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Maple Ridge council will review parking bids
Parkade needed near Haney bus exchange, once B-line to Coquitlam begins, says councillor
The city has asked for bids to run its four parking lots downtown
Any proposals for a new parking system in downtown Maple Ridge will have to get through council first.
The request for bids to operate five downtown parking lots, four of which are owned by the city, was issued in January with a request to show a "clear and concise overview" of how each bidder intends to grow revenues.
The Maple Ridge Downtown Parking Association has operated the lots for the last two and half decades on a non-profit basis, turning over the net proceeds of about $14,000 yearly to the city.
But there's no intent on removing parking lots, chief administrator Ted Swabey said at Monday's council workshop.
"Whether the municipality provides public parking in the future is something that council needs to decide," he added.
"I think the private sector, if there's a market, will pick up the slack."
In the request for proposals, the city could also expand any deal with a contractor later to include the 750 other city-owned parking stalls.
Coun. Gordy Robson said the purpose behind the bidding process seems to be about making money for the city. He's concerned about the city losing trust with downtown businesses.
The downtown parking association was told last summer about the request, a decision which stems from the previous council, legislative clerk Laurie Darcus told council.
Robson was one of the original founders of the parking association when it was formed in the 1990s, with the intent of providing downtown spaces for employees of businesses, leaving the prime spots for shoppers.
Downtown businesses, back then, paid a property tax levy to cover the $1.5 million the city borrowed to buy the lots, plus interest. Two parking lots were also donated by local businessmen.
Robson, along with Coun. Bob Masse, who has a business downtown, both said there's a shortage of downtown parking.
That shortage could worsen with the arrival in a few years of the B-line express bus connecting the bus loop at Haney Place Mall to the new Evergreen line in Coquitlam.
"I think we should be talking to TransLink right now," about funding a multi-storey parkade next to the bus loop, Robson said.
"Let's talk about what we're going to do in the future."
Masse added that Maple Ridge is not Vancouver.
"People can't get their head around paying $8.25 to park."
The parking issue concerns Downtown Maple Ridge Business Improvement Association executive-director Ineke Boekhorst, who says the issue is on the agenda for the next board meeting.
"It [the parking association] is a great source of parking in our downtown," Boekhorst said.
Bruce Fuller, who runs Fuller Watson BrandSource Home Furnishings, said parking is crucial for downtown businesses.
The business has its own parking lot in the back for employees and customers, but many people like to park on the street just outside the store.
However, every new building in the downtown increases the demand for parking, not just from permanent residents, but from visitors, while city parking enforcement on weekends is lacking, he added.
Megan Rowland, who works at MNP accountants on 224th Street, said her company leases four stalls from the city's 119th Avenue lot.
"Parking is already a nightmare over here."
She wonders about the benefit of privatizing the operation of the lots, saying they could end up with higher prices and shorter hours and said the current system is working.
"We need more parking, especially if they want people to shop local," she said.
"So why change? Leave well enough alone if it works."
One of the lots, on Brown Avenue, with 40 stalls, was sold in 2014 to the developer of the property who plans on building condos. The lot is still managed by the parking association and the city holds a long-term lease for the property.
Once new condos are built on the property, the parking lot will close.
The Maple Ridge-Pitt Meadows News leases several stalls on that property for its employees.
Proceeds from the sale of that property will be used in acquiring parking elsewhere.
According to city property manager Darrell Denton, net proceeds from the sales of parking rights will be set aside in support of public parking to ensure that there is no net loss of public parking spaces.
The city also requires the builder to create a certain number of public parking spaces as it builds the condo towers.
On Health: Debunking detox diets
Premier defends her 'real time' reporting of pay-for-play donations

redpajama_set_name: RedPajamaCommonCrawl


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Flávia Cavalcante Rebêlo (Salvador, Bahia, 23 de abril de 1971) é uma apresentadora e jornalista de televisão brasileira, que foi eleita Miss Brasil representando o Ceará quando tinha dezenove anos de idade. Entre as candidatas, derrotou a Miss São Paulo, Adriana Colin. Reinou por dois anos, pois em 1990 não houve concurso. No Miss Universo, realizado no dia 26 de maio de 1989 ganhou o prêmio de melhor traje típico, mas não ficou entre as 10 finalistas do concurso vencido pela holandesa Angela Visser.

Carreira

Anos 90
Foi apresentadora de programas jornalísticos e esportivos na extinta Rede Manchete e na TV Gazeta. Em 1999 e 2000 foi repórter da Rede Bandeirantes. Depois, foi apresentadora do programa Terceiro Milênio, na emissora católica Rede Vida. Em vários estados da Região Nordeste, participou de campanhas políticas. Na internet, foi repórter e editora do LogWeb, de logística.

Até 2008
Apresentou o programa São Paulo Acontece, da TV Band (no qual era apresentadora e repórter), quando a emissora decidiu trocar de apresentador e horário.

Ligações externas
Página oficial de Flávia Cavalcante

Vencedoras do Miss Brasil
Naturais de Salvador
Jornalistas da Bahia

redpajama_set_name: RedPajamaWikipedia


He was arrested after three minibuses were damaged at Wath Comprehensive School, Wath-upon-Dearne, at 2am on Monday.
Firefighters extinguished the blaze and the incident was reported to South Yorkshire Police.
A South Yorkshire Police spokesman said the teenage suspect has been released on bail pending further inquiries.


It's been a very sad week for Lincoln City in particular and football in general after the sad news that Graham Taylor passed away on Thursday.
In a way his legacy is living on through the work of Danny and Nicky Cowley who, like Graham, are true innovators.
Today their team reached the Third Round of the FA Trophy after coming from a goal behind to overpower Gateshead 3-1.
Goals from Habergham, Whitehouse and Hawkridge gave a much-changed Imps' line-up a deserved win.
This performance and the style of victory was for Graham Taylor who will be sadly missed not only by the footballing family but the people of Lincoln who jumped on board the fabulous, triumphant ride that Graham took them on back in the seventies.


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You are here: Home / Beyond Boulders Blog / Five Questions for a Fellow: Jamie Eastman Addresses Sexual Education for Adolescents With Disabilities
Five Questions for a Fellow: Jamie Eastman Addresses Sexual Education for Adolescents With Disabilities
July 8, 2010 ASF Leave a Comment
Each week, Beyond Boulders delivers a new installment of "Five Questions for a Fellow" – an interview series with Schweitzer Fellows across the country and in Gabon, Africa who are leading the movement to eliminate health disparities. For an archive of previous "Five Questions for a Fellow" interviews, click here.
It's a sad but true fact that due to a number of factors, children with disabilities are disproportionally vulnerable to sexual and other abuse.
As a Schweitzer Fellow, Jamie Eastman is doing her part to address that vulnerability. This University of Pittsburgh Graduate School of Public Health is collaborating with the Western Pennsylvania School for Blind Children to develop and administer a sexual education curriculum that empowers the school's students to safely navigate adolescence.
Why did you decide to develop your particular project?
I've been passionate about education my entire life. Before I decided to go into public health, I seriously considered becoming a teacher. Then, when I was studying in South Africa, I realized that I was really interested in doing sexual education. The area I volunteered in had an HIV rate of almost 40%. When I returned to the States, I felt like I needed to find some way to help. I started volunteering at an HIV/AIDS organization doing sexuality education.
The Western Pennsylvania School for Blind Children (WPSBC) has been in Pittsburgh since 1887 and serves 175 children ages 3-21. The students there are not only visually impaired, but have another disability that keeps them from being able to enter the public school system.
The curriculum and interdisciplinary environment that WPSBC has built is incredible and provides the students with an individualized plan to meet their needs.
When I first approached WPSBC about doing my project, they asked if I would be willing to do sexual education. Students at the school are at high risk for sexual abuse because of their medical status. In addition, the older students need education surrounding puberty in order to navigate their way through adolescence into adulthood.
My hope is that by collaborating with WPSBV I will be able to create a curriculum for sexual education that can be used for years to come.
What do you hope will be the lasting impact of your project on the community it serves?
I have two hopes for the lasting impact of my project. The first is that when I finish, I will be leaving the school with the ability to keep the sexual education courses running without me. The second is that I will leave the students with the ability to make smart decisions about their bodies.
What do you think is the most pressing health-related issue of our time, and how do you think it should be addressed?
I believe access is the most pressing health-related issue of our time. The US is one of the richest countries in the world, yet a significant portion of our population lacks access to basic health care. Health should not be a privilege. It should be a basic human right. We need to find a way to make basic health care available to everyone.
So many steps toward this goal have been taken in the last year, but we still have a long way to go. It is important that people continue to write to their congressmen and ask them to support equal access to healthcare.
What has been the most surprising element of your experience as a Schweitzer Fellow so far?
The most surprising thing to me has been how helpful and encouraging everyone has been. All the teachers and the staff at the school have been incredibly hospitable and helpful. I know that sometimes my presence in the classroom makes things more difficult for them, but they have always been incredibly patient.
The professors at my school have also been a tremendous resource. They have gone out of their way to put me in touch with people who might be able to help me with my project. In addition, they have been nothing but supportive. A mentor of mine has taken multiple afternoons out of her busy schedule to simply listen to me talk about my project and help me brainstorm. It really has been wonderful to feel so supported.
What does being a Schweitzer Fellow mean to you?
It's still early in my time as a Schweitzer Fellow, so I am still growing to understand more of what being a Fellow means. Right now, it means so much to me that I am able to be around so many other students who want to change the world. They are all incredible people and I've learned so much from them.
Jamie Eastman is a Schweitzer Fellow in Pittsburgh, PA. Click here to read more about The Pittsburgh Schweitzer Fellows Program and the Fellows like Eastman it supports in creating and carrying out yearlong direct service projects. To make a gift to The Albert Schweitzer Fellowship in honor of Eastman's efforts to empower children with disabilities, click here.
Filed Under: Beyond Boulders Blog Tagged With: children with disabilities, Disabilities, HIV/AIDS, Jamie Eastman, Pittsburgh, Pittsburgh Schweitzer Fellows Program, Sexual abuse, University of Pittsburgh Graduate School of Public Health, Western Pennsylvania School for Blind Children


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HomeFeatureAlice Coachman Davis, first Black Olympic gold medalist, dies
Alice Coachman Davis, first Black Olympic gold medalist, dies
July 28, 2014 Carma Henry Feature, Local News 0
By WALB Albany
Alice Coachman Davis, the first African American to win an Olympic gold medal, died Monday at the age of 90.
She was born in Albany, GA on Nov. 9, 1923 as Alice Coach-man. She attended Tuskegee University at the age of 16.
Davis won a gold medal at the 1948 Summer Olympic Games in London, in the high jump, setting a record of 5-feet-6 1/8 inches. She had won the AAU outdoor high jump title from 1939 to 1948, but was unable to compete in the Olympics until the London games. The 1940 Summer Olympics originally scheduled for Tokyo and the 1944 Summer Olympics, originally set for London, were cancelled due to World War II.
In 1952, Coca-Cola awarded Davis an endorsement contract, making her the first African-American to earn an endorsement deal. She created the Alice Coachman Track and Field Foundation to help support young athletes. At the 1996 Summer Olympic Games in A-lanta, she was recognized as one of the 100 greatest Olympians in history.
Albany's Coachman Elementary School is named in her honor.
first Black Olympic gold medalist
Gridiron dreams, false hope and false promises
Community shocked as a white man slaps 79-year-old judge and spits on her
New Atlanta's Founding Father, Herman J. Russell, dies
November 28, 2014 Carma Henry Feature, National News 0
Daughter of Whitney Houston, Bobbi Kristina Brown, dies
July 27, 2015 Dvon Entertainment, Video, WHATS TWEETIN 0
Bobbi Kristina Brown, 22, the only daughter of singers Whitney Houston and Bobby Brown, died Sunday, nearly six months after she was found unresponsive in a bathtub of her home, her family said. Brown has […]
Black Radio Pioneer, Hal Jackson, dies in New York City
May 31, 2012 keesto Obituaries, Religion 0
Radio pioneer Harold "Hal" Jackson, a staple of New York radio, has died. He was in his late 90s. Jackson died Wednesday (May 23, 2012) in a hospital […]


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Rise of the Phoenix: Ranbir owns GQ cover
New Delhi [India], May 31 (ANI): Bollywood actor Ranbir Kapoor has got another feather in his cap - he has been featured on the cover of GQ magazine.
Shared by GQ itself on its Twitter handle, the 35-year-old actor has been featured on the June issue of the Indian edition of the illustrious fashion magazine.
The tweet read, "He's BACK! June 2018 cover star #RanbirKapoor opens up about...Everything."
He's BACK! June 2018 cover star #RanbirKapoor opens up about...Everything. pic.twitter.com/kkQ9vl0qk9
— GQ India (@gqindia) May 31, 2018
Donning a floral attire with colourful flowers on a dark purple base, the 'Sanju' actor looked dashing, lounging in a relaxed state.
Career-wise, the Kapoor scion is making headlines for his performance in Dutt biopic, 'Sanju'. He has received critical acclamation from the audience and industry gurus since the release of the film's teaser.
More recently, the trailer of the film was released on Wednesday, post which Ranbir is being lauded for his performance. (ANI)
Rise of the Phoenix
Dutt biopic


We said we wouldn't do it again.... But we did. We pulled the camp trailer to Florida and back again. And, again we raced a winter storm on our way back home.
Our anniversary trip to the Black Hills of South Dakota and the Badlands.
Around the Corpus Christi area.
A day-trip to Caprock Canyons State Park near Quitaque, TX.
A challenging photo day with bright sunlight and a heavy haze hanging in the air.


Her story suggests the perils of becoming the subject of your work. As long as your creative fires burn, you are propelled forward. But if you are consuming yourself in the process, what is left when the fire begins to sputter?
That her posthumous celebrity now overshadows the reputations of her parents is inescapably rankling to her father, although he acknowledges her superior talent. If she hadn't been so gifted, he confesses, he would resent it. In an earlier scene he says her photos "made my work look kind of stupid." After her death he took up photography, and using young female models, made work that closely resembled his daughter's.
Francesca's tragedy casts a shadow over the larger portion of the film, in which the parents recall their family history, with Francesca's older brother, Charles, a video artist, offering only a few disengaged comments.
A stern believer in the Puritan work ethic, George states his belief that you should go to your studio every day, and if inspiration doesn't arrive, "sharpen pencils" until it does. Art to him is an almost sacred calling. As you watch scenes of the parents at work and play in their homes in Italy and the United States, their shared creativity is obviously the glue that has kept them together.
After settling in Boulder, Colo., the couple never consciously decided to have children. These "gift calamities," as George describes the appearances of Charles and Francesca, just occurred. Betty, two years older than George, devoted her energy to making "useful functional objects." After Francesca's death, Betty rejected mere functionality. She is shown in the film working on an enormous, brightly colorful installation for the new United States Embassy in Beijing. The most frequent comment that she hears about her work is that it makes people feel good.
Would "The Woodmans" have been made if Francesca hadn't killed herself? Probably not. Does it exploit her tragedy? Of course it does, but not in a cheap, sensationalistic way.


I wish everyone was healed when they asked for it. I wish cancer, HIV, AIDS, fibromyalgia, and other diseases didn't exist, but when people were diagnosed with something terrible and asked for God to remove it, He would.
I hate when friends or family members are sick and I can't do anything about it. I feel horrible when I hear of young children carrying diseases that will shorten their lifespan.
My heart hurts for these situations.
I wish I could answer why healing doesn't always come. I can't. I hate when I hear people who claim to be teachers of the Word who tell people, "Just have faith" or "Maybe you don't have enough faith". That's bogus!
In fact, I feel like those words cause more harm than good to those who are truly seeking to understand God and why He isn't doing something about their illness.
God uses everything for His glory- somehow. It is just hard to see it while we are going through hell on earth.
God's peace is what we should be seeking in our need for healing. Healing might not come while we are alive, but peace can be achieved. God's faithfulness is found in His peace.
I don't know if complete healing is ever attainable anyway. We might get rid of the disease, but there are still affects that linger. You will always remember the treatments. You will always remember the fear. You will always remember what you went through. Physical healing might be there, but the emotional and mental healing might not be.
We know and believe that Christ is the ultimate healer, which we will experience in the end of our life on earth. He will give us our new bodies, our new life, and our new home. Healing will be complete and we will be renewed in His presence.
Perhaps healing is something we should be seeking, but peace in our pain is what we should be asking for. Perhaps that peace will heal enough in our earthly experience?
I could be wrong, but I always have a hard time telling people, "You'll be healed", when God does not guarantee it the way we understand it. However, peace is something He guarantees if we seek it.
May we find peace in our inability to understand God's ways. May we find rest in the fact that God is God and we are not. May we trust that His glory can be found in all of our circumstances. May we experience the peace that heals.
QUESTION: Have you sought peace before seeking healing?
When Healing Doesn't Come was originally posted on bobbybenavides.me April 11, 2016.


I'm a little bit obsessed with India. I've been three times, and I'm counting down the days until my next visit in November. It's an incredible country where I've had some incredible experiences. I've laughed. I've cried. I've come very close to shitting myself in the back of a rickshaw. And I've loved (almost) every minute of it: the diverse and welcoming people, the vibrant streets, the stunning landscapes and landmarks. And the food. Of course I love the food.
In comparison, Indian food at home rarely comes close. It's all too often heavy, greasy and stodgy; the same boring sauces trotted out over and over again. So I'm welcoming the Indian street food movement currently sweeping Birmingham (and the rest of the UK) with open arms. Places like Indian Brewery, Raja Monkey and (arriving soon) Mowgli are repping Indian street food all over the city, but I'm lucky enough to have Zindiya – one of the very best "streatery"s – right on my doorstep in Moseley.
I loved this place as soon as I walked in. The decor is fun, colourful and kitschy, with the exposed brickwork, mismatched chairs and painted shutters throwing a not-even-trying-to-be-subtle nod to the Indian streets that inspired the place. It's brilliant, transporting the mind and spirit (though sadly not the body) far away from the grey streets of Birmingham.
The cocktail menu has been curated by Birmingham bartender extraordinaire Rob Wood: a sure sign of quality. Lime2 is the best of the four we try; tangy lime pickle cordial, fresh lemons and limes combine with Finlandia Vodka to give a sharp, complex flavour. The Chai Wala, an 'Indian' take on prosecco which adds Chai bitters, is – unsurprisingly, as I love chai tea – another favourite of mine. We find From Ooty with Love (rose, strawberries) and Coconut Bliss (coconut, lime) a little sweet, but the quirky presentation is sure to appeal to the Instagram masses.
So on to the food; the main reason we are here. The dishes are smaller, lighter and fresher than your standard curry house; the idea is that you order a selection and share. We order far too much of course – we always do – but the menu is so full of temptation that I challenge you not do the same if you visit. I skip over some old favourites – pani puri, bhajis, fish amritsari and seekh kebab, to name a few – in favour of trying some of the more unusual dishes.
I'm a committed carnivore but I only ate meat once when I spent a month in India, and because the veggie options are so good I didn't miss it at all. Sadly this is rarely the case in the UK, so it was a pleasant surprise to notice that we had inadvertently chosen an almost entirely vegetarian meal (bar one portion of chicken tikka which my boyfriend has dubbed "the best chicken tikka in Birmingham"). I'm pleased to say that this mostly-meat-free meal did not disappoint.
Papri Chaat is my highlight of the night. Pops of crispy dough wafers nestle amongst chickpeas, potatoes, yoghurt and tamarind chutney, giving a perfect contrast of texture. The spicing is beautiful – all depth rather than heat – giving the dish a complex flavour that demands another mouthful. It's absolute heaven in a dish, and a bargain at only £4.
A generous trio of dosas (£6.50) come with your choice of filling (Masala Aloo on this occasion), sambhar and coconut chutney. They're excellent, as are the deceptively named Hara Bara kebabs (£7) – not kebabs at all but vegetable patties often served as a streetside snack in the Punjab.
We're so full that I can't give the paneer kati roll (£6.50) the attention it deserves, but my boyfriend finishes it off with pleasure. He was reluctant to order the Okra fries (£3) – branding the vegetable "disgusting" – but when faced with Zindiya's crisp, beautifully seasoned rendition he was rapidly converted. He's wrong a lot; don't hold it against him. One thing he was right about was the chicken tikka though. It's moist, fragrant and thoroughly delicious.
Zindiya is really bloody brilliant, and I reckon it's pretty perfect for any occasion. It's versatile, vibrant food served up in a cool, relaxed setting, with the small plates meaning you can spend (and eat) as much or as little as you want. It may be 4 months until I land in my beloved India again, but until then I'm very, very lucky to have Zindiya's amazing street food less than half a mile from my door. Go there, and go soon – it's a lot cheaper than a plane ticket.
Zindiya is at 21 Woodbridge Rd, Moseley, B13 8EJ. WEBSITE.
Posted in Birmingham, Moseley, United Kingdom and tagged Birmingham, Indian, Indian street food, Moseley, street food, streetfood, Zindiya. Bookmark the permalink.
Wow! Great post. Your work and photos are improving!!
Amazing! I got excited by the glimpse of the paratha menu. Have you tried Bundobust? Their chaat is great and similar to this one. I'm from India and I'm not a fan at all, I've been a few times with family. Most other people who are tourists or Indian love it.
Yes, I've been to Bundobust once and really enjoyed it! They have so many awesome veggie dishes.


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Q: How to align rendered cell rows to the right? I have a CellTable with Columns. One of the column I align the text to the right programatically:
column.setHorizontalAlignment(HasHorizontalAlignment.ALIGN_RIGHT);

Further I want the cell to have a padding on the right. By cannot get it to work. This is what I tried:
column.setCellStyleNames("numberCell");

.numberCell {
  padding-right: 5px !important;
  margin-right: 5px !important;
  color: red !important;
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The text in the cell is rendered in red, but there is no padding.

A: As you applied style it should work. but let You try apply particular column style using getCellStyleNames. it may work.
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             @Override
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                  return "numberCell";
            }
            @Override
            public String getValue(Data object) {
                return "";
            }
        };

redpajama_set_name: RedPajamaStackExchange


The trip to Cracow & Wieliczka runs every Friday.
Cracow is one of the oldest Polish cities, with a history of over 1000 years, renowned for high cultural and architectural values. You will start sightseeing at the Barbican and follow the Royal Road to reach Wawel Royal Castle. The Florianska Street leads you to Main Square Market, where you will visit St Mary's Basilica with its famous and biggest in Europe gothic altar sculpted by master Veit Stoss. You will also listen to the Heynal, played each hour from the highest tour of the church.
You will see Cracow's trademark Sukiennice (Cloth Hall), Adam Mickiewicz Monument and historical houses surrounding the market. Afterwards, you will take Grodzka Street to reach Wawel Royal Castle, which is our national and cultural symbol. The Castle was the seat of Polish kings and leaders, their necropolis and heart of the Polish history. Individual tourists visiting Wawel Castle can see i.a.: Royal Apartments, Treasury, Armouries. Children always enjoy visiting Smocza Jama (The Dragon's Cave).
In the afternoon you will go to Wieliczka, where you will visit the Salt Mine, whose history dates back to the 13th century. It is the only mine in the world which has been continuously operating since the Middle Ages. Once the visit is over, you can rest in a unique complex of chambers situated 125 m under the ground where you will find souvenir shops, restaurant and post office.
It is worth reminding that in 1978 the Salt Mine in Wieliczka, Wawel Royal Castle, Old Town and Kazimierz district were listed World Heritage Sites by UNESCO.
entrance ticket to The Salt Mine: June and September - 64 PLN adults/ 46 PLN children 4-18 years, students up to 26 years, disabled and carers/ FREE- children up to 4 years. July and August - 69 PLN adults/ 51 PLN children 4-18 years, students up to 26 years, disabled and carers/ FREE- children up to 4 years.
Wawel Castle: Zygmunt Bell, the Royal Tombs: adults 19PLN (with audio-guide)/children, students, pensioners: 12PLN (with audio-guide).
Notice: Tour in Polish. For an extra fee, there is the possibility to tour Salt Mine in a foreign language. Salt Mine allows maximum bag or backpack size of 30cm x 20cm x 20cm.
We are not responsible for fee/price changes during the season.


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I believe in treating the individual as well as the whole family unit. I believe in being compassionate and giving the patient and family the time and attention they need and deserve.
visit was great and provider was very understanding.
all my visits have been good experience, I was treated with care and respect, everyone was friendly, and my questions were answered.
Good experience. Dr sent me for an X-ray and that helped find my underlieng help issue.
I just feel Dr could spend a little bit more time with me as a patient. But since this was my first time, I like her overall.
I was very happy after my visit, Dr. and staff were friendly and cared about my health.


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PSG complain to Uefa about German referee
Paris Saint-Germain have sent a letter of complaint to Uefa about the referee who oversaw their spectacular second-leg collapse to Barcelona last week, accusing him of ruining their Champions League campaign.
The Spanish champions' historic win was overshadowed by a series of controversial calls from German referee Deniz Aytekin and he awarded them two penalties on the way to a 6-1 hammering to surge into the quarter-finals.
A source with knowledge of the letter confirmed Tuesday, media reports about the dossier sent to Uefa president Aleksander Ceferin. Uefa said that it had received a letter from PSG, giving no details of its content.
The source told AFP that the letter was designed to "point out a series of errors" and "errors of judgment" by Aytekin.
PSG coach Unai Emery said after Barcelona's miracle revival from a 4-0 first-leg deficit: "We had chances to make it 3-2 and then the refereeing decisions, I don't know if they were right or not but, for sure, they damaged us."
PSG president Nasser Al-Khelaifi was less forgiving.
"The outcome of the match could have been different with more clear-sighted officiating," he told Le Parisien newspaper. "Everyone saw that Angel di Maria should have had a penalty that would have made the score 2-3... Not to mention that there was no penalty on (Luis) Suarez late in the game (which was awarded)." - AFP


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Master of Science in Mechanical Engineering
The Master of Science in Mechanical Engineering (MSME) program is designed to give engineering, mathematics and science graduates an opportunity to explore advanced and emerging topics in mechanical engineering.
Apply to Saint Martin'sRequest info
Why Saint Martin's University?
Curriculum tailored to both recent graduates and engineers with professional experience
Students obtain enhanced engineering analysis and design skill tailored to professional objectives
Evening and summer courses accommodate working professionals
Teaching and research paid assistantships are available and encouraged
Faculty have real-world experience in the subjects they teach
Classes are held in the most environmentally friendly and energy efficient building with LEED Platinum Certification in the Western Hemisphere
Federal financial aid is available for graduate students who will be enrolled at least half time and are citizens, or eligible non-citizens, of the USA. Most federal aid for graduate students is in the form of direct loans.
For more details about federal aid eligibility and the different types of aid available for graduate and professional studies visit studentaid.edu.gov.
Saint Martin's University offers a limited number of graduate assistantships to graduate students. Contact the Office of the Provost for more information.
Master of Science in Mechanical Engineering Requirements
Degree pathways
For the research-oriented student, a thesis option is offered, which requires a research project and thesis be completed with a faculty member. For the profession-oriented student, a non-thesis option is offered, which requires more coursework.
The thesis option requires students complete
all required MSME Core courses,
a total of 24 credits (including the MSME Core) of non-thesis coursework,
at least six credits of thesis (MME 599), and
a successful thesis defense.
If a student begins a thesis but fails to complete it, up to two thesis credits can count toward the coursework of an MSME degree.
Coursework must be completed with a minimum grade point average of 3.00 ('B'). Students must complete all work within seven years from the start of the program. Directed study (MSME 596) is limited to two credits per semester and six credits in total. A maximum of eight credit hours of equivalent coursework from other institutions may be transferred into the program if approved by the program director.
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text:
\section{\bf Introduction}
This paper is devoted to the study of blow-up solutions for the
following semilinear  wave equation:
\begin{equation}\label{gen}
\left\{
\begin{array}{l}
\partial_t^2 u =\Delta u+|u|^{p-1}u+f(u)+g(x,t,\nabla u,\partial_t u ),\\
\\
(u(x,0),\partial_tu(x,0))=(u_0(x),u_1(x))\in  H^{1}_{loc}(\er^N)\times L^{2}_{loc}(\er^N),\\
\end{array}
\right.
\end{equation}
in spatial dimensions $N\ge 2$, where $u(t):x\in \er^N \rightarrow
u(x,t)\in \er$ and $p_c<p<p_S$, where $ p_c\equiv 1+\frac4{N-1}$ is
the conformal critical exponent and  $p_S\equiv 1+\frac4{N-2}$
is the Sobolev critical exponent. Moreover, we take
 $f:\R\rightarrow \R $ and $g:\R^{2N+2}\rightarrow \R $
${\cal {C}}^1$ functions satisfying
\begin{eqnarray*}
(H_f)& |{f(u)}|\le M(1+|u|^q), \  &{\textrm {for all }}\ u\in \R \  \qquad{{\textrm {with}}}\ \ (q<p,\ \ M>0),\\
(H_g)& |{g(x,t,v,z)}|\le M(1+|v|+|z|), &  {\textrm {for all }}\
x,v\in \R^N, t,z\in \R \  \ {{\textrm {with}}}\  (M>0).
\end{eqnarray*}

\bigskip
\no We would like to mention that equation \eqref{gen}
encompasses the case of the following nonlinear Klein-Gordon
equation
\begin{equation}\label{KG}
 \partial^2_{t}u=\Delta u+ |u|^{p-1} u-u,\qquad  (x,t)\in \er^N\times [0,T).
\end{equation}
In order to keep our analysis clear, we only give the proof  for the following non perturbed equation
\begin{equation}
\label{1}
 \partial_t^2u=
\Delta u+ |u|^{p-1} u,\qquad  (x,t)\in \er^N\times [0,T),
\end{equation}
 and refer the reader  to  \cite{HZlyap09} and
\cite{HZJHDE} for straightforward adaptations to
 equation \eqref{gen}.

\bigskip

 \no The Cauchy problem of equation ({\ref{1}}) is solved in
$H^{1}_{loc}\times L^{2}_{loc}$. This follows from the finite speed
of propagation and the the wellposedness in $H^{1} \times L^{2}$,
valid whenever $ 1< p< p_S$.
The existence of
blow-up solutions for the associated ordinary differential equation
of (\ref{1})  is a classical result. By using the finite speed of
propagation, we conclude that there exists a
 blow-up solution $u(t)$ of (\ref{1}) which depends non trivially on the space variable.
In this paper, we consider  a blow-up solution $u(t)$ of (\ref{1}),
we define (see for example Alinhac \cite{Apndeta95} and
\cite{Afle02})  $\Gamma$ as the graph of a function $x \mapsto T(x)$
such that the domain of definition of  $u$  is given by
$$D_u=\{(x,t)\ \ \big |t<T(x)\}.$$
The set $D_u$ is called the maximal influence domain of $u$.
Moreover, from the finite speed of propagation, $T$ is a
$1$-Lipschitz function.
The
 graph
$\Gamma$  is called
the blow-up graph of $u$. \\
Let us first  introduce the following non-degeneracy condition for
$\Gamma$. If we introduce for all $x\in\er^N$, $t\le T(x)$ and
$\delta>0$, the cone
\begin{equation} \label{cone}
C_{x,t,\delta}=\{(\xi,\tau)\neq(x,t)|0\le \tau\le t-\delta|\xi-x|\},
\end{equation}
then our non degeneracy condition is the following: $x_0$ is a non
characteristic point if
\begin{equation} \label{cone1}
\exists\delta_0=\delta_0(x_0)\in (0,1) \ \textrm{ such that}\  u \
\textrm{ is defined on }\ C_{x_0,T(x_0),\delta_0}.
\end{equation}
\no We aim at studying  the growth estimate  of $u(t)$ near the
space-time blow-up graph in the  super-conformal   case (where
$p_c<p<p_S$).

Let us briefly mention some results concerning the blow-up  rate of
solutions of  semilinear wave equations.
  The first result valid for general solutions is due to Merle and Zaag   in \cite{MZimrn05}
(see also \cite{MZajm03} and \cite{MZma05}) who   proved, that  if
$1<p\le p_c$ and $u$ is a solution of (\ref{1}), then   the growth
estimate near the space-time blow-up graph is given by the
associated ODE. In \cite{HZlyap09} and \cite{HZJHDE}, we extend the
result of Merle and Zaag  to perturbed equations of type (\ref{gen})
under some reasonable growth estimates on $f$ and $g$ in (\ref{gen})
(see hypothesis $(H_f)$ and $(H_g)$).
 Note that, in all  these papers,
 the method  crucially relies on the existence of a Lyapunov
functional in similarity variables established by Antonini and Merle
\cite{AMimrn01}.
 Recently, Killip, Stovall and Vi\c san in \cite{KSV} have
shown, among other results, that the results of
 Merle and Zaag  remain valid for the semilinear Klein-Gordon equation (\ref{KG}).
 Moreover, they  consider also the case where $p_c<p<p_S$ and prove
 that, if $u $   is a solution of ({\ref{KG}}),  then for all  $x_0\in \er^N$, there
exists $K>0$ such that, for all $t\in [0,T(x_0))$,
\begin{eqnarray}\label{v1}
(T(x_0)-t)^{-\frac{(p-1)N}{p+3}}\int_{B(x_0,\frac{T(x_0)-t}2)} \! u^2(x,t)\mathrm{d}x &\le&
K,
\end{eqnarray}
and for all   $t\in (0,T(x_0)]$,
\begin{eqnarray}\label{v2}
\int_{T(x_0)-t}^{T(x_0)-\frac{t}2}\int_{B(x_0,\frac{T(x_0)-\tau }2)}
\!\Big(|\grad u(x,\tau )|^{2}+|\partial_tu(x,\tau
)|^2\Big)\mathrm{d}x\mathrm{d}\tau&\le&K.
\end{eqnarray}
Moreover, if   $x_0$ is a non characteristic point,  then they use
 a covering argument to obtain  the same
estimates with   the ball $B(x_0,\frac{T(x_0)-\tau}2)$ replaced by the ball
$B(x_0,T(x_0)-\tau )$ in the inequalities (\ref{v1}) and (\ref{v2}).

\bigskip

\no Here, we obtain a better result
 thanks to a
different method based  on the use of  self-similar variables. This
method allows us to improve the results of \cite{KSV} as we state in the following:
\begin{theo}\label{t0}\textbf{(Growth estimate near the blow-up surface for Eq. ({\ref{gen}}))}.\\
If $u $   is a solution of ({\ref{gen}}) with blow-up graph
$\Gamma:\{x\mapsto T(x)\}$, then for all  $x_0\in \er^N$  and  $t\in
[0,T(x_0))$, we have
\begin{eqnarray}\label{t1}
(T(x_0)-t)^{\frac{-(p-1)N}{p+3}}\int_{B(x_0,T(x_0)-t)} \!
u^2(x,t)\mathrm{d}x \ \ \rightarrow 0 ,\ \ \textrm{ as}\ \ \
t\rightarrow T(x_0).
\end{eqnarray}
Moreover, for all   $t\in (0,T(x_0)]$, we have
\begin{eqnarray}\label{t2}
\int_{T(x_0)-t}^{T(x_0)-\frac{t}2}\int_{B(x_0,\frac{T(x_0)-\tau }2)}
\!|\partial_tu(x,\tau )|^2\mathrm{d}x \mathrm{d}\tau&\le&K_1,
\end{eqnarray}and
\begin{eqnarray}\label{t22}
\int_{T(x_0)-t}^{T(x_0)-\frac{t}2}\int_{B(x_0,\frac{T(x_0)-\tau }2)}
\!|\grad u(x,\tau )|^{2}\mathrm{d}x \mathrm{d}\tau &\le&K_1.
\end{eqnarray}

If in addition  $x_0$ is a non characteristic point, then we have
for all $t\in (0,T(x_0)]$,
\begin{eqnarray}\label{t4}
\int_{T(x_0)-t}^{T(x_0)-\frac{t}2}\int_{B(x_0,{T(x_0)-\tau })}
\!\Big(|\grad u(x,\tau )|^{2}+|\partial_tu(x,\tau
)|^2\Big)\mathrm{d}x\mathrm{d}\tau\ \ \rightarrow 0, \ \textrm{ as}\
\ t\rightarrow 0.\qquad
\end{eqnarray}
Moreover, we have
\begin{eqnarray}\label{limener}
&&\frac{T(x_0)-t}2\int_{B(x_0,{T(x_0)-t
})} \!\Big( |\grad u(x,t )|^{2}- \big(\frac{x-x_0}{T(x_0)-t}.\grad u(x,t )\big)^2\nonumber\\
&&+|\partial_tu(x,t )|^2-\frac1{p+1} \!|
u(x,t)|^{p+1}\Big)\mathrm{d}x\rightarrow 0, \ \textrm{ as}\ \
t\rightarrow T(x_0).
\end{eqnarray}
\end{theo}
\begin{rem}{}
i)Let us remark that,  we have the following   lower bound which
follows  from standard techniques (scaling arguments, the
wellposedness in $H^1(\er^N )\times L^2(\er^N)$, the finite speed of
propagation and the fact that $x_0$ is a non characteristic point):
there exist
 $\varepsilon_0>0$, such that
\begin{eqnarray*}
&&0<\varepsilon_0\le (T(x_0)-t)^{\frac{2}{p-1}}\frac{\|u(t)\|_{L^2(B(x_0,{T(x_0)-t}))}}{ (T(x_0)-t)^{\frac{N}{2}}}\nonumber\\
&&+ (T(x_0)-t)^{\frac{2}{p-1}+1}\Big
(\frac{\|\partial_tu(t)\|_{L^2(B(x_0,{T(x_0)-t}))}}{
(T(x_0)-t)^{\frac{N}{2}}}+
 \frac{\|\grad u(t)\|_{L^2(B(x_0,{T(x_0)-t}))}}{ (T(x_0)-t)^{\frac{N}{2}}}\Big
 ).
\end{eqnarray*}
ii) In  Theorem 1, we improve recent results of
 Killip, Stovall and Vi\c san in \cite{KSV}. More precisely,
  we obtain a better estimate in (\ref{t1}) and  if $x_0$ is non
characteristic point we have the better estimate  (\ref{t4}).\\
iii) Up to a time dependent factor, the expression in (\ref{limener})
is equal to the main terms of the energy  in similarity variables
(see (\ref{E3})).

However, even with this improvement, we think that our estimates are still not optimal. \\
iv) The constant $K_1$, and the rate of convergence to $0$ of the
different quantities in the previous theorem  and in the whole
paper, depend only on $N$, $p$ and the upper bound on $T(x_0)$,
$1/{T(x_0)}$, and the initial data $(u_0,u_1)$ in $
H^{1}(B(x_0,2T(x_0)))\times L^{2}(B(x_0,2T(x_0)))$, together with
$\delta_0(x_0)$ if $x_0$ is non characteristic point.

\end{rem}

\bigskip
Our method relies on the estimates in similarity variables
introduced in \cite{AMimrn01} and used in \cite{MZajm03},
\cite{MZimrn05} and \cite{MZma05}. More precisely, given $(x_0,T_0)$
such that $0< T_0\le T(x_0)$, we introduce  the following
self-similar change of variables:
\begin{equation}\label{scaling}
y=\frac{x-x_0}{T_0-t},\qquad s=-\log (T_0-t),\qquad
u(x,t)=\frac{1}{(T_0-t) ^{\frac{2}{p-1}}}w_{x_0,T_0}(y,s).
\end{equation}
 This change of variables
transforms the backward light cone with vortex $(x_0,T_0)$ into the
infinite cylinder $(y,s)\in B\times [-\log T_0,+\infty)$. In the new
set of variables $(y,s),$ the behavior of $u$ as $t \rightarrow T_0$
is equivalent to the behavior of $w$ as $s \rightarrow +\infty$.

\bigskip

From (\ref{1}), the  function $w_{x_0,T_0}$  (we write $w$ for
simplicity) satisfies the following equation for all $y\in B\equiv
B(0,1)$ and $s\ge -\log T_0$:
\begin{eqnarray}\label{A}
\partial_{s}^2w+\frac{p+3}{p-1}\partial_sw+2y.\grad \partial_sw&=&
\sum_{i,j}(\delta_{i,j}-y_iy_j) \partial^2_{y_i,y_j} w-\frac{2p+2}{(p-1)^2}y.\grad
w\nonumber\\
&&-\frac{2p+2}{(p-1)^2}w+|w|^{p-1}w.
\end{eqnarray}
Putting this equation in the following form
\begin{eqnarray}\label{C}
\partial_{s}^2w&=&\div( \grad w-(y.\grad w)y)+2\eta
 y.\grad w
-\frac{2p+2}{(p-1)^2}w+|w|^{p-1}w\\
&&-\frac{p+3}{p-1}\partial_s w-2y.\grad \partial_sw,\
  \forall y\in B\ {\textrm{ and}} \ s\ge -\log T_0,\nonumber
\end{eqnarray}
where
\begin{equation}\label{eta}
\eta=\frac{N-1}{2}-\frac2{p-1}=\frac2{p_c-1}-\frac2{p-1}>0,
\end{equation}
 the key idea of our paper is to view this equation as a perturbation of the conformal case (corresponding to $\eta=0$) already treated in \cite{HZJHDE} with the term $2\eta y \cdot \nabla w$. Of course, this term is not a lower order term with respect to the nonlinearity. For that reason, we will have exponential growth rates in the $w$ setting. Let us emphasize the fact that our analysis is not just a trivial adpatation of our previous work \cite{HZJHDE}.

\bigskip

\no The equation (\ref{A}) will be studied in the Hilbert  space $\cal H$
$${\cal H}=\Big \{(w_1,w_2), |
\displaystyle\int_{B}\Big ( w_2^2 +|\grad w_1|^2(1-|y|^2)+w_1^2\Big
) {\mathrm{d}}y<+\infty \Big \}.$$
In the conformal case where $p=p_c$, Merle and Zaag \cite{MZimrn05}
proved that
\begin{eqnarray}
\label{f} E_0(w)=\displaystyle\int_{B}\Big (
\frac{1}{2}(\partial_sw)^2 +\frac{1}{2}|\grad
w|^2-\frac{1}{2}(y.\grad w)^2+\frac{p+1}{(p-1)^2}w^2
-\frac{|w|^{p+1}}{p+1}\Big ) {\mathrm{d}}y,\quad
\end{eqnarray}
 is a Lyapunov functional for equation (\ref{A}).
 When $p>p_c$, we introduce
\begin{eqnarray}\label{E1}
E(w)&=&\displaystyle {E_0(w)+I(w),}
\end{eqnarray}
where
\begin{eqnarray}
\label{E2}
 I(w)&=& -\eta\displaystyle\int_{B}w\partial_s w {\mathrm{d}}y+ \frac{\m N}2 \displaystyle\int_{B}w^2
 {\mathrm{d}}y,
\end{eqnarray}
and $\eta$ is defined in \eqref{eta}.
 Finally, we define the energy function as
\begin{eqnarray}
\label{E3}
 F(w,s)&=&\displaystyle {E(w)e^{-2\eta s}.}
\end{eqnarray}

The  proof of Theorem 1 crucially relies on the fact that  $F(w,s)$ is
 a Lyapunov functional for equation (\ref{A})  on the one hand, and on the other hand, on   a blow-up criterion involving
 $F(w,s)$.
 Indeed, with the  functional $F(w,s)$ and some
more work, we are able to adapt the analysis performed in
\cite{MZimrn05}. In the following, we show that $F(w,s)$  is a
Lyapunov functional:

\begin{pro}\textbf{(Existence  of  a decreasing  functional for Eq. ({\ref{A}}))}.\\
\label{pro} For all $s_2>s_1\ge -\log T_0=s_0$, the functional
$F(w,s)$ defined in (\ref{E3}) satisfies
\begin{eqnarray}\label{LE}
F(w(s_2),s_2)-F(w(s_1),s_1) &=&
- \int_{s_1}^{s_2} e^{-2\m s}\!\int_{\partial B}\!\Big(\partial_s w-\m w\Big)^2{\mathrm{d}}\sigma {\mathrm{d}}s\nonumber\\
&&-  \frac{\m (p-1)}{p+1}\int_{s_1}^{s_2} e^{-2\m
s}\int_{B}|w|^{p+1}{\mathrm{d}}y{\mathrm{d}}s.
\end{eqnarray}
Moreover, for all $s\ge s_0$, we have  $F(w,s)\ge0$.
\end{pro}

\bigskip

This paper is organized as follows:  In section 2, we prove
Proposition \ref{pro}. Using this result, we prove Theorem \ref{t0}
in section 3.

\section{Existence  of  a decreasing  functional for equation (\ref{A}) and a
blow-up criterion} Consider $u $ a solution of ({\ref{1}}) with
blow-up graph $\Gamma:\{x\mapsto T(x)\}$, and consider its
self-similar transformation $w_{x_0,T_0}$ defined at some scaling
point $(x_0,T_0)$ by (\ref{scaling}) where $T_0\le T(x_0)$.
 This section is devoted to the proof of
Proposition \ref{pro}. We proceed in two parts:
\begin{itemize}
\item  In subsection 2.1, we show the existence  of  a decreasing  functional for equation (\ref{A}).
\item In subsection 2.2, we prove a blow-up criterion involving this functional.
\end{itemize}
\subsection{Existence  of  a decreasing  functional for equation
(\ref{A})} In this subsection, we prove  that the functional $F(w,s)$
defined in (\ref{E3})  is decreasing. More precisely we prove that
the functional $F(w,s)$ satisfies the inequality (\ref{LE}).
Now we state two lemmas which are crucial for the proof. We
begin with   bounding the time derivative of $ E_0(w)$ defined in
(\ref{f}) in the following lemma.
\begin{lem}
\label{energylyap0} For all $s\ge -\log T_0$, we have
\begin{eqnarray}\label{energylem21}
 \frac{d}{ds}(E_0(w))&=& - \int_{\partial B} (\partial_s w)^2
{\mathrm{d}}\sigma + 2 \m \int_{B} (\partial_s w)^2 {\mathrm{d}}y
+2\m \int_{B} \partial_s w(y.\grad w){\mathrm{d}}y.
\end{eqnarray}
\end{lem}
{\it{Proof}}:  Multiplying $(\ref{C})$ by $\partial_s w $ and
integrating over the ball $B$, we obtain for all $s\ge -\log T_0$,
\begin{eqnarray}\label{energyE0}
\frac{d}{ds}(E_0(w))&=& -2 \int_{B} \partial_s w(y.\grad
\partial_s w){\mathrm{d}}y
-\frac{p+3}{p-1} \int_{B}(\partial_s w)^2{\mathrm{d}}y +2\m \int_{B}
\partial_s w(y.\grad w){\mathrm{d}}y.\nonumber
\end{eqnarray}
Since we see from  integration by parts that
\begin{eqnarray*}
 - 2\int_{ B} \partial_s w(y.\grad \partial_s w)  {\mathrm{d}}y&=& - \int_{ B}y.\grad (\partial_s w)^2  {\mathrm{d}}y=
 N
\int_{B} (\partial_s w)^2 {\mathrm{d}}y - \int_{\partial B}
(\partial_s w)^2{\mathrm{d}}\sigma,
\end{eqnarray*}
this concludes the proof of Lemma
 \ref{energylyap0}.
\Box

\vspace{0.3cm}

\no We are now going to prove the following estimate for the
functional $I(w)$:
\begin{lem}
\label{energylyap1} For all $s\ge -\log T_0$, we have
\begin{eqnarray}\label{energytheta1}
\frac{d}{ds}I(w)&=& 2\m E(w) -2\eta \int_{B}(\partial_s
w)^2{\mathrm{d}}y - \frac{\m
(p-1)}{p+1}\int_{B}|w|^{p+1}{\mathrm{d}}y\nonumber\\
&&-2\m \int_{B}
\partial_sw(y.\grad w) {\mathrm{d}}y  -\m^2 \int_{\partial B} w^2
{\mathrm{d}}\sigma+2\m \int_{\partial B} w\partial_s w
{\mathrm{d}}\sigma.
\end{eqnarray}
\end{lem}
{\it Proof:} Note that  $I(w)$ is a differentiable  function for all
 $s\ge   -\log T_0$ and that
\begin{eqnarray*}
\frac{d}{ds}I(w)&=&- \m\int_{B}(\partial_sw)^2{\mathrm{d}}y-
\m\int_{B}w\partial^2_{s}w{\mathrm{d}}y +\m N \int_{B}w\partial_s w
{\mathrm{d}}y.
\end{eqnarray*}
By using  equation (\ref{C}) and integrating by parts, we have
\begin{eqnarray*}
\frac{d}{ds}I(w)&=& -\eta \int_{B}(\partial_s w)^2{\mathrm{d}}y
+ \m\int_{B}(|\grad w|^2-(y.\grad w)^2){\mathrm{d}}y- \m\int_{B}|w|^{p+1}{\mathrm{d}}y\nonumber\\
&&-2\m^2 \int_{B} w(y.\grad w){\mathrm{d}}y +\m \frac{2p+2}{(p-1)^2}\int_{B}w^2{\mathrm{d}}y\\
&&+2\m \int_{B} w(y.\grad \partial_sw) {\mathrm{d}}y +\m
(\frac{p+3}{p-1}+N) \int_{B}w\partial_sw{\mathrm{d}}y.\nonumber
\end{eqnarray*}
Then by integrating by parts, we have
\begin{eqnarray}\label{e8}
\frac{d}{ds}I(w)&=& -\eta \int_{B}(\partial_s w)^2{\mathrm{d}}y
+ \m\int_{B}(|\grad w|^2-(y.\grad w)^2){\mathrm{d}}y- \m\int_{B}|w|^{p+1}{\mathrm{d}}y\nonumber\\
&&-\m^2 \int_{\partial B}  w^2{\mathrm{d}}\sigma +(\m^2N  +\m \frac{2p+2}{(p-1)^2})\int_{B}w^2{\mathrm{d}}y\\
&&+2\m \int_{\partial B} w \partial_sw {\mathrm{d}}\sigma -2\m
\int_{B} (y.\grad w)
\partial_sw {\mathrm{d}}y  + \m
(\frac{p+3}{p-1}-N) \int_{B}w\partial_sw{\mathrm{d}}y.\nonumber
\end{eqnarray}
By combining (\ref{f}), (\ref{E1}),  (\ref{E2}), (\ref{eta}) and
(\ref{e8}), we conclude the proof of Lemma \ref{energylyap1}.
\Box

\bigskip
\no From Lemmas \ref{energylyap0} and \ref{energylyap1}, we are in a
position to prove   the first part of Proposition \ref{pro}.

\vspace{0.2cm} \no {\it Proof of the first part of  Proposition
\ref{pro}:} From Lemmas \ref{energylyap0} and \ref{energylyap1}, we
obtain for all $s\ge -\log T_0$,
\begin{eqnarray*}
\frac{d}{ds}E(w)&=&2\m E(w)- \int_{\partial B} \Big(\partial_s w-\m
w\Big)^2 {\mathrm{d}}\sigma - \frac{\m
(p-1)}{p+1}\int_{B}|w|^{p+1}{\mathrm{d}}y.
\end{eqnarray*}
Therefore, using  the definition of the functional $F(w,s)$ in
(\ref{E3}), we  write
\begin{eqnarray}\label{m}
\frac{d}{ds}F(w,s)&=&-e^{-2\m s} \int_{\partial B} \Big(\partial_s
w-\m w\Big)^2 {\mathrm{d}}\sigma - \frac{\m (p-1)}{p+1}e^{-2\m s}
\int_{B}|w|^{p+1}{\mathrm{d}}y  .
\end{eqnarray}
By integration, we get (\ref{LE}). This concludes the first part of
the proof of Proposition \ref{pro}. \Box

\subsection{A  blow-up criterion}
\no We finish  the proof  of  Proposition \ref{pro} here. More
precisely, for all $ x_0\in \er^N$ and $ T_0\in (0,T(x_0)]$, we
prove that
\begin{equation}\label{254}
\forall \ s\ge -\log T_0,\ \ \ F(w_{x_0,T_0}(s),s)\ge 0.
\end{equation}
 We  give the proof only  in the case where $x_0$ is a non
characteristic point. Note that the case where $x_0$ is a
 characteristic point  can be  done  exactly as in Appendix A page 119 in \cite{MZjfa07}.

\bigskip

\no

{\it{Proof of the last point of Proposition 1.2}}:  The argument is the
same as in the corresponding part in \cite{AMimrn01}. We write the
proof for completeness. Arguing by contradiction, we assume that
there exists  a non characteristic point $ x_0\in \er^N$, $ T_0\in
(0,T(x_0)]$  and  $ s_1\ge -\log T_0$ such that $F(w(s_1),s_1)<0$, where
$w=w_{x_0,T_0}$. Since the energy $F(w(s),s)$ decreases in time, we
have $F(w(1+s_1),1+s_1)<0$.

\no Consider now for $\delta>0$ the function
$\widetilde{w}^{\delta}(y,s)=w_{x_0,T_0-\delta }(y,s)$. From
(\ref{scaling}), we see that for all $(y,s)\in B\times
[1+s_1,+\infty)$
\begin{equation*}
 \widetilde{w}^{\delta}(y,s)=\frac{1}{(1+\delta e^s)^{\frac2{p-1}}}
w( \frac{y}{1+\delta e^s},-\log(\delta+e^{-s})).
\end{equation*}
\begin{itemize}
\item  (A) Note that $\widetilde{w}^{\delta}$ is defined in  $ B\times [1+s_1,+\infty)$,
whenever $\delta>0$ is small enough such that
$-\log(\delta+e^{-1-s_1})\ge s_1.$
\item (B) By construction, $\widetilde{w}^{\delta}$
is also a solution of equation (\ref{A}).
\item (C) For $\delta$ small enough, we have
$F(\widetilde{w}^{\delta}(1+s_1),1+s_1)<0$ by continuity of the function
$\delta \mapsto F(\widetilde{w}^{\delta}(1+s_1),1+s_1)$.
\end{itemize}
Now, we fix $\delta=\delta_0>0$ such that (A), (B) and (C) hold. Since
 $F(\widetilde{w}^{\delta_0},s)$ is decreasing  in time, we have
 \begin{equation}
\label{255}
\liminf_{s\rightarrow +\infty}F(\widetilde{w}^{\delta_0}(s),s)\le F(\widetilde{w}^{\delta_0}(1+s_1),1+s_1)<0.
 \end{equation}
 Let
us note that we have
\begin{eqnarray}\label{c1}
-\m\int_{B}\widetilde{w}^{\delta_0}\partial_s\widetilde{w}^{\delta_0}
{\mathrm{d}}y&\ge&-\frac{1}{2}
\int_{B}(\partial_s\widetilde{w}^{\delta_0})^2{\mathrm{d}}y-\frac{\m^2}2
\int_{B}(\widetilde{w}^{\delta_0})^2{\mathrm{d}}y
\end{eqnarray}
By (\ref{f}), (\ref{E1}), (\ref{E2}), (\ref{c1}) and the fact that $\m \in [0,1]$,  we deduce
\begin{eqnarray}
E(\widetilde{w}^{\delta_0}(s))&\ge& (\frac{\m N}2-
\frac{\m^2}2)\int_{B}(\widetilde{w}^{\delta_0})^2
{\mathrm{d}}y-\frac{1}{p+1}\int_{B}|\widetilde{w}^{\delta_0}|^{p+1}
{\mathrm{d}}y\nonumber\\
&\ge& -\frac{1}{p+1}\int_{B}|\widetilde{w}^{\delta_0}|^{p+1}
{\mathrm{d}}y.
\end{eqnarray}
So, by (\ref{E3}), we have
\begin{eqnarray}
F(\widetilde{w}^{\delta_0}(s),s) &\ge& -\frac{e^{-2\m
s}}{p+1}\int_{B}|\widetilde{w}^{\delta_0}|^{p+1} {\mathrm{d}}y.
\end{eqnarray}
After a change of variables, we find that
\[
F(\widetilde{w}^{\delta_0}(s),s) \ge -\frac{e^{-2\m s}}
{(p+1)(1+\delta_0e^s)^{\frac{4}{p-1}+2-N}}\displaystyle\int_{B}|w(z,-\log
(\delta_0+e^{-s}))|^{p+1} {\mathrm{d}}z.
\]
Since we have $-\log (\delta_0+e^{-s})\rightarrow -\log \delta_0$
as $s\rightarrow +\infty$ and since $\|w(s)\|_{L^{p+1}(B)}$ is
locally bounded from the fact that $w=w_{x_0,T_0}$ and $x_0$ is  non
characteristic point, by a continuity argument, it follows that the
former integral remains bounded and
\begin{eqnarray}\label{ggg}
F(\widetilde{w}^{\delta_0}(s),s)&\ge& -\frac{Ce^{-2\m
s}}{(1+\delta_0e^s)^{\frac{4}{p-1}+2-N}}\rightarrow 0,
\end{eqnarray}
as $s\rightarrow +\infty$ (use the fact that $\frac{4}{p-1}+2-N-2\m
=1$ and $\m >0$). So, from (\ref{ggg}), it follows that
\begin{eqnarray}\label{d2}
\liminf_{s\rightarrow +\infty}F(\widetilde{w}^{\delta_0}(s),s)\ge 0 .
\end{eqnarray}
\no From (\ref{255}), this is a contradiction. Thus (\ref{254}) holds.
 This concludes  the  proof of Proposition \ref{pro}.
\Box

\section{Proof of Theorem \ref{t0}}
\no Consider $u $ a solution of ({\ref{1}}) with blow-up graph
$\Gamma:\{x\mapsto T(x)\}$. Translating Theorem \ref{t0} in the self-similar setting
$w_{x_0,T_0}$ (we write $w$ for simplicity) defined
by (\ref{scaling}), our goal becomes the following Proposition:
\begin{pro}
\label{final}
If $u $   is a solution of ({\ref{gen}}) with blow-up graph
$\Gamma:\{x\mapsto T(x)\}$, then for all  $x_0\in \er^N$  and  $T_0\le
T(x_0)$, we have
for all $s\ge s_0=-\log T_0$,
\begin{eqnarray}\label{cor1}
e^{-2\eta s}\int_{s}^{s+1} \!\int_{B}\!\Big((\partial_s
w(y,\tau))^2+|\grad
w(y,\tau)|^2(1-|y|^2)\Big){\mathrm{d}}y{\mathrm{d}}\tau\le K.
\end{eqnarray}
Moreover,
\begin{eqnarray}\label{cor2}
e^{-2\m s}\int_{ B}\!|w(y,s)|^{\frac{p+3}2}{\mathrm{d}}y \ \
\rightarrow 0 ,\ \ \textrm{ as}\ \ \ s\rightarrow +\infty ,
\end{eqnarray}
\begin{eqnarray}\label{cor3}
e^{\frac{-8\m s}{p+3}}\int_{ B}\!|w(y,s)|^2{\mathrm{d}}y \ \
\rightarrow 0 ,\ \ \textrm{ as}\ \ \ s\rightarrow +\infty .
\end{eqnarray}
If in addition  $x_0$ is a non characteristic point, then we have,
\begin{eqnarray}
e^{-2\eta s}\int_{s}^{s+1} \!\int_{B}\!\Big((\partial_s
w(y,\tau))^2
\Big){\mathrm{d}}y{\mathrm{d}}\tau\ \ \ &\rightarrow& 0\label{03bis} ,\\
e^{-2\eta s}\int_{s}^{s+1} \!\int_{B}\!\Big(
|\grad
w(y,\tau)|^2\Big){\mathrm{d}}y{\mathrm{d}}\tau\ \ \ &\rightarrow& 0 ,\label{03}
\end{eqnarray}
as $s\rightarrow +\infty$. Moreover, we have
\begin{eqnarray}\label{04}
F(w,s)\ \rightarrow 0 ,\ \ \textrm{ as}\ \ \ s\rightarrow +\infty .
\end{eqnarray}
\end{pro}
In this section, we prove  Proposition \ref{final} which directly implies Theorem \ref{t0},  as in  the proof of Theorem 1.1, (page 1145)  in \cite{MZimrn05}.

\bigskip

Let us first use Proposition \ref{pro} and the averaging technique of \cite{MZma05} and \cite{MZimrn05} to get the following bounds:
\begin{lem}\label{lem1}
For  all $s\ge s_0=-\log T_0$, we have
\begin{equation}\label{F1}
0\le F(w(s),s)\le F(w(s_0),s_0),
\end{equation}
\begin{eqnarray}\label{F2}
\int_{s_0}^{\infty}e^{-2\m s}
\!\!\int_{B}\!|w(y,s)|^{p+1}{\mathrm{d}}y{\mathrm{d}}s &\le&
\frac{p+1}{\m (p-1)} F(w(s_0),s_0),
\end{eqnarray}
\begin{eqnarray}\label{F3}
\int_{s_0}^{\infty}e^{-2\m s} \!\!\int_{\partial B}\!\Big(\partial
_sw(\sigma ,s)-\m w(\sigma ,s)\Big)^{2}{\mathrm{d}}\sigma
{\mathrm{d}}s &\le& F(w(s_0),s_0).
\end{eqnarray}
If in addition $x_0$ is non characteristic (with a slope $\delta_0\in (0,1)$), then
\begin{eqnarray}\label{cor00001}
e^{-2\m s}\int_{s}^{s+1}\int_{B}\Big (\partial_s w_{x_0,T_0}(y,\tau
)-\lambda(\tau ,s) w_{x_0,T_0}(y,\tau )\Big )^2dy d\tau \rightarrow
0, \ \ {\textrm {as}}\  s\rightarrow +\infty,\ \
\end{eqnarray}
where $0\le \lambda(\tau ,s) \le C(\delta_0)$, for all $\tau \in
[s,s+1]$.
\end{lem}
\no {\it{Proof:}}
  The first three estimates are  a direct
consequence of Proposition \ref{pro}. As for the last estimate, by introducing $f(y,s)=e^{-\m s}w(y,s)$, we see that  the
dispersion estimate (\ref{F3}) can be written as follows:
\begin{eqnarray}\label{F4}
\int_{s_0}^{\infty} \!\!\int_{\partial B}\!\Big(\partial _sf(\sigma
,s)\Big)^{2}{\mathrm{d}}\sigma {\mathrm{d}}s &\le& F(w(s_0),s_0).
\end{eqnarray}
In particular, we have
\begin{eqnarray}\label{F5}
\int_{s}^{s+1} \!\!\int_{\partial B}\!\Big(\partial _sf(\sigma
,\tau)\Big)^{2}{\mathrm{d}}\sigma {\mathrm{d}}\tau \rightarrow 0, \
\ {\textrm {as}}\ \ s\rightarrow +\infty.
\end{eqnarray}
 By exploiting (\ref{F5}) where the space integration is done over the unit sphere, one can use the averaging technique of Proposition 4.2 (page 1147) in   \cite{MZimrn05} to get the same estimate with the space variable integrated over the whole ball $B$.
\Box

\bigskip

From Lemma \ref{lem1}, we are in a position to prove
Proposition \ref{final}

\bigskip

\no {\it{Proof of
Proposition \ref{final}:}}\\
- {\it Proof of \eqref{cor1}}:
 By integrating  the functional $F(w,s)$  defined in  (\ref{E3}) in time between $s$ and $s+1$, we obtain:
\begin{eqnarray}\label{et}
&&\displaystyle\int_{s}^{s+1}\!\!e^{-2\m
\tau}\int_{B}\!\!\Big((\partial_sw)^2+|\grad w|^2(1-|y|^2)\Big)
{\mathrm{d}}y{\mathrm{d}}\tau=  -\frac{2(p+1)}{(p-1)^2}
\displaystyle\int_{s}^{s+1}\!\!e^{-2\m \tau}\int_{B}\!\! w^2
{\mathrm{d}}y{\mathrm{d}}\tau\nonumber \\
&&2\int_{s}^{s+1}\!\!F(w(\tau ),\tau )
d\tau-\displaystyle\int_{s}^{s+1}\!e^{-2\m \tau}\int_{B}\!\!\Big
(|\grad w|^2|y|^2-(y.\grad w)^2\Big )
{\mathrm{d}}y{\mathrm{d}}\tau\\
&&+\frac{2}{p+1}\displaystyle\int_{s}^{s+1}e^{-2\m
\tau}\int_{B}\!\!|w|^{p+1} {\mathrm{d}}y{\mathrm{d}}\tau
 +\underbrace{2\m \int_{s}^{s+1}\!e^{-2\m
\tau}\displaystyle\int_{B}\!\!\big(w\partial_s w
 -\frac{
N}2 w^2\big) {\mathrm{d}}y{\mathrm{d}}\tau }_{A(s)}.\nonumber
\end{eqnarray}
Now, we    control all the terms on the right-hand side of the
relation (\ref{et}):

\no Note that the first term is negative, while the second term is
bounded because of the  bound (\ref{F1}) on the energy $F(w,s)$.
 Since $|y.\grad
w|\le |y| |\grad w|$, we can say that the third is also negative.
Remark that  (\ref{F2}) implies that the fourth term is also bounded.
Finally, it remains only to control the  term  $A(s)$.

\medskip

\no Combining  the Cauchy-Schwarz inequality, the inequality  $ab\le
 \varepsilon a^2+ \frac1{4\varepsilon}b^2$, and the fact that $N\ge 2$ and $\m\in [0,1]$, we write
\begin{eqnarray}\label{et1}
 A(s)&\le& \frac{1}2 \int_{s}^{s+1}\!\!e^{-2\m
\tau}\displaystyle\int_{B}\!\!(\partial_s w)^2
{\mathrm{d}}y{\mathrm{d}}\tau.
\end{eqnarray}
Now, we are able to conclude the proof of the inequality
(\ref{cor1}). For this, we combine (\ref{et}), (\ref{et1}) and  the above-mentioned arguments for
the first four terms
to get
\begin{eqnarray}\label{et4}
\int_{s}^{s+1}\!\!\!e^{-2\m \tau}\!\!\int_{B}\!\!\!\Big (
(\partial_sw)^2+|\grad w|^2(1-|y|^2)\Big)
{\mathrm{d}}y{\mathrm{d}}\tau \le K + \frac{1}{2}
\!\int_{s}^{s+1}\!\!\!e^{-2\m \tau}\!\!\int_{B}\!\!\!(\partial_s
w)^2 {\mathrm{d}}y{\mathrm{d}}\tau.\quad
\end{eqnarray}
The desired bound in (\ref{cor1}) follows then from (\ref{et4}).

\medskip

\no - {\it Proof of \eqref{cor2}}:
 Using the mean value theorem, we derive the existence of $\sigma(s)\in [s,s+1]$ such that
\begin{eqnarray}\label{s1}
\int_{ B}\!|w(y,\sigma (s))|^{\frac{p+3}2}{\mathrm{d}}y
&=&\int_{s}^{s+1}\int_{ B}\!|w(y,\tau
)|^{\frac{p+3}2}{\mathrm{d}}y{\mathrm{d}}\tau.
\end{eqnarray}
By Jensen's inequality, we have
\begin{eqnarray}\label{s2}
\int_{s}^{s+1}\!\!\int_{ B}\!|w(y,\tau
)|^{\frac{p+3}2}{\mathrm{d}}y{\mathrm{d}}\tau&\le&C\Big(\int_{s}^{s+1}\!\!\int_{
B}\!\!|w(y,\tau
)|^{p+1}{\mathrm{d}}y{\mathrm{d}}\tau\Big)^{\frac{p+3}{2(p+1)}}
\end{eqnarray}
By combining  (\ref{s1}) and (\ref{s2}), we can write that
\begin{eqnarray}\label{s3}
\int_{ B}\!|w(y,\sigma (s))|^{\frac{p+3}2}{\mathrm{d}}y
&\le&C\Big(\int_{s}^{s+1}\!\!\int_{ B}\!\!|w(y,\tau
)|^{p+1}{\mathrm{d}}y{\mathrm{d}}\tau\Big)^{\frac{p+3}{2(p+1)}}.
\end{eqnarray}
Using   (\ref{s3}) and the fact that $ab\le a^2+b^2$, we have
\begin{eqnarray*}
\int_{ B}\!|w(y,s )|^{\frac{p+3}2}{\mathrm{d}}y&\le&\int_{
B}\!|w(y,\sigma (s))|^{\frac{p+3}2}{\mathrm{d}}y  +
C\int_{s}^{s+1}\int_{
B}\!|\partial_s w(y,\tau)||w(y,\tau)|^{\frac{p+1}2}{\mathrm{d}}y{\mathrm{d}}\tau\\
&\le& C\Big(\int_{s}^{s+1}\!\!\int_{ B}\!\!|w(y,\tau
)|^{p+1}{\mathrm{d}}y{\mathrm{d}}\tau\Big)^{\frac{p+3}{2(p+1)}}\\
&& + C\Big(\int_{s}^{s+1}\int_{
B}\!|w(y,\tau)|^{p+1}{\mathrm{d}}y{\mathrm{d}}\tau\Big)^{\frac12}
\Big(\int_{s}^{s+1}\int_{ B}\!|\partial_s
w(y,\tau)|^2{\mathrm{d}}y{\mathrm{d}}\tau\Big)^{\frac12}.
\end{eqnarray*}
Since $\ds{e^{-2\m s}\int_{s}^{s+1}\int_{ B}\!|w(y,\tau
)|^{p+1}{\mathrm{d}}y{\mathrm{d}}\tau\rightarrow 0}$ from
(\ref{F2}), we use (\ref{cor1}) to obtain (\ref{cor2}).

\medskip

\no - {\it Proof of \eqref{cor3}}: It follows from \eqref{cor2} through the Holder
inequality and (\ref{cor2}).

\medskip

\no - {\it Proof of \eqref{03bis}}:
Note that from now on, we assume that  $x_0$ is a non characteristic point with slope $\delta_0\in (0,1)$.
It is a direct consequence of (\ref{cor3})
and (\ref{cor00001}).
\medskip

\no - {\it Proof of \eqref{03}}:
Let $s\ge s_0+1$, $s_1=s_1(s)\in [s-1,s]$  and $s_2=s_2(s)\in
[s,s+1]$ to be chosen later. By integrating after multiplication by $e^{-2\m s}$ the
expression (\ref{e8}) of $I(w)$ in time between $s_1$ and $s_2$, we
obtain
\begin{eqnarray}\label{q2}
 &&\m\int_{s_1}^{s_2}e^{-2\m s}\int_{B}|\grad w|^2(1-|y|^2){\mathrm{d}}y{\mathrm{d}}s=
\underbrace{e^{-2\m s_2}I(w(s_2))-e^{-2\m s_1}I(w(s_1))}_{B_1(s)}\qquad \qquad \nonumber\\
&&+\underbrace{2\m \int_{s_1}^{s_2}e^{-2\m s}\int_{B} (y.\grad w)
\partial_sw {\mathrm{d}}y{\mathrm{d}}s}_{B_2(s)}
 +\underbrace{\eta \int_{s_1}^{s_2}e^{-2\m s}\int_{B}(\partial_s
w)^2{\mathrm{d}}y{\mathrm{d}}s}_{B_3(s)}\\
&&+ \underbrace{\m \int_{s_1}^{s_2}e^{-2\m
s}\int_{B}|w|^{p+1}{\mathrm{d}}y{\mathrm{d}}s}_{B_4(s)} \underbrace{
-(\m^2N  +\m \frac{2p+2}{(p-1)^2})\int_{s_1}^{s_2}e^{-2\m s}\int_{B}w^2{\mathrm{d}}y{\mathrm{d}}s}_{B_5(s)}\nonumber\\
&&\underbrace{- \m (\frac{p+3}{p-1}-N)\int_{s_1}^{s_2} e^{-2\m
s}\int_{B}w\partial_sw{\mathrm{d}}y{\mathrm{d}}s}_{B_6(s)}\underbrace{-
\int_{s_1}^{s_2}e^{-2\m s}\int_{\partial B}  (\partial_sw)^2
{\mathrm{d}}\sigma{\mathrm{d}}s}_{B_7(s)}\nonumber\\  &&+
\underbrace{\int_{s_1}^{s_2}e^{-2\m s}\int_{\partial B}
(\partial_sw-\m w)^2{\mathrm{d}}\sigma
{\mathrm{d}}s}_{B_8(s)}\underbrace{-\m\int_{s_1}^{s_2}e^{-2\m
s}\int_{B}(|y|^2|\grad w|^2-(y.\grad
w)^2){\mathrm{d}}y{\mathrm{d}}s}_{B_{9}(s)}.\nonumber
\end{eqnarray}
Now, we    control all the terms on the right-hand
 side of the
relation (\ref{q2}):

\no Note that, by (\ref{E2}) and using the Cauchy-Schwarz
inequality, we can write
\begin{equation}\label{I1}
e^{-2\m s_2}I(w(s_2))\le Ce^{-2\m s_2}
\displaystyle\int_{B}(\partial_s w(s_2))^2 {\mathrm{d}}y+Ce^{-2\m
s_2} \displaystyle\int_{B}w^2(s_2)
 {\mathrm{d}}y.
\end{equation}
By exploiting  (\ref{cor3}) and the fact that  $s_2\in [s,s+1]$, we
conclude that
\begin{equation}\label{I11}
e^{-2\m s_2} \displaystyle\int_{B}w^2(s_2)
 {\mathrm{d}}y\rightarrow 0\ \ {\textrm {as}}\ \ s\rightarrow
 +\infty,
\end{equation}
on the one hand. On the other hand, by   using the mean value
theorem, let us choose  $s_2=s_2(s)\in [s,s+1]$ such that
\begin{eqnarray}\label{q4}\displaystyle\int_{s}^{s+1}
e^{-2\m \tau}\displaystyle\int_{B}(\partial_s w(\tau))^2
{\mathrm{d}}y{\mathrm{d}}\tau&=&e^{-2\m
s_2}\displaystyle\int_{B}(\partial_s w(s_2))^2 {\mathrm{d}}y.
\end{eqnarray}
By combining  (\ref{03bis}) and (\ref{q4}) we
obtain
\begin{equation}\label{I111}
e^{-2\m s_2}\displaystyle\int_{B}(\partial_s w(s_2))^2
{\mathrm{d}}y\rightarrow 0\ \ {\textrm {as}}\ \ s\rightarrow
 +\infty.
\end{equation}
Then, by using (\ref{I1}), (\ref{I11}) and (\ref{I111}), we get
\begin{equation}\label{I1111}
e^{-2\m s_2}I(w(s_2))\rightarrow 0\ \ {\textrm {as}}\ \ s\rightarrow
+\infty.
\end{equation}
From (\ref{E2}) and the fact that $ab\le a^2+b^2$, we write
\begin{equation}\label{I02}
-e^{-2\m s_1}I(w(s_1))\le C e^{-2\m s_1}\displaystyle\int_{B}(\partial_s w(s_1))^2
{\mathrm{d}}y.
\end{equation}
Similarly, by   using the mean value
theorem, we choose $s_1=s_1(s)\in [s-1,s]$ such that
\begin{eqnarray}\label{q44}\displaystyle\int_{s-1}^{s}
e^{-2\m \tau}\displaystyle\int_{B}(\partial_s w(\tau))^2
{\mathrm{d}}y{\mathrm{d}}\tau&=&e^{-2\m
s_1}\displaystyle\int_{B}(\partial_s w(s_1))^2 {\mathrm{d}}y.
\end{eqnarray}
By  (\ref{03bis}), (\ref{I02}) and (\ref{q44}) we obtain
\begin{equation}\label{I2}
-e^{-2\m s_1}I(w(s_1))\rightarrow 0\ \ {\textrm {as}}\ \ s\rightarrow
+\infty.
\end{equation}
Note that by combining   (\ref{I1111}), (\ref{I2}) and the fact that $B_1(s)=e^{-2\m
s_2}I(w(s_2))-e^{-2\m s_1}I(w(s_1))$,
we deduce that
\begin{equation}\label{B1}
B_1(s)\rightarrow 0\ \ {\textrm {as}}\ \ s\rightarrow +\infty.
\end{equation}
To estimate $B_2(s)$, since $s_1\in [s-1,s]$ and $s_2\in [s,s+1]$, we write
\begin{eqnarray}\label{q01}
B_2(s)&\le&C\Big(\int_{s-1}^{s+1}e^{-2\m \tau}\int_{B} |\grad w|^2
{\mathrm{d}}y{\mathrm{d}}\tau \Big)^{\frac12}
\Big(\int_{s-1}^{s+1}e^{-2\m \tau}\int_{B} (
\partial_sw)^2 {\mathrm{d}}y{\mathrm{d}}\tau \Big)^{\frac12}.\qquad
\end{eqnarray}
By using (\ref{cor1}) and the covering argument of \cite{MZimrn05},
we have
\begin{eqnarray}\label{cor01}
e^{-2\eta s}\int_{s}^{s+1} \!\int_{B}\!|\grad
w|^2{\mathrm{d}}y{\mathrm{d}}\tau\le K.
\end{eqnarray}
Thus
\begin{eqnarray}\label{B02}
\int_{s-1}^{s+1}e^{-2\m \tau}\int_{B} |\grad w|^2
{\mathrm{d}}y{\mathrm{d}}\tau &\le C K.
\end{eqnarray}
Then, by (\ref{q01}), (\ref{B02}) and  (\ref{03bis}), we deduce
\begin{equation}\label{B2}
B_2(s)\rightarrow 0\ \ {\textrm {as}}\ \ s\rightarrow +\infty.
\end{equation}
By (\ref{03bis}), we can say that
\begin{equation}\label{B3}
B_3(s)\rightarrow 0\ \ {\textrm {as}}\ \ s\rightarrow +\infty.
\end{equation}
By (\ref{F2}), we also deduce that
\begin{equation}\label{B4}
B_4(s)\rightarrow 0\ \ {\textrm {as}}\ \ s\rightarrow +\infty.
\end{equation}
The terms $B_5(s)$ and $B_7(s)$  are negative. By
using  (\ref{cor3}) and (\ref{03bis}), we have
\begin{equation}\label{B6}
B_6(s)\rightarrow 0\ \ {\textrm {as}}\ \ s\rightarrow +\infty.
\end{equation}
By (\ref{F3}), we write that
\begin{equation}\label{B8}
B_8(s)\rightarrow 0\ \ {\textrm {as}}\ \ s\rightarrow +\infty.
\end{equation}
Finally, since $|y.\grad
w|\le |y| |\grad w|$, we can say that the term $B_9(s)$ is negative.
By combining (\ref{B1}), (\ref{B2}), (\ref{B3}), (\ref{B4}), (\ref{B6}), (\ref{B8})
and the fact that the terms $B_5(s)$, $B_7(s)$ and $B_9(s)$  are negative, we conclude that
\begin{eqnarray}\label{q9}
 \int_{s}^{s+1}e^{-2\m \tau}\int_{B}|\grad w|^2(1-|y|^2){\mathrm{d}}y{\mathrm{d}}\tau\rightarrow 0
 \ \ {\textrm {as}}\ \ s\rightarrow +\infty.
\end{eqnarray}
By using (\ref{q9}) and the covering argument of \cite{MZimrn05},
we deduce that  estimate (\ref{03})
holds.

\medskip

\no - {\it Proof of \eqref{04}}:
By integrating the functional $F(w,s)$ defined in
(\ref{E3}) in time between $s$ and $s+1$, we  write
\begin{eqnarray}\label{fin2}
\int_{s}^{s+1}\!\!F(w,\tau)
d\tau&=&\displaystyle\int_{s}^{s+1}\int_{B}\!\!e^{-2\eta \tau}\Big (
\frac{1}{2}(\partial_s w)^2
+\frac{p+1}{(p-1)^2}w^2-\frac{1}{p+1}|w|^{p+1}\Big ) {\mathrm{d}}y{\mathrm{d}}\tau\nonumber\\
&&+\frac{1}{2}\displaystyle\int_{s}^{s+1}\!\!e^{-2\eta
\tau}\int_{B}\!\!\Big (|\grad w|^2-(y.\grad w)^2\Big )
{\mathrm{d}}y{\mathrm{d}}\tau\\
&&-\eta\int_{s}^{s+1}\!\!e^{-2\eta
\tau}\displaystyle\int_{B}w\partial_s w
{\mathrm{d}}y{\mathrm{d}}\tau+ \frac{\m N}2
\int_{s}^{s+1}\!\!e^{-2\eta \tau}\displaystyle\int_{B}w^2
 {\mathrm{d}}y{\mathrm{d}}\tau.\nonumber
\end{eqnarray}
By using (\ref{cor3}),  (\ref{03}), (\ref{F2}) and  (\ref{fin2}), we conclude that
\begin{eqnarray}\label{fin3}
\int_{s}^{s+1}\!\!F(w,\tau)
d\tau\rightarrow 0,
 \ \ {\textrm {as}}\ \ s\rightarrow +\infty.
\end{eqnarray}
Combining The monotonicity of $F(w,s)$ proved in Proposition \ref{pro}, and (\ref{fin3}), we deduce the identity (\ref{04}).  This concludes
 the proof
 of  Proposition \ref{final}.

\Box

\no Since the derivation of Theorem \ref{t0}   from Proposition  \ref{final}  is the
same as in  \cite{MZimrn05} (up to some very minor changes),
this concludes the proof of Theorem \ref{t0}.
\Box

\def\cprime{$'$}

redpajama_set_name: RedPajamaArXiv


Here are some after pictures.
You and Joey did a great job on that one Bryan.
Also in regards to this thread, UB2SLOW went with myself and Matt... to bring his Gallardo and Ford GT up to par with where they should be.
On a side note....RIP Fast Eddy.
Contacted Bryan about my Ferrari California. He sent me VERY DETAILED emails about what opti-coat and gave me other advice on how I can clean the car myself and the dangers of having other hand washers doing it.
He then visited me in Midtown and checked out my car and noted the paint thickness in various spots and scoured it looking for faults, swirls, scratches, etc that I even hadn't noticed. Beyond being obviously extremely competent, he's also a very giving person. Kind-hearted, easy-to-laughter, and gregarious. Not in a salesman-type way, but in a this-is-a-good-human way.
I wanted him to finish early so I could drive down on Sunday to pick it up because I was busy on Monday and he worked all night to do this for me. Who does that?
When I came down to pick it up, part of me wanted to leave it in his garage because it was so stunning, I didn't want it to ever look dirty again. And he had the game on his garage and allowed my friend to watch the end of the Patriots-Miami game. Who does that?
This isn't a mash note. Getting opti-coat on your car isn't cheap. But it looks amazing. And it was so hassle free and Bryan said if I wasn't happy with it in the next 5 years, he'd fix it up for me for free. Who does that? Bryan does.
I've used Mike at Adams Classics for my 430, SL and Aventador. He does a great job at paint correction, and has applied opticoat and cquartz (I prefer c quartz).


text:
import {IAuthenticationService} from "./auth.service";
import {ConfigService} from "./config.service";
import {HttpClient, HttpEvent, HttpResponse} from "@angular/common/http";
import {User} from "../models/user.model";
import {HttpUtils} from "../util/common";

/**
 * Base class for all Hub-API based services.
 */
export abstract class AbstractHubService {

    private apiBaseHref: string;
    private editingBaseHref: string;

    /**
     * Constructor.
     * @param http
     * @param authService
     */
    constructor(protected http: HttpClient, protected authService: IAuthenticationService, protected config: ConfigService) {
        this.apiBaseHref = this.config.hubUrl();
        this.editingBaseHref = this.config.editingUrl();
    }

    /**
     * Gets the current user.
     */
    protected user(): User {
        return this.authService.getAuthenticatedUserNow();
    }

    /**
     * Creates a hub API endpoint from the api path and params.
     * @param path
     * @param params
     *
     */
    protected endpoint(path: string, params?: any, queryParams?: any): string {
        if (params) {
            for (let key in params) {
                let value: string = encodeURIComponent(params[key]);
                path = path.replace(":" + key, value);
            }
        }
        let rval: string = this.apiBaseHref + path;
        if (queryParams) {
            let first: boolean = true;
            for (let key in queryParams) {
                if (queryParams[key]) {
                    let value: string = encodeURIComponent(queryParams[key]);
                    if (first) {
                        rval = rval + "?" + key;
                    } else {
                        rval = rval + "&" + key;
                    }
                    if (value != null && value != undefined) {
                        rval = rval + "=" + value;
                    }
                    first = false;
                }
            }
        }
        return rval;
    }

    /**
     * Creates an editing endpoint from the given relative path and params.
     * @param path
     * @param params
     *
     */
    protected editingEndpoint(path: string, params?: any): string {
        if (params) {
            for (let key in params) {
                let value: string = encodeURIComponent(params[key]);
                path = path.replace(":" + key, value);
            }
        }
        return this.editingBaseHref + path;
    }

    /**
     * Creates the request options used by the HTTP service when making
     * API calls.
     * @param headers
     * @param authenticated
     *
     */
    protected options(headers: {[header: string]: string}, authenticated: boolean = true): any {
        let options = {
            headers: headers
        };
        if (authenticated) {
            this.authService.injectAuthHeaders(options.headers);
        }
        return options;
    }

    /**
     * Performs an HTTP GET operation to the given URL with the given options.  Returns
     * a Promise to the HTTP response data.
     * @param url
     * @param options
     */
    protected httpGet<T>(url: string, options: any, successCallback?: (value: T) => T): Promise<T> {
        options["observe"] = "response";
        return HttpUtils.mappedPromise(this.http.get<HttpResponse<any>>(url, options).toPromise(), response => {
            if (successCallback) {
                return successCallback(response.body);
            } else {
                return response.body;
            }
        });
    }

    /**
     * Performs an HTTP POST operation to the given URL with the given body and options.  Returns
     * a Promise to null (no response data expected).
     * @param url
     * @param body
     * @param options
     *
     */
    protected httpPost<I>(url: string, body: I, options: any, successCallback?: () => void): Promise<void> {
        options["observe"] = "response";
        return HttpUtils.mappedPromise(this.http.post<HttpResponse<any>>(url, body, options).toPromise(), () => {
            if (successCallback) {
                successCallback();
            }
            return null;
        });
    }

    /**
     * Performs an HTTP POST operation to the given URL with the given body and options.  Returns
     * a Promise to the HTTP response data.
     * @param url
     * @param body
     * @param options
     *
     */
    protected httpPostWithReturn<I, O>(url: string, body: I, options: any, successCallback?: (data: O) => O): Promise<O> {
        options["observe"] = "response";
        return HttpUtils.mappedPromise(this.http.post<HttpResponse<any>>(url, body, options).toPromise(), response => {
            let data: O = response.body;
            if (successCallback) {
                return successCallback(data);
            } else {
                return response.body;
            }
        });
    }

    /**
     * Performs an HTTP PUT operation to the given URL with the given body and options.  Returns
     * a Promise to null (no response data expected).
     * @param url
     * @param body
     * @param options
     *
     */
    protected httpPut<I>(url: string, body: I, options: any, successCallback?: () => void): Promise<void> {
        options["observe"] = "response";
        return HttpUtils.mappedPromise(this.http.put<HttpResponse<any>>(url, body, options).toPromise(), () => {
            if (successCallback) {
                successCallback();
            }
            return null;
        });
    }

    /**
     * Performs an HTTP PUT operation to the given URL with the given body and options.  Returns
     * a Promise to the HTTP response data.
     * @param url
     * @param body
     * @param options
     *
     */
    protected httpPutWithReturn<I, O>(url: string, body: I, options: any, successCallback?: (data: O) => O): Promise<O> {
        options["observe"] = "response";
        return HttpUtils.mappedPromise(this.http.put<HttpResponse<any>>(url, body, options).toPromise(), response => {
            let data: O = response.body;
            if (successCallback) {
                return successCallback(data);
            } else {
                return response.body;
            }
        });
    }

    /**
     * Performs an HTTP DELETE operation to the given URL with the given body and options.
     * @param url
     * @param options
     *
     */
    protected httpDelete(url: string, options: any, successCallback?: () => void): Promise<void> {
        options["observe"] = "response";
        return HttpUtils.mappedPromise(this.http.delete<HttpResponse<any>>(url, options).toPromise(), () => {
            if (successCallback) {
                successCallback();
            }
            return null;
        });
    }

}

redpajama_set_name: RedPajamaGithub


text:
Home Latest News IPL 2022 Retentions: 5 Players Who Got Biggest Pay Hikes
IPL 2022 Retentions: 5 Players Who Got Biggest Pay Hikes
Vignesh Rajagopal
players who got biggest pay hikes
We look at 5 players who received major pay jumps as compared to last year:
#5 Mayank Agarwal (+1200%)
#4 Abdul Samad (+2000%)
#3 Arshdeep Singh (+2000%)
#2 Ruturaj Gaikwad (+3000%)
#1 Venkatesh Iyer (+4000%)
The IPL retentions threw up plenty of surprises. Loyalty was damned and only the last season's performance was counted. Even a team like CSK were forced to choose a player like Moeen Ali over Faf du Plessis and Dwayne Bravo. Here is all about IPL 2022 Retentions.
The retentions saw few players receive major pay hikes. Not many had expected them to bag such rich contracts but that is the nature of the game.
Punjab Kings had bought Mayank Agarwal in 2018 mega auctions. They had got him for a steal at just INR 1 crore. Agarwal had a forgettable season in 2018 but he has been sensational since then. In the last 3 seasons, he has amassed 1197 runs at an average of 34.2 with a strike rate of 145.66.
Agarwal and KL Rahul were the force that kept Punjab driving. His retention was always on the cards but people were curious about the amount he would be paid. The Kings stunned everyone when they offered him a package of INR 12 crores. It marked a 1200% increase in his annual IPL salary.
Abdul Samad caught everyone's attention when he made his IPL debut in 2020. He had a fantastic first season where he smashed high-profile bowlers for fun. Samad took down Kagiso Rabada, Jasprit Bumrah, and Pat Cummins in different games.
He didn't get enough chances this season to build on his dream first season. But SRH have shown great faith in the youngster. They have retained uncapped Samad for INR 4 crores. This marks a jump of 2000% in his annual IPL contract.
Arshdeep Singh remains one of the finds of the IPL. He formed a potent combination with Mohammad Shami. Singh often always delivered for his team. He is someone who loves to counter challenges head-on.
The franchise believes he is very close to making his national debut. Punjab had no qualms in retaining Singh and offer him an annual contract of INR 4 crores. His salary too saw a spike of 2000%.
Ruturaj Gaikwad showed tremendous class and temperament in CSK's title-winning run this year. He was the winner of the orange cap. Gaikwad was the youngest to achieve this milestone.
There is very little doubt in anyone's mind that Gaikwad is a future India prospect. He was one of those rare CSK players who didn't have much international experience but he defied all odds to become the franchise's number 1 match-winner. CSK duly awarded him with a fat paycheque of INR 6 crores. Gaikwad's salary witnessed a hike of 3000%.
Venkatesh Iyer is well and truly living his dream. He made his IPL debut in just the 2nd phase. Iyer made an instant impression scoring attacking innings against IPL heavyweights, RCB and MI. He was consistent right throughout showcasing his skills with the ball too.
Iyer soon earned the national cap against New Zealand. KKR have now awarded him with an annual contract of INR 8 crores. His salary jump is the biggest amongst the lot. It stands at an unimaginable 4000%!
Also read: IPL 2022: We Have 'Gun Players', Release Them Is 'Heart-Breaking'- Rohit
Ruturaj Gaikwad
Venkatesh Iyer
Previous articleIPL 2022: 5 Controversial Non-Retentions
Next articleIPL 2022: We Have 'Gun Players', Release Them Is 'Heart-Breaking'- Rohit
I am a passionate cricket lover, who likes to express his views on the beautiful game and share insights and analysis about the finer aspects of the sport.
Gautam Gambhir Reveals Lucknow Strategies for IPL Auction


text:
Q: Pixel font renders incorrectly in firefox http://i.stack.imgur.com/zl3iE.png
Can anyone tell me why this pixel web-font is becoming blurred in Firefox? I'm using @font-face to render it using CSS. The font-size is 8px, which is identical to 50%, and it renders perfectly in Chrome; no blurry artifacts. But in Firefox, every other word is blurry.
EDIT: Here is the relevant code snippet:
<link rel="stylesheet" type="text/css" href="http://yui.yahooapis.com/2.9.0/build/reset/reset-min.css">
<style type="text/css">
@font-face {
 font-family: 'pokemon_rbyregular';
 src: url('pokemon_rby-webfont.eot');
 src: url('pokemon_rby-webfont.eot?#iefix') format('embedded-opentype'),
 url('pokemon_rby-webfont.woff') format('woff'), url('pokemon_rby-webfont.ttf') format('truetype');
 font-weight: normal;
 font-style: normal;
 font-size: 8px;
 text-rendering: optimizeSpeed;
 }

This works in every other browser except Firefox (incl. Opera, IE8, and Chrome). I am just wondering if there is some sort of CSS text-rendering setting I could implement that would fix this. I have a feeling it is related to how Firefox implements spacing, as it is only every other word. If I have a long paragraph of jumbled letters, they're all rendered perfectly until I put some spaces in-between them.


Your financial statements will remain accessible as usual. However, you will notice that we have made some small changes to the PDF which includes more information about your Gross and Net Remittance.
The columns in the transaction report have also been rearranged and some renamed.