sweet-point number lower (i.e., change swpt from 3 to 2, or from 2 to 1). If the swpt value is already #1, leave the swpt unchanged. Else, if n/m < 0.5 for the ith trial, shift the current sweet-point number higher (i.e., change swpt from 1 to 2, or from 2 to 3). If the swpt value is already 3, leave the swpt unchanged. Once the swpt value is updated, the SNR for the next trial is simply the SNR for that sweet point obtained from step 3. Print the SNR for the next trial on the screen.
5. If the direction of the track (i.e. from increase in swpt to decrease, or from decrease in swpt to increase), the current trial counts for a reversal and nrev has to be accumulated by 1 (i.e., nrev = nrev+1).
###Termination phase:
1. When nrev reaches 16, the current experiment is complete, the iteration stops.
2. Instead of print the SNR for the next trial, print a message informing the experimenter that the experiment is done. Also, the current estimate alpha', beta' are displayed alone with a plot of the psychometric function
```
pf(x,alpha',beta')
```
3. Calculate the slope of the psychometric function and display it.
That about it. This algorithm requires creating a lots of variables with considerable sizes. Start from the Initialization phase first. You might consider using separate files for various different parts. I usually have an initialization function and an iteration function. Give it a try, let me know if you have any questions.
| code |
I had been married for 20 years and we decided to separate. It was a good enough marriage but we simply grew apart. At 48 I met a man whom I thought was great and I fell in love again.
Looking back he was controlling but nothing that really worried me at the time – I thought it romantic he wanted to be with me all the time and turning up at work to take me out. He would come and get me if I had nights out, I thought he was caring. I see now it was control and jealousy. He asked me to marry him very quickly and I did. But good god, once we started living together he changed. It was like I was his property. He used to hurt me and continually told me he would kill me if I ever left him. I swear I was in shock for a year.
I had never experienced anything like this in my life before. I didn’t tell anyone, I was ashamed that I had got myself into this situation. I was 52 by this point. We’d bought a house together and all my money was in it. I felt so trapped. Outwardly we looked like the perfect couple but all the time I was terrified.
The waiting for the violence was one of the worst things. Once I smashed a can of lager into my own face. It was to get it over with. I couldn’t stand the waiting. I said to him ‘there it is done’. He replied he wasn’t going to do that and that I was mad and I should get myself ‘seen to’. Most of the time I did feel as if I was mad. I had a good job in the NHS but I took a redundancy package that was offered as by this point as I couldn’t face it.
I hadn’t told my family what was happening but my daughter and son had watched the dramatic change in me. They took me out for lunch one day and they told me how worried they were. My daughter told me that she had got some of her clients to go to the Freedom Programme and that she saw a big difference in them and maybe I should go. What a relief it was for it to be out in the open. We talked for hours about what I could do.
I decided to go the Freedom Programme. My husband worked so I could go to the group. I was terrified that my husband would find out. But it was was worth it.
I got strength and courage from all the wonderful, amazing women in my group. Knowing I was not the only woman who had lived with something similar was so incredible. Each session was an eye opener. You just don’t notice all the tactics they use to get you to change – to mould you, to get you to behave how they want. The course was so powerful, so empowering. I got confidence back, I started feeling like I could do something about my situation.
With my children’s help I left, it wasn’t easy. Life has changed and there are still many challenges for me personal and financially. But I feel like the person I used to be. | english |
STAFFORD — Southern took a major step toward securing its third division title in four years by holding off Jackson Memorial on Tuesday night.
Pins by Seon Bowker at 182 and Dan Lynch at heavyweight led the way as the Rams, ranked No. 2 in the Shore Sports Network Top 10, won eight bouts, including four of the final six, for a 35-24 Shore Conference Class A South victory over the No. 6 Jaguars. Nick O’Connell at 145 and Owen Kretschmer at 113 each won by major decision and John Stout won a key toss-up bout at 160 pounds.
Southern needs to defeat Brick on Saturday and Toms River North on Jan. 31 to clinch the outright A South title for the second year in a row and third time in four seasons.
Vin Miele (195) and Nick Pepe (126) added wins by decision and Matt Barnett clinched the win with a pin at 132 pounds.
Jackson was without starting 126-pounder Carsten DiGiantomasso (13-5), who was on crutches on the Jaguars bench after suffering an ankle injury during a match with DePaul on Saturday.
The teams traded wins over the first six bouts until Miele’s 5-2 victory over Freddy Michel at 195 pounds gave the Rams (8-1, 5-0) two straight wins and a 16-12 lead. In the previous bout Bowker pinned Brad Galassi in 44 seconds to give Southern a 13-12 lead it would not relinquish.
Jackson senior Matt McGowan, the Region VI champ at 126 pounds last season, started the match with a 15-0 technical fall over Ryan Thorn in 4:17 at 138 pounds. O’Connell’s 11-3 win over Nick Tomasiello got Southern on the board, but Jackson came back with a 7-3 decision from senior Tim Hamann over Dan Gorman at 152 pounds. At 160 it was Stout getting two takedowns in the third period to edge Kyle Epperly, 7-4, and cut Jackson’s lead to 8-7.
O’Connell’s major decision was secured with a third-period takedown and Stout had the only offensive points in the final period to win a bout that was 3-3 after two periods. At 113 points it was Kretschmer scoring three points in the third period for an 11-3 major decision and an extra team point.
Following Bowker’s pin and Miele’s 5-2 win, Jackson closed the gap to 16-15 with Dave Lemay’s 3-1 win over Matt Mackanic at 220. Lemay rode out Mackanic in the second period to keep the bout scoreless heading to the third. Up 1-0 after an intentional escape awarded by Mackanic, Lemay scored the only takedown of the bout in the final period.
Southern’s cushion grew at heavyweight when Lynch ran a half nelson to pin Naj’Zir Humbert in 2:48 and put the Rams up 22-15. Jackson sophomore Vincent Scollo edged Jayson Scerbo, 6-5, at 106 pounds to make it 22-18, but Kretschmer’s major decision at 113 put Southern up 26-18.
Jeremiah Nash made things interesting at 120 pounds when he pinned Sebastian Delligatti in 2:30, cutting Southern’s lead to 26-25 with two bouts left. But without DiGiantomasso Southern had a big advantage in the final two weights. Pepe won 5-0 over Tyler Mitrosky at 126 pounds for a 29-24 Southern lead and junior Matt Barnett, a state qualifier last season, pinned Hugh Mai in 2:49 at 132 pounds to close out the 11-point victory.
Records: at Southern (8-1, 5-0); Jackson Memorial (5-7, 3-1). | english |
अखिलेश ने बीजेपी पर साधा निशाना BJP से सावधान रहें, वोट की खातिर उसने कृषि कानून वापस लिए समाजवादी पार्टी के प्रमुख अखिलेश यादव ने शुक्रवार को कहा कि किसानों को भाजपा से सावधान रहना चाहिए क्योंकि सत्तारूढ़ दल की सरकार ने सिर्फ वोट की खातिर विवादास्पद कृषि कानून वापस लिए हैं। इसके साथ ही उन्होंने यह भी कहा कि उत्तर प्रदेश में सपारालोद गठबंधन की सरकार आने पर राज्य में इस तरह के किसी भी किसान विरोधी कानून को लागू नहीं होने दिया जाएगा। रालोद और सपा मिलकर लड़ रहे हैं उत्तर प्रदेश विधानसभा चुनाव पूर्व मुख्यमंत्री यहां राष्ट्रीय लोकदल रालोद के प्रमुख जयंत चौधरी के साथ एक संयुक्त संवाददाता सम्मेलन को संबोधित कर रहे थे। रालोद और सपा मिलकर उत्तर प्रदेश विधानसभा चुनाव लड़ रहे हैं। विधानसभा चुनाव से पहले, भाजपा पश्चिमी उत्तर प्रदेश में जाट मतदाताओं से संपर्क साध रही है। इस क्षेत्र से समुदाय के सदस्यों ने केंद्र के तीन कृषि कानूनों के खिलाफ दिल्ली की सीमाओं पर साल भर चले प्रदर्शन में हिस्सा लिया था। भाजपा नेताओं ने रालोद प्रमुख से अपनी पार्टी से हाथ मिलाने को भी कहा था और केंद्रीय गृह मंत्री अमित शाह ने बुधवार को दिल्ली में जाट नेताओं के साथ एक बैठक की थी। जयंत ने कहा कि सपा के साथ उनकी पार्टी का गठबंधन बहुत मजबूत है और किसानों के हितों के लिए काम करने का लक्ष्य रखता है। भाजपा एक ऐसी पार्टी है जो कोई चीज कहे बगैर कानून ले आती है अखिलेश अखिलेश ने कहा, भाजपा ने किसानों की आय दोगुनी करने का वादा किया था, लेकिन वह किसान विरोधी तीन कानून ले आई। किसानों ने सरकार को इन कानूनों को वापस लेने के लिए मजबूर कर दिया। भाजपा ने वोट की खातिर इन कानूनों को वापस लिया। भाजपा एक ऐसी पार्टी है जो कोई चीज कहे बगैर कानून ले आती है। पूर्व मुख्यमंत्री ने कहा कि सत्ता में आने पर उनकी पार्टी राज्य में इस तरह का कोई कानून लागू नहीं होने देगी। उन्होंने कहा कि जयंत, पूर्व प्रधानमंत्री चौधरी चरण सिंह, पूर्व मुख्यमंत्री मुलायम सिंह यादव और दिवंगत किसान नेता महेंद्र सिंह टिकैत के सिद्धांत पर चलकर किसानों के हितों से जुड़े मुद्दों की पैरोकारी को आगे बढ़ा रहे हैं। जयंत ने कहा कि पहले लोगों को संदेह था कि क्या दोनों दलों के बीच गठबंधन होगा। उन्होंने अपने दादा चौधरी चरण सिंह को याद करते हुए कहा, हमारा मेल काफी पहले हो गया था। हम उप्र का विकास करना चाहते हैं और गठबंधन किया क्योंकि हम चौधरी चरण सिंह की लड़ाई को आगे ले जाना चाहते हैं। अखिलेश संवाददाता सम्मेलन के लिए मुजफ्फरनगर देर से पहुंचे। उन्होंने इससे पहले दिन में दावा किया था कि वह दिल्ली में फंस गए थे क्योंकि उनके हेलीकॉप्टर को उड़ने की अनुमति नहीं दी गई। उन्होंने एक ट्वीट में पृष्ठभूमि में एक हेलीकॉप्टर के साथ अपनी तस्वीर साझा की। उन्होंने ट्वीट में कहा, मेरे हेलीकॉप्टर को अभी भी बिना कोई कारण बताए दिल्ली में रोककर रखा गया है और मुजफ्फरनगर नहीं जाने दिया जा रहा है। जबकि भाजपा के एक शीर्ष नेता अभी यहाँ से रवाना हुए हैं। हारती हुई भाजपा की ये हताशा भरी साजिश है। जनता सब समझ रही है। इसके घंटे भर बाद यादव ने उड़ान के लिए तैयार होने की बात कहते हुए एक और ट्वीट किया सत्ता का दुरुपयोग हारते हुए लोगों की निशानी है समाजवादी संघर्ष के इतिहास में ये दिन भी दर्ज होगा। हम जीत की ऐतिहासिक उड़ान भरने जा रहे हैं। अखिलेश ने संवाददाता सम्मेलन में कहा कि सपारालोद के सरकार में आने पर 300 यूनिट बिजली मुफ्त दी जाएगी और सिंचाई शुल्क भी माफ होगा। सपा प्रमुख ने कहा कि उनके गठबंधन की सरकार के आने पर किसानों के लिए न्यूनतम समर्थन मूल्य एमएसपी पर खरीद के लिए सरकार की तरफ से जो इंतजाम करने पड़ेंगे, वे किए जाएंगे। अखिलेश ने यह भी कहा कि उनकी सरकार आने पर ऐसी व्यवस्था की जाएगी कि किसानों को गन्ने के भुगतान के लिए इंतजार न करना पड़े। इस दौरान जयंत चौधरी ने आरोप लगाया कि प्रशासन रालोदसपा गठबंधन के प्रत्याशियों के साथ सहयोग नहीं कर रहा है। | hindi |
#!/bin/sh
# Using https://github.com/markogresak/cloud-ignore-files
if ! brew ls --versions unison 2>/dev/null >/dev/null
then
echo " Installing unison for you."
brew install unison > /tmp/unison-install.log
fi
cd $DOTFILES_ROOT/syncing/cloud-ignore-files
./install-cloud-ignore-files.sh
| code |
مے ہاو ماحولس مُطلِق نیوز | kashmiri |
ماہرین حیاتیات چِھ جینیٲتی کوڈ کہِ ہر جایہِ آسنہٕ کہِ تمام بیکٹیریا، archaea تہٕ یوکرائٹس خٲطرٕہ عالمگیر مشترک نزول کہِ ثبوت سمجٕنہٕ یوان۔ | kashmiri |
کنگسٹن لائن ویسٹ انڈیز ٹیم کے بلے باز شیو نرائن چندرپال نے 300 فرسٹ کلاس میچز کا سنگ میل عبور کر لیا جس پر ویسٹ انڈیز کرکٹ بورڈ نے انہیں مبارکباد دی 38 سالہ چندرپال نے یہ اعزاز انگلش کانٹی چیمپئن شپ میں سیسکس کیخلاف ڈربی شائر کی نمائندگی کرتے ہوئے کیا | urdu |
ಮಾಸ್ಕ್ ಹೆಸರಲ್ಲಿ ವಸೂಲಿ..ಜನ್ರಿಗೆ ಹೆದರಿಸಿ, ಬೆದರಿಸಿ ಕಿರುಕುಳ ಕೊಟ್ರೆ ಹುಷಾರ್ ಬೆಂಗಳೂರು: ಮಾಸ್ಕ್ ಧರಿಸದವರಿಗೆ ದಂಡ ಹಾಕುವ ನೆಪದಲ್ಲಿ ಸರ್ಕಾರ ಜನರಿಗೆ ಕಿರುಕುಳ ಕೊಡುತ್ತಿದೆ ಅಂತಾ ಕೆಪಿಸಿಸಿ ಅಧ್ಯಕ್ಷ ಡಿಕೆ ಶಿವಕುಮಾರ್ ರಾಜ್ಯ ಸರ್ಕಾರದ ವಿರುದ್ಧ ಗರಂ ಆಗಿದ್ದಾರೆ. ಮಾಸ್ಕ್ ಹಾಕದ ಜನರಿಗೆ 1000 ರೂಪಾಯಿ ದಂಡವನ್ನ ವಿಧಿಸಲಾಗುತ್ತಿದೆ. ಇದಕ್ಕೆ ಆಕ್ರೋಶ ವ್ಯಕ್ತಪಡಿಸಿ ಮಾತನಾಡಿದ ಡಿಕೆಎಸ್, ಸರ್ಕಾರ ಮೊದಲು ಮಾಸ್ಕ್ ಬಗ್ಗೆ ಸರಿಯಾಗಿ ಅರಿವು ಮೂಡಿಸಬೇಕು. ತುಂಬಾ ಹೊತ್ತು ಮಾಸ್ಕ್ ಹಾಕಿದ್ರೆ ಉಸಿರಾಟಡಲು ಸಮಸ್ಯೆ ಆಗುತ್ತದೆ. ಹಲವರಿಗೆ ಹಲವು ಸಮಸ್ಯೆಗಳು ಇರುತ್ತವೆ. ದಂಡ ಹಾಕುವ ಮೂಲಕ ಅಧಿಕಾರಿಗಳ ವಸೂಲಿಗೆ ದಾರಿ ಮಾಡಿಕೊಟ್ಟಿದೆ ಎಂದು ಆರೋಪಿಸಿದರು. ಜನರನ್ನ ಬೆದರಿಸೋದು ಸರಿಯಲ್ಲ, ಗುತ್ತಿಗೆದಾರರು ಮತ್ತು ಅಧಿಕಾರಿಗಳನ್ನು ಹೆದರಿಸುವ ರೀತಿಯಲ್ಲಿ ಜನರನ್ನ ಹೆದರಿಸುತ್ತಿದ್ದೀರಾ? ಮಾಸ್ಕ್ ಬಗ್ಗೆ ಯಾವ ತಿಳುವಳಿಕೆ ಕೊಡಬೇಕೋ ಕೊಡಿ. ಆದ್ರೆ ವಸೂಲಿಗೆ ಅವಕಾಶ ಬೇಡ ಅಂತಾ ರಾಜ್ಯ ಸರ್ಕಾರಕ್ಕೆ ಶಿವಕುಮಾರ್ ಆಗ್ರಹಿಸಿದ್ದಾರೆ. | kannad |
விவசாயிகள் போராட்டத்தை அரசியல் சூழ்ச்சியென முத்திரை குத்தக் கூடாது ராம்பூா்,: விவசாயிகள் நடத்தி வரும் போராட்டத்தை, அரசியல் சூழ்ச்சியென்று முத்திரை குத்தக் கூடாது எனக் கண்டித்துள்ள காங்கிரஸ் பொதுச் செயலாளா் பிரியங்கா, வேளாண் சட்டங்களை மத்திய அரசு உடனடியாகத் திரும்பப் பெற வேண்டும் என்று தெரிவித்துள்ளாா். மத்திய அரசு இயற்றிய வேளாண் சட்டங்களுக்கு எதிராக கடந்த மாதம் 26ஆம் தேதி விவசாயிகள் தில்லியில் டிராக்டா் பேரணி நடத்தினா். அப்பேரணியில் டிராக்டா் கவிழ்ந்து விபத்துக்குள்ளானதில் உத்தர பிரதேசத்தைச் சோந்த நவ்ரீத் சிங் என்ற விவசாயி உயிரிழந்தாா். ராம்பூா் மாவட்டத்தின் திப்திபா கிராமத்திலுள்ள நவ்ரீத் சிங்கின் உறவினா்களை வியாழக்கிழமை நேரில் சந்தித்த பிரியங்கா, அவா்களுக்கு ஆறுதல் கூறினாா். அதைத் தொடா்ந்து நவ்ரீத்துக்காக நடைபெற்ற சடங்கிலும் அவா் கலந்து கொண்டாா். அங்கு செய்தியாளா்களிடம் பிரியங்கா கூறியதாவது: வேளாண் சட்டங்களைத் திரும்பப் பெற வேண்டுமென்று கோரி விவசாயிகள் தொடா்ந்து போராடி வருகின்றனா். ஆனால், அவா்களின் கோரிக்கையை மத்திய அரசு ஏற்க மறுக்கிறது. அதைவிடப் பெரிய தவறாக, விவசாயிகள் போராட்டத்தை அரசியல் சூழ்ச்சியென மத்திய அரசு முத்திரை குத்தி வருகிறது. விவசாயிகளை பயங்கரவாதிகள் எனக் கூறி வருகிறது. இதுபோன்ற செயல்களை மத்திய அரசு தவிா்க்க வேண்டும். விவசாயிகள் போராட்டம் அரசியல் சாா்ந்தது அல்ல. விவசாயிகளும் நாட்டு மக்களும் தாமாக முன்வந்து போராடி வருகின்றனா். இந்த விவகாரத்தில் அரசியலை நுழைப்பது முற்றிலும் தவறானது. நாட்டு மக்கள் அனைவரும் நவ்ரீத் சிங்கின் குடும்பத்தினருக்கு ஆதரவாக உள்ளனா் என்பதைத் தெரிவிக்கவே வந்தேன். ஏழைகள், விவசாயிகளின் வலியை உணர முடியாத தலைவா்களால் நாட்டுக்கு எந்தப் பலனும் இல்லை. மத்திய அரசு இயற்றிய கருப்பு சட்டங்களுக்கு எதிராக காங்கிரஸ் தொடா்ந்து போராடும். அச்சட்டங்களை மத்திய அரசு உடனடியாகத் திரும்பப் பெற வேண்டும் என்றாா் பிரியங்கா. | tamil |
To reduce redness and draw the pus out of pimples, just put a pinch of toothpaste on the affected area for a few minutes then wash away. For dryness afterward, have some lotion handy. Again, avoid gels or pastes with unnecessary bleach additives, simple white abrasive toothpaste is all you need for maximum effect and minimal reaction risk.
Want to freshen up your nails cheaply and effectively? Simply brush them as you would your teeth. They are, after all, made from the same substance. The mild abrasive in the paste will work to exfoliate and shine up dull nails.
Simply applying toothpaste to wounds can be incredibly soothing. The vapor action of the minty paste feels cool while the action of the paste tightens skin and eases inflammation.
If you’ve ever dyed your hair and ended up with stained skin around the hairline, you can remove the stain by treating with toothpaste. As a pretreatment, you can also head off staining by simply smearing a little around the hairline before you begin applying dye.
The mild abrasive in toothpaste is gentle, yet effective at cleaning and polishing chrome. So feel free to shine up your towel racks, faucets, or even your motorcycle cheaply and easily!
Yes, it works on silver too! So the next time your cutlery, silverware, or silver jewelry is looking a little dull, skip the fancy polish and just buff it to brightness with toothpaste and a cloth.
Got a favorite disk that’s starting to skip? Simple toothpaste is mild enough to shine up your CD and get you back to bopping along in no time!
A lot of money gets wasted on buying up baby bottles to replace some that start to smell like sour milk. Instead of tossing your next not-so-fresh bottle, try adding a little toothpaste and some warm water and then shaking it vigorously. Dump it out and rinse and you should be left with a clean and minty fresh bottle. | english |
बजट पर दिल्ली सरकार को मिले 5,500 सुझाव मोहल्ला पुस्तकालय से लेकर इलेक्ट्रिक वाहनों के लिए सस्ती दर पर पार्किंग मुहैया कराने का सुझाव शिक्षा, स्वास्थ्य और अर्थव्यवस्था पर आए बड़ी संख्या में सुझाव नई दिल्ली । दिल्ली सरकार को अपने जन सहभागी बजट के लिए मोहल्ला पुस्तकालयों से लेकर इलेक्ट्रिक वाहनों के लिए सस्ती दर पर पार्किंग मुहैया कराने जैसे प्रस्तावों पर लोगों से 5,500 प्रतिक्रियाएं मिली हैं। सरकार ने वर्ष 202223 के बजट में जनता की भागीदारी सुनिश्चित करने के लिए सुझाव मांगे थे। लोगों को वेबसाइट के माध्यम से वित्त विभाग को सुझाव भेजने के लिए कहा था। दिल्ली के लोगों को जमीनी स्तर पर पेश आ रही समस्याएं दूर करने के बारे में सुझाव आए हैं। इसका एक बड़ा हिस्सा शिक्षा, स्वास्थ्य व आर्थिक वृद्धि से जुड़ा हुआ है। इसके अलावा पर्यटन को बढ़ावा देने, परिवहन व्यवस्था को सुधारने,ग्रामीण विकास, प्रदूषण, शहर के सौंदर्यीकरण और समाज कल्याण से जुड़ी नीतियों पर भी लोगों ने सुझाव दिए हैं। आप सरकार द्वारा विकसित शिक्षा प्रणाली से प्रभावित होकर एक व्यक्ति ने वयस्कों के लिए भी इसी तरह की सुविधा की मांग की है। उनका कहना है कि सरकार ने बेहतरीन बुनियादी ढांचा और कई तरह के पाठ्यक्रम विकसित किए हैं, जिनका लाभ वयस्कों को भी मिल सकता है। इन स्कूलों में वयस्कों के लिए शाम की कक्षाओं के संचालन की मांग की है। एक अन्य व्यक्ति ने मोहल्ला क्लीनिक की तर्ज पर मोहल्ला पुस्तकालय खोलने की मांग की है। उनका तर्क है कि यह उन लोगों की बहुत मदद कर सकती है जो घनी आबादी वाले क्षेत्रों में रहते हैं और उनके पास अध्ययन के लिए एकांत स्थान नहीं है। वहीं, बिजनेस ब्लास्टर्स की तर्ज पर छोटे व्यवसायों के लिए उद्यमीनिवेशक सम्मेलनों और कार्यक्रमों का सुझाव दिया गया है। एक व्यक्ति ने पोर्टर्स और कुरियर के लिए इलेक्ट्रिक वाहन अनिवार्य करने की नीति की मांग की है। उनका कहना है कि पोर्टर्स और ट्रांसपोर्टर थोक से लेकर खुदरा बाजारों तक बड़े स्तर पर काम करते हैं और इनको इलेक्ट्रिक वाहनों द्वारा आसानी से कवर किया जा सकता है। इसी तरह एक अन्य दिल्लीवासी ने इलेक्ट्रिक वाहनों के लिए सस्ती पार्किंग की मांग की है। सुझाव दिया कि इस कदम से लोगों को ग्रीन परिवहन को अपनाने के लिए प्रोत्साहित किया जा सकता है। एक युवा छात्र ने सुझाव दिया है कि शैक्षणिक संस्थानों और भीड़भाड़ वाली कॉलोनियों के पास ईबाइक रेंटल प्वाइंट लगाकर ईबाइक को बढ़ावा दिया जाए। इसके अलावा छोटे पैमाने के सामुदायिक सौर ऊर्जा संयंत्रों, स्थानीय बायोगैस संयंत्रों, सीवेज उपचार संयंत्रों, पार्कों की सिंचाई के लिए उपचारित पानी के उपयोग व घरघर से ईकचरा संग्रह का सुझाव मिला है। कई अन्य सुझाव भी मिले हैं। | hindi |
മണ്ഡല മകരവിളക്ക് ശബരിമലയില് പ്രതിദിനം 25,000 പേരെ പ്രവേശിപ്പിക്കും കൊറോണ വാക്സിന്, ആര്ടിപിസിആര് നെഗറ്റീവ് സര്ട്ടിഫിക്കറ്റ് നിര്ബന്ധം തിരുവനന്തപുരം : ശബരിമലയില് മണ്ഡല മകരവിളക്കിനോടനുബന്ധിച്ച് ആദ്യ ദിവസങ്ങളില് പ്രതിദിനം 25,000 പേരെ പ്രവേശിപ്പിക്കും. മുഖ്യമന്ത്രി പിണറായി വിജയന്റെ അദ്ധ്യക്ഷതയില് ചേര്ന്ന യോഗത്തിലാണ് ഇതുമായി ബന്ധപ്പെട്ട തീരുമാനം. എണ്ണത്തില് മാറ്റം വേണമെങ്കില് പിന്നീട് ചര്ച്ച ചെയ്ത് ആവശ്യമായ നടപടികള് സ്വീകരിക്കും. വെര്ച്വല് ക്യൂ സംവിധാനം തുടരാനാണ് തീരുമാനം. 10 വയസ്സിന് താഴെയും 65 വയസ്സിന് മുകളിലുമുള്ള തീര്ഥാടകര്ക്കും പ്രവേശനം അനുവദിക്കും. രണ്ട് ഡോസ് കൊറോണ പ്രതിരോധ വാക്സിന് എടുത്തവര് അല്ലെങ്കില് ആര്.ടി.പി. സി. ആര് നെഗറ്റീവ് സര്ട്ടിഫിക്കറ്റ് ഹാജരാക്കുന്നവര്ക്കാവും പ്രവേശനം നല്കുക. അഭിഷേകം ചെയ്ത നെയ്യ് എല്ലാവര്ക്കും കൊടുക്കുന്നതിന് ദേവസ്വംബോര്ഡ് സംവിധാനമൊരുക്കണമെന്ന് മുഖ്യമന്ത്രി നിര്ദ്ദേശിച്ചു. ദര്ശനം കഴിഞ്ഞ് സന്നിധാനത്ത് തങ്ങാന് അനുവദിക്കില്ല. ഇക്കാര്യത്തില് കഴിഞ്ഞ വര്ഷത്തെ നില തുടരും. എരുമേലി വഴിയുള്ള കാനനപാത, പുല്മേട് വഴി സന്നിധാനത്ത് എത്തുന്ന പരമ്ബരാഗത പാത എന്നിവയിലൂടെ തീര്ത്ഥാടകരെ അനുവദിക്കില്ല. പമ്ബയില് സ്നാനത്തിന് അനുമതി നല്കും. വാഹനങ്ങള് നിലക്കല് വരെ മാത്രമേ അനുവദിക്കുള്ളൂ. അവിടെ നിന്ന് പമ്ബ വരെ കെ.എസ്.ആര്.ടി.സി ബസ്സുകള് ഉപയോഗിക്കണം. അതിന് ആവശ്യമായ സൗകര്യങ്ങള് ഏര്പ്പെടുത്താന് മുഖ്യമന്ത്രി അധികൃതര്ക്ക് നിര്ദ്ദേശം നല്കി. കെ.എസ്.ആര്.ടി.സി. ബസ് സ്റ്റോപ്പുകളില് മതിയായ ശൗചാലയങ്ങള് ഉറപ്പാക്കണം. ശുചീകരണ തൊഴിലാളികളുടെ ശമ്ബളം വര്ദ്ധിപ്പിക്കും. അഗ്നി സുരക്ഷാ സംവിധാനങ്ങള് നിലവിലില്ലാത്ത കെട്ടിടങ്ങളില് സ്മോക്ക് ഡിറ്റക്ടറുകള് സ്ഥാപിക്കണം. കൊറോണ മുക്തരായ അനുബന്ധരോഗങ്ങള് ഉള്ളവര് ആരോഗ്യസ്ഥിതി പരിശോധിച്ച് മാത്രമേ ദര്ശനത്തിന് വരാന് പാടുള്ളൂ എന്ന് മുഖ്യമന്ത്രി നിര്ദേശിച്ചു. | malyali |
तालझारी में जेसीबी से काम कराने पर सात लोगों पर केस जाटी, तालझारीतीनपहाड़ साहिबगंज : तालझारी प्रखंड में मनरेगा योजना में जेसीबी मशाीन से काम कराने पर सात लोगों पर तीनपहाड़ थाने में प्राथमिकी दर्ज कराई गई है। इनमें चार अधिकृत मेट, दो बिचौलिया व एक रोजगार सेवक है। प्राथमिकी दर्ज होने के बाद पुलिस ने मामले की छानबीन शुरू कर दी है। बताया जाता है कि पिछले साल ग्रामीण विकास विभाग की सोशल ऑडिट टीम ने बड़ा दुर्गापुर पंचायत में चार मनरेगा योजनाओं में जेसीबी से काम करने का मामला पकड़ा था। इसकी रिपोर्ट जिला एवं राज्य स्तर पर भी की गई थी। इसे राज्य मुख्यालय ने गंभीरता से लिया और चारों योजनाओं में मनरेगा अधिनियम का उल्लंघन मामले में प्राथमिकी दर्ज करने को कहा। इसके आलोक में बीडीओ साइमन मरांडी ने तीनपहाड़ थाना में प्राथमिकी दर्ज करा दी है। परते पहाड़ में मंगली पहाड़िन की जमीन पर मेड़बंदी सह समतलीकरण कार्य की अधिकृत मेट नोमिता देवी, इसी गांव में जबरा पहाड़िया की जमीन पर मेड़बंदी सह समतलीकरण के अधिकृत मेट बिल्लू मुर्मू, अत्तरभिट्ठा में छोटा मैसा पहाड़िया की जमीन का मेड़बंदी सह समतलीकरण की अधिकृत मेट नोमिता देवी व परते पहाड़ पर सूरजी पहाड़िन की जमीन पर मेड़बंदी सह समतलीकरण कार्य की अधिकृत मेट अनिता देवी, बड़ा दुर्गापुर पंचायत के शहरपुर गांव निवासी बिचौलिया बालक साहा, संजय साहा व स्थानीय रोजगार सेवक मो. शमशाद अली पर प्राथमिकी दर्ज करायी गई है। स्थल जांच से पूर्व ही दे दी रिपोर्ट! साहिबगंज : तालझारी बीडीओ साइमन मरांडी ने इस मामले में प्राथमिकी दर्ज करने के लिए जो आवेदन दिया है उसमें कहा गया है कि ग्रामीण विकास विभाग की सोशल आडिट यूनिट ने पांच अक्टूबर 2021 को सूचना दी है कि छह अक्टूबर 21 को स्थलीय जांच के दौरान तालझारी प्रखंड की बड़ा दुर्गापुर पंचायत में चार योजनाओं में जेसीबी के उपयोग की बात कही गई है। अब सवाल यह उठता है कि जब छह अक्टूबर को निरीक्षण किया गया तो पांच अक्टूबर को ही जेसीबी के उपयोग की बात कहां से आ गई। इस संबंध में पूछे जाने पर बीडीओ साइमन मरांडी ने बताया कि वरीय अधिकारियों के निर्देश पर यह प्राथमिकी दर्ज करायी गई है। उन्होंने मामले को देखने की बात कही। | hindi |
ವಿಟ್ಲ: ಒಂಟಿ ಮಹಿಳೆಯ ಕಟ್ಟಿ ಹಾಕಿ, ಹಲ್ಲೆ ನಡೆಸಿ ದರೋಡೆ ಬಂಟ್ವಾಳ : ಒಂಟಿ ಮಹಿಳೆ ಇರುವ ಮನೆಗೆ ನುಗ್ಗಿದ ದುಷ್ಕರ್ಮಿಗಳು ಮಹಿಳೆಯನ್ನು ಕಟ್ಟಿ ಹಾಕಿ, ಹಲ್ಲೆ ನಡೆಸಿ ಚಿನ್ನಾಭರಣ ದರೋಡೆಗೈದ ಘಟನೆ ವಿಟ್ಲದ ಕಾನತ್ತಡ್ಕದಲ್ಲಿ ನಡೆದಿದೆ. ಕಾನತ್ತಡ್ಕ ಜುಮಾ ಮಸೀದಿ ಮುಂಭಾಗದಲ್ಲಿರುವ ಬಾಡಿಗೆ ಮನೆಯಲ್ಲಿ ಆಟೋ ಚಾಲಕ ರಫೀಕ್ ವಾಸ ಮಾಡುತ್ತಿದ್ದು, ರಫೀಕ್ ಅವರು ಶುಕ್ರವಾರ ಮಸೀದಿಗೆ ತೆರಳಿದ್ದರು. ಅವರ ಜತೆ ಅವರ 10 ವರ್ಷದ ಮಗ ಕೂಡಾ ತೆರಳಿದ್ದರು. ಅವರ ಪತ್ನಿ ಒಬ್ಬರೇ ಮನೆಯಲ್ಲಿ ಇದ್ದರು. ಈ ಬಗ್ಗೆ ಮಾಹಿತಿ ಅರಿತ ವ್ಯಕ್ತಿ ಮಹಿಳೆಯ ಕಟ್ಟಿ ಹಾಕಿ, ಬಾಯಿಗೆ ಬಟ್ಟೆ ತುರುಕಿ ಹಲ್ಲೆ ನಡೆಸಿದ್ದಾನೆ ಎಂದು ದೂರಲಾಗಿದೆ. ಇವರ ಮನೆಯ ಸುತ್ತಲೂ ಮನೆಗಳಿದ್ದು, ಯಾರ ಗಮನಕ್ಕೆ ಬಂದಿಲ್ಲ. ಪತಿ ಮಸೀದಿಯಿಂದ ಮನೆಗೆ ಬಂದ ವೇಳೆ ಘಟನೆ ಬೆಳಕಿಗೆ ಬಂದಿದೆ. ಸ್ಥಳಕ್ಕೆ ವಿಟ್ಲ ಪೊಲೀಸರು ಭೇಟಿ ನೀಡಿದ್ದು, ತನಿಖೆ ನಡೆಸುತಿದ್ದಾರೆ. | kannad |
## HISTORY:
with history of CML complicated by
graft-versus-host disease of the skin, mucous membranes and lungs with new
worsening hypoxia on top of his baseline O2// get CT scan to look for occult
PNA vs. inflammatory changes ISO GVHD.
## HEART AND VASCULATURE:
The thoracic aorta is normal in caliber. Minimal
coronary artery calcification is seen. The heart, pericardium, and great
vessels are otherwise within normal limits based on an unenhanced scan. No
pericardial effusion is seen.
AXILLA, HILA, MEDIASTINUM AND CHEST WALL:
No axillary or mediastinal
lymphadenopathy is present. No mediastinal mass or hematoma. No abnormal
findings in the soft tissue of the chest wall.
## PLEURAL SPACES:
No pleural effusion or pneumothorax.
## LUNGS/AIRWAYS:
Increased AP diameter of the chest wall. Multifocal bilateral
nodular and scattered ground-glass opacities in bronchovascular distribution
with some opacities throughout all lobes of the lung, new in the
interval, and compatible with infectious small airways disease/infectious
bronchiolitis. Diffuse cylindrical bronchiectasis with a lower-lobe
predominance appears similar however, there is increased degree of the
peribronchial thickening and mucous plugging.
## BASE OF NECK:
Visualized portions of the base of the neck show no abnormality.
## ABDOMEN:
2 mm nonobstructive renal calculus in the right upper pole;
aaotherwise unremarkable visualized structures.
## BONES:
Generalized bone demineralization without acute fractures or suspicious
bone lesions.
## IMPRESSION:
1. New multifocal nodular and ground-glass bronchocentric opacities throughout
all lobes of the lungs bilaterally, most likely due to infectious
bronchiolitis.
2. Diffuse bronchiectasis with worsening bronchitis and mucous plugging.
3. 2 mm nonobstructive renal calculus in the right upper pole.
| medical |
PUBG Mobile Ban: ಅತ್ಯಂತ ಜನಪ್ರಿಯ ಗೇಮ್ ಅನ್ನು ಭಾರತ ಸರ್ಕಾರ ನಿಷೇಧಿಸಲಿದೆಯೇ? ಪಬ್ಜಿ PUBG ಸಂಭಾವ್ಯ ನಿಷೇಧಕ್ಕಾಗಿ ಸರ್ಕಾರವು 250 ಕ್ಕೂ ಹೆಚ್ಚು ಚೀನೀ ಅಪ್ಲಿಕೇಶನ್ಗಳನ್ನು ನೋಡುತ್ತಿದೆ ಎಂದು ವರದಿಯಾಗಿದೆ. ಈಗಾಗಲೇ 100 ಕ್ಕೂ ಹೆಚ್ಚು ಅಪ್ಲಿಕೇಶನ್ಗಳ ಮೇಲೆ ನಿಷೇಧವನ್ನು ಆದೇಶಿಸಿದ ನಂತರ ಅಂತಿಮವಾಗಿ ಭಾರತದಲ್ಲಿ PUBG ಮೊಬೈಲ್ ಅನ್ನು ನಿಷೇಧಿಸಬಹುದೆಂದು ವದಂತಿಗಳಿವೆ. ಆದಾಗ್ಯೂ ನಿಷೇಧಿತ ಅಪ್ಲಿಕೇಶನ್ಗಳ ಯಾವುದೇ ಹೊಸ ಪಟ್ಟಿಯಲ್ಲಿ ಪಬ್ಜಿ PUBG ಮೊಬೈಲ್ ಅನ್ನು ಸೇರಿಸದಂತೆ ಆಟದ ಅನೇಕ ಅಭಿಮಾನಿಗಳು ಅಧಿಕಾರಿಗಳಿಗೆ ವಿನಂತಿಸುತ್ತಿದ್ದಾರೆ. ಸೋಮವಾರ 47 ಚೀನೀ ಆಯಪ್ಗಳನ್ನು ಸರ್ಕಾರ ನಿಷೇಧಿಸಿದೆ ಎಂದು ವರದಿಯಾದ ನಂತರವೇ ಇತ್ತೀಚಿನ ಬೆಳವಣಿಗೆ ಸಂಭವಿಸಿದೆ. ಮಾಹಿತಿ ತಂತ್ರಜ್ಞಾನ ಕಾಯ್ದೆಯ ಸೆಕ್ಷನ್ 69 ಎ ಯ ನಿಬಂಧನೆಗಳ ಅಡಿಯಲ್ಲಿ ರಾಷ್ಟ್ರೀಯ ಹಿತಾಸಕ್ತಿ ಮತ್ತು ಸುರಕ್ಷತೆಯನ್ನು ರಕ್ಷಿಸುವ ಪರವಾಗಿ 59 ಚೀನೀ ಅಪ್ಲಿಕೇಶನ್ಗಳನ್ನು ನಿಷೇಧಿಸಲಾಗಿತ್ತು. ಹೊಸ ನಿರ್ಧಾರದಡಿಯಲ್ಲಿ ಸರ್ಕಾರವು ನಿಷೇಧಿಸಿದ ಅಪ್ಲಿಕೇಶನ್ಗಳ ಪಟ್ಟಿಯನ್ನು ಸರ್ಕಾರ ಒದಗಿಸದಿದ್ದರೂ ಚೀನಾ ಬೆಂಬಲಿತ ಇತರ ಕೆಲವು ಅಪ್ಲಿಕೇಶನ್ಗಳು ಮತ್ತು ಆಟಗಳಲ್ಲಿ PUBG ಮೊಬೈಲ್ ಅನ್ನು ನಿಷೇಧಿಸುವುದನ್ನು ಪರಿಗಣಿಸುತ್ತಿದೆ ಎಂದು ಹೇಳಲಾಗಿದೆ. ಇದು ಯಾವುದೇ ಬಳಕೆದಾರರ ಗೌಪ್ಯತೆ ಮತ್ತು ರಾಷ್ಟ್ರೀಯ ಭದ್ರತಾ ಉಲ್ಲಂಘನೆಗಳಿಗಾಗಿ ಪರಿಶೀಲಿಸಲಾಗುವ 250 ಕ್ಕೂ ಹೆಚ್ಚು ಅಪ್ಲಿಕೇಶನ್ಗಳ ಪಟ್ಟಿಯ ಒಂದು ಭಾಗವಾಗಿರಬಹುದು. ಯಾವುದೇ ಅಧಿಕೃತ ಪ್ರಕಟಣೆಗೆ ಮುಂಚಿತವಾಗಿ ಸುದ್ದಿ ವರದಿಗಳಲ್ಲಿ ಪಿಬಿಜಿ ಮೊಬೈಲ್ ಹೆಸರಿನ ಆಗಮನವು ಸಾಮಾಜಿಕ ಮಾಧ್ಯಮವನ್ನು ಬಿರುಗಾಳಿಯಿಂದ ತಳ್ಳಿದೆ. ಭಾರತದಲ್ಲಿ ಮೊಬೈಲ್ ಆಟಗಳಿಗೆ ಬಂದಾಗ ಪಬ್ಜಿ PUBG ಮೊಬೈಲ್ ಪ್ರಸಿದ್ಧ ಹೆಸರು. ಚೀನಾದ ಟೆನ್ಸೆಂಟ್ ಗೇಮ್ಸ್ ಒಡೆತನದ ಲೈಟ್ಸ್ಪೀಡ್ ಮತ್ತು ಕ್ವಾಂಟಮ್ ಸ್ಟುಡಿಯೋ ಅಭಿವೃದ್ಧಿಪಡಿಸಿದ ಬ್ಯಾಟಲ್ ರಾಯಲ್ ಆಟವು ಭಾರತದಿಂದ 17.5 ಕೋಟಿ ಸ್ಥಾಪನೆಗಳನ್ನು ಪಡೆದಿದೆ ಇದು ವಿಶ್ವದಾದ್ಯಂತದ ಒಟ್ಟು ಸ್ಥಾಪನೆಗಳಲ್ಲಿ 24 ಪ್ರತಿಶತದಷ್ಟಿದೆ ಎಂದು ಸೆನ್ಸಾರ್ ಟವರ್ ವರದಿ ಮಾಡಿದೆ. ಇದು ಚೀನಾದಿಂದ ಪಡೆದದ್ದಕ್ಕಿಂತ ಹೆಚ್ಚಿನ ಸ್ಥಾಪನೆಗಳನ್ನು ಗಳಿಸಿದೆ ಇದು ತನ್ನ ಒಟ್ಟು ಸ್ಥಾಪನೆಗಳಲ್ಲಿ ಶೇಕಡಾ 16.7 ರಷ್ಟು ಪಾಲನ್ನು ಹೊಂದಿದೆ. ಸ್ಥಾಪನೆಗಳ ಹೊರತಾಗಿ ಟ್ವಿಚ್ ಮತ್ತು ಯೂಟ್ಯೂಬ್ನಲ್ಲಿ ತಮ್ಮ ಚಾನೆಲ್ಗಳ ಮೂಲಕ ಪಬ್ಜಿ PUBG ಮೊಬೈಲ್ ಅನ್ನು ಸಕ್ರಿಯವಾಗಿ ಸ್ಟ್ರೀಮ್ ಮಾಡುವ ಅನೇಕ ಭಾರತೀಯ ಗೇಮರ್ಗಳು ಇದ್ದಾರೆ. ನಿಕೊ ಪಾರ್ಟ್ನರ್ಸ್ ಹಿರಿಯ ವಿಶ್ಲೇಷಕ ಡೇನಿಯಲ್ ಅಹ್ಮದ್ ಅವರು ಭಾರತದಲ್ಲಿ ಆಟವು ದೊಡ್ಡ ನೆಲೆಯನ್ನು ಹೊಂದಿದ್ದರೂ ಅದೇ ಪ್ರಮಾಣದಲ್ಲಿ ಆದಾಯವನ್ನು ಗಳಿಸುವುದಿಲ್ಲ ಇದು ಮಾಸಿಕ ಆಧಾರದ ಮೇಲೆ ಸುಮಾರು 23 ಮಿಲಿಯನ್ ಸರಿಸುಮಾರು 1522 ಕೋಟಿ ರೂ. ಗಳಿಸುತ್ತದೆ. ಆಪಲ್ ಆಪ್ ಸ್ಟೋರ್ ಮತ್ತು ಗೂಗಲ್ ಪ್ಲೇ ಎರಡರಲ್ಲೂ PUBG ಮೊಬೈಲ್ಗೆ ಸಾಕಷ್ಟು ಪರ್ಯಾಯಗಳಿವೆ. ಕಾಲ್ ಆಫ್ ಡ್ಯೂಟಿ: ಮೊಬೈಲ್, ಬಟರ್ ರಾಯಲ್, ಗರೆನಾ ಫ್ರೀ ಫೈರ್, ಮತ್ತು ಫೋರ್ಟ್ನೈಟ್ ಇವುಗಳಲ್ಲಿ ಒಂದು ರೀತಿಯ ಅನುಭವವನ್ನು ಪಡೆಯಲು ಪಬ್ಜಿ PUBG ಮೊಬೈಲ್ ಅನುಪಸ್ಥಿತಿಯಲ್ಲಿ ಆಡಬಹುದಾದ ಆಟಗಳಾಗಿವೆ. ದೇಶದಲ್ಲಿ ಆಟವನ್ನು ನಿಷೇಧಿಸಿದರೆ ಇವೆಲ್ಲವೂ ಹೆಚ್ಚಿನ ಲಾಭ ಪಡೆಯಲು ನಿಲ್ಲುತ್ತವೆ ಎಂದು ಸಿಂಗ್ ಹೇಳಿದರು. ಸೆನ್ಸರ್ ಟವರ್ನ ಚಾಪಲ್ ವಿಶೇಷವಾಗಿ ಗರೆನಾ ಗೇಮ್ಸ್ನ ಗರೆನಾ ಫ್ರೀ ಫೈರ್ ಈಗಾಗಲೇ ಭಾರತದಲ್ಲಿ ಜನಪ್ರಿಯ ಶೂಟರ್ ಆಟವಾಗಿದೆ. ಅಂದಾಜು ಜೀವಿತಾವಧಿಯ ಆದಾಯ 15.3 ಮಿಲಿಯನ್ ಸರಿಸುಮಾರು 114 ಕೋಟಿ ರೂ. ಮತ್ತು 13.02 ಕೋಟಿ ಡೌನ್ಲೋಡ್ಗಳನ್ನು ಸಂಗ್ರಹಿಸಿದೆ. | kannad |
Nanaimo Foodshare Society is a registered non-profit charitable organization.
Strength in membership gives us the strength to achieve our goals. Your membership helps us and costs just $10 for a lifetime membership.
To become a Foodshare member, fill out the form below, or feel free to stop by the Foodshare Centre at 271 Pine Street.
Please take a moment to review the Nanaimo Foodshare Society draft Bylaws and Constitution.
Once you click "submit," you will be redirected to a page where you can purchase a lifetime membership for $10. | english |
திமுகவுக்கு 4வது தொடர் வெற்றி: ஸ்டாலின் தலைமைக்கு கிடைத்த அசுர பலம் தமிழக அரசியலில் 5 ஆண்டுகளுக்கு முன்னர், தொடர் தோல்விகளை சந்தித்த திமுக, கடந்த மக்களவைத் தேர்தல் முதல் மு.க.ஸ்டாலின் தலைமையில் தொடர் வெற்றிகளைப் பெற்று அசுர பலத்தை வெளிப்படுத்தியுள்ளது. முன்னாள் முதல்வர் மு.கருணாநிதி உயிருடன் இருந்தபோது, முதுமையின் காரணமாக தனது அரசியல் பயணங்களை சற்று குறைத்துக்கொண்டபோதே, அவருடைய மகன் மு.க.ஸ்டாலின் திமுகவில் முழு வீச்சில் செயல்படத் தொடங்கிட்டார். அப்போதே, திமுகவில் கருணாநிதிக்கு பிறகு மு.க.ஸ்டாலின் தலைவர் என்பது உறுதியாகிவிட்டது. 2011 சட்டமன்றத் தேர்தல் தோல்விக்கு பிறகு, திமுக 2016 சட்டமன்றத் தேர்தலை, மு.க.ஸ்டாலின் தலைமையில்தான் சந்தித்தது என்று கூறலாம். மு.க. ஸ்டாலின் நமக்கு நாமே திட்டம் என்று தீவிர சுற்றுப் பயணம் செய்தபோதும், அதிமுகதான் வெற்றி பெற்று ஆட்சி அமைத்தது. திமுக ஆட்சியைப் பிடிக்க முடியாவிட்டாலும் வலுவான எதிர்க்கட்சியாக அமைந்தது. அதற்கு முன்னதாக, 2014 மக்களவைத் தேர்தலில் திமுக படுதோல்வி அடைந்தது. இப்படி திமுக தொடர் தோல்விகளை சந்தித்து வந்த நிலையில், ஆகஸ்ட் 7, 2018ல் திமுக தலைவர் கருணாநிதி மறந்தார். அதே போல, அதிமுக பொது செயலாளர் ஜெயலலிதா டிசம்பர் 5, 2016ல் மறைந்தார். தமிழக அரசியலில் இருதுருவ பெரும் தலைவர்களாக இருந்த இருவரின் மறைவு ஒரு வெற்றிடத்தை ஏற்படுத்தியது. இப்படி 2011ம் ஆண்டு முதல் தொடர் தோல்விகளை சந்தித்து வந்த திமுக, மு.க.ஸ்டாலின் தலைவரான பிறகு, 2019ம் ஆண்டு மக்களவைத் தேர்தல் முதல் அதன் வெற்றி பயணம் தொடங்கியது. திமுக தலைவரான மு.க.ஸ்டாலின், 2019 மக்களவைத் தேர்தலில் நாடு முழுவதும் பாஜக மற்றும் மோடி அலை அடித்தாலும் தமிழகத்தில் மு.க.ஸ்டாலின் தலைமையிலான திமுக காங்கிரஸ் கூட்டணி 38 இடங்களை வென்றது. இது மு.க.ஸ்டாலின் தலைமையில் திமுக பெற்ற முதல் வெற்றி. தமிழகத்தில் மக்களவைத் தேர்தல் வெற்றி பாஜகவுக்கு எதிரான மனநிலை காரணமாக வெற்றி என்று அரசியல் நோக்கர்கள் கூறினார்கள். இப்படி சொல்வதற்கான காரணம், அப்போது நடைபெற்ற இடைத் தேர்தலில் அதிமுக போதுமான இடங்களில் வெற்றி பெற்று ஆட்சியை தக்கவைத்து கொண்டது. இதையடுத்து, 2020ம் ஆண்டு ஊரக உள்ளாட்சி தேர்தல் நடைபெற்றது. இதில், அப்போது ஆளும் கட்சியாக இருந்த அதிமுகவை விட அதிக இடங்களில் வெற்றி பெற்றது. இது மு.க.ஸ்டாலின் தலைமையில் திமுக பெற்ற 2வது வெற்றி. இதனைத் தொடர்ந்து, ஏப்ரல் மே, 2021ல் நடைபெற்ற தமிழ்நாடு சட்டமன்றத் தேர்தலில் திமுக கூட்டணி 159 இடங்களில் வெற்றி பெற்று ஆட்சியை பிடித்தது. இதில் திமுக மட்டும் 133 இடங்களில் வெற்றி பெற்றது. மு.க.ஸ்டாலின் தலைமையில் திமுக பெற்ற வெற்றி 3வது வெற்றி இது. முதலமைச்சரான மு.க.ஸ்டாலின், கடந்த காலங்களில் திமுக ஆட்சியில் இருந்தபோது நடந்த தவறுகள் மீண்டும் நடந்துவிடக் கூடாது என்ற எண்ணத்துடன் கட்சியையும் ஆட்சியையும் நடத்தி வருவதாக அரசியல் நோக்கர்கள் பலரும் கருத்து தெரிவித்து வருகின்றனர். இந்த சூழலில்தான், திமுக 9 மாவட்டங்களின் ஊரக உள்ளாட்சி தேர்தலை சந்தித்தது. நேற்று நடைபெற்ற வாக்கு எண்ணிக்கையில் திமுக ஒரு ஸ்வீப் அடித்து வெற்றி பெற்றுள்ளது. 9 மாவட்ட ஊரக உள்ளாட்சி தேர்தல் முடிவில் மாவட்ட கவுன்சிலர்கள் மொத்தம் 140 இடங்களில் 138 இடங்களை திமுக முன்னிலை வகிக்கிறது. அதிமுக 2 இடங்களில் முன்னிலை வகிக்கிறது. அதே போல, ஒன்றிய கவுன்சிலர்கள் 1381 இடங்களில் திமுக 994 வார்டுகளில் முன்னிலை பெற்றுள்ளது. அதிமுக 200 இடங்களில் முன்னிலை பெற்றுள்ளது. மற்றவை 139 இடங்களில் முன்னிலை வகிக்கின்றன. 9 மாவட்ட ஊரக உள்ளாட்சி தேர்தலில் திமுக பெற்றுள்ள இந்த பெரும் வெற்றி என்பது முதலமைச்சர் மு.க.ஸ்டாலின் தலைமையில் அசுரபலம் பெற்றுள்ளதைக் காட்டுகிறது. இந்த வெற்றி குறித்து திமுக அமைப்பு செயலாளர் ஆர்.எஸ்.பாரதி ஊடகங்களில் கூறுகையில், ஊரக உள்ளாட்சித் தேர்தல்களில் இந்த அளவுக்கு ஒரு பெரும் வெற்றியை நாங்கள் எதிர்பார்த்திருந்தாலும், மக்கள் எங்கள் மீது வைத்திருக்கும் நம்பிக்கையைக் கண்டு மகிழ்ச்சி அடைகிறோம். இது எங்கள் தலைவர் மு.க. ஸ்டாலின் தலைமையிலான திமுக அரசின் மீது அனைத்துத் தரப்பு மக்களின் நம்பிக்கையின் பிரதிபலிப்பாகும். என்று கூறினார். மேலும், அவர் உள்ளாட்சி அமைப்புகளின் வளர்ச்சிக்கான ஐந்தாண்டு திட்டத்தை ஊக்குவிப்பதற்கான தி.மு.க அரசின் முன்மொழிவு, அனைத்து வரித் துறைகளிலிருந்தும் முழுமையான ஒத்துழைப்பு ஆதரவு அளிப்பதாக உறுதியளித்தது. இந்த நடவடிக்கை உள்ளாட்சி அமைப்புகளுக்கு அதிகாரம் அளிப்பது மட்டுமல்லாமல், பொதுமக்களுக்கு சிறப்பாக சேவை செய்வதற்காக தேர்ந்தெடுக்கப்பட்ட பிரதிநிதிகளின் அணுகுமுறையில் ஒரு முன்னுதாரண மாற்றத்திற்கு வழிவகுக்கும் என்று கூறினார். திமுக தலைவர் மு.க.ஸ்டாலின் தலைமையில் திமுக 4வது தொடர் வெற்றி பெற்று மக்களின் நம்பிக்கையுடன் அசுர பலம் பெற்று நிற்கிறது. இந்த வெற்றி மு.க.ஸ்டாலின் தலைமைக்கு கிடைத்த அசுர பலம் என்றால் மிகையல்ல. | tamil |
کنگسٹن اردو پوائنٹ اخبار تازہ ترین14جون2015ء ویسٹ انڈیز اور سٹریلیا کے مابین دوسرا ٹیسٹ فیصلہ کن مرحلے میں داخل ہوگیا کالی ندھی کو میچ جیتنے کے لئے 376ر نز اور سٹریلیا کو 8وکٹیں درکار جبکہ میچ کے ابھی دو دن باقی ہیں ویسٹ انڈیز نے اپنی دوسری اننگز میں وکٹوں کے نقصان پر 16رنز بنا ئےقبل ازیں ویسٹ انڈیز اپنی پہلی اننگز میں 220 رنز بنا کر ہوگئی ویسٹ انڈیز کی جانب سے جیسن ہولڈر 82 اور بلیک ووڈ 51رنز بنا کر نمایاں بلے باز رہے کینگروز کی جانب سے جوش ہیزل وڈ نے 5وکٹیں حاصل کیں سٹریلیا نے اپنی پہلی اننگز میں 179 رنز کی مجموعی برتری حاصل کرنے کے بعد دوسری اننگز میں دو وکٹوں کے نقصان پر 212رنز پر ڈیکلیئر کردی اور میزبان ٹیم کو جیت کیلئے 392رنز کا ہدف دیا سٹریلیا کی جانب سے دوسری اننگز میں شون مارش69 ڈیوڈ وارنر62 اور اسٹیون اسمتھ نے 54 رنز سکور کئے تفصیلات کے مطابق کنگسٹن میں جاری ویسٹ انڈیز اور سٹریلیا کے مابین دوسرے ٹیسٹ میں تیسرے دن کا کھیل شروع ہوا تو ویسٹ انڈیز نے اپنی پہلی اننگز 143رنز 8وکٹوں پر دوبارہ شروع کی تو ان کی پوری ٹیم 595اوورز میں 220رنز کے اسکور پر ہوگئی ٹیم کی جانب سے جیسن ہولڈر نے 82رنز بنائے تاہم کینگروز کی جانب سے جوش ہیزل وڈ نے 5وکٹیں حاصل کیں اور کینگروز ٹیم کو کالی اندھی ٹیم پر 179رنز کی مجموعی برتری حاصل ہوگئی سٹریلیا نے اپنی دوسری اننگز کا غاز 179رنز کی برتری کے ساتھ دوبارہ شروع کیا تو شون مارش69 ڈیوڈ وارنر62 نے کالی اندھی کے بالرز کا بھرپور سامنا کیا اور 336اوورز میں 117رنز کی پارٹنرشپ پہلی وکٹ پر کر ڈالی اسٹیون اسمتھ نے بھی54 کے ساتھ نمایاں کردار ادا کیا سٹریلوی کپتان نے 65اوور کے اختتام پر اننگز ڈکلیر کردی اس وقت تک ٹیم کا اسکور 212رنز 2وکٹوں کے نقصان پر تھا اور ویسٹ انڈیز کو جیتنے اور سیریز برابر کرنے کے لیے 392رنز کا ہدف دے دیا تیسرے دن کا کھیل ختم ہونے تک ویسٹ انڈیز نے دو وکٹوں کے نقصان پر 16رنز بنا لئے ویسٹ انڈیز کے دونوں اوپنرز بغیر کوئی رنز بنائے پویلین لوٹ گئے سٹریلیا کی جانب سے مچل سٹارک نے دونوں وکٹیں حاصل کیں | urdu |
Budget Session: बिहार विधान परिषद का 200वां सत्र आज से शुरू, 31 मार्च तक होंगी कुल 22 बैठकें राज्यपाल के अभिभाषण के बाद विधानसभा और विधान परिषद की कार्यवाही अलगअलग शुरू होगी. बिहार विधान परिषद का यह 200वां सत्र है, जिसमें आज वित्तीय वर्ष 202122 का आर्थिक सर्वेक्षण पेश होगा.Click here to get the latest updates on State Elections 2022 बजट सत्र 31 मार्च तक चलेगा.पटना: बिहार विधानमंडल का बजट सत्र Budget session of Bihar Legislature आज से शुरू हो रहा है. बजट सत्र में कुल 22 Economic Survey 202122 बैठकें होंगी. आज दोनों सदनों में आर्थिक सर्वेक्षण 202122 पेश होगा. उसके पहले राज्यपाल फागू चौहान Governor Fagu Chauhan विधानमंडल के सेंट्रल हॉल में दोनों सदनों की संयुक्त बैठक को संबोधित करेंगे. वहीं विधान परिषद का ये 200वां सत्र होगा, जो आज से शुरू हो रहा है.: बिहार विधानसभा का बजट सत्र आज से शुरू, सर्वेक्षण रिपोर्ट पेश करेंगे वित्त मंत्रीबजट सत्र में राज्यपाल के अभिभाषण के बाद विधानसभा और विधान परिषद की कार्यवाही अलगअलग शुरू होगी. विधान परिषद के कार्यकारी सभापति के प्रारंभिक संबोधन के बाद राज्यपाल के अभिभाषण की प्रति सदन में रखी जाएगी, विधान परिषद के दूसरे सत्र के लिए अध्ययन सदस्यों की तालिका की घोषणा होगी. बिहार विधान परिषद का यह 200वां सत्र है, जिसमें आज वित्तीय वर्ष 202122 का आर्थिक सर्वेक्षण पेश होगा. बजट सत्र 31 मार्च तक चलेगा.: विधानसभा अध्यक्ष विजय सिन्हा की मुख्य सचिव और डीजीपी के साथ बैठक, बजट सत्र को लेकर चर्चावित्तीय वर्ष 202223 का बजट 28 फरवरी को पेश होगा. बजट पेश होने के बाद राज्यपाल के अभिभाषण के धन्यवाद प्रस्ताव पर वाद विवाद शुरू होगा. 2 मार्च को तीसरी बैठक के दिन विधानसभा में राज्यपाल के अभिभाषण पर वाद विवाद के बाद सरकार अपना उत्तर देगी. 3 मार्च को वित्तीय वर्ष 202223 के आयव्यय पर सामान्य विमर्श होगा. 3 मार्च को ही 202122 का तृतीय अनुपूरक बजट भी पेश किया जाएगा. 4 मार्च को वित्तीय वर्ष 202223 के आयव्यय पर सरकार का उत्तर होगा. 7 मार्च को तृतीय अनुपूरक बजट पर सरकार का उत्तर होगा. 8 से 25 मार्च तक वित्तीय वर्ष 202223 के अनुदानों की मांग पर वाद विवाद और मतदान होगा. 28 मार्च को राजकीय विधेयक, 29 को गैर सरकारी संकल्प, जबकि 30 मार्च को राजकीय विधेयक और 31 मार्च को अंतिम दिन गैर सरकारी सदस्यों के कार्य पूरे होंगे. पूरे बजट सत्र में 22 बैठकें होंगी.: बिहार विधानमंडल के बजट सत्र को लेकर कांग्रेस की बैठक, कहा जनहित के मुद्दे उठाने के लिए हम अकेले ही काफीइधर विपक्ष ने सरकार को घेरने की पूरी तैयारी कर ली है. राष्ट्रीय जनता दल के मुख्य प्रवक्ता भाई वीरेंद्र ने कहा कि जातीय जनगणना और विशेष राज्य के मुद्दे के अलावा बिहार में बढ़ते अपराध, भ्रष्टाचार और बेरोजगारी को लेकर सदन में हम जनता के सवालों को पुरजोर तरीके से रखेंगे. जिसका जवाब सरकार को देना होगा.: बजट सत्र में नीतीश सरकार की नैया डूब जाएगी, RJD प्रवक्ता का बड़ा बयान | hindi |
اسلام اباد اردو پوائنٹ اخبارتازہ ترین اے پی پی 17 اگست2020ء گزشتہ مالی سال 201920ء کے دوراں خطے کے ممالک کو کی جانے والی برامدات میں 24 فیصد کمی ہوئی ہے برامدات میں کمی کا بنیادی سبب مارچ 2020ء میں کوروناوائرس کی وبا کے پھیلائو پر قابو پانے کے لئے لگایا جانے والا لاک ڈائون ہے سٹیٹ بینک اف پاکستان ایس بی پی کے اعداد وشمار کے مطابق گزشتہ مالی سال میں خطے کے ممالک افغانستان چین بنگلہ دیش سری لنکا بھارت ایران نیپال بھوٹان اور مالدیپ کو کی جانے والی قومی برامدات 3738 ارب ڈالر تک کم ہو گئیں جبکہ مالی سال 201819ء کے دوران برامدات کاحجم 4678 ارب ڈالر رہا تھا اس طرح مالی سال 2019ء کے مقابلہ میں گزشتہ مالی سال 2020 کے دوران ہمسایہ ممالک کوکی جانے والی برامدات میں 92 کروڑ ڈالر یعنی 24 فیصد کی کمی ریکارڈ کی گئی ہے دوسری جانب درامدات میں کمی کے باعث ہمسایہ ممالک سے کی جانے والی تجارت کے خسارہ میں بھی کمی ہوئی ہے ایس بی پی کے مطابق مالی سال کے دوران افغانستان کو کی جانے والی برامدات 255 فیصد کی کمی سے 1192 ارب ڈالر کے مقابلہ میں 888913 ملین ڈالر تک کم ہو گئیں اسی طرح چین کو کی جانے والی برامدات بھی 105 فیصد کم ہو گئیں اور گزشتہ مالی سال 2020ء کے دوران برامدات کاحجم 1663 ارب ڈالر تک کم ہو گیا جبکہ مالی سال2019ء میں چین کو 1858 ارب ڈالر کی برامدات کی گئی تھیں سٹیٹ بینک کے مطابق گزشتہ مالی سال کے دوران بھارت کو کی جانے والی قومی برامدات میں سب سے زیادہ 908 فیصد کی کمی ریکارڈ کی گئی ہے مالی سال 2019ء کے دوران بھارت کو 311958 ملین ڈالر کی برامدات کی گئی تھیں جو گزشتہ مالی سال 2020ء کے دوران 28644 ملین ڈالر تک کم ہو گئیں اسی طرح ایران کو کی جانے والی برامدات 2942 ملین ڈالر کے مقابلہ میں 0055 ملین ڈالر تک کم ہو گئیں جبکہ بنگلہ دیش کو کی جانے والی برامدات کا حجم بھی 679 فیصد کی کمی سی 744720 ملین ڈالر کے مقسابلہ میں 694124 ملین ڈالر تک کم ہو گیا مزید براں سری لنکا کو کی جانے والی برامدات میں گزشتہ مالی سال کے دوران 549 فیصد کمی ریکارڈ کی گئی ہے اور برامدات کا حجم 303761 ملین ڈالر کے مقابلہ میں 287941 ملین ڈالر تک کم ہو گیا دوسری جانب نیپال کو کی جانے والی ملکی برامدات گزشتہ مالی سال میں 867 فیصد اضافہ سے 21679 ملین ڈالر تک پہنچ گئیں جبکہ مالی سال 2019ء میں نیپال کو 2872 ملین ڈالر کی برامدات کی گئی تھیں جبکہ مالدیپ کو کی جانے والی برامدات بھی 3736 فیصد اضافہ سے 6172 ملین ڈالر کے مقابلہ میں 8478 ملین ڈالر تک بڑھ گئیں اس طرح گزشتہ مالی سال کے دوران ہمسایہ ممالک کو کی جانے والی قومی برامدات میں مجموعی طور پر 24 فیصد کمی ریکارڈ کی گئی ہے | urdu |
ಬಿಗ್ ನ್ಯೂಸ್: ಶಾಲೆಗಳ ಭಾಗಶಃ ಪುನಾರಂಭಕ್ಕೆ ಸರ್ಕಾರದಿಂದ ಮಾರ್ಗಸೂಚಿ ಧಾರವಾಡ: ರಾಜ್ಯದಲ್ಲಿ ಭಾಗಶಃ ಶಾಲೆ ಪುನಾರಂಭಕ್ಕೆ ಶೀಘ್ರವೇ ಮಾರ್ಗಸೂಚಿ ಹೊರಡಿಸಲಾಗುವುದು. ಪ್ರಾಥಮಿಕ ಮತ್ತು ಪ್ರೌಢಶಿಕ್ಷಣ ಖಾತೆ ಸಚಿವ ಎಸ್. ಸುರೇಶ್ ಕುಮಾರ್ ಈ ಬಗ್ಗೆ ಮಾಹಿತಿ ನೀಡಿ, 9 ರಿಂದ 12 ನೇ ತರಗತಿವರೆಗಿನ ಮಕ್ಕಳು ಶಾಲೆಗೆ ಆಗಮಿಸಿ ಶಿಕ್ಷಕರು ಮಾರ್ಗದರ್ಶನ ಪಡೆಯಲು ಸೆಪ್ಟೆಂಬರ್ 21 ರಿಂದ ಅವಕಾಶ ನೀಡಲಾಗಿದೆ ಎಂದು ಹೇಳಿದ್ದಾರೆ. ಕೇಂದ್ರ ಸರ್ಕಾರ ಶಾಲೆಗಳಿಗೆ 9 ರಿಂದ 12ನೇ ತರಗತಿ ಮಕ್ಕಳು ಬಂದು ಹೋಗಲು ಅವಕಾಶ ನೀಡಿದ್ದು ನಂತರದಲ್ಲಿ ಶಾಲೆಕಾಲೇಜು ತರಗತಿ ಆರಂಭಿಸಲು ಹಸಿರು ನಿಶಾನೆ ಸಿಕ್ಕಲ್ಲಿ ಮುನ್ನೆಚ್ಚರಿಕೆ ಕ್ರಮ ಕೈಗೊಂಡು ಆರಂಭಿಸಲಾಗುವುದು. ಸದ್ಯಕ್ಕೆ 9 ರಿಂದ 12 ನೇ ತರಗತಿಯ ಮಕ್ಕಳು ಶಾಲೆಗೆ ಆಗಮಿಸಿ ಶಿಕ್ಷಕರ ಮಾರ್ಗದರ್ಶನ ಪಡೆಯಲು ಅವಕಾಶವಿದ್ದು ಶನಿವಾರ, ಭಾನುವಾರದ ವೇಳೆಗೆ ಭಾಗಶಃ ಶಾಲೆ ಆರಂಭಕ್ಕೆ ಮಾರ್ಗಸೂಚಿ ಸಿದ್ಧಪಡಿಸಿ ಪ್ರಕಟಿಸಲಾಗುವುದು ಎಂದು ಹೇಳಲಾಗಿದೆ. | kannad |
حال ہی میں حکومت کی جانب سے سٹیزن پروٹیکشن اگینسٹ لائن ہارم ایکٹ 2020 پر اسٹیک ہولڈرز سے مشاورت کا غاز کیا گیا تھااگرچہ زیادہ تر ڈیجیٹل رائٹس پلیٹ فارمز اور گروپوں نے حکومت سے مذکورہ بل پر مشاورت کا بائیکاٹ کیا تھا تاہم اس کے باوجود حکومت نے مشاورت کا غاز جون سے شروع کیاسٹیزن پروٹیکشن اگینسٹ لائن ہارم ایکٹ 2002 کو ابتدائی طور پر وفاقی کابینہ نے منظور کیا تھا تاہم انسانی حقوق کی تنظیموں ڈیجیٹل رائٹس گروپس اور ٹیکنالوجی کمپنیوں سمیت غیر ملکی اداروں نے بھی اس پر تنقید کی تھیجس کے بعد وزیر اعظم نے مذکورہ ضوابط کو معطل کرتے ہوئے اس پر پاکستان ٹیلی کمیونیکیشن اتھارٹی پی ٹی اے کو وسیع تناظر میں مشاورت کی ہدایت کی تھیوزیر اعظم کی ہدایت کے بعد پی ٹی اے نے مشاورتی کمیٹی تشکیل دے کر اسٹیک ہولڈرز سے مشاورت کا غاز کیا تھا اور اس کا پہلا اجلاس جون کو ہوایہ بھی پڑھیں حکومت نے متنازع لائن قوانین پر مشاورت شروع کردیمذکورہ کمیٹی میں ایڈیشنل سیکریٹری ئی ٹی اعزاز اسلم ڈار ڈیجیٹل پاکستان کی سربراہ تانیہ ایدروس اور ان کے ڈیجیٹل میڈیا کے فوکل پرسن ڈاکٹر ارسلان خالد انسانی حقوق کی وفاقی وزیر ڈاکٹر شیریں مزاری اور بیریسٹر علی ظفر شامل ہیںتاہم متعدد ڈیجیٹل رائٹس گروپس اور ٹیکنالوجی کمپنیوں نے مشاورتی کمیٹی کے ساتھ کسی بھی طرح کی مشاورت کرنے سے انکار کرتے ہوئے حکومت سے مطالبہ کیا کہ متنازع لائن قوانین کو واپس لیا جائےان قوانین پر مشاورت کے لیے جون کو ہونے والے اجلاس میں انکشاف ہوا کہ دراصل اس وقت پاکستان میں جعلی خبروں کو انٹرنیٹ اور سوشل میڈیا پلیٹ فارمز سے ہٹانا غیر قانونی ہےڈان اخبار کے مطابق جون کو صحافیوں اور سول سوسائٹی کے ارکان کے ساتھ ہونے والے الگ مشاورتی اجلاس میں پی ٹی اے کی مشاورتی کمیٹی نے انکشاف کیا کہ اس وقت ملک میں جعلی خبروں کو ہٹائے جانے کا کوئی قانون موجود نہیںاجلاس کے دوران بتایا گیا کہ سوشل میڈیا پر جعلی اکانٹس کے ذریعے فحش نازیبا مواد کی تشہیر سمیت نفرت انگیز گستاخانہ اور نسلی تفریق پر مبنی مواد کے پھیلا سمیت کسی کی ذاتی کردارکشی کرنے جیسے عوامل عام ہیںمزید پڑھیں ٹیکنالوجی ادارے متنازع لائن قوانین واپس لینے کے خواہاںپی ٹی اے چیئرمین عامر عظیم باجوہ نے اجلاس کے دوران بتایا کہ انفرادی طور پر جعلی خبریں قابل سزا نہیں پی ٹی اے صرف ان جعلی خبروں اور افواہوں کو رپورٹ کرتا ہے جو عوام میں خوف اور بے یقینی کی صورتحال پیدا کریںان کا کہنا تھا کہ اگر کسی حقائق کے بغیر رپورٹ کو بنیاد بناکر جعلی خبریں پھیلائی جا رہی ہوں تو وہ ایسی خبروں کو رپوٹ نہیں کرتے انہوں نے بتایا کہ پی ٹی اے کو زیادہ تر ایسی جعلی خبروں کو شکایات موصول ہوتی ہیں جن میں کسی کی ذاتی بدنامی یا کردار کشی کی جاتی ہےاجلاس میں بتایا گیا کہ پی ٹی اے کے تنظیم نو ایکٹ 1996 اور دی پریونشن الیکٹرانک کرائمز ایکٹ پی ای سی اے پیکا 2016 میں جعلی خبروں کو محدود رکھنے کے حوالے سے کوئی شق موجود نہیںاجلاس کے دوران انکشاف کیا گیا کہ ملک میں جعلی خبروں کو ہٹانا غیر قانونی ہوسکتا ہےاجلاس کو بتایا گیا کہ نئے ڈرافٹ میں سوشل میڈیا سائٹس سے اس بات کا مطالبہ کیا گیا ہے کہ وہ ایسی جعلی خبروں کو ہٹانے کی پابند ہوں گی جو کسی کی بدنامی پاکستان کی مذہبی ثقافتی نسلی یا قومی سلامتی کی حساسیت کی خلاف ورزی کرتی ہوں | urdu |
ઘેટાંબકરીના દૂધમાંથી દસ પ્રકારના ચીઝનું ઉત્પાદન સુરેન્દ્રનગરના બે માલધારી યુવાનોએ શરૂ કર્યો નવો જ વ્યવસાય ગુજરાતમાં જેની સૌથી ઓછી ડિમાન્ડ છે તેવા ઘેટા અને બકરીના દૂધમાંથી સુરેન્દ્રનગરનો બે યુવાનો અરફાન કલોત્રા અને ભીમસીભાઈ ધાંધલે ખાવામાટેના ચેવરે, ફેટા, બેરલ એજડ ફેટા સહિત દસ પ્રકારના ચીઝનું ઉત્પાદન શરૂ કર્યું છે. આ બન્ને યુવાનોએ ચેન્નાઈ ખાતે તાલિમ મેળવી વિવિધ પ્રકારના ચીઝનું ઉત્પાદન શરૂ કરી પરંપરાગત ગુજરાતી વાનગીઓ સાતે જોડવાનું શરૂ કરી એક નવા જ પ્રકારનો વ્યવસાય શરૂ કર્યો છે. ગુજરાતના રાબડી સમુદાયના પશુપાલક માલધારી પશુપાલકો દરરોજ ઘેટાં અને બકરીનું દૂધ પીવે છે. ગુજરાતમાં આ દૂધનું મોટા પ્રમાણમાં ઉત્પાદન થાય છે, પરંતુ તેની માંગ ઓછી છે તેથી દૈનિક સરપ્લસ છે, માલધારીઓ બગાડ ઘટાડવા માટે ઘણી રીતે તેનો ઉપયોગ કરે છે. અમે ચા બનાવીએ છીએ, દહીં બનાવીએ છીએ અને તેની સાથે ખોવા પણ બનાવીએ છીએ, તે કહે છે, અમે બાકીનું ગાયના દૂધમાં ભેળવીએ છીએ અને સ્થાનિકોને વેચીએ છીએ. ગયા વર્ષે માલધારીઓને ચીઝ બનાવવાની તાલીમ આપનાર ચેન્નાઈ સ્થિત કેસેના સહસ્થાપક નમ્રતા સુંદરેસન સમજાવે છે કે તેઓ જે ફેટા બનાવે છે તે સ્વાદિષ્ટ છે ટેરોરપ્રેરિત ચીઝ જ્યાં પશુધન અને જ્યાં દૂધ મેળવવામાં આવે છે તે પ્રદેશ દ્વારા તેનો સ્વાદ પ્રભાવિત થાય છે. ગુજરાતમાં પશુપાલકોની આજીવિકા માટે દૂધ કેન્દ્ર સ્થાને છે. સ્વદેશી દૂધના વધારા સાથે, અમને કારીગરી ચીઝ મેકિંગ દ્વારા પશુપાલન આજીવિકા વધારવાની એક અનોખી તક આપવામાં આવી છે, નમ્રતા કહે છે, તેમણે ઉમેર્યું કે આ પહેલ પશુપાલન સમુદાયના યુવાનો માટે એક સાહસિકતા તાલીમ કાર્યક્રમ તરીકે શરૂૂ થઈ હતી, અને અર્ફાનભાઈ અને ભીમસીભાઈ દ્વારા પસંદ કરવામાં આવ્યા હતા. આ બન્ને યુવાનો હવે 10 પ્રકારની ચીઝ બનાવે છે, જેમાં ચેવરે, ફેટા, બેરલ એજ્ડ ફેટા, પેકોરિનો ફ્રેસ્કો, ચેડર અને ટોમનો સમાવેશ થાય છે. તેઓએ તેમના ગામમાં 6 લાખના ખર્ચે સ્ટોરેજ સુવિધા સાથે પ્રોસેસિંગ યુનિટ સ્થાપ્યું છે. આટલા વર્ષોમાં ઘણું કિંમતી દૂધ વેડફાયું હતું. આજે, અમે તેનો શ્રેષ્ઠ ઉપયોગ કરવા માટે સશક્ત છીએ. અમારી પાસે 100 થી વધુ બકરીઓ છે અને અમે આ પ્રદેશના અન્ય પશુપાલકો પાસેથી બકરીનું દૂધ પણ ખરીદીએ છીએ કારણ કે અમને એક કિલોગ્રામ દૂધ બનાવવા માટે 100 લિટરની જરૂૂર છે. તેમ ભીમસીભાઈ કહે છે. ધો.10ની છાત્રાએ મોબાઈલ લેવાની જીદ સાથે કર્યો આપઘાત રિલાયન્સ જનરલ ઇન્સ્યુરન્સ કંપનીને 1.23 લાખનો ક્લેમ ચૂકવવા હુકમ | gujurati |
ಜಸ್ಟ್ 100ರೂ. ಒಳಗೆ ಲಭ್ಯವಿರುವ ಜಿಯೋ, ಏರ್ಟೆಲ್, ವಿ ಪ್ಲ್ಯಾನ್ಗಳ ಮಾಹಿತಿ! ಭಾರತೀಯ ಟೆಲಿಕಾಂ ವಲಯದಲ್ಲಿ ಜಿಯೋ, ಏರ್ಟೆಲ್, ವೊಡಾಫೋನ್ಐಡಿಯಾ ಟೆಲಿಕಾಂಗಳು ಆಕರ್ಷಕ ಪ್ರೀಪೇಯ್ಡ್ ಪ್ಲ್ಯಾನ್ಗಳಿಂದ ಗ್ರಾಹಕರ ಗಮನ ಸೆಳೆದಿದೆ. ಈ ಟೆಲಿಕಾಂಗಳು ಭಿನ್ನ ಪ್ರೈಸ್ಟ್ಯಾಗ್ನಲ್ಲಿ ಹಲವು ಪ್ರೀಪೇಯ್ಡ್ ಯೋಜನೆಗಳ ಲಿಸ್ಟ್ ಹೊಂದಿದ್ದು, ಅವುಗಳಲ್ಲಿ ಅಗ್ಗದ ಬೆಲೆಯಲ್ಲಿಯೂ ಬೆಸ್ಟ್ ಪ್ಲ್ಯಾನ್ಗಳನ್ನು ಹೊಂದಿವೆ. ಇನ್ನು ಕೆಲವು ಯೋಜನೆಗಳು 100ರೂ. ಒಳಗೂ ಲಭ್ಯ ಇವೆ. ವೊಡಾಫೋನ್ ಹೌದು, ಜಿಯೋ, ಏರ್ಟೆಲ್, ವೊಡಾಫೋನ್ಐಡಿಯಾ ಸಂಸ್ಥೆಗಳು 100ರೂ. ಒಳಗೆ ಕೆಲವು ಪ್ರೀಪೇಯ್ಡ್ ಯೋಜನೆಗಳನ್ನು ಹೊಂದಿವೆ. ಈ ಅಗ್ಗದ ಯೋಜನೆಗಳ ಡೇಟಾ ಸೌಲಭ್ಯ ಸೇರಿದಂತೆ ವಾಯಿಸ್ ಕರೆ, ಎಸ್ಎಮ್ಎಸ್ ನಂತಹ ಪ್ರಯೋಜನಗಳನ್ನು ಪಡೆದಿವೆ. ಹಾಗಾದರೇ ಜಿಯೋ, ಏರ್ಟೆಲ್, ವಿ ಟೆಲಿಕಾಂಗಳ 100ರೂ ಬೆಲೆಯ ಒಳಗೆ ಇರುವ ಪ್ರೀಪೇಯ್ಡ್ ಪ್ಲ್ಯಾನ್ಗಳ ಪ್ರಯೋಜನಗಳೆನು ಎಂಬುದನ್ನು ತಿಳಿಯಲು ಮುಂದೆ ಓದಿರಿ. ಏರ್ಟೆಲ್ ಅಗ್ಗದ ಯೋಜನೆಗಳು ಏರ್ಟೆಲ್ ಪ್ರಸ್ತುತ 100ರೂ ಬೆಲೆಯ ಒಳಗೆ ನಾಲ್ಕು ಯೋಜನೆಗಳನ್ನು ಹೊಂದಿದೆ. ಏರ್ಟೆಲ್ 19ರೂ. ಪ್ಲ್ಯಾನ್ ಎರಡು ದಿನಗಳ ಮಾನ್ಯತೆಗಾಗಿ 200 ಎಂಬಿ ಡೇಟಾವನ್ನು ನೀಡುತ್ತದೆ. ಏರ್ಟೆಲ್ 49ರೂ. ಮತ್ತು ಏರ್ಟೆಲ್ 79ರೂ. ಪ್ರಿಪೇಯ್ಡ್ ಯೋಜನೆಗಳು 28 ದಿನಗಳ ಮಾನ್ಯತೆಯೊಂದಿಗೆ ಬರುತ್ತವೆ. ಹಾಗೂ ಕ್ರಮವಾಗಿ 100MB ಮತ್ತು 200 MB ಡೇಟಾ ಸೌಲಭ್ಯ ಪಡೆದಿವೆ. ಹಾಗೆಯೇ ಏರ್ಟೆಲ್ 48ರೂ. ಪ್ಲ್ಯಾನ್ 28ದಿನಗಳ ವ್ಯಾಲಿಡಿಟಿ ಜೊತೆಗೆ 3GB ಡೇಟಾ ನೀಡುತ್ತದೆ. ಜಿಯೋದ ಅಗ್ಗದ ಯೋಜನೆಗಳು ಜಿಯೋ ಟೆಲಿಕಾಂ ಸಹ 100ರೂ. ಒಳಗೂ ಆಕರ್ಷಕ ಯೋಜನೆಗಳನ್ನು ಹೊಂದಿದೆ. ಆರಂಭಿಕ ಜಿಯೋ 10ರೂ. ಪ್ರಿಪೇಯ್ಡ್ ರೀಚಾರ್ಜ್ ಯೋಜನೆಯು 1 ಜಿಬಿ ಕಾಂಪ್ಲಿಮೆಂಟರಿ ಡೇಟಾದೊಂದಿಗೆ 124 ಐಯುಸಿ ನಿಮಿಷಗಳ ಟಾಕ್ಟೈಮ್ ಪ್ರಯೋಜನವನ್ನು ನೀಡುತ್ತದೆ. ಅದೇ ರೀತಿ ರೂ. 20 ಪ್ರಿಪೇಯ್ಡ್ ಯೋಜನೆ 249 ಐಯುಸಿ ನಿಮಿಷಗಳ ಟಾಕ್ಟೈಮ್ ಪ್ರಯೋಜನಗಳೊಂದಿಗೆ 2 ಜಿಬಿ 4 ಜಿ ಡೇಟಾವನ್ನು ನೀಡುತ್ತದೆ. ರೂ. 50 ಮತ್ತು ರೂ. 100 ಯೋಜನೆಗಳು ಕ್ರಮವಾಗಿ 5 ಜಿಬಿ ಮತ್ತು 10 ಜಿಬಿ ಹೈಸ್ಪೀಡ್ ಡೇಟಾವನ್ನು ನೀಡುತ್ತವೆ. ಡೇಟಾದ ಜೊತೆಗೆ, ಎರಡು ಯೋಜನೆಗಳು ಕ್ರಮವಾಗಿ 656 ಮತ್ತು 1362 ಐಯುಸಿ ನಿಮಿಷಗಳನ್ನು ಸಹ ತರುತ್ತವೆ. ವಿ ಟೆಲಿಕಾಂನ ಅಗ್ಗದ ಯೋಜನೆಗಳು ವಿ ಟೆಲಿಕಾಂನಲ್ಲಿಯೂ ಅಗ್ಗದ ಪ್ರಿಪೇಯ್ಡ್ ಯೋಜನಗಳ ಆಯ್ಕೆ ಇದೆ. ವಿ 16ರೂ, ಪ್ಲ್ಯಾನಿನಲ್ಲಿ 24 ಗಂಟೆಗಳ ವ್ಯಾಲಿಡಿಟಿ ಜೊತೆಗೆ 1 ಜಿಬಿ ಡೇಟಾವನ್ನು ನೀಡುತ್ತದೆ. ವಿ 19ರೂ. ರೀಚಾರ್ಜ್ ಯೋಜನೆ, ಬಳಕೆದಾರರು 2 ದಿನಗಳ ಮಾನ್ಯತೆಗಾಗಿ 200 ಎಂಬಿ ಡೇಟಾ ಮತ್ತು ಅನಿಯಮಿತ ಟಾಕ್ಟೈಮ್ ಪಡೆಯುತ್ತಾರೆ. ಹಾಗೆಯೇ ವಿ 98ರೂ. ಯೋಜನೆಯಲ್ಲಿ 6 ಜಿಬಿ ಡೇಟಾ ಲಭ್ಯ, ಆದರೆ ಪ್ರಸ್ತುತ ಈ ಯೋಜನೆಗೆ ಆಫರ್ ಇದ್ದು, ಹೀಗಾಗಿ 6GB ಬದಲಾಗಿ 12 GB ಹೈಸ್ಪೀಡ್ ಡೇಟಾವನ್ನು ಲಭ್ಯ. source: gizbot.com | kannad |
దీపావళి బాణసంచా ఎఫెక్ట్: ఢిల్లీలో ప్రమాదకరస్థాయికి చేరిన వాయు కాలుష్యం న్యూఢిల్లీ:Diwali సందర్భంగా crackery పేల్చడం ద్వారా Delhiలో ప్రమాదకర స్థాయికి వాయు కాలుష్యం చేరింది. టపాసులపై నిషేధం విధించినా కూడ టపాసులు కాల్చడం వల్ల Air Pollution మరింత పెరిగింది. ఢిల్లీలో గురువారం నాడు సాయంత్రం నాలుగు గంటలకు ఎయిర్ క్వాలిటీ ఇండెక్స్ 382గా నమోదైంది. దీపావళిని పురస్కరించుకొని టపాసులు పేల్చడం వల్ల గాలిలో వాయు కాలుష్యం పెరిగింది. ఢిల్లీలోని పలు చోట్ల ఎయిర్ క్వాలిటీ ఇండెక్స్ 500 గా రికార్డైంది. పూసా రోడ్డు వద్ద ఎయిర్ క్వాలిటీ ఇండెక్స్ 505కి చేరిందని అధికారులు చెప్పారు. also read:వాతావరణ కాలుష్యం: హైదరాబాద్లో ఆ రెండు ఏరియాల్లోనే స్వచ్ఛమైన గాలి శుక్రవారం నాడు ఉదయం నగరంలోని జవహర్లాల్ నెహ్రు స్టేడియంలో 999 క్యూబిక్ మీటర్కు పార్టిక్యులేట్ మ్యాటర్ పీఎం 2.5 గా నమోదైంది. పొరుగున ఉన్న ఫరీదాబాద్ లో 424, ఘజియాబాద్ లో 4421, నోయిడాలో 431 గా నమోదైందని అధికారులు తెలిపారు. టపాకాయలు కాల్చడం ద్దవారా గాలిలో కాలుష్యం పెరిగిందని అధికారులు తెలిపారు. ఢిల్లీతో పాటు పలు రాష్ట్రాల్లో టపాకాయలు కాల్చడంపై నిషేధం విధించినా కూడ దీపావళిని పురస్కరించుకొని పెద్ద ఎత్తున టపాకాయలు కాల్చడం వల్ల వాయు కాలుష్యం పెరిగింది.దక్షిణ ఢిల్లీలోని లజ్పత్నగర్, ఉత్తర ఢిల్లీలోని బురారీ, పశ్చిమ ఢిల్లీలోని పశ్చిమ విహార్, తూర్పు ఢిల్లీలోని షహదారా వాసులు రాత్రి 7 గంటల వరకే టపాకాయలను కాల్చారు.నగరం శివారు ప్రాంతాల నుండి వచ్చిన ప్రజలు గొంతు దురద, కళ్లలో నీరు కారుతున్నట్టుగా ఫిర్యాదులతో ప్రజలు ఆసుపత్రులకు చేరుతున్నారు. హర్యానా ప్రభుత్వం జాతీయ రాజధాని ప్రాంతంలోని 14 జిల్లాల్లో అన్ని రకాల టపాకాయల వినియోగంపై నిషేధం విధించింది. సెంటర్ రన్ సిస్టమ్ ఆన్ ఎయిర్ క్వాలిటీ, వెదర్ పోర్ కాస్టింగ్ రీసెర్చ్ ప్రకారం ఆదివారం నాడు సాయంత్రం వరకు అంటే నవంబర్ 7వ తేదీ వరకు గాలి నాణ్యత మెరుగుపడే అవకాశం లేదని అధికారులు తెలిపారు. ప్రతికూల వాతావరణ పరిస్థితులు, తక్కువ ఉష్ణోగ్రత, బాణసంచా కాల్చడం వల్ల గాలి నాణ్యత తగ్గిపోయిందని అధికారులు అభిప్రాయపడుతున్నారు. ఢిల్లీలో 24 గంటల వాయు నాణ్యత సూచిక బుధవారం ఉదయం 314 గా గురువారం నాడు ఉదయం 382 గా నమోదైంది. మంగళవారం నాడు 303, సోమవారం నాడు 281గా రికార్డైందని అధికారులు ప్రకటించారు. సున్నా నుండి 50 మంది అవరేజీ క్వాలిటీ ఇండెక్స్ AQI మంచిదిగా చెబుతారు. 51 నుండి 100 సంతృప్తికరంగా, 101 నుండి 200 వరకు మితమైందిగా, 201 నుండి 300 వరకు గాలిలో నాణ్యత లేనిదిగా చెబుతారు. 301 నుండి 400 వరకు గాలిలో కాలుష్యం తీవ్రమైందిగా అధికారులు చెబుతున్నారు. 401 నుండి 500 వరకు గాలిలో కాలుష్యం మోతాదు మించిందిగా అధికారులు పరిగణిస్తారు. టపాకాయల విక్రయాలతో పాటు కాల్చడంపై రాష్ట్రాలు నిషేధం విధించినా కూడా విచ్చలవిడిగా బాణసంచా కాల్చడం ద్వారా వాయు కాలుష్యం పెరిగింది. అయితే నిషేధం విధించి దాన్ని అమలుకు పాలకులు చర్యలు తీసుకోని కారణంగానే ఈ పరిస్థితి నెలకొందనే విమర్శలు కూడా లేకపోలేదు. | telegu |
डाक विभाग दिल्ली में 10वीं पास के लिए सीधी भर्ती, नई दिल्ली, । India Post Delhi Recruitment 2022: भारतीय डाक विभाग में नौकरी या ड्राइवर की सरकारी नौकरी के इच्छुक उम्मीदवारों के लिए काम की खबर। भारतीय डाक विभाग के नई दिल्ली स्थित मेल मोटर सर्विस में स्टाफ कार ड्राइवर ऑर्डिनरी ग्रेड के पदों पर सीधी भर्ती के लिए विज्ञापन हाल ही में जारी किया गया है। विभाग द्वारा 17 जनवरी 2022 को जारी विज्ञापन के अनुसार जनरल सेंट्रल सर्विस ग्रुप सी, नॉनगजेटेड, नॉनमिनिस्ट्रियल कटेगरी में स्टाफ कार ड्राइवर की 29 रिक्तियों के लिए योग्य उम्मीदवारों से आवेदन आमंत्रित किए जा रहे हैं। विज्ञापित रिक्तियों की संख्या में 15 अनारक्षित हैं, जबकि 8 ओबीसी, 3 एससी और 3 ईडब्ल्यूएस उम्मीदवारों के लिए आरक्षित हैं। डाक विभाग दिल्ली भर्ती 2022 के लिए आवेदन प्रक्रिया डाक विभाग दिल्ली भर्ती 2022 के लिए आवेदन के इच्छुक उम्मीदवार आधिकारिक वेबसाइट, indiapost.gov.in पर भर्ती सेक्शन में दिए गए लिंक या नीचे दिए गए डायरेक्ट लिंक पर क्लिक करके स्टाफ कार ड्राइवर सीधी भर्ती विज्ञापन डाउनलोड कर सकते हैं। आवेदन के लिए अप्लीकेशन फॉर्म विज्ञापन में ही दिया गया है। इस फॉर्म को पूरी तरह से भरकर और मांगे गये डॉक्यूमेंट्स को संलग्न करते हुए निर्धारित आखिरी तारीख यानि 15 मार्च 2022 तक इस पते पर जमा कराएं सीनियर मैनेजर, मेल मोटर सर्विस, सी121, नारायणा इंडस्ट्रियल एरिया फेज1, नारायणा, नई दिल्ली110028। उम्मीदवारों को ध्यान देना चाहिए कि किसी भी अन्य मोड में आवेदन स्वीकार नहीं किए जाएंगे। | hindi |
குவாரன்டைனை முடிச்சாச்சு... கொரோனா டெஸ்ட்லயும் நெகட்டிவ்... டெஸ்ட்டுக்கு இந்திய அணி ரெடி! சென்னை : இந்தியா இங்கிலாந்து இடையிலான டெஸ்ட் தொடரின் முதல் 2 போட்டிகள் சென்னையில் நடைபெறவுள்ளது. வரும் 5ம் தேதி முதல் போட்டி நடைபெறவுள்ளநிலையில், தற்போது இந்திய வீரர்கள் தங்களது 6 நாட்கள் குவாரன்டைனை முடித்துள்ளனர். பெரிய சூழ்ச்சி.. வரிந்து கட்டிக்கொண்டு வரும் ஜாம்பவான்கள்..இந்திய அணிக்கு எதிராக இப்படி ஒரு திட்டமா மேலும் அவர்களுக்கு எடுக்கப்பட்ட அனைத்து கொரோனா பரிசோதனைகளிலும் நெகட்டிவ் வந்துள்ள நிலையில், நாளை முதல் நெட் பயிற்சிகளை துவக்க உள்ளனர். 4 போட்டிகளை கொண்ட தொடர் இந்தியா மற்றும் இங்கிலாந்து இடையிலான 3 வடிவங்களிலான தொடர்கள் திட்டமிடப்பட்டுள்ளன. முதலில் 4 போட்டிகளை கொண்ட டெஸ்ட் தொடர் துவங்கவுள்ள நிலையில், முதல் இரண்டு போட்டிகள் சென்னையில் வரும் 5ம் தேதி துவங்கி நடைபெறவுள்ளது. இதையொட்டி இரு அணி வீரர்களும் சென்னையில் உள்ளனர். கொரோனா பரிசோதனைகள் கடந்த வாரத்தில் சென்னை வந்த இந்திய வீரர்களுக்கு 6 நாட்கள் குவாரன்டைனில் ஈடுபட்ட நிலையில், தற்போது குவாரன்டைன் முடிந்துள்ளது. மேலும் அவர்களுக்கு 5 முறை கொரோனா பரிசோதனைகள் மேற்கொள்ளப்பட்டு நெகட்டிவ் ரிசல்ட் வந்துள்ளது. நாளை முதல் துவக்கம் இந்நிலையில் அவர்கள் ஹோட்டலில் இருந்து வெளியில் வந்து இன்றைய தினம் பயிற்சிகளில் ஈடுபட உள்ளனர். மேலும் நாளை முதல் நெட் பயிற்சிகளில் வீரர்கள் ஈடுபட உள்ளதாக தெரிவிக்கப்பட்டுள்ளது. தொடர்ந்து பயிற்சிகளில் ஈடுபடும் வீரர்கள் பயோ பபள் முறையையும் கடுமையாக கடைபிடிக்க அறிவுறுத்தப்பட்டுள்ளனர். ரசிகர்கள் அனுமதிக்கப்படுவார்களா? வரும் 5ம் தேதி மற்றும் 13ம் தேதிகளில் அடுத்தடுத்த இரு டெஸ்ட் போட்டிகளில் இந்தியா மற்றும் இங்கிலாந்து வீரர்கள் விளையாட உள்ளனர். இந்த போட்டிகளில் 50 சதவிகித பார்வையாளர்கள் அனுமதிக்கப்படுவார்களா என்ற எதிர்பார்ப்பு எழுந்துள்ளது. அவ்வாறு அனுமதிக்கும்பட்சத்தில் ஒரு வருடத்திற்கு பிறகு லைவ் போட்டியில் ரசிகர்கள் அனுமதிக்கப்படுவார்கள் என்பது குறிப்பிடத்தக்கது. | tamil |
Redmi Note 11S and Note 11 launched in India check price specifications offers discounts Tech news hindi Xiaomi ने अपने पॉपुलर मिडबजट रेंज स्मार्टफोन Redmi Note 11 और Redmi Note 11S को लॉन्च कर दिया है। Redmi के ये लेटेस्ट स्मार्टफोन Redmi Note 10 सीरीज के सक्सेसर के रूप में पेश हुए हैं। Redmi Note 11 और Redmi Note 11S के साथ कंपनी ने Redmi स्मार्ट टीवी और Redmi स्मार्ट बैंड प्रो स्मार्टवॉच को भी लॉन्च किया है। लॉन्च इवेंट को Redmi India के आधिकारिक YouTube चैनल पर लाइव स्ट्रीम किया गया था। आइए जानते हैं इन दोनों स्मार्टफोन्स के सभी वैरिएंट की कीमत और स्पेसिफिकेशन से जुड़ी हर एक डिटेल: Redmi Note 11 की भारत में कीमत 4GB 64GB स्टोरेज ऑप्शन के साथ आने वाले Redmi Note 11 की कीमत भारत 13,499 रुपये है। 6GB 64GB वैरिएंट की कीमत 14,499 रुपये है, जबकि 6GB 128GB वैरिएंट की कीमत 15,999 रुपये है। फोन की सेल 11 फरवरी से Amazon India, Mi.com, Mi Home Stores पर शुरू होगी। Redmi Note 11 के स्पेसिफिकेशन Redmi Note 11 में 90Hz रिफ्रेश रेट के साथ 6.43इंच फुल HD AMOLED डिस्प्ले है। फोन में 6GB तक रैम के साथ स्नैपड्रैगन 680 SoC है। फोन में 33W फास्ट चार्जिंग सपोर्ट के साथ 5000 एमएएच की बैटरी है। नोट 11 में 50MP का प्राइमरी कैमरा, 8MP अल्ट्रावाइड सेंसर और दो 2MP सेंसर के साथ क्वाडकैमरा सेटअप है। सेल्फी के लिए फोन में 8MP का फ्रंट कैमरा दिया गया है। डिवाइस आउट ऑफ द बॉक्स Android 11आधारित MIUI 13 चलाता है। Redmi Note 11S की भारत में कीमत Xiaomi ने Redmi Note 11S को तीन स्टोरेज ऑप्शन में लॉन्च किया है। बेस 6GB 64GB वैरिएंट की कीमत 16,499 रुपये है। यह 6GB 128GB और 8GB 128GB विकल्पों में भी आता है, जिनकी कीमत 17,499 रुपये और 18,499 रुपये है। डिवाइस की सेल 16 फरवरी से Amazon India, Mi.com, Mi Home Stores पर शुरू होगी। Redmi Note 11S के स्पेसिफिकेशंस Redmi Note 11S में 90Hz रिफ्रेश रेट के साथ 6.43इंच फुल HD AMOLED डिस्प्ले है। 16MP के फ्रंट कैमरे के लिए टॉप सेंटर में होलपंच कटआउट है। इसमें हुड के नीचे MediaTek Helio G96 प्रोसेसर है। पीछे की तरफ क्वाडकैमरा सेटअप है। मैन 108MP सेंसर वही है जो Xiaomi 11i HyperCharge रिव्यू और Xiaomi 11T Pro रिव्यू में मिलता है। फोन में डेप्थ और मैक्रो के लिए 8MP का अल्ट्रावाइड कैमरा और 2MP के दो सेंसर हैं। डिवाइस में 33W फास्ट चार्जिंग सपोर्ट के साथ 5000 एमएएच की बैटरी है। यह एक साइडमाउंटेड फिंगरप्रिंट स्कैनर और एक 3.5 मिमी हेडफोन जैक के साथ आता है। फोन में स्टीरियो स्पीकर सेटअप भी है। Follow Us: Google News Dailyhunt News Facebook Instagram Twitter Pinterest Tumblr esarkariresult.info | hindi |
The below information is valid for all planning to visit Bhutan. But one major area of difference is the mechanism used for tourist from Bangladesh, India and Maldives and the rest of the world. The citizens from India, Bangladesh and Maldives receive visa on arrival and also do not need to used travel agent. Will other citizens need to use a travel agents and also use a tour operator to plan their trip to Bhutan. This is a requirement by the Government of Bhutan.
The best time to visit Bhutan is during spring (March – May) and autumn (September – November.) Expect dry but warm weather, sunny but chilly during mornings and night time.
January – February:Expect clear blue skies during the day, you will see snow in the mountain passes between valleys. There’s could be rare snow in the valleys where the towns are located during January and February. Day temperatures hover about 11°C to 12°C during the day and -5°C to 0°C at night.
March – May:Best time to visit with temperature ranging from 24°C to 27°C, at night it can drop to 18°C.
June – August:Expect some rain but most of the time the rain is light shower for an hour or two in the afternoon. The temperature ranges between 27°C to 15°C at day and night.
September – December:Very good time to visit as well. Expect European autumn weather, and less rain than summertime. The temperature ranges between 22°C to 10°C at day and night.
In order to attain a visa for Bhutan, travelers must work with a licensed, approved tour company who will prepare every part of the visa application process. Bhutanese law requires visitors to come through via a licensed local tour company. As explained by DAJ Expeditions, “The Bhutan Government Regulations require the deposit of Tour Payments in full prior to processing your visa. You need to identify a local tour operator who will help you program your tour and process all paper work required for you to granted the visa. The visa will be stamped at the entry point on your arrival. Bring two passport size photographs and there is a Visa Fee of USD 40 per person or you can pay in advance to your tour operator who will pay it for you.
The above visa process is not required for citizens of India, Bangladesh and Maldives as they get Visa on arrival and also do not need to travel with a tour operator. This is special concession by the Royal Government of Bhutan.
How are you – Gadaybay zhu ga?
How much – Gademchi mo?
Bhutan is a disciplined Buddhist traditional society and they follow a highly refined system of etiquette called “driglam namzha.” When you visit Bhutan, you’ll notice the Bhutanese’s respect for authority, devotion to marital and familial institutions. It’s shown in the way they behave toward one another, how to conduct themselves in public, even how to eat in public and especially how to dress.
Local clothing is conventional, and you’ll notice a national dress code that is authentic and beautiful.
Bhutan remains as one of the safest countries for tourists to visit.
In Bhutan, Bhutanese ngultrum (BTN) is the currency. There are no coins. For better exchange rates, use bigger bills such as USD 50 or USD 100. You’ll get 5% less if you were to exchange with a USD 20 bill or less.
There was a counterfeit which occurred in 1996, so do not use any USD 100 bills that were made in 1996. Try to stay away from using cash that has tears, holes or marks.
Visa and MasterCard are sometimes accepted in hotels and larger boutiques. But mostly, credit cards are not accepted in smaller stores. There is an American Express office in Thimphu so the cards are also accepted in selected locations.
If you need to exchange cash, at the airport, bank or the hotel most of whom use the same rate as the banks in town.
If you’re staying for two weeks, USD 400 is more than enough to cover tips (go down to “tips” section for more details.) You’ll only be spending money on shopping and souvenirs since lodging, meals, gasoline should be previously covered in your tour package price.
Besides BTN, travellers can also use Indian rupee.
Avoid tank tops and shorts. In other words, dress respectfully. Especially when you enter Women – Try to stay away from revealing clothing and always have your shoulders and knees covered. Men – Try not to tie your jackets around your waist and don’t leave your jackets unzipped.
Don’t hike up Tiger’s Nest without bottles of water. It’s a 10,000 feet hike, for those who need to bring hiking poles, do it.
Avoid bringing gifts for local children since the Bhutanese government want to discourage children from the culture of begging.
Avoid heading to Bhutan without a pre-paid arrangement since it’s not possible to attain an entry visa for Bhutan unless tour itineraries have been made with a licensed and pre-approved tour operator which includes the $250 tourist tariff.
Smart phone will work in Bhutan, but you do need to switch to local providers. When you arrive, you can purchase local SIM cards which can give you great Wi-Fi connectivity.
Bhutan’s country code is 975.
All 4 star and above hotel now have free Wi-Fi in the hotel lobby and in some cases in the room also but it’s not necessarily free in guest rooms. Other hotels have Wi-Fi for free for a few hours, and the connection may be poor. Don’t not expect internet connection in remote valleys/cities.
Voltage in Bhutan is 240V and AC 50Hz (cycles per second), power sockets are type D/F/G. Make sure your appliances like shavers, hairdryers, curling irons, camera chargers, laptops, etc. have a switch to change the voltage to 240.
Drinking bottled water is advised for travelers. Your guide will most likely prepare a huge supply of bottled water in the car for you during the trip.
Since your trip to Bhutan is pre-paid, tipping in restaurants and hotels is not necessary. But if you’re going on treks, you may want to tip your horseman (Tiger’s Nest) then ask your guide how much is adequate. | english |
నిరుద్యోగులకు శుభవార్త.. టెక్ మహీంద్రాలో ఉద్యోగ ఖాళీలు..? ప్రముఖ ఐటీ కంపెనీలలో ఒకటైన టెక్ మహీంద్రా నిరుద్యోగులకు తీపికబురు చెప్పింది. 100కు పైగా ఉద్యోగ ఖాళీల కొరకు దరఖాస్తులను ఆహ్వానిస్తోంది. సాఫ్ట్వేర్ ఇంజినీర్, టెక్నికల్ ఆర్కిటెక్ట్ ఉద్యోగ ఖాళీలతో పాటు టెక్ లీడ్, సీనియర్, ప్రోగ్రామ్ మేనేజర్ ఇతర ఉద్యోగ ఖాళీలు ఉన్నాయి. అర్హత, ఆసక్తి ఉన్న అభ్యర్థులు ఈ ఉద్యోగ ఖాళీల కోసం దరఖాస్తు చేసుకోవచ్చు. టెలీకాంతో పాటు ఐటీ, బ్యాంకింగ్ విభాగాల్లో ఈ ఉద్యోగ ఖాళీలు ఉన్నాయి. ఉద్యోగ ఖాళీలను బట్టి వేర్వేరు తేదీలు ఉండగా వెబ్ సైట్ ద్వారా ఈ ఉద్యోగ ఖాళీలకు సంబంధించిన పూర్తి వివరాలను తెలుసుకోవచ్చు. ఆన్ లైన్ లో ఈ ఉద్యోగ ఖాళీల కొరకు దరఖాస్తు చేసుకోవచ్చు. దరఖాస్తు చేసుకున్న అభ్యర్థులను షార్ట్ లిస్టింగ్ చేసి ఇంటర్వ్యూ ద్వారా ఎంపిక చేయడం జరుగుతుంది. సంబంధిత రంగంలో అనుభవం ఉన్నవాళ్లు ఈ ఉద్యోగ ఖాళీల కొరకు దరఖాస్తు చేసుకునే అవకాశం ఉంటుంది. కోతి వల్ల ప్రాణాలు కోల్పోయిన్ సాఫ్ట్ వేర్ ఉద్యోగి.. ఏం జరిగిందంటే..? డిసెంబర్ 31, 2020 అమెరికా తెలంగాణ సాఫ్ట్ వేర్ ఇంజినీర్ మృతి డిసెంబర్ 23, 2020 హయ్యర్ డిగ్రీ లేదా బ్యాచిలర్ డిగ్రీ పాసైన వాళ్లు ఈ ఉద్యోగ ఖాళీల కొరకు దరఖాస్తు చేసుకోవచ్చు. ఈ ఉద్యోగ ఖాళీలకు ఎంపికైన వారికి అర్హతకు తగిన వేతనం లభిస్తుంది. నిరుద్యోగులకు ఈ నోటిఫికేషన్ ద్వారా ప్రయోజనం చేకూరుతుందని చెప్పవచ్చు. ఉద్యోగ ఖాళీలకు సంబంధించి ఏవైనా సందేహాలు ఉంటే వెబ్ సైట్ ద్వారా సులభంగా నివృత్తి చేసుకునే అవకాశం ఉంటుంది. ఈ మధ్య కాలంలో ప్రముఖ ఐటీ కంపెనీలు భారీ సంఖ్యలో ఉద్యోగ ఖాళీలను భర్తీ చేయడానికి సిద్ధమైంది. ఐటీ కంపెనీలలో ఉద్యోగ ఖాళీల కోసం ఎదురు చూసేవాళ్లు ఈ ఉద్యోగాల కొరకు దరఖాస్తు చేసుకుంటే మంచిది. ప్రముఖ ఐటీ కంపెనీలలో ఒకటైన టెక్ మహీంద్రా నిరుద్యోగులకు తీపికబురు చెప్పింది. 100కు పైగా ఉద్యోగ ఖాళీల కొరకు దరఖాస్తులను ఆహ్వానిస్తోంది. సాఫ్ట్వేర్ ఇంజినీర్, టెక్నికల్ ఆర్కిటెక్ట్ ఉద్యోగ ఖాళీలతో పాటు టెక్ లీడ్, సీనియర్, ప్రోగ్రామ్ మేనేజర్ ఇతర ఉద్యోగ ఖాళీలు ఉన్నాయి. అర్హత, ఆసక్తి ఉన్న అభ్యర్థులు ఈ ఉద్యోగ ఖాళీల కోసం దరఖాస్తు చేసుకోవచ్చు. టెలీకాంతో పాటు ఐటీ, బ్యాంకింగ్ విభాగాల్లో ఈ ఉద్యోగ ఖాళీలు ఉన్నాయి. | telegu |
اسلام باد سپورٹس ڈیسک کامن ویلتھ گیمز کے چمپئن اطہر کامران بٹ نے ایشین کلاسک پاور لیفٹنگ چمپئن شپ میں چار سونے کے تمغے حاصل کرلئے یہ چمپئن شپ قازقستان میں کھیلی گئی ہے اطہر کامران بٹ نے سکوت 220 کلوگرام بینچ پریس 160کلوگرام ڈیڈ لیفٹ 225کلوگرام اردو ٹوٹل ویٹ 605کلوگرام میں سونے کے تمغے حاصل کئے ہیں اطہر کامران بٹ کامن ویلتھ گیمز میں 105کلو گرام کیٹگری میں سونے کا تمغہ حاصل کرچکے ہیں اس کے علاوہ اطہر کامران بٹ نے کئی ملکی اور غیر ملکی ایونٹس میں بھی میڈلز حاصل کررکھے ہیں اور وہ قازقستان میں ایک ہفتے کے ایلیٹ سپورٹس کوچنگ کورس میں شرکت کے بعد واپس وطن کراچی پہنچ گئے ہیں پاکستان کی طرف سے ایشین کلاسک پاور لیفٹنگ چمپئن شپ دو کھلاڑیوں نے شرکت کی جن میں غالی کاظم 74کلوگرام اور اطہر کامران بٹ شامل تھے | urdu |
സിനിമയില് വേഷം വാഗ്ദാനം ചെയ്ത് കാല് ലക്ഷം തട്ടിച്ചെന്ന് മുക്കം: സിനിമയില് അഭിനയിപ്പിക്കാമെന്ന് പറഞ്ഞ് കാല് ലക്ഷം രൂപ തട്ടിയതായി മുക്കം മണാശ്ശേരിയിലെ കലാകാരിയുടെ പരാതി. ഇടുക്കി സ്വദേശികളായ സിബി, മേരിക്കുട്ടി എന്നിവരടക്കം അഞ്ചു പേര്ക്കെതിരെയാണ് മുക്കം പൊലീസ് കേസ് രജിസ്റ്റര് ചെയ്തു. ഉണ്ണി മുകുന്ദനും മഞ്ജു വാര്യരും മുഖ്യകഥാപാത്രങ്ങളായി എത്തുന്ന കണ്ണീരും കിനാവും എന്ന സിനിമയില് അഭിനയിപ്പിക്കാം എന്നു പറഞ്ഞ് അന്പതിനായിരം രൂപയാണ് അഞ്ചംഗ സംഘ ആവശ്യപ്പെട്ടത്. വാഗ്ദാനം ചെയ്തത് മഞ്ജു വാര്യരുടെ അമ്മ വേഷമായിരുന്നു. 25,000 രൂപ കൈമാറിയ ശേഷമാണ് അങ്ങനെ ഒരു സിനിമയേ ഇല്ലെന്ന് കലാകാരി അറിഞ്ഞത്. | malyali |
ಎಸ್ಎಸ್ಎಲ್ಸಿ, ದ್ವಿತೀಯ ಪಿಯು ಪರೀಕ್ಷೆಗೆ ಸಿಬಿಎಸ್ಇ ಮಾನದಂಡ? ಬೆಂಗಳೂರು ಜೂ.01: ಕೊರೋನಾ ಹಿನ್ನೆಲೆಯಲ್ಲಿ ರಾಜ್ಯದಲ್ಲಿ ಮುಂದೂಡಲಾಗಿರುವ ಎಸ್ಎಸ್ಎಲ್ಸಿ ಮತ್ತು ಪಿಯುಸಿ ಪರೀಕ್ಷೆಗಳನ್ನು ಹೇಗೆ ನಡೆಸಬೇಕು ಎಂಬ ಬಗ್ಗೆ ಇನ್ನೂ ಗೊಂದಲದಲ್ಲಿರುವ ರಾಜ್ಯ ಶಿಕ್ಷಣ ಇಲಾಖೆ, ಈ ವಿಷಯದಲ್ಲಿ ಕೇಂದ್ರ ಸರ್ಕಾರದ ಹಾದಿಯನ್ನು ಕಾದುನೋಡುತ್ತಿದೆ ಎನ್ನಲಾಗಿದೆ. ಕೇಂದ್ರ ಸರ್ಕಾರವು ಸಿಬಿಎಸ್ಇ 12ನೇ ತರಗತಿ ವಿದ್ಯಾರ್ಥಿಗಳಿಗೆ ಪರೀಕ್ಷೆ ನಡೆಸಬೇಕೇ ಅಥವಾ ರದ್ದುಗೊಳಿಸಬೇಕೇ ಎಂಬ ಬಗ್ಗೆ ಶೀಘ್ರವೇ ತೀರ್ಮಾನ ಕೈಗೊಳ್ಳಲಿದ್ದು, ಅದೇ ಮಾನದಂಡವನ್ನೇ ರಾಜ್ಯದಲ್ಲೂ ಅನುಸರಿಸುವ ಸಾಧ್ಯತೆಗಳಿವೆ ಎಂದು ಶಿಕ್ಷಣ ಇಲಾಖೆ ಮೂಲಗಳು ತಿಳಿಸಿವೆ. ಸದ್ಯ ರಾಜ್ಯದಲ್ಲಿ ಕೊರೋನಾ ಎರಡನೇ ಅಲೆ ಸೋಂಕು ತೀವ್ರವಾಗಿರುವುದರಿಂದ ಎಸ್ಎಸ್ಎಲ್ಸಿ ಮತ್ತು ದ್ವಿತೀಯ ಪಿಯುಸಿ ವಿದ್ಯಾರ್ಥಿಗಳಿಗೆ ಪರೀಕ್ಷೆ ನಡೆಸಬೇಕೇ, ಮೌಲ್ಯಾಂಕನ ಪರೀಕ್ಷೆ ನಡೆಸಬೇಕೇ ಅಥವಾ ಬೇರಾರಯವ ಹೊಸ ಮಾರ್ಗ ಅನುಸರಿಸಬಹುದು ಎಂಬ ಗೊಂದಲವಿದೆ.ಕೋವಿಡ್ ಕುರಿತ ಎಲ್ಲಾ ಲೇಟೆಸ್ಟ್ ಅಪ್ಡೇಟ್ಸ್ ಓದಿ ಹೀಗಾಗಿ, ಕೇಂದ್ರ ಸರ್ಕಾರ ಯಾವ ರೀತಿಯಲ್ಲಿ ತೀರ್ಮಾನ ಕೈಗೊಳ್ಳಬಹುದು ಎಂದು ಕಾಯ್ದು ನೋಡುವ ತಂತ್ರವನ್ನು ಶಿಕ್ಷಣ ಇಲಾಖೆ ಮಾಡುತ್ತಿದೆ. ಶೈಕ್ಷಣಿಕ ವೇಳಾಪಟ್ಟಿ ಬದಲು, ಶಾಲೆ ಆರಂಭದ ದಿನಾಂಕ ಘೋಷಿಸಿದ ಇಲಾಖೆ .. ಒಂದು ವೇಳೆ ಕೇಂದ್ರದ ಮಾನದಂಡ ರಾಜ್ಯದಲ್ಲಿಯೂ ಅಳವಡಿಸಿಕೊಳ್ಳಬಹುದು. ವಿದ್ಯಾರ್ಥಿಗಳ ಹಿತದೃಷ್ಟಿಯಿಂದ ಯಾವುದೇ ರೀತಿಯಲ್ಲಿ ತೊಂದರೆಯಾಗುವುದಿಲ್ಲ ಎನಿಸಿದರೆ, ಅದೇ ಮಾನದಂಡವನ್ನು ರಾಜ್ಯದಲ್ಲಿ ಅಳವಡಿಸಿಕೊಳ್ಳುವ ಎಲ್ಲಾ ಸಾಧ್ಯತೆಗಳಿವೆ ಎಂದು ತಿಳಿದು ಬಂದಿದೆ. ರಾಜ್ಯದಲ್ಲಿ ಚರ್ಚೆಯಾಗುತ್ತಿರುವ ಮಾರ್ಗಗಳು ಆನ್ಲೈನ್ ಮೂಲಕ ಪರೀಕ್ಷೆ ನಡೆಸುವುದು, ಪೂರ್ವಸಿದ್ಧತಾ ಪರೀಕ್ಷೆ, ಮಧ್ಯವಾರ್ಷಿಕ ಪರೀಕ್ಷೆ ಸೇರಿದಂತೆ ಹಿಂದಿನ ತರಗತಿಗಳ ಫಲಿತಾಂಶ ಆಧರಿಸಿ ಫಲಿತಾಂಶ ಪ್ರಕಟಿಸುವುದು ಅಥವಾ ಮಕ್ಕಳಿಗೆ ಲಸಿಕೆ ಕೊಡಿಸಿ ಪರೀಕ್ಷೆ ನಡೆಸುವುದು ಸೇರಿದಂತೆ ಹಲವಾರು ಮಾರ್ಗಗಳು ಚರ್ಚೆಯಾಗುತ್ತಿವೆ. ಮತ್ತೊಂದೆಡೆ ಅಗತ್ಯ ಇರುವಷ್ಟುಲಸಿಕೆಗಳನ್ನು ತರಿಸಿಕೊಂಡು ವಿದ್ಯಾರ್ಥಿಗಳು ಹಾಗೂ 12 ರಿಂದ 18 ವರ್ಷದ ಮಕ್ಕಳಿಗೆ ಲಸಿಕೆ ಹಾಕಿಸುವುದು ಉತ್ತಮ. ಪರೀಕ್ಷೆ ನಡೆಸುವ ಮುನ್ನ ವಿದ್ಯಾರ್ಥಿಗಳಿಗೆ ಲಸಿಕೆ ಹಾಕಿಸುವಂತೆ ರಾಜ್ಯ ಸರ್ಕಾರಕ್ಕೆ ಕೊರೋನಾ ತಾಂತ್ರಿಕ ಸಮಿತಿಯ ಡಾ. ಗಿರಿಧರ್ ಬಾಬು ಸಲಹೆ ನೀಡಿದ್ದಾರೆ ಎನ್ನಲಾಗಿದೆ. ಆದರೆ, ಅಂತಿಮವಾಗಿ ಸರ್ಕಾರ ಯಾವ ತೀರ್ಮಾನ ಕೈಗೊಳ್ಳಲಿದೆ ಎಂಬುದನ್ನು ಕಾಯ್ದು ನೋಡಬೇಕಿದೆ. ಸೂಚನೆ: ಕೊರೋನಾ ಮಹಾಮಾರಿ ಎಲ್ಲೆಡೆ ಹರಡುತ್ತಿದೆ. ಹೀಗಿರುವಾಗ ಎಲ್ಲರೂ ತಪ್ಪದೇ ಮಾಸ್ಕ್ ಧರಿಸಿ, ಸಾಮಾಜಿಕ ಅಂತರ ಕಾಪಾಡಿ ಹಾಗೂ ಲಸಿಕೆ ಪಡೆಯಿರಿ ಎಂಬುವುದು ಏಷ್ಯಾನೆಟ್ ನ್ಯೂಸ್ ಕಳಕಳಿಯ ವಿನಂತಿ. ಒಗ್ಗಟ್ಟಿನಿಂದ ನಾವು ಈ ಕೊರೋನಾ ಸರಪಳಿ ಮುರಿಯೋಣ ANCares IndiaFightsCorona | kannad |
Ukraine Crisis: ইউক্রেনের বাঙ্কারে ছেলে, কীভাবে ফিরবে? জানেন না মোমিনুদ্দিন Bangla News একরাশ দুশ্চিন্তার মধ্যে দিন কাটছে মালদার হরিশ্চন্দ্রপুরের মিটনা হাইস্কুলের শিক্ষক মহম্মদ মোমিনুদ্দিনের ছেলে মাসুম হামিদ পারভেজ ইউক্রেনের কিভে ডাক্তারি পড়ুয়া যুদ্ধ শুরুর পর ছেলের সঙ্গে যোগাযোগ হয়েছিল চতুর্থ বর্ষের ওই মেডিক্যাল পড়ুয়া জানিয়েছেন, বাঙ্কারে আশ্রয় নিয়েছেন খাবারের ব্যবস্থা নেই বাইরে সাইরেন আর গোলাগুলির শব্দফলে বেরোতেও পারছেন না কীভাবে বাড়ি ফিরবে ছেলে, এখনও জানে না পরিবার | bengali |
ಅಂತರ್ ರಾಷ್ಟ್ರೀಯ ಡ್ರಗ್ಸ್ ಪೆಡ್ಲರ್ ಸೇರಿ ಮೂವರ ಬಂಧನ: ಲಕ್ಷಾಂತರ ರೂ. ಮೌಲ್ಯದ ಮಾದಕ ವಸ್ತು ಜಪ್ತಿ ಬೆಂಗಳೂರು, ಡಿ.24: ಅಂತರ್ರಾಷ್ಟ್ರೀಯ ಡ್ರಗ್ಸ್ ಪೆಡ್ಲರ್ ಸೇರಿದಂತೆ ಮೂವರನ್ನು ಬಂಧಿಸಿರುವ ಸಿಸಿಬಿ ಪೊಲೀಸರು, 5 ಲಕ್ಷ ರೂ. ಮೌಲ್ಯದ 100 ಗ್ರಾಂ ಎಂಡಿಎಂಎ, ಕಾರು, ಬೈಕ್, 5 ಸಾವಿರ ರೂ. ನಗದು ಜಪ್ತಿ ಮಾಡಿದ್ದಾರೆ. ನೈಜೀರಿಯಾ ಮೂಲದ ಉದೆಲುದೇಯುಜಾ33, ಕೇರಳ ಮೂಲದ ಪ್ರಸೂನ್27, ಆನಂದ್ ಚಂದನ್27 ಬಂಧಿತರು ಎಂದು ಪೊಲೀಸರು ತಿಳಿಸಿದ್ದಾರೆ. ಯಲಹಂಕದ ರಾಮಕೃಷ್ಣಪ್ಪ ಬಹುಮಹಡಿ ಕಟ್ಟಡ ಮುಂಭಾಗ 5ನೆ ಕ್ರಾಸ್ನಲ್ಲಿ ವಾಹನವೊಂದರಲ್ಲಿ ಮಾದಕ ಮತ್ತು ಎಂಡಿಎಂಎ ಇಟ್ಟುಕೊಂಡು ಮಾರಾಟ ಮಾಡಲು ಹೊಂಚು ಹಾಕುತ್ತಿದ್ದರು. ಈ ಬಗ್ಗೆ ಮಾಹಿತಿ ಸಂಗ್ರಹಿಸಿ ಸಿಸಿಬಿ ಪೊಲೀಸರು ಕಾರ್ಯಾಚರಣೆ ನಡೆಸಿದಾಗ ಪ್ರಕರಣ ಬೆಳಕಿಗೆ ಬಂದಿದೆ. ಆರೋಪಿಗಳ ವಿರುದ್ಧ ಯಲಹಂಕ ಪೊಲೀಸ್ ಠಾಣೆಯಲ್ಲಿ ಪ್ರಕರಣ ದಾಖಲಾಗಿದ್ದು, ಈ ಸಂಬಂಧ ತನಿಖೆ ಮುಂದುವರಿಸಲಾಗಿದೆ. | kannad |
10ம் வகுப்பு முடித்த நபர்களுக்கு நீதிமன்றத்தில் வேலை..! மாதம் ரூ.20,600 ஊதியம்.! தமிழ்நாடு நீதித்துறை சேவையின் கீழ் பெரம்பலூர் நீதித்துறை பிரிவில் காலியாக உள்ள பணியிடங்களை நிரப்ப அறிவிப்பு வெளியாகியுள்ளது. Steno Typist Gr III என 7 இடங்களும் Typist பணிக்கு என 4 இடங்கள் என மொத்தம் 11 பணியிடங்கள் ஒதுக்கப்பட்டுள்ளன. அரசு அங்கீகாரம் பெற்ற கல்வி நிறுவனத்தில் 10ம் வகுப்பு தேர்ச்சி பெற்றவர்கள் இந்த பணிக்கு விண்ணப்பிக்கலாம். விண்ணப்பிக்கும் நபர்கள் குறைந்தபட்ச வயதானது 18 அதிகபட்சம் 35 ஆகவும் இருக்க வேண்டும். Steno Typist Gr III பணிக்கு தேர்வு செய்யப்படும் நபர்களுக்கு மாதம் ரூ.20,600 முதல் ரூ.65,500 ஊதியம் வழங்கப்படவுள்ளது. Typist பணிக்கு தேர்வு செய்யப்படும் விண்ணப்பதாரர்களுக்கு மாதம் ரூ.19,500 முதல் ரூ.62,000 ஊதியம் வழங்கப்படவுள்ளது. விண்ணப்பிக்கும் நபர்களுக்கு நேர்காணல் மூலம் தேர்வு செய்யப்பட்டு பணி வழங்கப்படும். ஆர்வமுள்ளவர்கள் அதிகாரபூர்வ தளத்திற்குள் சென்று விண்ணப்ப படிவத்தை பதிவு செய்து அதை பூரித்து செய்து முதன்மை மாவட்ட நீதிபதி, பெரம்பலூர் என்ற முகவரிக்கு அனுப்ப வேண்டும். 20.12.2021 க்கு பிறகு பெறப்படும் விண்ணப்பங்கள் ஏற்றுக்கொள்ளப்படாது என தெரிவிக்கப்பட்டுள்ளது. Also Read: மீண்டும் ஆரம்பம்.! ஒரே கல்லூரியில் 281 பேருக்கு கோவிட் பாசிட்டிவ்.! இது தான் காரணம்..? அமைச்சர் தகவல்.! | tamil |
/********************************************************************************
* MiNya pjeject
* Copyright 2014 nyatla.jp
* https://github.com/nyatla/JMiNya
*
* This program is free software; you can redistribute it and/or
* modify it under the terms of the GNU General Public License
* as published by the Free Software Foundation; either version 2
* of the License, or (at your option) any later version.
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
*
********************************************************************************/
package jp.nyatla.minya;
import jp.nyatla.minya.connection.IMiningConnection;
import jp.nyatla.minya.connection.StratumMiningConnection;
import jp.nyatla.minya.worker.CpuMiningWorker;
import jp.nyatla.minya.worker.IMiningWorker;
public class EntryPoint
{
private static final String DEFAULT_URL = "stratum+tcp://stratum-eu.nyan.luckyminers.com:3320";
private static final String DEFAULT_USER = "user";
private static final String DEFAULT_PASS = "pass";
public static void main(String[] args)
{
try {
// IMiningConnection mc=new TestStratumMiningConnection(0);
IMiningConnection mc=new StratumMiningConnection(DEFAULT_URL,DEFAULT_USER,DEFAULT_PASS);
IMiningWorker imw=new CpuMiningWorker(4);
SingleMiningChief smc=new SingleMiningChief(mc,imw);
smc.startMining();
for(;;){
Thread.sleep(1000);
}
} catch (Exception e) {
// TODO Auto-generated catch block
e.printStackTrace();
}
}
}
| code |
दिल्ली सरकार ने यूरोप की तर्ज पर बनी दिल्ली फिल्म पॉलिसी2022 को दी मंजूरी, क्या है इसमें खास, क्या होंगे फायदे, यहां जानें सब कुछ दिल्ली को दुनियाभर में फिल्म निर्माण का केंद्र बनाने और वैश्विक पहचान देने के लिए दिल्ली कैबिनेट Delhi Cabinet ने दिल्ली फिल्म पॉलिसी को मंजूरी दी है Delhi Film Policy . इस मौके पर दिल्ली के उपमुख्यमंत्री मनीष सिसोदिया Manish Sisodia ने कहा कि दिल्ली फिल्म पॉलिसी 2022, दिल्ली में पर्यटन आकर्षित करने के साथसाथ दिल्ली में बड़े पैमाने पर रोजगार और आर्थिक विकास के अवसर पैदा करेगी. दिल्ली में फिल्म प्रोडक्शन के लिए दिल्ली की केजरीवाल सरकार 3 करोड़ रूपये तक की सब्सिडी देगी और फिल्म इंडस्ट्री में स्थानीय लोगों को काम पर रखने के लिए भी प्रोत्साहित करेगी. दिल्ली का जल्द ही अपना अंतर्राष्ट्रीय फिल्म फेस्टिवल होगा और दिल्ली फिल्म एक्सीलेंस अवार्ड्स भी शुरू किया जाएगा जिसमें न केवल फिल्म स्टार बल्कि क्रू के सदस्यों को भी सम्मानित किया जाएगा. फिल्म उद्योग से जुड़े लोगों की सहूलियत के लिए दिल्ली सरकार ईफिल्म क्लेअरेंस पोर्टल भी स्थापित करेगी जिसके माध्यम से फिल्मप्रोडक्शन के लिए प्रोड्यूसर्स को दिल्ली पुलिस और DDA समेत 25 से ज्यादा एजेंसियों की मंजूरी 15 दिनों के भीतर ऑनलाइन दी जाएगी. इस पॉलिसी के तहत शुरू किया गया दिल्ली फिल्म फंड, फिल्म निर्माताओं के लिए उत्पादन लागत कम करने में मदद करेगा साथ ही उन्हें दिल्ली फिल्म कार्ड से हॉस्पिटैलिटी क्षेत्र में भारी छूट भी मिलेगी. पॉलिसी की मदद से सिल्वर स्क्रीन पर ज्यादा कवरेज के साथ दिल्ली अपने आप में एक ब्रांड के तौर पर भी स्थापित होगा और दिल्ली के नागरिकों में दिल्ली की संस्कृति,कला के प्रति गर्व का भाव पैदा करेगा. ग्लोबल लेवल पर फिल्म प्रमोशन को स्टडी करने के बाद किया तैयार सिसोदिया ने कहा कि नई फिल्म पॉलिसी हॉस्पिटैलिटी, टूरिज्म, परिवहन, सिनेमा और कलाकारों की पूरी दुनिया को एक साथ लाएगी. इस पूरी प्रक्रिया में दिल्ली पर्यटन और परिवहन विकास निगम DTTDC नोडल एजेंसी के रूप में राज्य के सभी स्टेकहोल्डर्स और फिल्म प्रोडक्शन एजेंसियों के साथ कोआर्डिनेशन करने का काम करेगी. गुरुवार को दिल्ली कैबिनेट की बैठक दिल्ली फिल्म पॉलिसी 2022 को मंजूरी दी गई, पॉलिसी के बारे में बात करते हुए मनीष सिसोदिया ने कहा कि दिल्ली में पर्यटन के माध्यम से इंक्लूसिव ग्रोथ के लिए, दिल्ली फिल्म पॉलिसी 2022 को कैबिनेट ने मंजूरी दी. यह एक प्रगतिशील पॉलिसी है जिसे रोजगार के नए अवसर, अर्थव्यवस्था के अलग अलग क्षेत्रों को बढ़ावा देने और लोगों में गर्व की भावना पैदा करने में मदद करने के लिए ग्लोबल लेवल पर फिल्म प्रमोशन और टूरिज्म पॉलिसी को स्टडी करने के बाद तैयार किया गया है. दिल्ली की नई फिल्म पॉलिसी का क्या है उद्देश्य? दिल्ली सरकार के मुताबिक नई पॉलिसी लागू करने से दिल्ली के लोगों को दिल्ली के साथ गर्व से जुड़ाव होगा. नई फिल्म पॉलिसी से दिल्ली को एक वैश्विक ब्रांड के रूप में पहचान मिलेगी. राष्ट्रीय और अंतरराष्ट्रीय स्तर के फिल्म निर्माताओं को आकर्षित कर दिल्ली को फिल्म कैपिटल के रूप में प्रमोट किया जाएगा. टूरिज्म को बढ़ावा देना और रोजगार के नए अवसर तैयार करना. फिल्मों के माध्यम से दिल्ली की कला और संस्कृति को बढ़ावा मिलेगा. फिल्म प्रोड्यूसर्स को ऑनलाइन सिंगल विंडो सिस्टम से मिलेगी सभी मंजूरी फिल्म प्रोड्यूसर्स को आकर्षित करने के लिए दिल्ली सरकार एक सिंगल विंडो ऑनलाइन क्लेअरेंस सिस्टम तैयार करेगी. जिससे फिल्म प्रोड्यूसर्स को ऑनलाइन 15 दिनों के भीतर ही फिल्म शूटिंग से जुडी सभी मंजूरियां एक ही जगह पर मिल जाएगी. अगर किसी प्रोड्यूसर को 15 दिन के अंदर किसी एजेंसी से अप्रूवल नहीं मिलता है तो दिल्ली टूरिज्म विभाग बतौर नोडल एजेंसी खुद ही अप्रूवल देगी. अगर किसी प्रोड्यूसर को 15 दिनों से पहले अप्रूवल चाहिए तो उन्हें प्रीमियम पेमेंट का भुगतान करना होगा. आपको बता दें कि इससे पहले प्रोड्यूसर्स को 25 अलगअलग एजेंसीज से मंजूरी लेनी होती थी. प्रोड्यूसर्स को 3 करोड़ रूपये तक की सब्सिडी दिल्ली सरकार ने अंतराष्ट्रीय शूटिंग लोकेशन की तर्ज़ पर फिल्म ब्रांडिंग और प्रमोशन के लिए दिल्ली में फिल्म बनाने पर फिल्म निर्माण में आने वाले खर्चों में सरकार द्वारा प्रोड्यूसर्स को 3 करोड़ रूपये तक की सब्सिडी दी जाएगी, सब्सिडी का निर्धारण पॉइंट सिस्टम के आधार पर किया जाएगा और ये दिल्ली में फिल्म निर्माण में लगे कुल लागत का 5 से 25 तक होगा. फिल्म लोकेशन, लोकेशन में दिल्ली की ब्रांडिंग, फिल्म क्रू और सपोर्ट स्टाफ में लोकल आर्टिस्टों का इन्वोल्वेमेंट, दिल्ली में फिल्म के प्रीप्रोडक्शन, प्रोडक्शन और पोस्टप्रोडक्शन के दौरान खर्चों को ध्यान में रखते हुए फिल्म को पॉइंट्स दिए जाएंगे और उसके आधार पर सब्सिडी दी जाएगी. दिल्ली सरकार ने जर्मन, जापानी शहरों और मास्को के साथ कई सिस्टरसिटी समझौतों पर हस्ताक्षर किए हैं. ऐसे शहरों के प्रोड्यूसर्स को भी इस नीति के तहत प्रोत्साहन दिया जाएगा. साथ ही केजरीवाल सरकार ने दिल्ली फिल्म फंड के लिए 50 करोड़ रूपये का आवंटन किया जो फिल्म प्रचार के दृष्टिकोण से दिल्ली को राष्ट्रीय और अंतरराष्ट्रीय पर्यटन स्थल के रूप में ब्रांड करने में मदद करेगा. फिल्म प्रोड्यूसर्स को मिलेगा खास फिल्म कार्ड, हॉस्पिटैलिटी सर्विसेज में मिलेगी छूट फिल्म निर्माता एजेंसियों को खास डील और पैकेज देने के लिए दिल्ली फिल्म कार्ड दिया जाएगा जिसकी कीमत 1 लाख रूपये होगी, पॉलिसी के तहत टूरिज्म और हॉस्पिटैलिटी कंपनियों को पर्यटन विभाग के साथ पैनल में रखा जाएगा. दिल्ली फिल्म कार्ड रखने वालों को दिल्ली के भीतर यात्रा, लोजिस्टिक्स, होटल जैसे सुविधाओं में छूट मिलेगी. इसके साथ दिल्ली का जल्द ही अपना अंतर्राष्ट्रीय फिल्म फेस्टिवल होगा और दिल्ली फिल्म एक्सीलेंस अवार्ड्स भी शुरू किया जाएगा जिसमें न केवल फिल्म स्टार बल्कि क्रू के सदस्यों को भी सम्मानित किया जाएगा. दिल्ली फिल्म पॉलिसी 2022 के फायदे क्या होंगे? दिल्ली फिल्म पॉलिसी 2022, दिल्ली में, फिल्म इंडस्ट्री इकोसिस्टम को मजबूत करने, फिल्म प्रोडक्शन और उससे संबंधित क्षेत्रों में लगे स्किल्ड वर्कफ़ोर्स के लिए रोजगार के अवसर पैदा करने में मदद करेगा. इस पॉलिसी के माध्यम से दिल्ली को राष्ट्रीय और अंतर्राष्ट्रीय स्तर पर पहचान मिलेगा और पर्यटकों का दिल्ली को लेकर आकर्षण बढ़ेगा जिसके परिणामस्वरुप दिल्ली में टूरिज्म को भी बढ़ावा मिलेगा जिससे दिल्ली के होटल, रेस्टोरेंट, हॉस्पिटैलिटी इंडस्ट्री को भी फायदा होगा, रोजगार के नए अवसर पैदा होंगे और इकॉनमी बेहतर होगी. दिल्ली बनेगा फिल्म टूरिज्म डेस्टिनेशन, फिल्म शूटिंग प्रमोशन सेल किया जाएगा स्थापित दिल्ली को फिल्म टूरिज्म डेस्टिनेशन के रूप में प्रमोट करने के लिए यहां फिल्म प्रोडक्शन से संबंधित सुविधाओं और इंफ्रास्ट्रक्चर विकसित किया जाएगा ताकि फिल्म प्रोडक्शन एजेंसीज़ आकर्षित हो सके, इसके अलावा अलगअलग स्थानीय, राष्ट्रीय और अंतरराष्ट्रीय इंडस्ट्री इवेंट में भाग लेने और फिल्मसेक्टर के साथ कार्यक्रमों और इसी तरह के अन्य कार्यक्रमों की मेजबानी के माध्यम से फिल्म शूटिंग के लिए दिल्ली को बढ़ावा दिया जाएगा. दिल्ली में एक फिल्म शूटिंग प्रमोशन सेल की स्थापना की जाएगी. इसके अलावा, दिल्ली में फिल्म प्रोडक्शन को बढ़ावा देने के लिए एक डेवलपमेंट सेल और फिल्म एडवाइजरी बॉडी का भी गठन किया जाएगा यहां दिल्ली पर्यटन और परिवहन विकास निगम नोडल एजेंसी के रूप में राज्य के सभी स्टेकहोल्डर्स और फिल्म प्रोडक्शन एजेंसियों के साथ कोआर्डिनेशन करने का काम करेगी. Delhi School: दिल्ली स्कूल में हिजाब हटाने के आरोपों पर मनीष सिसोदिया ने ऐसे प्रतिबंध से किया इंकार Delhi: दिल्ली सरकार ने मानी कई दिनों से हड़ताल पर बैठी आंगनवाड़ी वर्कर और हेल्पर की मांगें, मानदेय 9600 से बढ़ाकर किया 11 हजार | hindi |
बेहतरीन स्कीम! कोरोना काल में नौकरी जाने पर भी मिलेगी सैलरी, जानिए नई स्कीम के बारे में.. डेस्क : अगर आप भी उन लोगों में से एक हैं जिनकी कोरोना काल में नौकरी चली गई है, तो अब आप लोगों को घबराने की कोई जरूरत नहीं है, क्योंकि देश के ऐसे 40 लाख लोगों को केंद्र सरकार ने चयनित किया है, अब उन्हें बेरोजगारी भत्ता आगमी जून माह तक मिलता रहेगा, बता दे की केंद्र सरकार ने इसके लिए ESIC की देखरेख में अटल बीमित व्यक्ति कल्याण योजना की शुरुआत की है। इस योजना के तहत केंद्र सरकार उन लोगों को बेरोजगारी भत्ता देगी, जिन्होंने कोविड काल में अपनी नौकरी गंवाई है। एक रिपोर्ट के मुताबिक, इस योजना में करीब 40 लाख लोगों को नौकरी मिलेगी, मालूम हो को यह योजना ESIC से बीमित कर्मचारियों को उनकी बेरोजगारी के दौरान नकद मुआवजे के रूप में राहत प्रदान करती है, हालाकि इससे इतनी सैलरी तो नहीं मिलेगी, जितनी वहां मिलती होगी, लेकिन सैलरी के 50 पैसे पात्र लोगो को जून तक देने का प्रावधान है। अगर आसान भाषा में इस स्कीम को समझाएं तो अगर कोई लोग हर महीने 30,000 कमाता है तो उसकी 90 दिन की औसत कमाई 90 हजार का 50 यानी 2 साल में लगभग 45 हजार रुपए उसे दिए जाएंगे, हालांकि केंद्र सरकार की अटल बीमित व्यक्ति कल्याण योजना का लाभ लेने के लिए कुछ शर्तें निर्धारित हैं, इस योजना का फायदा सिर्फ उन्हीं लोगों को मिलेगा जो असंगठित क्षेत्र में नौकरी कर रहे हों और उनका पैसा PF या ESIC में कटता हो अगर आप भी इस योजना का लाभ लेना चाहते हैं तो सबसे पहले आपको ESIC की वेबसाइट www.esic.nic.in पर जाना होगा, बेरोजगारी भत्ता का फायदा लेने के लिए फॉर्म डाउनलोड करें और उसे विधिवत भर दें, इसके बाद उसे ESIC की नजदीकी ब्रांच में जमा करा दें, फॉर्म के साथ नोटरी शपथ पत्र लगाना होगा जिसमें 20 रुपये का स्टांप पेपर अटैच करना अनिवार्य है। Previous 10 रुपये का ये सिक्का असली है नकली, सरकार ने दी बड़ी जानकारी.. जानें Next यदि आपके पास भी है 786 नंबर वाला नोट, तो मिलेंगे पूरे 3 लाख रुपए, यहां जानिए पूरा प्रोसेस. | hindi |
રાજકોટ: જેતપુરમાં ખેત વિજળીનાં ધાંધીયાથી કંટાળી ત્રસ્ત ખેડૂતોએ પીજીવીસીએલ કચેરી ખાતે આપ્યું આવેદન જેતપુર પીજીવીસીએલ કચેરી ખાતે આપ્યું આવેદન. ખેડૂતોએ પીજીવીસીએલ કચેરી ખાતે લગાવ્યા નારા, લોડ સેટિંગ કે સરકારના સેટિંગ લગાવ્યા, પીજીવીસીએલ હાય હાયનાં નારા લગાવામાં આવ્યા જેતપુર ગ્રામ્ય વિસ્તારમાં ખેતર માટે અપાતી વિજળીમાં ધાંધીયા થતા 15 જેટલા ગ્રામજનો પીજીવીસીએલ કચરી ખાતે આપ્યું આવેદન. ખેડૂતોએ આક્ષેપ કર્યા કે રાજય સરકાર ઉદ્યોગ માટે 24 કલાક વિજળી અપાઈ છે અને ખેડૂતો ને આઠ કલાક વિજળી દેવામા પણ ધાંધીયા કરે છે. વિજળી આપવામા પણ સમયસર તંત્ર દ્વારા વિજળી અપાતી નથી ખેડૂત મોડી રાત થી સવાર સુધી રાહ જોતો હોય છે વિજળી સમયસર ન અપાતા જેતપુર પંથકમાં ખેડૂતો ને ખેતર માટે વિજ ધાંધિયા થી ત્રસ્ત. અગાઉ પણ ખેડૂતોએ વિરોધ નોંધાવ્યો હતો. સમયસર વીજળી નહીં આપવામાં આવે તો ઉપવાસ ઉપર બેસવાની આપી ચીમકી. | gujurati |
सरकारी एजेंसियों में बढ़ रहा है इलेक्ट्रिक वाहनों का इस्तेमाल, 5,384 इलेक्ट्रिक वाहन हैं रजिस्टर भारत सरकार के साथसाथ देश के विभिन्न राज की सरकारों द्वारा बड़े पैमाने पर इलेक्ट्रिक वाहनों का उपयोग किया जा रहा है। केंद्रीय परिवहन मंत्री नितिन गडकरी ने संसद में दिए एक बयान में सरकारी एजेंसियों द्वारा इस्तेमाल में लाए जा रहे इलेक्ट्रिक वाहनों के आंकड़ों का खुलासा किया है। गडकरी ने गुरुवार को संसद को सूचित किया कि 4 फरवरी 2022 तक केंद्र और राज्य सरकारों और स्वायत्त निकायों सहित सरकारी एजेंसियों द्वारा उपयोग किए जा रहे 8,47,544 वाहनों में से कुल 5,384 इलेक्ट्रिक वाहन हैं। लोकसभा में एक लिखित उत्तर में, मंत्री नितिन गडकरी ने कहा कि स्थानीय अधिकारियों द्वारा सबसे ज्यादा 1,352 इलेक्ट्रिक वाहनों का उपयोग किया जा रहा है। वहीं सरकारी उपक्रमों द्वारा 1,273 वाहन और राज्य सरकारों द्वारा 1,237 इलेक्ट्रिक वाहन उपयोग किए जा रहे हैं। एक अलग सवाल का जवाब देते हुए गडकरी ने कहा कि जनवरी, 2022 में स्वीकृत किये गए भारतमाला परियोजना चरण I के तहत 34,800 किलोमीटर के राष्ट्रीय राजमार्ग विकास परियोजना एनएचडीपी के कुल लंबाई में से लगभग 19,363 किलोमीटर की परियोजनाओं को पूरा किया जा चुका है। एक अन्य सवाल के जवाब में उन्होंने कहा कि सरकार ने वित्तीय वर्ष के लिए भारतीय राष्ट्रीय राजमार्ग प्राधिकरण एनएचएआई के लिए 59,000 करोड़ रुपये की अतिरिक्त बजटीय सहायता आवंटित की है। मंत्री ने कहा कि राष्ट्रीय राजमार्गों की कुल लंबाई 2014 में लगभग 91,287 किमी से बढ़ाकर वर्तमान में लगभग 1,41,190 किमी हो गई है। एक अन्य सवाल के जवाब में गडकरी ने कहा कि शराब पीकर गाड़ी चलाने के मामले में देश भर में 48,144 ईचालान जारी किये गए हैं। देश में 31 मार्च 2019 तक 63.71 लाख किलोमीटर का सड़क नेटवर्क है, जिसमें राष्ट्रीय राजमार्ग, राज्य राजमार्ग, जिला सड़कें, ग्रामीण सड़कें और शहरी सड़कें शामिल हैं। यह अमेरिका के 66.45 लाख किमी के सड़कों के बाद दुनिया का दूसरा सबसे बड़ा सड़क नेटवर्क है। बता दें कि देश में 2025 तक राष्ट्रीय राजमार्गों की कुल लंबाई 2 लाख किलोमीटर तक हो जाएगी। राष्ट्रीय राजमार्ग और एक्सप्रेसवे यात्रा के समय और ईंधन लागत में कटौती के अलावा, आर्थिक विकास में भी मदद करते हैं। जानकारी के अनुसार, सरकार की प्राथमिकता लॉजिस्टिक्स की लागत को जीडीपी के मौजूदा 1416 फीसदी से घटाकर 10 फीसदी करना है। चीन में यह 810 फीसदी और यूरोपीय देशों में 12 फीसदी है। अगर इसे 1012 फीसदी तक लाया जाता है तो अंतरराष्ट्रीय बाजार में भारत अच्छी प्रतिस्पर्धा कर सकता है। बता दें कि राष्ट्रीय राजमार्गों के निर्माण और रखरखाव के लिए जिम्मेदार एजेंसी, भारतीय राष्ट्रीय राजमार्ग प्राधिकरण एनएचएआई ने लॉकडाउन के दौरान भी राजमार्गों के निर्माण का काम जारी रखा था। मार्च 2020 से लॉकडाउन के वजह से ट्रैफिक नहीं होने के कारण कई परियोजनों में निर्माण की रफ्तार को बढ़ाने में भी सफलता मिली थी। इस दौरान एजेंसी ने निर्माण में कुछ रिकॉर्ड उपलब्धियां भी हासिल कीं। पिछले साल, NHAI ने 25.54 किलोमीटर सिंगल लेन सड़क का विकास केवल 18 घंटों में पूरा कर विश्व रिकॉर्ड बनाया। यह सड़क NH52 पर विजयपुर और सोलापुर के बीच फोरलेन हाईवे पर स्थित है। एनएचएआई ने पिछले साल फरवरी में एक दिन के भीतर फोरलेन हाईवे पर सबसे अधिक मात्रा में कंक्रीट डालने का एक और विश्व रिकॉर्ड बनाया था। यह उपलब्धि ठेकेदार पटेल इंफ्रास्ट्रक्चर ने हासिल की और इंडिया बुक ऑफ रिकॉर्ड्स और गोल्डन बुक ऑफ वर्ल्ड रिकॉर्ड्स में अपना नाम दर्ज कराया। source: drivespark.com | hindi |
Our last day at the beach has finally come. I know I don't have to tell anyone that the time spent together has been wonderful and extra special, but it has.
When I married at nineteen, DH and I were still attending Lipscomb, both working and trying to learn how to live together. DH had to drive one and a half hours to Pulaski to work right after he got out of school and would not be home until 11:30 pm. This went on for about a year and a half until he graduated early. Then he worked full days and that would be leaving early and not getting home until 9:15. And this was with zero days off.
The business in Pulaski was sold and they started up in Franklin and Nashville. Savannah came in 1991 and I KNOW John worked every day of our marriage for about 15 years. We would take a five day trip once a year and go visit my family for a couple days at holidays.
Savannah and I would go visit my parents and they would come see me. Leslie usually came up about once a month. DH worked hard so I could stay home with Sav and always encouraged me to go visit my family.
In between there the cancer came three times and life went on. That is life altering in itself and I won't even go there.
About 6 or 7 years ago, DH decided to take one day off a week. Then a few years ago 2 days. This was huge! There were plenty of times when I went home that people didn't know what my husband looked like.
Well, a year and a half ago, DH had an epiphany and said, "Is this it?" And he realized that Savannah only had three more years home and she would be leaving for college.
Our lives changed that day. We all spend more time together and value every day.
DH has said twice, when he was last down here and last night, "I'm in my happy place. Right now." I wanted to cry with love. I don't know how many of you are married to a workaholic or a former workaholic, but it does and can get better.
I am so thankful for the place we are now in our lives. It has been long and hard, with lots of hours away from each other, but I know all along that God has a plan for us. He is so loving and giving to us if we just love Him and are faithful.
I will have to say that all through our marriage, being faithful to the Lord's church has been our strength. Without the love of our church family, it would have been doubly hard. So I cannot stress enough how important your Spiritual life is.
There are alot of things I know that we would do differently now, but we have always had the love of both sides of the families and most importantly, the love for each other and I thank God every day for my darling husband John.
We finished our Bruno's pizza, which is the best pizza down here and happened to see it was 7:15pm and there was a yell and we all hopped up and zoomed out to see the sunset.
Yesterday we went out and there wasn't much color because it was so hazy, but tonight it cleared enough for it to be glorious!!
Sometimes when there are some clouds, the sunset is more colorful because the sun shines around and through and over. Pinks, ambers, greys, blues, oranges, and golds. You could even see rain streaming down in the distant horizon.
We felt like we had really seen God tonight. The day had almost finished and I didn't really have anything to blog about. Just the Lowe's, barber shop, etc. kind of day. But what a finish!!
"Frankly scallop, I don't give a clam."
My love affair with books started with my mom taking my sister and I to the library each week. At that time the library was fairly small because Peachtree City was a relatively new city. The library was located in the only main building. The library, city hall, the police station and the doctor's office were all in the same building. Our new church even met in the city hall every Sunday. Anyway, we each picked approximately six or seven books a piece and off course I had them all read by the end of the day.
Every week my sister and I got $1.50 a week for allowance. We would ride our bikes first to Hudson's supermarket, pick out a treat (usually either a big dill pickle, some sort of fake squeezable cheeze or a candy bar) and then ride our bikes over the bridge to the library. There we would spend hours in the air conditioned library and read and read.
We could only carry one or two books a piece in my wire basket, so we had to choose wisely.
I think I must have read almost every book in the children's section first, then I moved up to the young adult fiction. I read the books with the attractive covers and then when I felt like there wasn't much left to read, I checked out a book with a worn out cover. There was no book jacket, the cover boards were bumped and rubbed. I don't remember now the title of that first worn out book that I checked out and read, but I was blown away with how wonderful it was!
From then on I realized that the most bumped and beat up books were the best books at the library! "To Kill a Mockingbird" was one of those books. I truly learned the meaning ,"Don't judge a book by it's cover!" Isn't that true in all things in life? Grandparents ? Don't they have the best stories?
I still tend to pull the book off the shelf that is most love-worn. They are the best.
Every time there is thunder, my dog Harvey gets scared and paws on my side of the bed until I pick him up, put him on our bed and cover him with a blanket. It never bothers his sibling Azalea and she continues to sleep just fine.
It's 3:00 am and Harvey has woken me up because of a thunderstorm, so I have had a hard time going back to sleep and have decided to come out here on the porch and listen to the rain and blog.
It sounds like camp out here with the rain hitting the leaves and the tree frogs singing. There is something peaceful about a good rain, isn't there? By late morning there probably will hardly be any sign that it was raining at all. Rain showers come quickly here and leave just as fast.
Just a couple more days with mom and dad here and DH will be coming to get us and it will be back to Tennessee. School will be starting and there will be a schedule to keep just about hourly.
But for now I am listening to rain on a tin roof, knowing all is at peace.
I get so frustrated with what I see on the television in regards to political journalism. I get so confused and mad! But never fear, when I need something explained in a logical, straight to the point, from the horses mouth, Christian point of view, I go to my friend Kimberly's husbands blog, www.trumanstake.blogspot.com.
Truman is always up on local happenings and usually goes right to the source if there is a question. He has some very excellent links on his site if you need a broader or generalized subject concerning politics.
So if you want to scream with what all that is happening right now and need some answers, now you know where to go for a trusted answer.
Okay, here is my crazy family. DH is practicing his swimming strokes in a concrete pool and my dad is hanging out working on getting some sun on his own beach.
It is so hot down here, they are probably seeing a mirage of water.
I will try to keep the diving to a minimum.
My sister has a brand new blog! It's wonderful(of course)! Now you will see why I love her so much and love hanging with her!
Isn't that a precious title for a blog? To be able to sit with Les is a treasure!
I am so thankful for the time I have had to spend with my parents here. It had been almost two months since I had seen them. Staying with them for extended time has been glorious. It has been great for Savannah too. I can remember the weeks I would go to Mississippi to spend with my grandparents.
The first time I got to drive by myself was with my granddad. I was probably 11 and my grandaddy told me to go get the pick-up truck. I stood there shocked, wondering if he was serious and he had to tell me again. I was SO excited. The three wheeler wouldn't start and he wanted me to pull him on it so he could pop the clutch to get it started. That is now one of my favorite memories, driving and seeing my granddaddy in the rear view mirror, just hoping I wouldn't jerk him off his seat.
Savannah is cooking, cleaning and playing with her Mimi. These are precious moments that she will cherish all her life.
This week my dad was telling me some stories about when he was in Korea during the war there...stories I had never heard before.
Treasure those moments with your loved ones.
Drama, intrigue, tragedy, history, secrecy, a love story and a 300 year old English country manor.
I have just finished a great summer read. While I was in London in June and we had a little time to spare at Victoria Station, we went looking for a decent copy of 'Pride and Prejudice' at a book store for Savannah. While there, I found a couple books on their best-seller list. The House at Riverton by Kate Morton was one of them.
Ninety-eight year old Grace decides to tell her life story and what a story it is. Upstairs/downstairs daily life, sisters, family, the aftershocks of WWI and many, many secrets.
It is a novel that reads like a mystery. Ms. Morton says it is "the haunting of the present by the past."
This morning we awoke to the yummy scent of Poppy's beignet's. Poppy has a standing tradition of making French donuts at least once a visit when we are all together. He uses the real Cafe du Monde mix from New Orleans and he has it down to a science.
DH's parents came down two days ago and they are enjoying the condo they are renting. We went over and visited with them for the afternoon. It's always fun to see how another person decorates their space and this one was different for a beach condo. They used the jungle theme. Elephants, monkeys, tiger prints, lots of brown accent walls and wrought iron. Mrs. C said she was going to have to spend a while to make sure they haven't left anything because of the darker colors and prints.
It is so very hot down here! It feels like you walk right into a sauna when you go out. You pretty much need to be in the water all the time.
DH leaves tomorrow and we have all had such a great time! He is going to leave us for a week and then come back giving us some more time with mom and dad. Sav is finishing up her required reading and is looking forward to seeing her friends at school, but not the schedule.
The pool has been dug and poured with concrete, but still has a long way to go. DH says maybe we can try to come back over labor day weekend and try it out.
Today we are all going to try to see the movie Mama Mia and hit the outlet mall. DH is creaming us all at Mexican dominoes and he is not being humble about winning. He's rubbing it in to much. So, I am off to try again to bring him down a little.
Take walks on the beach at 10pm to see what the moon looks like.
The other day I was flipping on the television and came across the movie "Babe." It had been a long time since I had seen it and I had forgotten what a precious movie it is and one of my all time favorite movie scenes is in it.
The mean cat has told Babe that he is just being loved on so he can be fattened up and eaten and he has no hope. Babe is heartbroken and runs away. The kind gentle giant farmer (James Cromwell?) finds Babe and the vet says he won't make it unless he eats. The kind farmer tries to feed Babe in his arms with a bottle. Babe doesn't start eating until the farmer starts singing this sweet, loving song.
This little song just makes me cry with happiness. It starts out sweet and Babe starts eating. It builds in tempo and the gentle farmer gets up and starts to dance for him right there in his living room while Babe watches him from the sofa, happy and loved.
I think of my heavenly Father in the role of the gentle farmer, desolate when he has lost one of his own, and then the joy of bringing us home, feeding us and then dancing with joy!
I love, love, love perfumes, colognes, body butters, lotions, hand cremes and free samples!!!! I am always looking for that "signature" scent that is the perfect one for me. My sister Leslie is the same. It must be a genetic thing because Sav seems to have her favorites too.
I thought I would list some of my favorites, some of Leslie's favorites and Sav's favorites.
Popy Moreui- (bought at Nordstrom in San Francisco years ago) -my friend Jodie has made this her signature scent and it smells fabulous on her!
Bond no. 9-Nuits de Noho- I absolutely LOVE this one! This whole line is incredible!
I told you I was crazy about the stuff!
Savannah just finished a week of working at Elaine's art day camp. She had an absolute blast! There were nine darling little girls between the ages of 3 and 9. She says they all wanted to be in her lap at some point in time. I am so proud of her! She loved listening and watching Elaine teach them and she always came home thinking about what she could do to help for the next day. It's wonderful when Savannah has been able to apprentice under someone like Elaine who is so creative, talented and imaginative and has so much to teach and pass on to Savannah. She said it brought back so many memories of her art classes as a child. And she says it was fun! It's a great lesson on working and doing what you love at the same time.
Sunday after church we will head to the cottage at the beach for a visit. After 20 years plus, mom and dad are putting in a swimming pool. We are so excited! Yes, we will still go swim in the ocean lots. DH just won't have to worry about sharks so much! They say it should take about 6 weeks or so. With this heat, it won't come soon enough. They are trying to decide what trees can stay and what trees have to go. That's the main reason one hasn't been put in before. The wind bent oaks are beautiful and only native to the area. I told them on the phone that I hope they can get all that worked out and all the big decisions done before we get there. Mom chuckled.
We have been having some of the best speakers at our church summer series on Wednesday nights. I will have to say I think the second speaker was the best and was outstanding!! Oh, did I mention it was my DH (darling husband)? I think he worked on it for 6 months. And you could tell that he had put a lot of time, thought and preparation. He sure was happy when he finished and it was over too!
DH, Sav, DH's parents and I all went to Kentucky to watch Savannah's best friend's little sister compete in a Saddlebred horse show. This is Holly's new horse Casper and he is GORGEOUS! This was Holly's first time showing him and she took 1st place! We were so proud of her and it was so beautiful to watch her confidence with him. I believe they are going to go a long way together. She and her sister Amy have already each earned a spot in the Grand Championship in August. Way to go girls!
Wow! What a read Beach Music has been! It is over 700 pages long and is quite a long summer read.
The first three chapters take place in Italy, so it was a beautiful beginning. Pat Conroy writes lyrically and it was a joy reading as he described the sights, smells and tastes wherever he is.
Beach music is the sound that the ocean makes that we love to listen to. I love that!
I liked reading the book, but I was really not in a place in my life where I could fully enjoy it like I might at another time. There are certain times that I can appreciate someone writing about an incredibly disfunctional family with a capital "D", and this summer is not one of them. I felt like I was reading about lots of people I know, have known and loved. This I believe is where his talent lies.
Mr. Conroy definitely knows how to write about the South and I hope he continues to write more books. I was exhausted by the time I finished this one. Back to my cozy mysteries for me!
The whole gang in Clare's bed for movie night and lots of silliness!
On our way to Monhegan, poor Elaine and Clare were a little woozy.
The whole group at Pickety Place for lunch. Beautiful luncheon plates served with herbs and edible flowers.
Beautiful gardens, with rows of lavender!
This week along with reading my book group book Beach Music, I am re-reading a favorite historical mystery series. I finished Crocodile on the Sandbank, and have started Curse of the Pharaoh's. They are the first couple of approximately eighteen Amelia Peabody mysteries. I am not sure how I found this series, but they are my all time favorite historical mystery series! Barbara Mertz writes as Elizabeth Peters and Barbara Michaels. Barbara Michaels writes more of the contemporary mysteries.
My friend Jodie and I have devoured these books through the years and have decided to collect them all. Jodie only collects a certain few authors, so it shows how much she loves them too.
Amelia Peabody is the heroine we all want to be. She is very intelligent and independently wealthy and travelling the world after her father passes when she meets her equal in Radcliffe Emerson. Radcliffe and his brother Walter are archaeologists excavating in Egypt. Amelia and Emerson marry and soon have a son named Ramses. He is as intelligent as his parents and many adventures ensue as they continue to do what they love.
I can't recommend these mysteries enough! Amelia makes me laugh and I come away having learned about the period and Egyptian history.
Jodie and I love trying to figure out who would be great playing Amelia and Emerson on screen. We have a hard time giving the honor to anyone because they are so special to us! So please keep writing Ms. Mertz! | english |
சல்மான் கானை பாம்பு கடித்தது பாலிவுட் சூப்பர் ஸ்டார் சல்மான் கானை பாம்பு கடித்த சம்பவம் பரபரப்பை ஏற்படுத்தியுள்ளது. பாலிவுட் சூப்பர் ஸ்டார் சல்மான் கான். இவர், நேற்றிரவு மும்பையில் உள்ள தனது பன்வெல் பண்ணை வீட்டில் தங்கியிருக்கிறார். அப்போது அதிகாலை 3.30 மணியளவில் இவரை பாம்பு கடித்ததாக கூறப்படுகிறது. உடனடியாக அவர், அருகிலுள்ள எம்ஜிஎம் மருத்துவமனையில் அனுமதிக்கப்பட்டார். இதையும் படிக்க கெங்கவல்லியில் ஆற்றின் நடுவே ரூ.2 லட்சம் சொந்த செலவில் தற்காலிக பாலம் அமைத்த மக்கள் அங்கு அவருக்கு மருத்துவர்கள் உரிய சிகிச்சை அளித்தனர். சல்மான் கானை கடித்த பாம்பு விஷமற்றது என்பதால் அவருக்கு பெரியளவில் எந்த பாதிப்பும் இல்லை எனக் கூறப்படுகிறது. சிகிச்சைக்குப் பிறகு, சல்மான் கான் இன்று காலை மருத்துவமனையில் இருந்து டிஸ்சார்ஜ் செய்யப்பட்டார். தற்போது அவர் தனது பண்ணை வீட்டில் ஓய்வெடுத்து வருகிறார். மேலும் அவரது உடல்நிலை நன்றாக இருப்பதாக கூறப்படுகிறது. சல்மான் கானை பாம்பு கடித்த சம்பவம் இந்தி திரையுலகில் பெரும் அதிர்ச்சியை ஏற்படுத்தியுள்ளது. | tamil |
सिलावन हादसे में दो और बरातियों की मौत ललितपुर। कोतवाली महरौनी अंतर्गत ग्राम सिलावन के पास ट्रक और पिकअप वाहन की टक्कर में मरने वालों की संख्या बढ़कर अब चार हो गई है। दो बरातियों की मौत शु्क्रवार रात में जिला अस्पताल ले जाते समय रास्ते में हो गई थी, जबकि दो अन्य घायलों की मौत उपचार के दौरान हो गई। पुलिस ने मृतकों के शवों को पोस्टमार्टम के लिए भेज दिया। कस्बा पाली निवासी बजेंद्र बुनकर की बरात शुक्रवार रात को महरौनी क्षेत्र के ग्राम खटौरा में जा रही थी। बरात के अधिकांश वाहन आगे निकल गए, जबकि पीछे एक पिकअप वाहन में बराती और कुछ बैंड बाजे वाले जा रहे थे। अभी पिकअप वाहन महरौनी क्षेत्र के ग्राम सिलावन मंडी के आगे ही पहुंची थी कि पिकअप वाहन सामने से आ रहे ट्रक से टकरा गया। हादसे में पिकअप वाहन बुरी तरह क्षतिग्रस्त हो गया। जबकि उसमें सवार बराती और बैंड बाजे वाले बुरी तरह घायल हो गए। राहगीरों ने यह देख तत्काल 108 एंबुलेंस को फोन किया और घायलों को जिला अस्पताल भेजा दिया, जहां चिकित्सकों ने परीक्षण उपरांत कस्बा पाली निवासी सियाराम 32 पुत्र चिंतामणी एवं कल्याण उर्फ कल्लू 50 पुत्र मठू बुनकर को मृत घोषित कर दिया था। जबकि पाली निवासी अंकित 18 पुत्र प्रकाश बरार, भरत 22 पुत्र बलदेव एवं रामसेवक पुत्र गोकुल, ग्राम गदनपुर निवासी मुन्ना बुनकर और पाली निवासी आनंद राज 22 पुत्र देशराज गंभीर रूप से घायल होने पर उनका उपचार जिला अस्पताल में चल रहा था। लेकिन देर रात में उक्त दुर्घटना में गंभीर रूप से घायल अंकित पुत्र प्रकाश बरार एवं ग्राम गदनपुर निवासी दीपक पुत्र मुन्नालाल बुनकर की भी उपचार के दौरान मौत हो गई। इससे परिजनों में कोहराम मच गया। कोतवाली पुलिस ने सभी मृतकों के शवों का पंचनामा भरकर पोस्टमार्टम के लिए भेज दिया। सियाराम और अंकित बैंड बजाकर चलाते थे घर खर्च बारात में ग्राम पाली निवासी मृतक सियाराम और अंकित बैंड बजाते थे। जिसमें सियाराम काफी अच्छा कलाकार था। जिसे बैंड बाजे वाले अपने बैंडों में अलग से बुलाते थे। सियाराम की तीन बेटियां हैं दो जुड़वा 33 वर्ष और वह पांच भाई व तीन बहनों में दूसरे नंबर का था। जबकि मृतक अंकित अविवाहित था और दो भाई और चार बहनों में दूसरे नंबर का था। वहीं मृतक कल्लू की एक बेटी है और मृतक दीपक तीन बहनों में इकलौता भाई था। | hindi |
കാട്ട് കുളവിയുടെ ആക്രമണത്തില് പശുവിന് ദാരുണാന്ത്യം പന്തളം: കാട്ട് കുളവിയുടെ ആക്രമണത്തില് പശുവിന് ദാരുണാന്ത്യം. പന്തളം കുരമ്ബാല തെക്കാണ് സംഭവം.കുരമ്ബാല മുകളയ്യത്ത് പ്രദേശത്ത് റബര്തോട്ടത്തില് പുല്ലു തിന്നുകൊണ്ടിരുന്ന വെച്ചൂര് ഇനത്തില്പെട്ട പശുവാണ് ചത്തത്. കുരമ്ബാല തെക്ക് ഇടത്തുണ്ടില് ചൈത്രം വീട്ടിലെ പശുവിനാണ് ദാരുണ അന്ത്യം സംഭവിച്ചത്. പശുവിനെ രക്ഷപ്പെടുത്താന് ശ്രമിച്ച അധ്യാപകനായ കുരമ്ബാല തെക്ക് ചൈത്രത്തില് രാജേഷ് 42 തിരുവല്ല ബിലീവേഴ്സ് ആശുപത്രിയില് ചികിത്സയിലാണ്. രാജേഷിന്റെ സഹോദരന് പ്രതീഷ് 38, രാജേഷിന്റെ അനന്തരവന് ദേവ്കൃഷ്ണ 11 എന്നിവര്ക്കും കുത്തേറ്റിട്ടുണ്ട്. കുളവിയുടെ സാന്നിധ്യം കണ്ട ഭാഗത്ത് രാത്രിയില് പ്രദേശവാസികള് തീയിട്ടു. ഇന്നലെയും സമീപപ്രദേശങ്ങളില് കുളവിയുടെ സാന്നിധ്യം കണ്ടിരുന്നു. | malyali |
BREAKING NEWS: ರಾಜ್ಯ ಸರ್ಕಾರದಿಂದ ಕೈಗಾರಿಕಾ ಕಟ್ಟಗಳಿಗೆ ಬಿಡಬೇಕಾದ ಖಾಲಿ ಜಾಗ ನಿಗದಿಪಡಿಸಿ ಗೆಜೆಟ್ ಅಧಿಸೂಚನೆ: ಹೀಗಿದೆ ಅಳತೆ ಬೆಂಗಳೂರು: ಕರ್ನಾಟಕ ಗ್ರಾಮ ಸ್ವರಾಜ್ ಮತ್ತು ಪಂಚಾಯತ್ ರಾಜ್ ಕಟ್ಟಡಗಳ ನಿರ್ಮಾಣದ ಮೇಲೆ ಜಿಲ್ಲಾ ಪಂಚಾಯತ್ ಗಳ, ತಾಲೂಕು ಪಂಚಾಯತ್ ಮತ್ತು ಗ್ರಾಮ ಪಂಚಾಯತ್ ಗಳ ನಿಯಂತ್ರಣ ಮಾದರಿ ಉಪವಿಧಿ 2015ಕ್ಕೆ ರಾಜ್ಯ ಸರ್ಕಾರ ತಿದ್ದುಪಡಿ ತಂದಿದೆ. ಅದರ ಹಿನ್ನಲೆಯಲ್ಲಿಯೇ ಕೈಗಾರಿಕಾ ಕಟ್ಟಗಳಿಗೆ ಬಿಡಬೇಕಾಗಿರುವಂತ ಖಾಲಿ ಜಾಗವನ್ನು ನಿಗದಿ ಪಡಿಸಿದೆ. ಉಡಪಿ: ಮಣಿಪಾಲದಲ್ಲಿ ಡಿವೈಡರ್ ಗೆ ಬೈಕ್ ಡಿಕ್ಕಿಯಾಗಿ ಇಬ್ಬರು ವಿದ್ಯಾರ್ಥಿಗಳು ಸಾವು ಈ ಸಂಬಂಧ ಗೆಜೆಟ್ ಅಧಿಸೂಚನೆಯನ್ನು ಹೊರಡಿಸಿರುವಂತ ಗ್ರಾಮೀಣಾಭಿವೃದ್ಧಿ ಮತ್ತು ಪಂಚಾಯತ್ ರಾಜ್ ಇಲಾಖೆಯ ಸರ್ಕಾರದ ಅಧೀನ ಕಾರ್ಯದರ್ಶಿಗಳು, ಕರ್ನಾಟಕ ಗ್ರಾಮ ಸ್ವರಾಜ್ ಮತ್ತು ಪಂಚಾಯತ್ ರಾಜ್ ಮಾದರಿ ಉಪ ವಿಧಿಗಳು 2015ರ 18ನೇ ಉಪವಿಧಿಯ ಖಂಡ 4ಕ್ಕೆ ತಿದ್ದುಪಡಿಯನ್ನು ತರಲಾಗಿದೆ ಎಂದು ತಿಳಿಸಿದ್ದಾರೆ. ಶಿವಮೊಗ್ಗ: ರಸ್ತೆ ಅವ್ಯವಸ್ಥೆಯ ಬಗ್ಗೆ ಪಾಲಿಕೆ ವಿರುದ್ದ ನಾಗರೀಕರ ಪ್ರತಿಭಟನೆ ತಿದ್ದುಪಡಿ ಅಧಿಸೂಚನೆಯಂತೆ ಕೈಗಾರಿಕಾ ಕಟ್ಟಡಗಳಿಗೆ ಬಿಡಬೇಕಾಗಿರುವಂತ ಖಾಲಿ ಜಾಗದ ಕನಿಷ್ಠ ಅಳತೆಯನ್ನು ಈ ಕೆಳಕಂಡಂತೆ ನಿಗದಿ ಪಡಿಸಿದೆ. 255 ಚದರ ಮೀಟರ್ ಗಳವರೆಗಿನ ಪ್ಲಾಟುಗಳಿಗೆ ಮುಂಭಾಗದಲ್ಲಿ 3 ಮೀಟರ್, ಎಡಬದಿ, ಬಲಬದಿ ಹಾಗೂ ಹಿಂಬದಿಯಲ್ಲಿ 1.5 ಮೀಟರ್ ಬಿಡಬೇಕು. 510 ಚದರ ಮೀಟರ್ ಪ್ಲಾಟ್ ಗಳಿಗೆ ಮುಂಭಾಗದಲ್ಲಿ 3 ಮೀಟರ್, ಎಡಬದಿ, ಬಲಬದಿ ಹಾಗೂ ಹಿಂಬದಿಯಲ್ಲಿ 2.5 ಮೀಟರ್ ಖಾಲಿ ಜಾಗ ಬಿಡುವುದು. 1020 ಚದರ ಮೀಟರ್ ಪ್ಲಾಟ್ ಗಳಿಗೆ ಮುಂಭಾಗದಲ್ಲಿ 4.5 ಮೀಟರ್, ಎಡಬದಿ, ಬಲಬದಿ ಹಾಗೂ ಹಿಂಬದಿಯಲ್ಲಿ 3 ಮೀಟರ್ ಖಾಲಿ ಜಾಗ ಬಿಡುವುದು. 2025 ಚದರ ಮೀಟರ್ ಪ್ಲಾಟುಗಳಿಗೆ ಮುಂಭಾಗದಲ್ಲಿ 8 ಮೀಟರ್, ಎಡಬದಿ, ಬಲಬದಿ ಹಾಗೂ ಹಿಂಬದಿಯಲ್ಲಿ 4.50 ಮೀಟರ್ ಖಾಲಿ ಬಿಡುವುದು. 4050 ಚದರ ಮೀಟರ್ ಪ್ಲಾಟ್ ಗಳಿಗೆ ಮುಂಭಾಗದಲ್ಲಿ 9 ಮೀಟರ್, ಎಡಬದಿ, ಬಲಬದಿ ಹಾಗೂ ಹಿಂಬದಿಯಲ್ಲಿ 6 ಮೀಟರ್ ಖಾಲಿ ಬಿಡಬೇಕು. 8100 ಕ್ಕಿಂತ ಹೆಚ್ಚು ಚದರ ಮೀಟರ್ ವಿಸ್ತೀರ್ಣದ ಪ್ಲಾಟ್ ಗಳಿಗೆ ಮುಂಭಾಗದಲ್ಲಿ 10 ಮೀಟರ್, ಎಡಬದಿ, ಬಲಬದಿ ಹಾಗೂ ಹಿಂಬದಿಯಲ್ಲಿ 8 ಮೀಟರ್ ಬಿಡುವುದು. ಈ ಕಟ್ಟಡಗಳು ಲಿಫ್ಟ್ ಗಳನ್ನು ಹೊಂದಿಲ್ಲದಿದ್ದರೇ ಅವುಗಳ ಗರಿಷ್ಠ ಎತ್ತರವು 2 ಮಹಡಿಗಳು ಆಗಿರಬೇಕು. ಕೈಗಾರಿಕಾ ಕಟ್ಟಗಳ ಗರಿಷ್ಠ ಎತ್ತರದ ಎಳತೆಯು 15 ಮೀಟರ್ ಗಳಾಗಿರಬೇಕು. ಇಂತಹ ಎತ್ತರವುಳ್ಳ ಕೈಗಾರಿಕಾ ಕಟ್ಟಡಗಳಿಗೆ ಸಂಬಂಧಿಸಿದಂತೆ ಅನುಮತಿಸಬಹುದಾದ ಗರಿಷ್ಠ ನೆಲದ ವ್ಯಾಪ್ತಿ ಶೇ.65ರಷ್ಟು ಆಗಿರಬೇಕು ಎಂದು ತಿಳಿಸಿದೆ. ಈ ಗೆಜೆಟ್ ಅಧಿಸೂಚನೆ ಪ್ರಕಟಗೊಂಡ ಮೂವತ್ತು ದಿನಗಳೊಳಗಾಗಿ ಆಕ್ಷೇಪಣೆಯನ್ನು ಸಲ್ಲಿಸುವಂತೆ ಸೂಚಿಸಿದೆ. | kannad |
یِہ چھُ نہٕ لورِ دستار بٔلِکہِ چھُ اکھ خودکار کھاتہٕ۔ یِہ کھاتہٕ چھُ واریاہ اؠڑِٹ کَم وَقتس مَنٛز کَرن یؠتھ اِنسان اؠڑِٹرس واریاہ وَقت لَگن چھُ۔ | kashmiri |
பட்டம்பாளையம் குளம் நிரம்பி வழியுது! திருப்பூர்: பட்டம்பாளையம், கருப்பராயன்கோவில் குளம், தொடர்ந்து இரண்டாவது நாளாக நிரம்பி வழிவதால், விவசாயிகள் மகிழ்ச்சி அடைந்துள்ளனர். திருப்பூர் ஒன்றியத்தின் வடக்கே உள்ள பட்டம்பாளையம், சொக்கனுார், தொரவலுார், வள்ளிபுரம், மேற்குபதி, ஈட்டிவீரம்பாளையம், பெருமாநல்லுார், காளிபாளையம் உள்ளிட்ட ஊராட்சிகள், வானம் பார்த்த பூமியாக உள்ளன.குளம், குட்டைகள் இருந்தும், துார்வாரி சுத்தம் செய்யாமல் இருப்பதால், மழைநீரை முழுமையாக சேகரிக்க முடிவதில்லை. கடந்த சில நாட்களாக, வடக்கு பகுதியில் மழை பெய்துகொண்டிருப்பதால், கருப்பராயன்கோவில் குளம் நிரம்பி, இரண்டாவது நாளாக நேற்றும் உபரிநீர் வெளியேறியது. பட்டம்பாளையம், கரிச்சி பாளையம், தைலாம்பாளையம் பகுதிகளில் பெய்யும் மழைநீர், சிற்றோடைகள் வழியாக, கருப்பராயன்கோவில் குளத்துக்கு வருகிறது. மொத்தம், ஏழு ஏக்கர் பரப்பில் உள்ள குளம், 2017ம் ஆண்டுக்கு பிறகு, மீண்டும் நிரம்பி வழிகிறது. குளத்தில் இருந்து வெளியேறும் உபரிநீர், வாய்க்கால் வழியாக நம்பியூர் சென்று, அங்கிருந்து எலத்துார் குளம் வரை செல்கிறது. கருப்பராயன்கோவில் குளத்தில், உபரிநீர் வெளியேறும் மதகு பகுதியில், கருங்கற்கள் பெயர்ந்து சேதமாகியுள்ளன. இதேநிலை தொடர்ந்தால், மதகு சீர்குலையும் அபாயம் உள்ளது.இனியும் காலம்கடத்தாமல், மதகு பகுதியை கான்கிரீட் மூலம் சீரமைக்க வேண்டுமென, விவசாயிகள் கோரிக்கை வைத்துள்ளனர். அப்பகுதியினர் கூறுகையில், கருப்பராயன்கோவில் குளம், இரண்டு நாட்களாக நிரம்பி வழிவதால், நிலத்தடி நீர் செறிவூட்டப்படும். ஆழ்துழை கிணறுகளுக்கு தண்ணீர் கிடைக்கும். உபரிநீர் வீணாக சென்று கொண்டிருக்கிறது. வாய்க்காலின் குறுக்கே, கூடுதலாக தடுப்பணைகள் அமைத்தால், ஆங்காங்கே, பல இடங்களில் உபரிநீரை தேக்கி வைக்க முடியும், என்றனர். | tamil |
फैक्ट्रीकर्मी के खाते से उड़ाए 45 हजार रुपये खुद को बजाज फाइनेंस कंपनी का कस्टमर केयर अधिकारी बताकर साइबर ठग ने फैक्ट्रीकर्मी के खाते से 45000 रुपये निकाल लिए। पीड़ित ने पुलिस को सूचना देकर कार्रवाई की मांग की है।टांडा उज्जैन निवासी गौरव कुमार पशुपति कंपनी में नौकरी करते हैं।कार और बाइक पर लेटेस्ट अपडेट पाने के लिए यहां क्लिक करें शुक्रवार को उनके मोबाइल पर फोन आया। फोन करने वाले ने बताया कि वह बजाज फाइनेंस कंपनी से बोल रहा है और क्रेडिट कार्ड बनाना है। इस पर उन्होंने एटीएम कार्ड की सारी डिटेल दे दी। जिसके बाद साइबर ठग ने खाते से 45 हजार रुपये निकाल लिए। पीड़ित ने पुलिस को सूचना देकर कार्रवाई की मांग की है। For Hindustan : हिन्दुस्तान ईसमाचार पत्र के लिए क्लिक करें epaper.livehindustan.com | hindi |
ಈಗ ಮತ್ತೊಂದು ಟ್ವೀಟ್ ಗಾಗಿ ಕಾಮ್ರಾ ವಿರುದ್ಧ ನ್ಯಾಯಾಂಗ ನಿಂದನೆ ಪ್ರಕರಣ ದಾಖಲಿಸಲು ಎಜಿ ಅಸ್ತು ನವದೆಹಲಿ: ಸುಪ್ರೀಂ ಕೋರ್ಟ್ ವಿರುದ್ಧದ ತಮ್ಮ ವಿವಾದಾತ್ಮಕ ಟ್ವೀಟ್ಗಳನ್ನು ಡಿಲೀಟ್ ಮಾಡುವುದಿಲ್ಲ ಎಂದು ನವೆಂಬರ್ 18 ಮಾಡಿದ್ದ ಹೊಸ ಟ್ವೀಟ್ ಗೆ ಸಂಬಂಧಿಸಿದಂತೆ ಕಾಮಿಡಿಯನ್ ಕುನಾಲ್ ಕಾಮ್ರಾ ಅವರ ವಿರುದ್ಧ ನ್ಯಾಯಾಂಗ ನಿಂದನೆ ಪ್ರಕರಣ ದಾಖಲಿಸಲು ಅಟಾರ್ನಿ ಜನರಲ್ ಕೆಕೆ ವೇಣುಗೋಪಾಲ್ ಅವರು ಶುಕ್ರವಾರ ಅನುಮತಿ ನೀಡಿದ್ದಾರೆ. ಕಳೆದ ವಾರ ರಿಪಬ್ಲಿಕ್ ಟಿವಿ ಸಂಪಾದಕ ಆರ್ನಬ್ ಗೋಸ್ವಾಮಿ ಜಾಮೀನು ವಿಚಾರಕ್ಕೆ ಸಂಬಂಧಿಸಿದಂತೆ ಸುಪ್ರೀಂಕೋರ್ಟ್ ಆದೇಶವನ್ನು ಟೀಕಿಸಿ ಕಾಮ್ರಾ ವಿರುದ್ಧ ಕ್ರಿಮಿನಲ್ ನ್ಯಾಯಾಂಗ ನಿಂದನೆ ಪ್ರಕರಣ ದಾಖಲಿಸಲು ಅಟಾರ್ನಿ ಜನರಲ್ ಕೆಕೆ ವೇಣುಗೋಪಾಲ್ ಒಪ್ಪಿಗೆ ನೀಡಿದ್ದರು. ಈಗ ಹೊಸ ಟ್ವೀಟ್ ಗೆ ಸಂಬಂಧಿಸಿದಂತೆ ಮತ್ತೊಂದು ಕೇಸ್ ದಾಖಲಿಸಲು ಅನುಮತಿ ನೀಡಿದ್ದಾರೆ. ಇದು ಅತ್ಯಂತ ಅಶ್ಲೀಲ ಮತ್ತು ಅಸಹ್ಯಕರ. ಇದು ಸುಪ್ರೀಂ ಕೋರ್ಟ್ನ ಅಧಿಕಾರವನ್ನು ಕಡಿಮೆ ಮಾಡುವ ಉದ್ದೇಶ ಹೊಂದಿದೆ. ನ್ಯಾಯಾಂಗದ ಮೇಲೆ ಆಕ್ರಮಣ ಮಾಡುವುದರಿಂದ ಶಿಕ್ಷೆಗೆ ಗುರಿಯಾಗಬೇಕಾಗುತ್ತದೆ ಎಂಬುದನ್ನು ಜನರು ಅರ್ಥಮಾಡಿಕೊಳ್ಳಬೇಕು ಎಂದು ಅಟಾರ್ನಿ ಜನರಲ್ ಹೇಳಿದ್ದಾರೆ. ಕಾಮಿಡಿಯನ್ ಕುನಾಲ್ ಕಾಮ್ರಾ ವಿರುದ್ಧ ನ್ಯಾಯಾಂಗ ನಿಂದನೆ ಪ್ರಕರಣ ದಾಖಲಿಸಲು ಅನುಮತಿ ಕೋರಿ ಪ್ರಯಾಗರಾಜ್ ಮೂಲದ ವಕೀಲ ಅನುಜ್ ಸಿಂಗ್ ಅವರು ಅರ್ಜಿ ಸಲ್ಲಿಸಿದ್ದರು. ಅದಕ್ಕೆ ವೇಣುಗೋಪಾಲ್ ಒಪ್ಪಿಗೆ ನೀಡಿದ್ದಾರೆ. | kannad |
یمہ نظمہِ ہند کردار چھہ بے بس۔ اسمان چھس نہ پلان نرِ تہ زمین چھس نہ پلان زنگہ۔ او ہ کِنۍ چھہ کردار پننٮن خوٲہشاتن ہند دامن وٹان تہ نرِ زنگہ ؤٹِتھ پانسے منز شرپان۔ | kashmiri |
لاہوراردو پوائنٹ اخبارتازہ ترین این این ئی 29 جنوری2019ءشہر میں پرچون سطح پر برائلر مرغی کے گوشت کی قیمت مزید 7روپے اضافے سے 178روپے زندہ برائلر مرغی 5روپے اضافے سے 121 روپے فی کلو رہی برائلر گوشت کی قیمت میں دو روز میں مجموعی طو رپر 21روپے فی کلو اضافہ ریکارڈ کیا گیا فارمی انڈوں کی قیمت 105روپے فی درجن پر مستحکم رہی | urdu |
ఇక రెండు రకాలుగా.. ఐటీఐ పరీక్షల నిర్వహణ ఓఎమ్మార్, కంప్యూటర్ ఆధారితంగా... విజయనగరం విద్యావిభాగం, న్యూస్టుడే: ఐటీఐ సప్లిమెంటరీ పరీక్షలు ఓఎమ్మార్, సీీబీటీ విధానంలో రెండు రకాలుగా నిర్వహించనున్నారు. 2017, అంతకు ముందు సంవత్సరాల్లో ప్రవేశాలు పొంది పరీక్షలో తప్పిన అభ్యర్థులకు ఓఎమ్మార్పై పరీక్ష నిర్వహిస్తారు. ఈ నెల 8 వరకు ఇవి జరగనున్నాయి. . వీరిలో ఎక్కువగా ప్రైవేటు విద్యార్థులే ఉన్నారు. మంగళవారం నుంచి ఓఎమ్మార్పై జరిగే పరీక్షలకు విజయనగరం, బొబ్బిలి ప్రభుత్వ ఐటీఐలను కేంద్రాలుగా ఎంపిక చేశారు. 22 పారిశ్రామిక శిక్షణ సంస్థలకు చెందిన విద్యార్థులు విజయనగరం ప్రభుత్వ ఐటీఐలో పరీక్ష రాయనున్నారు. బొబ్బిలిలో తొమ్మిది ఐటీఐల వారికి కేటాయించారు. బొబ్బిలిలోని రెండు ప్రైవేటు ఐటీఐల అభ్యర్థులకు విజయనగరం పరీక్షా కేంద్రం కేటాయించడంపై పలువురు విమర్శలు గుప్పిస్తున్నారు. వారి పర్యవేక్షణలో...: సీీబీటీ కంప్యూటర్ ఆధారిత పరీక్ష థర్ట్పార్టీ కనుసన్నల్లో జరగనుంది. 2018లో ప్రవేశాలు పొందిన వారికి సప్లిమెంటరీ పరీక్షలు సీీబీటీ విధానంలో నిర్వహించనున్నారు. పరీక్షలు ఈ నెల ఆరు నుంచి పది వరకు జరుగుతాయి. ఈ పరీక్షలను తొలిసారిగా థర్డ్పార్టీకి అప్పగించారు. ఓఎమ్మార్ పరీక్షలకు భిన్నంగా వాటిని గజపతినగరంలోని ఓ ప్రయివేటు పాలిటెక్నిక్ కళాశాలను కేంద్రంగా కేటాయించారు. కళాశాలకు అదే ప్రాంగణంలో ఐటీఐ కూడా ఉండటంతో కొందరు పెదవివిరుస్తున్నారు. వివిధ ఐటీఐలకు చెందిన విద్యార్థులు హాజరవుతున్నందున ఏర్పాట్లు పూర్తయ్యాయని కన్వీనరు గోపాలకృష్ణ న్యూస్టుడేకు తెలిపారు. మొత్తం రాయనున్న వారు 1254 సీబీటీ 720 ఓఎమ్మార్ 534 | telegu |
فیصل اباد23 نومبراردو پوائنٹ اخبارتازہ ترین اے پی پی 23 نومبر2017ءماہرین زراعت نے کہاہے کہ فیصل اباد جھنگ تھل خوشاب میانوالی لیہ ساہیوال ملتان بہاولپور اور بہاولنگر سمیت وسطی جنوبی پنجاب کے علاقوں میںموسم ربیع کی اہم پھلی دار فصل چنا کا شت کرکے بہترین پیداوار حاصل کی جاسکتی ہے جبکہ جدید ٹیکنالوجی سے استفادہ بھی چنے کی فی ایکڑ پیداورار میں اضافہ کا باعث بن سکتا ہے لہذا چنے کے کاشتکاروں کو چاہیے کہ وہ چنے کی دیسی اقسام بٹل 98 پنجاب 2008 بلکسر 2000 ونہار 2000 تھل 2006 سی ایم 98کابلی قسم نور 91اور سی ایم 2008 وغیرہ کی بروقت کاشت یقینی بنائیں تاکہ انہیں مالی منفعت حاصل ہو سکےایک ملاقات کے دوران انہوںنے بتایاکہ چنے کی پچھیتی کاشت 10دسمبر تک مکمل کی جا سکتی ہے انہوںنے کہا کہ کھادوں کا متناسب استعمال بھی پیداوار میں اضافہ کرتا ہے لہذا چنے کی بہتر پیداوار کیلئے 13کلو گرام نائٹروجن فی ایکڑ ضرور استعمال میں لائی جائے انہوںنے کہا کہ فاسفورس 36کلو گرام فی ایکڑ بھی موذوں ہے انہوںنے کہا کہ چنے کی بہترین پیداوار حاصل کرنے کیلئے پودوں کی تعداد 85سے 95 ہزار فی ایکڑ ہونی چاہیے انہوںنے کہا کہ اگر چنے کے کاشتکاروں کو کسی قسم کی مشکل یا پریشانی درپیش ہو تو وہ محکمہ کی فری ہیلپ لائن سے رابطہ کر سکتے ہیں | urdu |
\begin{document}
\begin{abstract}
The goal is to review the notion of a complete Segal space and how certain categorical notions behave
in this context.
In particular, we study functoriality in complete Segal spaces via fibrations.
Then we use it to define limits and adjunctions in a complete Segal space.
This is mostly expository and the focus is on examples and intuition.
\end{abstract}
\mathscr{M}aketitle
\addtocontents{toc}{\protect\mathscr{M}athscr{S}etcounter{tocdepth}{1}}
\tableofcontents
\mathscr{M}athscr{S}ection{Introduction}\label{Introduction}
\mathscr{M}athscr{S}ubsection{Motivation}\label{Motivation}
The theory of higher categories has become very important in modern mathematics. From topological quantum field theory to derived algebraic geometry
to symplectic geometry, higher categories have proven to be a good way study objects in their proper contexts.
The theory of higher categories or $(\infty,1)$-categories, as it is sometimes called, however, can be very intractable at times.
That is why there are now several models which allow us to understand what a higher category should be.
Among these models is the theory of {\it quasi-categories}, introduced by Bordman and Vogt (\cite{BV73}) and much studied by Joyal and Lurie
(\cite{Jo08}, \cite{Jo09} or \cite{Lu09}).
There are also other very prominent models such as simplicial categories, relative categories and Segal categories.
For a general survey on different models of ($\infty$,1)-categories see \cite{Be10}. \par
One of those models,
\emph{complete Segal spaces}, were introduced by \emph{Charles Rezk} in his seminal paper \emph{"A model for the homotopy theory of homotopy theory"}
(\cite{Re01}).
Later they were shown to be a model for \emph{$(\infty ,1)$-categories} (see \cite{JT07} for a direct proof and \cite{To05} for an axiomatic argument).
Despite their importance, most basic categorical constructions for complete Segal spaces have never been written out in detail.
\par
The goal of this note is to write an introduction to higher categorical concepts from the perspective of complete Segal spaces.
It focuses on examples and giving an understanding of the ideas rather than technical proofs.
\mathscr{M}athscr{S}ubsection{Outline}\label{Outline}
In the first section we start with the two concepts that motivated higher category theory: category theory and homotopy theory.
We show how these seemingly different ideas can be generalized to one general concept.
\par
In the second section we define Segal spaces which are our first approach to higher categories and show how they already have
many categorical properties. In particular, we can study object, morphisms and composition in a Segal space.
We end this section by discussing why that is not enough and why we need further conditions.
\par
In the third section we define complete Segal spaces and show that is a model of a higher category.
\par
In the fourth section we study functoriality in the realm of higher categories. In particular, we show how we can
use fibrations to study functors valued in spaces and functors valued in higher categories.
\par
In the fifth section we discuss colimits and adjunctions of complete Segal spaces.
\par
In the last section we show that complete Segal spaces have their own model structure and review some important features
of that model structure.
\mathscr{M}athscr{S}ubsection{Background}
The main background we assume is a general familiarity with category theory.
In particular, topics such as the definition of categories and functors, colimits and adjunctions are required.
Such material can be found in the first chapters of \cite{Ml98} or \cite{Ri17}.
\par
In addition, we assume some familiarity with homotopy theory. In particular, concepts such as topological spaces and
homotopy equivalences of spaces.
\par
Finally, it would be very helpful to have some background in the theory of simplicial sets and simplicial homotopy theory.
This in particular includes the definition of Kan complexes and homotopy equivalences.
\mathscr{M}athscr{S}ubsection{Acknowledgements} \label{Subsec Acknowledgements}
I want to thank my advisor Charles Rezk who has guided me through every step of the work.
I also want to thank Matt Ando for many fruitful conversations and suggestions.
\mathscr{M}athscr{S}ection{Category Theory \texorpdfstring{$\&$} \ \ Homotopy Theory: Two Paths towards Simplicial Spaces}
\label{Sec Category Theory Homotopy Theory Two Paths towards Simplicial Spaces}
In this section we take a quick look at categories and topological spaces to see how both of them can be thought of as special cases of
simplicial sets. This is an informal review of these subjects and serves as a motivation
for our definition of a higher category, rather than a thorough introductory text.
The section culminates in a introduction to simplicial spaces, which combines category theory and homotopy theory.
\mathscr{M}athscr{S}ubsection{Review of Category Theory} \label{Subsec First Definition of a Category}
The philosophy of categories is not to just focus on objects but also consider how they are related to each other.
This leads to following definition of a category.
\begin{defone} \label{Def Category set version}
A category $\mathscr{M}athcal{C}$ is a set of objects $\mathscr{M}athcal{O}$ and a set of morphisms $\mathscr{M}athcal{M}$ along with following functions:
\begin{enumerate}
\item An identity map $id: \mathscr{M}athcal{O} \to \mathscr{M}athcal{M}$.
\item A source-target map $(s,t): \mathscr{M}athcal{M} \to \mathscr{M}athcal{O} \times \mathscr{M}athcal{O}$.
\item A composition map $m: \mathscr{M}athcal{M} ^s\times_{\mathscr{M}athcal{O}}^t \mathscr{M}athcal{M} \to \mathscr{M}athcal{M}$.
\end{enumerate}
These functions have to make the following diagrams commute:
\begin{enumerate}
\item {\it Source-Target Preservation:}
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
\mathscr{M}athcal{M} \arrow[d, "s"] & \mathscr{M}athcal{M} \underset{\mathscr{M}athcal{O}}{\times} \mathscr{M}athcal{M} \arrow[r, "\pi_2"] \arrow[l, "\pi_1"] \arrow[d, "m"] &
\mathscr{M}athcal{M} \arrow[d, "t"] \\
\mathscr{M}athcal{O} & \mathscr{M}athcal{M} \arrow[r, "t"] \arrow[l, "s"] & \mathscr{M}athcal{O}
\end{tikzcd}
\end{center}
\item {\it Identity Relations:}
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
\mathscr{M}athcal{O} \arrow[r, "id"] \arrow[dr, "(id_{\mathscr{M}athcal{O}} , id_{\mathscr{M}athcal{O}})"'] & \mathscr{M}athcal{M} \arrow[d, "(s , t)"] \\
& \mathscr{M}athcal{O} \times \mathscr{M}athcal{O}
\end{tikzcd}
\end{center}
\item {\it Identity Composition:}
\begin{center}
\begin{tikzcd}[row sep =0.5in, column sep=0.5in]
\mathscr{M}athcal{O} \times \mathscr{M}athcal{M} \arrow[dr, "\pi_2"] \arrow[r, "id \times id_{\mathscr{M}athcal{M}}"] &
\mathscr{M}athcal{M} \times \mathscr{M}athcal{M} \arrow[d, "m"] &
\mathscr{M}athcal{M} \times \mathscr{M}athcal{O} \arrow[l, "id_{\mathscr{M}athcal{M}} \times id"'] \arrow[dl, "\pi_1"] \\
& \mathscr{M}athcal{M} &
\end{tikzcd}
\end{center}
\item {\it Associativity:}
\begin{center}
\begin{tikzcd}[row sep =0.5in, column sep=0.5in]
\mathscr{M}athcal{M} \underset{\mathscr{M}athcal{O}}{\times} \mathscr{M}athcal{M} \underset{\mathscr{M}athcal{O}}{\times} \mathscr{M}athcal{M}
\arrow[d, " m \times id_{\mathscr{M}athcal{M}}"]
\arrow[r, "id_{\mathscr{M}athcal{M}} \times m"] &
\mathscr{M}athcal{M} \underset{\mathscr{M}athcal{O}}{\times} \mathscr{M}athcal{M} \arrow[d, "m"] \\
\mathscr{M}athcal{M} \underset{\mathscr{M}athcal{O}}{\times} \mathscr{M}athcal{M} \arrow[r, "m"] &
\mathscr{M}athcal{M}
\end{tikzcd}
\end{center}
\end{enumerate}
\end{defone}
There are many examples of categories in the world of mathematics.
\begin{exone} \label{Ex Category of sets}
Let $\mathscr{M}athscr{S}et$ be the category which has as objects all sets and as morphisms all functions of sets.
Then the function $id$ assigns to each set the identity function and the source target maps $(s,t)$ assigns to each function
it's source and target. Finally $m$ is just the usual composition of functions.
\end{exone}
\begin{exone} \label{Ex Category of top}
We can repeat the same example as above by replacing sets with a set that has additional structure.
So, we can define the category $\mathscr{T}\text{op}$ of topological spaces and continuous maps, or groups and homomorphisms.
\end{exone}
\begin{remone} \label{Rem Hom sets}
Very often we care about the morphisms between two specific objects. Concretely, for two objects
$c, d \in \mathscr{M}athcal{C} = (\mathscr{M}athcal{O},\mathscr{M}athcal{M})$
we want to define the set of maps with source $c$ and target $d$ and denote it as $Hom_{\mathscr{M}athcal{C}}(c,d)$, which we define as the following pullback
$$ Hom_{\mathscr{M}athcal{C}}(c,d) = * ^c\underset{\mathscr{M}athcal{O}}{\times}^s \mathscr{M}athcal{M} ^t\underset{\mathscr{M}athcal{O}}{\times}^d * $$
\end{remone}
Using the philosophy of categories on categories themselves means we should consider studying maps between categories.
\begin{defone} \label{Def Functor}
A functor $F: \mathscr{M}athcal{C} \to \mathscr{M}athcal{D}$ is a tuple of two maps. One map for objects
$F_{\mathscr{M}athcal{O}}: \mathscr{M}athcal{O}_{\mathscr{M}athcal{C}} \to \mathscr{M}athcal{O}_{\mathscr{M}athcal{D}}$
and one map for morphisms $F_{\mathscr{M}athcal{M}}: \mathscr{M}athcal{M}_{\mathscr{M}athcal{C}} \to \mathscr{M}athcal{M}_{\mathscr{M}athcal{D}}$,
such that they satisfy following conditions:
\begin{enumerate}
\item {\it Respecting Identity}: $id_{\mathscr{M}athcal{D}} F_{\mathscr{M}athcal{O}} = F_{\mathscr{M}athcal{M}} id_{\mathscr{M}athcal{C}}$.
\item {\it Respecting Source/Target}: $s_{\mathscr{M}athcal{D}}F_{\mathscr{M}athcal{M}} = F_{\mathscr{M}athcal{O}}s_{\mathscr{M}athcal{C}}$
and $t_{\mathscr{M}athcal{D}}F_{\mathscr{M}athcal{M}} = F_{\mathscr{M}athcal{O}}t_{\mathscr{M}athcal{C}}$.
\item {\it Respecting Composition}: $F_{\mathscr{M}athcal{M}} m_{\mathscr{M}athcal{C}}= m_{\mathscr{M}athcal{D}}(F_{\mathscr{M}athcal{M}} \times F_{\mathscr{M}athcal{M}})$.
\end{enumerate}
\end{defone}
\begin{exone} \label{Ex Category of categories}
The definition above allows us to define the category $\mathscr{M}athcal{C}$at which has objects categories and morphisms functors.
\end{exone}
Repeating the philosophy of categories for functors leads us to the definition of a {\it natural transformation}.
\begin{defone} \label{Def Nat trans}
Let $F,G: \mathscr{M}athcal{C} \to \mathscr{M}athcal{D}$ be two functors. A natural transformation $\alpha: F \Rightarrow G$ is a collection of maps
$$\alpha_c : F(c) \to G(c)$$
for every object $c \in \mathscr{M}athcal{C}$ such that for every map $f: c \to d$ the diagram
\begin{center}
\comsq{F(c)}{F(d)}{G(c)}{G(d)}{F(f)}{\alpha_c}{\alpha_d}{G(f)}
\end{center}
commutes.
\end{defone}
Using natural transformations we can even build more categories.
\begin{theone} \label{The Category of functors}
Let $\mathscr{M}athcal{C}$ and $\mathscr{M}athcal{D}$ be two categories. The collection of functors from $\mathscr{M}athcal{C}$ to $\mathscr{M}athcal{D}$, denoted by $Fun(\mathscr{M}athcal{C},\mathscr{M}athcal{D})$ is a category with
objects functors and morphisms natural transformations.
\end{theone}
\begin{notone}
For two functors $F,G: \mathscr{M}athcal{C} \to \mathscr{M}athcal{D}$, we denote the hom set in this category as $Nat(F,G)$.
\end{notone}
This finally leads to the famous {\it Yoneda lemma}, which is one of the most powerful results in category theory.
\begin{defone}
Let $c \in \mathscr{M}athcal{C}$ be an object. There is a functor $\mathscr{M}athcal{Y}_c: \mathscr{M}athcal{C} \to \mathscr{M}athscr{S}et$ that send each object $d$ to the set
$Hom_{\mathscr{M}athcal{C}}(c,d)$. Functoriality follows from composition.
\end{defone}
\begin{lemone}
Let $F: \mathscr{M}athcal{C} \to \mathscr{M}athscr{S}et$ be a functor. For each object $c \in \mathscr{M}athcal{C}$, there is a bijection of sets
$$Nat(\mathscr{M}athcal{Y}_c,F) \cong F(c)$$
induced by the map that sends each natural transformation $\alpha$ to the value at the identity $\alpha_c(id_c)$.
\end{lemone}
The definitions given up to here are quite cumbersome and necessitate the reader to keep track of a lot of different information.
It would be helpful if we could package that same information and present it in a more elegant manner.
The way we can achieve this goal is by using {\it simplicial sets}.
\mathscr{M}athscr{S}ubsection{Simplicial Sets: A Second Look at Categories} \label{Subsec Simplicial Sets: A Second Look at Categories}
Simplicial sets are a very powerful tool that can help us study categories.
\begin{defone} \label{Def Delta category}
Let $\mathscr{M}athcal{D}elta$ be the category with objects all non-empty finite linearly ordered sets
$$ [0] = \{ 0 \}, \ [1] = \{ 0 \leq 1 \}, \ [2] = \{ 0 \leq 1 \leq 2 \}, \ ... $$
and morphisms order-preserving maps of linearly ordered sets.
\end{defone}
\begin{notone} \label{Not Boundary maps}
There are some specific morphisms in the category $\mathscr{M}athcal{D}elta$ that we will need later on.
\begin{itemize}
\item For each $n \geq 0$ and $0 \leq i \leq n+1$ there is a unique injective map
$$d_i:[n] \to [n+1]$$
such that $i \in [n+1]$ is not in the image. More explicitly $d_i(k) = k$ if $k < i$ and $d_i(k) = k+1$ if $k \geq i$.
\item For each $n \geq 1$ and $0 \leq i \leq n$ there is a unique surjective map
$$s_i:[n] \to [n-1]$$
defined as follows. $s_i(k) = k$ if $k \leq i$ and $s_i(k) = k-1$ if $k > i$.
Notice in particular that $s_i(i) = s_i(i+1) = i$ and that $s_i$ is injective for all other values.
\end{itemize}
\end{notone}
We have following amazing fact regarding these two classes of maps.
\begin{remone} \label{Rem Generators of Delta}
Every morphisms in $\mathscr{M}athcal{D}elta$ can be written as a finite composition of these two classes of maps stated above.
The maps satisfy certain relations that can be found in \cite[Page 4]{GJ09}.
\end{remone}
\begin{notone} \label{Not Representing Delta}
Because of this remark we can depict the category $\mathscr{M}athcal{D}elta$ as the following
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
[0]
\arrow[r, shift left=1.2, "d_0"]
\arrow[r, shift right=1.2, "d_1"']
&
[ 1 ]
\arrow[l, shorten >=1ex,shorten <=1ex, "s_0" near start]
\arrow[r] \arrow[r, shift left=2, "d_0"] \arrow[r, shift right=2, "d_2"']
& [ 2 ]
\arrow[l, shift right, shorten >=1ex,shorten <=1ex ]
\arrow[l, shift left, shorten >=1ex,shorten <=1ex]
\arrow[r, shift right=1] \arrow[r, shift left=1] \arrow[r, shift right=3] \arrow[r, shift left=3]
&
\cdots
\arrow[l, shorten >=1ex,shorten <=1ex] \arrow[l, shift left=2, shorten >=1ex,shorten <=1ex]
\arrow[l, shift right=2, shorten >=1ex,shorten <=1ex]
\end{tikzcd}
\end{center}
\end{notone}
Having studied $\mathscr{M}athcal{D}elta$ we can finally define a simplicial set.
\begin{defone} \label{Def Simp set}
A simplicial set is a functor $X: \mathscr{M}athcal{D}elta^{op} \to \mathscr{M}athscr{S}et$.
\end{defone}
\begin{remone} \label{Rem Opposite of Delta}
Recall that $\mathscr{M}athcal{D}elta^{op}$ is the opposite category of $\mathscr{M}athcal{D}elta$. It has the same objects but every morphism has reverse source and targets.
\end{remone}
\begin{remone} \label{Rem Diagram of Simp set}
Concretely a simplicial set is a choice of sets $X_0, X_1, X_2, ...$ which have the appropriate functions between them.
Using the diagram above, we can depict a simplicial set as:
\begin{center}
\mathscr{M}athscr{S}impset{X_0}{X_1}{X_2}{d_0}{d_1}{d_0}{d_2}
\end{center}
notice that all arrows are reversed because this functor is mapping out of the opposite category of $\mathscr{M}athcal{D}elta$.
\end{remone}
\begin{defone} \label{Def sSet}
A simplicial set is a functor and so the collection of simplicial sets is itself a category
with morphisms being natural transformations. We will denote this category by $s\mathscr{M}athscr{S}et$.
\end{defone}
A simplicial set is an amazing object of study. In the coming two sections we will see how, depending on which aspects we focus on,
a simplicial set can have a very interesting and diverse behavior. For now we focus on the categorical aspects of simplicial sets.
First we show how we can build a simplicial set out of a category.
\begin{constrone} \label{Constr Nerve}
Let $\mathscr{M}athcal{C} = (\mathscr{M}athcal{O},\mathscr{M}athcal{M})$ be a category. Then we define $N\mathscr{M}athcal{C}$ as the following simplicial set.
First we define it level-wise as
$$N\mathscr{M}athcal{C}_0 = \mathscr{M}athcal{O}$$
$$N\mathscr{M}athcal{C}_n = \mathscr{M}athcal{M} \underset{\mathscr{M}athcal{O}}{\times} ... \underset{\mathscr{M}athcal{O}}{\times} \mathscr{M}athcal{M}$$
where there are $n$ factors of $\mathscr{M}athcal{M}$ and $n \geq 1$. So, the $0$ level is the set of objects and at level $n$
we have the set of $n$ composable morphisms.
\par
Now we construct the maps between them.
It suffices to specify the maps $s_i$ and $d_i$. If $n=0$, then
$s_0: N\mathscr{M}athcal{C}_0 \to N\mathscr{M}athcal{C}_1$ is defined as $s_0 = id_{\mathscr{M}athcal{C}}$. Moreover, $d_0,d_1: N\mathscr{M}athcal{C}_1 \to N\mathscr{M}athcal{C}_0$ are defined as $d_0=s, d_1 = t$.
\par
Let $n \geq 1$ and let $(f_1, f_2, ... , f_n) \in N\mathscr{M}athcal{C}_n$ be an element.
For $ 0 \leq i \leq n+1$, we define $d_i: N\mathscr{M}athcal{C}_n \to N\mathscr{M}athcal{C}_{n-1}$ for the following $3$ cases:
\begin{itemize}
\item[(0)] $d_i((f_1, f_2, ... , f_n)) = (f_2, f_3, ... , f_n)$
\item[(1 to n)] $d_i((f_1, f_2,..., f_{i-1},f_i, ... , f_n)) = (f_1, f_2, ..., f_{i-1}f_{i} ,... , f_n)$
\item[(n+1)] $d_i((f_1, f_2, ... , f_n)) = (f_1, f_2, ... , f_{n-1})$
\end{itemize}
Similarly, for $0 \leq i \leq n$ we define $s_i: \mathscr{M}athcal{C}_n \to \mathscr{M}athcal{C}_{n+1}$ for the following two cases:
\begin{itemize}
\item[(0 to n)] $s_i((f_1, f_2, ...,f_i, ... , f_n)) = (f_1, f_2, ..., id_{s(f_i)},f_i, ... , f_n)$
\item[(n+1)] $s_i((f_1, f_2, ...,f_i, ... , f_n)) = (f_1, f_2, ...,f_i, ... , f_n, id_{t(f_n)})$
\end{itemize}
It is an exercise in diagram chasing to show that $N\mathscr{M}athcal{C}$ satisfies the relations of a simplicial set with the $d_i$ and $s_i$ defined above.
\end{constrone}
\begin{remone} \label{Rem Comp needed for nerve}
Notice in order to define $N\mathscr{M}athcal{C}$ it did not suffice to have a two sets with $3$ maps between them. We needed the existence of the
composition map to be able to make the definition work.
\end{remone}
This construction merits a new definition.
\begin{defone} \label{Def Nerve of C}
Let $\mathscr{M}athcal{C}$ be a category. The {\it nerve} of $\mathscr{M}athcal{C}$ is the simplicial set $N\mathscr{M}athcal{C}$ described above.
\end{defone}
The nerve construction fits well into our philosophy of category theory.
\begin{theone} \label{The Nerve functor}
The nerve construction is functorial. Thus we get a functor
$$N: \mathscr{C}\text{at} \to s\mathscr{M}athscr{S}et$$
\end{theone}
\begin{proof}
We already constructed the map on objects. For a functor $F: \mathscr{M}athcal{C} \to \mathscr{M}athcal{D}$, the simplicial map $NF: N\mathscr{M}athcal{C} \to N\mathscr{M}athcal{D}$ can be defined level-wise as
\begin{itemize}
\item $NF_0 = F_{\mathscr{M}athcal{O}}$
\item $NF_n = F_{\mathscr{M}athcal{M}} \underset{F_{\mathscr{M}athcal{O}}}{\times} ... \underset{F_{\mathscr{M}athcal{O}}}{\times} F_{\mathscr{M}athcal{M}}$.
\end{itemize}
From here on it is a diagram chasing exercise to see that $NF_n$ make all the necessary squares commute.
\par
Note that it clearly follows that if $I_{\mathscr{M}athcal{C}}: \mathscr{M}athcal{C} \to \mathscr{M}athcal{C}$ is the identity functor, then $NI_{\mathscr{M}athcal{C}}$ is the identity map.
Moreover, $N(F \circ G) = NF \circ NG$.
\end{proof}
\begin{exone} \label{Ex Delta Rep Functors}
We have already introduced the linearly ordered set $[n]$ before (Definition \ref{Def Delta category}).
We can think of $[n]$ as a category, where the objects are the elements and a morphism are ordered $2$-tuples $(i,j)$, where $i \leq j$.
The source of such map $(i,j)$ is $i$ and the target is $j$. The identity map of an element $i$ is the tuple $(i,i)$.
Finally, we can compose two morphisms $(i,j)$ and $(j,k)$ to the morphism $(i,k)$. This gives us a category,
which we will still denote by $[n]$. Notice in this case for each chosen objects $i, j$ there either is a unique morphism from $i$ to $j$
(if $i \leq j$) or there is no morphism at all.
\par
There is a more direct way to think about the set of morphisms. The ordered set $[1]$ has two ordered elements $0 \leq 1$.
Given that a morphism is a choice of two ordered elements, we can think of a morphism as an order preserving map
$[1] \to [n]$. But that is exactly a morphism in the category $\mathscr{M}athcal{D}elta$. Thus the set of morphisms also corresponds to
$Hom_{\mathscr{M}athcal{D}elta}([1],[n])$.
Let us compute $N([n])$. By definition $N([n])_0 = [n]$. Moreover, $N([n])_1 = Hom_{\mathscr{M}athcal{D}elta}([1],[n])$.
Next notice that $N([n])_m = N([n])_1 \times_{N([n])_0} ... \times_{N([n])_0} N([n])_1$, which corresponds to a choice of
$m$ ordered numbers $(i_1, i_2, ..., i_m)$. Using the same argument as the last paragraph, we see that
$N([n])_m = Hom_{\mathscr{M}athcal{D}elta}([m],[n])$. Thus, $N([n])$ is really just the representable functor
$$N([n]) = Hom_{\mathscr{M}athcal{D}elta}( - , [n]): \mathscr{M}athcal{D}elta^{op} \to \mathscr{M}athscr{S}et$$.
\end{exone}
This simplicial set is really special and thus deserves its own name.
\begin{defone} \label{Def Rep Functors}
For each $n$ there is a representable functor, which maps $[i]$ to $Hom_{\mathscr{M}athcal{D}elta}([i],[n])$.
We will denote this simplicial set by $\mathscr{M}athcal{D}elta[n]$.
By the Yoneda lemma, for any simplicial set $X$ we have following isomorphism of sets:
$$Hom_{s\mathscr{M}athscr{S}et}(\mathscr{M}athcal{D}elta[l], X) \cong X_n.$$
\end{defone}
By now we have shown that we can take a category and build a simplicial set out of it. But can we build every simplicial set this way?
If not then which ones do we get?
\begin{defone} \label{Def Segal condition}
A simplicial set $X$ satisfies the {\it Segal condition} if the map
$$X_n \xrightarrow{ \ \ \cong \ \ } X_1 \underset{X_0}{\times} ... \underset{X_0}{\times} X_1 $$
is a bijection for $n \geq 2$.
\end{defone}
The nerve $N\mathscr{M}athcal{C}$ satisfies the Segal condition by its very definition. Thus not every simplicial set is equivalent to
the nerve of a category. But what condition other than the Segal condition do we need?
\begin{theone} \label{The Segal condition image}
Let $X$ be a simplicial set that satisfies the Segal condition. Then there exists a category $\mathscr{M}athcal{C}$ such that $X$ is equivalent to $N\mathscr{M}athcal{C}$.
\end{theone}
\begin{proof}
We define the category $\mathscr{M}athcal{C}$ as follows. It has objects $\mathscr{M}athcal{O}_{\mathscr{M}athcal{C}} = X_0$ and morphisms $\mathscr{M}athcal{M}_{\mathscr{M}athcal{C}} = X_1$.
Then the source, target and identity maps are defined as $s_{\mathscr{M}athcal{C}} = d_1: X_1 \to X_0$, $t_{\mathscr{M}athcal{C}}= d_0: X_1 \to X_0$, $id_{\mathscr{M}athcal{C}} = s_0: X_0 \to X_1$
and the product map is defined as $m_{\mathscr{M}athcal{C}} = d_1:X_2 \to X_1$.
Here we are using the fact that
$X_2 \cong X_1 \times_{X_0} X_1$.
Thus we can think of $m$ as a map $m: \mathscr{M}athcal{M}_{\mathscr{M}athcal{C}} \times{\mathscr{M}athcal{O}_{\mathscr{M}athcal{C}}} \mathscr{M}athcal{M}_{\mathscr{M}athcal{C}} \to \mathscr{M}athcal{M}_{\mathscr{M}athcal{C}}$,
which is exactly what we wanted. The simplicial relations show that $\mathscr{M}athcal{C}$ satisfies the conditions stated in
Definition \ref{Def Category set version}.
\par
Finally, we have the following bijection.
$$(N\mathscr{M}athcal{C})_n = \mathscr{M}athcal{M}_{\mathscr{M}athcal{C}} \underset{\mathscr{M}athcal{O}_{\mathscr{M}athcal{C}}}{\times} ... \underset{\mathscr{M}athcal{O}_{\mathscr{M}athcal{C}}}{\times} \mathscr{M}athcal{M}_{\mathscr{M}athcal{C}} =
X_1 \underset{X_0}{\times} ... \underset{X_0}{\times} X_1 \cong X_n$$
This shows that $N\mathscr{M}athcal{C}$ is equivalent to $X$ and finished the proof.
\end{proof}
The upshot is that a simplicial set that satisfies the Segal condition has the same data as a category and so instead of keeping track
of all the necessary data and maps between them it packages everything very nicely and it gives us much better control.
This doesn't just hold for the categories themselves, but also carries over to functors.
\begin{theone} \label{The Nerve embed}
Let $\mathscr{M}athcal{C}$ and $\mathscr{M}athcal{D}$ be two categories. Then the functor $N$ induces a bijection of $hom$ sets
$$N: Hom_{\mathscr{C}\text{at}}(\mathscr{M}athcal{C},\mathscr{M}athcal{D}) \to Hom_{s\mathscr{M}athscr{S}et}(N\mathscr{M}athcal{C},N\mathscr{M}athcal{D})$$
\end{theone}
\begin{proof}
We prove the result by showing the map above has an inverse. Let $f: N\mathscr{M}athcal{C} \to N\mathscr{M}athcal{D}$ be a simplicial map.
Then we define $P(f)$ as the functor that is defined on objects as $f_0$ and defined on morphisms as $f_1$.
The simplicial identities then show that it satisfies the conditions of a functors. Finally, for any functor
$F: \mathscr{M}athcal{C} \to \mathscr{M}athcal{D}$, the composition $PN(F) = F$ by definition.
On the other hand for any simplicial map $f: N\mathscr{M}athcal{C} \to N\mathscr{M}athcal{D}$, $NP(f) = f$ as they agree at level $0$ and $1$ and that characterizes
the map completely.
\end{proof}
Up until now we have shown how we can use the data of a simplicial set to study categories and recover category theory.
The next goal is to show we can use the same ideas to study homotopy theory.
\mathscr{M}athscr{S}ubsection{Homotopy Theory of Topological Spaces} \label{Subsec Homotopy Theory of Topological Spaces}
Homotopy theory can now be found in many forms, but one of the most famous examples of homotopy theories is the homotopy theory
of spaces. Again, similar to the case of categories, there are various ways to study spaces. Let us
first review the more familiar one: topological spaces.
Recall the classical definition of homotopies of topological spaces.
\begin{defone} \label{Def Homotopic maps}
Two maps of topological spaces $f,g: X \to Y$ are called {\it homotopic} if there exists a map $H: X \times [0,1] \to Y$ such that
$H|_{X \times \{ 0\} } = f$ and $H|_{X \times \{ 1\} } = g$.
\end{defone}
\begin{defone} \label{Def Homotopy equiv}
A map $f: X \to Y$ is called a {\it homotopy equivalence} if there exists a map $g: Y \to X$ such that both $fg$ and $gf$ are homotopic to the
identity map.
\end{defone}
A key question in the homotopy theory of spaces is to determine whether a map is an equivalence or not.
However topological spaces can be quite pathological and so we often look for suitable "replacements" i.e. equivalent spaces which have
a simpler structure. One good example is a CW-complex.
\begin{theone} \label{The CW cofibrant rep}
For each topological space $X$ there exists a CW-complex $\tilde{X}$ and map $\tilde{X} \to X$ that is a homotopy equivalence.
\end{theone}
Thus from a homotopical perspective it often suffices to study CW-complexes rather than all spaces. However, a CW-complex is built out of
simplices. Thus what we really care about is how many simplices we have and how they are attached to each other.
This suggests that we can study spaces from the perspective of simplicial sets.
\mathscr{M}athscr{S}ubsection{Simplicial Sets: A Second Look at Spaces} \label{Subsec Simplicial Sets: A Second Look at Spaces}
Here we show how we can use simplicial sets to study the homotopy theory of topological spaces.
We have already defined simplicial sets in the previous section.
So, first we show how to construct a simplicial set out of any topological space.
\begin{defone} \label{Def Topo simplex}
Let $S(l)$ be the standard $l+1$-simplex. Concretely $S(l)$ is the convex hull of the $l+1$ points $(1,0,...,0), (0,1,...,0), ... , (0,0,...,1)$
in $\mathscr{M}athbb{R}^{l+1}$. In particular, $S(0)$ is a point, $S(1)$ is an interval and $S(2)$ is a triangle.
\end{defone}
\begin{remone} \label{Rem Cosimp simplex}
One important fact about those simplices is that the boundary is built out of lower dimensional simplices.
For example, the boundary of a line is the union of two points or the boundary of a triangle is the union of three lines.
This means we have two maps $d_0, d_1: S(0) \to S(1)$ that map to the two boundary points or we have
three maps $d_0,d_1,d_2: S(1) \to S(2)$. \par
On the other side, we can always collapse one boundary component to lower the dimension of our simplex. Thus there are two ways to collapse
our triangle $S(2)$ to a line $S(1)$, which gives us two maps $s_0,s_1: S(2) \to S(1)$.
It turns out these maps do satisfy the covariant version of the simplicial identities, which are also called the
{\it cosimplicial identities}.
This means we can thus define a functor
$$S: \mathscr{M}athcal{D}elta \to \mathscr{T}\text{op}$$
This functor can be depicted in the following diagram.
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
S(0)
\arrow[r, shift left=1.2, "d_0"]
\arrow[r, shift right=1.2, "d_1"']
& S(1)
\arrow[l, shorten >=1ex,shorten <=1ex, "s_0" near start]
\arrow[r] \arrow[r, shift left=2, "d_0"] \arrow[r, shift right=2, "d_2"']
& S(2)
\arrow[l, shift right, shorten >=1ex,shorten <=1ex ]
\arrow[l, shift left, shorten >=1ex,shorten <=1ex]
\arrow[r, shift right=1] \arrow[r, shift left=1] \arrow[r, shift right=3] \arrow[r, shift left=3]
&
\cdots
\arrow[l, shorten >=1ex,shorten <=1ex] \arrow[l, shift left=2, shorten >=1ex,shorten <=1ex]
\arrow[l, shift right=2, shorten >=1ex,shorten <=1ex]
\end{tikzcd}
\end{center}
\end{remone}
\begin{defone} \label{Def Sing Functor}
Let $X$ be a topological space. We define the simplicial set $S(X)$ as follows. Level-wise we define $S(X)$ as
$$S(X)_n = Hom_{\mathscr{T}\text{op}}(S(n),X).$$
The functoriality of $I$ as described in the remark above shows that this indeed gives us a simplicial set.
\end{defone}
Thus we can build a simplicial set out of every topological space. Each level indicates how many $n+1$-simplices can be mapped into our space.
However, we cannot build every kind of simplicial set this way. Rather the simplicial set we constructed is called a
{\it Kan complex}. In order to be able to give a definition we need to gain a better understanding of simplicial sets first.
\begin{defone} \label{Def Subsimp set}
We say $K$ is a subsimplicial set of $S$, if for any $l$ we have $K_l \mathscr{M}athscr{S}ubset S_l$.
\end{defone}
\begin{exone} \label{Ex Boundary and Horns}
There are two important classes of sub simplicial sets of $\mathscr{M}athcal{D}elta[l]$
(Definition \ref{Def Rep Functors}):
\begin{enumerate}
\item The first one is denoted by $\partial \mathscr{M}athcal{D}elta[l]$ and defined as follows:
$\partial \mathscr{M}athcal{D}elta[l]_i$ is the subset of all non-surjective maps in $Hom_{\mathscr{M}athcal{D}elta}([i],[l])$.
In particular, this implies that for $i < n$, we have $\partial \mathscr{M}athcal{D}elta[l]_i = \mathscr{M}athcal{D}elta[l]_i$ and for $i= l$ we have
$\partial \mathscr{M}athcal{D}elta[l]_l = \mathscr{M}athcal{D}elta[l]_l - \{ id_{[l]} \} $.
Intuitively it looks like the boundary of our convex space i.e. $\mathscr{M}athcal{D}elta[l]$ with the center $n$-dimensional cell removed.
\item The second is denoted by $\Lambda[l]_i $ ($0 \leq i \leq l$) and consists of non-surjective maps that satisfy
the following condition:
$(\Lambda[n]_i)_j$ is the subset of all maps in $Hom_{\mathscr{M}athcal{D}elta}([j],[l])$, that satisfy following condition.
If $i$ is not in the image of the map then at least one other elements also has to be not in the image.
Concretely, this means it is also a subspace of $\partial \mathscr{M}athcal{D}elta[l]$ and it excludes the face which is formed by all vertices except for $i$.
Intuitively, this one looks like a boundary where one of the faces (the one opposing the vertex $i$) has been removed as well.
Given the resulting shape it is very often called a "horn".
\end{enumerate}
\end{exone}
Having gone through these definitions we can finally define a Kan complex.
\begin{defone} \label{Def Kan complex}
A simplicial set $K$ is called a Kan complex if for any $l \geq 0$ and $0 \leq i \leq l$, the map
$$Hom_{\mathscr{M}athscr{S}}(\mathscr{M}athcal{D}elta[l], K) \to Hom_{\mathscr{M}athscr{S}}(\Lambda[l]_i, K)$$
is surjective.
\end{defone}
\begin{remone} \label{Rem Lift for Kan complex}
Basically the definition is saying that following diagram lifts:
\begin{center}
\begin{tikzcd}
\Lambda[l]_i \arrow[d] \arrow[r] & K \\
\mathscr{M}athcal{D}elta[l] \arrow[ur, dashed]
\end{tikzcd}
\end{center}
\end{remone}
\begin{exone} \label{Ex SX a Kan complex}
For every topological space $X$, the simplicial set $SX$ is a Kan complex. We will not prove this fact here.
It relies on the idea that a topological space has no sense of direction. Thus every path can be inverted.
Concretely, for any map $\gamma: I(1) \to X$, there is a map $\gamma^{-1}: I(1) \to X$ that is defined as
$\gamma^{-1}(t) = \gamma(1-t)$. Thus every element $\gamma \in S(X)_1$ has a reverse path.
A similar concept applies to higher dimensional maps.
\par
It is that idea that allows us to lift any map of the form above.
For a rigorous argument see \cite[Chapter 1]{GJ09}.
\end{exone}
\begin{exone} \label{Ex Delta not a Kan complex}
Contrary to the example above $\mathscr{M}athcal{D}elta[l]$ is not a Kan complex (if $l > 0$).
For example the map $\Lambda[2]_0 \to \mathscr{M}athcal{D}elta[l]$ that sends $0$ to $0$, $1$ to $2$ and $2$ to $1$ cannot be lifted.
\end{exone}
The definition above is a special case of a {\it Kan fibration}.
\begin{defone} \label{Def Kan fibration}
A map of simplicial sets $f:S \twoheadrightarrow T$ is a Kan fibration if any commutative square of the form
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
\Lambda[l]_i \arrow[r] \arrow[d] & S \arrow[d, twoheadrightarrow]\\
\mathscr{M}athcal{D}elta[l] \arrow[r] \arrow[ur, dashed] & T
\end{tikzcd}
\end{center}
lifts, where $n \geq 0$ and $ 0 \leq i \leq n$.
\end{defone}
\begin{remone} \label{Rem Kan complex is Kan fib}
This generalizes Kan complexes as $K$ is a Kan complex if and only if the map $K \to \mathscr{M}athcal{D}elta[0]$ is a Kan fibration.
As a result, if $K \twoheadrightarrow L$ is a Kan fibration and $L$ is Kan fibrant, then $K$ is also Kan fibrant
\end{remone}
Kan complexes share many characteristics with topological spaces. In particular, we can talk about equivalences and homotopies.
\begin{defone} \label{Def Homotopy of Kan complexes}
Two maps $f,g: L \to K$ between Kan complexes are called {\it homotopic} if there exists a map $H: L \times \mathscr{M}athcal{D}elta[1] \to K$ such that
$H|_0 = f$ and $H|_1 = g$.
\end{defone}
\begin{remone} \label{Rem Homotopy Equiv for Kan complex}
This definition can be made for any simplicial set, but it is only a equivalence relation for the case of Kan complex.
\end{remone}
\begin{exone} \label{Ex Homotopic Points in Kan complex}
One particular instance of this definition is when $L = \mathscr{M}athcal{D}elta[0]$. In this case we have two points $x,y: \mathscr{M}athcal{D}elta[0] \to K$.
We say $x$ and $y$ are homotopic or {\it equivalent} if there is a map $\gamma: \mathscr{M}athcal{D}elta[1] \to K$
such that $\gamma(0)=x$ and $\gamma(1) = y$.
\end{exone}
\begin{defone} \label{Def Homotopy equiv of Kan complex}
A map $f: L \to K$ between Kan complexes is called an {\it equivalence} if there are maps $g,h: K \to L$ such that
$fg: K \to K$ is homotopic to $id_K$ and $hf: L \to L$ is homotopic to $id_L$.
\end{defone}
Most importantly, in order to study equivalences of spaces it suffices to study equivalences of the analogous Kan complexes.
\begin{lemone} \label{Lemma S homotopic embed}
A map of topological spaces $f:X \to Y$ is a homotopy equivalence if and only if the map of Kan complexes
$Sf: SX \to SY$ is a homotopy equivalence.
\end{lemone}
Seeing how that result holds requires us to use much more machinery. One very efficient way is to use the language of
{\it model categories}.
A model structure can capture the homotopical data in the context of a category.
Using model categories we can show that topological spaces and simplicial sets
(if we focus on Kan complexes) have equivalent model structures.
For a better understanding of model structures see Section \ref{Sec Model Structures of Complete Segal Spaces}.
\begin{remone} \label{Rem Kan fib basechange}
Kan fibrations are important in the homotopy theory of simplicial sets. That is because base change along Kan fibrations is equivalence preserving.
By that we mean that in the following pullback diagram
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
K \underset{M}{\times} L \arrow[dr, phantom, "\ulcorner", very near start] \arrow[d, twoheadrightarrow] \arrow[r, "\mathscr{M}athscr{S}imeq", "g^*f"']
& K \arrow[d, "g", twoheadrightarrow] \\
L \arrow[r, "\mathscr{M}athscr{S}imeq", "f"'] & M
\end{tikzcd}
\end{center}
if $f$ is an equivalence and $g$ is a Kan fibration then $g^*f$ is also an equivalence. Moreover, the pullback of a Kan fibration is also a Kan
fibration. Thus we say such a pullback diagram is {\it homotopy invariant}.
\end{remone}
\begin{remone}
The homotopy invariance of base change by a Kan fibration implies in particular that we can define a {\it homotopy pullback}.
We say a diagram of Kan complexes
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
A \arrow[dr, phantom, "\ulcorner", very near start] \arrow[d] \arrow[r] & B \arrow[d, "g", twoheadrightarrow] \\
C \arrow[r, "f"'] & D
\end{tikzcd}
\end{center}
is a homotopy pullback if the induced map $A \to B \times_D C$ is a homotopy equivalence.
In other words, we demand a pullback ``up to homotopy" rather than a strict pullback.
The fact that $g$ is a Kan fibration implies that this definition is well-defined.
\end{remone}
Before we move on we will focus on one particular, yet very important instance of a homotopy equivalence.
\begin{defone} \label{Def Contractible Kan complex}
A Kan complex $K$ is {\it contractible} if the map $K \to \mathscr{M}athcal{D}elta[0]$ is a homotopy equivalence.
\end{defone}
\begin{remone} \label{Rem Contractible like unique}
The notion of a contractible Kan complex is central in homotopy theory. It is the homotopical analogue of uniqueness as it implies that every
two points in $K$ are equivalent. Moreover, any two paths are themselves equivalent in the suitable sense and this pattern
continues.
\end{remone}
A contractible Kan complex is again a special kind of Kan fibration.
\begin{defone}
We say a map $K \to L$ is a {\it trivial Kan fibration} if it is a Kan fibration and a weak equivalence.
\end{defone}
\begin{lemone}
A map $K \to L$ is a trivial Kan fibration if and only if it is a Kan fibration and for every map $\mathscr{M}athcal{D}elta[0] \to L$,
the fiber $\mathscr{M}athcal{D}elta[0] \times_L K$ is contractible.
\end{lemone}
\begin{remone}
Thus a trivial Kan fibration not only has lifts, but the space of lifts is contractible, meaning there is really only
one choice of lift up to homotopy.
\end{remone}
Having a homotopical notion of an isomorphism, namely an equivalence, we can also define the homotopical version of an injection,
namely a {\it ($-1$)-truncated map}.
\begin{defone} \label{Def Neg one trunc}
A Kan fibration $K \to L$ is ($-1$)-truncated if for every map $\mathscr{M}athcal{D}elta[0] \to L$, the fiber
$\mathscr{M}athcal{D}elta[0] \times_L K$ is either contractible or empty.
\end{defone}
Before we move on there is one last property of Kan complexes that we need, namely that they are Cartesian closed.
\begin{remone}
The category of simplicial sets is Cartesian closed. For every two simplicial sets $X,Y$ there is a mapping simplicial set,
$Map(X,Y)$ defined level-wise as
$$Map(X,Y)_n = Hom(X \times \mathscr{M}athcal{D}elta[n],Y).$$
\end{remone}
\begin{propone}
If $K$ is a Kan complex, then for every simplicial set $X$, the simplicial set $Map(X,K)$ is also a Kan complex.
\end{propone}
\begin{notone}
As we have established a well functioning homotopy theory with Kan complexes, we will henceforth
exclusively use the word space to be a Kan complex.
\end{notone}
\mathscr{M}athscr{S}ubsection{Two Paths Coming Together} \label{Two Paths Coming Together}
Until now we showed that we can think of categories as a simplicial set that satisfies the Segal condition and a topological space
as a Kan complex. Thus simplicial sets have two different aspects to them. \par
We can either think of simplicial sets that have a notion of direction and allow us to do category theory.
When we think of simplicial sets this way we denote them by $s\mathscr{M}athscr{S}et$ and pictorially we can depict them as:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
\mathscr{M}athcal{C}_0 \arrow[r, shorten >=1ex,shorten <=1ex, "id" very near end]
& \mathscr{M}athcal{C}_1
\arrow[l, shift left=1.2, "s"] \arrow[l, shift right=1.2, "t"']
\arrow[r, shift right, shorten >=1ex,shorten <=1ex ] \arrow[r, shift left, shorten >=1ex,shorten <=1ex]
& \mathscr{M}athcal{C}_2
\arrow[l, "m" very near start] \arrow[l, shift left=2] \arrow[l, shift right=2]
\arrow[r, shorten >=1ex,shorten <=1ex] \arrow[r, shift left=2, shorten >=1ex,shorten <=1ex]
\arrow[r, shift right=2, shorten >=1ex,shorten <=1ex]
& \cdots
\arrow[l, shift right=1] \arrow[l, shift left=1] \arrow[l, shift right=3] \arrow[l, shift left=3]
\end{tikzcd}
\end{center}
On the other side, we can think of simplicial sets that have homotopical properties. In this case we call them spaces and denote
that very same category as $\mathscr{M}athscr{S}$. This time we depict it as:
\mathscr{M}athscr{S}trut \hspace{1.4in}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
K_0 \arrow[d, shorten >=1ex,shorten <=1ex, "s_0" very near end] \\
K_1
\arrow[u, shift left=1.2, "d_0"] \arrow[u, shift right=1.2, "d_1"']
\arrow[d, shift right, shorten >=1ex,shorten <=1ex ] \arrow[d, shift left, shorten >=1ex,shorten <=1ex] \\
K_2
\arrow[u] \arrow[u, shift left=2, "d_0"] \arrow[u, shift right=2, "d_2"']
\arrow[d, shorten >=1ex,shorten <=1ex] \arrow[d, shift left=2, shorten >=1ex,shorten <=1ex]
\arrow[d, shift right=2, shorten >=1ex,shorten <=1ex] \\
\vdots
\arrow[u, shift right=1] \arrow[u, shift left=1] \arrow[u, shift right=3] \arrow[u, shift left=3]
\end{tikzcd}
A higher category should generalize categories and spaces at the same time.
Thus our goal is it to embed both versions of simplicial sets (categorical and homotopical) into a larger setting.
We need to start with a category which can house two versions of simplicial sets in itself independent of each other
so that we can give each the properties we desire and make sure one part has a categorical behavior
and one part has a homotopical behavior.
This point of view leads us to the study of simplicial spaces.
\mathscr{M}athscr{S}ubsection{Simplicial Spaces} \label{Subsec Simplicial Spaces}
In this section we define and study objects that have enough room to fit two versions of simplicial sets inside of it.
We will call this object a {\it simplicial space}, although they are also known as {\it bisimplicial sets}.
The next subsection will justify why we have decided to use the term simplicial space.
\begin{defone} \label{Def Simp Space}
We define the category of {\it simplicial spaces} as $Fun(\mathscr{M}athcal{D}elta^{op},\mathscr{M}athscr{S})$ and denote it by $s\mathscr{M}athscr{S}$.
\end{defone}
\begin{remone} \label{Rem Depict Simp Space}
We have the adjunction
$$Fun(\mathscr{M}athcal{D}elta^{op} \times \mathscr{M}athcal{D}elta^{op} , Set) \cong Fun(\mathscr{M}athcal{D}elta^{op},Fun(\mathscr{M}athcal{D}elta^{op},Set)) = Fun(\mathscr{M}athcal{D}elta^{op},\mathscr{M}athscr{S}).$$
Thus on a categorical level a simplicial space is a bisimplicial set. Therefore, we can depict it at the same
time as a bisimplicial set or as a simplicial space. We an depict those two as follows:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
X_{00}
\arrow[d, shorten >=1ex,shorten <=1ex] \arrow[r, shorten >=1ex,shorten <=1ex]
& X_{10}
\arrow[d, shorten >=1ex,shorten <=1ex]
\arrow[l, shift left=1.2] \arrow[l, shift right=1.2]
\arrow[r, shift right, shorten >=1ex,shorten <=1ex ] \arrow[r, shift left, shorten >=1ex,shorten <=1ex]
& X_{20}
\arrow[d, shorten >=1ex,shorten <=1ex]
\arrow[l] \arrow[l, shift left=2] \arrow[l, shift right=2]
\arrow[r, shorten >=1ex,shorten <=1ex] \arrow[r, shift left=2, shorten >=1ex,shorten <=1ex] \arrow[r, shift right=2, shorten >=1ex,shorten <=1ex]
& \cdots
\arrow[l, shift right=1] \arrow[l, shift left=1] \arrow[l, shift right=3] \arrow[l, shift left=3]
\\
X_{01}
\arrow[d, shift right, shorten >=1ex,shorten <=1ex ] \arrow[d, shift left, shorten >=1ex,shorten <=1ex]
\arrow[u, shift left=1.2] \arrow[u, shift right=1.2] \arrow[r, shorten >=1ex,shorten <=1ex]
& X_{11}
\arrow[d, shift right, shorten >=1ex,shorten <=1ex ] \arrow[d, shift left, shorten >=1ex,shorten <=1ex]
\arrow[u, shift left=1.2] \arrow[u, shift right=1.2]
\arrow[l, shift left=1.2] \arrow[l, shift right=1.2]
\arrow[r, shift right, shorten >=1ex,shorten <=1ex ] \arrow[r, shift left, shorten >=1ex,shorten <=1ex]
& X_{21}
\arrow[d, shift right, shorten >=1ex,shorten <=1ex ] \arrow[d, shift left, shorten >=1ex,shorten <=1ex]
\arrow[u, shift left=1.2] \arrow[u, shift right=1.2]
\arrow[l] \arrow[l, shift left=2] \arrow[l, shift right=2]
\arrow[r, shorten >=1ex,shorten <=1ex] \arrow[r, shift left=2, shorten >=1ex,shorten <=1ex] \arrow[r, shift right=2, shorten >=1ex,shorten <=1ex]
& \cdots
\arrow[l, shift right=1] \arrow[l, shift left=1] \arrow[l, shift right=3] \arrow[l, shift left=3]
\\
X_{02}
\arrow[d, shorten >=1ex,shorten <=1ex] \arrow[d, shift left=2, shorten >=1ex,shorten <=1ex] \arrow[d, shift right=2, shorten >=1ex,shorten <=1ex]
\arrow[u] \arrow[u, shift left=2] \arrow[u, shift right=2]
\arrow[r, shorten >=1ex,shorten <=1ex]
& X_{12}
\arrow[d, shorten >=1ex,shorten <=1ex] \arrow[d, shift left=2, shorten >=1ex,shorten <=1ex] \arrow[d, shift right=2, shorten >=1ex,shorten <=1ex]
\arrow[u] \arrow[u, shift left=2] \arrow[u, shift right=2]
\arrow[l, shift left=1.2] \arrow[l, shift right=1.2]
\arrow[r, shift right, shorten >=1ex,shorten <=1ex ] \arrow[r, shift left, shorten >=1ex,shorten <=1ex]
& X_{22}
\arrow[d, shorten >=1ex,shorten <=1ex] \arrow[d, shift left=2, shorten >=1ex,shorten <=1ex] \arrow[d, shift right=2, shorten >=1ex,shorten <=1ex]
\arrow[u] \arrow[u, shift left=2] \arrow[u, shift right=2]
\arrow[l] \arrow[l, shift left=2] \arrow[l, shift right=2]
\arrow[r, shorten >=1ex,shorten <=1ex] \arrow[r, shift left=2, shorten >=1ex,shorten <=1ex] \arrow[r, shift right=2, shorten >=1ex,shorten <=1ex]
& \cdots
\arrow[l, shift right=1] \arrow[l, shift left=1] \arrow[l, shift right=3] \arrow[l, shift left=3]
\\
\ \vdots \
\arrow[d, leftrightsquigarrow]
\arrow[u, shift right=1] \arrow[u, shift left=1] \arrow[u, shift right=3] \arrow[u, shift left=3]
& \ \vdots \
\arrow[d, leftrightsquigarrow]
\arrow[u, shift right=1] \arrow[u, shift left=1] \arrow[u, shift right=3] \arrow[u, shift left=3]
& \ \vdots \
\arrow[d, leftrightsquigarrow]
\arrow[u, shift right=1] \arrow[u, shift left=1] \arrow[u, shift right=3] \arrow[u, shift left=3]
& \ \ddots \
\arrow[d, leftrightsquigarrow]
\\
X_{0\bullet} \arrow[r, shorten >=1ex,shorten <=1ex]
& X_{1\bullet}
\arrow[l, shift left=1.2] \arrow[l, shift right=1.2]
\arrow[r, shift right, shorten >=1ex,shorten <=1ex ] \arrow[r, shift left, shorten >=1ex,shorten <=1ex]
& X_{2\bullet}
\arrow[l] \arrow[l, shift left=2] \arrow[l, shift right=2]
\arrow[r, shorten >=1ex,shorten <=1ex] \arrow[r, shift left=2, shorten >=1ex,shorten <=1ex] \arrow[r, shift right=2, shorten >=1ex,shorten <=1ex]
& \cdots
\arrow[l, shift right=1] \arrow[l, shift left=1] \arrow[l, shift right=3] \arrow[l, shift left=3]
\end{tikzcd}
\end{center}
Notice that $X_{0 \bullet}, X_{1 \bullet}, ...$ are themselves simplicial sets.
\end{remone}
\begin{remone} \label{Rem Two embed from sSet to sS}
There are two ways to embed simplicial sets into simplicial spaces.
\begin{enumerate}
\item There is a functor
$$i_F: \mathscr{M}athcal{D}elta \times \mathscr{M}athcal{D}elta \to \mathscr{M}athcal{D}elta$$
that send $([n], [m])$ to $[n]$. This induces a functor
$$i^*_F: s\mathscr{M}athscr{S}et \to s\mathscr{M}athscr{S}$$
that takes a simplicial set $S$ to the simplicial space $i^*_F(S)$ defined as follows.
$$i^*_F(S)_{kl} = S_{k}$$
We call this embedding the {\it vertical embedding}.
\item There is a functor
$$i_{\mathscr{M}athcal{D}elta}: \mathscr{M}athcal{D}elta \times \mathscr{M}athcal{D}elta \to \mathscr{M}athcal{D}elta$$
that send $([n], [m])$ to $[m]$. This induces a functor
$$i^*_{\mathscr{M}athcal{D}elta}: s\mathscr{M}athscr{S}et \to s\mathscr{M}athscr{S}$$
that takes a simplicial set $S$ to the simplicial space $i^*_{\mathscr{M}athcal{D}elta}(S)$ defined as follows.
$$i_F(S)_{kl} = S_{l}$$
We call this embedding the {\it horizontal embedding}.
\end{enumerate}
\end{remone}
Given that there are two embeddings there are two ways to embed generators.
\begin{defone} \label{Def of F}
We define $F(n) = i_F^*(\mathscr{M}athcal{D}elta[n])$ and $\mathscr{M}athcal{D}elta[l] = i_{\mathscr{M}athcal{D}elta}^*(\mathscr{M}athcal{D}elta[l])$.
Similarly, we define $\partial F(n) = i_F^*(\partial \mathscr{M}athcal{D}elta[n])$ and $L(n)_i = i_F^*(\Lambda[n]_i)$.
\end{defone}
The category of simplicial spaces has many pleasant features that we will need later on.
\begin{defone} \label{Def SS Cart closed}
The category of simplicial spaces is Cartesian closed.
For any two objects $X$ and $Y$ we define the simplicial space $Y^X$ as
$$(Y^X)_{nl} = Hom_{s\mathscr{M}athscr{S}}(F(n) \times \mathscr{M}athcal{D}elta[l] \times X, Y)$$
\end{defone}
\begin{remone} \label{Rem SS enriched over S}
In particular, the previous statement implies that $s\mathscr{M}athscr{S}$ is enriched over simplicial sets, as for every $X$ and $Y$, we have a
mapping space $Map_{s\mathscr{M}athscr{S}}(X,Y) = (Y^X)_0$.
\end{remone}
\begin{remone} \label{Rem Sn generated by F}
Using the enrichment, by the Yoneda lemma, for any simplicial space $X$ we have following isomorphism of simplicial sets:
$$Map_{s\mathscr{M}athscr{S}}(F(n), X) \cong X_n.$$
\end{remone}
\mathscr{M}athscr{S}ection{Segal Spaces} \label{Sec Segal Spaces}
The goal of this section is to introduce Segal spaces and show how the conditions we impose on it actually allows
us to do interesting higher categorical constructions.
In the previous section we described that in order to study category theory and homotopy theory at the same time
we need to expand our playing field and use simplicial spaces. However, clearly we cannot just use any simplicial space,
but rather need to impose the right set of conditions to be able to develop a proper theory.
We will achieve this goal in three steps:
\begin{enumerate}
\item Reedy fibrant simplicial spaces
\item Segal spaces
\item Complete Segal spaces
\end{enumerate}
In this section we focus on the first two conditions. The next section will discuss the last condition.
\mathscr{M}athscr{S}ubsection{Reedy Fibrant Simplicial Spaces} \label{Subsec Reedy Fibrant Simplicial Spaces}
First we have to make sure that the vertical axis actually behave like a space as described in
Subsection \ref{Subsec Simplicial Sets: A Second Look at Spaces}. This is achieved by the Reedy fibrancy condition.
\begin{defone} \label{Def Reedy Fibrant}
A simplicial space $X \in s\mathscr{M}athscr{S}$ is {\it Reedy fibrant} if for every $n \geq 0$, the induced map of spaces
$$Map_{s\mathscr{M}athscr{S}}(F(n),X) \twoheadrightarrow Map_{s\mathscr{M}athscr{S}}(\partial F(n),X)$$
is a Kan fibration.
\end{defone}
\begin{remone} \label{Rem Reedy fib more than levelwise Kan}
By the Yonda lemma $Map_{s\mathscr{M}athscr{S}}(F(n),X) \cong X_n$. Moreover,
$$Map_{s\mathscr{M}athscr{S}}(\partial F(0),X) = Map_{s\mathscr{M}athscr{S}}(\emptyset ,X) = \mathscr{M}athcal{D}elta[0]$$
so an inductive argument shows that $X_n$ is a Kan complex. Notice, the opposite does not necessarily hold. In other words,
a level-wise Kan fibrant simplicial space is not necessarily Reedy fibrant.
\end{remone}
\begin{remone}
At the beginning of Subsection \ref{Subsec Simplicial Spaces} we stated that we chose to use the word simplicial space
rather than bisimplicial set. The reason is exactly because we will focus on the Reedy fibrancy condition
which guarantees to us that we have level-wise spaces.
\end{remone}
\begin{remone} \label{Rem Lifting for Reedy fib}
Concretely, $X$ is Reedy fibrant if for every $n \geq 0, l \geq 0, 0 \leq i \leq n$ the following diagram lifts.
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
\displaystyle \partial F(n) \times \mathscr{M}athcal{D}elta[l] \coprod_{\partial F(n) \times \Lambda[l]_i} F(n) \times \Lambda[l]_i \arrow[r] \arrow[d] & X \\
F(n) \times \mathscr{M}athcal{D}elta[l] \arrow[ur, dashed]
\end{tikzcd}
\end{center}
As is the case for Kan fibrations there is an analogous Reedy fibration. A map $Y \twoheadrightarrow X$ is a Reedy fibration if the
following square lifts.
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
\displaystyle \partial F(n) \times \mathscr{M}athcal{D}elta[l] \coprod_{\partial F(n) \times \Lambda[l]_i} F(n) \times \Lambda[l]_i \arrow[r] \arrow[d] &
Y \arrow[d, twoheadrightarrow] \\
F(n) \times \mathscr{M}athcal{D}elta[l] \arrow[ur, dashed] \arrow[r] & X
\end{tikzcd}
\end{center}
\end{remone}
\begin{remone} \label{Rem Reedy fibrant is Reedy fibration}
Note in particular $X$ is Reedy fibrant if and only if $X \to F(0)$ is a Reedy fibration.
So, if $Y \twoheadrightarrow X$ is a Reedy fibration and $X$ is Reedy fibrant then $Y$ is also Reedy fibrant.
\end{remone}
\begin{remone} \label{Rem Reedy fib cartesian closed}
\cite[2.5]{Re01}
If $X$ is Reedy fibrant and $Y$ is any simplicial space then $X^Y$ is also Reedy fibrant.
\end{remone}
As Reedy fibrant simplicial spaces are just level-wise spaces they have their own homotopy theory.
\begin{defone} \label{Def Homotopy of sS}
A map of $X \to Y$ of Reedy fibrant simplicial spaces is an equivalence if and only if for any $n \geq 0$, the map of spaces
$X_n \to Y_n$ is an equivalence of spaces.
\end{defone}
\begin{remone} \label{Rem Reedy fib gives good pullbacks}
One important reason we use the Reedy fibrancy condition is outlined in Remark \ref{Rem Kan fib basechange}.
We want all definitions to be homotopy invariant and as many of those definitions have pullback conditions involved,
Reedy fibrancy is an effective way to guarantee that all definitions have that condition.
\end{remone}
\begin{remone}
There is a more conceptual reason for the Reedy fibrancy condition. It allows us to use the tools from the theory of model categories
to study higher categories (Theorem \ref{The Reedy Model Structure}).
\end{remone}
\mathscr{M}athscr{S}ubsection{Defining Segal Spaces} \label{Subsec Defining Segal Spaces}
In the last subsection we made sure that the vertical axis of the simplicial space actually has the behavior of spaces, by adding
the Reedy fibrancy condition. In the next step we will add the necessary condition to the horizontal axis to make sure it has
the proper categorical behavior, which will lead to the definition of a Segal space.
Segal spaces were originally defined in \cite[Section 4]{Re01}
\par
Recall in Subsection \ref{Subsec Simplicial Sets: A Second Look at Categories} we showed how a category is really a simplicial set
that satisfies the Segal condition. We want to repeat that argument for simplicial spaces.
\begin{constrone} \label{Constr G}
Let $\alpha_i \in F(n)_1 = Hom_{\mathscr{M}athcal{D}elta}([1],[n])$ be defined by $\alpha_i(k) = k+i$, where $ 0 \leq i \leq n-1$.
Concretely it is the map $[1] \to [n]$, which takes $0$ to $i$ and $1$ to $i+1$.
Let $G(n)$ be the simplicial subspace of $F(n)$ generated by $A = \{ \alpha_i : 0 \leq i \leq n-1 \}$.
This means that
$$G(n) = F(1) \coprod_{F(0)} ... \coprod_{F(0)} F(1) $$
where there are $n$ factors of $F(1)$.
Let $\varphi^n: G(n) \to F(n)$ be the natural inclusion map. Thus we think of $G(n)$ as a subobject of $F(n)$.
This is commonly known as the {\it ``spine"} of $F(n)$.
\end{constrone}
\begin{remone} \label{Rem G into X}
It easily follows that for a simplicial space $X$ we have
$$ Map_{s\mathscr{M}athscr{S}}(G(n), X) \cong X_1 \prescript{d_0}{}{\underset{X_0}{\times}}^{d_1} X_1 \prescript{d_0}{}{\underset{X_0}{\times}}^{d_1} \ ... \
\prescript{d_0}{}{\underset{X_0}{\times}}^{d_1} X_1 \prescript{d_0}{}{\underset{X_0}{\times}}^{d_1} X_1$$
where there are $n$ factors of $X_1$ and $d_0, d_1: X_1 \to X_0$
are the simplicial maps. This gives us the following canonical Kan fibration:
$$\varphi_n = (\varphi^n)^*: X_n \cong Map_{s\mathscr{M}athscr{S}}(F(n),X) \to Map_{s\mathscr{M}athscr{S}}(G(n),X ) \cong X_1 \underset{X_0}{\times} ... \underset{X_0}{\times} X_1.$$
\end{remone}
Having properly defined our map we can now make following definition.
\begin{defone} \label{Def Segal space}
A \emph{Segal space} $T_n \in s\mathscr{M}athscr{S}$ is a Reedy fibrant simplicial space such that the canonical map
$$ \varphi_n : T_n \xrightarrow{ \ \ \mathscr{M}athscr{S}imeq \ \ } T_1 \underset{T_0}{\times} ... \underset{T_0}{\times} T_1$$
is a Kan equivalence for every $n \geq 2$.
\end{defone}
Note that Reedy fibrancy implies that the maps $\varphi_n$ are actually fibrations
and so, for Segal spaces, these maps are trivial fibrations. Also, Reedy fibrancy tells us that $d_1$ and $d_0$ are fibrations of spaces and so the
pullbacks are already homotopy pullbacks.
\begin{intone} \label{Int Segal Spaces}
What is the idea of a Segal space? If $T$ is a Segal space then it is a simplicial space, thus has spaces $T_0$, $T_1$, $T_2$, ... .
The Segal condition tells us that we should think of $T_0$ as the ``space of objects", $T_1$ the ``space of morphisms", $T_2$
the ``space of compositions".
Let us see how we can manifest those ideas in a more concretely: \par
{\bf n=2:} The first condition states that $T_2 \xrightarrow{ \ \ \mathscr{M}athscr{S}imeq \ \ } T_1 \times_{T_0} T_1$.
Note that $T_2$ is the space of $2$-cells.
Concretely, we depict a $2$-cell $\mathscr{M}athscr{S}igma$ as follows:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
& y \arrow[dr, "g"]& \\
x \arrow[rr, ""{name=U, above}] \arrow[ur, "f"] & & z
\arrow[start anchor={[xshift=-10ex, yshift=10ex]}, to=U, phantom, "\mathscr{M}athscr{S}igma"]
\end{tikzcd}
\end{center}
Similarly, we think of $T_1 \times_{T_0} T_1$ as the space of {\it two composable arrows}, which we depict as:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
& y \arrow[dr, "g"]& \\
x \arrow[ur, "f"] & & z
\end{tikzcd}
\end{center}
The Segal condition states that every such diagram can be filled out to a complete $2$-cell:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
& y \arrow[dr, "g"]& \\
x \arrow[rr, ""{name=U, above}, "h"', dashed] \arrow[ur, "f"] & & z
\arrow[start anchor={[xshift=-10ex, yshift=10ex]}, to=U, phantom, "\mathscr{M}athscr{S}igma"]
\end{tikzcd}
\end{center}
From this point of view, we think of $h$ as the composition of $f$ and $g$, and we think of $\mathscr{M}athscr{S}igma$ as a witness of that composition.
Thus we often depict $h$ as $g \circ f$.
\par
Right here we can already notice the difference between the classical setting and the higher categorical setting.
In the classical setting every two composable maps have a {\it unique} composition, whereas in this case neither $h$ nor $\mathscr{M}athscr{S}igma$ are unique.
Rather we only know such lift exists. In the next part we will explain how we are still justified in using a specific name
(``$g \circ f$") despite its non-uniqueness.
{\bf n=3:} The second condition is that $T_3 \xrightarrow{ \ \ \mathscr{M}athscr{S}imeq \ \ } T_1 \times_{T_0} T_1 \times_{T_0} T_1$.
$T_3$ is the space of $3$-cells.
We depict a $3$-cell as a tetrahedron.
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
& w & \\
& & \\
x \arrow[uur] \arrow[dr] \arrow[rr, ""{name=U, above}, "f" near start] & & y \arrow[dl, "g"] \arrow[uul] \\
& z \arrow[uuu, "h" near end]
\arrow[start anchor={[xshift=-2ex, yshift=2ex]}, to=U, phantom, "\mathscr{M}athscr{S}igma"]
\arrow[start anchor={[xshift=10ex, yshift=20ex]}, to=U, phantom, "\gamma"]
\end{tikzcd}
\end{center}
where all $2$-cells and the middle $3$-cell are filled out.
On the other hand $T_1 \times_{T_0} T_1 \times_{T_0} T_1$ can be depicted as three composable arrows.
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
& w & \\
& & \\
x \arrow[rr, ""{name=U, above}, "f" near start] & & y \arrow[dl, "g"] \\
& z \arrow[uuu, "h" near end]
\end{tikzcd}
\end{center}
The Segal condition then implies that this diagram can be completed to a diagram of the form:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
& w & \\
& & \\
x \arrow[uur, dashed, "h \circ (g \circ f)" near start, "(h \circ g) \circ f" near end]
\arrow[dr, dashed, "g \circ f"'] \arrow[rr, ""{name=U, above}, "f" near start] & &
y \arrow[dl, "g"] \arrow[uul, dashed, "h \circ g"'] \\
& z \arrow[uuu, "h" near end]
\arrow[start anchor={[xshift=-2ex, yshift=2ex]}, to=U, phantom, "\mathscr{M}athscr{S}igma"]
\arrow[start anchor={[xshift=10ex, yshift=20ex]}, to=U, phantom, "\gamma"]
\end{tikzcd}
\end{center}
In the diagram above
the $2$-cell $\mathscr{M}athscr{S}igma$ witnesses the composition of $f$ and $g$ and
the $2$-cell $\gamma$ witnesses the composition of $g$ and $h$. The middle $3$-cell (and the two other $2$-cells) witness an
equivalence between the composition $h \circ (g \circ f)$ and $(h \circ g) \circ f$.
So, the existence of this $3$-cell witnesses the {\it associativity} of the composition operation.
\par
The Segal condition for $3$-cells has even more interesting implications.
In the previous part we discussed that composition is not unique and that any lift is a possible composition.
Using the Segal condition we can show that every two choices of equivalences are equivalent.
Let $f:x \to y$ $g: y \to z$, then the Segal condition implies we can fill in following diagram:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
& z & \\
& & \\
x \arrow[uur, "c_1"] \arrow[dr, "f"'] \arrow[rr, ""{name=U, above}, "id_x" near start] & & x \arrow[dl, "f"] \arrow[uul, "c_2"] \\
& y \arrow[uuu, "g" near end]
\arrow[start anchor={[xshift=-10ex, yshift=20ex]}, to=U, phantom, "\mathscr{M}athscr{S}igma"]
\arrow[start anchor={[xshift=10ex, yshift=20ex]}, to=U, phantom, "\gamma"]
\arrow[start anchor={[xshift=-4ex, yshift=2ex]}, to=U, phantom, "f"]
\end{tikzcd}
\end{center}
The two cells $\mathscr{M}athscr{S}igma$ and $\gamma$ represent two possible compositions, which we denoted by $c_1$ and $c_2$.
The Segal condition (which fills in a $3$-cell) tells us that those
two compositions are equivalent.
\end{intone}
Up to here the goal has been to give the reader an intuition on how a Segal space take familiar concepts from category theory and
generalize them to a homotopical setting. The goal of the next subsection is to actually give precise definitions and make those
intuitive arguments rigorous.
\mathscr{M}athscr{S}ubsection{Category Theory of Segal Spaces} \label{Subsec Category Theory of Segal Spaces}
The goal of this subsection is to make our first steps towards studying the category theory of Segal spaces.
In particular, we define objects, morphisms, compositions and homotopy equivalences.
A lot of the concepts here are guided by the ideas introduced in Intuition \ref{Int Segal Spaces}.
The work here was originally developed in \cite[Section 5]{Re01}.
\begin{defone} \label{Def Objects of SS}
Let $T$ be a Segal space. We define the {\it objects of $T$} as the set of objects of $T_0$.
Thus, $Obj(T) = T_{00}$.
\end{defone}
\begin{intone} \label{Int Objects of SS}
In Intuition \ref{Int Segal Spaces} we discussed how $T_0$ should be the ``space of objects". Consistent with that philosophy
the points in $T_0$ are exactly the objects.
\end{intone}
\begin{notone} \label{Not Objects in SS}
Following standard conventions we often use the notation $x \in T$ to denote an object $x$ in the Segal space $T$.
\end{notone}
\begin{defone} \label{Def Mapping space}
For two objects $x,y \in T$, we define the \emph{mapping space} by the following pullback:
\begin{center}
\pbsq{ map_{T} (x,y) }{T_1}{\mathscr{M}athcal{D}elta[0]}{T_0 \times T_0}{}{}{(d_0 , d_1)}{(x , y)}
\end{center}
or in other words the fiber of $(d_0,d_1)$ over the point $(x,y)$.
\end{defone}
\begin{intone} \label{Int Mapping space}
Again this definition is consistent with Intuition \ref{Int Segal Spaces}. $T_1$ should be thought of as the ``space of morphism".
From that point of view the boundary map $d_0$ gives us the source of the morphism and $d_1$ is the target of the morphism.
The space of morphisms that start at a certain object $x$ and end with the object $y$ is exactly the pullback constructed above.
\end{intone}
\begin{remone}
We can compare the definition of a mapping space to the definition of a Hom set in a usual category as introduced in Remark \ref{Rem Hom sets}.
The overall definitions are very similar, giving one more evidence that this is a good way to define the space of maps.
\end{remone}
Again by Reedy fibrancy the map $(d_0,d_1)$ is a fibration and so our pullback diagram is actually homotopy invariant.
\begin{notone} \label{Not Map in SS}
In order to simplify our notation instead of writing $f \in map_{T}(x,y)$ we will use the more familiar $f: x \to y$.
\end{notone}
\begin{remone} \label{Rem Mapping space a space}
The fact that $(d_0,d_1)$ is a Kan fibration also implies that $map_{T} (x,y)$ is a Kan complex,
which justifies using the word mapping space.
\end{remone}
Our main goal now is to make the idea of composition, which we discussed in the last subsection, precise.
In order to do so we need to define the {\it space of compositions}.
\begin{defone} \label{Def Space of Comp}
Let $x_0,x_1, ..., x_n \in Ob(T)$ be objects in $T$.
We define the space of composition $map_{T}(x_0,...,x_n)$ as the pullback:
\begin{center}
\pbsq{ map_{T}(x_0,...,x_n) }{T_n}{*}{(T_0)^{n+1} }{}{}{}{(x_0 , x_1 , ... , x_{n+1} )}
\end{center}
\mathscr{M}athscr{N}oindent
or, in other words, the fiber of the map $T_n \to T_0^{n+1}$ over the point
$(x_0,x_1, ..., x_n)$.
\end{defone}
\begin{intone} \label{Int Space of Comp}
We can think of a point in $map_{T}(x_0,...,x_n)$ as an $n+1$-simplex which has vertices $x_0,x_1,...,x_n$.
In particular, a point in $map_{T}(x_0,x_1,x_2)$ is the following triangle.
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
& x_1 \arrow[dr]& \\
x_0 \arrow[rr, ""{name=U, above}] \arrow[ur] & & x_2
\arrow[start anchor={[xshift=-10ex, yshift=10ex]}, to=U, phantom, "\mathscr{M}athscr{S}igma"]
\end{tikzcd}
\end{center}
\end{intone}
\begin{remone} \label{Rem Space of comp is maps}
We have the commuting triangle
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
T_n \arrow[rr, twoheadrightarrow, "\mathscr{M}athscr{S}imeq", "\varphi_n"'] \arrow[dr] & & T_1 \underset{T_0}{\times} ... \underset{T_0}{\times} T_1 \arrow[dl] \\
& T_0 \times ... \times T_0 &
\end{tikzcd}
\end{center}
where the top map is a trivial Kan fibration. Pulling back this equivalence along the point $(x_0, ... ,x_n): \mathscr{M}athcal{D}elta[0] \to (T_0)^{n+1}$
we get the trivial fibration
$$ \displaystyle \left. \varphi_n \right|_{map_{T}(x_0,...,x_n)}: map_{T}(x_0,...,x_n) \xrightarrow{ \ \ \mathscr{M}athscr{S}imeq \ \ }
map_{T}(x_0,x_1) \times ... \times map_{T}(x_{n-1},x_n).$$
\end{remone}
\begin{intone} \label{Int Space of comp is maps}
Intuitively, this map takes an $n+1$-simplex with vertices $x_0, x_1, ... ,x_n$ and restricts it to the spine.
For example the triangle above we will be restricted to the diagram:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
& x_1 \arrow[dr]& \\
x_0 \arrow[ur] & & x_2
\end{tikzcd}
\end{center}
\end{intone}
\begin{constrone} \label{Constr Comp of maps}
Using this map we can give a rigorous definition of the composition map. Let us fix three objects $x,y$ and $z$ in $T$.
This gives us following diagram:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
map(x,y,z) \arrow[d, "\mathscr{M}athscr{S}imeq", "(\alpha_0 , \alpha_1 )"', twoheadrightarrow] \arrow[r, "d_1"] & map(x,z) \\
map(x,y) \times map(y,z)
\end{tikzcd}
\end{center}
Here $(\alpha_0,\alpha_1):T_2 \to T_1 \times_{T_0} T_1$ and $d_1: W_2 \to W_1$ are both restrictions of the actual maps.
Now picking two morphisms $f \in map_T(x,y)$, $g \in map_T(y,z)$ is the same as picking a map $ \mathscr{M}athcal{D}elta[0] \to map_T(x,y) \times map_T(y,z)$.
That allows us to expand our diagram to following pullback diagram:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
Comp(f,g) \arrow[dr, phantom, "\ulcorner", very near start] \arrow[d, "\mathscr{M}athscr{S}imeq", twoheadrightarrow] \arrow[r, hookrightarrow] &
map(x,y,z) \arrow[d, "\mathscr{M}athscr{S}imeq", "(\alpha_0 , \alpha_1 )"', twoheadrightarrow] \arrow[r, "d_1"] & map(x,z) \\
\mathscr{M}athcal{D}elta[0] \arrow[r, "(f , g )"] & map(x,y) \times map(y,z)
\end{tikzcd}
\end{center}
We can now take any point $\mathscr{M}athscr{N}u \in Comp(f,g)$, then $d_1\mathscr{M}athscr{N}u \in map_T(x,z)$ is a composition morphism for $(f,g)$.
At first this definition might seem arbitrary as we are choosing a point.
However, the map $Comp(f,g) \to \mathscr{M}athcal{D}elta[0]$ is an equivalence, which means $Comp(f,g)$ is contractible.
Thus any two points $\mathscr{M}athscr{N}u_1$ and $\mathscr{M}athscr{N}u_2$ are equivalent.
Thus we are justified in naming any such choice $d_1 \mathscr{M}athscr{N}u = g \circ f$ as our composition map, with the understanding that there
is a contractible space of such choices.
This is the precise argument for why compositions are unique, which was already outlined in Intuition \ref{Int Segal Spaces}.
\end{constrone}
\begin{intone} \label{Int Comp of maps}
The way to think about it is that $f$ and $g$ are the morphisms we are trying to compose $d_1 \mathscr{M}athscr{N}u = g \circ f$ is the composition and $\mathscr{M}athscr{N}u$
is the witness for that composition. This can be captured in the following diagram:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
& y \arrow[dr, "g"]& \\
x \arrow[rr, ""{name=U, above}, "g \circ f"'] \arrow[ur, "f"] & & z
\arrow[start anchor={[xshift=-10ex, yshift=10ex]}, to=U, phantom, "\mathscr{M}athscr{N}u"]
\end{tikzcd}
\end{center}
This exactly ties back to our discussion in Intuition \ref{Int Segal Spaces}. It is by using the properties of Kan fibrations
and contractibility that can make those ideas into precise definitions.
\end{intone}
In this subsection we focused on the categorical aspects of a Segal space in the sense that it has objects, morphisms
and identity maps. Moreover, those satisfy a homotopical analogue of composition, identity rule and associativity.
In the next subsection we will focus on homotopical aspects. However, there is one homotopical definition
that we need right now.
\begin{defone} \label{Def Homotopic maps in SS}
Let $x,y \in T$ be two objects. We say two morphisms $f,g \in map_T(x,y)$ are homotopic and denote it by $f \mathscr{M}athscr{S}im g$, if the two maps
$f,g: \mathscr{M}athcal{D}elta[0] \to map(x,y)$ are homotopic maps in the space $map_T(x,y)$, as discussed in Example \ref{Ex Homotopic Points in Kan complex}.
\end{defone}
\begin{defone} \label{Def Identity Map}
For every object there is a notion of an \emph{identity map}. It is the image of $x$ under the degeneracy map
$s_0: T_0 \to T_1$. Following standard notation, we denote the identity map of $x$ as $id_x$.
\end{defone}
Having a definition of a composition and identity map in a Segal space, we can show they satisfy the right homotopical properties.
\begin{propone} \label{Prop Comp Ass and Id}
Let $f \in map_{T}(x,y)$, $g \in map_{T}(y,z)$ and $h \in map_{T}(z,w)$. Then, $h \circ (g \circ f) \mathscr{M}athscr{S}im (h \circ g ) \circ f$
and $f \circ id_x \mathscr{M}athscr{S}im id_y \circ f \mathscr{M}athscr{S}im f$ i.e. composition is associative and has units up to homotopy.
\end{propone}
\begin{proof}
We have following commutative diagram
\begin{center}
\begin{tikzcd}[row sep =0.5in, column sep=0.3in]
& map_T(x,y) \times map_T(y,z) \times map_T(z,w) & \\
& map_T(x,y,z,w) \arrow[dl, "d_1"] \arrow[dr, "d_2"] \arrow[u, twoheadrightarrow, "\mathscr{M}athscr{S}imeq"] & \\
map_T(x,y,w) \arrow[dr, "d_1"] & & map_T(x,z,w) \arrow[dl, "d_1"] \\
& map_T(x,w) &
\end{tikzcd}
\end{center}
If we take $(f,g,h) \in map_T(x,y) \times map_T(y,z) \times map_T(z,w)$ we can lift it to a $\mathscr{M}athscr{S}igma \in map_T(x,y,z,w)$.
Going the left hand map gives us $(h \circ g) \circ f$, but the right hand map gives us $h \circ (g \circ f)$.
This proves associativity.
\par
For the identity relation, let $f \in map_T(x,y)$, this gives us a $2$-cell $s_0(f) \in map_T(x,x,y)$, which satisfies
$\varphi_2(s_0(f)) = (id_x,f)$. Moreover, $d_1(s_0(f)) = f$. This proves one side of the identity relation.
The other side follows similarly.
\end{proof}
\begin{remone}
For a different (but similar) way to prove the same proposition see \cite[Proposition 5.4]{Re01}.
\end{remone}
This proposition allows us construct an ordinary category out of every Segal space, confirming the connection between Segal spaces
and categories.
\begin{constrone} \label{Constr Homotopy Category of SS}
Let $T$ be a Segal space. We will define the category, called the \emph{homotopy category} and denoted by $HoT$, as follows.
We let the objects of $HoT$ to be the objects of $T$. For two objects $x,y \in HoT$ we define the mapping space as
$$Hom_{HoT}(x,y) = \pi_0(map_{T}(x,y))$$
in other words $Hom_{HoT}(x,y)$ is the set of path components of the space $map_{T}(x,y)$.
\par
For three objects $x,y,z \in HoT$, the composition map $ map_{T}(x,y) \times map_{T}(y,z) \to map_{T}(x,z)$ gives us a composition map
$$ Hom_{HoT}(x,y) \times Hom_{HoT}(y,z) \to Hom_{HoT}(x,z).$$
\end{constrone}
This construction gives us following theorem.
\begin{theone}
Let $T$ be a Segal space. Then $HoT$ is a category.
\end{theone}
\begin{proof}
We already specified the objects, morphisms and composition. The previous proposition shows that this composition has identity maps
and is associative.
\end{proof}
\mathscr{M}athscr{S}ubsection{Homotopy Equivalences} \label{Homotopy Equivalences}
A Segal space should be amalgamation of category theory and homotopy theory. Up to here we mostly focused on basic categorical phenomena.
In this subsection we point to some homotopical aspects of a Segal space. In particular, we will discuss homotopic maps and homotopy
equivalences in a Segal space. Most of the work in this subsection follows \cite[Section 5]{Re01}.
In the last subsection we already discussed the definition of {\it homotopic morphisms}.
That was made possible by the fact that we have mapping spaces (rather than sets)
combined with the homotopy theory of spaces.
This naturally leads to the next definition.
\begin{defone} \label{Def Homotopy Equiv in SS}
Let $T$ be a Segal space. A morphism $f \in map_{T}(x,y)$ is a homotopy equivalence if
there exist maps $g,h \in map_{T}(y,x)$ such that $g \circ f \mathscr{M}athscr{S}im id_x$ and $f \circ h \mathscr{M}athscr{S}im id_y$.
\end{defone}
\begin{intone} \label{Int Homotopy Equiv in SS}
Although it might appear as the definition only involves the existence of three maps, but in reality the nature of a
Segal space demands that there are several other important pieces of information. In particular, each composition $g \circ f $
and $f \circ h$ has a $2$-cell that witnesses the composition. Moreover, there are homotopies between the compositions
and identities. The information can be captured in a diagram of the following form
\begin{center}
\begin{tikzcd}[row sep =0.5in, column sep=0.5in]
x \arrow[r, "id_x"] \arrow[dr, "f"] & x \\
y \arrow[u, "h"] \arrow[r, "id_y"'] & y \arrow[u, "g"']
\end{tikzcd}
\end{center}
\end{intone}
\begin{remone} \label{Rem Uniqueness of Inverse}
Note that Proposition \ref{Prop Comp Ass and Id} implies that $$g \mathscr{M}athscr{S}im g \circ id_y \mathscr{M}athscr{S}im g \circ f \circ h \mathscr{M}athscr{S}im id_x \circ h \mathscr{M}athscr{S}im h$$
and so the inverse is unique (as always only up to homotopy).
\end{remone}
There is another way to define homotopy equivalences.
\begin{constrone} \label{Constr Z3 and maps}
Let $Z(3)$ be the simplicial space defined by the colimit of the following diagram.
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
F(1) & & F(1) & & F(1) \\
& F(0) \arrow[ur, "1"'] \arrow[ul, "1"]& & F(0) \arrow[ur, "0"'] \arrow[ul, "0"]&
\end{tikzcd}
\end{center}
Thus the space $Map(Z(3), T)$ is the limit of the following diagram.
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
T_1 \arrow[dr, "d_1"] & & T_1 \arrow[dl, "d_1"] \arrow[dr, "d_0"] & & T_1 \arrow[dl, "d_0"]\\
& T_0 & & T_0 &
\end{tikzcd}
\end{center}
which we can also express as.
$$Map(Z(3), T) \cong T_1 \prescript{d_1}{}{\times}^{d_1}_{T_0} T_1 \prescript{d_0}{}{\times}^{d_0}_{T_0} T_1$$.
This construction comes with the following map.
$$ (d_1d_3,d_0d_3,d_1d_0) :T_3 \to T_1 \prescript{d_1}{}{\times}^{d_1}_{T_0} T_1 \prescript{d_0}{}{\times}^{d_0}_{T_0} T_1$$
It follows from the simplicial identities that this map is well-defined, by which we mean that
$$ d_1d_1d_3 = d_1d_0d_3 \ \ \ \text{and} \ \ \ d_0d_0d_3 = d_0d_1d_0.$$
Also, for any $f \in T_1$ we have $(s_0d_0f,f,s_0d_1f) \in T_1 \prescript{d_1}{}{\times}^{d_1}_{T_0} T_1 \prescript{d_0}{}{\times}^{d_0}_{T_0} T_1$
\end{constrone}
\begin{lemone} \label{Lemma Homotopy Equiv tetra version}
A map f is an equivalence if and only if the element
$$(s_0d_0f,f,s_0d_1f) \in T_1 \prescript{d_1}{}{\times}^{d_1}_{T_0} T_1 \prescript{d_0}{}{\times}^{d_0}_{T_0} T_1$$
lifts to an element $H \in T_3$.
\end{lemone}
\begin{proof}
Let $f:x \to y$ be a homotopy equivalence and $g$ be its inverse. Then $(g,f,g) \in T_1 \times_{T_0} T_1 \times_{T_0} T_1 \mathscr{M}athscr{S}imeq T_3$
and $(d_1d_3,d_0d_3,d_1d_0) (g,f,g) = (id_x , f , id_y)$, which implies that $(g,f,g)$ is our lift.
On the other side, assume that $(id_x , f , id_y)$ lifts to $H \in T_3$. Let us denote $d_2d_1H = g$ and $d_0d_0 H = h$.
Now, $d_3H$ gives a homotopy from the map $gf$ to $id_y$ and $d_0H$ gives a homotopy from the map $hf$ to $id_x$.
This means that $f$ is a homotopy equivalence and so we are done.
\end{proof}
\begin{intone} \label{Int Homotopy Equiv Tetra version}
The description here can seem quite confusing and so a more detailed breakdown can be quite helpful.
The element in $(id_x , f , id_y) \in T_1 \prescript{d_1}{}{\times}^{d_1}_{T_0} T_1 \prescript{d_0}{}{\times}^{d_0}_{T_0} T_1$
can be represented by the diagram:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
& x & \\
& & \\
y \arrow[dr, "id_y"'] & & x \arrow[dl, "f"] \arrow[uul, "id_x"'] \\
& y
\end{tikzcd}
\end{center}
A lift to an element in $T_3$ would imply the existence of a diagram of the following form.
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
& x & \\
& & \\
y \arrow[uur, dashed, "g"] \arrow[dr, "id_y"'] \arrow[rr, ""{name=U, above}, "g" near start, dashed] & & x \arrow[dl, "f"] \arrow[uul, "id_x"'] \\
& y \arrow[uuu, dashed, "h" near end]
\arrow[start anchor={[xshift=-10ex, yshift=20ex]}, to=U, phantom, "g"]
\arrow[start anchor={[xshift=10ex, yshift=20ex]}, to=U, phantom, "\iota_2"]
\arrow[start anchor={[xshift=-4ex, yshift=2ex]}, to=U, phantom, "\iota_1"]
\end{tikzcd}
\end{center}
The data in this diagram is exactly that of two morphisms $g,h: y \to x$ and two $2$-cells $\iota_1, \iota_2$
that give us the right and left inverses.
\end{intone}
\begin{remone} \label{Rem Homotopy Equiv Square vs Tetra}
On a first look Definition \ref{Def Homotopy Equiv in SS} and Lemma \ref{Lemma Homotopy Equiv tetra version} might seem different,
but we can use the Segal condition to show that they are actually the same. Note that we have two maps
$$\beta_1: F(2) \to F(3)$$
$$\beta_1: F(2) \to F(3)$$
defined as $\beta_1(0) = 0, \beta_1(1) = 1, \beta_1(2) = 2$ and $\beta_2(0) = 1, \beta_2(1) = 2, \beta_2(2) = 3$.
Notice that $d_2 \beta_1 = d_0\beta_2: F(1) \to F(3)$, namely the map that sends $0$ to $1$ and $1$ to $2$, so we call it $\beta_3$.
Thus this gives us following map:
$$\beta_1 \coprod_{\beta_3} \beta_2 : F(2)^{d_2} \coprod_{F(1)}^{d_2} F(2) \to F(3)$$
This gives us following commuting triangle.
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
T_3 \arrow[rr, "\displaystyle \beta_1 \coprod_{\beta_3} \beta_2"] \arrow[dr, "\mathscr{M}athscr{S}imeq", "\varphi_3"'] & &
T_2 \underset{T_1}{\times} T_2 \arrow[dl, "\mathscr{M}athscr{S}imeq"', "(\varphi_2 , \varphi_2)"] \\
& T_1 \underset{T_0}{\times} T_1 \underset{T_0}{\times} T_1 &
\end{tikzcd}
\end{center}
By the Segal condition the down left map is an equivalence. Using the Segal condition twice also implies that the right map is an equivalence.
By the Segal condition the two diagonal maps are equivalences which means the top horizontal map is an equivalence as well.
\par
But we also have the equivalence $F(2) \coprod_{F(1)} F(2) \cong F(1) \times F(1)$, which gives us equivalences
$$T_3 \mathscr{M}athscr{S}imeq T_2 \times_{T_1} T_2 \cong Map(F(1) \times F(1), T)$$
This means that in a Segal space there is an equivalence between $T_3$ and maps of squares into $T$.
Therefore, we can impose the right conditions on either shape to define a notion of equivalence.
\end{remone}
This second method gives us an easy way of showing that the notion of a homotopy equivalence is homotopy invariant.
\begin{lemone} \label{Lemma Hoequiv embeds in arrows}
Let $\gamma: \mathscr{M}athcal{D}elta[1] \to T_1$ be a path from $\gamma(0) = f \in T_1$ to $\gamma(1) = f' \in T_1$
and $f'$ is a homotopy equivalence. Then $f$ is also an homotopy equivalence.
\end{lemone}
\begin{proof}
We have the following diagram.
\begin{center}
\begin{tikzcd}[row sep=0.7in, column sep=0.7in]
\mathscr{M}athcal{D}elta[0] \arrow[d, " 1 "] \arrow[rr] & & T_3 \arrow[d, "(d_1d_3 , d_0d_3 , d_1d_0)", twoheadrightarrow] \\
\mathscr{M}athcal{D}elta[1] \arrow[urr, dashed, "\tilde{\gamma}"'] \arrow[r, "\gamma"'] & T_1 \arrow[r, "(s_0d_0 , id_{T_1} , s_0d_1)"'] &
T_1 \prescript{d_1}{}{\underset{T_0}{\times}}^{d_1} T_1 \prescript{d_0}{}{\underset{T_0}{\times}}^{d_0} T_1
\end{tikzcd}
\end{center}
This diagram lifts because the right-hand map is a fibration. Thus $\tilde{\gamma}(1)$ is the lift of $(s_0d_0g,g,s_0d_1g)$ which we were looking for.
\end{proof}
\begin{defone}
We define $i: T_{hoequiv} \hookrightarrow T_1$ as the subspace generated by the set of homotopy equivalences
and call it the \emph{space of homotopy equivalences}.
\end{defone}
\begin{remone}
Lemma \ref{Lemma Hoequiv embeds in arrows} implies that this map is $(-1)$-truncated (Definition \ref{Def Neg one trunc}).
\end{remone}
There are other, yet equivalent ways to understand the space $T_{hoequiv}$ and its ($-1$)-truncated map from $T_{hoequiv} \to T_1$.
\begin{theone}
In the following pullback diagram
\begin{center}
\begin{tikzcd}[row sep=0.4in, column sep=0.4in]
& T_{hoeqchoice} \arrow[dd, "j"] \arrow[rr] \arrow[dl, "U", dashed] \arrow[ddrr, phantom, "\ulcorner", very near start] & &
T_3 \arrow[dd, "(d_1d_3 , d_0d_3 , d_1d_0)"] \\
T_{hoequiv} \arrow[dr, "i"] & & & \\
& T_1 \arrow[rr, "(s_0d_0 , id_{T_1} , s_0d_1)"] & &
T_1 \prescript{d_1}{}{\underset{T_0}{\times}}^{d_1} T_1 \prescript{d_0}{}{\underset{T_0}{\times}}^{d_0} T_1
\end{tikzcd}
\end{center}
the map $j$ factors through $i$ and the resulting map $U$ is a homotopy equivalence.
\end{theone}
\begin{proof}
In order to prove our result we have to show that for every homotopy equivalence $f:x \to y \in T_{hoequiv}$ the fiber of $U$ over
$f$, which we call $HEquiv(f)$, is a contractible space. This means we are looking at following pullback squares.
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.7in]
HEquiv(f) \arrow[d] \arrow[rr] \arrow[dr, phantom, "\ulcorner", very near start] & &
T_{hoeqchoice} \arrow[d] \arrow[r] \arrow[dr, phantom, "\ulcorner", very near start] &
T_3 \arrow[d, "(d_1d_3 , d_0d_3 , d_1d_0)"] \\
\mathscr{M}athcal{D}elta[0] \arrow[r, "f"] & T_{hoequiv} \arrow[r, "i"] & T_1 \arrow[r, "(s_0d_0 , id_{T_1} , s_0d_1)"] &
T_1 \prescript{d_1}{}{\underset{T_0}{\times}}^{d_1} T_1 \prescript{d_0}{}{\underset{T_0}{\times}}^{d_0} T_1
\end{tikzcd}
\end{center}
$HEquiv(f)$ is really the subspace of $T_3$ generated by all point $H$ such that
$(d_1d_3 , d_0d_3 , d_1d_0)(H) = (id_x,f,id_y)$.
We already know that $f$ is an equivalence which means there exists maps $g,h$ such that $f \circ g$ and $h \circ f$ are
equivalent to identity maps, which implies that $HEquiv(f)$ is non-empty. Finally, we also have following homotopy pullback square:
\begin{center}
\begin{tikzcd}[column sep=0.5in, row sep=0.5in]
HEquiv(f) \arrow[r] \arrow[d, "\mathscr{M}athscr{S}imeq"] & T_3 \arrow[d, "\mathscr{M}athscr{S}imeq", "\varphi_3"', twoheadrightarrow] \\
\mathscr{M}athcal{D}elta[0] \arrow[r, "( h , f , g)"] & T_1 \underset{T_0}{\times} T_1 \underset{T_0}{\times} T_1
\end{tikzcd}
\end{center}
This follows from the fact that $\varphi_3$ is a Kan fibration and so the pullback is homotopy invariant combined with
the fact that any choice of inverses for $f$ are themselves equivalent maps (Remark \ref{Rem Uniqueness of Inverse}).
But the fiber of each trivial Kan fibration is itself contractible and hence we are done.
\end{proof}
\begin{intone}
The idea of the proof is that $T_{hoequiv}$ is the space consisting of all maps that are equivalences in the sense that some inverses exist,
whereas $T_{hoeqchoice}$ is the space of all maps with specifically chosen inverses. The map $U: T_{hoeqchoice} \to T_{hoequiv}$ forgets
the specific chosen inverse and only remembers the map that is an equivalence. The proof above basically says that up to homotopy
there is only one way to find inverses for an equivalence. This is in line with the philosophy we layed out in Remark
\ref{Rem Contractible like unique}.
\end{intone}
\begin{remone}
The concept of $T_{hoequiv}$ as the subspace of $T_1$ was introduced in \cite[Subsection 5.7]{Re01}.
We used that definition to define and study $T_{hoeqchoice}$.
\end{remone}
Viewing the space of equivalences as a pullback gives us a more systematic way to study it.
We can even simplify the pullback diagram to make computations of $T_{hoeqchoice}$ simpler.
\begin{lemone} \label{Lemma T hoeqchoice Second Def}
The following is a pullback square:
\begin{center}
\pbsq{T_1}{T_1 \prescript{d_1}{}{\underset{T_0}{\times}}^{d_1} T_1 \prescript{d_0}{}{\underset{T_0}{\times}}^{d_0} T_1}{
T_0 \times T_0}{T_1 \times T_1}{(s_0d_0 , id_{T_1} , s_0d_1)}{( d_0 , d_1)}{( \pi_1 , \pi_3)}{( s_0 , s_0)}
\end{center}
Thus the following is a pullback square:
\begin{center}
\pbsq{T_{hoeqchoice}}{T_3}{
T_0 \times T_0}{T_1 \times T_1}{}{}{( d_1d_3 , d_1d_0)}{( s_0 , s_0)}
\end{center}
\end{lemone}
There are two last definitions related to homotopy equivalences that play important roles.
\begin{defone} \label{Def Segal space groupoid}
We say a Segal space $T$ is a {\it Segal space groupoid} if every map is a homotopy equivalence.
\end{defone}
We can also have a local definition of homotopy equivalences.
\begin{defone} \label{Def Hoequiv xy}
For every two objects $x$ and $y$ we can define
the \emph{space of homotopy equivalences between $x$ and $y$},
$hoequiv_{T}(x,y)$, as the pullback
\begin{center}
\pbsq{hoequiv_{T}(x,y)}{T_{hoequiv}}{\mathscr{M}athcal{D}elta[0]}{T_0 \times T_0}{}{}{(d_0 , d_1)}{(x , y)}
\end{center}
as the fiber of the map $(d_0,d_1): T_{hoequiv} \to T_0 \times T_0$
over the point $(x,y)$ and the lemma implies that the natural inclusion map $hoequiv_{T}(x,y) \hookrightarrow map_{T}(x,y)$
is ($-1$)-truncated.
\end{defone}
\mathscr{M}athscr{S}ubsection{Examples of Segal Spaces} \label{Subsec Examples of Segal Spaces}
Now that we spent some time developing the category theory of Segal spaces, it is a good idea to see some examples and realize how each of
the previous definitions manifest in those particular examples.
\begin{exone} \label{Ex Nerve a SS}
Let $\mathscr{M}athcal{C}$ be a category.
Then $i_F^*(N\mathscr{M}athcal{C})$ is a discrete simplicial space. The discreteness implies that it is Reedy fibrant.
Moreover, it satisfies the Segal condition by definition. Thus $i_F^*(\mathscr{M}athcal{C})$ is a Segal space.
\par
Fortunately, the definitions we are used to from category theory perfectly match up with the ones for a Segal space.
In particular, as object in the Segal space $i_F^*(N\mathscr{M}athcal{C})$ is just an object in the category $\mathscr{M}athcal{C}$. Same is true for morphisms.
\par
However, as $i_F^*(N\mathscr{M}athcal{C})_1$ is just a set, the mapping space is actually just a set as well.
This in particular implies that composition is well-defined not just up to homotopy. In fact for any collection of objects $x_0, ... x_n \in i_F^*(N\mathscr{M}athcal{C})$.
The space $map(x_0,...,x_n)$ is bijective to $map(x_0,x_1) \times ... \times map(x_{n-1},x_n)$. Thus the pullback
\begin{center}
\pbsq{Comp(f,g)}{map(x_0,x_1,x_2)}{\mathscr{M}athcal{D}elta[0]}{map(x_0,x_1) \times map(x_1,x_2)}{}{}{}{(f , g)}
\end{center}
is not just contractible, but actually just a point.
\par
In addition to all of these, as $map(x_0,x_1)$ is just a set, two maps are homotopic if and only if they are equal to each other.
This implies that a map is a homotopy equivalence if and only if it is an isomorphism.
\par
In particular, the homotopy category of the Segal space, $Ho \mathscr{M}athscr{N} \mathscr{M}athcal{C}$ is exactly $\mathscr{M}athcal{C}$,
as the two categories have the same objects and same morphisms.
\end{exone}
\begin{remone}
As expected in the case of an ordinary category, the corresponding Segal space has all the category theory we desire, but has no
valuable homotopical information.
\end{remone}
\begin{exone} \label{Ex Spaces are Segal spaces}
Let $K$ be a space. Our first guess might be to take $i_{\mathscr{M}athcal{D}elta}^*(K)$. While it does satisfy the Segal condition, but it is not Reedy fibrant!
Fortunately, there is an equivalent simplicial space that is Reedy fibrant.
Namely, let $K^{\mathscr{M}athcal{D}elta[\bullet]}$ be the simplicial space defined as
\begin{center}
\mathscr{M}athscr{S}impset{K}{K^{\mathscr{M}athcal{D}elta[1]}}{K^{\mathscr{M}athcal{D}elta[2]}}{}{}{}{}
\end{center}
where the boundary maps are induced by the maps between the simplices. This simplicial space is actually Reedy fibrant.
Moreover it satisfies the Segal condition as in the diagram
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
K^{\mathscr{M}athcal{D}elta[n]} \arrow[rr] \arrow[dr, "\mathscr{M}athscr{S}imeq"] & &
K^{\mathscr{M}athcal{D}elta[1]} \underset{K^{\mathscr{M}athcal{D}elta[0]}}{\times} ... \underset{K^{\mathscr{M}athcal{D}elta[0]}}{\times} K^{\mathscr{M}athcal{D}elta[1]} \arrow[dl, "\mathscr{M}athscr{S}imeq"] \\
& K &
\end{tikzcd}
\end{center}
the vertical maps are equivalences, which means the horizontal map is also an equivalence.
\par
How does the category theory of a Segal space look like in this case? An object in this Segal space is a point in $K$.
A morphism is now a point in the path space $K^{\mathscr{M}athcal{D}elta[1]}$ and so is just a path in the space.
Composition of morphisms corresponds to concatenation of paths in the space. Notice here we really the contractibility condition.
In other words, when we concatenate two paths then we get another path that is determined only up to homotopy.
\par
Two paths are homotopic in the mapping space if they are homotopic in the usual sense for spaces.
As every path in a space is reversible, we see that every morphism is an equivalence.
Thus a space is an example of a Segal space groupoid (Definition \ref{Def Segal space groupoid}).
\par
Notice that the homotopy category of this Segal space is the category which has objects the points in $K$ and has morphisms
homotopy classes of paths in $K$. This category is commonly called the {\it fundamental groupoid} of $K$ and is denoted by
$\displaystyle \Pi (K)$.
\end{exone}
\begin{exone} \label{Ex Gn not a Segal space}
Let us see one non-example. The simplicial space $G(n)$ is not a Segal space, although it is Reedy fibrant.
For the case $n=2$ we can see this directly as $G(2)$ is the following simplicial space:
\begin{center}
\mathscr{M}athscr{S}impset{\{0,1,2 \}}{\{00,01,11,12,22 \}}{\{ 000,001,011,111,112,122,222 \} }{d_0}{d_1}{d_0}{d_2}
\end{center}
where the numbers indicate how the simplicial maps act. Thus $d_i$ drops the $i$th digit. So, we have
$$G(2)_1 \underset{G(2)_0}{\times} G(2)_1 = \{ (00,00),(00,01),(01,11),(01,12),(11,11),(11,12),(12,22),(22,22)\}$$
So, clearly $G(2)$ is not equivalent to $G(2)_1 \times_{G(2)_0} G(2)_1$ as $G(2)$ has $7$ elements and the other has $8$ elements.
\par
Concretely, $G(2)_1 \times_{G(2)_0} G(2)_1$ has the element $(01,12)$ which wants to be composed to a $012$ in $G(2)_2$, which is
the element in $F(2)_2$ that is missing in $G(2)_2$.
\end{exone}
\mathscr{M}athscr{S}ubsection{Why are Segal Spaces not Enough?} \label{Subsec Why are Segal Spaces not Enough}
Until now we defined Segal spaces and showed how we can use them to define
all kinds of categorical concepts. In the last subsection we will see where a Segal space falls short
of what we expect.
\par
The problem with a Segal space can be summarized thusly: A Segal space has a category theory and has a homotopy theory,
however, they are not compatible with each other which causes major problems.
We will lay out the case in several examples that focus on a central theme.
Before we can do so we have to discuss one important construction.
\begin{constrone} \label{Constr E one}
Let $I(1)$ be the category which has two objects and one invertible arrow.
We want to carefully understand the Segal space $i_F^*(NI(1))$, which we will denote as $E(1)$. Clearly it is a discrete simplicial space.
We can describe it explicitly as
$$E(1)_n = \{ x , y \}^{[n-1]}.$$
More concretely an element in the set $E(1)_n$ is a map from the set $\{ 0, ..., n-1 \}$ to the set $\{ x,y \}$.
Thus $E(1)_n$ has exactly $2^n$ elements. At the lower levels we can give a more explicit description.
\par
$E(1)_0$ has two elements, $x$ and $y$, which correspond to the two objects in $I(1)$. $E(1)_1$ has four elements which can be depicted as
$\{xx,xy, yx, yy \} $, where $xx$ and $yy$ correspond to the identity map, and $xy$ is a morphism from $x$ to $y$ that has inverse $yx$.
This trend continues in the higher levels.
\end{constrone}
\begin{exone}
The category $I(1)$ is equivalent to the category $[0]$, which has only one object.
However, the corresponding Segal spaces $i_F^*(I(1)) = E(1)$ and $i_F^*(\mathscr{M}athcal{D}elta[0]) = F(0)$ are clearly not
equivalent Segal spaces, as $E(1)$ is not level-wise contractible.
\end{exone}
\begin{intone}
What essentially happened here is that the category theory has an underlying homotopy theory of groupoids ($I(1)$ is a groupoid),
which is completely ignored and thus missed by the Segal space.
\end{intone}
\begin{exone}
Let us go back to $E(1)$ once more. It is a discrete Segal space, with two objects $x,y$.
Moreover, it has two morphisms, $xy, yx$ which are inverses of each other. Thus the two objects are equivalent to each other in the sense that there
is a homotopy equivalence between them. However, they are NOT equivalent in the space $E(1)_0$, as there is no path between them.
\end{exone}
\begin{intone}
Here we see a clear mismatch between homotopy theory and category theory. Categorically the two points are equivalent, but homotopically
they are not.
\end{intone}
\begin{exone}
One familiar fact from category theory is the following. A functor $F: \mathscr{M}athcal{C} \to \mathscr{M}athcal{D}$ is an equivalence if and only if:
\begin{enumerate}
\item $F$ is fully faithful, meaning that for any two objects $x,y$
$$Hom_{\mathscr{M}athcal{C}}(x,y) \to Hom_{\mathscr{M}athcal{D}}(Fx,Fy)$$
is a bijection.
\item $F$ is essentially surjective, meaning that for any object $d \in \mathscr{M}athcal{D}$, there is an object $c \in \mathscr{M}athcal{C}$ such that $Fc$ is equivalent to $d$.
\end{enumerate}
However, this condition does not behave well for Segal spaces. As a clear example, any map $F(0) \to E(1)$ satisfies both conditions
stated above, however we do not get an equivalence of Segal spaces.
\end{exone}
\begin{intone}
As in the previous example, the problem is that $x$ and $y$ are equivalent in the Segal space $E(1)$, but as points in the space
$E(1)_0$ they are not homotopic.
\end{intone}
\begin{exone}
In Definition \ref{Def Segal space groupoid} we defined a Segal groupoid as a Segal space in which every morphism is an equivalence.
In Example \ref{Ex Spaces are Segal spaces} we discussed how every spaces gives us a Segal groupoid.
However, the opposite is not true, as indicated by the existence of $E(1)$, which is a Segal groupoid, but
not equivalent to a space.
\end{exone}
\begin{intone} \label{Int Homotopy Hypothesis}
This example is contrary to our understanding of higher category theory.
Intuitively, a higher category has homotopical data and categorical data. However, in a groupoid every morphism
is invertible, which means it does not contain any non-trivial categorical data. Therefore, our notion of
groupoid should really correspond to just a space. The idea we just explained is commonly called the
{\it homotopy hypothesis} and is one guiding idea in the realm of higher category theory.
\end{intone}
Seeing those examples we realize that we need to impose one additional condition to make sure the homotopy and category theory
of groupoid work well with each other.
\mathscr{M}athscr{S}ection{Complete Segal Spaces} \label{Section Complete Segal Spaces}
The goal of this section is to define and study complete Segal spaces.
The notion of a complete Segal space relies on the notion of {\it completeness} that was defined in \cite[Section 6]{Re01}.
\mathscr{M}athscr{S}ubsection{Defining Complete Segal Spaces} \label{Subsec Defining Complete Segal Spaces}
In order to define a complete Segal space we first need to review some concepts related to homotopy equivalences.
\begin{constrone}
Recall that for any Segal space $T$ we get the space $T_{hoequiv}$, which is the subspace of $T_1$ consisting of
all homotopy equivalences. There is a natural map $s_0: T_0 \to T_1$, which takes each object to the identity map.
However an identity map is a homotopy equivalence. Thus the map will factor through $T_{hoequiv}$.
which gives us following diagram
$$T_0 \xrightarrow{ \ \ s_0 \ \ } T_{hoequiv} \xrightarrow{ \ \ i \ \ } T_1.$$
\end{constrone}
The map $s_0$ is always an injection, however, it does not have to be surjective.
\begin{exone}
In the Segal space $E(1)$, we have two objects and so $E(1)_0 = \{ x,y \}$, but four homotopy equivalences
$\{ xx, xy, yx, yy \}$ (Construction \ref{Constr E one}).
Thus the map from objects to equivalences
is clearly not surjective.
\end{exone}
In order to fix this problem we give the next definition.
\begin{defone}
A \emph{complete Segal space (CSS)} is a Segal space $W$ for which the map
$$s_0 : W_0 \to W_{hoequiv}$$
described above is an equivalence.
\end{defone}
There are several other equivalent ways to define a complete Segal space.
\begin{lemone} \label{Lemma Completeness Conditions}
Let $W$ be a Segal space. The following are equivalent.
\begin{enumerate}
\item $W$ is a complete Segal space.
\item The following is a homotopy pullback square.
\begin{center}
\pbsq{W_0}{W_3}{W_1}{W_1 \underset{W_0}{\times} W_1 \underset{W_0}{\times} W_1}{}{}{}{}
\end{center}
\item The map of spaces
$$Map(E(1),W) \to Map(F(0),W)$$
is a weak equivalence.
\item For any two objects $x,y$ the natural map
$$ \mathscr{M}athcal{D}elta[0] \underset{W_0}{\times} W_0^{\mathscr{M}athcal{D}elta[1]} \underset{W_0}{\times} \mathscr{M}athcal{D}elta[0] \to hoequiv_W(x,y)$$
is a equivalence of spaces.
\end{enumerate}
\end{lemone}
\begin{proof}
The proof is mostly comparing various definitions.
{\it $(1 \Longleftrightarrow 2)$}
The actual pullback is the space $W_{hoeqchoice}$, which is equivalent to $W_{hoequiv}$.
Thus the diagram is a homotopy pullback if and only if $W_0$ is equivalent to $W_{hoequiv}$, which is exactly the completeness condition.
{\it $(1 \Longleftrightarrow 3)$}
The space $W_{hoequiv}$ is equivalent to $Map(E(1),W)$, thus $W$ is complete if and only if
$$W_{hoequiv} \mathscr{M}athscr{S}imeq Map(E(1),W) \to Map(F(0),W) = W_0.$$
{\it $(1 \Longleftrightarrow 4)$}
The map $W_0 \to W_{hoequiv}$ is an equivalence if and only if for each two points $x,y$ the map
$$ \mathscr{M}athcal{D}elta[0] \underset{W_0}{\times} W_0^{\mathscr{M}athcal{D}elta[1]} \underset{W_0}{\times} \mathscr{M}athcal{D}elta[0] \to
\mathscr{M}athcal{D}elta[0] \underset{W_{hoequiv}}{\times} (W_{hoequiv})^{\mathscr{M}athcal{D}elta[1]} \underset{W_{hoequiv}}{\times} \mathscr{M}athcal{D}elta[0]= hoequiv_W(x,y).$$
\end{proof}
\begin{intone} \label{Int CSS}
The completeness condition exactly addresses the problems we raised in Subsection \ref{Subsec Why are Segal Spaces not Enough}.
By adding the condition that every equivalence in the Segal space in $W$ can be represented by a path in $W_1$, we are making sure
that homotopic points in $W_0$ correspond to equivalent points in the Segal space $W_1$.
\par
Thus a complete Segal space is now a bisimplicial set where
\begin{enumerate}
\item The vertical axis has a homotopical behavior (Reedy fibrancy condition)
\item The horizontal axis has a categorical behavior (Segal condition)
\item The two interact well with each other (Completeness condition)
\end{enumerate}
This is exactly the definition we had been working towards from the start.
\end{intone}
\begin{notone}
Henceforth we will use the short form CSS to describe a complete Segal space.
\end{notone}
CSS satisfy several helpful conditions, some of which are analogues to the conditions a Segal space satisfied and some of which
correct the problems we brought up in Subsection \ref{Subsec Why are Segal Spaces not Enough}.
\begin{remone}
Let $W$ be a CSS and $X$ any simplicial space. Then $W^X$ is also a CSS.
\end{remone}
\begin{theone}
Let $f: W \to V$ be a map of CSS. The following are equivalent:
\begin{enumerate}
\item $f$ is a level-wise equivalence, meaning $f_n: W_n \to V_n$ is an equivalence of spaces.
\item $f$ is fully faithful and essentially surjective.
\begin{itemize}
\item[(I)] Fully Faithful: For any two objects $x,y \in W$ the induced map of spaces
$$map_W(x,y) \to map_V(fx,fy)$$
is an equivalence of spaces.
\item[(II)] Essentially Surjective: For any object $y \in V$ there is an object $x \in W$ such that $fx$ is equivalent to $y$ in $V$.
\end{itemize}
\end{enumerate}
\end{theone}
In Definition \ref{Def Segal space groupoid} we defined a Segal space groupoid. Analogously we can define a complete Segal space groupoid
as a CSS in which every morphism is an equivalence. The next proposition confirms the homotopy hypothesis
we discussed in Intuition \ref{Int Homotopy Hypothesis}.
\begin{propone}
A CSS W is a CSS groupoid if and only if $W$ is homotopically constant.
\end{propone}
\begin{proof}
If $W$ is homotopically constant then $s_0:W_0 \to W_1$ is an equivalence of spaces. This means that every map is an equivalence.
On the other side, if $W$ is a CSS groupoid then $s_0: W_0 \to W_1$ is an equivalence. This means that the maps $d_0,d_1: W_1 \to W_0$
are also equivalences. This implies that $W_1 \times_{W_0} ... \times_{W_0} W_1 \mathscr{M}athscr{S}imeq W_1$.
By the Segal condition this implies that $W_n \mathscr{M}athscr{S}imeq W_n$ and so $W$ is homotopically constant.
\end{proof}
Our next goal is to discuss how we can build a CSS out of a category. Until now we used the horizontal embedding of the nerve
, $i^*_FN$, however, while it does give us a Segal space, it might not be complete.
As we already discussed before $E(1)$ is not a CSS as the two objects are equivalent in the Segal space, but not connected by a path in
$E(1)_0$. Thus we have to completely change our approach.
\par
The problem is that the embedding functor $i_F^*$ only considers the categorical aspect of the underlying category, but completely ignores
the homotopy theory. Thus there is no way to get a CSS. The way to adjust things is to consider the category and homotopy theory at the
same time. In order to achieve that we need a completely new construction.
Before we can do so we need several important definitions.
\begin{defone} \label{Def Relative Category}
A relative category $(\mathscr{M}athcal{C},W)$ is a category $\mathscr{M}athcal{C}$ along with a subcategory $W$ that satisfies following conditions:
\begin{enumerate}
\item Every object in $\mathscr{M}athcal{C}$ is an object in $W$.
\item Every isomorphism in $\mathscr{M}athcal{C}$ is a morphism in $W$.
\end{enumerate}
\end{defone}
\begin{intone} \label{Int Relative Category}
Intuitively, a relative category is a category that has some homotopical information in the sense that morphisms that are in $W$
play the role of ``weak equivalences". Notice, there is no notion of homotopy and these maps are not invertible up to some homotopy condition.
Rather, this is an intuition on how think about relative categories.
\end{intone}
\begin{defone}
Let $\mathscr{M}athcal{C}$ be a category. We define $\mathscr{M}athcal{C}^{core}$ as the category which has the same objects, but only has invertible morphisms between any two objects.
By definition it is the maximal subcategory of $\mathscr{M}athcal{C}$ that is a groupoid, or, in other words, the maximal subgroupoid of $\mathscr{M}athcal{C}$.
\end{defone}
\begin{exone}
Let $\mathscr{M}athcal{C}$ be any category, then $(\mathscr{M}athcal{C},\mathscr{M}athcal{C}^{core})$ is a relative category.
\end{exone}
\begin{defone} \label{Def WE definition}
Let $(\mathscr{M}athcal{C},W)$ be a relative category and $\mathscr{M}athcal{D}$ any category. Then we define the category $we(\mathscr{M}athcal{C}^{\mathscr{M}athcal{D}})$ as the category
which has as objects functors $F: \mathscr{M}athcal{D} \to \mathscr{M}athcal{C}$ and as morphisms natural transformations $\alpha$ such that
for every object $d$, the morphisms $\alpha_d \in \mathscr{M}athcal{C}$ is actually a morphism in $W$.
\end{defone}
\begin{constrone} \label{Constr Classification Diagram}
We are now in a position to construct a new and improved version of the nerve construction that takes the homotopy theory
of a category into account. Right now we will do our construction for a relative category.
\par
Let $(\mathscr{M}athcal{C},W)$ be a relative category. We define a simplicial space $\mathscr{M}athscr{N}(\mathscr{M}athcal{C},W)$ as follows.
$$\mathscr{M}athscr{N}(\mathscr{M}athcal{C},W)_n = N(we(\mathscr{M}athcal{C}^{[n]}))$$
The necessary simplicial maps of the simplicial space follow from the maps between the various $[n]$.
We call $\mathscr{M}athscr{N}(\mathscr{M}athcal{C},W)$ the {\it classification diagram} of the relative category $(\mathscr{M}athcal{C},W)$.
\end{constrone}
\begin{intone} \label{Int Classification Diagram}
This construction is a good illustration of what a CSS is and so it is worth dwelling over.
For a relative category $(\mathscr{M}athcal{C},W)$ we use following notation:
\begin{enumerate}
\item $\bullet$: For objects
\item $\longrightarrow$: For morphisms
\item $ \xrightarrow{ \ \ \mathscr{M}athscr{S}im \ \ }$: For morphisms that are in the subcategory $W$.
\end{enumerate}
With this notation the classification diagram has the form of the following simplicial space:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
\left \{
\bullet
\right \}
\arrow[d, shorten >=1ex,shorten <=1ex] \arrow[r, shorten >=1ex,shorten <=1ex]
&
\left \{
\begin{tabular}{c}
$\bullet$
\end{tabular}
\begin{tabular}{c}
$\rightarrow$
\end{tabular}
\begin{tabular}{c}
$\bullet$
\end{tabular}
\right \}
\arrow[d, shorten >=1ex,shorten <=1ex]
\arrow[l, shift left=1.2] \arrow[l, shift right=1.2]
\arrow[r, shift right, shorten >=1ex,shorten <=1ex ] \arrow[r, shift left, shorten >=1ex,shorten <=1ex]
&
\left \{
\begin{tabular}{c}
$\bullet$
\end{tabular}
\begin{tabular}{c}
$\rightarrow$
\end{tabular}
\begin{tabular}{c}
$\bullet$
\end{tabular}
\begin{tabular}{c}
$\rightarrow$
\end{tabular}
\begin{tabular}{c}
$\bullet$
\end{tabular}
\right \}
\arrow[d, shorten >=1ex,shorten <=1ex]
\arrow[l] \arrow[l, shift left=2] \arrow[l, shift right=2]
\arrow[r, shorten >=1ex,shorten <=1ex] \arrow[r, shift left=2, shorten >=1ex,shorten <=1ex] \arrow[r, shift right=2, shorten >=1ex,shorten <=1ex]
&
\cdots
\arrow[l, shift right=1] \arrow[l, shift left=1] \arrow[l, shift right=3] \arrow[l, shift left=3]
\\
\left \{ \begin{tabular}{c}
$\bullet$ \\
$\downarrow$ \\
$\bullet$
\end{tabular}
\hspace{-0.15in}
\begin{tabular}{c}
\mathscr{M}athscr{S}trut \\
$\mathscr{M}athscr{S}im$ \\
\mathscr{M}athscr{S}trut
\end{tabular}
\hspace{-0.1in}
\right \}
\arrow[d, shift right, shorten >=1ex,shorten <=1ex ] \arrow[d, shift left, shorten >=1ex,shorten <=1ex]
\arrow[u, shift left=1.2] \arrow[u, shift right=1.2] \arrow[r, shorten >=1ex,shorten <=1ex]
&
\left \{ \begin{tabular}{c}
$\bullet$ \\
$\downarrow$ \\
$\bullet$
\end{tabular}
\hspace{-0.15in}
\begin{tabular}{c}
\mathscr{M}athscr{S}trut \\
$\mathscr{M}athscr{S}im$ \\
\mathscr{M}athscr{S}trut
\end{tabular}
\hspace{-0.1in}
\begin{tabular}{c}
$\rightarrow$ \\
\\
$\rightarrow$
\end{tabular}
\begin{tabular}{c}
$\bullet$ \\
$\downarrow$ \\
$\bullet$
\end{tabular}
\hspace{-0.15in}
\begin{tabular}{c}
\mathscr{M}athscr{S}trut \\
$\mathscr{M}athscr{S}im$ \\
\mathscr{M}athscr{S}trut
\end{tabular}
\hspace{-0.1in}
\right \}
\arrow[d, shift right, shorten >=1ex,shorten <=1ex ] \arrow[d, shift left, shorten >=1ex,shorten <=1ex]
\arrow[u, shift left=1.2] \arrow[u, shift right=1.2]
\arrow[l, shift left=1.2] \arrow[l, shift right=1.2]
\arrow[r, shift right, shorten >=1ex,shorten <=1ex ] \arrow[r, shift left, shorten >=1ex,shorten <=1ex]
&
\left \{ \begin{tabular}{c}
$\bullet$ \\
$\downarrow$ \\
$\bullet$
\end{tabular}
\hspace{-0.15in}
\begin{tabular}{c}
\mathscr{M}athscr{S}trut \\
$\mathscr{M}athscr{S}im$ \\
\mathscr{M}athscr{S}trut
\end{tabular}
\hspace{-0.1in}
\begin{tabular}{c}
$\rightarrow$ \\
\\
$\rightarrow$
\end{tabular}
\begin{tabular}{c}
$\bullet$ \\
$\downarrow$ \\
$\bullet$
\end{tabular}
\hspace{-0.15in}
\begin{tabular}{c}
\mathscr{M}athscr{S}trut \\
$\mathscr{M}athscr{S}im$ \\
\mathscr{M}athscr{S}trut
\end{tabular}
\hspace{-0.1in}
\begin{tabular}{c}
$\rightarrow$ \\
\\
$\rightarrow$
\end{tabular}
\begin{tabular}{c}
$\bullet$ \\
$\downarrow$ \\
$\bullet$
\end{tabular}
\hspace{-0.15in}
\begin{tabular}{c}
\mathscr{M}athscr{S}trut \\
$\mathscr{M}athscr{S}im$ \\
\mathscr{M}athscr{S}trut
\end{tabular}
\hspace{-0.1in}
\right \}
\arrow[d, shift right, shorten >=1ex,shorten <=1ex ] \arrow[d, shift left, shorten >=1ex,shorten <=1ex]
\arrow[u, shift left=1.2] \arrow[u, shift right=1.2]
\arrow[l] \arrow[l, shift left=2] \arrow[l, shift right=2]
\arrow[r, shorten >=1ex,shorten <=1ex] \arrow[r, shift left=2, shorten >=1ex,shorten <=1ex] \arrow[r, shift right=2, shorten >=1ex,shorten <=1ex]
&
\cdots
\arrow[l, shift right=1] \arrow[l, shift left=1] \arrow[l, shift right=3] \arrow[l, shift left=3]
\\
\left \{
\begin{tabular}{c}
$\bullet$ \\
$\downarrow$ \\
$\bullet$ \\
$\downarrow$ \\
$\bullet$
\end{tabular}
\hspace{-0.15in}
\begin{tabular}{c}
\mathscr{M}athscr{S}trut \\
$\mathscr{M}athscr{S}im$ \\
\mathscr{M}athscr{S}trut \\
$\mathscr{M}athscr{S}im$ \\
\mathscr{M}athscr{S}trut
\end{tabular}
\hspace{-0.1in}
\right \}
\arrow[d, shorten >=1ex,shorten <=1ex] \arrow[d, shift left=2, shorten >=1ex,shorten <=1ex] \arrow[d, shift right=2, shorten >=1ex,shorten <=1ex]
\arrow[u] \arrow[u, shift left=2] \arrow[u, shift right=2]
\arrow[r, shorten >=1ex,shorten <=1ex]
&
\left \{
\begin{tabular}{c}
$\bullet$ \\
$\downarrow$ \\
$\bullet$ \\
$\downarrow$ \\
$\bullet$
\end{tabular}
\hspace{-0.15in}
\begin{tabular}{c}
\mathscr{M}athscr{S}trut \\
$\mathscr{M}athscr{S}im$ \\
\mathscr{M}athscr{S}trut \\
$\mathscr{M}athscr{S}im$ \\
\mathscr{M}athscr{S}trut
\end{tabular}
\hspace{-0.1in}
\begin{tabular}{c}
$\rightarrow$ \\
\\
$\rightarrow$ \\
\\
$\rightarrow$
\end{tabular}
\begin{tabular}{c}
$\bullet$ \\
$\downarrow$ \\
$\bullet$ \\
$\downarrow$ \\
$\bullet$
\end{tabular}
\hspace{-0.15in}
\begin{tabular}{c}
\mathscr{M}athscr{S}trut \\
$\mathscr{M}athscr{S}im$ \\
\mathscr{M}athscr{S}trut \\
$\mathscr{M}athscr{S}im$ \\
\mathscr{M}athscr{S}trut
\end{tabular}
\hspace{-0.1in}
\right \}
\arrow[d, shorten >=1ex,shorten <=1ex] \arrow[d, shift left=2, shorten >=1ex,shorten <=1ex] \arrow[d, shift right=2, shorten >=1ex,shorten <=1ex]
\arrow[u] \arrow[u, shift left=2] \arrow[u, shift right=2]
\arrow[l, shift left=1.2] \arrow[l, shift right=1.2]
\arrow[r, shift right, shorten >=1ex,shorten <=1ex ] \arrow[r, shift left, shorten >=1ex,shorten <=1ex]
&
\left \{
\begin{tabular}{c}
$\bullet$ \\
$\downarrow$ \\
$\bullet$ \\
$\downarrow$ \\
$\bullet$
\end{tabular}
\hspace{-0.15in}
\begin{tabular}{c}
\mathscr{M}athscr{S}trut \\
$\mathscr{M}athscr{S}im$ \\
\mathscr{M}athscr{S}trut \\
$\mathscr{M}athscr{S}im$ \\
\mathscr{M}athscr{S}trut
\end{tabular}
\hspace{-0.1in}
\begin{tabular}{c}
$\rightarrow$ \\
\\
$\rightarrow$ \\
\\
$\rightarrow$
\end{tabular}
\begin{tabular}{c}
$\bullet$ \\
$\downarrow$ \\
$\bullet$ \\
$\downarrow$ \\
$\bullet$
\end{tabular}
\hspace{-0.15in}
\begin{tabular}{c}
\mathscr{M}athscr{S}trut \\
$\mathscr{M}athscr{S}im$ \\
\mathscr{M}athscr{S}trut \\
$\mathscr{M}athscr{S}im$ \\
\mathscr{M}athscr{S}trut
\end{tabular}
\hspace{-0.1in}
\begin{tabular}{c}
$\rightarrow$ \\
\\
$\rightarrow$ \\
\\
$\rightarrow$
\end{tabular}
\begin{tabular}{c}
$\bullet$ \\
$\downarrow$ \\
$\bullet$ \\
$\downarrow$ \\
$\bullet$
\end{tabular}
\hspace{-0.15in}
\begin{tabular}{c}
\mathscr{M}athscr{S}trut \\
$\mathscr{M}athscr{S}im$ \\
\mathscr{M}athscr{S}trut \\
$\mathscr{M}athscr{S}im$ \\
\mathscr{M}athscr{S}trut
\end{tabular}
\hspace{-0.1in}
\right \}
\arrow[d, shorten >=1ex,shorten <=1ex] \arrow[d, shift left=2, shorten >=1ex,shorten <=1ex] \arrow[d, shift right=2, shorten >=1ex,shorten <=1ex]
\arrow[u] \arrow[u, shift left=2] \arrow[u, shift right=2]
\arrow[l] \arrow[l, shift left=2] \arrow[l, shift right=2]
\arrow[r, shorten >=1ex,shorten <=1ex] \arrow[r, shift left=2, shorten >=1ex,shorten <=1ex] \arrow[r, shift right=2, shorten >=1ex,shorten <=1ex]
&
\cdots
\arrow[l, shift right=1] \arrow[l, shift left=1] \arrow[l, shift right=3] \arrow[l, shift left=3]
\\
\ \vdots \
\arrow[u, shift right=1] \arrow[u, shift left=1] \arrow[u, shift right=3] \arrow[u, shift left=3]
& \ \vdots \
\arrow[u, shift right=1] \arrow[u, shift left=1] \arrow[u, shift right=3] \arrow[u, shift left=3]
& \ \vdots \
\arrow[u, shift right=1] \arrow[u, shift left=1] \arrow[u, shift right=3] \arrow[u, shift left=3]
&
\end{tikzcd}
\end{center}
Thus the vertical direction focuses on the subcategory $W$, which we think of as the homotopical direction, whereas the horizontal
direction focuses on the whole category which is clearly the categorical direction.
\end{intone}
In particular, we have following important special case.
\begin{exone}
Let $\mathscr{M}athcal{C}$ be a category. We define the {\it classifying diagram} of $\mathscr{M}athcal{C}$,
$$\mathscr{M}athscr{N} \mathscr{M}athcal{C} = \mathscr{M}athscr{N}(\mathscr{M}athcal{C}, \mathscr{M}athcal{C}^{core}) . $$
\end{exone}
\begin{remone}
There is a more concrete way to define the classifying diagram of a category $\mathscr{M}athcal{C}$.
Let $I(n)$ be the category with $n+1$ objects and exactly one isomorphism between every two objects.
We can define the $(n,m)$-simplices directly as follows:
$$ \mathscr{M}athscr{N}(\mathscr{M}athcal{C})_{(n,l)} = Hom_{\mathscr{C}\text{at}}([n] \times I(l), \mathscr{M}athcal{C}).$$
\end{remone}
\begin{intone}
It is very helpful to consider the diagram above for this case. In particular, at the zero level $\mathscr{M}athscr{N} (\mathscr{M}athcal{C} )_0$ is just the core $\mathscr{M}athcal{C}^{core}$
as it is just the subcategory of all isomorphisms. Thus two objects are equivalent in the category if and only if there is a path
in between them at the zero level.
\end{intone}
\begin{remone}
The notion of a classification diagram was introduced in \cite[Section 3]{Re01} as an improvement to the nerve construction.
\end{remone}
Having improved our definition of a nerve, we have following result.
\begin{theone}
\cite[Equation 3.6, Lemma 3.9, Proposition 6.1]{Re01}
Let $\mathscr{M}athcal{C}$ be a category, then $\mathscr{M}athscr{N}(\mathscr{M}athcal{C})$ is a CSS.
\end{theone}
\begin{proof}
The Reedy fibrancy condition is a technical condition and the proof can be found in \cite[Lemma 3.6]{Re01}.
The Segal condition follows from the fact that the simplicial set $N\mathscr{M}athcal{C}$ satisfies the Segal condition.
Thus $\mathscr{M}athscr{N}(\mathscr{M}athcal{C})$ satisfies the Segal condition level-wise.
\par
For the last part notice that a morphism in $\mathscr{M}athscr{N}(\mathscr{M}athcal{C})$ is an equivalence if it is an isomorphism in $\mathscr{M}athcal{C}$.
Thus $\mathscr{M}athscr{N}(\mathscr{M}athcal{C})_{hoequiv}$ inside $\mathscr{M}athscr{N}(\mathscr{M}athcal{C})_1 = N((\mathscr{M}athcal{C}^{[1]})^{core})$ is equivalent to the subcategory $N(\mathscr{M}athcal{C}^{I(1)})^{core})$.
Moreover, we know that $\mathscr{M}athcal{C}^{core}$ is categorically equivalent to $(\mathscr{M}athcal{C}^{I(1)})^{core}$.
Thus we have following diagram of equivalences
$$\mathscr{M}athscr{N}(\mathscr{M}athcal{C})_0 = N(\mathscr{M}athcal{C}^{core}) \cong N(\mathscr{M}athcal{C}^{I(1)})^{core}) \cong (\mathscr{M}athscr{N} \mathscr{M}athcal{C})_{hoequiv} $$
Hence, $\mathscr{M}athscr{N}(\mathscr{M}athcal{C})$ satisfies the completeness condition.
\end{proof}
\mathscr{M}athscr{S}ection{Functoriality in Higher Categories: Fibrations}
\label{Sec Functoriality in Higher Categories Fibrations}
When studying categories, it does not suffice to just study them in isolation, rather we also want to understand how
they relate to each other. That is why we introduce functors. We want to do the same in the realm of higher category theory.
The goal of this section is to study functoriality of higher categories via the theory of fibrations.
We start by motivating and reviewing the theory of fibrations and then move on to see some interesting examples.
\mathscr{M}athscr{S}ubsection{Why Functors Fail in Higher Categories}
\label{Subsec Why Functors Fail in Higher Categories}
In category theory functors are used effectively to understand the relation between categories.
However, as is often the case, the approach we are familiar with does not work for higher categories and needs to be refined.
This is best witnessed by the following example.
\begin{exone} \label{Ex Comp prevents Func}
Let $W$ be a CSS and $x$ an object in $W$. From our experience with classical category theory, we expect a functor
$$W \to Spaces$$
$$y \mathscr{M}apsto map_W(x,y)$$
Clearly, we can define the map above on an object level, but we also need a way to deal with the functoriality.
In particular, for any map $f: y_1 \to y_2$ we need a corresponding map
$$map_W(x,y_1) \to map_W(x,y_2).$$
But all we have is the following diagram
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
& map_W(x,y_1,y_2) \arrow[d, twoheadrightarrow, "\mathscr{M}athscr{S}imeq"] \arrow[r] & map_W(x,y_2) \\
map_W(x,y_1) \arrow[r, "(id , f)"] & map_W(x,y_1) \times map(y_1,y_2)
\end{tikzcd}
\end{center}
which means we have no direct map from $map_W(x,y_1)$ to $map_W(x,y_2)$, but rather a zig-zag.
Thus we cannot just define a functor from $W$ to the CSS of $Spaces$.
\end{exone}
\begin{intone}
It's clear where the problem lies: composition. In a classical category composition is unique and leaves us with no choice, which allows us
to define a functor. In a CSS composition is only defined up to contractible choice and that choice prevents us from actually getting a
functor.
\end{intone}
\begin{remone}
Note we never actually defined the CSS of spaces. However, the problem we described above arises regardless of how we define this CSS.
\end{remone}
\begin{remone}
Because of this issue we also cannot just generalize the Yoneda lemma to higher categories, as we need a proper notion of
representable functors first.
\end{remone}
When we can relax our composition condition in the world of higher categories,
not why not relax the functoriality condition as well?
This leads us to the study of fibrations.
\mathscr{M}athscr{S}ubsection{Fibrations in Categories} \label{Subsec Fibrations in Categories}
Our goal is to relax the functoriality condition in order to get a functioning notion of a functor suited for higher categories.
Unfortunately, that is quite difficult for functors. Fortunately, however, there is a different way to think about functors,
which can readily be generalized to a more general setting, namely the theory of {\it fibrations}. Thus in this
subsection we review the theory of fibrations for ordinary categories.
\begin{defone}
A functor $p:\mathscr{M}athcal{D} \to \mathscr{M}athcal{C}$ is cofibered in sets if for any map $f: x \to y$ in $\mathscr{M}athcal{C}$ and any object $x' \in \mathscr{M}athcal{D}$ such that $p(x') = x$,
there exists a {\it unique} lift $f':x' \to y'$ in $\mathscr{M}athcal{D}$, such that $p(f') = f$.
\end{defone}
\begin{remone}
If we let $f = id_x \in \mathscr{M}athcal{C}$ and choose a target $x' \in D$ such that $p(x') = x$, then clearly $p(id_{x'}) = id_x$.
Thus by uniqueness, the fiber over each $x$ has to be a set.
This in particular implies that any functor $p: \mathscr{M}athcal{D} \to \mathscr{M}athcal{D}elta[0]$ is cofibered in sets if and only if $\mathscr{M}athcal{D}$ is a set.
\end{remone}
\begin{intone} \label{Int Cat Fib in Set}
How does this definition model functoriality? We can see this by taking $\mathscr{M}athcal{C} = [1]$ and analyzing a functor
$p: \mathscr{M}athcal{D} \to [1]$ that is cofibered over sets.
First some notation. Let $S_0$ be the subcategory of $\mathscr{M}athcal{D}$ that maps to $0 \in [1]$ and $S_1$ be the subcategory
that maps to $1 \in \mathscr{M}athcal{D}elta[1]$. By the previous remark both of those are sets.
\par
Now, the fibration conditions says that for every choice of point $x \in S_0$, there is a unique map in the category $\mathscr{M}athcal{D}$ that starts at $x$
and ends with an object $y \in S_1$. Thus we have a unique way to assign a value in $S_1$ for every point in $S_0$.
That is the definition of a function of sets $f: S_0 \to S_1$.
We can depict this situation as follows.
\begin{center}
\begin{tikzcd}[row sep=0.1in, column sep=0.1in]
& x \bullet \arrow[drrrrrrrrrrrrrrrrr, "f", mapsto]
& \bullet & \bullet & & & & & & & & & & & & & & & \hspace{0.05in}\bullet & \bullet \\
\bullet & \hspace{0.05in}\bullet & & & & & & & & & & & & & & & & \bullet & y\bullet & \\
& \hspace{0.05in}\bullet \arrow[dddddd, Rightarrow] & & & & & & & & & & & & & & & & \bullet & \hspace{0.05in}\bullet \arrow[dddddd, Rightarrow] & \\
\\ \\ \\ \\ \\
& 0 \arrow[rrrrrrrrrrrrrrrrr, "01"] & & & & & & & & & & & & & & & & & 1 & \\
\end{tikzcd}
\end{center}
Having a function between sets, we can now define the functor
$$[1] \to \mathscr{M}athscr{S}et$$
that maps $0$ to the set $S_0$, $1$ to the set $S_1$ and morphism $01$ to the map $f: S_0 \to S_1$.
\end{intone}
\begin{exone}
The key example for a functor that is fibered in sets is the notion of an under-category.
For each object $c \in \mathscr{M}athcal{C}$, we have the over category $\mathscr{M}athcal{C}_{c/}$, which has objects all maps $c \to d$ (maps with domain $c$)
and morphisms commuting triangles of the form:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
& c \arrow[dr] \arrow[dl] & \\
d_1 \arrow[rr, "f"] & & d_2
\end{tikzcd}
\end{center}
The natural projection map $p: \mathscr{M}athcal{C}_{c/} \to \mathscr{M}athcal{C}$, which takes $c \to d$ to $d$, is fibered in sets. Indeed, if we take any map
$f: d_1 \to d_2$ in $\mathscr{M}athcal{C}$ and chosen lift $c \to d_1$ there is a unique arrow in $C_{c/}$ namely
the triangle depicted above that lifts $f$ to the category $C_{c/}$.
\end{exone}
There is a much more rigorous way to see how to get a category cofibered sets out of a functor.
This method is commonly called the {\it Grothendieck construction}.
\begin{defone}
Let $F: \mathscr{M}athcal{C} \to \mathscr{M}athscr{S}et$ be a functor. We define the category $\displaystyle \int_{\mathscr{M}athcal{C}} F$, called the Grothendieck construction, in the following way:
\begin{itemize}
\item {\bf Objects}: An object is a tuple $(c, x)$ such that $c \in \mathscr{M}athcal{C}$ is an object and $x \in F(c)$.
\item {\bf Morphisms}: For two objects $(c,x)$ and $(d,y)$, we define the maps as
$$Hom_{\int_{\mathscr{M}athcal{C}} F}((c,x),(y,d)) = \{ f \in Hom_{\mathscr{M}athcal{C}}(c,d): F(f)(x) = y \}. $$
Note that $F$ is a functor, thus we get a map of sets $F(f): F(c) \to F(d)$.
\end{itemize}
The category comes with a natural functor $\displaystyle p: \int_{\mathscr{M}athcal{C}} F \to \mathscr{M}athcal{C}$, which maps each tuple $(c,x)$ to $c$.
\end{defone}
Let us see one important example
\begin{exone} \label{Ex under cat cofibered}
Let $\mathscr{M}athcal{Y}_x: \mathscr{M}athcal{C} \to \mathscr{M}athscr{S}et$ be the functor represented by $c$. Applying the Grothendieck construction we get
$$\int_{\mathscr{M}athcal{C}} \mathscr{M}athcal{Y}_x = C_{x/}.$$
Thus, we can in some sense think of $C_{x/}$ as the {\it representable functor cofibered in sets}.
\end{exone}
The example suggests following result.
\begin{theone}
For any functor $\displaystyle F: \mathscr{M}athcal{C} \to \mathscr{M}athscr{S}et$, the Grothendieck construction $\displaystyle p: \int_{\mathscr{M}athcal{C}} F \to \mathscr{M}athcal{C}$ is a functor cofibered in sets.
\end{theone}
\begin{proof}
Let $f: c \to d$ be a map in $\mathscr{M}athcal{C}$ and $(c,x)$ a chosen lift of $c$ in $\displaystyle \int_{\mathscr{M}athcal{C}} F$. Then there is a unique lift of $f$,
namely $f: (c,x) \to (d,F(f)(x))$.
\end{proof}
This theorem shows that we can transfer the whole concept of set valued functors into the language of functors fibered in sets and
we even have a notion of a representable functor. This perspective on functoriality even comes with its own Yoneda lemma:
\begin{lemone}
({\it Yoneda Lemma for cofibered Categories}).
For any functor cofibered in sets $p: \mathscr{M}athcal{D} \to \mathscr{M}athcal{C}$, we have following equivalence:
$$ (F_c)^*: Fun_{\mathscr{M}athcal{C}}(\mathscr{M}athcal{C}_{c/}, \mathscr{M}athcal{D}) \xrightarrow{ \ \cong \ } Fun_{\mathscr{M}athcal{C}}( [0], \mathscr{M}athcal{D})$$
where the map comes from precomposing by the functor $F_c : [0] \to C_{c/}$, which maps the point
to the identity map $id_c: c \to c$.
\end{lemone}
The notion of functoriality introduced in this section relies on the existence of {\it unique lifts}.
But we know exactly how to adjust the notion of uniqueness to get a functioning definition for higher categories,
namely by replacing uniqueness with contractibility. The goal of the next subsection is to take that idea to show
we this gives us a well-defined notion of a functor valued in spaces.
\mathscr{M}athscr{S}ubsection{Left Fibrations} \label{Subsec Left Fibration}
The idea of a left fibration is a way to generalize functors fibered in sets we discussed above and models
functors valued in spaces.
They key is to have {\it unique lifts up to homotopy} instead of demanding unique lifts.
In other words there should be a space of lifts which is contractible.
The material in this section is a summary of \cite{Ra17a}, which is a rigorous treatment of the theory of left fibrations.
\begin{defone}
A Reedy fibration $p: L \to W$ between CSS is a left fibration if the following is a homotopy pullback square
\begin{center}
\pbsq{L_1}{L_0}{W_1}{W_0}{p_1}{s}{s}{p_0}
\end{center}
where the map $s:W_1 \to W_0$ is the source map that takes each arrow to its source.
\end{defone}
\begin{remone}
This is equivalent to saying that the map
$$L_1 \xrightarrow{ \ \ \mathscr{M}athscr{S}imeq \ \ } L_0 \underset{W_0}{\times} W_1$$
is a trivial Kan fibration.
\end{remone}
\begin{intone} \label{Int Left Fib}
How is this the proper generalization of functors cofibered in sets described in the previous subsection?
We have following pullback diagram of spaces:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
Lift(x',f) \arrow[d, "\mathscr{M}athscr{S}imeq"] \arrow[r] & L_1 \arrow[d, twoheadrightarrow, "\mathscr{M}athscr{S}imeq"] \\
\mathscr{M}athcal{D}elta[0] \arrow[r, "(x' , f)"] & L_0 \underset{W_0}{\times} W_1
\end{tikzcd}
\end{center}
The map $(x', f): \mathscr{M}athcal{D}elta[0] \to L_0 \underset{W_0}{\times} W_1$ picks a map $f: x \to y$ in $W$
(which is exacly what a point in $W_1$ is, an object $x' \in L$ (a point in $L_0$) such that $p(x') = x$.
\par
The space $Lift(x',f)$ consists of all morphisms in $f': x' \to y'$ in $L$ (points in $L_1$) such that $p(f')= f$.
The cofibered of sets condition from the previous subsection exactly stated that such a map $f'$ is unique.
In the higher categorical situation $Lift(x',f)$ is contractible as it is a pullback of an equivalence.
Thus the definition of a left fibration exactly replaces the unique lifting condition with contractible lifting condition.
\end{intone}
Let us see some important examples.
\begin{exone} \label{Lfib over point}
Let $W = F(0)$ (i.e. the point). Let $L \to F(0)$ be a left fibration over $F(0)$. Then the left fibration condition implies that
$$L_1 \xrightarrow{ \ \mathscr{M}athscr{S}imeq \ } L_0 \times_{(F(0))_0} F(0)_n = L_0.$$
So, $L$ is a just a {\it homotopically constant CSS} or in other words only has the data of a space.
Thus a left fibration over the point is really equivalent to a map from the point into spaces, which is just a space.
\end{exone}
\begin{exone} \label{Lfib over arrow}
Let us a see a more interesting example. Let $L \to F(1)$ be a left fibration over $F(1)$.
We think of $F(1)$ as the discrete simplicial space for which $F(1)_n = Hom([n],[1])$, thus for the first levels we have
\begin{center}
\mathscr{M}athscr{S}impset{\{ 0,1\}}{\{ 00,01,11\}}{\{ 000,001,011,111\}}{}{}{}{}.
\end{center}
Given that $F(1)$ is a discrete simplicial space (each level is a set), the map $L \to F(1)$ can be expressed as a disjoint union
of the fibers over each point in the underlying set.
For the purposes of this example, we express the fiber over the point $i$ as $L_{/i}$. So, the fiber over the point $001$
is expressed as $L_{/001}$.
With this notation, $L$ can be expressed as following simplicial space:
$$\mathscr{M}athscr{S}impset{L_{/0} \coprod L_{/1}}{ L_{/00} \coprod L_{/01} \coprod L_{/11} }{ L_{/000} \coprod L_{/001} \coprod L_{/011} \coprod L_{/111} }{}{}{}{}.$$
The left fibration condition implies that the following is a homotopy pullback square.
\begin{center}
\pbsq{L_{/00} \coprod L_{/01} \coprod L_{/11}}{L_{/0} \coprod L_{/1}}{\{ 00,01,11\}}{\{ 0,1\}}{}{}{}{}
\end{center}
As the spaces are themselves disjoint unions of smaller spaces, the equivalence breaks down into following three equivalences:
$$L_{/00} \xrightarrow{ \ \ \mathscr{M}athscr{S}imeq \ \ } L_{/0} \underset{ \{ 00 \} }{\times} \{ 0 \} \cong L_{/0}$$
$$L_{/01} \xrightarrow{ \ \ \mathscr{M}athscr{S}imeq \ \ } L_{/0} \underset{ \{ 01 \} }{\times} \{ 0 \} \cong L_{/0}$$
$$L_{/11} \xrightarrow{ \ \ \mathscr{M}athscr{S}imeq \ \ } L_{/1} \underset{ \{ 11 \} }{\times} \{ 1 \} \cong L_{/1}$$
Thus we can disregard $L_{/00}$ and $L_{/11}$ as they are equivalent to $L_{/0}$ and $L_{/1}$ respectively.
What remains is the following zig-zag
\begin{center}
\begin{tikzcd}[row sep=0.3in, column sep=0.5in]
& L_{/01} \arrow[dl, "s"', "\mathscr{M}athscr{S}imeq"] \arrow[dr, "t"] & \\
L_{/0} & & L_{/1}
\end{tikzcd}
\end{center}
where the left map is an equivalence by the left fibrancy condition.
So we have one non-equivalent map in this diagram
$$L_{/01} \xrightarrow{ \ \ t \ \ } L_{/1}.$$
This justifies why we think of a left fibration over $F(1)$ as a functor from the category
$\mathscr{M}athcal{D}elta[1]$ into spaces, which is exactly what we had hoped for.
\end{exone}
\begin{intone}
This example is very instructive in how a left fibration really is the appropriate notion of a functor in a higher categorical setting.
Recall in Example \ref{Ex Comp prevents Func} we described how the composition map really gives us the following zig-zag:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
& map(x,y_1,y_2) \arrow[dl, "s"', "\mathscr{M}athscr{S}imeq"] \arrow[dr, "t"] & \\
map_W(x,y_1) \times map_W(y_1,y_2) & & map_W(x,y_2)
\end{tikzcd}
\end{center}
A left fibration allows us to define functoriality while taking this zig-zag structure into account.
\end{intone}
\begin{exone} \label{Lfib over simplex}
The same example can be expanded to show that every right fibration over $F(n)$ is just the data of a functor from
$[n]$ into spaces.
\end{exone}
In Example \ref{Ex under cat cofibered} we discussed how an under-category gives us the appropriate notion of a representable functor cofibered in sets.
In the next example we will generalize this to the setting of higher categories.
\begin{defone} \label{Def UnderCSS}
Let $x \in W$ be an object. We define the {\it under-CSS} $W_{x/}$ as
$$W_{x/} = F(0) ^x\underset{W}{\times}^s W^{F(1)}.$$
\end{defone}
\begin{intone}
Let us see why we are justified in calling $W_{x/}$ the under CSS. This is easiest by looking at $(W_{x/})_0$.
We have following equivalences
$$(W_{x/})_0 = \mathscr{M}athcal{D}elta[0] \underset{W_0}{\times} W_1$$
Thus a point in $W_{x/}$ corresponds to a point in $W_1$, which is just a morphism, such that the source of that morphism is $x$.
That is exactly what we expected.
\end{intone}
\begin{exone}
For an object $x$ have following pullback square.
\begin{center}
\pbsq{W_{x/}}{W^{F(1)}}{W}{W \times W}{}{p_x}{(s , t)}{i_x}
\end{center}
this gives us a map $p_x: W_{x/} \to W$.
\end{exone}
We have following fact about this map.
\begin{propone}
\cite[Example 3.11]{Ra17a}
The map $p_x:W_{x/} \to W$ is a left fibration.
\end{propone}
For the proof we need a technical understanding of the equivalences of Segal spaces.
For more details see \cite[Example 3.11, Theorem 7.1]{Ra17a}
Note that there is a map $x:F(0) \to W_{x/}$, which maps the point to the identity map $id_x$.
We will call this left fibration the {\it representable left fibration} represented by $x$.
The following theorem justifies our naming convention:
\begin{lemone}
\cite[Theorem 4.2]{Ra17a}
({\it Yoneda Lemma for Left Fibrations})
Let $L \to W$ be a left fibration. Then the map induced by $F(0) \to W_{x/}$ over $W$
$$ Map_{W}(W_{x/},L) \xrightarrow{ \ \ \mathscr{M}athscr{S}imeq \ \ } Map_{W}(F(0),L) = \mathscr{M}athcal{D}elta[0] \underset{W_0}{\times} L_0$$
is a trivial fibration of Kan complexes.
\end{lemone}
The proof of this theorem needs a serious treatment of the theory of left fibrations, which is far beyond the scope of this note.
For a detailed account of the theory of left fibrations see \cite{Ra17a}.
Before we move on we should point out that we here focused on covariant functors and there is a way to define
fibrations that model contravariant functors.
\begin{defone}
A Reedy fibration $p: R \to W$ between CSS is a left fibration if the following is a homotopy pullback square
\begin{center}
\pbsq{R_1}{R_0}{W_1}{W_0}{p_1}{s}{s}{p_0}
\end{center}
where the map $s:W_1 \to W_0$ is the source map that takes each arrow to its source.
\end{defone}
Using this definition everything we have done until here can now be adjusted to the contravariant setting and give us the same results.
\mathscr{M}athscr{S}ubsection{CoCartesian Fibrations} \label{Subsec CoCartesian Fibrations}
Up until now we gave a accurate description for functors valued in spaces. However, we also have functors that are valued in higher categories.
Using the definition of a left fibration as a guide, we need a certain lifting condition.
This lifting condition must have two differences compared to left fibrations. First, it must relax the conditions on each fiber as
they could be a CSS rather than just a space. Second, the fact that each fiber is a CSS implies that
the lifting condition cannot be restricted to points, but also has to take arrows into account.
More on this concept can be found in \cite{Ra17b}.
\par
The right way to generalize the lifting condition to arrows is via the language of coCartesian morphism.
\begin{defone} \label{Def Map from f to z}
Let $W$ be a CSS, $f: x \to y$ a morphism in $W$ and $z$ an object in $W$.
We define the space $map_W(f,z)$ as the following pullback:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
map_W(f,z) \arrow[dr, phantom, "\ulcorner", very near start] \arrow[r] \arrow[d, "\mathscr{M}athscr{S}imeq"] & map_W(x,y,z) \arrow[d, "\mathscr{M}athscr{S}imeq", twoheadrightarrow] \\
map_W(y,z) \arrow[r] & map_W(x,y) \times map_W(y,z)
\end{tikzcd}
\end{center}
\end{defone}
\begin{intone} \label{Int Map from f to z}
A point in $map_W(f,z)$ is a triangle of the form:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
& y \arrow[dr] & \\
x \arrow[ur, "f"] \arrow[rr, ""{name=U, above}] & & z
\arrow[start anchor={[xshift=-10ex, yshift=10ex]}, to=U, phantom, "\mathscr{M}athscr{S}igma"]
\end{tikzcd}
\end{center}
The reason we care about this space is because it fits into the following zig zag of spaces where the down left map
is an equivalence.
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
& map_W(f,z) \arrow[dl, "\mathscr{M}athscr{S}imeq", twoheadrightarrow] \arrow[dr, "f^*"] & \\
map_W(y,z) & & map_W(x,z)
\end{tikzcd}
\end{center}
The map labeled $f^*$ plays the role of ``pre-composing with f", however, unlike regular category theory there is no direct map
from $map_W(y,z) \to map_W(x,z)$ and so we use $map_W(f,z)$ as the appropriate replacement for $map_W(y,z)$.
\end{intone}
\begin{defone} \label{Def CoCart Mor}
Let $p: V \to W$ be a map of CSS. We say the morphism $f: x \to y$ in $V$ is a $p$-coCartesian morphism if for each object $z \in V$,
the following diagram is a homotopy pullback square.
\begin{center}
\pbsq{map_V(f,z)}{map_V(x,z)}{map_W(pf,pz)}{map_W(px,pz)}{f^*}{p}{p}{pf^*}
\end{center}
\end{defone}
\begin{intone}
We can depict the situation as follows:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
& y \arrow[dr, dashed, "\hat{g}"] & \\
x \arrow[ur, "f"] \arrow[rr, ""{name=U, above}] & \textcolor{white}{.} \arrow[dd, Rightarrow, "p", shorten >=1ex,shorten <=1ex] & z
\arrow[start anchor={[xshift=-10ex, yshift=10ex]}, to=U, phantom, "\hat{\mathscr{M}athscr{S}igma}"] \\
\\
& py \arrow[dr, "g"] & \\
px \arrow[ur, "pf"] \arrow[rr, ""{name=U, above}] & & pz
\arrow[start anchor={[xshift=-10ex, yshift=10ex]}, to=U, phantom, "\mathscr{M}athscr{S}igma"]
\end{tikzcd}
\end{center}
The coCartesian condition stipulates that $\mathscr{M}athscr{S}igma$ can be lifted to a $\hat{\mathscr{M}athscr{S}igma}$. This in particular implies that $g$ lifts to an arrow
$\hat{g}$. Notice how this condition tells us something about
the existence of certain $2$-cells that lift the diagram. Thus we can really think of it as a higher dimensional version of the
lifting condition for left fibrations.
\end{intone}
\begin{exone} \label{Ex Id is coCart}
Let $p: V \to W$ be a map of CSS. For any object $x \in V$, the map $id_x$ is $p$-coCartesian.
\end{exone}
\begin{exone}
Let $p: V \to W$ be map of CSS. Let $f \in V$ be a coCartesian morphism such that $p(f) = id_y$ for an object $y \in W$.
Then $f$ is an equivalence in $V$.
\end{exone}
With the notion of a $p$-coCartesian morphism we finally define a fibration that models functors valued in CSS.
\begin{defone}
A Reedy fibration of CSS $p: C \to W$ is a {\it coCartesian fibration} if for every morphism $f: x \to y$ in $W$ and chosen lift
$x' \in C$ there exists a $p$-coCartesian lift $f': x' \to y'$ of $f$.
\end{defone}
\begin{intone} \label{Int CoCart Fib}
The best way to gain some intuition on this example is to review the graph we used in Intuition \ref{Int Cat Fib in Set}.
Let $C \to F(1)$ be a coCartesian fibration. Intuitively it corresponds to a functor from the category $[1]$ into CSS,
thus it corresponds to a map of CSS. Following Intuition \ref{Int Cat Fib in Set} we think of the fiber over the object $0$ as
the domain $CSS$ and the fiber over $1$ as the target CSS. The lifting condition should help us get a map from the fiber over $0$ to the
fiber over $1$. We already know how to get a map on objects, so let us understand how the condition of a coCartesian fibration
gives us a map on morphisms. Let $g: x_1 \to x_2$ be an arrow in the fiber over $0$.
\begin{center}
\begin{tikzcd}[row sep=0.1in, column sep=0.1in]
& \bullet
& x_2 \bullet \arrow[drrrrrrrrrrrrrrrrr, "f_2", mapsto] & \bullet & & & & & & & & & & & & & & & \hspace{0.05in}\bullet & \bullet \\
x_1 \bullet \arrow[urr, "g"] \arrow[drrrrrrrrrrrrrrrrr, "f_1", mapsto] & \hspace{0.05in}\bullet & & & & & & & & & & & & & & & & \bullet &
\bullet & y_2 \bullet \\
& \hspace{0.05in}\bullet \arrow[dddddd, Rightarrow] & & & & & & & & & & & & & & & & y_1 \arrow[urr, dashed, "\hat{g}"] \bullet &
\hspace{0.05in}\bullet \arrow[dddddd, Rightarrow] & \\
\\ \\ \\ \\ \\
& 0 \arrow[rrrrrrrrrrrrrrrrr, "01"] & & & & & & & & & & & & & & & & & 1 & \\
\end{tikzcd}
\end{center}
As $p$ is a coCartesian fibration, this gives us two $p$-coCartesian lifts,
namely $f_1:x_1 \to y_1$ and $f_2:x_2 \to y_2$. The fact that $f_1$ is $p$-coCartesian implies that this diagram lifts
to a map $\hat{g}: y_1 \to y_2$.
So the lifting condition gives us the necessary map on morphisms that we need.
Thus we think of the morphism $\hat{g}$ as the ``target" of the morphism $g$ under the ``functor" $C$.
\end{intone}
There is a second way of defining coCartesian fibrations that aligns more closely to our work with left fibrations.
For that one we need some definitions and lemmas. First, we can generalize the definition of a $p$-coCartesian morphism.
\begin{defone}
Let $p: V \to W$ be a map of CSS. An $n$-simplex $\mathscr{M}athscr{S}igma \in V_n$ is $p$-coCartesian if for every map $s: V_n \to V_1$, $s(\mathscr{M}athscr{S}igma)$ is $p$-coCartesian.
\end{defone}
\begin{exone}
In light of Example \ref{Ex Id is coCart}, every point in $V_0$ is $p$-coCartesian.
\end{exone}
\begin{defone}
Let $p: V \to W$ be a Reedy fibration between CSS. We define $LFib_W(V)$ as the subsimplicial space generated by all $p$-coCartesian arrows
in $V$. Based on the previous example $LFib_W(V)_0 = V_0$.
\end{defone}
Here is a very crucial lemma that exemplifies the importance of this construction.
\begin{lemone}
The map $LFib_W(V)_1 \to LFib_W(V)_0 \times_{W_0} W_1$ is $(-1)$-truncated.
\end{lemone}
\begin{intone}
The lemma is telling us that for a chosen point $\mathscr{M}athcal{D}elta[0] \to LFib_W(V)_0 \times_{W_0} W_1$, the space
$coCartLift(f)$ defined by the following pullback diagram is either empty or contractible.
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep = 0.5in]
coCartLift(f) \arrow[d] \arrow[r] & LFib_W(V)_1 \arrow[d, "\mathscr{M}athscr{S}imeq", twoheadrightarrow] \\
\mathscr{M}athcal{D}elta[0] \arrow[r, "(x' , f)"] & LFib_W(V)_0 \times_{W_0} W_1
\end{tikzcd}
\end{center}
However, this space is just the space of $p$-coCartesian lifts of the map $f: x \to y$ with given lift $x'$.
So, this lemma shows that either no such lift exists or there is a contractible space of choices of such lifts.
In other words, if a $p$-coCartesian lift exists it is unique up to homotopy.
\end{intone}
This gives us following way of identifying coCartesian fibrations.
\begin{propone}
A Reedy fibration of CSS $p:C \to W$ is a coCartesian fibration if and only if the map of simplicial spaces
$LFib_W(V) \to W$ is a left fibration.
\end{propone}
\begin{proof}
By definition we have to show that the condition of being a coCartesian fibration is equivalent to the following
being a homotopy pullback square:
\begin{center}
\pbsq{LFib_W(V)_1}{LFib_W(V)_0}{W_1}{W_0}{}{}{}{}
\end{center}
But this is just equivalent to the following map being an equivalence.
$$LFib_W(V)_1 \to LFib_W(V)_0 \times_{W_0} W_1$$
We already know it is always $(-1)$-truncated, thus it is an equivalence if and only if it is a surjection. But being a surjection is exactly
the condition that every arrow with a given lift for the source has a $p$-coCartesian lift. That gives us the desired result.
\end{proof}
\begin{intone}
If $C \to W$ is a coCartesian fibration, then it models a functor valued in CSS. However, every CSS $W$ has an underlying maximal subgroupoid,
namely $W_0$. The left fibration $LFib_W(C)$ exactly models the functor valued in spaces that maps each point to the maximal subgroupoid of its
image. The lemma above says that analyzing the functoriality of that underlying fibration already suffices.
Why is that?
\par
In order to build a fibration that models a certain functor two conditions are necessary. First, we must make sure
that each point has the right fiber i.e. the fiber has the value we desire such as a space or a CSS. Second, we must make sure
that we have the right lifting conditions to get a good notion of functoriality. For a coCartesian fibration between CSS the first
condition is already given without any extra condition. The only thing that we need to add is a lifting condition to get functoriality.
However, this can already be achieved at the level of spaces, if the lifts are coCartesian morphisms, which is because
such morphisms will give us the necessary functoriality property.
\end{intone}
Let us complement this intuition by looking at some examples.
\begin{exone} \label{Ex CoCart over point}
Let $p:C \to F(0)$ be a coCartesian fibration. There is only one map in $F(0)$, namely the identity map, and for any chosen source $x \in C$
the identity map lifts to the identity map on that source $id_x: x \to x$, which is always $p$-coCartesian.
Thus the map $p$ does not imposes any condition $C$ and $C$ is just a given CSS, which is exactly what we expected.
\end{exone}
\begin{intone}
Notice in this situation $LFib_{F(0)}(C) \mathscr{M}athscr{S}imeq C_0$. This corroborates our point that $LFib_W(C)$ gives us the functor valued in the underlying
maximal subgroupoids.
\end{intone}
\begin{exone} \label{Ex CoCart over arrow}
Let $p: C \to F(1)$ be a coCartesian fibration. The fibers $C_{/0}$ and $C_{/1}$ are two CSS.
Let
$$C_{/01} = C^{F(1)} \underset{F(1)^{F(1)}}{\times} F(0).$$
The two maps $0,1: F(0) \to F(1)$ give us a diagram of CSS.
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
& C_{/01} \arrow[dl, "s"] \arrow[dr, "t"] & \\
C_{/0} & & C_{/1}
\end{tikzcd}
\end{center}
We had a similar diagram when we worked with left fibrations. In that case the source map was an equivalence. However, here this is not the case
as $C_{/01}$ has functoriality information that cannot be recovered from $C_{/0}$. However, there is a way to salvage the equivalence.
We define
$$LFib_{/01} (C) = (LFib_{F(1)}(C))^{F(1)} \underset{F(1)^{F(1)}}{\times} F(0)$$
This gives us a map $LFib_{/01}C \to C_{/01}$. Now, the composition map
$$LFib_{/01}(C) \xrightarrow{ \ \ \mathscr{M}athscr{S}imeq \ \ } C_{/0}$$
is an equivalence of CSS, by the left fibration property.
\end{exone}
As in the case of left fibrations, coCartesian fibrations also have a contravariant analogue.
\begin{defone} \label{Def Cart Mor}
Let $p: V \to W$ be a map of CSS. We say the morphism $f: x \to y$ in $V$ is a $p$-Cartesian morphism if for each object $z \in V$,
the following diagram is a homotopy pullback square.
\begin{center}
\pbsq{map_V(z,f)}{map_V(z,y)}{map_W(pz,pf)}{map_W(pz,py)}{f_*}{p}{p}{pf_*}
\end{center}
\end{defone}
\begin{defone}
A Reedy fibration of CSS $p: C \to W$ is a {\it Cartesian fibration} if for every morphism $f: x \to y$ in $W$ and chosen lift
$y' \in C$ there exists a $p$-Cartesian lift $f': x' \to y'$ of $f$.
\end{defone}
\mathscr{M}athscr{S}ection{Colimits and Adjunctions} \label{Sec Colimits and Adjunctions}
In this section we discuss colimits and adjunctions
in CSS using the work we have done before.
\mathscr{M}athscr{S}ubsection{Colimits in Complete Segal Spaces} \label{Limits and Colimits in Complete Segal Spaces}
The goal of this subsection is to study colimits. We will proceed in two steps. First we study initial objects, which are the
simplest example of a colimit. Then we generalize from an initial object to an arbitrary colimit using cocones.
For this subsection let $W$ be a fixed CSS.
\begin{defone} \label{Def Initial Obj}
Let $i \in W$. We say $i$ is {\it initial} in $W$ if the projection map $p_i:W_{i/} \to W$ is an equivalence of $CSS$.
\end{defone}
\begin{intone} \label{Int Initial Obj}
There is a way to make this definition look more familiar. Let $y \in W$ be an object in $W$.
Then we have the following pullback diagram:
\begin{center}
\pbsq{map_W(i,y)}{W_{i/}}{F(0)}{W}{}{}{p_i}{y}
\end{center}
The fact that $p_i$ is an equivalence implies that $map(i,y)$ is contractible. This means that there is a unique map from $i$ to $y$,
up to homotopy. This is correct generalization of an initial object in a classical category.
\end{intone}
In the classical case we know that if an initial object exists then it is unique. We need to show the same thing holds for CSS.
\begin{defone}
Let $W$ be a CSS. Let $W_{init}$, called \emph{the space of final objects},
be the subspace of $W_0$ generated by all initial objects.
\end{defone}
\begin{lemone}
The space $W_{init}$ is $(-1)$-truncated.
\end{lemone}
\begin{proof}
$W_{init}$ is (-1)-truncated if and only if the map $\mathscr{M}athcal{D}elta : W_{init} \to W_{init} \times W_{init}$ is an equivalence.
This is true if for every map $(x,y): * \to W_{init} \times W_{init}$ the following square is a homotopy pullback.
\begin{center}
\pbsq{*}{*}{W_{init}}{W_{init} \times W_{init} }{}{}{( x , y )}{\mathscr{M}athcal{D}elta}
\end{center}
Now the problem is $\mathscr{M}athcal{D}elta: W_{init} \to W_{init} \times W_{init}$ is not a Kan fibration and thus we need to replace this map with an
equivalent map that is a Kan fibration. Fortunately that is not too hard. Concretely, we know that $W_{init} \to (W_{init})^{\mathscr{M}athcal{D}elta[1]}$
is a Kan equivalence and $(s,t): (W_{init})^{\mathscr{M}athcal{D}elta[1]} \to W_{init} \times W_{init}$ is a Kan fibration.
Thus we need following to be a homotopy pullback.
\begin{center}
\pbsq{*}{*}{(W_{init})^{\mathscr{M}athcal{D}elta[1]}}{W_{init} \times W_{init} }{}{}{( x , y )}{(s , t)}
\end{center}
The actual pullback is the space of paths inside the space $W_{fin}$ which start at $x$ and end at $y$. By the completeness condition
this space is equivalent to $hoequiv_{/W}(x,y) \hookrightarrow map_{/W}(x,y) \mathscr{M}athscr{S}imeq *$ (Lemma \ref{Lemma Completeness Conditions}).
As we know that $hoequiv_{/W}(x,y) \hookrightarrow *$ is (-1)-truncated all we have to do
is to show that $hoequiv_{/W}(x,y) \mathscr{M}athscr{N}eq \emptyset$ and we can conclude that $hoequiv_{/W}(x,y) \mathscr{M}athscr{S}imeq *$.
Indeed, let $f \in map_{/W}(x,y) = *$ and $g \in map_{/W}(y,x) = *$. Then, $g \circ f \mathscr{M}athscr{S}imeq id_x \in map_{/W}(x,x) = *$
and $f \circ g \mathscr{M}athscr{S}imeq id_y \in map_{/W}(y,y)= *$ and so $x$ and $y$ are equivalent and so $hoequiv_{/W}(x,y) \mathscr{M}athscr{N}eq \emptyset$
and we are done.
\end{proof}
This means that if $W_{init} \mathscr{M}athscr{N}eq \emptyset$ then it is contractible, which implies that if an initial object exists
then it is unique up to homotopy.
Using initial objects we can define colimits. But first we have to define the complete Segal space of cocones.
\begin{defone} \label{Def Cocones}
Let $f: I \to W$ be a map of simplicial spaces.
We define $W_{f/}$ as the following CSS
$$ W_{f/} = F(0) \underset{W^I}{\times} W^{I \times F(1)} \underset{W^I}{\times} W.$$
\end{defone}
\begin{intone} \label{Int Cocones}
Let us look at $(W_{f/})_0$. We have
$$ (W_{f/})_0 = F(0) \underset{(W^I)_0}{\times} (W^{I \times F(1)})_0 \underset{(W^I)_0}{\times} W_0 = $$
$$ F(0) \underset{Map(I,W)}{\times} Map(I \times F(1),W) \underset{Map(I,W)}{\times} W_0. $$
Thus a point in $W_{f/}$ is a map $\tilde{f}: I \times F(1) \to W$ such that $\displaystyle \tilde{f}|_{I \times \{ 0\} } = f$ and
$\displaystyle \tilde{f}|_{I \times \{ 1\} } = x$, where $x$ is an object in $W$. That is very similar to the definition of a cocone
that we have seen in category theory.
\end{intone}
\begin{exone}
As an example, if $I = F(0)$ and $x: F(0) \to W$, then the CSS of cocones, $W_{x/}$ is exactly the under-CSS, $W_{x/}$ as defined in
Definition \ref{Def UnderCSS}.
\end{exone}
\begin{defone}
Let $f: I \to W$ be map of CSS. The \emph{diagram $f$ has a colimit} if $(W_{/f})_{init} \mathscr{M}athscr{N}eq \emptyset$.
We will denote any choice of point in this space by $colim_I f$ and call it the \emph{colimit of f} or {\it colimit cocone of f}.
\end{defone}
\begin{defone}
A CSS is \emph{cocomplete} if for any map $f:I \to W$ the diagram $f$ has a colimit.
\end{defone}
\begin{remone}
Let us give a more detailed description of the colimit of a diagram $f:I \to W$. It is an initial object in $W_{f/}$ and
as such it is a map $\bar{f}: I \times F(1) \to W$ such that $\bar{f}|_{ \{ 0 \} } = f$ and $\bar{f}|_{ \{ 1 \} } = v$,
where $v$ is an object in $W$.
We very often abuse notation and call this point in $W$ the \emph{colimit of f} and also denote it by $colim_{I} f$.
\end{remone}
Here are some examples of important colimits.
\begin{exone}
Let $I = \emptyset$ be the empty CSS. Then there is a unique map $e:\emptyset \to W$.
In this case $W_{e/} = W$ and so the colimit of the diagram $e$ is just the initial object.
\end{exone}
\begin{exone}
Let $S$ be a set, thought of as a discrete simplicial space.
Concretely, $S$ is a disjoint union of $F(0)$.
Let $f: S \to W$ be any map. Notice we have following equivalence
$$W^S \cong \prod_{ s \in S} W.$$
So
$$W_{f/} = F(0) \underset{W^S}{\times} W^{S \times F(1)} \underset{W^S}{\times} W = $$
$$ F(0) \underset{\displaystyle \prod_{s \in S} W}{\times} \prod_{s \in S} W^{F(1)} \underset{\displaystyle \prod_{s \in S} W}{\times} W. $$
Thus a point in $W_{f/}$ is a choice of object $x \in W$ and morphisms $i_s: f(s) \to x$.
This means the data of a colimit of a set diagram is the same as in ordinary categories.
\end{exone}
\begin{exone}
In a similar fashion, let $\mathscr{M}athcal{C}$ be the following category:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
1 & 0 \arrow[r, "r"] \arrow[l, "l"'] & 2
\end{tikzcd}
\end{center}
and let $I=\mathscr{M}athscr{N} \mathscr{M}athcal{C}$. Concretely, $\displaystyle I = F(1)^s \coprod_{F(0)} \mathscr{M}athscr{S}trut^s F(1)$.
A map $f:I \to W$ is of the form:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
x_1 & x_0 \arrow[l, "f"'] \arrow[r, "g"]& x_2
\end{tikzcd}
.
\end{center}
where $f:x_0 \to x_1$ and $g:x_0 \to x_2$ are two arrows in $W$.
An object in the CSS of cocones, $W_{f/}$, is a diagram of the form:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
x_1 \arrow[d] & x_0 \arrow[l, "f"', ""{name=U, above}] \arrow[r, "g", ""{name=V, above}] \arrow[d] & x_2 \arrow[d] \\
v & v \arrow[l, "id_v"'] \arrow[r, "id_v"]& v
\arrow[start anchor={[xshift=-17.5ex, yshift=-2.5ex]}, to=U, phantom, "\mathscr{M}athscr{S}igma"]
\arrow[start anchor={[xshift=-5ex, yshift=-2.5ex]}, to=V, phantom, "\psi"]
\end{tikzcd}
.
\end{center}
Thus a colimit a diagram of the form above that is an initial.
In such diagram, the object $v$ is often expressed as $x_1 \coprod_{x_0} x_2$.
\par
Considering the fact the bottom horizontal maps are identity maps, we can reduce the diagram to the more familiar form:
\begin{center}
\begin{tikzcd}[row sep=0.5in, column sep=0.5in]
x_1 \arrow[dr, bend right=20] & x_0 \arrow[l, "f"', ""{name=U, above}] \arrow[r, "g", ""{name=V, above}] \arrow[d] & x_2 \arrow[dl, bend left=20] \\
& \displaystyle x_1 \coprod_{x_0} x_2 &
\arrow[start anchor={[xshift=-20ex, yshift=2.5ex]}, to=U, phantom, "\mathscr{M}athscr{S}igma"]
\arrow[start anchor={[xshift=-12.5ex, yshift=2.5ex]}, to=V, phantom, "\psi"]
\end{tikzcd}
\end{center}
\end{exone}
On classic fact about colimits is that a map out of a colimit is determined by a map out of the diagram that formed the colimit.
The same result holds for complete Segal spaces.
\begin{theone}
\cite[Theorem 5.13]{Ra17a}
Let $f : I \to W$ be a map of CSS which has colimit cocone $\tilde{f} : F(0) \to X_{p/}$
with vertex point $v$. Let $y$ be any object in $W$. This gives us a constant map $\mathscr{M}athcal{D}elta y : I \to F(0) \to W$.
There is a Kan equivalence of spaces
$$map_W(v, y) \xrightarrow{ \ \ \mathscr{M}athscr{S}imeq \ \ } map_{W_{f/}}(\tilde{f}, \mathscr{M}athcal{D}elta y).$$
\end{theone}
\mathscr{M}athscr{S}ubsection{Adjunctions} \label{Subsec Adjunctions}
In this subsection we discuss adjunctions of complete Segal spaces.
Recall that for two ordinary categories $\mathscr{M}athcal{C}$ and $\mathscr{M}athcal{D}$ an adjunction are two functors $F: \mathscr{M}athcal{C} \to \mathscr{M}athcal{D}$
and $G:\mathscr{M}athcal{D} \to \mathscr{M}athcal{C}$ along with a choice of bijections
$$Hom_{\mathscr{M}athcal{D}}(Fc,d) \cong Hom_{\mathscr{M}athcal{C}}(c, Gd)$$
for two objects $c$ in $\mathscr{M}athcal{C}$ and $d$ in $\mathscr{M}athcal{D}$.
\par
When we want to generalize this to the higher categorical setting we again run into coherence issues when trying
to specify the equivalences. Fortunately, there is a way to manage this difficulty,
namely by using (co)Cartesian fibrations.
\begin{defone} \label{Def Adj}
A map $p:A \to F(1)$ is an adjunction if it is a coCartesian and Cartesian fibration.
\end{defone}
\begin{intone} \label{Int Adj}
How does this definition relate to a notion of an adjunction that we are familiar with?
First notice that $F(1)$ has two objects, namely $0$ and $1$, that have fibers
$$W = A \underset{F(1)}{\times} F(0)$$
$$V = A \underset{F(1)}{\times} F(0)$$
which are both CSS.
\par
Next as $A \to F(1)$ is a coCartesian fibration it classifies a map $f: W \to V$ as described in Example \ref{Ex CoCart over arrow}.
Similarly, as $A \to F(1)$ is a Cartesian fibration it classifies a map $g: V \to W$.
The key is now to use the existence of (co)Cartesian lifts to get the structure of an adjunction.
\par
If $x$ is an object in $W$ then there exists an object $fx$ in $V$ along with a map $f: x \to fx$ in $A$ that is coCartesian.
The coCartesian property implies that for any other object $y$ in $V$ a morphism $x \to y$ in $A$ will factor through
$fx$.
We can depict the situation as follows:
\begin{center}
\begin{tikzcd}[row sep=0.1in, column sep=0.1in]
& x \bullet \arrow[rrrrrrrrrrrrrrrrrr, "f", mapsto] \arrow[ddrrrrrrrrrrrrrrrrr, mapsto]
& & & & & & & & & & & & & & & & & & \bullet f(x) \arrow[ddl, dashed] \\
& & & & & & & & & & & & & & & & & & & \\
& \mathscr{M}athscr{S}trut \arrow[dddddd, Rightarrow] & & & & & & & & & & & & & & & & & \hspace{0.05in}\bullet y \arrow[dddddd, Rightarrow] & \\
\\ \\ \\ \\ \\
& 0 \arrow[rrrrrrrrrrrrrrrrr, "01"] & & & & & & & & & & & & & & & & & 1 & \\
\end{tikzcd}
\end{center}
This gives us an equivalence
$$map_{V}(fx,y) \mathscr{M}athscr{S}imeq map_A(x,y).$$
On the other side for an object $y$ in $V$ there exists a Cartesian lift $gy \to y$. Using a similar argument to the one above,
for each object $x$ in $W$ we have an equivalence
$$map_W(x,gy) \mathscr{M}athscr{S}imeq map_A(x,y).$$
Combining these two equivalences we get that
$$map(fx,y) \mathscr{M}athscr{S}imeq map_V(x,gy).$$
This is exactly the familiar format of an adjunction that we would have expected.
It should be noted that as always the equivalence is not direct but rather through a zig-zag of equivalences.
\end{intone}
Having a definition of an adjunction we can recover some computational methods for how to determine adjunctions.
Recall that in classical category theory a map of categories $F: \mathscr{M}athcal{C} \to \mathscr{M}athcal{D}$ is a left adjoint if and only if for each object $d$ the functor
$$Hom_{\mathscr{M}athcal{D}}(F(-),d): \mathscr{M}athcal{C}^{op} \to \mathscr{M}athscr{S}et$$
is representable. The representing object will then be $G(d)$ and we can use the universality of a representing objects
to define a functor $G: \mathscr{M}athcal{D} \to \mathscr{M}athcal{C}$ that is the right adjoint.
\par
We want a similar result for higher categories. However, in this case representable functors are
modeled by representable right fibrations or, in other words, over-categories.
Using that we can adjust the result from above thusly.
\begin{theone}
\cite[Theorem 7.57]{Ra17b}
Let $f: W \to V$ be a functor of CSS. Then $f$ is a left adjoint if and only if
for each object $y$ in $V$ the CSS $W_{/y}$ defined by the pullback
\begin{center}
\pbsq{W_{/y}}{V_{/y}}{W}{V}{}{}{\pi_y}{f}
\end{center}
has a final object, which implies that $W_{/y} \to W$ is a representable right fibration.
\end{theone}
This gives us a very helpful computational method to determine whether a functor is a left adjoint.
We can use this computational method to redefine limits and colimits.
\begin{theone}
Let $I$ and $W$ be CSS and let $\mathscr{M}athcal{D}elta_I: W \to W^I$ be the natural inclusion induced by the map $I \to F(0)$.
Then $\mathscr{M}athcal{D}elta_I$ has a left adjoint if and only if each map $f: I \to W$ has a colimit and has a right adjoint if each map
$f: I \to W$ has a limit.
\end{theone}
\begin{proof}
$\mathscr{M}athcal{D}elta_I: W \to W^I$ has a right adjoint if and only if for each map $h: I \to W$ the pullback $W \times_{W^I} (W^I)_{/h}$
has a final object. However, this is just the category of cones over $f$, namely, $W_{/h}$, which by definition means
$h$ has a limit. The case for colimits follows similarly.
\end{proof}
Let us trace through the steps of the proof in one concrete example.
\begin{exone}
Let $I = \emptyset$ and $W$ be a CSS. Then $W^I = F(0)$ and $\mathscr{M}athcal{D}elta_I: W \to F(0)$ is just the map to the point.
In this case a map $i:F(0) \to W$ is a left adjoint if the fiber formed by the pullback diagram is a representable right fibration over $F(0)$.
\begin{center}
\pbsq{map(i,x)}{W_{/x}}{F(0)}{W}{}{}{}{i}
\end{center}
However, $F(0)$ has only one object and so the only representable right fibration over $F(0)$ is itself.
Thus $map(i,x)$ needs to be contractible for $i: F(0) \to W$ to be a left adjoint. But this is exactly
the condition of being an initial object.
\end{exone}
\mathscr{M}athscr{S}ection{Model Structures of Complete Segal Spaces} \label{Sec Model Structures of Complete Segal Spaces}
One very efficient way of studying higher categories is via the language of model categories. A model category is an ordinary
category with several distinguished classes of maps that allows us to recover some of the important homotopical data.
In particular, complete Segal spaces have a model structure that allows us to study them very efficiently.
The goal of this section is to define and study the model structure for complete Segal spaces.
\par
Most results about this model structure can be found in \cite{Re01}. We will not prove any of them, but rather provide proper references.
The goal of this section is only to give the reader an overview of the complete Segal space model structure.
\mathscr{M}athscr{S}ubsection{Review of Model Structures} \label{Subsec Review of Model Categories}
We are not going to develop the whole theory of model structures, but rather focus on several important properties
that will come up in the coming subsections.
For a good introduction to the theory of model structures see \cite{Ho98} or \cite{DS95}.
\begin{defone}
A model structure on a category is a complete and cocomplete category $\mathscr{M}athcal{M}$ along with three classes of maps:
\begin{enumerate}
\item Fibrations $\mathscr{M}athcal{F}$.
\item Cofibrations $\mathscr{M}athcal{C}$.
\item Weak Equivalences $\mathscr{M}athcal{W}$.
\end{enumerate}
that satisfies following conditions.
\begin{itemize}
\item {\it ($2$-out-of-$3$)} If two out of the three maps $f$, $g$ and $g \circ f$ are weak equivalences then so is the third.
\item {\it (Retracts)} If $f$ is a retract of $g$ and $g$
is a weak equivalence, cofibration, or fibration, then so is $f$.
\item {\it (Lifting)} Let $i: A \to B$ be a cofibration and $p: Y \to X$ be a fibration. The diagram in $\mathscr{M}athcal{M}$
\begin{center}
\liftsq{A}{Y}{B}{X}{}{i}{p}{}
\end{center}
lifts if either $i$ or $p$ are weak equivalences.
\item {\it (Factorization)} Any morphism $f$ can be factored into $f = pi$, where $p$ is a fibration and $i$ is a cofibration and weak equivalence
and $f = qj$, where $q$ is a fibration and weak equivalence and $j$ is a cofibration.
\end{itemize}
\end{defone}
One primary example of a model structure is the {\it Kan model structure}:
\begin{theone}
\cite[Theorem I.11.3]{GJ09}
There is a model structure on simplicial sets, called the Kan model structure and defined as follows:
\begin{itemize}
\item[C] A map is a cofibration if it is an inclusion.
\item[F] A map is a fibration if it is a Kan fibration.
\item[W] A map $X \to Y$ is an equivalence if and only if the for every Kan complex $K$ the induced map of Kan complexes
$$Map(Y,K) \to Map(X,K)$$
is a weak equivalence.
\end{itemize}
\end{theone}
Model structures can simplify many computations. For that reason there are many important properties that model structures can satisfy.
For the rest of this subsection we will review some of the more important properties.
\begin{defone}
We say a model structure is {\it left proper} if for every weak equivalence $A \to X$ and for every cofibration $A \to B$,
the pushout map $X \to X \coprod_{A} B$ is a weak equivalence.
\end{defone}
\begin{defone}
We say a model structure is {\it right proper} if for every weak equivalence $A \to X$ and for every cofibration $Y \to X$,
the pullback map $Y \times_X A \to Y $ is a weak equivalence.
\end{defone}
\begin{defone}
A model structure is proper if it is left proper and right proper at the same time.
\end{defone}
Sometimes we have a model structure on a category that is Cartesian closed. In this case we might wonder how well they interact
with each other.
\begin{defone}
Let $\mathscr{M}athcal{M}$ be a model structure on a category that is Cartesian closed. We say the model structure is compatible with Cartesian closure
if for every
cofibration $i:A \to B$, $j: C \to D$ and fibration $p:Y \to X$, the map
$$i: A \times D \coprod_{A \times B} B \times C \to B \times D$$
is a cofibration and
$$Y^B \to Y^A \underset{X^A}{\times} X^B$$
is a fibration either of which is a weak equivalence if any of the maps involved is a weak equivalence.
\end{defone}
\begin{remone}
Notice that the Kan model structure satisfies all of the conditions stated above. It is proper and compatible with Cartesian closure.
\end{remone}
\mathscr{M}athscr{S}ubsection{Reedy Model Structure} \label{Subsec Reedy Model Structure}
We defined a complete Segal space as a simplicial space that satisfied three conditions.
\begin{enumerate}
\item Reedy fibrancy
\item Segal condition
\item Completeness condition
\end{enumerate}
Accordingly we will define three model structures, which correspond to these three conditions.
Thus we first start with the {\it Reedy model structure}.
\begin{theone} \label{The Reedy Model Structure}
\cite{ReXX}
\cite[Subsection 2.4-2.6]{DKS93}
There is a model structure on the category $s\mathscr{M}athscr{S}$ called the Reedy model structure,
defined as follows.
\begin{itemize}
\item[C] A map $f: X \to Y$ is a cofibration if it is an inclusion.
\item[W] A map $f: X \to Y$ is a weak equivalence if it is a level-wise equivalence.
\item[F] A map $f: X \to Y$ is a fibration if the map
$$Map(F(n),X) \to Map(\partial F(n), X) \underset{Map(\partial F(n), Y)}{\times} Map( F(n), Y)$$
is a Kan fibration.
\end{itemize}
\end{theone}
The Reedy model structure has many amazing features. In particular it satisfies following properties:
\begin{enumerate}
\item Cofibrantly generated.
\item Proper.
\item Compatible with Cartesian closure.
\item Simplicial.
\end{enumerate}
One important property of the Reedy model structure on simplicial spaces is that we can use techniques of Bousfield
localizations on it.
\begin{theone}
\cite[Proposition 9.1]{Re01}
Let $f: A \to B$ be an inclusion.
There is a unique, simplicial, cofibrantly generated model structure on $s\mathscr{M}athscr{S}$ called the $f$-local model structure,
characterized as follows.
\begin{enumerate}
\item Cofibrations are inclusions.
\item A simplicial space $W$ is fibrant if it is Reedy fibrant and the map
$$f^*: Map(B, W) \to Map(A,W)$$
is a Kan equivalence.
\item A map $g: X \to Y$ is a weak equivalence if for every fibrant object $W$ the induced map
$$g^*: Map(Y,W) \to Map(X,W)$$
is a Kan equivalence.
\item A map between fibrant objects $W \to V$ is a weak equivalence (fibration) if and only if it is a
Reedy equivalence (Reedy fibration).
\end{enumerate}
\end{theone}
Using this property of Bousfield localizations we can define model structures for Segal spaces and complete Segal spaces.
\mathscr{M}athscr{S}ubsection{Segal Space Model Structure} \label{Subsec Segal Space Model Structure}
The Segal space model structure is a Bousfield localization of the Reedy model structure.
\begin{theone}
\cite[Theorem 7.1]{Re01}
There is a unique, simplicial model structure on $s\mathscr{M}athscr{S}$ called the Segal space model structure,
characterized as follows.
\begin{enumerate}
\item Cofibrations are inclusions.
\item A simplicial space $T$ is fibrant if it is Reedy fibrant and the map
$$\varphi_n^*: Map(F(n), T) \to Map(G(n),T)$$
is a Kan equivalence for $n \geq 2$.
\item A map $g: Y \to X$ is a weak equivalence if for every Segal space $T$ the induced map
$$g^*: Map(Y,T) \to Map(X,T)$$
is a Kan equivalence.
\item A map between Segal Spaces $T \to U$ is a weak equivalence (fibration) if and only if it is a
Reedy equivalence (Reedy fibration).
\end{enumerate}
\end{theone}
The Segal space model structure also satisfies some crucial properties.
In particular it satisfies following properties:
\begin{enumerate}
\item Cofibrantly generated.
\item Left proper.
\item Compatible with Cartesian closure.
\end{enumerate}
Notice we did not say the Segal space model structure is right proper. Here is a counter-example:
\begin{exone}
Let $c: F(1) \to F(2)$ be the unique map that sends $0$ to $0$ and $1$ to $2$. We have pullback square
\begin{center}
\pbsq{F(0) \coprod F(0) }{F(1)}{G(2)}{F(2)}{}{}{c}{\varphi_2}
\end{center}
The map $c$ is a Segal fibration as it is a Reedy fibration between Segal spaces. Moreover, $\varphi_2$ is a Segal equivalence.
However, the pullback is clearly not a Segal equivalence, as $F(0) \coprod F(0)$ is not equivalent to $F(1)$.
\end{exone}
\mathscr{M}athscr{S}ubsection{Complete Segal Space Model Structure} \label{Subsec Complete Segal Space Model Structure}
The Segal space model structure is a Bousfield localization of the Reedy model structure.
\begin{theone}
\cite[Theorem 7.2]{Re01}
There is a unique, simplicial model structure on $s\mathscr{M}athscr{S}$ called the complete Segal space model structure,
characterized as follows.
\begin{enumerate}
\item Cofibrations are inclusions.
\item A simplicial space $W$ is fibrant if it is a Segal space and the map
$$0^*: Map(E(1), W) \to Map(F(0),W)$$
is a Kan equivalence.
\item A map $g: X \to Y$ is a weak equivalence if for every complete Segal space $W$ the induced map
$$g^*: Map(Y,W) \to Map(X,W)$$
is a Kan equivalence.
\item A map between complete Segal Spaces $W \to V$ is a weak equivalence (fibration) if and only if it is a
Reedy equivalence (Reedy fibration).
\end{enumerate}
\end{theone}
The complete Segal space model structure also satisfies some crucial properties.
In particular it satisfies following properties:
\begin{enumerate}
\item Cofibrantly generated.
\item Left proper.
\item Compatible with Cartesian closure.
\end{enumerate}
Using the completeness condition we can characterize CSS equivalences in a more simple manner.
\begin{theone}
\cite[Theorem 7.7]{Re01}
A map $f: T \to U$ of Segal space is a CSS equivalence if and only if it is fully faithful and essentially surjective.
\end{theone}
The CSS model structure is also not right proper, as the example above still holds in this case, however, there are some
restricted cases where right properness condition holds
\begin{theone}
\cite[Theorem 7.26]{Ra17b}
Let $C \to W$ be a coCartesian fibration and $V \to W$ be a CSS (Segal) equivalence.
Then the pullback map $V \times_W C \to C$ is a CSS (Segal) equivalence.
\end{theone}
\end{document} | math |
ممبئی بالی وڈ اداکار سلمان خان کے بھائی ارباز خان نے انڈین پریمیئر لیگ میں سٹے بازی کا اعتراف کرلیاممبئی پولیس نے گزشتہ دنوں سٹے باز سونو جالان کو گرفتار کیا تھا جس نے انڈین پریمیئر لیگ میں بالی وڈ سمیت کئی نامور شخصیات کی جانب سے جوا کھیلنے کا اعتراف کیا تھا ممبئی پولیس نے سونو جالان کے اعتراف کے بعد ارباز خان کو انڈین پریمیئر لیگ میں سٹے بازی میں ملوث ہونے پر سمن جاری کیا تھا اور بیان ریکارڈ کرانے کی ہدایت کی تھیارباز خان پر ائی پی ایل میں کروڑ 75 لاکھ بھارتی روپے ہارنے کا الزام ہے جس کا اعتراف اج انہوں نے تفتیش کے دوران کیا بھارتی میڈیا کے مطابق ممبئی پولیس کے انسداد بھتہ سیل نے ارباز خان اور سونو جالان کو امنے سامنے بٹھا کر بیان ریکارڈ کرایا جس پر اداکار نے سٹے بازی کا اعتراف کرتے ہوئے کہا کہ وہ طلاق اور خراب مالی حالات کی وجہ سے ڈپریشن کا شکار تھے جس کے باعث وہ شوقیہ طور پر سٹے بازی کرتے رہے اداکار نے کہا کہ ان کی شادی ٹوٹنے کی ایک وجہ سٹے بازی ہی ہے جس کی وجہ سے ان کی اہلیہ سے متعدد بار لڑائی ہوئی جب کہ بڑے بھائی سلمان خان بھی اس عادت سے اس قدر ناراض تھے کہ ایک بار انہیں مارنے کے لیے اٹھ کھڑے ہوئے تھےارباز خان نے بتایاکہ سٹے بازی کی وجہ سے انہیں گھر میں بھی روک ٹوک کی جاتی تھی ان کی سابقہ اہلیہ اور والد بھی مسلسل اس عمل سے روکتے تھےبالی وڈ اداکار نے بیان میں کہا کہ وہ سالوں سے ائی پی ایل میں سٹے بازی کررہے ہیں اور گزشتہ سال کروڑ 75 لاکھ بھارتی روپے ہارنے کے بعد بھی سونو کے ذریعے پیسے جیتتے رہے ہیںیاد رہے کہ گزشتہ سال ارباز خان اور ملائکہ اروڑہ کے درمیان طلاق ہوگئی تھی اور دونوں کی شادی 18 سال تک قائم رہی تھی | urdu |
చిరంజీవి పవన్ ల స్పీడ్ వెనుక ఆంతర్యం ! చిరంజీవి రాజకీయాలలోకి వెళ్ళి 9 సంవత్సరాలు ఫిలిం ఇండస్ట్రీకి దూరంగా ఉండి తిరిగి సినిమాల వైపు యూటర్న్ తీసుకుని ఖైదీ నెంబర్ 150 తో సంచలనాలు సృష్టించి వరసపెట్టి సినిమాలు చేస్తున్నాడు. పవన్ ఆంధ్రప్రదేశ్ ఎన్నికల ముందు జనసేన కసం సినిమాలకు విరామం ప్రకటించి ఎవరు ఊహించని విధంగా యూటర్న్ తీసుకుని వకీల్ సాబ్ తో తిరిగి తన సెకండ్ ఇన్నింగ్స్ ప్రారంభించడమే కాకుండా వరసపెట్టి అతడు చేస్తున్న సినిమాలు అందర్నీ ఆశ్చర్య పరుస్తున్నాయి.చిరంజీవి లేటెస్ట్ మూవీ ఆచార్య ఇంకా విడుదల కాకుండానే జయం రాజా దర్శకత్వంలో గాడ్ ఫాదర్ మూవీలో నటిస్తున్నాడు. ఈమూవీ షూటింగ్ జరుగుతూ ఉండగానే చిరంజీవి చేతికి ఆపరేషన్ జరగడంతో ప్రస్తుతం విశ్రాంతి తీసుకుంటున్నాడు. ఈసినిమా కనీసం 30 శాతం కూడా పూర్తి కాకుండానే మెహర్ రమేష్ దర్శకత్వంలో రూపొందింపబడుతున్న భోళాశంకర్ మూవీ షూటింగ్ ను నవంబర్ 15 నుంచి ప్రారంభించబోతున్నాడు.ఇది చాలదు అన్నట్లుగానే దర్శకుడు బాబి దర్శకత్వంలో చిరంజీవి నటించబోయే మూవీ షూటింగ్ వచ్చే నవంబర్ లోనే పూజా కార్యక్రమాలు జరుగుతాయి అని తెలుస్తోంది. ఇక పవన్ కళ్యాణ్ నటిస్తున్న భీమ్లా నాయక్ షూటింగ్ అదేవిధంగా క్రిష్ హరిహర వీరమల్లు మూవీలు పూర్తి కాకుండానే వచ్చేనెల పవన్ నటించబోయే మరో రీమేక్ మూవీ ప్రారంభోత్సవం జరగబోతోంది అంటూ వార్తలు వస్తున్నాయి.పవన్ హరీష్ శంకర్ దర్శకత్వంలో నటించవలసి ఉన్న భవదీయుడు భగత్ సింగ్ మూవీ షూటింగ్ ప్రారంభం కాకుండానే పవన్ వేరొక సినిమాను ప్రారంభిస్తున్నాడు అని వస్తున్న వార్తలు చూసి పవన్ చిరంజీవి లు ఇన్ని సినిమాలను ఎప్పటికి పూర్తి చేయగలరు అంటూ ఇండస్ట్రీ వర్గాలు ఆశ్చర్యపోతున్నాయి. టాప్ హీరోలు అంతా ఒక సినిమా తరువాత మరొక సినిమాను చేస్తుంటే దీనికి భిన్నంగా చిరంజీవి పవన్ లు అనుసరిస్తున్న ఈ కొత్త వ్యూహం ఇండస్ట్రీ హాట్ టాపిక్ గా మారింది అంటున్నారు..మండలి కొత్త చైర్మన్ ఎవరు.. జగన్ మనసులో ఎవరున్నారు..! మంచిమాట : మనం ఏ గింజ వేస్తే ఆ మొక్కే మొలుస్తుంది..!! స్టార్ హీరో ఫామ్హౌస్లో పేకాట...! చంద్రబాబుకు వైసీపీ త్రిమూర్తుల ఫీవర్...! కేసీఆర్ గారు టీచర్ పోస్టులు వేయండి సారు..? ఏపీలో ఆకలి కేకలు.. ఏం జరిగింది..? నాగశౌర్య ఫామ్ హౌస్ లో పేకాట కేసులో బిగ్ ట్విస్ట్..! పాదయాత్రకు నో బ్రేక్ అంటున్న షర్మిల...? బ్రేకింగ్: ఆర్టీసీ ఎండీగా సజ్జనార్ సక్సెస్...? సోర్స్: ఇండియాహెరాల్డ్.కామ్ Seetha Sailaja | telegu |
● Enjoy up to 20% off up to seven-day packages by using online coupon codes available from external coupon sites.
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The ARTbar Studio/ARTstarters - Dream Weaving Retreat & Open Studio!
Dream Weaving Retreat & Open Studio!
Bring your own projects or loom; or help a friend with theirs!
Make a weaving that represents your life or a special memory. A fun session to attend with a family member or friend!
$79 (includes a goody bag worth over $50 plus use of our fiber buffet: your own reusable wood loom, tapestry needle, weaving comb, fiber kit, charms, art bag and more!
We will have TONS of fibers, yarns and trims that you can use. You can also bring your own special items, memorabilia (think buttons, charms, special ribbons, fabrics, etc.). Or purchase charms and ribbons from our studio store.
Loom weavers will learn how to put their loom together, as well as several weaving techniques for a variety of patterns in our artwork, using fibers, fabric, ribbon, charms and other memorabilia! | english |
નવી સરકારે વચન નિભાવ્યું : પેટ્રોલડીઝલના ભાવમાં ઘટાડો મુંબઈ,તા.14 જુલાઈ 2022,ગુરૂવારભારે રાજકીય ઉથલપાથલ વચ્ચે નવી નિમાયેલ સરકારે ઈંધણના ભાવમાં ઘટાડો કરવાનું આપેલ વચન નિભાવ્યું છે. પેટ્રોલડીઝલના ભાવમાં ઘટાડો કરવાની બાંહેધરી બાદ આજે મહારાષ્ટ્રની નવી સરકારે ભાવમાં ઘટાડાની જાહેરાત કરી છે.મહારાષ્ટ્રમાં સરકારે ગુરૂવારે મોંઘવારીના વિષચક્રમાં પિસાયેલ સામાન્ય જનતાને રાહત આપતા પેટ્રોલડીઝલના ઘટાડાનો નિર્ણય કર્યો છે. બળવાખોર ધારાસભ્યોના સહારે બીજેપી સાથે બનેલ એકનાથ શિંદેની નવી સરકારે પેટ્રોલના ભાવમાં 5 રૂપિયા પ્રતિ લિટર અને ડીઝલના ભાવમાં 3 રૂપિયા પ્રતિ લિટર ઘટાડો કરવાની જાહેરાત કરી છે. નવી સરકારે રાજ્ય દ્વારા લેવાતા VATમાં ઘટાડો કરતા આ ભાવ ઘટશે. આ અગાઉ વડાપ્રધાન મોદીએ અને ગત મહિને એક્સાઈઝ ડ્યૂટી કાપની સાથે નાણામંત્રી નિર્મલા સીતારમણે જે રાજ્યોએ VATમાં કાપ નહોતો મુક્યો તેમને અરજી કરી હતી કે દેશમાં મોંઘવારીને કાબૂમાં લેવા માટે તેમના દ્વારા લેવાતા ટેક્સમાં ઘટાડો કરે. ત્યારે મહારાષ્ટ્રની ઉદ્ધવ સરકારે કોરોના મહામારીને કારણે રાજ્યને થયેલ આર્થિક નુકશાનની ભરપાઈ આ ટેક્સ દ્વારા થતો હોવાનું કહીને કેન્દ્ર સરકારને જ તેમના દ્વારા વસૂલાતી એક્સાઈઝ ડ્યૂટી ઘટાડવા માટે અરજી કરાઈ હતી અને કેન્દ્ર સરકારની અરજી ફગાવી હતી. જોકે ઉદ્ધવ સરકાર સાથે બળવો કરીને બીજેપી સાથે મળીને સત્તા હાંસલ કરનાર શિવસેનાના જ એકનાથ શિંદેએ આ VAT ઘટાડાનો નિર્ણય કર્યો છે. | gujurati |
A local citizen and former employee of Brunei Shell Petroleum (BSP) Company Sdn Bhd, Dayang Hajah Asnah binti Haji Sairan, was sentenced to four (4) years six (6) months imprisonment after being found guilty by the Magistrate’s Court Bandar Seri Begawan on 16 charges under section 6 (a) of the prevention of Corruption Act, Chapter 131 and punishable under section 7 (1) of the same act.
Investigation conducted by Anti Corruption Bureau revealed that between February 2008 and May 2009, Dayang Hjh Asnah bte Hj Sairan, who at that time held the position of Supply Chain Support (Seismic/Geomatics) at the Supply Chain Management Department in Brunei Shell Petroleum (BSP) Company Sdn Bhd had corruptly received gratification in a total sum amounting to BND$28,300.00 (Twenty Eight Thousand and Three Hundred Brunei Dollars) from the Manager of Musfada Enterprise as a reward for creating purchase orders for industrial goods and equipment for Brunei Shell Petroleum Sdn Bhd.
Senior Magistrate Azrimah binti Haji Abdul Rahman sentenced the defendant to four (4) years six (6) months imprisonment for all charges to be run concurrently. The Defendant is also to pay a penalty under Section 17 of the Prevention of Corruption Act of BND$28,300.00 (Brunei Dollars Twenty Eight Thousand and Three Hundred Brunei dollars) within 3 months from today and in default of payment, to serve 6 months of imprisonment which is to be served consecutively. The Defendant is also to pay the prosecution costs of BND$200,000.00 (Brunei Dollars Two Hundred Thousand) within 3 months from today and in default of payment, to serve an additional 3 months imprisonment consecutively. The defendant’s imprisonment sentence takes effect from 14th July 2018.
The Prosecution team was represented by Sharon Yeo of the Attorney General Chambers. | english |
## INDICATION:
year old woman with large dominant left mid thyroid nodule. //
Cytology?
## OPERATORS:
Dr. trainee and Dr. radiologist.
Dr. supervised the trainee during the entire procedure and
reviewed and agrees with the trainee's findings.
## FINDINGS:
Limited scanning of the thyroid was performed. Again identified is a large
ill-defined left lobe nodule which corresponds to the nodule recommended for
fine needle aspiration on prior diagnostic ultrasound. This nodule was
selected for fine needle aspiration.
## PROCEDURE:
The risks and benefits of the procedure were explained to the
patient, and written informed consent was obtained. The preprocedure time out
was performed per protocol. An entrance site for the FNA was determine
over the left thyroid lobe. The patient was prepped and draped in usual
sterile fashion. 1% lidocaine was injected subcutaneously for local
anesthesia.
Using ultrasound guidance, 3 fine needle aspirates were obtained from the
thyroid nodule. One aspirate was obtained utilizing a 25 gauge needle. 2
aspirates were obtained utilizing 27 gauge needles. Samples were submitted in
Cytolyt. No periprocedural complications were encountered. The patient
tolerated the procedure well and was discharged in stable condition.
## IMPRESSION:
Technically successful fine needle aspiration of the left thyroid nodule. No
periprocedural complications. Cytology is pending.
| medical |
TISS Mumbai में परियोजना प्रबंधक के पदों पर भर्ती सरकारी नौकरीटाटा सामाजिक विज्ञान संस्थान मुबंई ने परियोजना प्रबंधक के पदों पर भर्तियां निकाली हैँ। जिन उम्मीदवारों के पास संबंधित विषय में स्नातकोत्तर डिग्री हैं और अनुभव हैँ। तो आज ही इन पदों के लिए आवेदन करें और पाए सरकारी नौकरी। इस मौके को अपने हाथ से जाने दें आज ही आवेदन करें। महत्वपूर्ण तिथि व सूचनाएं पद का नाम परियोजना प्रबंधक कुल पद 1 अंतिम तिथि 25 2 202 2 स्थान मुबंई टाटा सामाजिक विज्ञान संस्थान पद भर्ती विवरण 2021 आयु सीमा उम्मीदवारों की अधिकतम आयु विभाग के नियमानुसार मान्य होगी और आरक्षित वर्ग को आयु सीमा में छूट दी जाएगी। वेतन जिन उम्मीदवारों का चयन इन पदों के लिए किया जाएगा उन्हें 40000 वेतन मिलेगा। योग्यता उम्मीदवारों को किसी भी मान्यता प्राप्त संस्थान से सोशल वर्क में स्नातकोत्तर डिग्री प्राप्त हो और 3 साल अनुभव हो। आवेदन शुल्क: कोई आवेदन शुल्क नही है। चयन प्रक्रिया उम्मीदवार का साक्षात्कार के आधार पर चयन होगा। कैसे करें आवेदन योग्य और इच्छुक उम्मीदवार आवेदन पत्र के निर्धारित प्रारूप पर आवेदन करते हैं , साथ ही शिक्षा और अन्य योग्यता , जन्मतिथि की तिथि और अन्य आवश्यक जानकारी और दस्तावेजों के साथ स्वयं प्रतिबंधात्मक प्रतियां और नियत तारीख से पहले भेजते हैं। | hindi |
ஜனவரி 14 முதல் 18 வரை டாஸ்மாக் கடைகள் செயல்பட அனுமதி ரத்து? நீதிமன்றத்தின் முடிவு என்ன? ஜனவரி 14 முதல் 18 வரை டாஸ்மாக் கடைகள் செயல்பட அனுமதி ரத்து? நீதிமன்றத்தின் முடிவு என்ன? வரும் நாட்களில் தமிழகத்தில் பொங்கல் பண்டிகை வர உள்ளது. பண்டிகை முன்னிட்டு தமிழக அரசு பல கட்டுப்பாடுகளை தொடர்ந்து அமல்படுத்தி வருகிறது. நேற்று தலைமை செயலகத்தில் நடைபெற்ற ஆலோசனைக் கூட்டத்தில் வரும் வெள்ளிக்கிழமை முதல் 18ஆம் தேதி வரை அனைத்து ஆலயங்களிலும் வழி தடை தடை செய்யப்பட்டுள்ளது. அதேபோல இந்த இரவு ஊரடங்கு ஆனது இம்மாதம் 30ஆம் தேதி வரை நீடித்து உள்ளனர். இவ்வாறு இருக்கையில் தமிழகத்தில் ஜல்லிக்கட்டு போட்டி நடத்த தமிழக அரசு அனுமதி அளித்துள்ளது. தற்போது தொற்று அதிக அளவு பரவி வருவதால் தான் தமிழக அரசு ஊரடங்கு அமல் படுத்தி வருகிறது. இந்த சூழலில் தமிழகத்தில் பண்டிகை நாட்களில் ஜல்லிக்கட்டு நடத்த அனுமதி அளித்துள்ளதால் அதிகளவு தொற்று பரவும் அபாயம் ஏற்படும். அதனால் கொரோனா தொற்று குறைந்த பிறகு ஜல்லிக்கட்டு நடத்த கோரி உயர் நீதிமன்றம் உத்தரவிட வேண்டும் என்று ராம்குமார் ஆதித்தன் என்பவர் மனு அளித்துள்ளார். அதாவது தமிழகத்தில் ஜல்லிக்கட்டு ஜனவரி 10ஆம் தேதி நடைபெறுவதாக இருந்த இதற்கு இடைக்கால தடை விதிக்குமாறு மனுதாரர் கேட்டுள்ளார். அதுமட்டுமின்றி பண்டிகை நாட்களில் செயல்பட உள்ள டாஸ்மாக் கடைகளை அனுமதி ரத்து செய்யும்படி கோரிக்கை விடுத்துள்ளார். ஜனவரி 14 முதல் ஜனவரி 18ஆம் தேதி வரை விழா காலங்களில் டாஸ்மாக் செயல்பட்டால் மக்கள் அதிகளவு கூட்டம் கூடுவர். அதனை தடுக்க உயர் நீதிமன்றம் டாஸ்மாக் செயல்படும் அனுமதி ரத்து செய்ய வேண்டும் என மனுதாரர் தனது மனுவில் கூறியுள்ளார்.இந்த மனுவை குறித்து உயர்நீதிமன்றத்தின் முடிவுகளை மக்கள் பெருமளவு எதிர்பார்த்து காத்துக்கொண்டுள்ளனர். | tamil |
أمس چھ کھڑا گژھنس تہٕ لاٹھی ہٕنٛد مدتہٕ سۭتۍ پکنس منٛز دشوٲری | kashmiri |
Gujarat : हाई कोर्ट ने माना विवाहेत्तर संबंध के चलते पुलिसकर्मी को नहीं कर सकते बर्खास्त, मुआवजे के साथ नौकरी बहाली का आदेश गुजरात उच्च अदालत ने व्यभिचार के आरोप में पुलिस सेवा से बर्खास्त किये गए पुलिसकर्मी की याचिका पर सुनवाई करते हुए बड़ा फैसला सुनाया है। हाई कोर्ट ने कहा समाज की नजरों में भले ही विवाहेत्तर संबंध Extramarital Relationship अनैतिक काम हो सकता है, लेकिन इसे पुलिस सेवा नियमों के चश्मे से दुराचार का रूप नहीं माना जा सकता। गुजरात हाई कोर्ट ने व्यभिचार के आरोप में बर्खास्त अहमदाबाद कांस्टेबल को बहाल करने का आदेश देते हुए कहा कि जस्टिस संगीता विशेन ने बर्खास्तगी के आदेश को खारिज करते हुए कहा कि, याचिकाकर्ता के मामले में तथ्य को देखते हुए कहा जा सकता है कि यह उसका निजी मामला था। इसमें किसी भी तरह की जोरजबरदस्ती या शोषण नहीं किया गया था। साथ ही अदालत ने शहर की पुलिस को एक महीने के भीतर कांस्टेबल को फिर से नियुक्त करने और 25 प्रतिशत वेतन का भुगतान करने का निर्देश दिया। लोकप्रिय खबरें मोदी जी 20 साल और राज करेंगे तब भी नेहरू को ही दोष देंगे डॉ. मनमोहन सिंह का जिक्र कर PM पर भड़क गए बॉलीवुड एक्टर योगी आदित्यनाथ की जीत हुई तो इंडिया कभी नहीं लौटूंगा बॉलीवुड एक्टर ने खाई कसम, लोग करने लगे ऐसे सवाल मोदी जी जुमले छोड़ रहे हैं तो ये पीछे क्यों रहते. नितिन गडकरी पर भड़के बॉलीवुड एक्टर, BJP को बताया इतिहास की सबसे भ्रष्ट सरकार UP Election: मुख्तार के बाद बड़े भाई सिबगतुल्लाह को भी हटाया, फूटफूटकर रोए अब्दुल अंसारी, पूर्वांचल में बारबार क्यों प्रत्याशी बदल रही सपा दरअसल, याचिकाकर्ता पुलिसकर्मी अपने परिवार के साथ शाहीबाग में पुलिस क्वार्टर में रहता था, जहां उसकी मुलाकात उसी कॉलोनी में रहने वाली एक विधवा से हुई थी। महिला के परिवार ने रिश्ते के सबूत इकट्ठा करने के लिए क्वार्टर में सीसीटीवी कैमरे लगाए, जिसके आधार पर उन्होंने 2012 में पुलिस आलाकमान से शिकायत की थी। मामले में शिकायत के बाद दोनों ने अपने रिश्ते को स्वीकार किया, जिसके बाद पुलिस आलाकमान ने पुलिसकर्मी को कारण बताओ नोटिस जारी किया था। इसके बाद साल 2013 में उसे नैतिक पतन के आधार पर सेवा से बर्खास्त कर दिया, जो कि पुलिस में जनता के विश्वास को कम करने के लिए था। ऐसे में पुलिसकर्मी ने अपनी बर्खास्तगी के खिलाफ अदालत का रुख करते हुए कहा कि उसे बर्खास्त करने से पहले जांच की किसी प्रक्रिया का पालन नहीं किया गया था। जबकि उसने यह कहा था कि दोनों में संबंध सहमति से बने थे, इसलिए उसके द्वारा महिला का शोषण करने का कोई सवाल ही नहीं उठता है। न्यायमूर्ति विशेन ने कहा पुलिसकर्मी के मामले में व्यभिचार के आरोप में सरकारी कर्मचारियों पर विभिन्न निर्णयों का हवाला दिया। साथ ही कहा, वहीं अदालत ने पुलिस विभाग की भी खिंचाई की और कहा कि मामले में शर्मिंदगी से बचने के बहाने जांच शुरू की गई। साथ ही पुलिस आलाकमान ने बिना कानूनी कारणों के बर्खास्तगी का आदेश जारी किया और सेवा नियमों का हवाला देते हुए केवल को अपनाया। | hindi |
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During the UFC 225 post-fight press conference, UFC president Dana White said the official weigh-ins would be moved back to late afternoon instead of the mid-morning time slot. The weigh-ins used to take place in the late afternoon, but were moved to the morning (generally 9 a.m. locally) in 2016 citing fighter safety.
White stated that the fight promotion has interviewed multiple fighters about the planned move and that's what the majority want.
“Believe me, we’ve studied it. The numbers don’t lie. We talked to fighters. A ton of fighters want to go back to 4 o’clock. And there’s a lot of fighters that don’t," he said. “No matter what, there’s no debate about this. There’s no debate. We’re going to 4 o’clock."
One fighter that wasn't asked about moving the weigh-ins back to late afternoon was former lightweight title contender Al Iaquinta. Not only was Iaquinta not talked to about the policy change, he doesn't believe White spoke to as many fighters as he claimed.
“His sense of reality is gone. It’s gone. It’s shot. It’s absolutely shot. It’s scary. Now he’s putting our health at risk," said Iaquinta during an interview on The MMA Hour.
“Why did you move the weigh-ins in the first place? Because he (White) wanted more exciting, in his mind, he wants more exciting fights. How are you going to get more exciting fights? Give the fighters more time to recover, which is also beneficial to us because it’s healthy. It’s healthy for us to replenish. But what he realizes is now people aren’t making weight more, so this was actually a bad move.
"Even though it was healthier for the fighters, we’re going to move it back to the nighttime because he interviewed a million fighters. I haven’t seen one that wants it at night. You do an interview with him now; it’s not about the truth. It’s about his agenda. I’m weighing in in the morning,” said Iaquinta.
In July 2016, UFC sold to WME–IMG (which was later rebranded as Endeavor) for a staggering $4.2 billion. White remained president of the fight promotion, but Iaquinta believes White has been unleashed since the sale.
“Lorenzo (Fertitta) kept him in check. Now, I think that he’s going to run this thing into the ground. Now, he’s just saying whatever he wants to say. No one is keeping him under wraps. He’s saying whatever he wants to say. You’re just going to move the weigh-in? What fighters did he interview? He said, ‘we interviewed – we’ve got 550 fighters on the roster. We interviewed more of them than you.’ I don’t think so. I haven’t talked to a fighter that was interviewed or was talked to about moving the weigh-ins back to the nighttime. I think it’s about him. It’s all about him. It’s always about him, and if it’s not about him, he wants to get right in there," said Iaquinta.
"Ragin' Al" would like to see White part with the organization. He believes that White has one foot out the door and should step through with the other one.
“He did great. He’s got one foot out the door. Just go man. Go. Go. He’s done," he said. "And I don’t care. I’m going to say it. I think 50 percent of the people don’t see it and 50 percent of people are just too scared to say it." | english |
అమెరికా: ఆర్ఎస్ఎస్ అనుబంధ హెచ్ఎస్ఎస్కు సన్మానం.. టెక్సాస్ పోలీస్ శాఖపై జర్నలిస్ట్ ఆగ్రహం గత నెల 21న రాష్ట్రీయ స్వయం సేవక్ సంఘ్ ఆర్ఎస్ఎస్ అంతర్జాతీయ విభాగమైన హిందూ స్వయం సేవక్ సంఘ్ హెచ్ఎస్ఎస్ ఇర్వింగ్ శాఖను టెక్సాస్ పోలీస్ డిపార్ట్మెంట్ సత్కరించింది. అయితే పోలీసుల చర్యను ప్రముఖ అమెరికన్ జర్నలిస్ట్, కార్యకర్త పీటర్ ఫ్రీడ్రిక్ తప్పుబట్టారు. గతంలో హిందుత్వకు వ్యతిరేకంగా గళం విప్పిన ఈయన.. నగర కౌన్సిల్ సమావేశానికి హాజరయ్యారు. హెచ్ఎస్ఎస్కు పోలీస్ శాఖ సత్కారం.. అమెరికా సంయుక్త రాష్ట్రాలలో నివసిస్తున్న భారతీయ మైనారీటీను ఎలా కలవరపెడుతోందో వివరించారు. 2002లో భారతదేశంలోని గుజరాత్ వీధుల్లో ఆర్ఎస్ఎస్ శ్రేణులు వేలాది మంది ముస్లింలను చంపడం, అత్యాచారాలకు పాల్పడటం, సజీవదహనం వంటి అకృత్యాలకు పాల్పడుతున్నా పోలీసులు బొమ్మల్లా చూస్తూ నిలబడ్డారని .. దీంతో ఈ మారణకాండ నిరాంతరాయంగా సాగిందని ఫ్రీడ్రిక్ వ్యాఖ్యానించారు. భారతదేశంలో ఆర్ఎస్ఎస్ పాత్ర గురించి ఆయన చెబుతూ.. ఇండియాలోని పోలీసులు ఆర్ఎస్ఎస్తో కుమ్మక్కయ్యారని ఆరోపించారు. ఆర్ఎస్ఎస్ వ్యవస్థాపకులు భారతీయ ముస్లింలను, క్రైస్తవులను దేశద్రోహులుగా చిత్రీకరించారని ఫ్రీడ్రిక్ మండిపడ్డారు. యూరోపియన్ ఫాసిస్ట్ ఉద్యమాలను ఆదర్శంగా తీసుకోవాలని... ప్రధానంగా యూదుల పట్ల నాజీలు వ్యవహరించిన విధానాన్ని అనుసరించాలనే ఉద్దేశాన్ని గతంలోనే ఆర్ఎస్ఎస్ వెలిబుచ్చిందని పీటర్ గుర్తుచేశారు.. గడిచిన 20 ఏళ్లలో భారతదేశంలో ఆర్ఎస్ఎస్ నిర్వహించిన ప్రతి ప్రధాన మైనారిటీ వ్యతిరేక కార్యక్రమాలకు పోలీసుల సహకారం వుందని పీటర్ ఆరోపించారు. 2002లో గుజరాత్ అల్లర్ల నుంచి 2008లో ఒడిషా క్రైస్తవ సన్యాసినుల సామూహిక అత్యాచారం, 2020లో ఢిల్లీలో పోలీసులే అల్లరి మూకలకు తలుపులు బద్ధలుకొట్టి మరి ముస్లింలను అప్పగించడం వరకు రక్షకభటులు ఆర్ఎస్ఎస్కు మద్ధతుగా నిలిచారని ఆయన చెప్పారు.. తాజాగా అమెరికాలోని ఇర్వింగ్ పోలీసులు నిలబడి హెచ్ఎస్ఎస్ని గౌరవించినప్పుడు.. లెక్కలేనంతమంది భారతీయ అమెరికన్ క్రైస్తవులు, దళితులు, ముస్లింలు, సిక్కులు.. ఎవరైతే ఆర్ఎస్ఎస్ వల్ల నష్టపోయారో వారికి ఈ చర్య ఎలాంటి సందేశాన్ని పంపుతుందని ఫ్రీడ్రిక్ ప్రశ్నించారు. కాగా, గతంలో హిందుత్వకు వ్యతిరేకంగా మాట్లాడిన సమయంలో తనకు హిందుత్వ వాదుల నుంచి తనకు సోషల్ మీడియాలో బెదిరింపులు వస్తున్నాయని చెప్పారు. ఈ ఏడాది ఆగస్టులో రాష్ట్రీయ స్వయం సేవక్ సంఘ ఆర్ఎస్ఎస్ మద్ధతున్న ఎన్జీవో సంస్థ సేవా ఇంటర్నేషనల్కు ట్విట్టర్ సీఈవో జాక్ డోర్సే 2.5 మిలియన్ డాలర్లు విరాళం ఇచ్చినందుకు నిరసనగా ఫ్రీడ్రిక్ నిరాహార దీక్ష చేసిన సంగతి తెలిసిందే. ప్రతి రోజు ఇలాంటి తెలుగు వార్త విశేషాలు కోసం తెలుగుస్టాప్ ని ఫాలో అవ్వండి , తెలుగుస్టాప్ వెబ్ సైట్ ని చూడండి.ఈ ఆర్టికల్ ని తోటి తెలుగు మిత్రులకి షేర్ చేయండి. Source : TeluguStop.com , Author : Shiva Naga Prasad | telegu |
ബില് പേയ്മെന്്റ് എളുപ്പത്തിലാക്കാന് ക്ലിക്ക് പേ മുംബൈ: എന്പിസിഐ ഭാരത് ബില്പേ ലിമിറ്റഡുമായി NBBL സഹകരിച്ച് ഉപഭോക്താക്കള്ക്കായി ക്ലിക്ക്പേ ആരംഭിക്കുമെന്ന പ്രഖ്യാപനവുമായി ഇന്ത്യയിലെ മുന്നിര ഡിജിറ്റല് പേയ്മെന്റ് പ്ലാറ്റ്ഫോമായ ഫോണ്പേ. വൈദ്യുതി, വെള്ളം, ഗ്യാസ്, വായ്പ തിരിച്ചടവ് തുടങ്ങിയ ആവര്ത്തിച്ചുള്ള ഓണ്ലൈന് ബില് പേയ്മെന്റുകള് നടത്താന് ഉപഭോക്താക്കള്ക്ക് ക്ലിക്ക് പേ കൂടുതല് സൗകര്യമേകും. ക്ലിക്ക് പേ പേയ്മെന്റ് ലിങ്ക് ഉപയോഗിക്കുന്നതിലൂടെ, ഓരോ ബില്ലര് സേവനവുമായി ബന്ധപ്പെട്ട അക്കൗണ്ട് വിശദാംശങ്ങള് ഓര്മിക്കേണ്ടതിന്റെ ആവശ്യകതയും ഇല്ലാതാകും. ബില്ലര് അയയ്ക്കുന്ന ഈ ലിങ്ക് ഉപഭോക്താവിനെ നേരിട്ട് പേയ്മെന്റ് പേജിലേക്ക് നയിക്കുകയും ബില് തുക ഉടനടി തന്നെ കാണിക്കുകയും ചെയ്യും. | malyali |
2,600 டன் யூரியா இருப்பு: விற்பனையாளர்கள் கண்காணிப்பு பொள்ளாச்சிகாரைக்காலுக்கு கப்பலில் வந்த யூரியா உரம், அங்கிருந்து சரக்கு ரயில் வாயிலாக பொள்ளாச்சிக்கு கொண்டு வரப்பட்டது. ரயிலில் வந்த, 1,245 டன் யூரியாவில், கோவை மாவட்டத்துக்கு, 740 டன், திருப்பூர் மாவட்டத்துக்கு, 438 டன், நீலகிரி மாவட்டத்துக்கு, 31 டன் ஒதுக்கப்பட்டுள்ளது.இது குறித்து, மாவட்ட வேளாண் இணை இயக்குனர் சித்ராதேவி அறிக்கை வருமாறு:கோவை மாவட்டத்துக்கு ஒதுக்கப்பட்டுள்ள, 740 டன் யூரியாவில், தொடக்க வேளாண் கூட்டுறவு சங்கங்களுக்கு, 325 டன் தனியார் உர விற்பனையாளர்களுக்கு, 214 டன் வினியோகிக்கப்படும்.மீதமுள்ள, 201 டன், இப்கோ நிறுவன சேமிப்பு கிடக்கில் இருப்பு வைக்கபட்டு, தேவைக்கேற்ப வினியாகிக்கப்படும். இந்த உரத்துடன் சேர்த்து, தற்போது கோவை மாவட்டத்தில், 2,600 டன் யூரியா இருப்பு உள்ளது.விவசாயிகளின் ஆதார் அட்டை அடிப்படையில், நில பரப்பு, பயிர் சாகுபடிக்கு ஏற்ப ஒருவருக்கு, 10 மூட்டை வரை உரம் விற்பனை செய்யலாம். சிட்டா ஆவணம் பெற்று, கூடுதல் உரம் விற்பனை செய்யலாம். ஒரு விவசாயிக்கு, ஒரு மாதத்துக்கு, டி.ஏ.பி., பொட்டாஷ், காம்பளக்ஸ், யூரியா உரங்களை, 50 மூட்டைக்கு மிகாமல் விற்பனை செய்ய வேண்டும். யூரியா உர விற்பனை, மாவட்ட கலெக்டர் அலுவலகம் வாயிலாக கண்காணிக்கப்படுகிறது. மேலும், உர கண்காணிப்பாளர்கள் வாயிலாக, உர விற்பனை தீவிரமாக கண்காணிக்கப்படுகிறது. ஒரு விவசாயியின் பெயரில் கூடுதல் உரம் விற்பனை, விவசாயம் அல்லாத பயன்பாட்டுக்கு மானிய விலை உரம் விற்பனை, வேறு மாநில, மாவட்ட விவசாயிகளுக்கு உர விற்பனை உள்ளிட்ட முறைகேடுகள் கண்டறியப்பட்டால், உரிமம் ரத்து செய்யப்படுவதுடன், கடும் நடவடிக்கை மேற்கொள்ளப்படும்.இவ்வாறு, தெரிவித்துள்ளார். | tamil |
ഭര്ത്താവിനെയും മക്കളെയും ഉപേക്ഷിച്ച് യുവതി പോയത് 28കാരനായ കാമുകനൊപ്പം കോടതി വിധിയും യുവതിക്ക് അനുകൂലം അഞ്ചലില് യുവാവ് വീടുകയറി ആക്രമിച്ചത് കുടുംബ ജീവിതം തകര്ന്നതോടെ കൊല്ലം: ഭാര്യയുടെ കാമുകനെയും അമ്മയേയും യുവാവ് വീട്ടില് കയറി ആക്രമിച്ചത് ഭാര്യ ഉപേക്ഷിച്ച് പോയതിന്റെ ദേഷ്യത്തില്. കഴിഞ്ഞ ദിവസമാണ് അഞ്ചലില് ഏറം കളീലിക്കട പ്ലാവിള പുത്തന്വീട്ടില് കൃഷ്ണകുമാരി 50, മകന് അഖില് 28 എന്നിവരെ വെട്ടിക്കവല സ്വദേശിയായ സജി വെട്ടി പരിക്കേല്പ്പിച്ചത്. കഴിഞ്ഞദിവസം സ്വന്തം വീട്ടില് വെച്ചാണ് അഖിലിനും മാതാവിനും വെട്ടേറ്റത്. രണ്ട് മാസം മുന്പ് സജിയുടെ ഭാര്യ അയാളെയും മക്കളെയും ഉപേക്ഷിച്ച് അഖിലിനോടൊപ്പം പോവുകയും ഏറത്തെ വീട്ടിലെത്തി താമസം തുടങ്ങുകയും ചെയ്തിരുന്നു. ഇത് സംബന്ധിച്ച് സജി കൊട്ടാരക്കര പൊലീസ് സ്റ്റേഷനില് പരാതി നല്കിയിരുന്നു. എന്നാല് ഇത് സംബന്ധിച്ച കേസില് യുവതിക്ക് അനുകൂലമായിട്ടായിരുന്നു കോടതി വിധി വന്നത്. കഴിഞ്ഞ ദിവസം യുവതിയുടെ വസ്ത്രങ്ങളും മറ്റ് സാധനങ്ങളും നല്കാനെന്ന വ്യാജേന സജി അഖിലിന്റെ വീട്ടിലെത്തിയത്. എന്നാല് നേരത്തെ തന്നെ ആക്രമണത്തിന് പദ്ധതിയിട്ടിരുന്ന സജി യുവതിയെ ആക്രമിക്കാന് ഒരുങ്ങിയപ്പോള് തടയുന്നതിനിടെയാണ് അഖിലിനും മാതാവിനും വെട്ടേറ്റത്. അഖിലിന്റെ ഇടത് കൈയിനും കൃഷ്ണകുമാരിയുടെ വലത് കൈയിനുമാണ് വെട്ടേറ്റത്. പ്രതിയെ പൊലീസ് അറസ്റ്റ് ചെയ്തു. സജിയെ ഇന്ന് പുനലൂര് കോടതിയില് ഹാജരാക്കുമെന്ന് അഞ്ചല് പോലീസ് അറിയിച്ചു. Tags | malyali |
\begin{document}
\special{papersize=8.5in,11in}
\setlength{\pdfpageheight}{\paperheight}
\setlength{\pdfpagewidth}{\paperwidth}
\title{Towards Parallel Boolean Functional Synthesis}
\author{S. Akshay\inst{1} \and Supratik Chakraborty\inst{1} \and Ajith K. John\inst{2} \and Shetal Shah\inst{1}}
\institute{IIT Bombay, India
\and
HBNI, BARC, India}
\maketitle
\begin{abstract}
Given a relational specification $\varphi(X, Y)$, where $X$ and $Y$
are sequences of input and output variables, we wish to synthesize
each output as a function of the inputs such that the specification
holds. This is called the Boolean functional synthesis problem and has
applications in several areas. In this paper, we present the first
parallel approach for solving this problem, using compositional and
CEGAR-style reasoning as key building blocks. We show by means of
extensive experiments that our approach outperforms existing tools on
a large class of benchmarks.
\end{abstract}
\keywords{Synthesis, Skolem functions, Parallel algorithms, CEGAR}
\section{Introduction}
\label{sec:introduction}
Given a relational specification of input-output behaviour,
synthesizing outputs as functions of inputs is a key step in
several applications, viz. program repair~\cite{JGB05}, program
synthesis~\cite{Gulwani}, adaptive control~\cite{RW87} etc. The
synthesis problem is, in general, uncomputable. However, there are
practically useful restrictions that render the problem solvable, e.g., if all inputs and outputs are Boolean, the problem is
computable in principle. Nevertheless, functional synthesis may still
require formidable computational effort, especially if there are a
large number of variables and the overall specification is
complex. This motivates us to investigate techniques for Boolean
functional synthesis that work well in practice.
Formally, let $X$ be a sequence of $m$ input Boolean variables, and
$Y$ be a sequence of $n$ output Boolean variables. A relational
specification is a Boolean formula $\varphi(X, Y)$ that expresses a
desired input-output relation. The goal in Boolean functional
synthesis is to synthesize a function
$F: \{0,1\}^m\rightarrow \{0,1\}^n$ that satisfies the specification.
Thus, for every value of $X$, if there exists some value of $Y$ such
that $\varphi(X,Y)=1$, we must also have $\varphi(X,F(X)) = 1$. For
values of $X$ that do not admit any value of $Y$ such that
$\varphi(X,Y) = 1$, the value of $F(X)$ is inconsequential. Such a
function $F$ is also refered to as a \emph{Skolem function} for $Y$ in
$\varphi(X,Y)$~\cite{bierre,fmcad2015:skolem}.
An interesting example of Boolean functional synthesis is the problem
of integer factorization. Suppose $Y_1$ and $Y_2$ are $n$-bit
unsigned integers, $X$ is a $2n$-bit unsigned integer and
$\times_{[n]}$ denotes $n$-bit unsigned multiplication. The
relational specification $\varphi_{\ensuremath{\mathsf{fact}}}(X, Y_1, Y_2) \equiv
((X=Y_1\times_{[n]} Y_2) \wedge (Y_1\neq 1) \wedge (Y_2\neq 1))$
specifies that $Y_1$ and $Y_2$ are non-trivial factors of $X$. This
specification can be easily encoded as a Boolean relation.
The corresponding synthesis problem requires
us to synthesize the factors $Y_1$ and $Y_2$ as functions of $X$,
whenever $X$ is non-prime. Note that this problem is known to be hard,
and the strength of several cryptographic systems rely on this
hardness.
Existing approaches to Boolean functional synthesis vary widely in
their emphasis, ranging from purely theoretical treatments
(viz.~\cite{boole1847,lowenheim1910,deschamps1972,boudet1989,martin1989,baader1999})
to those motivated by practical tool development
(viz.~\cite{macii1998,bierre,Jian,Trivedi,fmcad2015:skolem,rsynth,KS00,BCK09,Gulwani,SRBE05,KMPS10}). A
common aspect of these approaches is their focus on sequential
algorithms for synthesis. In this paper, we present, to the best of
our knowledge, the first parallel algorithm for Boolean functional
synthesis. A key ingredient of our approach is a technique for solving
the synthesis problem for a specification $\varphi$ by composing
solutions of synthesis problems corresponding to sub-formulas in
$\varphi$. Since Boolean functions are often represented using
DAG-like structures (such as circuits, AIGs~\cite{aiger},
ROBDDs~\cite{akers1978,bryant1986}), we assume w.l.o.g. that $\varphi$
is given as a DAG. The DAG structure provides a natural decomposition
of the original problem into sub-problems with a partial order of
dependencies between them. We exploit this to design a parallel
synthesis algorithm that has been implemented on a message passing
cluster. Our initial experiments show that our algorithm
significantly outperforms state-of-the-art techniques on several
benchmarks.
\paragraph{Related work:}
The earliest solutions to Boolean functional synthesis date back to
Boole~\cite{boole1847} and Lowenheim~\cite{lowenheim1910}, who
considered the problem in the context of Boolean unification.
Subsequently, there have been several investigations into theoretical
aspects of this problem (see
e.g.,~\cite{deschamps1972,boudet1989,martin1989,baader1999}). More
recently, there have been attempts to design practically efficient
synthesis algorithms that scale to much larger problem
sizes. In~\cite{bierre}, a technique to synthesize $Y$ from a proof of
validity of $\forall X \exists Y \varphi(X,Y)$ was proposed. While
this works well in several cases, not all specifications admit the
validity of $\forall X \exists Y \varphi(X,Y)$. For example, $\forall
X \exists Y \varphi_{\ensuremath{\mathsf{fact}}}(X,Y)$ is not valid in the factorization
example. In~\cite{Jian,Trivedi}, a synthesis approach based on
functional composition was proposed. Unfortunately, this does not
scale beyond small problem instances~\cite{fmcad2015:skolem,rsynth}.
To address this drawback, a CEGAR based technique for synthesis
from \emph{factored} specifications was proposed
in~\cite{fmcad2015:skolem}. While this scales well if each factor in
the specification depends on a small subset of variables, its
performance degrades significantly if we have a few ``large'' factors,
each involving many variables, or if there is significant sharing of
variables across factors. In~\cite{macii1998}, Macii et al
implemented Boole's and Lowenheim's algorithms using ROBDDs and
compared their performance on small to medium-sized benchmarks. Other
algorithms for synthesis based on ROBDDs have been investigated
in~\cite{KS00,BCK09}. A recent work~\cite{rsynth} adapts the
functional composition approach to work with ROBDDs, and shows that
this scales well for a class of benchmarks with pre-determined
variable orders. However, finding a good variable order for an
arbitrary relational specification is hard, and our experiments show
that without prior knowledge of benchmark classes and corresponding
good variable orders, the performance of~\cite{rsynth} can degrade
significantly. Techniques using \emph{templates}~\cite{Gulwani}
or \emph{sketches}~\cite{SRBE05} have been found to be effective for
synthesis when we have partial information about the set of candidate
solutions. A framework for functional synthesis, focused on unbounded
domains such as integer arithmetic, was proposed in~\cite{KMPS10}.
This relies heavily on tailor-made smart heuristics that exploit specific
form/structure of the relational specification.
\section{Preliminaries}
\label{sec:preliminaries}
Let $X = (x_1, \ldots x_m)$ be the sequence of input variables, and $Y
= (y_1, \ldots y_n)$ be the sequence of output variables in the
specification $\varphi(X,Y)$. Abusing notation, we use $X$
(resp. $Y$) to denote the set of elements in the sequence $X$
(resp. $Y$), when there is no confusion. We use $1$ and $0$ to denote
the Boolean constants {\textsf{true}} and {\textsf{false}}, respectively.
A \emph{literal} is either a variable or its complement. An assignment
of values to variables \emph{satisfies} a formula if it makes the formula {\textsf{true}}.
\begin{wrapfigure}[12]{r}{0.6\textwidth}
\vspace*{-0.3in}
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child {
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child{node (x1){$x_1$}}
child{node (y1){$y_1$}}
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child{
node[main node] (6) {$\vee$}
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child { node[main node] (3) {$\wedge$}
child {node[main node] (6) {$\vee$}
child{node (x2){$\neg x_2$}}
child{node (y2){$0$}}
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child {node[main node] (7) {$\wedge$}
child {node (x3){$x_3$}}
child {node (y3){$\neg y_3$}}
child{node (t) {$\ldots$}}
}
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child { node {$\ldots$}[no edge below]}
child {
node[main node] (4) {$\wedge$}
child{node (s){$1$}}
child {node (8) {$x_{m-1}$}}
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child{node (x4){$x_m$}}
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\caption{DAG representing $\varphi(X,Y)$}
\label{fig:decomp}
\end{wrapfigure}
We assume that the specification $\varphi(X,Y)$ is represented as a
rooted DAG, with internal nodes labeled by Boolean operators and
leaves labeled by input/output literals and Boolean constants. If
the operator labeling an internal node $N$ has arity $k$, we assume
that $N$ has $k$ ordered children. Fig.~\ref{fig:decomp} shows an
example DAG, where the internal nodes are labeled by AND and
OR operators of different arities. Each node $N$ in such a DAG
represents a Boolean formula $\formula{N}$, which is inductively
defined as follows. If $N$ is a leaf, $\formula{N}$ is the label of
$N$. If $N$ is an internal node labeled by $\ensuremath{\mathsf{op}}$ with arity $k$, and
if the ordered children of $N$ are $c_1, \ldots c_k$, then
$\formula{N}$ is $\ensuremath{\mathsf{op}}(\formula{c_1}, \ldots \formula{c_k})$. A DAG
with root $R$ is said to represent the formula $\formula{R}$. Note
that popular DAG representations of Boolean formulas, such as AIGs,
ROBDDs and Boolean circuits, are special cases of this representation.
A $k$-ary Boolean function $f$ is a mapping from $\{0,1\}^k$ to
$\{0,1\}$, and can be viewed as the semantics of a Boolean formula
with $k$ variables. We use the terms ``Boolean function'' and
``Boolean formula'' interchangeably, using
formulas mostly to refer to specifications.
Given a Boolean formula $\varphi$ and a Boolean function $f$, we use
$\varphi[\subst{y}{f}]$ to denote the formula obtained by substituting
every occurrence of the variable $y$ in $\varphi$ with $f$. The set
of variables appearing in $\varphi$ is called the \emph{support} of
$\varphi$. If $f$ and $g$ are Boolean functions, we say that $f$
\emph{abstracts} $g$ and $g$ \emph{refines} $f$, if
$g \rightarrow f$, where $\rightarrow$ denotes logical implication.
Given the specification $\varphi(X, Y)$, our goal is to synthesize the
outputs $y_1, \ldots y_n$ as functions of $X$. Unlike some earlier
work~\cite{bierre,skizzo,jiang2}, we \emph{do not assume the validity
of $\forall X \exists Y~ \varphi(X,Y)$}. Thus, we allow the
possibility that for some values of $X$, there may be no value of $Y$
that satisfies $\varphi(X, Y)$. This allows us to accommodate some
important classes of synthesis problems, viz. integer factorization.
If $y_1=f_1(X), \ldots y_n=f_n(X)$ is a solution to the synthesis
problem, we say that $(f_1(X), \ldots f_n(X))$ \emph{realizes} $Y$ in
$\varphi(X, Y)$. For notational clarity, we simply use $(f_1, \ldots
f_n)$ instead of $(f_1(X), \ldots f_n(X))$ when $X$ is clear from the
context.
In general, an instance of the synthesis problem may not have a unique
solution. The following proposition, stated in various forms in the
literature, characterizes the space of all solutions, when we have one
output variable $y$.
\begin{proposition}
\label{prop:soln-space}
A function $f(X)$ realizes $y$ in $\varphi(X, y)$ iff the following holds:\\
$\varphi[\subst{y}{1}] \wedge \neg \varphi[\subst{y}{0}]
~\rightarrow~ f(X)$ and $f(X) ~\rightarrow~
\varphi[\subst{y}{1}] \vee \neg \varphi[\subst{y}{0}]$.
\end{proposition}
As a corollary, both $\varphi[\subst{y}{1}]$ and
$\neg \varphi[\subst{y}{0}]$ realize $y$ in $\varphi(X,y)$.
Proposition~\ref{prop:soln-space} can be easily extended when we have
multiple output variables in $Y$. Let $\sqsubseteq$ be a total
ordering of the variables in $Y$, and assume without loss of generality that $y_1 \sqsubseteq
y_2 \sqsubseteq \cdots y_n$.
Let $\ensuremath{\overrightarrow{F}}$ denote the vector of Boolean functions $(f_1(X), \ldots
f_n(X))$. For $i \in \{1, \ldots
n\}$, define $\varphi^{(i)}$ to be $\exists y_1 \ldots
\exists y_{i-1}\, \varphi$, and $\varphi^{(i)}_{\ensuremath{\overrightarrow{F}}}$ to be
$(\cdots (\varphi^{(i)}[\subst{y_{i+1}}{f_{i+1}}]) \cdots
)[\subst{y_n}{f_n}]$, with the obvious modifications for $i = 1$ (no
existential quantification) and $i = n$ (no substitution). The
following proposition, once again implicit in the literature,
characterizes the space of all solutions ${\ensuremath{\overrightarrow{F}}}$ that realize $Y$ in
$\varphi(X,Y)$.
\begin{proposition}
\label{prop:soln-space-multi}
The function vector $\ensuremath{\overrightarrow{F}} = (f_1(X), \ldots f_n(X))$ realizes $Y = (y_1, \ldots
y_n)$ in $\varphi(X, Y)$ iff the following holds for every
$i \in \{1, \ldots n\}$:\\
$\varphi^{(i)}_{\ensuremath{\overrightarrow{F}}}[\subst{y_i}{1}] \wedge \neg \varphi^{(i)}_{\ensuremath{\overrightarrow{F}}}[\subst{y_i}{0}]
\rightarrow f_i(X)$, and
$f_i(X) \rightarrow
\varphi^{(i)}_{\ensuremath{\overrightarrow{F}}}[\subst{y_i}{1}] \vee \neg \varphi^{(i)}_{\ensuremath{\overrightarrow{F}}}[\subst{y_i}{0}].
$
\end{proposition}
Propositions~\ref{prop:soln-space} and \ref{prop:soln-space-multi} are
effectively used in~\cite{Jian,Trivedi,fmcad2015:skolem,rsynth} to
sequentially synthesize $y_1, \ldots y_n$ as functions of $X$.
Specifically, output $y_1$ is first synthesized as a function $g_1(X,
y_2, \ldots y_n)$. This is done by treating $y_1$ as the sole output
and $X \cup \{y_2,\ldots y_n\}$ as the inputs in $\varphi(X, Y)$. By
substituting $g_1$ for $y_1$ in $\varphi$, we obtain
$\varphi^{(2)} \equiv \exists y_1 \varphi(X, Y)$. Output $y_2$ can
then be synthesized as a function $g_2(X, y_3, \ldots y_n)$ by
treating $y_2$ as the sole output and $X \cup \{y_3, \ldots y_n\}$ as
the inputs in $\varphi^{(2)}$. Substituting $g_2$ for $y_2$ in
$\varphi^{(2)}$ gives $\varphi^{(3)}
\equiv \exists y_1 \exists y_2\, \varphi(X, Y)$. This process is then
repeated until we obtain $y_n$ as a function $g_n(X)$. The desired
functions $f_1(X), \ldots f_n(X)$ realizing $y_1, \ldots y_n$ can now
be obtained by letting $f_n(X)$ be $g_n(X)$, and $f_i(X)$ be $(\cdots
(g_i[\subst{y_{i+1}}{f_{i+1}(X)}]) \cdots)[\subst{y_n}{f_n(X)}]$, for
all $i$ from $n-1$ down to $1$. Thus, given $\varphi(X,Y)$, it
suffices to obtain $(g_1, \ldots g_n)$, where $g_i$ has support
$X \cup \{y_{i+1}, \ldots y_n\}$, in order to solve the synthesis
problem. We therefore say that $(g_1, \ldots g_n)$
\emph{effectively realizes} $Y$ in $\varphi(X, Y)$, and focus on obtaining
$(g_1, \ldots g_n)$.
Proposition~\ref{prop:soln-space} implies that for every
$i \in \{1, \ldots n\}$, the function
$g_i \equiv \varphi^{(i)}[\subst{y_i}{1}]$ realizes $y_{i}$ in
$\varphi^{(i)}$. With this choice for $g_i$, it is easy to see that
$\exists y_i\, \varphi^{(i)}$ (or $\varphi^{(i+1)}$) can be obtained
as $\varphi^{(i)}[\subst{y_i}{g_i}]
= \varphi^{(i)}[\subst{y_i}{\varphi^{(i)}[\subst{y_i}{1}]}]$. While
synthesis using quantifier elimination by
such \emph{self-substitution}~\cite{rsynth} has been shown to scale
for certain classes of specifications with pre-determined optimized
variable orders, our experience shows that this incurs significant
overheads for general specifications with unknown ``good'' variable
orders. An alternative technique for \emph{factored} specification
was proposed by John et al~\cite{fmcad2015:skolem}, in which initial
abstractions of $g_1, \ldots g_n$ are first computed quickly, and then
a CEGAR-style~\cite{CEGAR-JACM} loop is used to refine these
abstractions to correct Skolem functions. We use John et al's refinement
technique as a black-box module in our work; more on this is discussed
in Section~\ref{sec:simple}.
\begin{definition}
\label{def:cb0-cb1}
Given a specification $\varphi(X, Y)$, we define $\cba{y_i}(\varphi)$
to be the formula $\left(\neg \exists y_1 \ldots
y_{i-1}\, \varphi\right)[y_i \mapsto 0]$, and $\cbb{y_i}(\varphi)$ to
be the formula $\left(\neg \exists y_1
\ldots y_{i-1}\, \varphi\right)[y_i \mapsto 1]$,
for all $i \in \{1, \ldots n\}$\footnote{In~\cite{fmcad2015:skolem},
equivalent formulas were called $Cb0_{y_i}(\varphi)$ and
$Cb1_{y_i}(\varphi)$.}. We also define $\cbva{\varphi}$ and $\cbvb{\varphi}$
to be the vectors $(\cba{y_1}(\varphi), \ldots \cba{y_n}(\varphi))$
and $(\cbb{y_1}(\varphi), \ldots \cbb{y_n}(\varphi))$ respectively.
\end{definition}
If $N$ is a node in the DAG representation of the specification,
we abuse notation and use $\cba{y_i}(N)$ to denote
$\cba{y_i}(\formula{N})$, and similarly for $\cbb{y_i}(N)$, $\cbva{N}$
and $\cbvb{N}$. Furthermore, if both $Y$ and $N$ are clear from the
context, we use $\cba{i}$, $\cbb{i}$, $\overrightarrow{\Delta}$ and $\overrightarrow{\Gamma}$
instead of $\cba{y_i}(N)$, $\cbb{y_i}(N)$, $\cbva{N}$ and $\cbvb{N}$,
respectively. It is easy to see that the supports of both $\cbb{i}$
and $\cba{i}$ are (subsets of) $X \cup \{y_{i+1}, \ldots y_n\}$.
Furthermore, it follows from Definition~\ref{def:cb0-cb1} that whenever
$\cbb{i}$ (resp. $\cba{i}$) evaluates to $1$, if the output $y_i$
has the value $1$ (resp. $0$), then $\varphi$ must evaluate to $0$.
Conversely, if $\cbb{i}$ (resp. $\cba{i}$) evaluates to $0$, it
doesn't hurt (as far as satisfiability of $\varphi(X,Y)$ is concerned)
to assign the value $1$ (resp. $0$) to output $y_i$. This suggests
that both $\neg \cbb{i}$ and $\cba{i}$ suffice to serve as the
function $g_i(X, y_{i+1}, \ldots y_n)$ when synthesizing functions for
multiple output variables.
The following proposition, adapted from~\cite{fmcad2015:skolem},
follows immediately, where we have abused notation and used
$\neg\overrightarrow{\Gamma}$ to denote $(\neg \cbb{1}, \ldots \neg \cbb{n})$.
\begin{proposition}
\label{prop:cb1-realizes}
Given a specification $\varphi(X, Y)$,
both $\overrightarrow{\Delta}$ and
$\neg\overrightarrow{\Gamma}$ effectively realize $Y$ in
$\varphi(X,Y)$.
\end{proposition}
Proposition~\ref{prop:cb1-realizes} shows that it suffices to compute
$\overrightarrow{\Delta}$ (or $\overrightarrow{\Gamma}$) from $\varphi(X,Y)$ in order to solve the
synthesis problem. In the remainder of the paper, we show how to
achieve this compositionally and in parallel by first computing
refinements of $\cba{i}$ (resp. $\cbb{i}$) for all $i \in \{1,\ldots
n\}$, and then using John et al's CEGAR-based
technique~\cite{fmcad2015:skolem} to abstract them to the desired
$\cba{i}$ (resp. $\cbb{i}$). Throughout the paper, we use
$\cbar{i}$ and $\cbbr{i}$ to denote refinements of $\cba{i}$ and
$\cbb{i}$ respectively.
\section{Exploiting compositionality}
\label{sec:compose}
Given a specification $\varphi(X,Y)$, one way to synthesize $y_1,
\ldots y_n$ is to decompose $\varphi(X,Y)$ into
sub-specifications, solve the synthesis problems for the
sub-specifications in parallel, and compose the solutions to the
sub-problems to obtain the overall solution. A DAG representation of
$\varphi(X,Y)$ provides a natural recursive decomposition of the
specification into sub-specifications. Hence, the key technical
question relates to compositionality: how do we compose solutions to
synthesis problems for sub-specifications to obtain a solution to the
synthesis problem for the overall specification? This problem is not
easy, and no state-of-the-art tool for Boolean functional synthesis
uses such compositional reasoning.
Our compositional solution to the synthesis problem is best explained
in three steps. First, for a simple, yet representationally complete,
class of DAGs representing $\varphi(X,Y)$, we present a lemma that
allows us to do compositional synthesis at each node of such a DAG.
Next, we show how to use this lemma to design a parallel synthesis
algorithm. Finally, we extend our lemma, and hence the scope of our
algorithm, to significantly more general classes of DAGs.
\subsection{Compositional synthesis in AND-OR DAGs}
\label{sec:simple}
For simplicity of exposition, we first consider DAGs with internal
nodes labeled by only AND and OR operators (of arbitrary arity).
Fig.~\ref{fig:decomp} shows an example of such a DAG. Note that this
class of DAGs is representationally complete for Boolean
specifications, since every specification can be expressed in negation
normal form (NNF). In the previous section, we saw that computing
$\cba{i}(\varphi)$ or $\cbb{i}(\varphi)$ for all $i$ in $\{1, \ldots
n\}$ suffices for purposes of synthesis.
The following lemma shows the relation between $\cba{i}$ and $\cbb{i}$
at an internal node $N$ in the DAG and the corresponding formulas at
the node's children, say $c_1, \ldots c_k$.
\begin{lemma}[Composition Lemma]
\label{prop:comp}
Let $\Phi(N) = \ensuremath{\mathsf{op}}(\Phi(c_1),\ldots, \Phi(c_k))$, where $\ensuremath{\mathsf{op}}=\vee$ or
$\ensuremath{\mathsf{op}}=\wedge$. Then, for each $1\leq i\leq n$:
\begin{align}
\left(\bigwedge_{j=1}^k \cba{i}(c_j)\right) \leftrightarrow \cba{i}(N) \quad\mbox{and}\quad
\left(\bigwedge_{j=1}^k \cbb{i}(c_j)\right) \leftrightarrow \cbb{i}(N) \mbox{ if } \ensuremath{\mathsf{op}} = \vee\label{eq:vee}\\
\left(\bigvee_{j=1}^k \cba{i}(c_j)\right) \rightarrow \cba{i}(N) \quad\mbox{and}\quad
\left(\bigvee_{j=1}^k \cbb{i}(c_j)\right) \rightarrow \cbb{i}(N) \mbox{ if } \ensuremath{\mathsf{op}} = \wedge\label{eq:wedge}
\end{align}
\end{lemma}
\begin{proof}
The proof of this lemma follows from Definition~\ref{def:cb0-cb1}.
Consider the case of disjunction $\ensuremath{\mathsf{op}}=\vee$, i.e., Equation~(\ref{eq:vee}) for $\cba{}$ (the case for $\cbb{}$ is similar). Then
\begin{align*}
\cba{i}(N) &= \neg \exists y_1\ldots y_{i-1} (\Phi(c_1)\vee \ldots \vee\Phi(c_k))[y_i\mapsto 0]\\
&\longleftrightarrow \forall y_1\ldots y_{i-1}(\neg\Phi(c_1)\wedge \ldots \wedge \neg \Phi(c_k))[y_i\mapsto 0]\\
&\longleftrightarrow (\forall y_1\ldots y_{i-1}\neg\Phi(c_1))[y_i\mapsto 0]\wedge \ldots \wedge (\forall y_1\ldots y_{i-1}\neg \Phi(c_k))[y_i\mapsto 0]\\
&\longleftrightarrow \cba{i}(c_1)\wedge\ldots \wedge\cba{i}(c_k)
\end{align*}
On the other hand for conjunction $\ensuremath{\mathsf{op}}=\wedge$, i.e., Equation~(\ref{eq:wedge}) (and similarly for $\cbb{}$), we only have one direction:
\begin{align*}
\cba{i}(N) &= \neg \exists y_1\ldots y_{i-1} (\Phi(c_1)\wedge \ldots \wedge\Phi(c_k))[y_i\mapsto 0]\\
&\longleftrightarrow \forall y_1\ldots y_{i-1}(\neg\Phi(c_1)\vee \ldots \vee \neg \Phi(c_k))[y_i\mapsto 0]\\
&\longleftarrow(\forall y_1\ldots y_{i-1}\neg\Phi(c_1))[y_i\mapsto 0]\vee \ldots \vee (\forall y_1\ldots y_{i-1}\neg \Phi(c_k))[y_i\mapsto 0]\\
&\longleftrightarrow \cba{i}(c_1)\vee\ldots \vee\cba{i}(c_k)
\end{align*}
This completes the proof.\qed
\end{proof}
Thus, if $N$ is an OR-node, we obtain $\cba{i}(N)$ and $\cbb{i}(N)$
directly by conjoining $\cba{i}$ and $\cbb{i}$ at its
children. However, if $N$ is an AND-node, disjoining the $\cba{i}$ and
$\cbb{i}$ at its children only gives refinements of $\cba{i}(N)$ and
$\cbb{i}(N)$ (see Equation~(\ref{eq:wedge})). Let us call these
refinements $\cbar{i}(N)$ and $\cbbr{i}(N)$ respectively. To obtain
$\cba{i}(N)$ and $\cbb{i}(N)$ exactly at AND-nodes, we must use the
CEGAR technique developed in~\cite{fmcad2015:skolem} to iteratively
abstract $\cbar{i}(N)$ and $\cbbr{i}(N)$ obtained above. More on
this is discussed below.
A CEGAR step involves constructing, for each $i$ from $1$ to $n$, a
Boolean \emph{error formula} $\ensuremath{\mathsf{Err}}_{\cbar{i}}$
(resp. $\ensuremath{\mathsf{Err}}_{\cbbr{i}}$) such that the error formula is unsatisfiable
iff $\cbar{i}(N) \leftrightarrow \cba{i}(N)$ (resp. $\cbbr{i}(N)
\leftrightarrow \cbb{i}(N)$). A SAT solver is then used to check the
satisfiability of the error formula. If the formula is unsatisfiable,
we are done; otherwise the satisfying assignment can be used to
further abstract the respective refinement. This check-and-abstract
step is then repeated in a loop until the error formulas become
unsatisfiable. Following the approach outlined
in~\cite{fmcad2015:skolem}, it can be shown that if we use
$\ensuremath{\mathsf{Err}}_{\cbar{i}} ~\equiv~ \neg \cbar{i}
~\wedge~ \bigwedge_{j=1}^{i}\left(y_j \leftrightarrow \cbar{j}\right)
~\wedge~ \neg \varphi$ and $\ensuremath{\mathsf{Err}}_{\cbbr{i}} ~\equiv~ \neg \cbbr{i}
~\wedge~ \bigwedge_{j=1}^{i}\left(y_j \leftrightarrow \neg\cbbr{j}\right)
~\wedge~ \neg \varphi$, and perform CEGAR in order from $i = 1$ to
$i=n$, it suffices to gives us $\cba{i}$ and $\cbb{i}$. For details
of the CEGAR implementation, the reader is referred
to~\cite{fmcad2015:skolem}. The above discussion leads to a
straightforward algorithm {\textsc{Compute}} (shown as
Algorithm~\ref{alg:composeatop}) that computes $\cbva{N}$ and
$\cbvb{N}$ for a node $N$, using $\cbva{c_j}$ and $\cbvb{c_j}$ for its
children $c_j$. Here, we have assumed access to a black-box function
{\textsc{Perform\_Cegar}} that implements the CEGAR step.
\begin{algorithm}[h]
\caption{\textsc{Compute(Node $N$)}}
\label{alg:composeatop}
\KwIn{A DAG Node $N$ labelled either AND or OR \\
\textbf{Precondition:} Children of $N$, if any, have their $\overrightarrow{\Delta}$ and $\overrightarrow{\Gamma}$ computed.}
\KwOut{ $\cbva{N}, \cbvb{N}$}
\If {$N$ is a leaf
\tcp{$\formula{N}$ is a literal/constant; use Definition~\ref{def:cb0-cb1}}}
{
for all $y_i\in Y$, $\cba{i}(N)= \neg \exists y_1\ldots y_{i-1} (\formula{N}) [y_i\mapsto 0]$\;
for all $y_i\in Y$, $\cbb{i}(N)= \neg \exists y_1\ldots y_{i-1} (\formula{N}) [y_i\mapsto 1]$\;
}
\Else{\tcp{$N$ is an internal node; let its children be $c_1, \ldots c_k$}
\If {$N$ is an OR-node}
{
\For {\textbf {each} $y_i \in Y$}
{
$\cba{i}(N) := \cba{i}(c_1) \wedge \cba{i}(c_2)\ldots \wedge \cba{i}(c_k)$\;
$\cbb{i}(N) := \cbb{i}(c_1) \wedge \cbb{i}(c_2)\ldots \wedge \cbb{i}(c_k)$\;
}
}
\If {$N$ is an AND-node}
{
\For {\textbf{ each} $y_i \in Y$}
{
$\cbar{i}(N) := \cba{i}(c_1) \vee \cba{i}(c_2)\ldots \vee \cba{i}(c_k)$;~~~~~~ \tcc{ $\cbar{i}(N)\rightarrow \cba{i}(N)$}
$\cbbr{i}(N) := \cbb{i}(c_1) \vee \cbb{i}(c_2)\ldots \vee \cbb{i}(c_k)$;~~~~~~~~ \tcc{$\cbbr{i}(N) \rightarrow \cbb{i}(N)$}
}
$\left(\cbva{N}, \cbvb{N}\right)= \textsc{Perform\_Cegar}(N, (\cbar{i}(N), \cbbr{i}(N))_{y_i\in Y})$\;
}
}
\Return $\left(\cbva{N}, \cbvb{N}\right)$\;
\end{algorithm}
\subsection{A parallel synthesis algorithm}
\label{sec:booalgo}
The DAG representation of $\varphi(X,Y)$ gives a natural, recursive
decomposition of the specification, and also defines a partial order
of dependencies between the corresponding synthesis sub-problems.
Algorithm {\textsc{Compute}} can be invoked in parallel on nodes in
the DAG that are not ordered w.r.t. this partial order, as long as
{\textsc{Compute}} has already been invoked on their children. This
suggests a simple parallel approach to Boolean functional synthesis.
Algorithm {\textsc{ParSyn}}, shown below, implements this approach,
and is motivated by a message-passing architecture. We consider a
standard manager-worker configuration, where one out of available $m$
cores acts as the manager, and the remaining $m-1$ cores act as
workers. All communication between the manager and workers is assumed
to happen through explicit {\bfseries send} and {\bfseries receive}
primitives.
\begin{algorithm}[h!]
\caption{{\textsc{ParSyn}}}
\label{alg:cparcegar}
\KwIn{AND-OR DAG with root $Rt$ representing $\varphi(X,Y)$ in NNF form}
\KwOut{$(g_1, \ldots g_n)$ that effectively realize $Y$ in $\varphi(X,Y)$ \\\nonl \hrulefill}
\tcc{Algorithm for Manager }
Queue $Q$ \; \tcc{Invariant: Q has nodes that can be processed in parallel, i.e., leaves or nodes whose children have their $\overrightarrow{\Delta}$, $\overrightarrow{\Gamma}$ computed.}
Insert all leaves of DAG into $Q$\;
\While {all DAG nodes not processed}
{
\While {a worker $W$ is idle and $Q$ is not empty}
{
Node $N := Q$.front()\;
{\bfseries send} node $N$ for processing to $W$\;
{\bfseries if} $N$ has children $c_1, \ldots c_k$ {\bfseries then} {\bfseries send} $\cbva{c_j},\cbvb{c_j}$ for $1\le j \le k$ to $W$\;
}
{\bf wait until} some worker $W'$ processing node $N'$ becomes free\;
{\bfseries receive} $\left(\overrightarrow{\Delta}, \overrightarrow{\Gamma}\right)$ from
$W'$, and store as $\left(\cbva{N'}, \cbvb{N'}\right)$\;
Mark node $N'$ as processed\;
\For {{\bf each} parent node $N''$ of $N'$}
{
{\bfseries if} all children of $N''$ are processed {\bfseries then}
insert $N''$ into $Q$
}
} \tcc{All DAG nodes are processed; return $\neg\overrightarrow{\Gamma}$ or
$\overrightarrow{\Delta}$ from root $Rt$}
\Return $\left(\neg\cbb{1}(Rt), \ldots \neg\cbb{n}(Rt)\right)$
\tcp{or alternatively $\left(\cba{1}(Rt), \ldots \cba{n}(Rt)\right)$}
{\nonl \hrulefill}
\tcc{Algorithm for Worker $W$}
{\bfseries receive} node $N$ to process, and $\cbva{c_j}$, $\cbvb{c_j}$ for every child $c_j$ of $N$, if any\;
$\left(\overrightarrow{\Delta}, \overrightarrow{\Gamma}\right)$ := \textsc{Compute}($N$) \;
{\bfseries send} $\left(\overrightarrow{\Delta}, \overrightarrow{\Gamma}\right)$ to Manager \;
\end{algorithm}
The manager uses a queue $Q$ of ready-to-process nodes.
Initially, $Q$ is initialized with the leaf nodes in the DAG, and we
maintain the invariant that all nodes in $Q$ can be processed in
parallel. If there is an idle worker $W$ and if $Q$ is not empty, the
manager assigns the node $N$ at the front of $Q$ to worker $W$ for
processing. If $N$ is an internal DAG node, the manager also sends
$\cbva{c_j}$ and $\cbvb{c_j}$ for every child $c_j$ of $N$ to $W$. If
there are no idle workers or if $Q$ is empty, the manager waits for a
worker, say $W'$, to finish processing its assigned node, say $N'$.
When this happens, the manager stores the result sent by $W'$ as
$\cbva{N'}$ and $\cbvb{N'}$. It then inserts one or more parents
$N''$ of $N'$ in the queue $Q$, if all children of $N''$ have been processed. The above steps are
repeatedly executed at the manager until all DAG nodes have been
processed. The job of a worker $W$ is relatively simple: on being
assigned a node $N$, and on receiving $\cbva{c_j}$ and $\cbvb{c_j}$
for all children $c_j$ of $N$, it simply executes Algorithm
{\textsc{Compute}} on $N$ and returns $\left(\cbva{N},
\cbvb{N}\right)$.
Note that Algorithm {\textsc{ParSyn}} is guaranteed to progress as
long as all workers complete processing the nodes assigned to them in
finite time. The partial order of dependencies between nodes ensures
that when all workers are idle, either all nodes have already been
processed, or at least one unprocessed node has $\overrightarrow{\Delta}$ and
$\overrightarrow{\Gamma}$ computed for all its children, if any.
\subsection{Extending the Composition Lemma and algorithms}
\label{sec:extensions}
So far, we have considered DAGs in which all internal nodes were
either AND- or OR-nodes. We now extend our results to more general
DAGs. We do this by generalizing the Composition Lemma to arbitrary Boolean
operators. Specifically, given the refinements $\cbar{i}(c_j)$ and
$\cbbr{i}(c_j)$ at all children $c_j$ of a node $N$, we show how to
compose these to obtain $\cbar{i}(N)$ and $\cbbr{i}(N)$, when $N$ is
labeled by an arbitrary Boolean operator. Note that the CEGAR
technique discussed in Section~\ref{sec:simple} can be used to
abstract the refinements $\cbar{i}$ and $\cbbr{i}$ to $\cba{i}$ and
$\cbb{i}$ respectively, at any node of interest. Therefore, with our
generalized Composition Lemma, we can use compositional synthesis for
specifications represented by general DAGs, even without computing
$\cba{i}$ and $\cbb{i}$ exactly at all DAG nodes. This gives an
extremely powerful approach for parallel, compositional synthesis.
Let $\formula{N} = \ensuremath{\mathsf{op}}(\formula{c_1}, \ldots \formula{c_r})$, where
$\ensuremath{\mathsf{op}}$ is an $r$-ary Boolean operator. For convenience of notation, we
use $\neg N$ to denote $\neg \formula{N}$, and similarly for other
nodes, in the subsequent discussion. Suppose we are given
$\cbar{i}(c_j)$, $\cbbr{i}(c_j)$, $\cbar{i}(\neg c_j)$ and
$\cbbr{i}(\neg c_j)$, for $1 \le j \le r$. We wish to compose these
appropriately to compute $\cbar{i}(N)$, $\cbbr{i}(N)$,
$\cbar{i}(\neg N)$ and $\cbbr{i}(\neg N)$ for $1 \le i \le n$. Once we
have these refinements, we can adapt
Algorithm~\ref{alg:composeatop} to work for node $N$, labeled by an arbitrary Boolean operator
$\ensuremath{\mathsf{op}}$.
\begin{figure}
\caption{An $\ensuremath{\mathsf{op}
\label{fig:decomp2}
\end{figure}
To understand how composition works for $\ensuremath{\mathsf{op}}$, consider the formula
$\ensuremath{\mathsf{op}}(z_1, \ldots z_r)$, where $z_1, \ldots z_r$ are fresh Boolean
variables, as shown in Figure \ref{fig:decomp2}.
Clearly, $\formula{N}$ can be viewed as $(\cdots(\ensuremath{\mathsf{op}}(z_1, \ldots
z_r)[\subst{z_1}{\formula{c_1}}])
\cdots)[\subst{z_r}{\formula{c_r}}]$. For simplicity of notation, we
write $\ensuremath{\mathsf{op}}$ instead of $\ensuremath{\mathsf{op}}(z_1,\ldots,z_r)$ in the following
discussion. W.l.o.g., let $z_1 \prec z_2 \prec \cdots \prec z_r$ be a
total ordering of the variables $\{z_1, \ldots z_r\}$. Given $\prec$,
suppose we compute the formulas $\cbar{z_l}(\ensuremath{\mathsf{op}})$, $\cbbr{z_l}(\ensuremath{\mathsf{op}})$,
$\cbar{z_l}(\neg\ensuremath{\mathsf{op}})$ and $\cbbr{z_l}(\neg\ensuremath{\mathsf{op}})$ in negation normal
form (NNF), for all $l \in \{1, \ldots r\}$. Note that these formulas
have support $\{z_{l+1}, \ldots z_r\}$, and do not have variables in
$X \cup Y$ in their support. We wish to ask if we can compose these
formulas with $\cbar{i}(c_j)$, $\cbbr{i}(c_j)$, $\cbar{i}(\neg c_j)$
and $\cbbr{i}(\neg c_j)$ for $1 \le j \le r$ to compute $\cbar{i}(N)$,
$\cbbr{i}(N)$, $\cbar{i}(\neg N)$ and $\cbbr{i}(\neg N)$, for all $i
\in \{1, \ldots n\}$. It turns out that we can do this.
Recall that in NNF, negations appear (if at all) only on literals.
Let $\Upsilon_{l, \ensuremath{\mathsf{op}}}$ be the formula obtained by replacing every
literal $\neg z_s$ in the NNF of $\cbbr{z_l}(\ensuremath{\mathsf{op}})$ with a fresh
variable $\overline{z_s}$. Similarly, let $\Omega_{l, \ensuremath{\mathsf{op}}}$ be
obtained by replacing every literal $\neg z_s$ in the NNF of
$\cbar{z_l}(\ensuremath{\mathsf{op}})$ with the fresh variable $\overline{z_s}$. The
definitions of $\Upsilon_{l, \neg\ensuremath{\mathsf{op}}}$ and $\Omega_{l, \neg\ensuremath{\mathsf{op}}}$ are
similar. Replacing $\neg z_s$ by a fresh variable $\overline{z_s}$
allows us to treat the literals $z_s$ and $\neg z_s$ independently in
the NNF of $\cbbr{z_l}(\ensuremath{\mathsf{op}})$ and $\cbar{z_l}(\ensuremath{\mathsf{op}})$. The ability to
treat these independently turns out to be important when formulating
the generalized Composition Lemma.
Let $\left(\Upsilon_{l,\ensuremath{\mathsf{op}}}\left[\subst{z_s}{\cbar{i}(\neg
c_{s})}\right]\left[\subst{\overline{z_s}}{\cbar{i}(c_{s})}\right]\right)_{s=l+1}^r$
denote the formula obtained by substituting $\cbar{i}(\neg c_s)$ for
$z_s$ and $\cbar{i}(c_s)$ for $\overline{z_s}$, for every $s \in
\{l+1, \ldots r\}$, in $\Upsilon_{l,\ensuremath{\mathsf{op}}}$. The interpretation of
$\left(\Omega_{l,\ensuremath{\mathsf{op}}}\left[\subst{z_s}{\cbar{i}(\neg
c_{s})}\right]\left[\subst{\overline{z_s}}{\cbar{i}(c_{s})}\right]\right)_{s=l+1}^r$
is analogous. Our generalized Composition Lemma can now be stated as follows.
\begin{lemma}[Generalized Composition Lemma]
\label{prop:gen}
Let $\formula{N} = \ensuremath{\mathsf{op}}(\formula{c_1}, \ldots \formula{c_r})$, where $\ensuremath{\mathsf{op}}$
is an $r$-ary Boolean operator. For each $1\leq i\leq n$ and $1\leq \ell\leq
r$:
\begin{align*}
1.~&\cbar{i}(c_l) \wedge
\left(\Omega_{l,\ensuremath{\mathsf{op}}}\left[\subst{z_s}{\cbar{i}(\neg c_{s})}\right]\left[\subst{\overline{z_s}}{\cbar{i}(c_{s})}\right]\right)_{s=l+1}^r ~\rightarrow~
\cba{i}(N) \nonumber\\
2.~&\cbar{i}(\neg c_l) \wedge
\left(\Upsilon_{l,\ensuremath{\mathsf{op}}}\left[\subst{z_s}{\cbar{i}(\neg c_{s})}\right]\left[\subst{\overline{z_s}}{\cbar{i}(c_{s})}\right]\right)_{s=l+1}^r
~\rightarrow~ \cba{i}(N) \nonumber\\
3.~&\cbbr{i}(c_l) \wedge
\left(\Omega_{l,\ensuremath{\mathsf{op}}}\left[\subst{z_s}{\cbbr{i}(\neg c_{s})}\right]\left[\subst{\overline{z_s}}{\cbbr{i}(c_{s})}\right]\right)_{s=l+1}^r ~\rightarrow~
\cbb{i}(N) \nonumber\\
4.~&\cbbr{i}(\neg c_l) \wedge
\left(\Upsilon_{l,\ensuremath{\mathsf{op}}}\left[\subst{z_s}{\cbbr{i}(\neg c_{s})}\right]\left[\subst{\overline{z_s}}{\cbbr{i}(c_{s})}\right]\right)_{s=l+1}^r
~\rightarrow~ \cbb{i}(N) \nonumber
\end{align*}
If we replace $\ensuremath{\mathsf{op}}$ by $\neg\ensuremath{\mathsf{op}}$ above, we get refinements
of $\cba{i}(\neg N)$ and $\cbb{i}(\neg N)$.
\end{lemma}
\begin{proof}
We provide a proof for the first implication. The proofs for the other implications are similar.
Consider an assignment $\eta$ of values to $X \cup \{y_1, \ldots
y_n\}$ such that $\eta$ satisfies the left hand side of implication
(1). We show below that $\eta$ satisfies $\cba{i}(N)$ as
well.
Let $\eta^\star$ denote an assignment of values to variables that
coincides with $\eta$ for all variables, except possibly for $y_i$.
Formally, $\eta^\star(v) = \eta(v)$ for $v \in X \cup \{y_1, \ldots
y_{i-1}, y_{i+1}, \ldots y_n\}$ and $\eta^\star(y_i) = 0$. We use
$\eta^\star(v)$ to denote the value assigned to variable $v$ in
$\eta^\star$. If $\psi$ is a Boolean formula, we abuse notation and
use $\eta^\star(\psi)$ to denote the value that $\psi$ evaluates to,
when variables are assigned values according to $\eta^\star$.
Since the right hand side of implication (1) does not have
$y_i$ in its support, it suffices to show that $\eta^\star$ satisfies
$\cba{i}(N)$. Furthermore, since neither side of implication (1)
depends on $\{y_1, \ldots y_{i-1}\}$, our arguments work for all
values of $\{y_1, \ldots y_{i-1}\}$. Hence, it suffices to show that
$\eta^\star(N) = 0$.
\begin{claim}[1] $\eta^\star(c_l) = 0$
\end{claim}
\begin{proof}
To see why this is true, note that $\eta^\star$ satisfies the left
hand side of implication (1), and hence it satisfies
$\cbar{i}(c_l)$. Since $\eta^\star(y_i) = 0$, it follows
from the definition of $\cbar{i}(\cdot)$ that
$\eta^\star(c_l) = 0$.
\end{proof}
Note that $\eta^\star$ also satisfies
$\left(\Omega_{l,\ensuremath{\mathsf{op}},\prec}\left[\subst{z_s}{\cbar{i}(\neg c_{s})}\right]\left[\subst{\overline{z_s}}{\cbar{i}(c_{s})}\right]\right)_{s=l+1}^r$.
Define $\rho$ to be an assignment of values to
$\{z_{l+1}, \overline{z_{l+1}}, \ldots z_r, \overline{z_r}\}$ such
that $\rho(z_s) = \eta^\star\left(\cbar{i}(\neg c_s)\right)$
and $\rho(\overline{z_s})
= \eta^\star\left(\cbar{i}(c_s)\right)$, for all
$s \in \{l+1, \ldots r\}$. It follows from the definition above that
$\rho$ is a satisfying assignment of $\Omega_{l,\ensuremath{\mathsf{op}},\prec}$.
\begin{claim}[2]
For every $s \in \{l+1, \ldots r\}$, either $\rho(z_s) = 0$ or $\rho(\overline{z_s}) = 0$. Further, for every $s \in \{l+1, \ldots r\}$, if $\rho(z_s) = 1$, then $\eta^\star(c_s) = 1$, and if $\rho(\overline{z_s}) = 1$, then $\eta^\star(c_s) = 0$.
\end{claim}
\begin{proof}
The proof of the first statement is by contradiction. If possible, let $\rho(z_s)= 1$ and $\rho(\overline{z_s}) = 1$ for some $s \in \{l+1, \ldots r\}$. By definition of $\rho$, we have $\eta^\star\left(\cbar{i}(\neg c_s)\right) = 1$ and
$\eta^\star\left(\cbar{i}(c_s)\right) = 1$. By definition
of $\cbar{i}(\cdot)$, it follows that both $\neg c_s$ and
$c_s$ evaluate to $0$ for the assignment $\eta^\star$ -- a
contradiction!
For the second statement, note that by definition, if $\rho(z_s) = 1$,
then $\eta^\star\left(\cbar{i}(\neg c_s)\right) = 1$. Since
$\eta^\star(y_i) = 0$, it follows from the definition of
$\cbar{i}(\cdot)$ that $\eta^\star(\neg(c_s) = 0$.
Equivalently, we have $\eta^\star(c_s) = 1$. Similarly, if
$\rho(\overline{z_s}) = 1$, then by definition,
$\eta^\star\left(\cbar{i}(c_s)\right) = 1$, and hence
$\eta^\star(c_s) = 0$.
\end{proof}
Finally, we define an assignment $\widehat{\rho}$ of values to
$\{z_{l+1}, \overline{z_{l+1}}, \ldots z_r, \overline{z_r}\}$ as
follows: for all $s \in \{l+1, \ldots r\}$, $\widehat{\rho}(z_s)
= \rho(z_s)$ if either $\rho(z_s) = 1$ or $\rho(\overline{z_s}) = 1$,
and $\widehat{\rho}(z_s) = \eta^\star(c_s)$ otherwise;
$\rho(\overline{z_s})$ is always equal to $\neg \rho(z_s)$. From
Claim~(2), we can now infer that $\widehat{\rho}(z_s)
= \eta^\star(c_s)$ if either $\rho(z_s) = 1$ or
$\rho(\overline{z_s}) = 1$. Therefore, we obtain the following claim,
\begin{claim}[3]
$(\widehat{\rho}(z_s), \widehat{\rho}(\overline{z_s}) =$
$\left(\eta^\star(c_s), \neg \eta^\star(c_s)\right)$ for
all $s \in \{l+1, \ldots r\}$.
\end{claim}
With the above claims, we can now prove the first implication/statement of the lemma. From Claim~(1), the values of $(\rho(z_s), \rho(\overline{z_s}))$
are either $(0,1), (1,0)$ or $(0,0)$, for all $s \in \{l+1,\ldots
r\}$. Therefore, $\rho(v) \rightarrow \widehat{\rho}(v)$ for all
$v \in \{z_{l+1}, \overline{z_{l+1}}, \ldots z_r, \overline{z_r}\}$.
Furthermore, since $\Omega_{l,\ensuremath{\mathsf{op}},\prec}$ is obtained by replacing all
literals $\neg z_s$ with $\overline{z_s}$ in the NNF of
$\cbar{z_l}(\ensuremath{\mathsf{op}})$, $\Omega_{l,\ensuremath{\mathsf{op}}}$ is positive unate in
$\{z_{l+1}, \overline{z_{l+1}}, \ldots z_r, \overline{z_r}\}$. It
follows that since $\rho$ satisfies $\Omega_{l,\ensuremath{\mathsf{op}},\prec}$ and
$\rho(v) \rightarrow \widehat{\rho}(v)$ for all $v$, $\widehat{\rho}$
also satisfies $\Omega_{l,\ensuremath{\mathsf{op}},\prec}$.
From the definition of $\Omega_{l,\ensuremath{\mathsf{op}},\prec}$, it is easy to see that
$\left(\Omega_{l,\ensuremath{\mathsf{op}},\prec}[\subst{\overline{z_s}}{\neg
z_s}]\right)_{s=l+1}^r$ is exactly $\cba{z_l}(\ensuremath{\mathsf{op}})$. Therefore,
$\Omega_{l,\ensuremath{\mathsf{op}},\prec}$ evaluated at $\widehat{\rho}$ has the same value,
i.e. $1$, as $\cba{z_l}(\ensuremath{\mathsf{op}})$ evaluated at $\widehat{\rho}$. From
the definition of $\cba{z_l}(\ensuremath{\mathsf{op}})$, it follows that $\ensuremath{\mathsf{op}}(z_1, \ldots
z_r)$ evaluates to $0$ for every assignment $\zeta$ of values to
$\{z_1, \ldots z_r\}$ such that $\zeta(z_l) = 0$, and $\zeta(z_s)
= \widehat{\rho}(z_s)$ for all $s \in \{l+1, \ldots r\}$.
Now, Claims (1) and (3) imply that $\zeta(z_s) = \eta^\star(c_s)$ for all $s \in \{l, \ldots r\}$. Therefore, $\eta^\star\left(\ensuremath{\mathsf{op}}[\subst{z_1}{\formula{c_1}}] \cdots
[\subst{z_r}{\formula{c_r}}]\right) = 0$. Since $\formula{N}
= \ensuremath{\mathsf{op}}[\subst{z_1}{\formula{c_1}}] \cdots [\subst{z_r}{\formula{c_r}}]$,
we have $\eta^\star(N) = 0$. This proves the first statement/implication of the lemma.
The other implications can be similarly proved following this same template.\qed
\end{proof}
We simply illustrate the idea behind the lemma
with an example here. Suppose $\formula{N} = \formula{c_1} \wedge \neg
\formula{c_2} \wedge (\neg \formula{c_3} \vee \formula{c_4})$, where
each $\formula{c_j}$ is a Boolean function with support $X \cup \{y_1,
\ldots y_n\}$. We wish to compute a refinement of $\cba{i}(N)$, using
refinements of $\cba{i}(c_j)$ and $\cba{i}(\neg c_j)$ for $j \in
\{1,\ldots 4\}$.
Representing $N$ as $\ensuremath{\mathsf{op}}(c_1, c_2, c_3, c_4)$, let $z_1, \ldots z_4$
be fresh Boolean variables, not in $X \cup \{y_1, \ldots y_n\}$; then
$\ensuremath{\mathsf{op}}(z_1, z_2, z_3, z_4) = z_1 \wedge \neg z_2 \wedge (\neg z_3 \vee
z_4)$. For ease of exposition, assume the ordering $z_1 \prec z_{2}
\prec z_{3} \prec z_{4}$. By definition, $\cba{z_2}(\ensuremath{\mathsf{op}}) = \left(\neg
\exists z_1\, (z_1 \wedge \neg z_2 \wedge (\neg z_3 \vee
z_4))\right)[\subst{z_2}{0}]$ $=$ $z_3 \wedge \neg z_4$, and suppose
$\cbar{z_2}(\ensuremath{\mathsf{op}}) = \cba{z_2}(\ensuremath{\mathsf{op}})$. Replacing $\neg z_{4}$ by
$\overline{z_{4}}$, we then get
$\Omega_{2,\ensuremath{\mathsf{op}}}=z_{3}\wedge\overline{z_{4}}$.
Recalling the definition of $\cbar{z_2}(\cdot)$, if we set $z_{3}=1$,
$z_{4}=0$ and $z_2=0$, then $\ensuremath{\mathsf{op}}$ must evaluate to $0$ regardless of
the value of $z_1$. By substituting $\cbar{i}(\neg c_{3})$ for
$z_{3}$ and $\cbar{i}(c_{4})$ for $\overline{z_{4}}$ in
$\Omega_{2,\ensuremath{\mathsf{op}}}$, we get the formula $\cbar{i}(\neg c_{3}) \wedge
\cbar{i}(c_{4})$. Denote this formula by $\chi$ and note that its
support is $X \cup \{y_{i+1}, \ldots y_n\}$. Note also from the
definition of $\cbar{i}(\cdot)$ that if $\chi$ evaluates to $1$ for
some assignment of values to $X \cup \{y_{i+1}, \ldots y_n\}$ and if
$y_i = 0$,
evaluates to $0$ and $\formula{c_{4}}$ evaluates to $0$, regardless of
the values of $y_1, \ldots y_{i-1}$. This means that $z_{3} = 1$ and
$z_{4} = 0$, and hence $\cbar{z_2}(\ensuremath{\mathsf{op}}) = 1$. If $z_2$ (or
$\formula{c_2}$ can also be made to evaluate to $0$ for the same
assignment of values to $X \cup \{y_i, y_{i+1}, \ldots y_n\}$, then $N
= \ensuremath{\mathsf{op}}(c_1, \ldots c_r)$ must evaluate to $0$, regardless of the values
of $\{y_1, \ldots y_{i-1}\}$. Since $y_i = 0$,
values assigned to $X \cup \{y_{i+1}, \ldots y_n\}$ must therefore be
a satisfying assignment of $\cba{i}(N)$. One way of achieving this is
to ensure that $\cba{i}(c_2)$ evaluates to $1$ for the same assignment
of values to $X \cup \{ y_{i+1}, \ldots y_n\}$ that satisfies $\chi$.
Therefore, we require the assignment of values to $X \cup \{ y_{i+1}, \ldots
y_n\}$ to satisfy $\chi \wedge \cba{i}(c_2)$, or even $\chi \wedge
\cbar{i}(c_2)$. Since $\chi = \cbar{i}(\neg c_{3}) \wedge
\cbar{i}(c_{4})$, we get $\cbar{i}(c_2) \wedge {\cbar{i}(\neg c_{3})}
\wedge {\cbar{i}(c_{4})}$ as a refinement of $\cba{i}(N)$.
\paragraph{Applying the generalized Composition Lemma:} Lemma~\ref{prop:gen}
suggests a way of compositionally obtaining $\cbar{i}(N)$,
$\cbbr{i}(N)$, $\cbar{i}(\neg N)$ and $\cbbr{i}(\neg N)$ for an
arbitrary Boolean operator $\ensuremath{\mathsf{op}}$. Specifically, the disjunction of
the left-hand sides of implications (1) and (2) in
Lemma~\ref{prop:gen}, disjoined over all $l \in \{1, \ldots r\}$ and
over all total orders ($\prec$) of $\{z_1, \ldots z_r\}$, gives a refinement of
$\cba{i}(N)$. A similar disjunction of the left-hand sides of
implications (3) and (4) in Lemma~\ref{prop:gen} gives a refinement of
$\cbb{i}(\varphi)$. The cases of $\cba{i}(\neg N)$ and $\cbb{i}(\neg
N)$ are similar. This suggests that for each operator $\ensuremath{\mathsf{op}}$ that
appears as label of an internal DAG node, we can pre-compute a
template of how to compose $\cbar{i}$ and $\cbbr{i}$ at the children of
the node to obtain $\cbar{i}$ and $\cbbr{i}$ at the node itself. In
fact, pre-computing this template for $\ensuremath{\mathsf{op}} = \vee$ and $\ensuremath{\mathsf{op}} = \wedge$
by disjoining as suggested above, gives us exactly the left-to-right
implications, i.e., refinements of $\cba{i}(N)$ and $\cbb{i}(N)$, as
given by Lemma~\ref{prop:comp}. We present
templates for some other common Boolean operators like \emph{if-then-else} in the next subsection.
Once we have pre-computed templates for composing $\cbar{i}$ and
$\cbbr{i}$ at children of a node $N$ to get $\cbar{i}(N)$ and
$\cbbr{i}(N)$, we can use these pre-computed templates in
Algorithm~\ref{alg:composeatop}, just as we did for AND-nodes. This
allows us to apply compositional synthesis on general DAG
representations of Boolean relational specifications.
\paragraph{Optimizations using partial computations:}
Given $\cbar{i}$ and $\cbbr{i}$ at children of a node $N$, we have
shown above how to compute $\cbar{i}(N)$ and $\cbbr{i}(N)$. To
compute $\cba{i}(N)$ and $\cbb{i}(N)$ exactly, we can use the CEGAR
technique outlined in Section~\ref{sec:simple}. While this is
necessary at the root of the DAG, we need not compute $\cba{i}(N)$ and
$\cbb{i}(N)$ exactly at each intermediate node. In fact, the
generalized Composition Lemma allows us to proceed with $\cbar{i}(N)$
and $\cbbr{i}(N)$. This suggests some optimizations: (i) Instead of
using the error formulas introduced in Section~\ref{sec:simple}, that
allow us to obtain $\cba{i}(N)$ and $\cbb{i}(N)$ exactly, we can use
the error formula used in~\cite{fmcad2015:skolem}. The error formula
of~\cite{fmcad2015:skolem} allows us to obtain some Skolem function
for $y_i$ (not necessarily $\cba{i}(N)$ or $\neg\cbb{i}(N)$) using the
sub-specification $\formula{N}$ corresponding to node $N$. We have
found CEGAR based on this error formula to be more efficient in
practice, while yielding refinements of $\cba{i}(N)$ and $\cbb{i}(N)$.
In fact, we use this error formula in our implementation. (ii) We
can introduce a \emph{timeout} parameter, such that
$\cbva{N},\cbvb{N}$ are computed exactly at each internal node until
we timeout happens. Subsequently, for the nodes still under process,
we can simply combine $\cbar{i}$ and $\cbbr{i}$ at their children
using our pre-computed composition templates, and not invoke CEGAR at
all. The only exception to this is at the root node of the DAG where
CEGAR must be invoked.
\subsection{Application of Lemma \ref{prop:gen} through Examples}
In this subsection, with the help of examples, we present the computation of the $\cba{.}$ and $\cbb{.}$ templates for various boolean formulae. We also compare these templates with those obtained by using Algorithm \ref{alg:composeatop}.
\begin{example}
To begin with, we demonstrate how to derive the templates for the $\wedge$ operator.
Let $\varphi = c_1 \wedge c_2$ where $c_1, c_2$ are arbitrary boolean formulae and have (a subset of) $X \cup Y$ in their support. As shown in Figure \ref{fig:decomp2}, let $z_1$ and $z_2$ be fresh Boolean variables. We first compute the relevant $\cbbr{}(.)$ and $\cbar{}(.)$'s for $z_1 \wedge z_2$ and then use Lemma \ref{prop:gen} to compute the template for $\cba{i}(\varphi)$ and $\cbb{i}(\varphi)$.
Let $F = z_1 \wedge z_2$. Without loss of generality, let the ordering be : $z_1 \prec z_{2} $. Given an ordering and boolean function $F$, we compute the $\cba{}(.)$ and $\cbb{}(.)$ sets for each variable as follows: $F[ \subst{z_1}{1}] = z_2$; $F[\subst{z_1}{0}] = 0$
Therefore, by definition~\ref{def:cb0-cb1}, $\cba{1}(F) = 1$ and $\cbb{1}(F) = \neg z_2$. On existentially quantifying $z_1$ from $F$, we get:
$(\exists z_1 F) = (z_2 \vee 0) = z_2$, i.e., $\exists z_1 F[ \subst{z_2}{1}] = 1$; $F[\subst{z_2}{0}] = 0$. Therefore, we have $\cba{2}(F) = 1$ and $\cbb{2}(F) = 0$.
Using the Generalized Compositional lemma, (Lemma \ref{prop:gen}), we get:
\begin{equation*}
\begin{aligned}
\cbar{i}(c_1) \wedge 1 \rightarrow~ \cba{i}(\varphi) ; \\
\cbar{i}(\neg c_1) \wedge 0 \rightarrow~ \cba{i}(\varphi) ; \\
\cbar{i}( c_2) \wedge 1 \rightarrow~ \cba{i}(\varphi) ; \\
\cbar{i}( \neg c_2) \wedge 0 \rightarrow~ \cba{i}(\varphi) ;
\end{aligned}
\end{equation*}
On disjunction of the terms on the LHS, we get the following template
\(\cbar{i}(c_1) \wedge \cbar{i}(c_2) ~\rightarrow~ \cba{i}(\varphi) \) ;
which is exactly the same as that proved in Lemma \ref{prop:comp}.
\end{example}
\begin{example}
We now consider the $\vee$ operator. Let $\varphi = c_1 \vee c_2$ where $c_1, c_2$ are arbitrary boolean formulae and have (a subset of) $X \cup Y$ in their support.
As beforelet $z_1$ and $z_2$ be fresh Boolean variables.
Let $F = z_1 \vee z_2$. Without loss of generality, let the ordering be : $z_1 \prec z_{2} $.
Then, $F[ \subst{z_1}{1}] = 1$; $F[\subst{z_1}{0}] = z_2$. Therefore, by definition~\ref{def:cb0-cb1}, $\cba{1}(F) = \neg z_2$ and $\cbb{1}(F) = 0$. On existentially quantifying $z_1$ from $F$, we get: $\exists z_1 F = z_2 \vee 1 = 1$, and so $\cba{2}(F) = 0$ and $\cbb{2}(F) = 0$.
Again, using the Generalized Compositional lemma, (Lemma \ref{prop:gen}), we get:
\begin{equation*}
\begin{aligned}
\cbar{i}(c_1) \wedge \cbar{i}(c_2) \rightarrow~ \cba{i}(\varphi) ; \\
\cbar{i}(\neg c_1) \wedge 0 \rightarrow~ \cba{i}(\varphi) ; \\
\cbar{i}( c_2) \wedge 0 \rightarrow~ \cba{i}(\varphi) ; \\
\cbar{i}( \neg c_2) \wedge 0 \rightarrow~ \cba{i}(\varphi) ;
\end{aligned}
\end{equation*}
\end{example}
By disjuncting the terms in the LHS, we get the following template
\(\cbar{i}(c_1) \wedge \cbar{i}(c_2) ~\rightarrow~ \cba{i}(\varphi) \) ;
which is, again, the same as that proved in Lemma \ref{prop:comp}.
\begin{example}
Now consider the \emph{if-then-else} or the $\ensuremath{\mathsf{ite}}$ operator. Let $\varphi = \ensuremath{\mathsf{ite}}(c_1, c_2, c_3)$ where $c_1, c_2, c_3$ are arbitrary boolean formulae and have (a subset of) $X \cup Y$ in their support. As before, let $z_1$, $z_2$ and $z_3$ be fresh Boolean variables. We first compute the relevant $\cbbr{}(.)$ and $\cbar{}(.)$'s for $\ensuremath{\mathsf{ite}}(z_1, z_2, z_3)$ and then use Lemma \ref{prop:gen} to compute the template for $\cba{i}(\varphi)$ and $\cbb{i}(\varphi)$.
Let $F = \ensuremath{\mathsf{ite}}(z_1, z_2, z_3)$. This means that if $z_1$ evaluates to true, then $F = z_2$ else $F = z_3$. Let the ordering be : $z_1 \prec z_{2} \prec z_{3}$.
Then, $F[ \subst{z_1}{1}] = z_2$; $F[\subst{z_1}{0}] = z_3$ and by definition, $\cba{1}(F) = \neg z_3$ and $\cbb{1}(F) = \neg z_2$. On existentially quantifying $z_1$ from $F$, we get: $\exists z_1 F = z_2 \vee z_3$.
To compute $\cba{2}(F)$ and $\cbb{2}(F)$, $\exists z_1 F [\subst{z_2}{1}] = 1 ; \exists z_1 F[ \subst{z_2}{0}] = z_3$, and so $\cba{2}(F) = \neg z_3$ and $\cbb{2}(F) = 0$.
Finally, on existentially quantifying $z_1$ and $z_2$ from $F$, we get:
$\exists z_1 \exists z_2 F = 1$; which gives $\cba{3}(F) = \cbb{3}(F) = 0$
From Lemma \ref{prop:gen}, we have:
\begin{equation*}
\begin{aligned}
\cbar{i}(c_1) \wedge \cbar{i}(c_3) ~\rightarrow~ \cba{i}(\varphi) \\
\cbar{i}(\neg c_1) \wedge \cbar{i}(c_2) ~\rightarrow~ \cba{i}(\varphi) \\
\cbar{i}( c_2) \wedge \cbar{i}(c_3) ~\rightarrow~ \cba{i}(\varphi)
\end{aligned}
\end{equation*}
Combining these terms, we get the template for $\cba{i}(\varphi)$ as:\\
$(\cbar{i}(c_1) \wedge \cbar{i}(c_3)) \vee (\cbar{i}(\neg c_1) \wedge \cbar{i}(c_2)) \vee ( \cbar{i}( c_2) \wedge \cbar{i}(c_3)) \rightarrow \cba{i}(\varphi)$ \\
Similarly, the template for $\cbb{i}(\varphi)$ is: \\
$(\cbbr{i}(c_1) \wedge \cbbr{i}(c_3)) \vee (\cbbr{i}(\neg c_1) \wedge \cbbr{i}(c_2)) \vee ( \cbbr{i}( c_2) \wedge \cbbr{i}(c_3)) \rightarrow \cbb{i}(\varphi)$ \\
Note that, we can also represent $\ensuremath{\mathsf{ite}}(c_1, c_2, c_3)$ as a formula G containing only AND, OR and NOT operators and derive the template directly using Lemma \ref{prop:comp}. That is, let $G = (c_1 \wedge c_2) \vee (\neg c_1 \wedge c_3)$. Using Algorithm \ref{alg:composeatop} (and not doing Step 14 of \textsc{Perform\_Cegar}), we get the following:
\(\cbar{i}(c_1) \vee \cbar{i}(c_2)) \wedge (\cbar{i}(\neg c_1) \vee \cbar{i}(c_3)) \rightarrow \cba{i}(\varphi) \)
On simplication we get,
$(\cbbr{i}(c_1) \wedge \cbbr{i}(c_3)) \vee (\cbbr{i}(\neg c_1) \wedge \cbbr{i}(c_2)) \vee ( \cbbr{i}( c_2) \wedge \cbbr{i}(c_3)) \rightarrow \cbb{i}(\varphi)$
Here again, the templates given by Lemma \ref{prop:comp} and Lemma \ref{prop:gen} are the same. However, in the next example we consider a boolean function where the two differ.
\end{example}
\begin{example}
With the help this example, we demonstrate that Lemma \ref{prop:gen} can give better underapproximations than the approach presented in Algorithm \ref{alg:composeatop}.
Let $\varphi = (c_1 \ensuremath{\mathsf{op}}lus c_2) \wedge (c_1 \ensuremath{\mathsf{op}}lus c_3)$ where $c_1, c_2, c_3$ are arbitrary boolean formulae and have (a subset of) $X \cup Y$ in their support. As before, let $z_1$, $z_2$ and $z_3$ be fresh Boolean variables. Let $F = (z_1 \ensuremath{\mathsf{op}}lus z_2) \wedge (z_1 \ensuremath{\mathsf{op}}lus z_3)$
Without loss of generality, let the ordering be : $z_1 \prec z_{2} \prec z_{3}$. Then, $F[ \subst{z_1}{1}] = \neg z_2 \wedge \neg z_3$; $F[\subst{z_1}{0}] = z_2 \wedge z_3$. By definition, $\cba{1}(F) = \neg z_2 \vee \neg z_3$ and $\cbb{1}(F) = z_2 \vee z_3$. On existentially quantifying $z_1$ from $F$, we get:
$\exists z_1 F = (\neg z_2 \wedge \neg z_3) \vee (z_2 \wedge z_3)$. To compute $\cba{2}(F)$ and $\cbb{2}(F)$, $\exists z_1 F [\subst{z_2}{1}] = z_3 ; \exists z_1 F[ \subst{z_2}{0}] = \neg z_3$. Therefore, $\cba{2}(F) = z_3$ and $\cbb{2}(F) = \neg z_3$
Again, on existentially quantifying $z_1$ and $z_2$ from $F$, we get:
$\exists z_1 \exists z_2 F = 1$ and $\cba{3}(F) = \cbb{3}(F) = 0$.
Using the compositional lemma we get:
\begin{equation*}
\begin{aligned}
\cbar{i}(c_1) \wedge (\cbar{i}(c_2) \vee \cbar{i}(c_3))~\rightarrow~ \cba{i}(\varphi) \\
\cbar{i}(\neg c_1) \wedge (\cbar{i}(\neg c_2) \vee \cbar{i}( \neg c_3))~\rightarrow~ \cba{i}(\varphi) \\
\cbar{i}( c_2) \wedge \cbar{i}(\neg c_3) ~\rightarrow~ \cba{i}(\varphi) \\
\cbar{i}( \neg c_2) \wedge \cbar{i}(c_3) ~\rightarrow~ \cba{i}(\varphi) \\
\end{aligned}
\end{equation*}
Disjunction of the terms on the LHS, allows us to get:
\( (\cbar{i}(c_1) \wedge (\cbar{i}(c_2)) \vee ( \cbar{i}(c_1) \wedge \cbar{i}(c_3)) \vee (\cbar{i}(\neg c_1) \wedge \cbar{i}(\neg c_2)) \vee ( \cbar{i}(\neg c_1) \wedge \cbar{i}(\neg c_3)) \\ \vee (\cbar{i}( c_2) \wedge \cbar{i}(\neg c_3) ) \vee (\cbar{i}( \neg c_2) \wedge \cbar{i}(c_3)) ~\rightarrow~ \cba{i}(\varphi) \)
However, if we represent $\varphi = (c_1 \ensuremath{\mathsf{op}}lus c_2) \wedge (c_1 \ensuremath{\mathsf{op}}lus c_3)$ as a boolean formula containing AND's and OR's as $\varphi = ((\neg c_1 \wedge c_2) \vee (c_1 \wedge \neg c_2)) \wedge (((\neg c_1 \wedge c_3) \vee (c_1 \wedge \neg c_3))$. Using Algorithm \ref{alg:composeatop} (and not doing Step 14 of \textsc{Perform\_Cegar}), we only get the following: \( (\cbar{i}(c_1) \wedge \cbar{i}(c_2)) \vee (\cbar{i}(c_1) \wedge \cbar{i}(c_3)) \vee (\cbar{i}(\neg c_1) \wedge \cbar{i}(\neg c_2)) \vee (\cbar{i}(\neg c_1) \wedge \cbar{i}(\neg c_3))~\rightarrow~ \cba{i}(\varphi) \);
Note that in addition to the terms above, the template for $\cba{i}(\varphi)$ derived using Lemma \ref{prop:gen} also has the additional terms
\( (\cbar{i}( c_2) \wedge \cbar{i}(\neg c_3)) \vee (\cbar{i}( \neg c_2) \wedge \cbar{i}(c_3)) \). It can easily be seen that these two terms are necessary, if $c_2 \neq c_3$ then $\varphi$ will not evaluate to true.
This example shows Lemma \ref{prop:gen} can give better underapproximations than Lemma \ref{prop:comp} for complex boolean formulae.
\end{example}
\section{Experimental results} \label{results}
\label{sec:expt}
\subsubsection*{Experimental methodology.}
We have implemented Algorithm~\ref{alg:cparcegar} with the error
formula from~\cite{fmcad2015:skolem} used for CEGAR in
Algorithm~\ref{alg:composeatop} (in function
{\textsc{Perform\_Cegar}), as described at the end of
Section~\ref{sec:extensions}. We call this implementation $\mathsf{ParSyn}$
in this section, and compare it with the following algorithms/tools:
$(i)$ $\textsc{CegarSkolem}sk$: This is based on the sequential algorithm for
conjunctive formulas, presented in~\cite {fmcad2015:skolem}. For
non-conjunctive formulas, the algorithm in~\cite{fmcad2015:skolem},
and hence $\textsc{CegarSkolem}sk$, reduces to~\cite{Jian,Trivedi}. $(ii)$
$\textsf{RSynth}$: The {\it RSynth} tool as described in \cite{rsynth}.
$(iii)$ $\textsf{Bloqqer}$: As prescribed in~\cite{bierre}, we first generate
special QRAT proofs using the preprocessing tool \textsf{bloqqer}, and
then generate Boolean function vectors from the proofs using the
$\textsf{qrat-trim}$ tool.
Our implementation of $\mathsf{ParSyn}$, available online
at~\cite{tacas2017:benchmarks}, makes extensive use of the
ABC~\cite{abc-tool} library to represent and manipulate Boolean
functions as AIGs. We also use the default SAT solver provided by
ABC, which is a variant of MiniSAT. We present our evaluation on
three different kinds of \emph{benchmarks}.
\begin{enumerate}
\ensuremath{\mathsf{ite}}m{\em Disjunctive Decomposition Benchmarks}: Similar to~\cite{fmcad2015:skolem}, these benchmarks were generated
by considering some of the larger sequential circuits in the HWMCC10 benchmark suite, and formulating
the problem of disjunctively decomposing each circuit into components as a problem of synthesizing a vector of Boolean functions. Each generated benchmark is of the form $\exists Y \varphi(X,Y)$ where $\exists X (\exists Y \varphi(X,Y))$ is $\textsf{true}$. However, unlike \cite{fmcad2015:skolem}, where each benchmark (if not already a conjunction of factors) had to be converted into factored form using Tseitin encoding (which introduced additional variables), we have used these benchmarks without Tseitin encoding.
\ensuremath{\mathsf{ite}}m {\em Arithmetic Benchmarks}:
These benchmarks were taken from the work described in~\cite{rsynth}. Specifically, the benchmarks considered are {\it floor}, {\it ceiling}, {\it decomposition}, {\it equalization} and {\it intermediate} (see~\cite{rsynth} for details).
\ensuremath{\mathsf{ite}}m {\em Factorization Benchmarks}:
We considered the integer factorization problem for different bit-widths,
as discussed in Section \ref{sec:introduction}.
\end{enumerate}
For each arithmetic and factorization benchmark, we first specified
the problem instance as an { SMT} formula and then used {\it
Boolector}~\cite{boolector} to generate the Boolean version of the
benchmark. For each arithmetic benchmark, three variants were
generated by varying the bit-width of the arguments of arithmetic
operators; specifically, we considered bit-widths of $32$, $128$ and
$512$. Similarly, for the factorization benchmark, we generated four
variants, using $8$, $10$, $12$ and $16$ for the bit-width of the
product. Further, as $\textsf{Bloqqer}$ requires the input to be
in \texttt{qdimacs} format and $\textsf{RSynth}$ in \texttt{cnf} format, we
converted each benchmark into \texttt{qdimacs} and $\texttt{cnf}$
formats using Tseitin encoding~\cite{tseitin68}. All benchmarks and
the procedure by which we generated them are detailed
in \cite{tacas2017:benchmarks}.
\paragraph{Variable ordering:}
We used the same ordering of variables for all algorithms. For each
benchmark, the variables are ordered such that the variable which
occurs in the transitive fan-in of the least number of nodes in the
AIG representation of the specification, appears at the top. For
$\textsf{RSynth}$ this translated to an interleaving of most of the input and
output variables.
\paragraph{Machine details}: All experiments were performed on a message-passing cluster, where each node had 20 cores and $64$ GB main memory, each core being a 2.20 GHz Intel Xeon processor.
The operating system was Cent OS 6.5.
For $\textsc{CegarSkolem}sk$, $\textsf{Bloqqer}$, and $\textsf{RSynth}$, a single core on the
cluster was used. For all comparisons, $\mathsf{ParSyn}$ was executed on
$4$ nodes using $5$ cores each, so that we had both intra-node and
inter-node communication.
The maximum time given for execution was 3600 seconds, i.e., 1 hour. We also
restricted the total amount of main memory (across all cores) to be 16GB.
The metric used to compare the different algorithms was the time taken to synthesize Boolean functions.
\noindent{\bf Results.}
Our benchmark suite consisted of
$27$ disjunctive decomposition benchmarks, $15$ arithmetic benchmarks and $4$ factorization benchmarks. These benchmarks are fairly comprehensive in size i.e., the number of AIG nodes ($|SZ|$) in the benchmark, and the number of variables ($|Y|$) for which Boolean functions are to be synthesized.
Amongst disjunctive decomposition benchmarks, $|SZ|$ varied from $1390$ to $58752$ and $|Y|$ varied from $21$ to $205$.
Amongst the arithmetic benchmarks, $|SZ|$ varied from $442$ to $11253$ and $|Y|$ varied from $31$ to $1024$.
The factorization benchmarks are the smallest and the most complex of the benchmarks, with
$|SZ|$ varying from $122$ to $502$ and $|Y|$ varying from $8$ to $16$.
We now present the performance of the various algorithms. On $4$ of
the $46$ benchmarks, none of the tools succeeded. Of these, $3$
belonged to the {\em intermediate} problem type in the arithmetic
benchmarks, and the fourth one was the $16$ bit factorization
benchmark.
\begin{figure*}
\caption{$\mathsf{ParSyn}
\caption{$\mathsf{ParSyn}
\caption{Legend: \texttt{Ar}
\label{fig:parcegarcores}
\label{fig:parsk}
\end{figure*}
\noindent {\it Effect of the number of cores}.
For this experiment, we chose $5$ of the larger benchmarks. Of these,
two benchmarks belonged to the disjunctive decomposition category, two
belonged to the arithmetic benchmark category and one was the 12 bit
factorization benchmark. The number of cores was varied from $2$ to
$25$. With $2$ cores, $\mathsf{ParSyn}$ behaves like a sequential algorithm
with one core acting as the manager and the other as the worker with
all computation happening at the worker core. Hence, with $2$ cores,
we see the effect of compositionality without parallelism. For number
of cores $>$ 2, the number of worker cores increase, and the
computation load is shared across the worker cores.
Figure \ref{fig:parcegarcores} shows the results of our
evaluation. The topmost points indicated by \texttt{FL} are instances
for which $\mathsf{ParSyn}$ timed out. We can see that, for all $5$
benchmarks, the time taken to synthesize Boolean function vectors when
the number of cores is $2$ is considerable; in fact, $\mathsf{ParSyn}$ times
out on three of the benchmarks. When we increase the number of cores
we observe that (a) by synthesizing in parallel, we can now solve
benchmarks for which we had timed out earlier, and (b) speedups of
about $4-5$ can be obtained with $5-15$ cores. From $15$ cores to $25$
cores, the performance of the algorithm, however, is largely invariant
and any further increase in cores does not result in further speed up.
To understand this, we examined the benchmarks and found that their
AIG representatation have more nodes close to the leaves than to the
root (similar to the DAG in Figure \ref{fig:decomp}). The time taken
to process a leaf or a node close to a leaf is typically much less
than that for a node near the root. Furthermore, the dependencies
between the nodes close to the root are such that at most one or two
nodes can be processed in parallel leaving most of the cores
unutilized. When the number of cores is increased from $2$ to $5 -
15$, the leaves and the nodes close to the leaves get processed in
parallel, reducing the overall time taken by the algorithm. However,
the time taken to process the nodes close to the root remains more or
less the same and starts to dominate the total time taken. At this
point, even if the number of cores is further increased, it does not
significantly reduce the total time taken. This behaviour limits the
speed-ups of our algorithm. For the remaining experiments, the number
of cores used for $\mathsf{ParSyn}$ was 20.
\noindent\textit{$\mathsf{\mathbf{ParSyn}}$ vs $\mathsf{\mathbf{CSk}}$:}
As can be seen from Figure \ref{fig:parsk}, $\textsc{CegarSkolem}sk$ ran
successfully on only $12$ of the $46$ benchmarks, whereas $\mathsf{ParSyn}$
was successful on $39$ benchmarks, timing out on $6$ benchmarks and
running out of memory on $1$ benchmark. Of the benchmarks that
$\textsc{CegarSkolem}sk$ was successful on, $9$ belonged to the arithmetic category,
$2$ to the factorization and $1$ to the disjunctive decomposition
category. On further examination, we found that factorization and
arithmetic benchmarks (except the {\em intermediate} problems) were
conjunctive formulae whereas disjunctive decomposition benchmarks were
arbitrary Boolean formulas. Since $\textsc{CegarSkolem}sk$ has been specially
designed to handle conjunctive formulas, it is successful on some of
these benchmarks. On the other hand, since disjunctive decomposition
benchmarks are not conjunctive, $\textsc{CegarSkolem}sk$ treats the entire formula
as one factor, and the algorithm reduces to~\cite{Jian,Trivedi}. The
performance hit is therefore not surprising; it has been shown
in~\cite{fmcad2015:skolem} and~\cite{rsynth} that the algorithms
of~\cite{Jian,Trivedi} do not scale to large benchmarks. In fact,
$\textsc{CegarSkolem}sk$ was successful only on the smallest disjunctive
decomposition benchmark.
\begin{figure*}
\caption{$\mathsf{ParSyn}
\caption{$\mathsf{ParSyn}
\caption{Legend: \texttt{Ar}
\label{fig:cegarrsynth}
\label{fig:cegarbloqqer}
\end{figure*}
\noindent \textit{$\mathsf{\mathbf{ParSyn}}$ vs $\textsf{RSynth}$:}
As seen in Figure \ref{fig:cegarrsynth}, $\textsf{RSynth}$ was successful only
on $3$ of the $46$ benchmarks; it timed out on $37$ and ran out of
memory on $6$ benchmarks. The $3$ benchmarks that RSynth was
successful on were the smaller factorization benchmarks. Note that
the arithmetic benchmarks used in \cite{rsynth} are semantically the
same. In~\cite{rsynth}, custom variable orders were used to construct
the ROBDDs, which resulted in compact ROBDDs. In our case, we use the
variable ordering heuristic mentioned above (see Sec. 4.1), and
include the considerable time taken to build BDDs from \texttt{cnf}
representation. As mentioned in Section~\ref{sec:introduction}, if we
know a better variable ordering, then the time taken can potentially
reduce. However, we do not know the optimal variable order for an
arbitrary specification in general. We also found the memory
footprint of $\textsf{RSynth}$ to be higher as indicated by the
memory-outs. This is not surprising, as $\textsf{RSynth}$ uses BDDs to
represent Boolean formula and it is well-known that BDDs can have
large memory requirements.
\noindent {\it $\mathsf{\mathbf{ParSyn}}$ vs $\textsf{Bloqqer}$:}
Since $\textsf{Bloqqer}$ cannot synthesize Boolean functions for formulas
wherein $\forall X \exists Y \varphi(X,Y)$ is not {\em valid}, we
restricted our comparison to only the disjunctive decomposition and
arithmetic benchmarks, totalling $42$ in number. From
Figure \ref{fig:cegarbloqqer}, we can see that $\textsf{Bloqqer}$ successfully
synthesizes Boolean functions for $25$ of the $42$ benchmarks. For
several benchmarks for which it is successful, it outperforms
$\mathsf{ParSyn}$. In line 14 of
Algorithm \ref{alg:composeatop}, \textsc{Perform\_Cegar} makes
extensive use of the SAT solver, and this is reflected in the time
taken by $\mathsf{ParSyn}$.
However, for the remaining $17$ benchmarks, $\textsf{Bloqqer}$ gave a
{\em Not Verified} message indicating that it could not synthesize Boolean functions for these benchmarks.
In comparison, $\mathsf{ParSyn}$ was successful on most of these benchmarks.
\noindent{\bf Effect of timeouts on $\mathsf{\mathbf{ParSyn}}$.}
Finally, we discuss the effect of the timeout optimization discussed
in Section~\ref{sec:extensions}. Specifically, for $60$ seconds
(value set through a \emph{timeout} parameter), starting from the
leaves of the AIG representation of a specification, we synthesize
exact Boolean functions for DAG nodes. After timeout, on the
remaining intermediate nodes, we do not invoke the CEGAR step at all,
except at the root node of the AIG.
This optimization enabled us to handle $3$ more benchmarks, i.e.,
$\mathsf{ParSyn}$ with this optimization synthesized Boolean function
vectors for all the {\em equalization} benchmarks (in $<$ 340
seconds). Interestingly, $\mathsf{ParSyn}$ without timeouts was unable to
solve these problems. This can be explained by the fact that in these
benchmarks many internal nodes required multiple iterations of the
CEGAR loop to compute exact Boolean functions, which were, however, not
needed to compute the solution at the root node.
\section{Conclusion and future work}
\label{sec:discussion}
In this paper, we have presented the first parallel and compositional
algorithm for complete Boolean functional synthesis from a relational
specification. A key feature of our approach is that it is agnostic
to the semantic variabilities of the input, and hence applies to a
wide variety of benchmarks. In addition to the disjunctive
decomposition of graphs and the arithmetic operation benchmarks, we
considered the combinatorially hard problem of factorization and
attempted to generate a functional characterization for it. We found
that our implementation outperforms existing tools in a variety of
benchmarks.
There are many avenues to extend our work. First, the ideas for
compositional synthesis that we develop in this paper could
potentially lead to parallel implementations of other synthesis tools,
such as that described in~\cite{rsynth}. Next, the
factorization problem can be generalized to synthesis of inverse functions for
classically hard one-way functions, as long as the function can be
described efficiently by a circuit/AIG.
Finally, we would like to explore improved ways of parallelizing
our algorithm, perhaps exploiting features of specific classes of
problems.
\end{document} | math |
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