Sugarloaf is a 10.4 mile out and back hike. It follows traditional single track hiking except for the last ~1/2 mile which is a steep uphill scramble. Given the terrain and elevation I'd suggest going on a cool day. The final scramble would be a bear in the heat. Also, while there is no traditional huge elevation gain there is considerable up and down that cumulatively will have you climbing over 3100′ during the hike.
The trailhead is located at the turnout on the left side of the road just as you approach Blue Jay campground off Ortega Highway. See the GPX file for details. You'll need an adventure pass to park.
I did the hike on an overcast day in May trying to get it in between two rain storms. Since rain was not predicted again until late afternoon I decided to go for it. I hit the trail at 11:30 am. It was a nice cool day with no one else on the trail. You usually see mountain bikers or hikers out here, but they probably stayed home given the rain prediction.
If you are not familiar with the San Juan Trail make sure you have a trail map or follow a gpx file. There are multiple "four corners," shortcuts and trail splits. It is not difficult to get turned around so please pay attention.
The big trick with W6/SC-405 "Sugarloaf" is that there are actually two Sugarloaf peaks in the area with various maps labeling the peaks differently. The SOTA database labels the peak for activation as "Sugarloaf" (this matches some map sources and not others.) However, the summit is most commonly referred to as Old Sugarloaf (3326') with the neighboring shorter summit being Sugarloaf (3227'). The SOTA database has the correct information for coordinates and elevation, key in on those – not the map name(s). I've included a picture to help you distinguish the summits. The basic rule is that if upon approaching Sugarloaf you think it looks easy, it is the wrong one.
At roughly 4.0 miles you'll pass the shorter non-SOTA Sugarloaf and descend down another half mile to the base of the SOTA summit. The entrance to the use trail to the top is a bit overgrown so keep an eye out it is easy to miss. I've put a waypoint marker in the GPX file. The path to the summit is similar to the terrain I encountered activating Iron Mountain, albeit much shorter. Here you have a little over 550 feet of elevation gain packed into roughly one half mile. It is a steep use trail that heads pretty much straight up to the summit.
I reached the summit just after 1:15 pm and busied myself signing the register and taking in the views – which by this point was mainly fast approaching rain clouds. I set up for 2-meters and quickly logged 2 contacts. Then it slowed. The Thursday-mid-afternoon-low-elevation-summit was working against me on 2-meters. Given the approaching weather I was hoping to get at least four contacts on 2-meters before deciding what to do about HF. I jumped on the Catalina repeater and successfully enticed a station to QSY over to 146.52 for the simplex contact and then picked up a friend of his who had heard our QSO. I had my four, all was good. The pressure was off.
I started getting out my HF gear when I heard Bob KB6CIO calling to let me know that Charles KM6CEM was trying to reach me on 2-meters. It didn't help that I had broken my main 2-meter HT antenna at the trailhead when packing my bag. The antenna I use while hiking was not letting me reach quite as far as my 1/2 wave collapsable that I had broken earlier. I decide to go to my back up Ed Fong rollup j-pole. Setting that up quickly I was able to log Charles was well as Jerry N6GR.
During all this commotion I had finally turned around towards the west and noticed that those dark clouds had now opened up with large swathes of rain coming towards the summit. Just as I finished up on 2-meters the rain finally hit. I had hiked with all my HF gear and if the weather had been nicer I had a great place to set up. But here it was May in Southern California and I was wearing all my layers including a down jacket under my rain gear standing on a cold, windy, rainy summit. I decided to pull the plug on the HF portion and pack up and go.
After that it was back on the trail and I was back to the car at 4:50 pm. Thank you to all the chasers who helped to make this a successful activation.
2-meter activation note: The location is a bit tricky for the normal 2-meter chasers as surrounding peaks narrow your line of site options. For example I could hear our QTH in Newport Beach but they could not hear me. However, I was able to hit Huntington Beach at 59. Given the proximity to Santiago and the great line of site for Catalina you should have no problem making a successful 2-meter activation by enticing folks on the repeaters over to simplex in the central and southern OC and coastal LA areas if needed.


text:
.. routine.php generated using docpx on 02/01/13 07:27am

Class - XPSPL\\Routine
**********************

Routine

The routine class is used by the processor during the routine calculation for
storing the idle functions and the signals which should be triggered in the
loop.

This was added due to the current loop not providing a simple means for
objects inside the loop determining what has happened within the routine
calculation and the functionality required for the upgraded idle required
more complex algorithms which would not fit well inside the entire routine
loop method.

Methods
-------

get_signals
+++++++++++

.. function:: get_signals()

    Returns the signals to trigger in the loop.

    :rtype: array

get_idle
++++++++

.. function:: get_idle()

    Returns the object to idle the processor.

    This will only return a single object which has the greatest priority.

    :rtype: integer

get_idles_available
+++++++++++++++++++

.. function:: get_idles_available()

    Returns the objects createed to idle the processor.

    :rtype: integer

add_idle
++++++++

.. function:: add_idle($idle)

    Adds a new function to idle the engine.

    :param object: Idle

    :rtype: void

set_idle
++++++++

.. function:: set_idle($idle)

    Sets a new function to idle the processor.

    :param object: Idle

    :rtype: void

add_signal
++++++++++

.. function:: add_signal($signal, [$vars = false, [$event = false]])

    Adds a signal to trigger in the loop.

    :rtype: array

reset
+++++

.. function:: reset()

    Resets the routine after the processor has used it.

    :rtype: void

Last updated on 02/01/13 07:27am

redpajama_set_name: RedPajamaGithub


text:
NewsPhoenix Metro NewsCentral Phoenix News
Record-setting rainfall leaves behind increased mosquito population around the Valley
By: John Patrick
The monsoon may be long gone, but the mosquitoes are still hanging around thanks to record-setting rainfall the Valley received last month.
A lot of rain and pleasant fall temperatures has mosquito season buzzing right along, but there's not much we can do about it, according to Dr. Kirk Smith with Maricopa County Environmental Services.
"As this county continues to grow and we put down more grass and more water, we're just kind of exacerbating our own problem," said Smith.
The kind of mosquito lingering around the Valley right now is known as the ankle biter, Aedes aegypti, it's also known to carry things like dengue fever and Zika virus.
"The State Department and County is worried about those diseases someday making it into Arizona," said Smith. "If that happens, then this ankle biter is going to be the vector for these diseases."
With a Zika outbreak currently ongoing just to our south in Sonora Mexico, the County Vector Control Lab is on high alert and already equipped to test trapped mosquitoes for the dangerous virus.
"Right now there are more than 100 cases of Zika down in Sonora, so there is some concern," said Smith.
So far, no mosquitoes in Maricopa County have tested positive for Zika, but the Vector Control Lab has counted over 250 mosquitoes that have been carrying either West Nile Virus or St. Louis Encephalitis.
"We've had a lot of positive mosquitoes with the disease, unfortunately, all it takes it that one little mosquito biting you and you may or may not react to it," said Smith.
Relief will soon be on the way. As overnight temperatures start to drop into the 40s and 50s, the mosquito population should begin to die off.

redpajama_set_name: RedPajamaCommonCrawl


About: Vertical Air Limited is owned by Rob Kelly and Geoff Storey and is based in Kendal, Cumbria.
VerticalAir has a dedicated outdoor High Ropes centre in the grounds of the Castle Green Hotel in Kendal Cumbria but they do so much more than that!
VerticalAir has a team of instructors, trainers and consultants who have a wide range of experience in the education sector, from primary teaching through to work on graduate and post-graduate programmes. This staff team enables the configuration of appropriate teams and the selection of the right people for each programme, using the skills and experience available to fit the exact requirements of the client.
We specialise in personal development, teamwork and leadership training using experiential outdoor and creative activities to integrate with the curriculum that you run, or to stand-alone as events in their own right that add real value to your provision.
We can run events in the Lake District and Yorkshire Dales, that is our home territory but we can also operate at schools and colleges nationally or abroad. If you have a need to provide that essential aspect of development that can differentiate your offering then we would like to talk to you to see if we can help.


Garden Photo of the Day: Is this my garden?
For at least two weeks, I have been in heaven. The white Lady Banks rose I planted several years ago finally topped my hemlock hedge and spilled over the sides and rambled across to an iron arbor covered in trumpet vine (not visible here). I couldn't believe that this was actually my garden.
For years, this area was a disaster. What had started out 30 years ago as a croquet lawn with sodded zoysia grass surrounded by a hemlock hedge and English boxwoods, eventually disintegrated into a sparse patch of weeds that got mown only occasionally. It was just too shady for any kind of grass. I finally gave the croquet set away, as the mallets and balls had languished unused after a few games in the early days.
In exasperation and trying to buy myself some time to figure out what to do, I covered the area with black landscape cloth. That was fine for a few weeks, but the weeks turned into months, then to years, and weeds popped up between the seams. I put my head in the sand, so to speak, and wouldn't even look down from the terrace above if I could help it.
In 2013, my daughter became engaged and wanted to have the wedding outdoors at our house -something I said I would never, ever do. I gave in, and the decision forced me to go into high gear. There was no place for about 20 people to gather for the small ceremony. We could have a tent in front for a bigger reception later, but to line up chairs and make an altar (her childhood minister agreed to do the ceremony) and have room for a cello and violin was problematic. I didn't have a room large enough inside, and we needed the back terrace for a bar and the cake and for people to just get away from the band at the reception that followed.
I had already made a small, narrow arch garden on the opposite side of the house, but there wasn't room for twenty people to stand, much less sit. For that area, we had used tiny pea gravel with sharp edges that were not slippery like the rounded gravel I had in the front parking court. What if we could take up that terrible black, messy cloth, edge the rectangle with cobblestones and cover the ground with this small pea gravel? I already had 4,000 Belgian blocks - very heavy ballast stones - stored at the farm; the cobblestones had been in the front parking area, but kept sinking. A few years before, at great trauma and expense, I had them dug up and hauled the 40 miles to Chattahoochee Hills to deal with later.
Long story short, we created the "lower garden" in back, and that's where we held the small ceremony. The wire arch you see was used along with two handsome wire obelisks I borrowed to make a sort of altar. For the greenery, Elizabeth Dean contributed long strands of Italian ruscus (Danae racemosa) to go with smilax vine from Alabama. A bigger reception took place under a tent in front, and guests wandered all through the house. Most important of all, it was a perfect October evening, although I had already lost a good ten years off my life worrying about the weather.
There's still much to be done - perhaps plant foxgloves behind the boxwoods to the right and add a climbing rose to cover the wire arch which needs to be stabilized. Maybe put out some French cafe chairs and a small round table and more benches to enjoy a cocktail before dinner (as if I ever throw any dinner parties!).
Still, I'm ecstatic. On a recent evening, just at dusk, I stepped out onto the terrace and looked down. A stretch of cool weather had preserved the roses that glowed in the low light. For a moment it seemed that finally, after years and years, I had, at least temporarily, a dreamy garden scene in a place that had once been a nightmare.


Auctioneer Mark Drilling gets people fired up about bidding at the 2012 Promise Ball, benefitting the Juvenile Diabetes Research Foundation. Gavin Foyle and Sam Martin show off a Drew Brees jersey while Darrell Greene, of Fox 13 TV, emcees the event. The Promise Ball raised over $350,000 this year.
How do you get to the Mud Island stage? Practice!
The Mud Island Amphitheater is ready for the 2012 season. So far they have not let me play there, but I did get to photograph the venue.


Legal requirements for your automobile insurance may vary slightly from state to state, and circumstances such as financing may dictate certain levels of coverage. It is worth investing in coverage that provides maximum protection for you and your family. McManamon Insurance can assist you in determining the details of your coverage, and always puts your best interests first.
Liability coverage offers protection against claims, property damage or bodily injury to a third party that you or a member of your household accidentally caused, and is legally responsible for, while driving your car.
The Medical Payment portion of your policy provides you, your family members, and passengers, the assurance of necessary medical and dental services resulting from an auto accident, regardless of who caused the accident.
Uninsured Motorist coverage and Underinsured Motorist coverage address the expense of injury incurred should you, your family members or passengers be involved in an accident with an uninsured or underinsured motorist, or even a hit and run accident. In some states property damage coverage is also available in these circumstances.
Collision coverage addresses the cost of damages to your vehicle during an accident, and Comprehensive coverage applies to damages to your car resulting from a situation other than a collision. Comprehensive coverage often applies to breakage of glass, falling objects, fire, theft, windstorm, hail, water, flood, malicious mischief, vandalism or impact with a bird or animal.
Personal Injury Protection (No Fault) is required by some states to cover you, your family and any passengers under your policy regardless of who is responsible for the accident. This coverage includes medical, surgical, dental, hospital, work loss and certain other related expenses.
Additional optional coverage includes items such as Towing and Labor, Rental Reimbursement, Tapes and CD's, CB radio and/or Car Phone coverage.


If you have an MSc degree, a Certificate of Proficiency in English, a curious mind and ambition, you may find LUMC the spot for obtaining a PhD based on in-depth (scientific) knowledge.
Of course you are looking for a PhD program that fits your scientific interests. Equally important however is the match with your Thesis Advisor and the people in the research group. Given it will take you at least 4 years to conduct your PhD research and finalize your thesis with (peer-reviewed) articles. If you're interested, have a look at our departmental research programmes and feel free to contact one of our researchers for more information, or talk to one or two current PhD students in the group.
The LUMC offers people opportunities to discover, develop and use their talents. Our ambition is to strive for continuous improvement in the quality of healthcare.
Leiden University has a number of scholarships available for prospective PhD students to allow them to study in Leiden. Also Study in Holland provides information about scholarships and offers practical information about studying here.
Moving to a new country to work involves many challenges. These pages and the International Desk (Loket Buitenland) intend to help international LUMC employees find their way in the LUMC and get to know the Dutch regulations.


Impressed? You should be, for the figure above is the average number of goals per game scored by opponents of the Bowling Green boys' soccer team over the past two seasons.
If that's not eye-popping enough, try this: heading into the 4th Region Tournament, the Purples haven't allowed the ball to find the back of the net since September 24. That's a streak of five straight shutouts, if you're keeping tabs, and it's also a pretty strong indication that the Purples defense is playing at a level that could put BGHS in the hunt for a second consecutive state championship.
And defense, soccer aficionados will tell you, usually begins and often ends with the guy playing one of the loneliest positions in sports: goalkeeper. In the case of the Purples, that lonely guy is senior Ryan Hughes. A smallish 5-foot-8, Hughes throws around that undersized body to make saves and uses a man-sized voice to direct and exhort his teammates. How does he view the position he has played so well the past two seasons?
BGHS coach Craig Widener has a slightly different take on the qualities of a good goalkeeper, but he's quick to say that Hughes lives up to his standards.
Maybe all that hard work is how Hughes compensates for a lack of stature.
The pros have definitely outweighed the cons during Hughes' tenure as the Purples keeper. With hard work, athleticism and smarts (he carries a 3.75 GPA), Hughes has become a team leader.
Mostly, says coach Widener, those teammates are relying on the goalkeeper to communicate to them so they can be in the right positions.
Those demands will be needed as Bowling Green enters the regional tournament. While not overlooking any 4th Region opponents, Hughes says the goal for the Purples all season has been to get back to the state championship game.


text:
HomeClub NewsDave's Diary
11/02/2020 sedgleyparktigers Club News, First XV 0
I blame it all on Calon Lân. Two weeks ago, 2 pm Saturday, I was all ready to go I just needed my outside shoes on, and I would be here, Sedgley Park v Chester.
I still had the TV on and, at that very moment, the band in Cardiff struck up Calon Lân, and I had a startling vision from the past. I saw Barry John, in grainy black and white. The ball had gone loose behind the Wales backline, but he scooped it up before floating and ghosting past white-shirted defenders, one, two, three, four, to score a try of unbelievable brilliance.
You would have to be of a certain age to know just how bad England were in those days. The Welsh game was always the Five Nations opener, and it was the kiss of death to be selected, because ten of the team would inevitably be dropped following the ritual humiliation. The Welsh would beat us up in the first half, run us ragged in the second, and their supporters would gloat for twelve months until the whole sorry scene was repeated next season.
I'm not even Welsh, not the teeniest fraction, but this was lump in the throat stuff, watching that great side. Here at Park Lane (into the 1970s now) we had Leighton Hughes and Dick Hellings, who could be relied upon to put the boot in following England's annual walloping.
So, as this one tune soared over the stadium and through my speakers, fifty-thousand Welshmen telling us about their pure and honest hearts, I made a very stupid decision. I put my slippers back on and settled down to an afternoon of Six Nations on the box.
What was I thinking of? Wales v Italy, with Dan Biggar at 10? And Sedge v Chester is invariably a good game. Italy 2020 are probably about as good as England 1970, but fifty years from now you won't hear wrinkled old men telling you that back in their day they saw Dan Biggar play, worthy and skilful though he may be.
I was actually bored within ten minutes as Italy never remotely looked like making a game of it. I suspect shortage of quality players is their problem, the reverse of England's situation fifty years ago, where the talent was there, but spread thinly amongst perhaps forty first-class clubs. The selectors didn't know where to start, though a good performance in the Varsity match was often enough to propel one youngster into the team.
The change happened in the late 1970s, when a group of talented northern players came together to defeat the All Blacks' full-strength XV at Otley. Bill Beaumont, Fran Cotton, Roger Uttley, John Carleton and others were to become household names as Beaumont's England team won the 1980 Grand Slam. A giant had been awakened.
Like all teams (except the All Blacks) England had their ups and downs in the years that followed, but never again were they abysmal. Will Carling's side was brilliant, and pretty violent, it must be said, and they were denied the 1991 World Cup only by some dodgy Welsh refereeing (Campese deliberate knock-on, penalty try every day of the week). Martin Johnson's boys went the whole way in 2003, and the current team is handy, at worst, sensationally good at best.
I enjoy watching Jonny May as much as I ever enjoyed Barry John – more, because he's English – which is where we came in. Swing low is not quite Calon Lân though, is it?
Coach Talk – A report from Steve Hazzelby
"Notes from behind the Posts"


January 9, 2018 at 8:07 am PDT May contain affiliate links. Read disclosure.
Calling all Couponers! Scroll down below for the complete 2018 Sunday Coupon Insert Schedule – now you know what to expect in each week's paper! Keep in mind that there are usually no inserts on holiday weekends.


text:
IFJ deplores the killing of third blogger in Bangladesh
The International Federation of Journalists (IFJ) deplores the brutal murder of a Bangladeshi blogger in Dhaka today. The brutal attack is the third killing of a secular blogger in three months, as freedom of expression comes under attack across the country.
Earlier today, Ananta Bijoy Das was killed by an armed group near his home in Sylhet in north-eastern Bangladesh. Das, who worked at a local bank, was attacked by four men with machetes as he walked to work. According to reports, the group chased Das, attacking his head and body with the machetes. He was rushed to hospital but died upon arrival.
Das was a well-known atheist blogger in Bangladesh, who regularly wrote for Mukto-Mona (free mind) blog, run by US-Bangladeshi blogger Avijit Roy who was brutally murdered in a similar attack in March in Dhaka. In his writings, Das was critical of fundamentalism and the attack on secular thinkers, as well as championing science. In 2006, he was awarded with the Rationalist Award, by Mukto-Mona, for 'his deep and courageous interest in spreading secular and humanist ideals and messages in a place which is not only remote, but doesn't have a handful of rationalists.'
As news of his murder broke on social media, hundreds of protesters took to the streets demanding government action and for the killers to be brought to justice.
In March, following the murder of Avijit Roy, blogger Md Oyasiqur Rahman Babu was murdered on his way to work in Dhaka. Rahman was critical of irrational religious beliefs, superstitions and radical Islamists. Shortly after his murder, four assailants were charged with his murder.
The IFJ Asia Pacific said: "This brutal murder is a hideous attack on freedom of expression and the IFJ strongly urge the Bangladeshi government to put all necessary efforts into investigating those behind the attack, and bring them to justice."
"The attack against secular bloggers in Bangladesh is reaching epidemic levels with the third blogger killed in less than three months. The increase in press freedom attacks and the decline of safety for media workers is a primary concern. Bloggers are coming under attack from all sides and inaction on behalf of the government is a worrying trend. The failure on behalf of the government to stop these attacks has cultivated a culture of impunity."


text:
Q: What does "attribution is dusting for fingerprints" mean in this context?
Kathie Dello, a climate scientist said the study helps to pin the blame for the heat wave on climate change.
"If searching for a culprit for the intensity of these recent European heat waves, climate change is the obvious culprit," Dello said in an email. "Attribution is just dusting for fingerprints. Climate change will continue to be a menace when it comes to extreme heat, making these events more likely and more intense."

I was reading a Washington Post article and couldn't understand the sentence marked in bold.
I looked up all the words in dictionary and found out the meaning of each word, but still couldn't get what it means in this context.
Was it used in a figurative sense? What does it mean here?

A: When trying to understand this passage, the key word to understand is culprit. From Oxford:

culprit (noun) a person who is responsible for a crime or other misdeed.
  • the cause of a problem or defect: viruses could turn out to be the culprit.

So, a culprit can be a criminal, or a cause. Ms. Dello begins her quote with:

If searching for a culprit... climate change is the obvious culprit.

Clearly this relates to that secondary meaning of culprit: "The cause of a problem or defect". However, Dello then twists the word to its primary meaning – a criminal – when she adds (metaphorically):

Attribution is just dusting for fingerprints.

Dusting for fingerprints is often an early step in criminal investigations. Dello is attempting to cleverly say that figuring out the cause of the recent heat waves is merely a first step, implying that more needs to be done besides merely identifying the root cause of the problem.

redpajama_set_name: RedPajamaStackExchange


text:
Home Products Shoreditch Wild Life (Photo Book 4)
Shoreditch Wild Life (Photo Book 4)
Book 4 from the series 'East London Photo Stories'.
'Shoreditch Wild Life' by Dougie Wallace. 176pp hardcover, quarter-bound, gold foiled,199 x 139 mm
Wild, brash, outrageous and laced with humour, Dougie Wallace's unflinching photos capture the extreme variety of street life in one of London's most iconic and colourful areas: Shoreditch.
East London-based photographer Dougie Wallace grew up in Glasgow (hence his moniker 'Glasweegee'). Internationally recognised for his long-term social documentary projects and a distinct style of expressive street photography, his work has extensively been featured in international publications including The New York Times and Germany's Stern Magazine.
East London Food £28.00
East London Food
**Please note: you will receive EITHER a dark grey or pink book. I'm afraid we are unable to accommodate specific colour requests. We assure you that both are equally beautiful.**
"A celebration of the energy and creativity at work in the East End. This is a joyous, inspirational, heart-warming book." Nigel Slater
East London has become the most exciting place to eat in the UK. Home to a vibrant and diverse community, it has attracted a wave of fresh gastronomic talent, including top chefs, young producers and bold entrepreneurs. This book follows forty people at the heart of a culinary phenomenon — from Michelin-starred chefs and specialist butchers to wild foragers and urban beekeepers. With in-depth interviews and lavish photography, East London Food guides you on where to go, what to eat and how to cook it at home.
'East London Food' by Rosie Birkett and Helen Cathcart
288pp hardcover plus 20pp pull-out recipe booklet, copper foiled, 255 x 196 mm, available in 2 colours randomly selected.
Rosie Birkett is a food writer, stylist and author based in Hackney. She has written features on restaurants, chefs and food trends for magazines and newspapers including the Independent on Sunday, Condé Nast Traveller, the Guardian and Red magazine. Rosie's debut cookbook A Lot On Her Plate was published in 2015 to critical acclaim and she was named a rising star by the Observer the same year.
After studying photography and art direction in Manchester, Helen Cathcart started her career as a photo director in London working on various publications. In 2010 she moved to Sydney to kickstart a transition to being a full-time photographer and was soon photographing for publications such as Condé Nast Traveller, Monocle and Australian Gourmet Traveller. She currently lives in London and shoots for magazines, brands and cookbooks.
London Coffee £20.00
As featured on Huck Magazine and The Londonist.
It's no secret that London is addicted to coffee. Today's flat white drinkers are well versed - with a taste for single origins and light roasts - but what is often forgotten is that the city has a rich history of coffeehouses going back to the 1650s.
Bestselling author Lani Kingston and photographer David Post have captured the stories of the most inspirational modern-day artisans, roasters, growers, innovators and entrepreneurs who make London, once again, a world capital of coffee.
This book comes with a fold-out map featuring 50 of the best London coffee shops.
'London Coffee' by Lani Kingston and David Post
256pp hardback, black foiled, 222 x 167mm. Plus fold-out map.
Lani Kingston is an Australian food writer, consultant and management professional who proudly calls London home. Her first book, 'How to Make Coffee', was an international bestseller, and has been released in five languages and eight editions. Lani holds Masters degrees in Food Studies and in Education, and pastry chef and barista qualifications, and judges coffee for the Great Taste Awards.
David Post is a Canadian photographer whose portfolio includes work for a number of prominent London brands including a recent project for Mast Chocolate. David is also a respected character in the coffee world: he owned and operated an award-winning coffee bar in his home province of Newfoundland before moving to London, where he joined the vibrant coffee scene and worked with cafés across the city.
This is the first edition.
Startup London £28.00
Startup London
As featured in The Culture Trip and Creative Boom
Over 200,000 startups launch in London every year. Here are the stories from 30 of the most creative of them. From King's Cross to New Cross, from digital apps to distilleries, these small businesses have big ambitions to change the way we eat, shop, play and view the world. This book is an inspirational guide for anyone dreaming of starting their own business, whether it be in London or beyond.
This book comes with a 16pp 'Tips and Advice' booklet written by business writer Rebecca Burn-Callander, with illustrations by Daniel Frost.
'Startup London' by Christina Hopkinson and Rick Pushinsky
256pp hardback, gold and white foiled, 255 x 196mm. Plus 16pp booklet.
Christina Hopkinson is a novelist and journalist specialising in examining contemporary issues through both fiction and non-fiction. Her articles have appeared in a range of UK publications including the Guardian, The Times, The Sunday Times, Grazia and Red. Her five novels have been translated into more than 12 languages.
Rick Pushinsky is a British photographer based in East London, with a background in fine art and architecture. He shoots portraits internationally and his work has been featured in Vogue, The Telegraph and the Financial Times.
Book design by Matt Young.
Bored Tourists £9.95
Bored Tourists
'Bored Tourists' by Laurence Stephens. 112pp, paperback with French flaps, fine art paper, 162mm x 127mm.
As featured in Huck Magazine
Discover a lost civilisation.
Gaze up at a towering cathedral.
Take a selfie in front of a kebab shop.
Ever had a holiday that wasn't what you hoped for?
What began as a chance visit to Barcelona Cathedral to escape the mid-August sun, led photographer Laurence Stephens to spend the next three summers traversing the cultural landscapes of Spain and Portugal. From stuffy museums to breathtaking skylines, with camera in hand, he went in search of the not-so-elusive bored tourist.
Laurence Stephens is a British documentary photographer. Inspired by colour photographers of the 1980s, his approach is anthropological, exploring aspects of modern life with a satirical eye.
Book design by Make Good Print Co.


The young woman on the left was born on March 31, 1865 in Poonah, India, to a high-caste Brahmin Hindu family of the Maharashtra people. Her birth name was Yamuna, daughter of the sun.
When she was 9, she married Gopalrao Joshi, a distant relative almost three times her age. He strongly believed in women's education, so he taught her. Despite giving her the bride name of Anandibai, joy of my heart, Gopalrao was not a kind teacher. If she played hooky or helped her grandmother with the cooking, he'd beat her with canes or wooden chairs.
At age fourteen, her only child, a boy, died at ten days old. After this, the couple fixated on her studying medicine. Gopalrao wrote to many missionaries for help, but they all wanted the couple to convert to Christianity first.
In the spring of 1880, Theodosia Carpenter read one such exchange in the January, 1879 issue of The Missionary Review in her dentist's waiting office in Elizabeth, New Jersey. She was intrigued and wrote to them offering assistance. Anandibai could read and write English, so she and Theodosia became regular correspondents, writing about their lives and customs—and about Anandibai coming to America.
In 1882, she and Gopalrao decided that she would go to America alone, upsetting everyone who knew them.
Brahmins were forbidden to cross the ocean, to eat food prepared by non-Brahmins, and to drink water that had been touched by foreigners. Doing any one of these things meant ex-communication.
Hindu women needed female Hindu doctors.
She wanted to study Western medicine both because it was the most complete, and because, in America, she wouldn't be mocked in the street and spat at for seeking an education—as she already was in India.
The speech was a rousing success. The Director General of the Post Offices of India not only gave them 100 rupees, but also administered a fund for her education.
Finally, in April 1883, the 18-year-old left India and arrived in New York in June, spending the summer with Theodosia Carpenter in New Jersey.
Here is part of her letter requesting admittance to the Women's Medical College of Pennsylvania.
On October 3, 1883 she began classes.
Anandibai developed a reputation for her calm demeanor, talking down a mental patient who had a knife, and enduring through to the end of autopsies of babies and children, but her health did not remain strong. She had several bouts of illness and by the time Gopalrao joined her in the summer of 1885, was probably already suffering from tuberculosis. She graduated on March 11, 1886 and accepted a position by June 12: Physician-in-Charge of the female wards of Prince Edward Hospital in Kolhapur, India.
When they arrived home in mid-November she was very sick, but glad to discover they were celebrities, not outcasts. Wherever they went people of every caste prayed for her and newspapers printed updates about her health.
She died on February 26, 1887, just shy of her 22nd birthday. Her ashes were sent to America to rest in Theodosia Carpenter's family plot.
The first Brahmin woman to leave India to obtain an education.
The first foreign student at the Women's Medical College of Pennsylvania who hadn't converted to Christianity.
The first Hindu woman to be granted a Western medical degree.
Natalie Hart is a writer of biblical fiction, a children's worship leader, dancer, mother of two teenagers, and wife to a rock n roll musician. She has lived in Canada, Australia, New York City and the mountains of Oregon. Family and cheap real estate drove her back to Michigan.
This entry was posted in Career, Church, Faith and tagged Anandibai Joshi, Baptist College Hall, Brahmin Hindu, Gopalrao Joshi, Maharashtra, The Missionary Review, Theodosia Carpenter, Women's Medical College of Pennsylvania. Bookmark the permalink.


text:
Q: How can i get the first entity from Object class Updating the question,
How can i get the first object(TaskInstance) from Object, attached image below(obj)
obj
though TaskInstance is present in obj Control is not going inside of if check
for(Object obj : taskInstanceEntities) {
  if(obj instanceof TaskInstance ){
    procurementDTOs.add(bidPreviewMapper.map((TaskInstance)obj));
  }
        }

I am getting a class cast exception for the below code, I have two Objects in taskList, say FirstTask and SecondTask.
How can I get the FirstTask object from obj. When i try to cast the taskList to FirstTask getting a class cast exception, because obj contains two objects(FirstTask and SecondTask), requesting the help to resolve my challenge.
List<Object> taskList = findByNamedQuery("<select query with multiple table join>");
List<FirstTask> firstTaskList = new ArrayList<FirstTask>();
for(Object obj : taskList){
    firstTaskList.add((FirstTask)obj);
}

A: First of all: you don't need to write down the generic type twice; starting with Java 7 we have the diamond operator; that allows you to write:
List<FirstTask> firstTasks = new ArrayList<>();

Then your problem is that you are doing an unconditional cast to FirstTask. Instead, do:
for (Object someTask : taskList) {
    if (someTask instanceof FirstTask) {
        firstTasks.add((FirstTask) someTask);
    }
}

for example. And of course, to take the most out of generics, the assumption would be that all your Task classes probably extend some base class, so taskList should more look like:
List<BaseTask> taskList = ...

Having SomeCollection<Object> is almost always a smell of bad design!


text:
\section{Introduction}

The purpose of this paper is to study the probability law of the real-valued solution of the following general class of stochastic partial differential equations:
\begin{equation}
L u(t,x) = \sigma(u(t,x)) \dot{W} (t,x) + b(u(t,x)),
\label{1}
\end{equation}
$t \geq 0$, $x\in \mathbb{R}^{d}$, where $L$ denotes a second order differential operator, and we impose the
initial conditions
\begin{equation}
u(0,x) = \frac{\partial u}{\partial t}(0, x) = 0. \label{1a}
\end{equation}
The coefficients $\sigma$ and $b$ are some real-valued functions and $\dot{W} (t,x)$ is the formal notation for some Gaussian random perturbation
defined on some probability space. We will assume that it is white in time and with a homogeneous spatial correlation given by a function $f$,
and we denote by $(\mathcal{F}_t)_t$ the filtration generated by $W$ (see Section \ref{prel} for the precise definition of this noise).  \\

By definition, the solution to Equation (\ref{1}) is an adapted stochastic process $\{u(t,x), (t,x)\in [0,T]\times \mathbb{R}^{d}\}$
satisfying
\begin{align}
u(t,x)=& \int_0^t \int_{\mathbb{R}^{d}} \Gamma(t-s,x-y) \sigma(u(s,y)) W(ds,dy) \nonumber\\
&+ \int_0^t \int_{\mathbb{R}^{d}} b(u(t-s,x-y)) \Gamma(s,dy) ds,
\label{3}
\end{align}
where $\Gamma$ denotes the fundamental solution associated to $Lu=0$. As it is going to be clarified later on, for all $t\in [0,T]$,
we suppose that $\Gamma(t)$ is a non-negative measure on $\mathbb{R}^d$ (see Hypothesis A).
The stochastic integral appearing in formula (\ref{3}) requires some care because the integrand
is a measure. For integrands that are real-valued functions, that is $(t,x)\mapsto \Gamma(t,x)\in \mathbb{R}$, the stochastic integral was defined by Walsh in \cite{walsh}. Then, in order to deal with SPDEs whose associated fundamental solution is a generalised function, Dalang \cite{Da} extended Walsh's stochastic integral and covered, for instance, the case of the wave equation in dimension three. \\

However, in the first part of this paper we will give, in a general setting, a definition of the stochastic integral with respect to $W$ using the techniques of the stochastic integration with respect to a cylindrical Brownian motion (see, for instance, \cite{dz}). As we will see, this integral will turn out to be equivalent to Dalang's extension (\cite{Da}) when the integrand is of the form $G:=\Gamma(t-\cdot,x-*) Z(\cdot,*)$, for certain stochastic processes $Z$. More precisely, we will show that random elements of this latter form may be integrated with respect to $W$ using a localising procedure: first we will assume that $Z$ has bounded trajectories and then we will identify $G$ as the weak limit of some sequence $(\Gamma Z_N)_N$, where any $Z_N$ have bounded paths, almost surely (see Lemma \ref{ggamma} and Proposition \ref{prop1}).\\

We should mention at this point that solutions to stochastic partial differential equations of the form (\ref{1}) have been largely studied during the last two decades. For instance, for the case of the stochastic heat and wave equations, we refer to \cite{walsh,carmona,DF,MS,Da,PZ,Pe}.\\

The second part of the paper is devoted to study the probability law of the random variable $u(t,x)$, for any fixed $(t,x)\in (0,T]\times \mathbb{R}^d$. This will be done using the techniques of the so-called Malliavin calculus. The aim is two-fold: \\

First, we prove that $u(t,x)$ has an absolutely continuous law with respect to Lebesgue measure on $\mathbb{R}$, provided that, among other assumptions, the differential operator $L$ and the spatial correlation $f$ are related as follows:
\begin{equation}
\int_0^T \int_{\mathbb{R}^{d}} |\mathcal{F} \Gamma(t)(\xi)|^2 \mu(d\xi) dt< \infty,
\label{5bis}
\end{equation}
where $\mu$ is a non-negative tempered measure such that its Fourier transform is $f$
(see Theorem \ref{densitat} for the precise statement). This result provides a generalisation of Theorem 3 in \cite{QS}, where the authors deal with the three-dimensional stochastic wave equation and a slightly stronger condition than (\ref{5bis}) is assumed. In order to prove that the law of $u(t,x)$ has a density, we apply Bouleau-Hirsch's criterion. Indeed, to prove the Malliavin regularity of the solution, we take advantage of the results in
\cite{QS} and we fully identify the initial condition of the stochastic equation satisfied by the Malliavin derivative of $u(t,x)$. This is an important point in order to show that the Malliavin matrix is invertible almost surely. Eventually, we point out that (\ref{5bis}) is also a sufficient condition to have the stochastic integral in (\ref{3}) well defined. \\

Secondly, we prove that the law of $u(t,x)$ has an infinitely differentiable density with respect to Lebesgue measure on $\mathbb{R}$ (see
Theorem \ref{regularitatdensitat}). To obtain this result, we show that $u(t,x)$ is infinitely differentiable in the Malliavin sense and that the inverse of the Malliavin matrix has moments of all order. For the latter to be achieved, we need to impose a lower bound of the integral in (\ref{5bis})   in terms of a certain power of $T$ (see (\ref{lowbound})).
We should mention that
Theorem \ref{regularitatdensitat} in Section \ref{regularitat} provides an improvement of Theorem 3 in \cite{qs2} for the case of the three-dimensional stochastic wave equation, since in this latter reference the integrability conditions concerning the Fourier transform of $\Gamma$ are much more involved (see also \cite{marta})). Moreover, it is worth mentioning that Theorem \ref{regularitatdensitat} also generalises known results on existence and smoothness of the density for the case of the stochastic heat and wave equation with dimensions $d\geq 1$ and $d=1,2$, respectively (see
\cite{carmona,MS,mms,marta}).\\

The paper is organised as follows. In the next Section \ref{prel}, we present some preliminaries concerning the random perturbation in Equation (\ref{1}) as well as the main hypothesis on the fundamental solution $\Gamma$ and the space correlation $f$. We extend Walsh's stochastic integral in order to cover the case of measure-valued integrands in Section \ref{integral}; we also caracterise some measure-valued random elements that can be integrated with respect to $W$ and we sketch the construction of the integral in a Hilbert-valued setting. In Section \ref{ex}, a theorem on existence and uniqueness of solution for Equation (\ref{1}) is stated; we also deal with some particular examples of differential operators $L$, namely the heat and wave equations.   Sections \ref{existencia} and \ref{regularitat} are devoted, respectively, to the existence and regularity of the density for the probability law of the solution to (\ref{1}).  At the very beginning of Section \ref{existencia}, we introduce the main tools of the Malliavin calculus needed along the paper (we refer to \cite{nualart} for a complete account on the topic).\\

Throughout the paper we use the notation $C$ for any positive real constant, independently of its value.

\section{Preliminaries}
\label{prel}

Recall that we are interested in the following general class of stochastic partial differential equations:
$$
L u(t,x) = \sigma(u(t,x)) \dot{W} (t,x) + b(u(t,x)),
$$
with $t \geq 0$, $x\in \mathbb{R}^{d}$, $L$ denotes a second order differential operator, and we consider vanishing initial conditions (\ref{1a}).
The Gaussian random perturbation is described as follows: $W$ is a zero mean Gaussian family of random variables
$\{W(\varphi), \varphi \in \mathcal{C}_0^\infty (\mathbb{R}^{d+1})\}$,
defined in a complete probability space $(\Omega, \mathcal{F},P)$,  with covariance
\begin{equation}
E(W(\varphi) W(\psi)) = \int_0^\infty \int_{\mathbb{R}^{d}} \int_{\mathbb{R}^{d}} \varphi(t,x) f(x-y) \psi(t,y) dx dy dt,
\label{2}
\end{equation}
where $f$ is a non-negative continuous function of $\mathbb{R}^{d}\setminus \{0\}$ such that it is the Fourier
transform of a non-negative definite tempered measure $\mu$ on $\mathbb{R}^{d}$.
That is,
$$f(x)=\int_{\mathbb{R}^{d}} \exp(-2\pi i\; x\cdot \xi) \mu(d\xi)$$
and there is an integer $m\geq 1$ such that
$$\int_{\mathbb{R}^{d}} (1+|\xi|^2)^{-m}\mu(d\xi) <\infty.$$
Then, the covariance (\ref{2}) can also be written, using Fourier transform, as
$$E(W(\varphi) W(\psi)) = \int_0^\infty \int_{\mathbb{R}^{d}} \mathcal{F} \varphi(t)(\xi) \overline{\mathcal{F} \psi(t)(\xi)} \mu(d\xi) dt.$$
The main assumption on the differential operator $L$ may be summarised as follows:\\

\noindent {\bf Hypothesis A}.
The fundamental solution to $Lu = 0$, denoted by $\Gamma$, is a non-negative measure of
the form $\Gamma(t, dx)dt$ such that for all $T>0$
\begin{equation}
\sup_{0\leq t\leq T} \Gamma(t,\mathbb{R}^{d})\leq C_T<\infty
\label{e2}
\end{equation}
and
\begin{equation}
\int_0^T \int_{\mathbb{R}^{d}} |\mathcal{F} \Gamma(t)(\xi)|^2 \mu(d\xi) dt< \infty.
\label{5}
\end{equation}

\vspace{0.5cm}

The completion of the Schwartz space $\mathcal{S} (\mathbb{R}^{d})$ of rapidly decreasing $\mathcal{C}^\infty$ functions, endowed
with the inner product
$$\langle \varphi,\psi\rangle_\mathcal{H}
=\int_{\mathbb{R}^{d}} \int_{\mathbb{R}^{d}} \varphi(x) f(x-y) \psi(y) dx dy=\int_{\mathbb{R}^{d}} \mathcal{F} \varphi(\xi) \overline{\mathcal{F} \psi(\xi)} \mu(d\xi),$$
$\varphi, \psi\in \mathcal{S} (\mathbb{R}^{d})$, is denoted by $\mathcal{H}$. Notice that $\mathcal{H}$ may contain distributions. Set $\mathcal{H}_T = L^2([0,T];\mathcal{H})$.

\section{Stochastic integrals}
\label{integral}

In this section Walsh's stochastic integral with respect to martingale measures will be extended to more general integrands, namely the class of square integrable $\mathcal{H}$-valued predictable processes. The extension will be performed in the infinite dimensional setting described by Da Prato and Zabczyk in \cite{dz}.
Then, we will give non-trivial examples of integrands, which will be some measure-valued random elements. We will briefly recall the extension of the stochastic integral in a Hilbert-valued setting. This will be needed to give a rigorous meaning to the stochastic evolution equations satisfied by the Malliavin derivatives of the solution of (\ref{3}).\\

\noindent {\bf Extension of Walsh's stochastic integral}\\

Fix a time interval $[0,T]$. The Gaussian family $\{W(\varphi ),\varphi \in
\mathcal{C}_{0}^{\infty }([0,T]\times \mathbb{R}^{d})\}$ can be extended to the
completion $\mathcal{H}_T=L^{2}([0,T];\mathcal{H})$ of the space $\mathcal{C}_{0}^{\infty
}([0,T]\times \mathbb{R}^{d})$ under the scalar product%
\[
\left\langle \varphi ,\psi \right\rangle =\int_{0}^{T}\int_{\mathbb{R}^{d}}%
\mathcal{F}\varphi (t)(\xi )\overline{\mathcal{F}\psi (t)(\xi )}\mu (d\xi
)dt.
\]%
We will also denote by $W(g)$ the Gaussian random variable associated with
an element $g\in L^{2}([0,T];\mathcal{H})$.

Set $W_{t}(h)=W(1_{[0,t]}h)$ for any $t\geq 0$ and $h\in \mathcal{H}$. \
Then, $\{W_{t},t\in \lbrack 0,T]\}$ is a cylindrical Wiener process in the
Hilbert space $\mathcal{H}$. That is, for any $h\in \mathcal{H}$, $%
\{W_{t}(h),t\in \lbrack 0,T]\}$ is a Brownian motion with variance $\left\|
h\right\| _{\mathcal{H}}^{2}$, and%
\[
E(W_{t}(h)W_{s}(g)=\left( s\wedge t\right) \left\langle h,g\right\rangle _{%
\mathcal{H}}.
\]%
Let $\mathcal{F}_t$ be the $\sigma$-field generated by the random variables
$\{W_s(h), h\in \mathcal{H}, 0\le s\le t\}$ and the $P$-null sets. We define the predictable
$\sigma$-field as the $\sigma$-field in $\Omega\times [0,T]$ generated by
the sets $\{ (s,t]\times A, 0\le s<t\le T, A\in \mathcal{F}_s\}$.

Then (see, for instance, \cite{dz}), we can define the
stochastic integral of $\mathcal{H}$-valued square integrable predictable
processes. For any predictable process $g\in $ $L^{2}(\Omega \times \lbrack
0,T];\mathcal{H)}$ we denote its integral with respect to the cylindrical
Wiener process $W$ by%
\begin{equation}
\int_{0}^{T}\int_{\mathbb{R}^{d}}gdW=g\cdot W,  \label{e1}
\end{equation}
and we have the isometry property%
\[
E\left( \left| g\cdot W\right| ^{2}\right) =E\left( \int_{0}^{T}\left\|
g_{t}\right\| _{\mathcal{H}}^{2}dt\right) .
\]

\begin{remark}
Under the standing assumptions, using an approximation procedure by means of test functions,
one proves that the space $\mathcal{H}$ contains the indicator functions of bounded Borel sets (for details see \cite{DF} or
\cite{quer}, p. 13). Then, $M_{t}(A):=W(1_{[0,t]}1_{A})$ defines a martingale measure associated to the noise $%
W $ in the sense of Walsh (see \cite{walsh} and \cite{Da}) and the stochastic integral  (\ref{e1})
coincides with the integral defined in the work of Dalang \cite{Da}.
\end{remark}

\noindent {\bf Example of integrands}\\

We aim now to provide useful examples of  random distributions which belong to the space $L^2(\Omega\times [0,T]; \mathcal{H})$.
Before stating the result, we consider the following lemma:

\begin{lemma}\label{ggamma}
Assume that $\Gamma$ satisfies Hypothesis A. Let $g$ be a bounded Borel function on $[0,T]\times \mathbb{R}^{d}$. Then $%
g\Gamma \in \mathcal{H}_T$, and%
\[
\left\| g\Gamma \right\|_{\mathcal{H}_T}^2 \leq \left\| g\right\|^2
_{\infty }\int_{0}^{T}\int_{\mathbb{R}^{d}}\left| \mathcal{F}\Gamma (t)(\xi
)\right| ^{2}\mu (d\xi )dt.
\]
\end{lemma}

\noindent {\it Proof.}
We can decompose $g$ into the difference $g^{+}-g^{-}$ of two nonnegative
bounded Borel functions. Thus, without any loss of generality we can assume
that $g$ is nonnegative. Moreover, we observe that $g\Gamma $ also satisfies \ conditions (%
\ref{e2}) and (\ref{5}). Indeed, to prove the latter condition, we consider
an approximation of the identity $(\psi _{n})_n$ defined as follows: let $\psi\in \mathcal{C}_0^{\infty}(\mathbb{R}^d)$ such that $\psi\geq 0$,
the support of $\psi$ is contained in the unit ball of $\mathbb{R}^d$ and $\int_{\mathbb{R}^d} \psi(x)dx=1$.
Set $\Gamma
_{n}(t)=\psi _{n}\ast \Gamma (t)$ and $J_{n}(t)=\psi _{n}\ast (g\Gamma (t))$.
Then, for all $t\in \lbrack 0,T]$,  $\Gamma _{n}(t)$ and  $J_n(t)$ belong to $\mathcal{S}(\mathbb{R}^{d})\subset \mathcal{H}$, and \ $\left| \mathcal{F}%
\Gamma _{n}(t)\right| \leq \left| \mathcal{F}\Gamma (t)\right|$. Besides, since $\Gamma$ is non-negative, we have that
$J_n(t)(\xi)\leq \|g\|_\infty \Gamma_n(t)(\xi)$, for any $\xi\in \mathbb{R}^d$.  Thus, by Fatou's lemma
\begin{align*}
& \int_0^T \int_{\mathbb{R}^{d}} |\mathcal{F} (g\Gamma(t))(\xi)|^2 \mu(d\xi) dt \\
&\quad \leq \liminf_{n\rightarrow \infty} \int_0^T  \int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}} J_n(t,x) f(x-y) J_n(t,y) dx dy dt \\
&\quad \leq \|g\|_{\infty}^2 \liminf_{n\rightarrow \infty} \int_0^T \int_{\mathbb{R}^{d}} |\mathcal{F} \Gamma_n(t)(\xi)|^2 \mu(d\xi) dt\\
& \quad \leq \|g\|_{\infty}^2 \int_0^T \int_{\mathbb{R}^{d}} |\mathcal{F} \Gamma(t)(\xi)|^2 \mu(d\xi) dt < \infty.
\end{align*}
The fact that $g\Gamma$ satisfies conditions (\ref{e2}) and (\ref{5}) let us reduce the proof to the case where $g=1$.

We consider the regularisation $(\Gamma_n)_n$ of $\Gamma$ defined above. Condition (\ref{5}) implies that $\Gamma _{n}(t)$ belongs to
$\mathcal{H}_T$ and it has a uniformly bounded norm, so it converges weakly to some
element $h\in \mathcal{H}_T$. We claim that $h=\Gamma $, and this is a consequence of the
fact that, owing to the definition of $\Gamma_n(t)$, for any $\varphi \in \mathcal{S}(\mathbb{R}^{d})$, and for any $%
0\leq s<t\leq T$ we have%
\[
\int_{s}^{t}\left\langle h,\varphi \right\rangle _{\mathcal{H}%
}dr=\lim_{n\rightarrow \infty }\int_{s}^{t}\left\langle \Gamma
_{n}(r),\varphi \right\rangle _{\mathcal{H}}dr=\int_{s}^{t}\left\langle
\Gamma (r),\varphi \right\rangle _{\mathcal{H}}dr.
\]
More precisely, it holds that
$$
\int_{s}^{t}\left\langle \Gamma
_{n}(r),\varphi \right\rangle _{\mathcal{H}}dr  =  \int_{s}^{t} \int_{\mathbb{R}^d}
\Gamma(r,dz) \left( \int_{\mathbb{R}^d} \psi_n(x-z) F(x) dx \right)dr,$$
where $F(x):=\int_{\mathbb{R}^d} f(x-y)\varphi(y) dy$. Observe that the hypothesis on $f$ and $\varphi$ imply that $F$ is continuous and
$\lim_{x\rightarrow \infty} F(x)=0$. Hence, it turns out that we can apply the Bounded Convergence Theorem, so that we end up with
$$
\lim_{n\rightarrow \infty } \int_{s}^{t}\left\langle \Gamma
_{n}(r),\varphi \right\rangle _{\mathcal{H}}dr   =
\int_{s}^{t} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \Gamma(r,dz)  f(z-y) \varphi(y) dy dr,$$
and this let us identify $h$ with $\Gamma$ in $\mathcal{H}_T$. \hfill
$\square$

\medskip
This lemma allows us to prove the following result, which give examples of random distributions that can be integrated with respect to $W$.

\begin{proposition}\label{prop1}
Assume that $\Gamma$ satisfies Hypothesis A.
Let $Z=\{Z(t,x),(t,x)\in \lbrack 0,T]\times \mathbb{R}^{d}\}$ be a
predictable process such that%
\[
C_{Z}:=\sup_{(t,x)\in \lbrack 0,T]\times \mathbb{R}^{d}}E(|Z(t,x)|^{2})<%
\infty .
\]%
Then, the random element $G=G(t,dx)=Z(t,x)\Gamma (t,dx)$ is a predictable
process in the space $L^{2}(\Omega \times \lbrack 0,T];\mathcal{H)}$.
\end{proposition}

\noindent {\it Proof.}
For any $N\geq 1$ define%
\[
Z_{N}(t,x)=Z(t,x){\bf 1}_{\{|Z(t,x)|\leq N\}}.
\]%
Clearly, $Z_{N}$ is a predictable process with bounded trajectories. Thus,
by the previous lemma, $G_{N}(t,x):=Z_{N}(t,x)\Gamma (t,dx)$ is a
predictable process in $L^{2}(\Omega \times \lbrack 0,T];\mathcal{H)}$.
Let $\left( J_{N,n}(t) \right)_n \subset \mathcal{S}(\mathbb{R}^d) $ be the regularisation of $G_N(t)$ by means of an approximation of the identity $(\psi_n)_n$, as it has been defined in the proof of Lemma \ref{ggamma}. This let us prove that the norm of $G_N$ in $L^2(\Omega\times [0,T];\mathcal{H})$
is uniformly bounded because%
\begin{align*}
& E\left( \left\| G_{N}\right\| _{L^{2}([0,T];\mathcal{H})}^{2}\right)
= E\left( \lim_{n\rightarrow \infty} \|J_{N,n}\|_{L^{2}([0,T];\mathcal{H})}^{2}\right)\\
&\quad \leq \liminf_{n\rightarrow \infty} E\left( \int_0^T \int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}} J_{N,n}(t,x) f(x-y)J_{N,n}(t,y) dx dy dt\right)\\
&\quad \leq C_Z \liminf_{n\rightarrow \infty}  E\left( \int_{0}^{T}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\Gamma_n
(t,x) f(x-y) \Gamma_n(t,y) dx dy dt \right)\\
&\quad \leq C_{Z}\int_{0}^{T}\int_{\mathbb{R}^{d}}\left| \mathcal{F}\Gamma
(t)(\xi )\right| ^{2}\mu (d\xi )dt < \infty.
\end{align*}%
Recall that $\Gamma_n(t)=\psi_n * \Gamma(t)$.
Therefore, $G_{N}$ converges weakly to a predictable process $\widetilde{G}$
in $L^{2}(\Omega \times \lbrack 0,T];\mathcal{H)}$. We claim that $%
\widetilde{G}=G$. In fact, for any $\varphi \in \mathcal{S}(\mathbb{R}^{d})$%
,  for any $0\leq s<t\leq T$ and for any $B\in \mathcal{F}_{s}$, we can argue similarly as in the very last part of the proof of Lemma \ref{ggamma} and obtain
\begin{align*}
& E\left( {\bf 1}_{B}\int_{s}^{t}\left\langle \widetilde{G}(r),\varphi \right\rangle
_{\mathcal{H}}dr\right) \\
&\quad = \lim_{N\rightarrow \infty }E\left(
{\bf 1}_{B}\int_{s}^{t}\left\langle G_{N}(r),\varphi \right\rangle _{\mathcal{H}%
}dr\right)  \\
&\quad =\lim_{N\rightarrow \infty }E\left( {\bf 1}_{B}\int_{s}^{t}\left\langle
Z(r,x){\bf 1}_{\{|Z(r,x)|\leq N\}}\Gamma (r),\varphi \right\rangle _{\mathcal{H}%
}dr\right)  \\
&\quad =\lim_{N\rightarrow \infty }E\left( {\bf 1}_B \int_{s}^{t}\int_{\mathbb{R}^{d}}\int_{%
\mathbb{R}^{d}}\Gamma (r,dx)Z(r,x){\bf 1}_{\{|Z(rt,x)|\leq N\}}f(x-y)\varphi
(y)dxdydr\right)  \\
&\quad =E\left( {\bf 1}_B \int_{s}^{t}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\Gamma
(r,dx)Z(r,x)f(x-y)\varphi (y)dxdydr\right),
\end{align*}
so we can identify $\tilde G$ with $G$. \hfill$\square$

\begin{remark}
As a consequence of Proposition \ref{prop1}, we are able to define the stochastic integral of $G=Z \Gamma$ with respect to $W$:
$$G\cdot W =\int_0^T \int_{\mathbb{R}^{d}} G(s,y) W(ds,dy)=\int_0^T \int_{\mathbb{R}^{d}} \Gamma(s,y) Z(s,y) W(ds,dy).$$
Moreover, using the same ideas as in \cite{QS} (see also \cite{quer}, Theorem 1.2.5), one can obtain bounds for the $L^p(\Omega)-$norm of
$G\cdot W$. More precisely, suppose that
$$\sup_{(t,x)\in [0,T]\times \mathbb{R}^{d}} E(|Z(t,x)|^p)< \infty,$$
for some $p\geq 2$. Then
\begin{align}
& E(|G\cdot W|^p) = E\left( \left| \int_0^T \int_{\mathbb{R}^{d}} G(s,y) W(ds,dy)\right|^p\right) \nonumber \\
& \quad
\leq C_p (\nu_T)^{\frac{p}{2}-1} \int_0^T \left( \sup_{x\in \mathbb{R}^{d}} E(|Z(s,x)|^p) \right) \int_{\mathbb{R}^{d}} |\mathcal{F} \Gamma (s)(\xi)|^2 \mu(d\xi) ds,
\label{pbound}
\end{align}
where
$$\nu_T =\int_0^T \int_{\mathbb{R}^{d}} |\mathcal{F} \Gamma (t)(\xi)|^2 \mu(d\xi) dt.$$
\label{Lpbound}
\end{remark}

\begin{remark}
Since the noise's correlation is of the form $(x,y)\mapsto f(x-y)$, it is natural to be interested in spatially homogeneous situations. Indeed, suppose that we add the following hypothesis on the process $Z$: for all $s\in [0,T]$ and $x,y\in \mathbb{R}^{d}$ we have
$$E(Z(s,x)Z(s,y))=E(Z(s,0)Z(s,y-x)).$$
Then, owing to \cite{Da}, p. 10, we may construct a non-negative tempered measure $\mu_s^Z$ on $\mathbb{R}^{d}$ such that
$$\|G\|_{L^2(\Omega; \mathcal{H}_T)}= E(|G\cdot W|^2)=\int_0^T \int_{\mathbb{R}^{d}} |\mathcal{F} \Gamma(s)(\xi)|^2 \mu_s^Z(d\xi) ds.$$
\label{hom}
\end{remark}

As we will see in the next section, the main examples of deterministic measures $\Gamma$ will correspond to fundamental solutions associated to second order differential operators. First, we sketch the construction of the stochastic integral in a Hilbert-valued setting, that is when the process $Z$ takes values in some Hilbert space, usually different from $\mathcal{H}$. \\

\noindent {\bf Hilbert-valued stochastic integrals}\\

Let $\mathcal{A}$ be a separable real Hilbert space with inner-product and norm denoted
by
$\langle\cdot,\cdot\rangle_{\mathcal{A}}$ and $\Vert \cdot\Vert_{\mathcal{A}}$,
respectively.
Let $K=\{K(t,x),  (t,x)\in[0,T]\times \mathbb{R}^{d}\}$
be an $\mathcal{A}-$valued predictable process satisfying the following condition:
\begin{equation}
\sup_{(t,x)\in[0,T]\times \mathbb{R}^{d}}
E\left(||K(t,x)||_{\mathcal{A}}^{2}\right)<\infty.
\label{10}
\end{equation}
Our purpose is to define the stochastic integral of elements of the form $\Gamma  K=\Gamma(t,dx)  K(t,x) \in L^2(\Omega \times  [0,T]; \mathcal{H} \otimes \mathcal{A})$.

Let $(e_{j},j\geq 0)$ be a complete orthonormal  system of $\mathcal{A}$. Set
$K^j(t,x) = \langle K(t,x),e_{j}\rangle_{\mathcal{A}}$,  $(t,x)\in [0,T]\times
\mathbb{R}^{d}$.
According to Proposition \ref{prop1}, for any $j\geq 0$ the element $G^j= G^j(t,x)= \Gamma(t,dx) K^j (t,x)$ belongs to
$L^2(\Omega \times [0,T]; \mathcal{H})$ and, therefore, we may integrate it with respect to the noise $W$:
$$G^j \cdot W=\int_0^T \int_{\mathbb{R}^{d}} \Gamma(s,y) K^j(s,y) W(ds,dy).$$
We define, for $G=\Gamma  K$,
$$ G\cdot W := \sum_{j\geq 0} G^j \cdot W.$$
Owing to (\ref{10}) and Proposition \ref{prop1}, it can be proved that the above series is convergent and therefore $G\cdot W$ defines an element of $L^{2}(\Omega ;\mathcal{A})$ (see also
\cite{QS}, Remark 1). Moreover, using the same arguments as for the proof of (\ref{pbound}), we have the following bound for the moments of $G\cdot W$ in $\mathcal{A}$:
\begin{align}
& E\big(|| G\cdot W||_{\mathcal{A}}^{p}\big) \nonumber \\
& \quad \le C_p (\nu_T)^{\frac{p}{2}-1} \int_0^T  \sup_{x\in \mathbb{R}^{d}} E(||K(s,x)||^p_ {\mathcal{A}})
\int_{\mathbb{R}^{d}}  |\mathcal{F} \Gamma(s)(\xi)|^{2} \mu(d\xi) ds,
\label{11}
\end{align}
for all $p\geq 2$.

\section{Existence and uniqueness of solutions}
\label{ex}

Recall that a solution to Equation (\ref{1}) is a real-valued adapted stochastic process $u = \{u(t,x), (t,x)\in [0,T]\times \mathbb{R}^{d}\}$
satisfying
\begin{align}
u(t,x)=& \int_0^t \int_{\mathbb{R}^{d}} \Gamma(t-s,x-y) \sigma(u(s,y)) W(ds,dy) \nonumber \\
&+ \int_0^t \int_{\mathbb{R}^{d}} b(u(t-s,x-y)) \Gamma(s,dy) ds.
\label{eq}
\end{align}
We assume that   $Z=Z(s,y)=\sigma(u(s,y))$ satisfy the hypothesis of Proposition \ref{prop1}, so that the stochastic integral on the right hand-side is well-defined.

We suppose that $\sigma$ and $b$ are real-valued Lipschitz functions. Under these conditions, we may state an existence and uniqueness of solution's theorem:

\begin{theorem}
Suppose that the fundamental solution $\Gamma$ of $Lu=0$ satisfies Hypothesis A.
Then, Equation (\ref{eq}) has a unique solution $\{u(t,x), (t,x)\in [0,T]\times \mathbb{R}^{d}\}$ which is continuous in $L^2$ and satisfies
$$\sup_{(t,x)\in [0,T]\times \mathbb{R}^{d}} E(|u(t,x)|^p) <\infty,$$
for all $T>0$ and $p\geq 1$.
\label{sol}
\end{theorem}
For the proof we refer to \cite{Da}, Theorem 13, where the Walsh-Dalang equivalent setting is used.

Let us now enumerate some examples of differential operators whose associated fundamental solution fulfills the hypothesis of
Theorem \ref{sol}.

\begin{example}
{\it The wave equation}. Let $\Gamma_d$ be the fundamental solution of the wave equation in $\mathbb{R}^{d}$, that is, $\Gamma_d$ is the solution of
$$\frac{\partial^2 \Gamma_d}{\partial t^2} - \Delta \Gamma_d =0,$$
with vanishing initial conditions. It is known that for $d=1,2,3$, $\Gamma_d$ is given, respectively, by
\begin{align*}
\Gamma_1(t)&=\frac{1}{2} {\bf 1}_{\{|x|<t\}},\\
\Gamma_2 (t)& = C (t^2-|x|^2)_+^{-1/2},\\
\Gamma_3(t) &= \frac{1}{4\pi t}\sigma_t,
\end{align*}
where $\sigma_t$ denotes the surface measure on the three-dimensional sphere of radius $t$. In particular, for each $t$, $\Gamma_d(t)$ has compact support. It is important to remark that only in these cases $\Gamma_d$ defines a non-negative measure. Furthermore, for all dimensions $d\geq 1$, we have a unified expression for the Fourier transform of $\Gamma_d(t)$:
$$\mathcal{F} \Gamma_d (t)(\xi)= \frac{\sin (2\pi t |\xi|)}{2\pi |\xi|}.$$
Elementary estimates show that there are positive constants $c_1$ and $c_2$ depending on $T$ such that
$$\frac{c_1}{1+|\xi|^2}\leq \int_0^T \frac{ \sin^2 (2\pi t\xi)}{4\pi^2 |\xi|^2} dt \leq \frac{c_2}{1+|\xi|^2}.$$
Therefore, $\Gamma_d$ satisfies condition (\ref{5}) if and only if
\begin{equation}
\int_{\mathbb{R}^{d}} \frac{\mu(d\xi)}{1+|\xi|^2}<\infty.
\label{12}
\end{equation}
\label{wave}
\end{example}

\begin{example}
{\it The heat equation}. Let $\Gamma$ be the fundamental solution of the heat equation in $\mathbb{R}^{d}$ and with vanishing initial conditions, that is
$$\frac{\partial \Gamma}{\partial t} -\frac{1}{2}\Delta \Gamma =0.$$
Then, $\Gamma$ is given by the Gaussian density:
$$\Gamma(t,x)=(2\pi t)^{-d/2} \exp\left(-\frac{|x|^2}{2 t}\right)$$
and
$$\mathcal{F} \Gamma (t)(\xi)=\exp(-4\pi^2 t|\xi|^2).$$
Because
$$\int_0^T \exp(-4\pi^2 t|\xi|^2) dt =\frac{1}{4\pi^2 |\xi|^2} (1-\exp(-4\pi^2 T|\xi|^2)),$$
we conclude that condition (\ref{5}) holds if and only if (\ref{12}) holds.
\label{heat}
\end{example}

Let us express condition (\ref{12}) in terms of the covariance function $f$. Indeed, as it is pointed out in \cite{Da}, condition (\ref{12}) is always true when $d=1$; for $d=2$, (\ref{12}) holds if and only if
$$\int_{|x|\leq 1} f(x) \log\frac{1}{|x|} dx <\infty,$$
and for $d\geq 3$, (\ref{12}) holds if and only if
$$\int_{|x|\leq 1} f(x) \frac{1}{|x|^{d-2}} dx <\infty.$$

\section{Existence of density}
\label{existencia}

In this section we aim to prove that the solution to Equation (\ref{3}), at any point $(t,x)\in (0,T]\times \mathbb{R}^{d}$, is a random variable whose law admits a density with respect to Lebesgue measure on $\mathbb{R}$. For this, we will make use of the techniques provided by the Malliavin calculus and, more precisely, we will apply Bouleau-Hirsch's criterion (see \cite{bh} or Theorem 2.1.2 in \cite{nualart}).\\

First of all, we describe the Gaussian context in which we will use the tools of the Malliavin calculus. Namely, we consider the Hilbert space $\mathcal{H}_T=L^2([0,T];\mathcal{H})$ and the Gaussian family of random variables
$(W(h), h\in \mathcal{H}_T)$ defined at the very beginning of Section \ref{integral}.
Then
$(W(h), h\in \mathcal{H}_T)$ is a centered Gaussian process such that $E(W(h_1)W(h_2))=\langle h_1,h_2\rangle_{\mathcal{H}_T}$,
$h_1,h_2\in \mathcal{H}_T$,
and we can use the differential Malliavin calculus based on it (see, for instance, \cite{nualart}). The Malliavin derivative is denoted by $D$ and, for any $N\geq 1$, the domain of the iterated derivative $D^N$ in $L^p(\Omega; \mathcal{H}_T^{\otimes N})$ is denoted by
$\mathbb{D}^{N,p}$, for any $p\geq 2$. We shall also use the notation
$$\mathbb{D}^\infty =\cap_{p\geq 1}\cap_{k\geq 1} \mathbb{D}^{k,p}.$$

The first step in order to apply Bouleau-Hirsch's criterion is to study the Malliavin differentiability of $u(t,x)$, for all fixed $(t,x)\in (0,T]\times \mathbb{R}^{d}$. Recall that, for any random variable $X$ in the domain of the derivative operator $D$, $DX$ defines an $\mathcal{H}_T-$valued random variable. In particular, for some fixed $r\in [0,T]$, $DX(r)$ is an element of $\mathcal{H}$, which will be denoted by $D_rX$. In the sequel we will use the notation $\cdot$ and $*$ to denote, respectively, the time and $\mathcal{H}$ variables.

\begin{proposition}
Assume that $\Gamma$ satisfies Hypothesis A. Suppose also that the coefficients $b$ and $\sigma$ are $\mathcal{C}^1$ functions with bounded Lipschitz continuous derivatives. Then, for any $(t,x)\in [0,T]\times \mathbb{R}^{d}$, $u(t,x)$ belongs to
$\mathbb{D}^{1,p}$, for any $p\in [1,\infty)$.

Moreover, the Malliavin derivative $Du(t,x)$ defines an $\mathcal{H}_T-$valued process that satisfies the following linear stochastic differential equation:
\begin{align}
D_r u(t,x) = & \sigma(u(r,*))\Gamma(t-r,x-*) \nonumber\\
& + \int_r^t \int_{\mathbb{R}^{d}} \Gamma(t-s,x-y) \sigma'(u(s,y)) D_ru(s,y)W(ds,dy)\nonumber\\
& + \int_r^t \int_{\mathbb{R}^{d}}  b'(u(s,x-y)) D_r u(s,x-y)\Gamma(t-s,dy) ds,
\label{eqmal}
\end{align}
for all $r\in [0,T]$.
\label{difmal}
\end{proposition}

The stochastic integral on the right hand-side of Equation (\ref{eqmal}) must be understood by means of the Hilbert-valued integration setting described at the very final part of Section \ref{integral}.

Concerning the Hilbert-valued pathwise integral, it is defined as follows: let $\mathcal{A}$ be a Hilbert space, $(e_j)_{j\geq 1}$ a complete orthonormal system of $\mathcal{A}$ and $\{Y(s,y), (s,y)\in [0,T]\times \mathbb{R}^{d}\}$ an $\mathcal{A}-$valued stochastic process such that
$$\sup_{(s,y)\in [0,T]\times \mathbb{R}^{d}} E(\|Y(s,y)\|^2_{\mathcal{A}})<+\infty.$$
Then, the $\mathcal{A}-$valued integral
$$\mathcal{I}_t=\int_0^t  \int_{\mathbb{R}^{d}} Y(s,y)\Gamma(s,dy) ds$$
is determined by the components $\left( \int_0^t  \int_{\mathbb{R}^{d}} \langle Y(s,y),e_j\rangle_{\mathcal{A}}\; \Gamma(s,dy) ds, j\geq 1\right)$, which are real-valued integrals. Moreover, one can obtain an upper bound for the moments of the above integral (see \cite{quer}, p. 24):
\begin{equation}
E(|\mathcal{I}_t|^p)\leq \int_0^t \sup_{z\in \mathbb{R}^{d}} E(\|Y(s,z)\|_\mathcal{A}^p \int_{\mathbb{R}^{d}} \Gamma(s,dy) ds,\; p\geq 2.
\label{deter}
\end{equation}
Eventually, notice that owing to Proposition \ref{prop1}, the first term on the right hand-side of (\ref{eqmal}) is fully defined.

\medskip
\noindent {\it Proof of Proposition \ref{difmal}}. The statement is almost an immediate consequence of Theorem 2 in \cite{QS}.  Indeed, the authors of this latter reference prove that, under the standing hypothesis and for any fixed $(t,x)\in [0,T]\times \mathbb{R}^{d}$, the random variable $u(t,x)$
belongs to $\mathbb{D}^{1,p}$, for any $p\in [1,\infty)$. In addition, they show that there exists an $\mathcal{H}_T-$valued stochastic process
$\{\Theta(t,x), (t,x)\in [0,T]\times \mathbb{R}^{d}\}$ satisfying
$$\sup_{(t,x)\in [0,T]\times \mathbb{R}^{d}} E(\|\Theta(t,x)\|_{\mathcal{H}_T}^p)<\infty$$
and such that, in $\mathcal{H}_T$,
\begin{align*}
D u(t,x) = &
\Theta(t,x)+ \int_0^t \int_{\mathbb{R}^{d}} \Gamma(t-s,x-y) \sigma'(u(s,y)) Du(s,y)W(ds,dy)\\
& + \int_0^t \int_{\mathbb{R}^{d}}  b'(u(s,x-y)) D u(s,x-y)\Gamma(t-s,dy) ds.
\end{align*}
Moreover, it holds that
$$E(\|\Theta(t,x)\|_{\mathcal{H}_T}^2)=E(\| \Gamma(t-\cdot,x-*) \sigma(u(\cdot,*))\|_{\mathcal{H}_T}^2).$$
Hence, in order to conclude the proof, we only need to  show that the Hilbert-valued random variables $\Theta(t,x)$ and
$\Gamma(t-\cdot,x-*) \sigma(u(\cdot,*))$ coincide  as elements of $\mathcal{H}_T$.

Let $(\Gamma_n)_{n\geq 1}$ be the family of smooth functions defined in the proof of Lemma \ref{ggamma}. Then,
in the proof of Theorem 2 in \cite{QS} the process $\Theta$ is defined by the following limit in $\mathcal{H}_T$:
$$\Theta(t,x)=\mathcal{H}_T-\lim_{n\rightarrow \infty} \Gamma_n(t-\cdot,x-*) \sigma(u_n(\cdot,*)),$$
where $\{u_n(t,x), (t,x)\in [0,T]\times \mathbb{R}^{d}\}$ is the unique mild solution to an equation of the form (\ref{3}) but replacing $\Gamma$ by $\Gamma_n$.

As a consequence of the proof of Proposition 3 from \cite{QS}, it is readily checked that
$$
\lim_{n\rightarrow 0}  E(\| \Gamma_n(t-\cdot,x-*)[\sigma(u_n(\cdot,*))-\sigma(u(\cdot,*))]\|_{\mathcal{H}_T}^2)=0
$$
and
$$\lim_{n\rightarrow 0}  E(\| [\Gamma_n(t-\cdot,x-*)-\Gamma(t-\cdot,x-*)]\sigma(u(\cdot,*))\|_{\mathcal{H}_T}^2)=0.$$
Thus, we get that $\Theta(t,x)=\sigma(u(t-\cdot,x-*)) \Gamma(t-\cdot, x-*)$. \hfill $\square$

\vspace{0.3cm}

The main result of the section is the following:

\begin{theorem}
Assume that $\Gamma$ satisfies Hypothesis A. Suppose also that the coefficients $b$ and $\sigma$ are $\mathcal{C}^1$ functions with bounded Lipschitz continuous derivatives and that $|\sigma(z)|\geq c>0$, for all $z\in \mathbb{R}$ and some positive constant $c$. Then, for all $t>0$ and $x\in \mathbb{R}^{d}$, the random variable $u(t,x)$ has an absolutely continuous law with respect to Lebesgue measure on $\mathbb{R}$.
\label{densitat}
\end{theorem}

\noindent
{\it Proof}. Owing to Bouleau-Hirsch's criterion and Proposition \ref{difmal}, it suffices to show that $\|Du(t,x)\|_{\mathcal{H}_T}>0$ almost surely.

To begin with,  from Equation (\ref{eqmal})  we obtain
\begin{equation}
\int_0^t \|D_s u(t,x)\|_\mathcal{H}^2 ds\geq \frac{1}{2}\int_{t-\delta}^t \|\Gamma(t-s,x-*)\sigma(u(s,*))\|_\mathcal{H}^2 ds - I(t,x;\delta),
\label{15}
\end{equation}
for any $\delta>0$ sufficiently small, where
\begin{align}
I(t,x;\delta)= & \int_{t-\delta}^t \left\| \int_s^t \int_{\mathbb{R}^{d}} \Gamma(t-r,x-z) \sigma'(u(r,z)) D_s u(r,z) W(dr,dz) \right.\nonumber\\
&\quad + \left. \int_s^t \int_{\mathbb{R}^{d}}\Gamma(t-r,dz) b'(u(r,x-z)) D_s u(r,x-z) dr\right\|_\mathcal{H}^2 ds.
\label{15.5}
\end{align}
The above term $I(t,x;\delta)$ may be bounded by $2(I_1(t,x;\delta)+I_2(t,x;\delta)$, with
\begin{align}
I_1(t,x;\delta)= &  \int_0^\delta \left\| \int_{t-s}^t \int_{\mathbb{R}^{d}} \Gamma(t-r,x-z) \sigma'(u(r,z)) D_{t-s} u(r,z) W(dr,dz) \right\|_\mathcal{H}^2 ds, \label{15.6}\\
I_2(t,x;\delta)= & \int_0^\delta \left\| \int_{t-s}^t \int_{\mathbb{R}^{d}}\Gamma(t-r,dz) b'(u(r,x-z)) D_{t-s} u(r,x-z) dr\right\|_\mathcal{H}^2 ds.
\label{15.7}
\end{align}
In order to bound from below the term in the left hand-side of (\ref{15}), let us first
obtain a lower bound for the first one on the right hand-side. For this, we will make use of the family of smooth functions $(\Gamma_n)_n$ and
$(J_n^{t,x})_n$, considered in the proof of Lemma \ref{ggamma}, that
regularise the measures $\Gamma$ and $\Gamma(\cdot,x-*)\sigma(u(t-\cdot,*))$, respectively. Then, by the proof of Lemma \ref{ggamma}, the very definition of the norm in $\mathcal{H}_\delta$ and the non-degeneracy assumption on $\sigma$, we have
\begin{align}
& \int_{t-\delta}^t \|\Gamma(t-s,x-*)\sigma(u(s,*))\|_\mathcal{H}^2 ds = \|\Gamma(\cdot,x-*)\sigma(u(t-\cdot,*))\|_{\mathcal{H}_\delta}^2 \nonumber\\
&\quad = \lim_{n\rightarrow \infty} \|  J_n^{t,x} \|_{\mathcal{H}_\delta}^2\nonumber\\
& \quad = \lim_{n\rightarrow \infty} \int_0^\delta \int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}} J_n^{t,x}(s,y) f(y-z) J_n^{t,x}(s,z)
dydz ds\nonumber \\
&\quad \geq c^2 \lim_{n\rightarrow \infty} \int_0^\delta \int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}} \Gamma_n(s,x-y) f(y-z)\Gamma_n(s,x-z)\nonumber \\
& \quad = c^2 \lim_{n\rightarrow \infty} \|\Gamma_n(\cdot,x-*) \|_{\mathcal{H}_\delta}^2 = c^2 \|\Gamma(\cdot,x-*)\|_{\mathcal{H}_\delta}^2=c^2 g(\delta),
\label{16}
\end{align}
where
\begin{equation} \label{gedelta}
g(\delta):=\int_0^\delta \int_{\mathbb{R}^{d}} |\mathcal{F} \Gamma(s)(\xi)|^2 \mu(d\xi) ds.
\end{equation}

Now we find out upper bounds for the expectation of the terms $I_1(t,x;\delta)$ and $I_2(t,x;\delta)$. First, in order to deal with
the former term, one can use the bound (\ref{11}). Thus, taking into account that $\sigma'$ is bounded, we get the following estimate:
\begin{align}
& E(I_1(t,x;\delta))\nonumber \\
&\quad  \leq C  \sup_{(\tau,y)\in (0,\delta)\times \mathbb{R}^{d}} E\left( \|D_{t-\cdot} u(t-\tau,y)\|_{\mathcal{H}_\delta}^2\right) \int_0^\delta \int_{\mathbb{R}^{d}} |\mathcal{F} \Gamma(s)(\xi)|^2 \mu(d\xi) ds\nonumber \\
& \quad= C \sup_{(\tau,y)\in (0,\delta)\times \mathbb{R}^{d}} E\left( \|D_{t-\cdot} u(t-\tau,y)\|_{\mathcal{H}_\delta}^2\right) g(\delta).
\label{17}
\end{align}
On the other hand, by (\ref{deter}) the term $E(I_2(t,x;\delta))$ corresponding to the Hilbert-valued pathwise integral can be bounded by
\begin{equation}
E(I_2(t,x;\delta))\leq C \sup_{(\tau,y)\in (0,\delta)\times \mathbb{R}^{d}} E\left( \|D_{t-\cdot} u(t-\tau,y)\|_{\mathcal{H}_\delta}^2\right) h(\delta),
\label{17.2}
\end{equation}
where $h(\delta)=\int_0^\delta \int_{\mathbb{R}^{d}} \Gamma(s,dy) ds$.\\

At this point, we will make use of the following fact (see Lemma 5 in \cite{qs2} and \cite{quer}, p. 53):
\begin{equation}
\sup_{(\tau,y)\in (0,\delta)\times \mathbb{R}^{d}} E\left( \|D_{t-\cdot} u(t-\tau,y)\|_{\mathcal{H}_\delta}^{2q}\right)
\leq C (g(\delta))^q,
\label{17.5}
\end{equation}
for any $q\geq 1$. Hence, by (\ref{17}) and (\ref{17.2}) the terms $E(I_1(t,x;\delta))$ and $E(I_2(t,x;\delta))$ may be bounded, up to constants, respectively, by $(g(\delta))^2$ and $g(\delta) h(\delta)$, which implies that
\begin{equation}
E(I(t,x;\delta))\leq C g(\delta)(g(\delta)+h(\delta)).
\label{18}
\end{equation}
For any fixed small $\delta>0$, let $n$ be a sufficiently large positive integer such that $\frac{1}{n}\leq \frac{c^2}{2}g(\delta)$. Then, owing to
(\ref{16}) and (\ref{18}) and applying Chebyshev's inequality, we obtain
\begin{align}
P\left( \int_0^t \|D_s u(t,x)\|_\mathcal{H}^2 ds <\frac{1}{n}\right) & \leq P\left( I(t,x;\delta)\geq \frac{c^2}{2} g(\delta)-\frac{1}{n}\right) \nonumber\\
& \leq \left(\frac{c^2}{2}g(\delta)-\frac{1}{n}\right)^{-1} E(I(t,x;\delta))\nonumber \\
& \leq \left(\frac{c^2}{2}g(\delta)-\frac{1}{n}\right)^{-1} g(\delta)(g(\delta)+h(\delta)).
\label{19}
\end{align}
Therefore
$$\lim_{n\rightarrow \infty} P\left( \int_0^t \|D_s u(t,x)\|_\mathcal{H}^2 ds <\frac{1}{n}\right) \leq C (g(\delta)+h(\delta)),$$
and the latter term converges to zero as $\delta$ tends to zero. Hence,
$$P\left( \int_0^t \|D_s u(t,x)\|_\mathcal{H}^2 ds =0\right)=0,$$
which concludes the proof.   \hfill$\square$

\begin{remark}
In the particular case of the three-dimensional stochastic wave equation,
the above Theorem \ref{densitat} generalises  Theorem 3 in the reference \cite{QS}.
\end{remark}

\section{Smoothness of the density}
\label{regularitat}

This section is devoted to prove that, for any fixed $(t,x)\in (0,T]\times \mathbb{R}^{d}$, the law of the random variable $u(t,x)$ has an infinitely differentiable density with respect to Lebesgue measure on $\mathbb{R}$. This will be achieved by showing that $u(t,x)$ belongs to the space $\mathbb{D}^\infty$ and that the inverse of the Malliavin matrix of $u(t,x)$ has moments of all order (see, for instance, Theorem 2.1.4 in \cite{nualart}).

Recall that for any {\it differentiable} random variable $X$ and any $N\geq 1$, the iterated Malliavin derivative $D^N X$ defines an element of the Hilbert space $L^2(\Omega; \mathcal{H}_T^{\otimes N})$. As for the case $N=1$, for any $r=(r_1,\dots,r_N)\in [0,T]^N$, the element $D X(r)$ of $\mathcal{H}^{\otimes N}$ will be denoted by $D_rX$. We will also use the notation
$$D^{N}_{((r_{1},\varphi_{1}),\dots,(r_{N},\varphi_{N}))}X=
\langle D^{N}_{(r_{1},\dots,r_{N})} X, \varphi_{1}\otimes \dots \otimes
\varphi_{N}
\rangle_{\mathcal{H}^{\otimes N}},$$
for $r_{i}\in [0,T]$, $\varphi_{i}\in\mathcal{H}$, $i=1,\dots,N$.
In particular, we have that
$$
\|D^{N} X\|^{2}_{\mathcal{H}_T^{\otimes N}}=\int_{[0,T]^{N}}dr_{1}\dots dr_{N}
\sum_{j_{1},\dots,j_{N}}
|D_{((r_{1},e_{j_{1}}),\dots,(r_{N},e_{j_{N}}))} X|^{2},
$$
where $(e_{j})_{j\geq 0}$ is a complete orthonormal system of $\mathcal{H}$. Let
$$\Delta_{\alpha}^N (g,X):= D^{N}_{\alpha} g(X) - g'(X) D_{\alpha}^{N} X,$$
where $\alpha=((r_1,\varphi_1),\dots,(r_N,\varphi_N))$, $r_i\in [0,T]$ and $\varphi_i\in \mathcal{H}$.
Notice that $\Delta_{\alpha}^N (g,X)=0$ if $N=1$ and it only depends on the
Malliavin
derivatives up
to the order $N-1$ if $N>1$.\\

We now state the main result concerning the Malliavin regularity of the solution $u(t,x)$.

\begin{proposition}
Assume that $\Gamma$ satisfies Hypothesis A. Suppose also that the coefficients $\sigma$ and $b$ are $\mathcal{C}^\infty$ functions with bounded derivatives of any order greater than or equal to one. Then, for every $(t,x)\in [0,T]\times \mathbb{R}^{d}$, the random variable $u(t,x)$ belongs to the space $\mathbb{D}^\infty$.

The iterated Malliavin derivative $D^Nu(t,x)$ satisfies the following equation in $L^p(\Omega; \mathcal{H}_T^{\otimes N})$, for any $p\geq 1$ and
$N\geq 1$:
\begin{align}
& D^{N}u(t,x)  = Z^{N}(t,x) \nonumber \\
& + \int_{0}^{t}\int_{\mathbb{R}^{d}} \Gamma(t-s,x-z) [\delta^N(\sigma,u(s,z))
+ D^{N}u(s,z)\sigma'(u(s,z))] W(ds,dz) \nonumber \\
& + \int_{0}^{t}ds \int_{\mathbb{R}^{d}} \Gamma(s,dz) [\delta^N(b,u(t-s,x-z))\nonumber \\
&\quad \quad
+ D^{N}u(t-s,x-z)b'(u(t-s,x-z))],
\label{21}
\end{align}
where $Z^N(t,x)$ is the element of $L^p(\Omega; \mathcal{H}_T^{\otimes N})$ defined by
\begin{equation}
\langle Z^N_r(t,x), e_{j_1}\otimes\dots \otimes e_{j_N}\rangle_{\mathcal{H}^{\otimes N}} =
\sum_{i=1}^{N} \langle \Gamma(t-r_i,x-*)
D^{N-1}_{\hat{\alpha}_i}
\sigma(u(r_i,*)),e_{j_i}\rangle_{\mathcal{H}},
\label{ci}
\end{equation}
for any $r=(r_1,\dots,r_N)\in [0,T]^N$ and $j_1,\dots,j_N\in \{1,\dots,N\}$.

Moreover, it holds that
$$\sup_{(s,y)\in [0,T]\times \mathbb{R}^{d}} E(\|D^{N}u(s,y)\|^{p}_{\mathcal{H}_T^{\otimes N}}
)<+\infty,$$
for all $p\ge 1$.
\label{malinf}
\end{proposition}

\noindent
{\it Proof}. It is almost an immediate consequence of Theorem 1 in \cite{qs2} and Proposition \ref{difmal} from the preceding Section
\ref{existencia}.

Namely, we just need to check that, using the same notation as in the proof of Theorem 1 in \cite{qs2}, the sequence of
$L^2(\Omega; \mathcal{H}_T^{\otimes N})-$valued random variables $(Z^{N,n}(t,x))_{n\geq 1}$ converges to $Z^N(t,x)$ as $n$ tends to infinity. We should mention that in that reference, $Z^{N,n}(t,x)$ is constructed using a regularisation procedure, that is smoothing the measure $\Gamma$, and by means of a similar expression to (\ref{ci}). In \cite{qs2}, the objective was to define the initial condition of the linear stochastic equation satisfied by the iterated Malliavin derivative  $D^N u(t,x)$ as the limit of $Z^{N,n}(t,x)$. We claim that this limit equals to $Z^N(t,x)$, defined in the present proposition's statement (see (\ref{ci})).

Indeed, the convergence of $Z^{N,n}(t,x)$ to $Z^N(t,x)$ in $L^2(\Omega; \mathcal{H}_T^{\otimes N})$ can be easily studied using the same arguments as in the proof of Lemma 3 in \cite{qs2}.   \hfill $\square$

\vspace{0.3cm}

We are now in position to state and prove the main result of the paper.

\begin{theorem}
Assume that $\Gamma$ satisfies Hypothesis A, the coefficients $\sigma$ and $b$ are $\mathcal{C}^\infty$ functions with bounded derivatives of any order greater than or equal to one and $|\sigma(z)|\geq c>0$, for all $z\in \mathbb{R}$.
Moreover, suppose that there exist $\gamma >0$ such that for all $\tau\in (0,1]$,
\begin{equation}
   \int_0^\tau \int_{\mathbb{R}^{d}} |\mathcal{F} \Gamma(s)(\xi)|^2 \mu(d\xi) ds
   \ge C_1 \tau^{\gamma},
\label{lowbound}
\end{equation}
for some positive constant  $C_1$. Then, for all $(t,x)\in [0,T]\times \mathbb{R}^{d}$, the law of $u(t,x)$ has a $\mathcal{C}^\infty$ density with respect to Lebesgue measure on $\mathbb{R}$.
\label{regularitatdensitat}
\end{theorem}

\noindent
{\it Proof.} In view of Proposition \ref{malinf}, we need to show that the inverse of the Malliavin matrix of $u(t,x)$ has moments of all order, that is
$$E\left( \left| \int_0^T \|D_s u(t,x)\|_\mathcal{H}^2  ds \right|^{-q}  \right)<+\infty,$$
for all $q\geq 2$.
It turns out (see, for instance, Lemma 2.3.1 in \cite{nualart}) that it suffices to check that for any $q\geq 2$,
there exists an $\varepsilon _{0}(q)>0$ such that  for all $\varepsilon \leq
\varepsilon _{0}$
\begin{equation}
P\left( \int_{0}^{t}\left\| D_s u(t,x) \right\|_{\mathcal{H}}^{2} ds<\varepsilon \right) \leq C \varepsilon ^{q}.
\end{equation}
Proceeding as in the proof of Theorem \ref{densitat}, for any $\delta>0$ sufficiently small we obtain the following estimate:
\begin{align}
P\left( \int_0^t \|D_s u(t,x)\|^2_\mathcal{H} ds<\varepsilon\right) & \leq  P\left( I(t,x;\delta) \geq \frac{c^2}{2} g(\delta)-\varepsilon\right)\nonumber\\
& \leq \left( \frac{c^2}{2}g(\delta) -\varepsilon\right)^{-p} E(|I(t,x;\delta)|^p),
\label{23}
\end{align}
for any $p>0$, where we recall that $I(t,x;\delta)$ is defined by (\ref{15.5}) and
$g(\delta)$ is given by (\ref{gedelta}).

We decompose now the term $I(t,x;\delta)$ as in the proof of Theorem \ref{densitat}, so that we need to find upper bounds for $E(|I_i(t,x;\delta)|)^p$, $i=1,2$ (see (\ref{15.6}) and (\ref{15.7})). On one hand, owing to H\"older's inequality and (\ref{11}) we get
\begin{align}
& E(|I_1(t,x;\delta)|^p)\nonumber  \\
&\quad  = E\left( \int_0^\delta \left\| \int_{t-s}^t \int_\mathbb{R}^{d} \Gamma(t-r,x-z)\sigma'(u(r,z)) D_{t-s} u(r,z)W(dr,dz) \right\|_\mathcal{H}^2 ds\right)^p \nonumber \\
& \quad \leq \delta^{p-1} E\left( \int_0^\delta \left\| \int_{t-s}^t \int_\mathbb{R}^{d} \Gamma(t-r,x-z)\sigma'(u(r,z)) D_{t-s} u(r,z)W(dr,dz) \right\|_\mathcal{H}^{2p} ds\right) \nonumber\\
&\quad \leq \delta^{p-1} (g(\delta))^p \sup_{(\tau,y)\in [0,T]\times \mathbb{R}^{d}} E\left( \|D_{t-\cdot} u(t-\tau,y)\|_{\mathcal{H}_T}^{2p}\right).
\label{23.4}
\end{align}
The above estimate (\ref{23.4})   let us conclude that
$$
E(|I_1(t,x;\delta)|^p) \leq C \delta^{p-1} (g(\delta))^{p}.
$$
On the other hand, using similar arguments but for the  Hilbert-valued pathwise integral (see (\ref{deter})), one proves that $E(|I_2(t,x;\delta)|^p)$ may be bounded, up to some positive constant, by $\delta^{p-1} (g(\delta))^p  $.
Thus, we have proved that
\begin{equation}
 P\left( \int_0^t \|D_s u(t,x)\|^2_\mathcal{H} ds  < \epsilon\right)
  \leq C \left( \frac{c^2}{2}g(\delta) -\epsilon\right)^{-p} \delta^{p-1} (g(\delta))^p .
\label{24}
\end{equation}
At this point, we choose $\delta=\delta(\epsilon)$ in such a way that $g(\delta)=\frac{4}{c^2} \epsilon$. By (\ref{lowbound}), this implies that
$\frac{4}{c^{2}}\varepsilon \geq C\delta
^{\gamma }$, that is $\delta \leq C\varepsilon ^{\frac{1}{\gamma }}$. Hence,%
\[
P\left( \int_{0}^{t}\left\| D_{s}u(t,x)\right\| _{\mathcal{H}%
}^{2}ds<\varepsilon \right) \leq C\varepsilon ^{\frac{p-1}{\gamma }},
\]%
and it suffices to take $p$ sufficiently large such that $  \frac{p-1}{\gamma } \geq q$.
\hfill $\square$

\begin{remark}
As it is pointed out in \cite{QS}, Appendix A (see also \cite{lev}), when $\Gamma$ is the fundamental solution of the wave equation in $\mathbb{R}^d$, with
$d=1,2,3$, then condition  (\ref{lowbound}) is satisfied with $\gamma=3$.

On the other hand, if $\Gamma$ is the fundamental solution of the heat equation on $\mathbb{R}^d$, $d\geq 1$, then
condition (\ref{lowbound}) is satisfied  with  any $\gamma\geq 1$   (see Lemma 3.1 in \cite{mms}).
\end{remark}

\begin{remark}
Theorem \ref{regularitatdensitat} provides a generalisation of Theorem 3 in \cite{qs2} for the case of the three-dimensional wave equation (see also \cite{marta}). Moreover, it also generalises the results in \cite{mms} for the stochastic wave equation with space dimension $d=1,2$ and the stochastic heat equation in any space dimension.
\end{remark}

redpajama_set_name: RedPajamaArXiv


Discussion in 'Long Range Scopes and Other Optics' started by sakofan, Nov 24, 2003.
Anybody tried the IOR line of mounts/rings yet??..sakofan..Thanks in advance.
Do they compare with Badgers??
I tried their rings and had mixed feelings. Mike Lau at TBA built my tactical rifle and mounted the first IOR scope I got on that rifle and used there rings.
They're like the badgers, but not as much bearing surface. They are around 3/8" wide whereas the badgers are a full 1/2". Also the screws are metric and we couldn't find any trox screws to fit the rings. Two of the hex screws that came with the rings stripped as Mike tried to tighten them down to 20" pounds.
They came with my scope so I thought I'd give them a try. Mike and I decided to put badgers on. The IOR's rings will probably work great.


ThinkGeek has released its new line of Star Trek swimwear just in time for summer. The new "Trekini,"as they are called, are pretty adorkable looking, and they have updated their TNG One-Piece swimsuits.
Although we use the word for our entire Star Trek swimwear line, this two-piece form factor can properly be termed a "Trekini." Available in sizes small through 4X, these swimsuits come in Blue (with Science insignia), Gold (Command insignia), and Red (Engineering insignia). You can choose your top and bottom separately, because we know every body is unique, although we do advise you choose the same color for top and bottom to avoid confusing the rest of the crew.
Just be careful if you choose red. Pretty sure that even if you bring along a bottle of SPF 90, it ain't gonna block phaser rays.
We worked hard to address fit concerns on a range of body types early in development this time around, and we hope you're happy with the results you helped make happen!
Whether your destination is the ocean world of Pacifica, the pristine beaches of Risa, or your local Y, take the plunge with these Star Trek:TNG One-Piece Swimsuits modeled after the ST:TNG uniform. Choose Blue, Gold, or Red, whichever suits you. (There will be a test later.) The suit comes in sizes small through 4X. It also features soft bust cups for shape which are removable if you prefer the look without them. And, of course, because we've all tried on bathing suits we loved when we set eyes on them that turned out to be unflattering, they're returnable so long as you haven't worn it. See above for full details.


Kendrick Lamar has taken out the top spot on this year's Hottest 100 countdown for his hit single Humble.
After Triple J tallied a record 2.38 million votes, the American rapper narrowly pipped Gang of Youths and Angus and Julia Stone, who ended the day in second and third place respectively. Gang of Youths dominated this year's countdown, with four songs in the Hottest 100 and three in the top 10.
"We thought it would be universally panned and we would be marched out the back and shot," he joked. "This is so weird. Thank you, thank you. We're a little bit overwhelmed."
This countdown was the first in two decades that the Hottest 100 hadn't coincided with Australia Day (save for the 2004 Hottest 100, which was held on January 25). The station kicked off its biggest day of the year with an acknowledgement of Australia's first peoples, pointing out that many were listening for the first time.
The new format sparked outrage among some listeners and conservatives politicians, who accused the youth broadcaster of taking a stance on the debate surrounding whether or not Australia Day should be celebrated on January 26 out of respect for Indigenous Australians.
An overwhelming majority of Triple J listeners supported the change and – despite some calling for a boycott – around 130,000 more votes were cast in this year's countdown compared to last year.
The ABC has remained tight-lipped about the timing of next year's countdown. Under the new format, the Hottest 100 is to be broadcast on the final Saturday each January. However, in 2019 January 26 falls on a Saturday.
Content director Ollie Wards previously told Fairfax Media an announcement will be made soon.
"The way the weekend falls does change, which is out of everyone's control," he said. "We'll review how it all goes and make an announcement about the 2019 dates in the near future. We don't want the Hottest 100 to be part of the big political debate happening now."