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THE ARANSAS PROJECT
New Year Update
Jan 9, 2014 | The Aransas Project | TAP Updates
The Aransas Project (TAP) team wanted to provide a New Year's update on the Whooping Crane litigation, TAP v. Shaw, against the Texas Commission on Environmental Quality (TCEQ). The following is a summary of the highlights from the past year.
As most everyone knows, we had a tremendous victory in Corpus Christi federal court before Judge Janis Jack back in March 2013. The opinion by Judge Jack was 124 pages in length and held, among other things, that the water management actions of the TCEQ commissioners and executive director caused the death of at least 23 Whooping Cranes. Judge Jack also ordered the TCEQ officials to apply for an Incidental Take Permit which requires the development and approval of a Habitat Conservation Plan (HCP) that should lead to the provision of additional freshwater inflow to San Antonio Bay.
It was a decisive victory for TAP, the Whooping Cranes, and San Antonio and Aransas Bays.
TCEQ Appeals to Fifth Circuit
After the victory, the TCEQ appealed the decision to the federal Fifth Circuit Court of Appeals. Immediately, the Defendants asked the appeals court for an emergency stay of the district court's judgment so that they would not have to take steps towards implementation of the district court's order.
The Court of Appeals granted the emergency stay, and the district court's decision was put on hold. At the same time, however, at the request of TAP, the Court of Appeals also granted an expedited appellate schedule—and the oral argument before a three-judge panel of the Fifth Circuit was scheduled for August 2013. The written briefs were submitted by the parties in May and June.
In the written briefs, it was apparent that the TCEQ Defendants, as well as the Intervenors—the Guadalupe-Blanco River Authority (GBRA), the San Antonio River Authority (SARA) and the Texas Chemical Council—would be challenging nearly every aspect of the district court's decision. The Texas Solicitor General submitted a brief on behalf of the TCEQ, and GBRA took the lead on behalf of the Intervenors. In GBRA's brief, the Intervenors largely mounted a factual attack, challenging whether TAP had proven the various facts at trial. GBRA also claimed that the Senate Bill 3 process would provide sufficient water for the Bays, and that therefore the district court should have abstained from rendering a decision. In the TCEQ's brief, the State argued that they, as the entity regulating water permits, could not be held liable for downstream harms to the Bay. The State also argued that the Tenth Amendment acted as a barrier to their possible liability.
Oral Argument in New Orleans
Oral argument was heard in New Orleans on August 8, 2013 by a three-judge panel comprised of Judges Edith Jones, Jerry Smith and Emilio Garza. It was a very engaged panel, with many questions to counsel for all parties. A full transcript of the hearing can be found here. It is too difficult to predict which way the panel may rule based on the questions during oral argument.
As of today, December 31, 2013, no decision has been issued by the Fifth Circuit. Sometimes, with complicated cases, the Fifth Circuit may take a few months to issue a decision. (A time period of greater than 6 months to issue a decision is unusual for the Fifth Circuit). Depending upon the ruling, this case could be either headed back to the District Court, or headed for further appellate review, either for en banc (full court) review by all of the judges of the Fifth Circuit, or potentially to the U.S. Supreme Court.
TAP Opposes New GBRA Water Rights Permits
In the meantime, TAP has become engaged in additional legal action regarding the Guadalupe River and water rights. GBRA has applied for a permit to withdraw water from the Guadalupe near Gonzales to fill an off-channel reservoir and has applied for a permit to take water "created" behind the salt water barrier through withdrawal points anywhere up to 200 miles upriver from the barrier. TAP's attorneys have long been concerned about the operation of the salt water barrier that exists on the Guadalupe River just above the SH 35 bridge crossing the Guadalupe delta and are particularly concerned about the impact of the new proposed withdrawal permits. TAP has requested a contested case hearing on the Gonzales new water withdrawal permit and will be actively opposing both of these permit applications when they become ripe for hearing.
The one thing that is clear from our experience in the TAP litigation is that those concerned about the future of the health of our bays will have to fight for these bays. There are many competing demands for water in our state. That is just a fact of life in Texas today, and if we don't understand that and if we don't fight back hard and creatively, we may lose our coastal bays and estuaries.
The Whooping Crane is not only a magnificent white bird but also a symbol of a healthy ecosystem and economy. The San Antonio Bay and Estuary ecosystem can and does produce blue crabs and wolfberries for the Whoopers – this ecosystem also produces shrimp for the trout and shrimp for the shrimper and of course trout for the recreational fishermen and women – and it supports tackle sales, home and motel rentals and house sales. Ecology and economy go together, and we need to emphasize that linkage every way that we can.
At this time, TAP is initiating a major fundraising effort to support these administrative law hearings, as well as upcoming legal proceedings in TAP v. Shaw. If you would like to help TAP protect whooping cranes and the bay, please visit this page and make a donation. Or, if you prefer, you can send a check to The Aransas Project, PO Box 1839. Rockport, TX 78381-1839, or you can email us for instructions. TAP is a 501(c)(3).
From Jim Blackburn & the TAP legal team at Blackburn Carter
Filed in: Guadalupe Basin Water Management, TAP Progress, Whooping Cranes.


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redpajama_set_name: RedPajamaCommonCrawl


Summertime in Martha's Vineyard without delicious gluten-filled cookie crusted key lime pie has been a silent and sad loss. Until I realised I could probably make it myself with regular store bought gluten free cookie crust! And then realised just how insanely easy it is to make an impressive key lime pie. It's one of those recipes that looks hard until you realise there are about three ingredients in the filling. This pie is particularly good because the crust caramelises and becomes like a crispy toffee cookie cup filled with green citrus cream. Not exactly French... but let's call it an alteration on Tarte Citron... lemon tart. And every time I brought this pie to a party, it was destroyed in a particularly savage way. Always a good sign.
Mix cookies, room-temperature butter, honey and vanilla in a food processor and press dough into pie pan (preferably with removable bottom).
Whip egg yolks until thick and pale with an electric mixer.
Add the sweetened condensed milk and blend.
Add zest and slowly add lime juice and blend (the mixture will thicken) and pour into pie shell.
Bake for 10-15 minutes at 350 degrees until the filling sets (you will see tiny bubbles form at top) and make sure not to overcook our filling will separate and curdle.
Add all filling ingredients to a blender and blend on high until creamy and smooth. Add more lime zest/juice for more tart or more agave for added sweetness.
Pour filling into crust and tap on counter to release air bubbles. Top with a bit more lime zest and loosely cover.
Freeze for 2-4 hours or until firm. Remove from the freezer for 10-15 minutes to thaw before serving. Will keep covered in the freezer for up to 2 weeks, though best when fresh.


This wonderful Brazilian Cowhide Rug #498 measure's in at 8' long by 7'5" wide (approximately 4.30 meter). This stunning tri-color cowhide rug has rich browns and deep chocolates and brilliant whites. We are sure you will enjoy this hair-on South American cowhide from Brazil.


We know that active children need active adult role models, so we provide free fitness classes for parents and staff at our partner schools as well.
Community Projects within the gyms: We offer a number of free and low-cost activities at our gyms to make fitness accessible for the groups that will benefit the most. This includes fitness and dance classes for seniors; exercise classes for women and girls; community Soccer Tots, youth football, netball and tennis programmes and hosting a range of fitness and weight management classes in partnership with local councils and community groups.
Active Events: We recognise that the gym is not for everyone, so we try to encourage people to move, be fit and have fun wherever they are most comfortable. Projects include the Big Dance Bus, charity running, and outdoor gym projects. We also participate in large scale campaigns like Sport Relief and National Fitness Day to provide motivation and opportunity to be more active.
Concessionary Membership: We try to make fitness accessible to everyone by financial financial burdens where possible. More than 23% of our total memberships across our four gyms are concession – for students, elderly, disabled people, unemployed or juniors.
GP Referral Programmes: At our clubs in Hampstead and Westminster we run successful schemes where people referred by their doctor can receive free one-to-one supervised exercise sessions to help treat specific medical conditions.
Read about all our achievements in our latest annual review.


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Update on Tough Mudder/Spartan Deal
Just under two weeks ago I wrote about an odd series of events that suggested Spartan was going to acquire Tough Mudder. Or maybe they weren't. As I said, it was odd. Things got odder still, but long story short, the Tough Mudder founders said they will not contest an involuntary bankruptcy filing petition that some of Tough Mudders vendors had filed. This is progress.
So what's next? According to a joint announcement on Instagram, Joe De Sena and former Tough Mudder CEO Kyle McLaughlin have been working together on a plan. What this plan looks like remains to be seen but Joe seemingly remains committed to keeping Tough Mudder. I'll share more info when it's posted, but in the meantime, head over to OCR Media to read more.
Joe De Sena and His Thoughts on Work-Life Balance
Look at These New Benefits But Don't Use Them Because We'll Take Notice...


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Attacking The Spanish
The Spanish opening, 1.e4 e5 2.Nf3 Nc6 3.Bb5, is so respected that many Black players immediately start defending. Brunello shows that Black can engage in a tense fight for the win right from the start. The three gambits considered in this book have all been recently tested by top 10 players.
1.e4 e5 2.Nf3 Nc6 3.Bb5 f5 The Schliemann, with its reversed King's Gambit flair, has always been popular with club players, but grandmasters knew it was unsound. Then Teimour Radjabov started playing it successfully against the elite and sparked a revival.
1.e4 e5 2.Nf3 Nc6 3.Bb5 a6 4.Ba4 Nf6 5.0-0 Ne7 6.Re1 b5 7.Bb3 d6 8.c3 0-0 9.h3 Na5 10.Bc2 d5 The Gajewski variation is brand new. The position after White's tenth move had been reached a thousand times with 10...c5 universally played, when in the summer of 2007 the Polish grandmaster Gajewski revealed that Black has a fascinating gambit at his disposal.
1.e4 e5 2.Nf3 Nc6 3.Bb5 a6 4.Ba4 Nf6 5.0-0 Be7 6.Re1 b5 7.Bb3 0-0 8.c3 d5 The Marshall Attack is sound and aggressive, so it is no surprise that it is favoured by the world's best. The theory is so complex that an expert guide is required, and Brunello is well suited to the task, having played the Marshall for most of his life.
003 Preface
004 Key to symbols used & Bibliography
The Schliemann 007 1 Schliemann: 4.d3
037 2 Schliemann: 4.Sc3
069 3 Schliemann: Minor Lines
The Gajewski 093 4 Gajewski 11.d3 and 11.exd5
119 5 Gajewski 11.d4
The Marshall 153 6 Anti-Marshall
183 7 Marshall: 12.d4
257 9 Marshall: Modern Ideas


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Claude Bourque, C.M., est un journaliste canadien.

Biographie
Claude Bourque naît le  à Moncton, au Nouveau-Brunswick. Il fréquente des écoles de Moncton et de Saint-Anselme, désormais un quartier de Dieppe. Il obtient un baccalauréat en arts de l'Université de Moncton en 1966. En 1967, il est diplômé en études littéraires pratiques, mention journalisme, de l'Université de Strasbourg.

Il devient courriériste parlementaire à l'Assemblée législative du Nouveau-Brunswick, pour le compte de L'Évangéline, dès son retour au Canada. Il devient agent d'information à la Société d'aménagement régionale de Fredericton en 1968. Il est ensuite directeur adjoint du Service d'information du Nouveau-Brunswick de 1969 à 1971. Il occupe aussi la fonction de chargé de secrétaire adjoint et chargé des relations publiques du Comité d'étude sur le développement social du Nouveau-Brunswick.

Claude Bourque est réembauché par le quotidien L'Évangéline en 1971, à titre de rédacteur en chef. Il est président de l'Imprimerie acadienne en 1972, directeur-général de l'Imprimerie acadienne de 1975 à 1976 et éditeur du journal de 1977 à 1980. Il devient la même année chef du service des nouvelles de Radio-Canada Acadie, alors appelé Radio-Canada Atlantique. Il devient directeur des services en français de Radio-Canada Atlantique en 1982. Il quitte Radio-Canada en 1996 pour s'impliquer dans le milieu communautaire. Il publie en 1997 Rêves de visionnaires : historique de l'Hôtel-Dieu /Hôpital Dr Georges-L. Dumont.

Au cours de sa carrière, Claude Bourque fait la promotion de la culture et de l'histoire de l'Acadie, tout en travaillant à l'inclusion en favorisant le rapprochement entre les communautés francophones et anglophones du Nouveau-Brunswick. Il est fait membre de l'ordre du Canada en 1999.

Notes et références

Bibliographie

Naissance en novembre 1945
Naissance à Moncton
Journaliste néo-brunswickois
Journaliste acadien
Personnalité liée au comté de Westmorland
Membre de l'Ordre du Canada

redpajama_set_name: RedPajamaWikipedia


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Dmonta Harris take MLive.com Detroit Player of the Week honors
Dmonta Harris
MLive Detroit Slam Dunk for Jan. 27
By Jared Purcell | japurcell@mlive.com
on January 27, 2015 2:20 PM, updated January 27, 2015 2:34 PM
Each week, we'll honor MLive Detroit area basketball players and a coach for their efforts during the prior week.
Girls Player of the Week: Erika Davenport, Sr., Clarkston
Davenport, as expected, has been dominant throughout the season so far. In a blowout win over Avondale on Tuesday last week, Davenport had 19 points and nine rebounds. She capped the week on Thursday with a 29-point effort in a crucial 63-60 win over North Farmington.
Boys Player of the Week: Dmonta Harris, Sr., New Haven
Harris has been a threat in nearly every aspect of the game this season. He had 18 points, eight rebounds and three steals in a win over Warren Woods Tower to start last week. He then followed that up with arguably his most dominant performance of the season with 26 points and 20 rebounds in a win over Sterling Heights.
Coach of the Week: Dan Fife, Clarkston boys basketball
Fife is getting this honor for the second week in a row. Why? On Tuesday last week when his undefeated Wolves took on Oak Park, he got his 616th win as a head coach. That number puts him all alone at No. 5 on Michigan's all-time wins list.
He passed Ray Lauwers, Monroe St. Mary Catholic Central (615-303, 1967-09).
"I was lucky to get 616, to be honest with you," Fife said, who has been coaching at Clarkston since 1983. "I was watching Duke a little bit last night and coach Krzyzewski is getting (win) 999. You know, it's a number but I've weathered the storm through issues to last as long as I have."
He then picked up win 617 as Clarkston defeated Southfield on Friday, 55-39. In his career, Fife is 617-164.


The service: From the moment we stepped on site, we were quickly greeted with Lei's for the adults and little Tahitian necklaces for the kids. We were offered water and escorted right to the front desk for check in. From the smiles to the Aloha's the service and professional attitude from ALL staff members made us feel extra special. We were allowed to start enjoying all the amenities of the resort, left our bags at the bell desk, and were texted when our room was ready, super convenient.
The Pool: At the time, the hotel was at 98% capacity, which means yes, it was crowded. The resort includes many different pool options. There were water slides, a kid splash zone/water bridge, lazy river, infinity pool, hot tubs, and even an adult's only pool. Also, was a private resort only access right to the beach lagoon with complimentary chair and umbrellas for you to enjoy. I was also quite happy with the amount of lifeguards on hand, which I think is very important especially among children during water play. The whole family enjoyed the water slides and especially the lazy river.
Kids club: INCLUDED WITH YOUR STAY. Yes, you read that right; there is a Kids club where kids 3-12 can enjoy an array of different activities, all complimentary. While I was excited for our kids to try out Aunty's Beach house, I was still hesitant to leave them in there with out my supervision. Would they get bored? Would they eat? Would they be nice to them in there? These were just some of my thoughts going through my head while signing them into the center. All I can say is that all of these thoughts and concerns were quickly reassured and calmed from the moment we started the check in process.
Each child is given a green band that acts as its check in ticket/security tracking. Everyone entering the house is required to wash their hands; I loved this, such a good idea to help fight germs. As soon as you walk in, it is quite large, very clean, very tidy, with cast members standing and observing all around. Let's just say I never had to worry that my kids were unattended. With a 5,200 square foot house, there was a game room, movie room, arts and crafts room, eating room, and a huge outdoor play ground.
I was comforted by the cast member that the kids would be taken care of, would have a blast, and for us adults to go have a date night! I was honestly surprised at how calm she was, reassuring that they would be fine, and not to worry and HAVE FUN! As soon as we walked out of the kids club, my husband and I both looked at each other and were at lost about what to do with out the kids around, lol. Because of this great amenity, we were able to have some adult time, date nights and great reconnecting time. You can even call or go into the kids club anytime to check on the kids and see how they were doing. The kids center is open daily from 9:30 am – 9:00 pm for open play, they can come and go any time as long as they weren't at max capacity.
Parking: The overnight parking fee is $37 per night. This is outrageous in my opinion. We never left the resort, so our car literally just sat there paying rent for the 3 days we were there. Parking options are non-existent around that area, so we really had no choice but to pay it.
Food Options: While the food itself is good on site, the options in my opinion are limited. I would've liked to see a casual dining restaurant that wasn't pool based, but still quick and easy. I heard from guest that they would walk next door over to the Four Seasons because their prices were substantially lower than Aulani's.
Characters: This is a Disney Hotel, and they do have scheduled character meetings, but I just wish they had more. My kids would've liked to see MORE characters walking around interacting in different type of activities rather than just standing in line to take a picture with them.
Summary: This is definitely one of the best family vacations we've ever had, and would recommend it as a a great family vacation spot. It's a great place for the whole family to come together. There were many grandparents who were vacationing with their kids and grandkids . With the high price tag per night, I would say it's worth it, with all the amenities for kids and things to do as a family. We didn't want to leave by the end of our stay and will for sure make it back for at least one more stay during our time in Hawaii!


Coding.Cookies - A gentle introduction to functional programming, part 2.
A gentle introduction to functional programming, part 2.
// Simply call the method, which will show the alert box.
// to interact with it.
// Finally, we can also pass arguments to call.
// We store the logging function in the context.
// Remember that we can pass around functions as variables, yay!
// the other values are set to undefined.
// and a new value. We are passing the element to update as the context!
As you can see, the apply function is much more powerful and allows us to pass a variable number of arguments. If we wanted to implement the sum function using call, our function would have to accept an array and iterate through that, rather than the arguments object. One interesting side note: the pattern of iterating through an array and adding the result of a function to a total count (in our case our function is a basic addition) is an extremely common pattern in functional programming known as folding, reducing or accumulating. I encourage you to check it out here as well as to check out the built-in array function reduce and to try and implement it as an exercise.
//      10 + b + c, where b and c are the two arguments.
//     10 + 25 + c, where c is the one argument.
As you can see, currying allows us to continuously create a 'copy' of our original function with the value of the first argument being declared. What's kind of cool about this is that we can represent any function as a series of curried functions, a result which is emphasized in Haskell. But now you are probably wondering how the curry function is actually implemented! There are two approaches to this: one is to create a curry function which accepts a function to curry like shown above, and the second is add a curry function to the Function prototype so that we do not have to pass a function. For the sake of simplicity, we will be taking the first approach, however if you have any questions about implementing it the second way feel free to post on the comments! You will also notice that we convert the arguments object into an actual array by using the Array prototype's slice function, a trick I picked up here.
// the arguments in the function signature.
And there you go! Go ahead, test it out and play around with it! Create more than one curried version of the same base function but different fixed variables to see what you get. Now you may have noticed a limitation with currying: by definition, currying fixes only the first variable. This is where partial application comes in! Partial application is essentially like currying, however rather then being able to fix only the first parameter, we can now fix as many parameters as we want (in continuous order of course). We can actually easily modify our curry function to handle this, and again it can also be done via the Function prototype.
// with the firstArgs prepended to the arguments array.
// for specific elements and events.


Meet Iesodo, (YAY-sa-doe) a highly acclaimed Telly and Accolade Award winning series featuring a cast of animated birds who live in a Cypress Tree on the shores of Galilee.
As soon as I saw the cover of the DVD for this series, I immediately thought of Rio! My kids loved that movie so I knew they would be interested in Iesodo. The movie has cutting edge imagery that is somewhat unexpected for a Christian DVD series. (If you've seen/heard of Gaither's Pond, you'll know what I mean…).
Iesodo, means "The way of Jesus", and the name of cute white dove that is the star of the show. He is a wise and gentle bird that reminds me of a kind, old grandfather. What I like best about the series is the solid biblical teaching, which is what kids need more of these days!
With each and every story in the series, Iesodo and his cute feathered friends "present the essence of the Gospel of Jesus Christ in a winsome, compelling way" through music and adventure (MovieGuide.org).
Meet Iesodo (pronounced Yay-Sa-Doe), a wise dove whose name means "The Way of Jesus"! He and his lovable feathered friends live in the Holy Land in a beautiful Cypress Tree on the shores of a vast Lake (or as we know it, the Sea of Galilee). The birds are an unlikely group of friends, and though they have their differences, they have one thing in common: since they met Iesodo, their lives have never been the same!
I was definitely interested in this series after reading The Examiner's Review of Iesodo. Definitely take a moment to read it if you haven't seen it yet!
You can find more information about the series and the characters on their website www.iesodo.com. You'll also find fun stuff for the kids like printable bird masks, coloring pages and more in their Fun & Games section on the website.
I highly recommend these DVDs to any family, but especially those that find Christian values and morals to be important. You won't have to "pre-screen" these!
Want it? Buy it! Get Iesodo DVDs on Amazon!
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text:
Regional Lifestyles
Housing crisis in P.E.I.
Prince Edward Island > Opinion
OPINION: Finally, CRA obeys law
Canada Revenue Agency cannot be trusted to produce accurate, unbiased estimate of tax gap
SaltWire Network | Posted: March 29, 2018, 9 a.m. | Updated: March 29, 2018, 9 a.m. | 4 Min Read
P.E.I. Sen. Percy Downe's pressure on the Canada Revenue Agency to crack down on off-shore tax shelters is finally paying dividends. (File Photo) - The Guardian
BY PERCY DOWNE
Since 2012, the Parliamentary Budget Officer (PBO) has been trying to fulfil my request to measure the tax gap, the difference between what is owed in taxes and what has actually been collected, but has been stonewalled by the Canada Revenue Agency (CRA). Therefore, the announcement that the CRA has finally provided that information is welcome, if belated, news. It is disappointing, however, that it took so long and only happened because the PBO threatened to take CRA to court.
The PBO now has what he needs to conduct his study. That said, why should the PBO produce an estimate of the tax gap, when the CRA has committed to conducting its own study, and has released three minor reports on the subject since 2016? The reason an independent analysis is required is quite simple: the Agency has been repeatedly caught trying to mislead Canadians. The result is the lack of credibility facing any sort of CRA-produced tax gap report.
Below are just a few examples over the last years, from a very long list, of the CRA misleading Canadians:
The Claim: The Auditor General's recent report on the CRA's call centres cited the Agency's claim that 90 per cent of calls were successfully connected.
The Reality: According to that same Auditor General Report, however, it was revealed that the CRA achieved this high success rate by blocking 28.9 million of the 53.5 million calls received and excluding these blocked calls from their calculations. When blocked calls and other factors were considered, "the Agency's overall success rate was 36 percent."
DISABILITY TAX CREDIT
The Claim: Historically, 80% of applications for the benefit by Canadians with diabetes were approved by the CRA. However in recent months, almost all previously approved claims have been rejected. Meanwhile, the revenue agency publicly claimed that there has been no change to the eligibility criteria.
The Reality: Documents obtained by Diabetes Canada show a CRA email dated May 2nd, 2017 modifying the "variable for (Life Sustaining Therapy) clarification verses with a new variable." In effect, diabetics who previously qualified for the disability tax credit were disallowed.
When confronted with the evidence of their misleading statement, CRA backed down.
The Claim: In February and March of 2017, articles appeared in newspapers and online across Canada extolling the work of the CRA. Bearing titles like "Federal programs in place to address offshore tax avoidance and evasion" and "How Canada is cracking down on offshore tax evasion and aggressive tax avoidance", it was likely the best press the Agency had ever received.
The Reality: It was subsequently revealed that CRA—operating under the premise that if you can't earn good press, buy it—paid almost $300,000 for this positive "sponsored content" in six print and digital newspapers across Canada.
BILLION DOLLAR INVESTMENT
The Claim: A number of individuals, from CRA spokespersons to the Revenue Minister herself, have spoken of the "billion dollars" that has been invested in the fight against tax evasion, implying that this money was already at work.
The Reality: The billion dollars is the amount the government has promised to spend over six years finishing in 2022. As of the end of the 2016-2017 fiscal year, less than $40 million of that $1 billion had actually been spent.
To this litany of exaggeration, misinformation and outright falsehood, can be added the fake tough talk that comes from the Agency every time there is a public leak of information from some Bank or law firm operating in a tax haven.
Clearly, the Canada Revenue Agency cannot be trusted to produce an accurate and unbiased estimate of the tax gap, and Canadians are owed an independent analysis of the scale of uncollected taxes in our country. When the CRA fails to do its job and collect those taxes, the rest of us have to make up the shortfall. Canadians deserve an honest assessment of the extent of that problem. The only way to get one is through the work of the Parliamentary Budget Officer, and the fact that the CRA has finally agreed to cooperate with him means that, after six long years of delays and a threat of court action, that work can finally begin.
- Charlottetown Sen. Percy Downe has also tabled a Bill in the Senate requiring CRA to co-operate with the PBO in its independent study of the tax gap, and to report on all convictions for tax evasion, including a separate report listing all convictions for overseas tax evasion.
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text:
\section{Introduction}

Existence and uniqueness questions for the 2D Euler equations
\begin{equation}
\frac{\partial u}{\partial t}+u\cdot\nabla u+\nabla p=0,\quad\mathrm{div}\,%
u=0,\quad u|_{t=0}=u_{0} \label{Eulerian for velocity}%
\end{equation}
are well understood in suitable functions spaces (see, for instance,
\cite{MajdaBert} and \cite{PL Lions} for a review of several results). One of
the classical results is the existence of solutions when $u_{0}$ is in the
Sobolev space $W^{1,2}$ and the uniqueness when the (scalar) vorticity
$\xi=\nabla^{\perp}\cdot u=\partial_{2}u_{1}-\partial_{1}u_{2}$ is bounded.

The case when the vorticity is a signed measure received also a lot
of attention, due to the interest in the evolution of vortex
structures like sheets or points of vorticity concentration. See
\cite{MajdaBert} for a review. Deep existence and stability results
for distributional   vorticities which do  not change sign have
been proved, first for a class of distributions which includes
vortex sheets but not vortex points, then also for point vortices
 (see among others \cite{Delhort}, \cite{Poup}). Uniqueness is an open
problem in all such cases. When the vorticity has variable sign and
is, for instance, pointwise distributed, even a reasonable
formulation of the Euler equations is missing. However, in the case
of point vortices, there are good reasons to replace the Eulerian
formulation with a Lagrangian one, based on the autonomous motion of
a finite number of point vortices.

The Lagrangian formulation of point vortex motion gives rise to a
finite dimensional ordinary differential equation, which is well
posed only for almost all initial configurations with respect to
Lebesgue measure. One can give explicit examples of initial
configurations such that different vortices coalesce in finite time.
In such a case the Lagrangian equations loose meaning. Perhaps a
proper Eulerian description could be meaningful also after the
coalescence time, but a rigorous formulation of this fact is not
known.

The purpose of this paper is to show that the previous pathology, namely the
existence of initial configurations which coalesce in finite time, is
prevented by the presence of suitable noise in the system. The point vortex
motion is well defined for all times and all initial configurations, under
suitable noise perturbations (which may be arbitrarily small). Let us describe
our aim in more detail.

As shortly recalled in the next section, Euler equations can be recast in
terms of vorticity as the system
\begin{equation}
\frac{\partial\xi}{\partial
t}+u\cdot\nabla\xi=0,\quad\xi|_{t=0}=\xi_{0},
\label{vorticity eq}%
\end{equation}%
\begin{equation}
u=-\nabla^{\perp}\Delta^{-1}\xi. \label{Biot Sav}%
\end{equation}
Concepts of weak solutions of the Euler equations are meaningful
even for distributional vorticity $\xi$, when $u$ is sufficiently
regular; square integrable is sufficient, see the theory of vortex
sheet solutions, where there are at least some existence theorems
(see \cite{MajdaBert}). The limit case when $\xi$ is the sum of
finite number of delta Dirac masses
\begin{equation}
\xi( .,t) =\sum_{i=1}^{n}\omega_{i}\delta_{x_{t}^{i}}
 \label{vorticity field}%
\end{equation}
is unfortunately too singular:\ the velocity field is not square
integrable (so even the weakest form of (\ref{Eulerian for
velocity}) is not meaningful) and its singularity coincides with the
delta Dirac points of the vorticity (so also (\ref{vorticity eq}) is
not meaningful). In spite of this, there are good arguments, based
on the limit of regular solutions supported around the ideal point
vortices \cite{March Pulv}, to accept that a certain finite
dimensional differential equation for the position of the point
vortices is the correct physical description of the evolution of
 $\xi $ in  (\ref{vorticity field}). The equations for the evolution
of the positions of point vortices have the
form%
\begin{equation}
\frac{dx_{t}^{i}}{dt}=\sum_{j\neq i}\omega_{j}K( x_{t}^{i}-x_{t}^{j}) ,\quad
i=1,...,n. \label{det point vort intro}%
\end{equation}
A few more details are explained in the next section  (see also \cite{Newton}).

If we call $X_{0}=( x_{0}^{1},...,x_{0}^{n}) $ the initial condition in
$\mathbb{R}^{2n}$ of the system of $n$ point vortices, a result of existence
and uniqueness for \emph{Lebesgue almost every} $X_{0}$ is known, see
\cite{March Pulv}. But there are examples of initial condition $X_{0}$ such
that vortices collide in finite time and a global solution does not exist.

The purpose of this research is to investigate the effect of a multiplicative
noise on the Euler equations, in the form
\begin{equation}
d\xi+u\cdot\nabla\xi dt+\sum_{k=1}^{N}\sigma_{k}( x) \cdot\nabla\xi\circ
d\beta_{t}^{k}=0,\quad\xi|_{t=0}=\xi_{0}, \label{stoch vorticity eq}%
\end{equation}
 where  $\sigma_{k}( x) $ are suitable  2d vector fields
and $\{ \beta^{k}_t\} _{k=1,...,N}$ are independent Brownian
motions. Note that   $u$ is again reconstructed from $\xi$ by
means of Biot-Savart law (\ref{Biot Sav}). Linear transport
equations are regularized by multiplicative noise, see
\cite{FlaGubPriola}: non-uniqueness phenomena of the deterministic
case disappear under the random perturbation. Our aim,  in
principle, is to prove a similar regularizing effect  for the
nonlinear problem (\ref{stoch vorticity eq}). However this is a very
difficult problem and at the moment we are not able to solve it. The
result we present in this work is in some sense a first  step and
concerns  the stochastic point vortex dynamics which corresponds to
equation (\ref{stoch vorticity eq}) and has the
form%
\begin{equation}
dx_{t}^{i}=\sum_{j\neq i}\omega_{j}K( x_{t}^{i}-x_{t}^{j}) dt+\sum_{k=1}%
^{N}\sigma_{k}( x_{t}^{i})\circ d\beta_{t}^{k},\quad i=1,...,n.
\label{stoch point eq introduction}%
\end{equation}
We prove that, under suitable assumptions on the fields
$\sigma_{k}( x) $ (those of Section \ref{section assumptions}), this
stochastic point vortex dynamics is globally well posed (in
particular coalescence of point vortices disappear) for \emph{all}
initial conditions. This is a stochastic improvement of the
deterministic theory, as in \cite{FlaGubPriola}.

For the improvements in well posedness of the linear transport equations
considered in \cite{FlaGubPriola}, it was sufficient to take constant fields
$\sigma_{k}$. Here, to avoid point vortex coalescence, we need
space-dependent  fields with a high degree of hypoellipticity, a technically
complex condition (see Hypothesis~\ref{r})  which however is generically
satisfied  (see Section~\ref{sec:generic}). Notice that in the trivial case when $\omega
_{j}=0$ for every $j=1,...,n$, system (\ref{stoch point eq introduction}) is
the so called $n$-point motion associated to the vector fields $\sigma_{k}$. A
priori this is highly degenerate; for this reason we need suitable
hypoellipticity conditions.

It would be trivial to improve the regularity of the deterministic system
(\ref{det point vort intro}) by adding independent Brownian motions to each
component, but this would not correspond to a Lagrangian point vortex
formulation of stochastic Euler equations.

Let us mention that Kotelenez~\cite[Chapter 8]{Kotelenez} considered a
similar stochastic perturbation of Euler equation and the associated point
vortex dynamics with the aim to understand the physical interest and properties of
the model. However he is not concerned with the regularizing properties of such kinds of noises.

\medskip
\noindent\textbf{Plan of the paper.} In Section~\ref{sec:two} we
explain the formal relation between  the stochastic  Euler
equation and the SDE for the point vortices;  this relation fixes
the form of the noise  allowed in the SDE. In Section~\ref{sec:main}
we state our hypothesis and prove the main result about
well-posedness of point vortex dynamics for all initial conditions.
In Section~\ref{sec:generic} we give a self-contained proof of the
fact that our hypothesis is  generically satisfied. Last we gather
in the appendix a series of well-known result on the density of the
law of SDEs under various conditions on the vector fields.

\medskip
\noindent\textbf{Acknowledgement.}  The authors would like to thank
Olivier Glass for an interesting discussion about genericity and
the transversality theorem.

\section{Stochastic 2D Euler equations and vortex dynamics}
\label{sec:two}

The aim of this section is to provide an heuristic motivation for system
(\ref{stoch point eq introduction}). A rigorous link between it and the
original stochastic Euler equations is not given in this work. It is already a
difficult problem in the deterministic case, where one of the best available
justifications is the result which states that unique solutions of Euler
equations corresponding to smoothing of the distributional initial vorticity,
converge in the weak sense of measures to (\ref{vorticity field}), see
\cite{March Pulv}. The result holds as far as point vortices do not coalesce.

Due to the difficulty of this subject, we do not aim here to give rigorous
results on the link between (\ref{stoch point eq introduction}) and
(\ref{stoch vorticity eq}), but only to provide an heuristic motivation. For
this reason, the rest of this section is not always written in rigorous terms
and we intentionally miss important details like functions spaces, regularity
of functions, etc.

\subsection{Deterministic case}

In dimension 2, the vorticity field $\xi=\nabla^{\perp}\cdot u=\partial
_{2}u_{1}-\partial_{1}u_{2}$ satisfies equation (\ref{vorticity eq}) where
$\xi_{0}=\nabla^{\perp}\cdot u_{0}$. If $\varphi$ (called potential) solves
the equation
\[
\Delta\varphi=-\xi
\]
then $u=\nabla^{\perp}\varphi$ satisfies $\xi=\nabla^{\perp}\cdot u$. Hence,
formally speaking, $u$ can be reconstructed from $\xi$ by the so called
Biot-Savart law (\ref{Biot Sav}).

Depending on the fact that we consider the equations in full space with square
integrable conditions at infinity, or on a torus with periodic boundary
conditions or other cases, one can make rigorous and uniquely defined the
previous procedure of reconstruction of $u$ from $\xi$. Let us work on the
2D-torus $\mathbb{T}= \mathbb{R}^{2}/( 2\pi\mathbb{Z}^{2}) $. Denote by $G$
the Green's function of $-\Delta$ on $\mathbb{T}$, then
$G\left(  x\right)  =\sum_{k\in\mathbb{Z}^{2}\backslash\{0\}}\left\Vert k\right\Vert
^{-2}e^{ik\cdot x}$.
The distribution $G$ is in fact a function, with a logarithmic divergence at
$x=0$, smooth everywhere else and satisfies (cf.
page 18 of \cite{March Pulv}):
\begin{align*}
C_{1}\log\vert x\vert-C_{3}  &  \leq G( x) \leq C_{2}\log\vert x\vert
+C_{3}\quad\\
\vert DG( x) \vert &  \leq C_{3}\vert x\vert^{-1},\quad\vert D^{2}G( x)
\vert\leq C_{3}\vert x\vert^{-2}%
\end{align*}
for all $x\in[ -\pi,\pi] ^{2}$, for some positive constants $C_{1},C_{2}%
,C_{3}$.

Given a periodic field $\xi$ with suitable regularity, a periodic
(distributional) solution of $\Delta\varphi=-\xi$ is given by
$
\varphi\left(  x\right)  = \int_{\mathbb{T}}  G\left(  x-y\right)  \xi\left(
y\right)  dy
$.
All other solutions differ by constants. The vector field $u=\nabla^{\perp
}\varphi$ is thus uniquely defined from $\xi$:%
\begin{equation}
u\left(  x\right)  =\int_{\mathbb{T}} K\left(  x-y\right)  \xi\left(
y\right)  dy \label{integral Biot Sav}%
\end{equation}
where
\[
K( x) =\nabla^{\perp}G( x) = \sum_{k= (k_{1},k_{2})\in\mathbb{Z}^{2}\backslash\{0\}}\frac{i
k^{\perp}}{\left\Vert k\right\Vert ^{2}}e^{ik\cdot x},\;\;\;  k^{\perp}=
(k_{2}, -k_{1}).
\]
Equation (\ref{integral Biot Sav}) is the integral form of Biot Savart law,
used throughout the paper. Since $u$ is the unique field such that $\xi
=\nabla^{\perp}\cdot u$, it is the velocity field associated to $\xi$.

A limit case of vorticity field is the distributional one given by
(\ref{vorticity field}) where $x_{t}^{i}$ is the position of point vortex $i$
at time $t$ and $\omega_{i}$ is its intensity (independent of time because
vorticity is just transported). This distributional field does not satisfy
Euler equations in the usual distributional sense: the nonlinear term
($\varphi$ is a smooth test function)%
\[
\int\xi(x,t)u(x,t)\cdot\nabla\varphi(x)dx
\]
is not well defined a priori, because the velocity field $u(x,t)$ associated
to (\ref{vorticity field}) is singular exactly at the delta Dirac points of
$\xi$. However, there are limit arguments, see \cite{March Pulv} which
rigorously motivate the following closed set of equations for the positions of
point vortices:
\begin{equation}
\frac{dx_{t}^{i}}{dt}=\sum_{j\neq i}\omega_{j}K(x_{t}^{i}-x_{t}^{j}),\quad
i=1,...,n\label{ODE}%
\end{equation}
Let us briefly explain this equation. Formally, point vortices are transported
by the fluid, hence they should satisfy $\frac{dx_{t}^{i}}{dt}=u(x_{t}^{i},t)$
where $u(x,t)$ is the velocity field associated to the vorticity field
(\ref{vorticity field}). If we put (\ref{vorticity field}) in
(\ref{integral Biot Sav}) we get
\[
u(x,t)=\sum_{j=1}^{n}\omega_{j}K(x-x_{t}^{j}).
\]
However, this expression is correct in all points $x$ different from the
vortex points themselves; notice that $K(x)$ diverges at $x=0$ so $u(x,t)$
should be properly interpreted at $x=x_{t}^{i}$. It turns out (see \cite{March
Pulv}) that the correct interpretation is
\begin{equation}
u(x_{t}^{i},t)=\sum_{j\neq i}\omega_{j}K(x_{t}^{i}-x_{t}^{j}%
).\label{velocity of point vortices}%
\end{equation}
Giving us eq.~(\ref{ODE}).

\subsection{Lack of full well-posedness}

Equations (\ref{ODE}) are not trivial since the vector field is
divergent when two particles collide; and there is no repulsion (but
also no attraction) when particles approach each other.
Nevertheless, in the periodic case, equations (\ref{ODE}) are
well-posed for almost every initial condition $X_{0}=(
x_{0}^{1},...,x_{0}^{n}) $ with respect to Lebesgue measure on the
product space. One can interpret this result by saying that the
system is almost surely well-posed when initial conditions are
chosen at random in a uniform way.

In the whole space the same result is known under the additional assumption that $\{
\omega_{i}\} _{i=1,...,n}$ satisfy $\sum_{i\in\pi}\omega_{i}\neq0$ for all
$\pi\subset\{ 1,...,n\} $.

The restriction to almost every initial condition is not just a weakness of
the technique:  there are counterexamples in the form of explicit initial configurations which
collide in finite time, see~\cite{March Pulv}.

\subsection{Stochastic case}

We shall prove well-posedness for every initial condition under a
suitable random perturbation. The idea behind is simply that the
noise makes the same effect of a randomization of the initial
conditions. However, there is a subtle difference between these two
randomizations. In the case of initial conditions, Lebesgue measure
in product space is used, thus the initial positions of particles
are perturbed independently one of each other. In the noise case, it
would be trivial to perturb each particle independently: this
produces immediately a strong regularization which ultimately would
imply well-posedness. Such kind of noise, however, has no meaning in
terms of Euler equation. What we want is a noisy version of Euler
equation which is solvable in the case of distributional point
vortex fields. When we write the noise at the level of the point
vortex dynamics, \emph{the noise is the same for all vortices} (but
since it is a space dependent noise, it is computed ad different
spatial points). This is in principle  a  source of difficulties.

\vskip 1mm We consider the stochastic equation with multiplicative
noise (\ref{stoch vorticity eq}) where $\sigma_{k}( x) $ are 2D
smooth vector fields and $\{ \beta^{k}_t\} _{k=1,...,N}$  are a
finite sequence  of independent Brownian motions defined  on a
stochastic basis $( \Omega,F,( F_{t}) ,P) $ (fixed once and for
all).

The associated dynamics
of $n$ point vortices is the stochastic system in $\mathbb{R}^{2n}$
\begin{equation}
dx_{t}^{i}=\sum_{j\neq i}\omega_{j}K( x_{t}^{i}-x_{t}^{j}) dt+\sum_{k=1}%
^{N}\sigma_{k}( x_{t}^{i}) \circ d\beta_{t}^{k},\quad x_{0}^{i}=x^{i} \label{SDE}%
\end{equation}
for $i=1,...,n$, with each single $x_{t}^{i}$ in $\mathbb{R}^{2}$ and where $\circ$ denote Stratonovich integration.

Let us formally show that the measure valued vorticity field
(\ref{vorticity field}), with $x_{t}^{i}$ given by a solution of equations
(\ref{SDE}), solve (\ref{stoch vorticity eq}). The weak form of the stochastic
Euler equation is (assuming $\mathrm{div}\,\sigma_{k}=0$, $k =1, \ldots, N$)
\[
d\langle\xi,\varphi\rangle=\langle\xi,u\cdot\nabla\varphi\rangle dt+\sum
_{k=1}^{N}\langle\xi,\sigma_{k}\cdot\nabla\varphi\rangle\circ d\beta_{t}^{k}%
\]
where we have denoted by $\varphi$ a smooth test function and by
$\langle.,.\rangle$ the dual pairing. Let $\xi$ be the distribution defined by
(\ref{vorticity field}). From It\^{o} formula for $\varphi( x_{t}^{i}) $ in
Stratonovich form we get
\begin{align*}
d\langle\xi,\varphi\rangle &  =\sum_{i=1}^{n}\omega_{i}d\varphi( x_{t}^{i})
=\sum_{i=1}^{n}\omega_{i}\nabla\varphi( x_{t}^{i}) \sum_{j\neq i}\omega_{j}K(
x_{t}^{i}-x_{t}^{j}) dt\\
&  +\sum_{i=1}^{n}\omega_{i}\sum_{k=1}^{N}\nabla\varphi( x_{t}^{i}) \sigma
_{k}( x_{t}^{i}) \circ d\beta_{t}^{k}.
\end{align*}
The second term is the right one:
\[
\sum_{k=1}^{N}\langle\xi,\sigma_{k}\cdot\nabla\varphi\rangle\circ d\beta
_{t}^{k}=\sum_{i=1}^{n}\omega_{i}\sum_{k=1}^{N}\sigma_{k}( x_{t}^{i})
\cdot\nabla\varphi( x_{t}^{i}) \circ d\beta_{t}^{k}.
\]
As to the first term, recall that the velocity field $u$ associated to point
vortices is given by (\ref{velocity of point vortices}). Then
\begin{align*}
\langle\xi,u\cdot\nabla\varphi\rangle &  =\sum_{i=1}^{n}\omega_{i}u( x_{t}%
^{i}) \cdot\nabla\varphi( x_{t}^{i})
 =\sum_{i=1}^{n}\sum_{j\neq i}\omega_{i}\omega_{j}K( x_{t}^{i}-x_{t}^{j})
\cdot\nabla\varphi( x_{t}^{i})
\end{align*}
and thus also the first term is the right one. This completes the heuristic proof.

A very important remark,  already mentioned in the previous section,  is that
the noise in this system is the same for all particles. Thus this is very
similar to the so called $n$-point motion of a single SDE. The regularizing
effect of the noise at the level of the $n$-point motion is a very non-trivial
fact. For instance, the easiest non-degenerate noise, namely the simple
additive one ($\sigma_{k}$ are 2D vectors)
\[
dx_{t}^{i}=\sum_{j\neq i}\omega_{j}K( x_{t}^{i}-x_{t}^{j}) dt+\sum_{k=1}%
^{N}\sigma_{k}d\beta_{t}^{k}%
\]
cannot yield any better result than the deterministic case, because the change
of variables  $y_{t}^{i}=x_{t}^{i}-\sum_{k=1}^{N}\sigma_{k}d\beta_{t}^{k}$
leads to the equation
\[
dy_{t}^{i}=\sum_{j\neq i}\omega_{j}K( y_{t}^{i}-y_{t}^{j}) dt
\]
which is exactly the deterministic one. If collapse happens for an initial
condition of this equation, the same initial condition produces collapse in
the previous SDE.

On the contrary, a strongly space dependent noise may contrast collapse. When
point vortices come close one to the other, the noise should be sufficiently
un-correlated (at small distances)\ to perturb in a generic way the motion of
the two vortices and produce the same effect of a random perturbation of
initial conditions.

\section{Main results and proofs}
\label{sec:main}
\subsection{Regularization by noise}\label{section assumptions}

We consider system (\ref{SDE}) on the 2D-torus
$\mathbb{T}=\mathbb{R}^{2}/( 2\pi\mathbb{Z}^{2}) $. It is a
$C^{\infty}$  compact connected Riemannian manifold with the smooth
metric induced by Euclidean metric of $\mathbb{R}^{2}$. In fact, for
simplicity, we may assume we work on the full space $\mathbb{R}^{2}$
and all the vector fields and functions are $2\pi $-periodic, but
sometimes the interpretation as a compact manifold is more
illuminating. We will consider a fixed choice $\{ \omega_{j}\}
_{j=1,...,n}\subset$ $\mathbb{R}$ of vortex intensities.

Let $\Gamma$ be the set of all $\left(  x^{1},...,x^{n}\right)  \in
{\mathbb{T}}^{n}$ such that $x^{i}=x^{j}$ for some $i\neq j$  ($\Gamma$  is
the union of the generalized diagonals of $\mathbb{T}^{n}$).
Let $\left\{  \sigma_{k}\right\}  _{k=1,...,N}$ be a finite number of smooth
vector fields on ${\mathbb{T}}$. Introduce the associated vector fields on
$\mathbb{T}^{n}$:
\begin{equation}
A_{\sigma_{k}}(x^{1},\ldots,x^{n})=A_{k}(x^{1},\ldots,x^{n})=\big(\sigma
_{k}(x^{1}),\ldots,\sigma_{k}(x^{n})\big)\label{dopo}%
\end{equation}
Recall that given vector fields $A,B$ in $\mathbb{R}^{m}$, their Lie bracket $[A,B]$ is the vector fields in
$\mathbb{R}^{m}$ defined by
\[
[ A,B] =( A\cdot\nabla) B-( B\cdot\nabla) A.
\]

We assume that $\left\{  \sigma_{k}\right\}  _{k=1,...,N}$ satisfies:

\begin{hypothesis}
\label{r}
\parbox{0cm}{}
\begin{enumerate}
\item The vector fields $\sigma_{k}$ are periodic, infinitely differentiable and $\mathrm{div}%
\sigma_{k}=0$ ;

\item (Bracket  generating condition) The vector space spanned by the vector fields
\[
A_{1},...,A_{N},\qquad[ A_{i},A_{j}] ,1\leq i,j\leq N,\qquad[ A_{i},[
A_{j},A_{k}] ] ,1\leq i,j,k\leq N,...
\]
at every point $x\in\Gamma^{c}$ is $\mathbb{R}^{2n}$.
\end{enumerate}
\end{hypothesis}

The second assumption is a form of H\"{o}rmander's condition. It
will ensures that the law at  any time $t>0$
of the solution of a
regularized stochastic  equation is absolutely continuous with respect to
Lebesgue measure if we start outside the diagonal $\Gamma$ (see also
the appendix).

Under this hypothesis we are able to prove the following result of
well-posedness of the dynamics for  all initial $n$-point
configurations.

\begin{theorem}
\label{main theorem}Under Hypothesis~\ref{r},
for all
$X_{0}=(x_{0}^{1},...,x_{0}^{n})\in{\mathbb{T}}^{n}\backslash\Gamma$
equation (\ref{SDE}) has one and only one global strong solution.
\end{theorem}

Before going to the proofs,
let us make some remarks on our
hypothesis. The bracket generation condition appears already in  the
papers \cite{BaxStroock} and \cite{DolgKK} which study the
asymptotic behavior of stochastic flows (among
other properties, the
Lyapunov exponents and the large deviations for additive
functionals). In \cite[Section 2]{DolgKK} it is stated that the
bracket generating condition in Hypothesis~\ref{r} is \emph{generic}
among smooth vector fields. We have been unable to find a proof of
this statement in the literature (both stochastic or more
dynamical-system oriented) and so in Section~\ref{sec:generic} we give a
self-contained and elementary argument which justifies  this
statement.

\begin{remark}
Instead of the noise taken from \cite{BaxStroock},\cite{DolgKK}, it is
natural to consider an infinite dimensional noise $W( x,t) =\sum_{k=1}%
^{\infty}\sigma_{k}( x) \beta_{t}^{k}$, for instance the isotropic
divergence free Brownian field which generates the isotropic
Brownian motion, see~\cite{BaxHarris},\cite{LeJan},\cite{LeJan Ray}.
Here we restrict our attention to finite dimensional noise for which
we already know results about absolute continuity of fixed time
marginals.  The interesting fact about infinite dimensional
noise is that it is easy to constructs explicit noises which are
``full'' outside $\Gamma$ and for which it is reasonable to expect
the validity of density results on the law.
\end{remark}

The proof of Theorem~\ref{main theorem} goes through  the study of a
regularized problem where the singular Biot-Savart kernel is
replaced by a smooth one. We first prove well-posedness for almost
every initial conditions (as in the deterministic setting) and then,
exploiting the existence   of a density for the law at fixed time,
we improve to well-posedness for all initial conditions.

\subsection{Regularization}

For sufficiently small $\delta$, let $G^{\delta}( x) $ be a smooth $2\pi
$-periodic function (hence bounded with its derivatives) such that, on $[
-\pi,\pi] ^{2}$,
\[
G^{\delta}( x) =G( x) \text{ for }\vert x\vert>\delta.
\]
Set $K^{\delta}=\nabla^{\perp}G^{\delta}$. We shall use the following
quantitative properties, beside smoothness:
\begin{align*}
C_{1}\log( \vert x\vert\vee\delta) -C_{3}  &  \leq G^{\delta}( x) \leq
C_{2}\log( \vert x\vert\vee\delta) +C_{3}\quad\\
\vert DG^{\delta}( x) \vert &  \leq C_{3}( \vert x\vert\vee\delta) ^{-1}%
,\quad\vert D^{2}G^{\delta}( x) \vert\leq C_{3}( \vert x\vert\vee\delta) ^{-2}%
\end{align*}
for all $x\in[ -\pi,\pi] ^{2}$, for some positive constants $C_{1},C_{2}%
,C_{3}$ (possibly different from those of the same inequalities for $G$ but independent of $\delta$).
We consider the regularized equation
\begin{equation}
dx_{t}^{i,\delta}=\sum_{j\neq i}\omega_{j}K^{\delta}( x_{t}^{i,\delta}%
-x_{t}^{j,\delta}) dt+\sum_{k=1}^{N}\sigma_{k}( x_{t}^{i,\delta})\circ d\beta
_{t}^{k}. \label{delta SDE}%
\end{equation}
which in It\^o form reads
\begin{equation}
dx_{t}^{i,\delta}=\sum_{j\neq i}\omega_{j}K^{\delta}( x_{t}^{i,\delta}%
-x_{t}^{j,\delta}) dt+\frac12 \sum_{k=1}^{N}(\sigma_{k}\cdot \nabla\sigma_{k})( x_{t}^{i,\delta}) dt +\sum_{k=1}^{N}\sigma_{k}( x_{t}^{i,\delta}) d\beta
_{t}^{k}. \label{delta SDE-Ito}%
\end{equation}

We immediately have: for every $X_{0}=( x_{0}^{1},...,x_{0}^{n}) \in
\mathbb{R}^{2n}$, there exists a unique strong solution $( X_{t}^{X_{0}})
_{t\geq0}$ to\ this equation in $\mathbb{R}^{2n}$. We even have a smooth
stochastic flow $\varphi_{t}^{\delta}$ on ${\mathbb{T}}^{n}$, see
\cite{Kunita}.

\subsection{Measure conservation}

Denote the divergence in $\mathbb{R}^{2}$ by $\mathrm{div}_{2}$\ and in
$\mathbb{R}^{2n}$ by $\mathrm{div}_{2n}$.We have
\[
\mathrm{div}_{2n}[ \sum_{j\neq i}\omega^{i}K^{\delta}( x_{i}-x_{j}) ]
_{i=1,...,n}=\sum_{i=1}^{n}\mathrm{div}_{2}[ \sum_{j\neq i}\omega^{i}%
K^{\delta}( x_{i}-x_{j}) ] =0
\]
because $K^{\delta}=\nabla^{\perp}G^{\delta}$. The same is true without
regularization. Moreover,
\[
\mathrm{div}_{2n}[ \sum_{k=1}^{N}\sigma_{k}( x^{i}) \beta_{t}^{k}]
_{i=1,...,n}=\sum_{i=1}^{n}\mathrm{div}_{2}\sigma_{k}( x^{i}) \beta_{t}^{k}=0
\]
because $\mathrm{div}_{2}\sigma_{k}=0$. By classical computations on the
smooth flow $\varphi_{t}^{\delta}$, one can check that its Jacobian
determinant is equal to one, as a consequence of the previous divergence free
conditions. Hence we have:

\begin{lemma}
\label{conservation}For every integrable function $h$ on ${\mathbb{T}}^{n}$,
we have
\[
\int_{{\mathbb{T}}^{n}}h( \varphi_{t}^{\delta}( X_{0}) ) dX_{0}=\int
_{{\mathbb{T}}^{n}}h( Y) dY.
\]

\end{lemma}

\subsection{Estimates about coalescence}

Denote by $[ x,y] _{t}$ the mutual quadratic covariation of two continuous
semimartingales $( x_{t}) _{t\geq0}$ and $( y_{t}) _{t\geq0}$. Denote by
$x^{\alpha}$, $\alpha=1,2$, the two components of an element of ${\mathbb{T}}$
in the coordinate frame coming from Euclidean coordinates. If $(
x_{t}^{i,\delta}) $ is a solution of equation (\ref{delta SDE}), then
\begin{align*}
&  [ ( x^{i,\delta}-x^{j,\delta}) ^{\alpha},( x^{i,\delta}-x^{j,\delta})
^{\beta}] _{t}
=\sum_{k=1}^{N}\int_{0}^{t}( \sigma_{k}^{\alpha}( x_{s}^{i,\delta})
-\sigma_{k}^{\alpha}( x_{s}^{j,\delta}) ) ( \sigma_{k}^{\beta}( x_{s}%
^{i,\delta}) -\sigma_{k}^{\beta}( x_{s}^{j,\delta}) ) ds.
\end{align*}

\begin{lemma}
Let $\varphi_{t}^{\delta}$, be the flow on ${\mathbb{T}}^{n}$ associated to
(\ref{delta SDE}). Let $g^{\delta}:{\mathbb{T}}^{n}\rightarrow\mathbb{R}$ be
the function
\[
g^{\delta}( X) =-\sum_{i,j=1...,n,\;i\neq j}G^{\delta}( x^{i}-x^{j}) ,\quad
X=( x^{1},...,x^{n}) .
\]
Then there exists a non negative integrable function $h$ on ${\mathbb{T}}^{n}$
such that
\[
E[ \sup_{t\in[ 0,T] }g^{\delta}( \varphi_{t}^{\delta}( X_{0}) ) ] \leq
g^{\delta}( X_{0}) +\int_{0}^{T}E[ h( \varphi_{t}^{\delta}( X_{0}) ) ] dt.
\]

\end{lemma}

\begin{proof}
From It\^{o} formula we have
\[
g^{\delta}( \varphi_{t}^{\delta}( X_{0}) ) =g^{\delta}( X_{0}) -\sum
_{i,j=1...,n,\;i\neq j}( I_{1a}^{ij}( t) +I_{1b}^{ij}( t) +I_{2}^{ij}( t)
+I_{3}^{ij}( t)+I_{4}^{ij}( t) )
\]
where
\[
I_{1a}^{ij}( t) =\int_{0}^{t}\sum_{i^{\prime}\neq i}\omega_{i^{\prime}%
}K^{\delta}( x_{s}^{i,\delta}-x_{s}^{i^{\prime},\delta}) \cdot\nabla
G^{\delta}( x_{s}^{i,\delta}-x_{s}^{j,\delta}) ds
\]%
\[
I_{1b}^{ij}( t) =  - \int_{0}^{t}\sum_{j^{\prime}\neq j}\omega_{j^{\prime}%
}K^{\delta}( x_{s}^{j,\delta}-x_{s}^{j^{\prime},\delta}) \cdot\nabla
G^{\delta}( x_{s}^{i,\delta}-x_{s}^{j,\delta}) ds
\]%
\[
I_{2}^{ij}( t) =\sum_{k=1}^{N}\int_{0}^{t}( \sigma_{k}( x_{s}^{i,\delta})
-\sigma_{k}( x_{s}^{j,\delta}) ) \cdot\nabla G^{\delta}( x_{s}^{i,\delta
}-x_{s}^{j,\delta}) d\beta_{s}^{k}%
\]%
\[
I_{3}^{ij}( t) =\frac{1}{2}\sum_{\alpha,\beta=1}^{2}\int_{0}^{t}\frac
{\partial^{2}G^{\delta}}{\partial x^{\alpha}\partial x^{\beta}}(
x_{s}^{i,\delta}-x_{s}^{j,\delta}) d[ ( x^{i,\delta}-x^{j,\delta}) ^{\alpha},(
x^{i,\delta}-x^{j,\delta}) ^{\beta}] _{s}
\]
\[
I_{4}^{ij}( t) =\frac12\int_{0}^{t}\nabla G^{\delta}(
x_{s}^{i,\delta}-x_{s}^{j,\delta})\cdot [ (\sigma_k\cdot \nabla\sigma_k)(x_s^{i,\delta})-(\sigma_k\cdot\nabla \sigma_k)(x_s^{j,\delta})] d s.
\]
Since $K^{\delta}=\nabla^{\perp}G^{\delta}$ and $\nabla^{\perp}G^{\delta}( x)
$ is orthogonal to $\nabla G^{\delta}( x) $, we have
\[
I_{1a}^{ij}( t) =\int_{0}^{t}\sum_{i^{\prime}\neq i,\;i^{\prime}\neq j}%
\omega_{i^{\prime}}K^{\delta}( x_{s}^{i,\delta}-x_{s}^{i^{\prime},\delta})
\cdot\nabla G^{\delta}( x_{s}^{i,\delta}-x_{s}^{j,\delta}) ds
\]%
\[
I_{1b}^{ij}( t) = - \int_{0}^{t}\sum_{j^{\prime}\neq j,\;j^{\prime}\neq
i}\omega_{j^{\prime}}K^{\delta}( x_{s}^{j,\delta}-x_{s}^{j^{\prime},\delta})
\cdot\nabla G^{\delta}( x_{s}^{i,\delta}-x_{s}^{j,\delta}) ds.
\]
Hence
\[
| I_{1a}^{ij}( t) | \leq C\int_{0}^{t}\sum_{i^{\prime}\neq i,\;i^{\prime}\neq
j}( | x_{s}^{i,\delta}-x_{s}^{i^{\prime},\delta}| \vee\delta) ^{-1}( |
x_{s}^{i,\delta}-x_{s}^{j,\delta}| \vee\delta) ^{-1}ds
\]%
\[
| I_{1b}^{ij}( t) | \leq C\int_{0}^{t}\sum_{j^{\prime}\neq j,\;j^{\prime}\neq
i}( | x_{s}^{j,\delta}-x_{s}^{j^{\prime},\delta}| \vee\delta) ^{-1}( |
x_{s}^{i,\delta}-x_{s}^{j,\delta}| \vee\delta) ^{-1}ds
\]%
\[
\sum_{i,j=1...,n,\;i\neq j}( | I_{1a}^{ij}( t) | +| I_{1b}^{ij}( t) | ) \leq
C\int_{0}^{t}h_{1}^{\delta}( \varphi_{s}^{\delta}( X_{0}) ) ds
\]
where
\[
h_{1}^{\delta}( X) =\sum_{\substack{i,j,l=1...,n\\i\neq j,\;l\neq i,\;l\neq
j}}( | x^{i}-x^{l}| \vee\delta) ^{-1}( | x^{i}-x^{j}| \vee\delta) ^{-1}%
\]
with $X=( x^{1},...,x^{n}) $. Setting
\[
h_{1}( X) =\sum_{\substack{i,j,l=1...,n\\i\neq j,\;l\neq i,\;l\neq j}}( |
x^{i}-x^{l}| ) ^{-1}( | x^{i}-x^{j}| ) ^{-1}%
\]
we have that $h_{1}$ is integrable over ${\mathbb{T}}^{n}$, and $h_{1}%
^{\delta}( X) \leq h_{1}( X) $ for all $X\in{\mathbb{T}}^{n}$. Moreover, By
BDG inequality and the smoothness of $\sigma_{k}$ we have
\begin{align*}
E\left[  \sup_{t\in\left[  0,T\right]  }\left|  I_{2}^{ij}\left(  t\right)
\right|  \right]   &  \leq CE\left[  \left[  I_{2}^{ij},I_{2}^{ij}\right]
^{1/2} _{T}\right] \\
&  \leq C E \Big [ \Big(\int_{0}^{T}\left(  \left|  x_{s}^{i,\delta}%
-x_{s}^{j,\delta}\right|  \vee\delta\right)  ^{-2}\left|  x_{s}^{i,\delta
}-x_{s}^{j,\delta}\right|  ^{2}ds \Big)^{1/2} \Big]
\end{align*}
hence
$
\sum_{i,j=1...,n,\;i\neq j}E[ \sup_{t\in[ 0,T] }| I_{2}^{ij}( t) | ] \leq C
$.
Finally,
\[
 \sup_{t\in [  0,T ]  }  |  I_{3}^{ij}(  t)  |  \leq C\int_{0}^{T}(  |
x_{s}^{i,\delta}-x_{s}^{j,\delta}|  \vee\delta)  ^{-2}|
x_{s}^{i,\delta}-x_{s}^{j,\delta}|  ^{2}ds
\]
and
\[
 \sup_{t\in [  0,T ]  } |  I_{4}^{ij}(  t)  |  \leq C\int_{0}^{T}(  |
x_{s}^{i,\delta}-x_{s}^{j,\delta}|  \vee\delta)  ^{-1}|
x_{s}^{i,\delta}-x_{s}^{j,\delta}| ds
\]
so again
$
\sum_{i,j=1...,n,\;i\neq j}(| I_{3}^{ij}( t) |+| I_{4}^{ij}( t) |) \leq C
$.
Summarizing,
\[
E\left[  \sup_{t\in\left[  0,T\right]  }g^{\delta}\left(  \varphi_{t}^{\delta
}\left(  X_{0}\right)  \right)  \right]  \leq g^{\delta}\left(  X_{0}\right)
\]%
\[
+\sum_{i,j=1...,n,\;i\neq j}E [  \sup_{t\in [  0,T ]  } (
 |  I_{1a}^{ij} (  t )   |  +  |  I_{1b}^{ij} (
t )   |  + |  I_{2}^{ij} (  t )   |  + |
I_{3}^{ij} (  t )   |  + |
I_{4}^{ij} (  t )   |   )   ]
\]%
\[
\leq g^{\delta}\left(  X_{0}\right)  + C\int_{0}^{T}E\left[  h_{1}\left(
\varphi_{s}^{\delta}\left(  X_{0}\right)  \right)  \right]  ds+C.
\]
The proof is complete.
\end{proof}

\begin{corollary}
There is a constant $C>0$ such that for every $\delta>0$%
\[
E\int_{{\mathbb{T}}^{n}}\sup_{t\in[ 0,T] }[ g^{\delta}( \varphi_{t}^{\delta}(
X_{0}) ) ] dX_{0}\leq C<\infty.
\]

\end{corollary}

\begin{proof}
From the previous lemma and lemma \ref{conservation} we have
\begin{align*}
E\int_{{\mathbb{T}}^{n}}\sup_{t\in[ 0,T] }[ g^{\delta}( \varphi_{t}^{\delta}(
X_{0}) ) ] dX_{0}%
& \leq\int_{{\mathbb{T}}^{n}}g^{\delta}( X_{0}) dX_{0}+E\int_{0}^{T}[
\int_{{\mathbb{T}}^{n}}h( \varphi_{t}^{\delta}( X_{0}) ) dX_{0}] dt\\
&  =\int_{{\mathbb{T}}^{n}}g^{\delta}( X_{0}) dX_{0}+T\int_{{\mathbb{T}}^{n}%
}h( Y) dY.
\end{align*}
The proof is complete.
\end{proof}

\begin{corollary}
There exists a constant $C>0$ such that for all $\varepsilon,\delta>0$ we
have
\[
( \lambda_{{\mathbb{T}}^{n}}\otimes P) ( \inf_{i\neq j}\inf_{t\in[ 0,T] }|
x_{t}^{i,\delta}-x_{t}^{j,\delta}| \leq\varepsilon) \leq-\frac{C}{\log(
\varepsilon\vee\delta) }.
\]

\end{corollary}

\begin{proof}
We have%
\begin{align*}
g^{\delta}( \varphi_{t}^{\delta}( X_{0}) )  &  =-\sum_{i,j=1...,n,\;i\neq
j}G^{\delta}( x_{t}^{i,\delta}-x_{t}^{j,\delta})\\
&  \geq-\sum_{i,j=1...,n,\;i\neq j}C_{2}\log( \vert x_{t}^{i,\delta}%
-x_{t}^{j,\delta}\vert\vee\delta) -\frac{n( n-1) }{2}C_{3}.
\end{align*}
Given $\varepsilon,\delta>0$, smaller than one, if
$
\inf_{i\neq j}\inf_{t\in[ 0,T] }\vert x_{t}^{i,\delta}-x_{t}^{j,\delta}%
\vert\leq\varepsilon
$
namely if there are $t_{0}\in[ 0,T] $ and $i_{0}\neq j_{0}$ such that $\vert
x_{t_{0}}^{i_{0},\delta}-x_{t_{0}}^{j_{0},\delta}\vert\leq\varepsilon$ then%
\[
g^{\delta}( \varphi_{t_{0}}^{\delta}( X_{0}) ) \geq-C_{2}\log( \varepsilon
\vee\delta) -\frac{n( n-1) }{2}( C_{2}\log2\pi\sqrt{2}+C_{3})
\]
(we have used the fact that $\log( \vert x_{t}^{i,\delta}-x_{t}^{j,\delta
}\vert\vee\delta) \leq\log2\pi\sqrt{2}$) and thus
\[
\sup_{t\in[ 0,T] }g^{\delta}( \varphi_{t}^{\delta}( X_{0}) ) \geq-C_{2}\log(
\varepsilon\vee\delta) -C_{4}n^{2}.
\]
By Chebyshev inequality (notice that $-C_{2}\log( \varepsilon\vee\delta) >0$)
and the previous lemma,
\begin{align*}
&  ( \lambda_{{\mathbb{T}}^{n}}\otimes P) ( \inf_{i\neq j}\inf_{t\in[ 0,T]
}\vert x_{t}^{i,\delta}-x_{t}^{j,\delta}\vert\leq\varepsilon)\\
&  \leq( \lambda_{{\mathbb{T}}^{n}}\otimes P) ( \sup_{t\in[ 0,T] }g^{\delta}(
\varphi_{t}^{\delta}( X_{0}) ) +C_{4}n^{2}\geq-C_{2}\log( \varepsilon
\vee\delta) )\\
&  \leq-\frac{C_{5}n^{2}}{\log( \varepsilon\vee\delta) }.
\end{align*}
The proof is complete.
\end{proof}

\begin{remark}
The function $h$ and the constants $C$ of the previous statements depend on
the number $n$ of point vortices and the time interval $[ 0,T] $.
\end{remark}

\subsection{Well-posedness for Lebesgue almost every initial condition}

As a first consequence of the previous estimates, we prove the same result of
the deterministic case.

Recall that $\Gamma$ is the singular set in ${\mathbb{T}}^{n}$ for the vortex
dynamics, namely the set of all $( x^{1},...,x^{n}) \in{\mathbb{T}}^{n}$ such
that $x^{i}=x^{j}$ for some $i\neq j$. The drift of equation (\ref{SDE}) is
well defined only on $\Gamma^{c}$. Thus the notion of strong solution $(
X_{t}) _{t\geq0}$ to equation (\ref{SDE}) is the classical one for SDEs with
the addition of the condition that%
\[
P( X_{t}\in\Gamma^{c}\text{ for all }t\geq0) =1.
\]

\begin{theorem}
\label{teo 1}For Lebesgue almost every $X_{0}=( x_{0}^{1},...,x_{0}^{n})
\in{\mathbb{T}}^{n}$, equation (\ref{SDE}) has one and only one global strong solution.
\end{theorem}

\begin{proof}
Denote by $\Gamma_{\delta}$ the closed $\delta$-neighbor of $\Gamma$ in
${\mathbb{T}}^{n}$. Given $X_{0}\in\Gamma_{\delta}^{c}$, denote by
$\tau_{X_{0}}^{\delta}( \omega) $ the first instant when $\varphi_{t}^{\delta
}( X_{0}) \in\Gamma_{\delta}$ and set it equal to $+\infty$ if this fact never
happens. We have $P\left(  \tau_{X_{0}}^{\delta}>0\right)  =1$ by continuity
of trajectories. The solution $\varphi_{t}^{\delta}\left(  X_{0}\right)  $, on
the random interval $\left[  0,\tau_{X_{0}}^{\delta}\right]  $, is also the
unique solution $\left(  X_{t}\right)  $ of equation (\ref{SDE}).  Thus
$\tau_{X_{0}}^{\delta}( \omega) $ is also the first instant when $X_{t}%
\in\Gamma_{\delta}$. Set%
\[
\tau_{X_{0}}( \omega) =\sup_{\delta\in( 0,1) }\tau_{X_{0}}^{\delta}( \omega)
.
\]
By localization, we have a unique solution of equation (\ref{SDE}) on
$[0,\tau_{X_{0}})$. If we prove that $P( \tau_{X_{0}}=\infty) =1$ for a.e.
$X_{0}$, we have proved the theorem. Given $T>0$ and $\delta^{\ast}>0$, it is
sufficient to prove that for Lebesgue a.e. $X_{0}\in\Gamma_{\delta^{\ast}}%
^{c}$, we have $P( \tau_{X_{0}}\geq T) =1$.

Form the last corollary of the previous section we know that%
\[
( \lambda_{{\mathbb{T}}^{n}}\otimes P) ( \inf_{i\neq j}\inf_{t\in[ 0,T] }|
x_{t}^{i,\delta}-x_{t}^{j,\delta}| \leq\delta) \leq-\frac{C}{\log\delta}.
\]
Let $\{ \delta_{k}\} _{k\in\mathbb{N}}$ be a sequence such that the series
$\sum_{k=1}^{\infty}\frac{1}{\log\delta_{k}}$ converges. Take it such that
$\delta_{k}\leq\delta^{\ast}$ for all $k\in\mathbb{N}$. By Borel-Cantelli
lemma, there is a measurable set $N\subset{\mathbb{T}}^{n}\times\Omega$ with
$( \lambda_{{\mathbb{T}}^{n}}\otimes P) ( N) =0$, such that for all $(
X_{0},\omega) \in N^{c}$ there is $k_{0}=k_{0}( X_{0},\omega) \in\mathbb{N}$
such that for all $k\geq k_{0}( X_{0},\omega) $%
\[
\inf_{i\neq j}\inf_{t\in[ 0,T] }| \varphi_{t}^{i,\delta_{k}}( X_{0}) ( \omega)
-\varphi_{t}^{j,\delta_{k}}( X_{0}) ( \omega) | >\delta_{k}%
\]
where $\varphi_{t}^{i,\delta_{k}}( X_{0}) $ is $x_{t}^{i,\delta_{k}}$ when the
initial condition is $X_{0}$. If we restrict ourselves to $( X_{0},\omega) \in
N^{c}\cap( \Gamma_{\delta^{\ast}}^{c}\times\Omega) $, the previous statement
implies $\tau_{X_{0}}^{\delta_{k}}( \omega) \geq T$ for all $k\geq k_{0}(
X_{0},\omega) $. This implies
\[
\tau_{X_{0}}( \omega) \geq T.
\]
We have proved this inequality for all $( X_{0},\omega) \in N^{c}\cap(
\Gamma_{\delta^{\ast}}^{c}\times\Omega) $, namely for almost every $(
X_{0},\omega) $ in $\Gamma_{\delta^{\ast}}^{c}\times\Omega$. By Fubini-Tonelli
theorem, there is a measurable set $\Delta\subset\Gamma_{\delta^{\ast}}^{c}$
with $\lambda_{{\mathbb{T}}^{n}}( \Delta) =1$, such that for all $X_{0}%
\in\Delta$ we have $\tau_{X_{0}}( \omega) \geq T$ with $P$-probability one.
The proof is complete.
\end{proof}

\subsection{Improvement due to the noise}

We may now prove our main result,
Theorem~\ref{main theorem}.

\begin{proof}  (Theorem~\ref{main theorem})
Given $X_{0}\in\Gamma^{c}$, a strong unique local solution on $[0,\tau_{X_{0}%
})$ exists ($\tau_{X_{0}}$ defined in the proof of theorem \ref{teo 1}). Let
us add a point $\Delta$ to ${\mathbb{T}}^{n}$ and set $\varphi_{t}( X_{0})
=\Delta$ for $t\geq\tau_{X_{0}}$, where $\tau_{X_{0}}<\infty$. The family of
processes $\varphi_{t}( X_{0}) $, $X_{0}\in\Gamma^{c}$, so defined, lives in
$\Gamma^{c}\cup\Delta$ for positive times and is Markov.  Then
\[
P( \varphi_{[ \varepsilon,T] }( X_{0}) \in\Gamma^{c}) \newline=\int
_{\Gamma^{c}\cup\{ \Delta\} }P( \varphi_{[ 0,T-\varepsilon] }( Y) \in
\Gamma^{c}) \mu_{\varphi_{\varepsilon}( X_{0}) }( dY)
\]
where $\{  \varphi_{[ \varepsilon,T] }( X_{0}) \in\Gamma^{c} \}  = \{
\omega\in\Omega\; :\; \varphi_{t} (X_{0})(\omega) $ $\in\Gamma^{c},$ for any
$\; t \in[ \varepsilon,T] \}$ and $\mu_{\varphi_{\varepsilon}( X_{0}) }$ is
the law of $\varphi_{\varepsilon}( X_{0}) $. Denote by $N\subset{\mathbb{T}%
}^{n}$ a measurable set such that all initial conditions in $N^{c}$ give rise
to a well posed Cauchy problem. We have
\[
P( \varphi_{[ 0,T-\varepsilon] }( Y) \in\Gamma^{c}) =1
\]
for all $Y\in N^{c}$. Then%
\begin{align*}
&  P( \varphi_{[ \varepsilon,T] }( X_{0}) \in\Gamma^{c}) \geq\int_{N^{c}}P(
\varphi_{[ 0,T-\varepsilon] }( Y) \in\Gamma^{c}) \mu_{\varphi_{\varepsilon}(
X_{0}) }( dY)\\
&  =1-\mu_{\varphi_{\varepsilon}( X_{0}) }( N) .
\end{align*}

Now, assume $X_{0}\in\Gamma_{\delta^{\ast}}^{c}$ for some $\delta^{\ast}>0$.
We have, for all $\delta\in(0,\delta^{\ast})$,%
\begin{align*}
\mu_{\varphi_{\varepsilon}(X_{0})}(N) &  =P(\varphi_{\varepsilon}(X_{0})\in
N)\\
&  =P(\varphi_{\varepsilon}(X_{0})\in N,\tau_{X_{0}}^{\delta}>\varepsilon
)+P(\varphi_{\varepsilon}(X_{0})\in N,\tau_{X_{0}}^{\delta}\leq\varepsilon)\\
&  \leq P(\varphi_{\varepsilon}^{\delta}(X_{0})\in N,\tau_{X_{0}}^{\delta
}>\varepsilon)+P(\tau_{X_{0}}^{\delta}\leq\varepsilon)\\
&  \leq P(\varphi_{\varepsilon}^{\delta}(X_{0})\in N)+P(\tau_{X_{0}}^{\delta
}\leq\varepsilon)\\
&  =P(\tau_{X_{0}}^{\delta}\leq\varepsilon).
\end{align*}
To say that $P(\varphi_{\varepsilon}^{\delta}(X_{0})\in N)=0$ we have used two
facts: $N$ is Lebesgue-negligible, the law of $\varphi_{t}^{\delta}(X_{0})$ on
${\mathbb{T}}^{n}$ is absolutely continuous with respect to Lebesgue measure,
for each $X_{0}\in\Gamma^{c}$, $\delta>0$, $t>0$. The latter property is a
consequence of the second main assumption of section \ref{section assumptions}%
. See the appendix~\ref{sec:remarks} for details; we apply, in particular,
Theorem~\ref{Ito}.

Just by continuity of trajectories, we have $\lim_{\varepsilon\rightarrow
0}P(\tau_{X_{0}}^{\delta}\leq\varepsilon)=0$. Hence%
\[
\lim_{\varepsilon\rightarrow0}P(\varphi_{\lbrack\varepsilon,T]}(X_{0}%
)\in\Gamma^{c})=1.
\]
The family of events $(\varphi_{\lbrack\frac{1}{n},T]}(X_{0})\in\Gamma^{c})$
is decreasing in $n$, hence $P(\varphi_{\lbrack\frac{1}{n},T]}(X_{0})\in
\Gamma^{c})$ is also decreasing. This implies $P(\varphi_{\lbrack
\varepsilon,T]}(X_{0})\in\Gamma^{c})=1$ for every $\varepsilon$ giving
$P(\varphi_{\lbrack0,T]}(X_{0})\in\Gamma^{c})=1$.
\end{proof}

\subsection{Variations on the result  }

Let us complete this section with a variant of the previous result.
Next section is devoted to the proof that the assumptions of Section
\ref{section assumptions} are generic. But in fact we prove more,
namely that generically it happens that the vector fields
$A_{1},...,A_{N}$ themselves span $\mathbb{R}^{2n}$ at every point
$x\in\Gamma^{c}$ is $\mathbb{R}^{2n}$ (no Lie brackets are needed).
It is thus meaningful to investigate the problem under the following
assumption: $\left\{  \sigma_{k}\right\}  _{k=1,...,N}$ satisfies:\

\begin{hypothesis}
\label{r 2}
\parbox{0pt}{}
\begin{enumerate}
\item $\sigma_{k}$ are periodic, $C^2$ and
$\mathrm{div}\sigma_{k}=0$

\item the vector space spanned by the vector fields $A_{1},...,A_{N}$ at every
point $x\in\Gamma^{c}$ is $\mathbb{R}^{2n}$.
\end{enumerate}
\end{hypothesis}

Item 2 of this assumption is more restrictive than the corresponding
one of Hypothesis~\ref{r}, but smoothness of the fields is no more
needed. Under
Hypothesis~\ref{r 2}, we still have that the law of $\varphi_{t}^{\delta}%
(X_{0})$ on ${\mathbb{T}}^{n}$ (see the notations of the previous sections) is
absolutely continuous with respect to Lebesgue measure, for each $X_{0}%
\in\Gamma^{c}$, $\delta>0$, $t>0$; we use now Corollary~\ref{Corollary 2 Appendix}.
Let us also remark that the proof of absolute
continuity of the law under this assumption is more elementary than under
Hypothesis~\ref{r}. For all these reasons it is worth to state also the
following variant of Theorem \ref{main theorem} (the proof is the same, based
on the previous remark on the absolute continuity).

\begin{theorem}
Under Hypothesis~\ref{r 2}, for all $X_{0}=(x_{0}^{1},...,x_{0}^{n}%
)\in{\mathbb{T}}^{n}\backslash\Gamma$ equation (\ref{SDE}) has one and only
one global strong solution.
\end{theorem}

\section{Generic $n$-point motions are hypoelliptic}
\label{sec:generic}

In this section we are going to provide a self-contained proof of the following statement which stipulates that $n$-point motions satisfying our assumptions are \emph{generic}.

\begin{theorem}
\label{th:npoint-genericity}
For all $M > 2 n$ there exists a residual set $Q \subset(C^{\infty})^{2 n M}$
such that  for every $(f_{a, i})_{a = 1, \ldots, M, i = 1, \ldots, 2 n} \in Q$
we have  $\operatorname{span}  \{A_{f_{a, i}} (x)\}_{a = 1, \ldots, M, i = 1,
\ldots, 2 n}  =\mathbb{R}^{2 n}$ for every $x \in\Gamma^{c}$.
\end{theorem}

The parametric Sard's theorem (or Thom's transversality theorem) are  general
tools which allow to prove generic properties of geometric objects (see, for
instance~\cite{GuPol}); here we intend properties valid for almost all objects
with respect to some natural measure, or valid in a residual set (countable
intersection of open dense sets). For some applications of transversality to control theory the reader could look at~\cite{KSD} where some interesting examples are worked out in a quite explicit setting.

Here we consider an easy version of the theorem which
we are going to use to show that for a sufficiently large but otherwise
generic family of vector fields on the torus, the associated $n$-point motion
generate, as a Lie algebra, the full tangent space in each point outside the diagonals.
The basic idea is simple,  unfortunately we haven't found a reference to an equivalent statement which do not require some background in differential topology to be understood, so
we provide here the easy proof for reader sake.

\begin{theorem}
\label{th:transv}  Let $\ell< n$ and let $X \subset\mathbb{R}^{\ell}$ and $Y
\subset\mathbb{R}^{m}$ be open sets.  Consider a $C^{1}$ function $F : X
\times Y \rightarrow\mathbb{R}^{n}$  and assume that $0$ is a \emph{regular
value} for $F$ (i.e. the Jacobian matrix $D F  (x, y)$ is surjective for all
$(x, y) \in F^{- 1} (0)$). Then the set  $\mathcal{X}_{y} =  \{x \in X : F (x,
y) = 0\}$ is empty for Lebesgue almost every  $y \in Y$.
\end{theorem}

\begin{proof}
 Consider a point $(x_0, y_0) \in F^{- 1} (0)$. By the implicit function theorem
 and the fact that $\tmop{dim} \tmop{Im} DF (x_0, y_0) = n$ there exists open sets
$U, V$ such that  $x_0 \in U \subset X$ and $y_0 \in V \subset Y$ and for which
the set $(U \times V) \cap F^{- 1} (0)$ is the graph of a $C^1$ function defined on the open set
$W \subset \mathbb{R}^{\ell + m - n}$. In particular there exists an differentiable
homeomorphism $\psi : W \rightarrow
 (U \times V) \cap F^{- 1} (0) \subset \mathbb{R}^{\ell} \times
 \mathbb{R}^m$. Let $\pi_2 : X \times Y \rightarrow
 Y$ be the canonical projection over the second factor and consider the differentiable map $\pi_1 \circ \psi : W \rightarrow
 \mathbb{R}^m$: the image $W' = \pi_1 \circ \psi (W) \subset V$ of $W$ is a set of dimension
 $m + \ell - n < m$ and then of zero measure with respect to the $m$-dimensional Lebesgue measure.
 From an covering of $F^{- 1}
 (0)$ by open sets of the form $U \times V$ we can then obtain a finite subcover and form the union of the associated $W'$s which we call
 $\tilde{Y} \subset \mathbb{R}^m$ and which is still a negligible set. Now
 $\tilde{Y}$ contains exactly the points $y \in Y$ such that there exists $x \in X$ for which
  $F (x, y) = 0$, so we conclude that $y \in Y \backslash \tilde{Y} \Rightarrow
 \mathcal{X}_y = \emptyset$.
\end{proof}

\medskip

For every $d \in\mathbb{N}$ define the finite-dimensional real vector space
$\mathcal{F}_{d}$ of the solenoidal  vector fields $f : \mathbb{T}
\rightarrow\mathbb{R}^{2}$ on the torus $\mathbb{T}$ of the form
$
f (x) = \sum_{|k_{1} |,|k_2| \leqslant d}  k^{\bot} e^{i
\langle k, x \rangle} \hat{f} (k)
$.
Fix $n \geqslant1$ and recall that $\Gamma=\{(x^{1}, \ldots, x^{n}) : \min_{i
\neq j} |x^{i} - x^{j} | = 0\} \subseteq\mathbb{T}^{n}$.  Let $D =
\dim(\mathcal{F}_{d}) = (2d + 1)^{2}$.
According to \eqref{dopo}, for every vector field $f : \mathbb{T}
\rightarrow\mathbb{R}^{2}$ on $\mathbb{T}$ define $A_{f}$ as the vector field
on $\mathbb{T}^{n}$ given by $A_{f} (x) = (f (x^{1}), \ldots, f (x^{n}))$.

To understand how to use Theorem~\ref{th:transv}  to prove
genericity results for vector fields let us give a simple result
which helps in understanding the main argument.

\begin{lemma}
Let $d \in\mathbb{N}$. Fix a point $x \in\mathbb{T}^{2 n}$ and assume that
there exist vector fields $h_{1}, \ldots, h_{2 n} \in\mathcal{F}_{d}$ such
that the family $\{A_{h_{i}}  (x)\}_{i = 1, \ldots, 2 n}$ span all
$\mathbb{R}^{2 n}$. Then the same is true for Lebesgue almost every vector
fields $\sigma_{1}, \ldots, \sigma_{2 n} \in\mathcal{F}_{d}$ (i.e., we have
that  $\{A_{\sigma_{i}} (x)\}_{i = 1, \ldots, 2 n}$ spans all $\mathbb{R}^{2}$,
for  a.e. $\sigma_{1}, \ldots, \sigma_{2 n} \in\mathcal{F}_{d}$).
\end{lemma}

\begin{proof}
Consider the map $\Psi:  \mathcal{F}_{d}^{2 n} \times
\mathbb{R}^{2n} \rightarrow\mathbb{R}^{2 n}\times \mathbb{R}$
\[
\Psi(\sigma_{1}, \ldots, \sigma_{2 n}, u) = (u_{1} A_{\sigma_{1}} (x)
+ \cdots+ u_{2 n} A_{\sigma_{2 n}} (x) , \sum_{i=1}^{2n} u_i^2 -1).
\]
If we show that $\operatorname{rank}  (D \Psi(\sigma_{1}, \ldots, \sigma_{2
n}, u)) = 2  n+1$ for every $(\sigma_{1}, \ldots, \sigma_{2 n}, u)\in \Psi^{-1}(0,0)$  then we have
that the set  of vector fields $\sigma_{1}, \ldots, \sigma_{2 n}$ such that
$\Psi(\sigma_{1}, \ldots,  \sigma_{2 n}, u) = 0$ for some $u \in
\mathbb{R}^{2n}$ such that  $|u|=1$ is of zero Lebesgue measure  which allows us to conclude that
$\{A_{\sigma_{i}} (x)\}_{i = 1, \ldots, 2 n}$ span all
$\mathbb{R}^{2 n}$ for almost every choice  $\sigma$ in
$\mathcal{F}_{d}^{2n}$. Let us then compute $D \Psi$.  Taking
$q = (q_1, q_2)
\in\mathbb{Z}^2$, $ |q_1| \le d,\; |q_2| \le d$,
we have
\[
D_{\widehat{\sigma_{}}_{i} (q)} \Psi(\sigma_{1}, \ldots, \sigma_{2 n}, u) = (
u_{i} D_{\widehat{\sigma_{}}_{i} (q)} A_{\sigma_{i}} (x),0) = (u_{i}   ( q^{\bot} e^{i \langle q, x^{1} \rangle}, \ldots,q^{\bot} e^{i \langle q, x^{n} \rangle}), 0)
\]
and
\[
D_{u_{i}} \Psi(\sigma_{1}, \ldots, \sigma_{2 n}, u) = (A_{\sigma_i}(x),2 u_i)
\]
Since $|u|^2=1$ at least one of the components $u_i \neq 0$ so that the span of these vectors contains all the elements of the form
$
(h (x^{1}), \ldots, h (x^{n})) =  (A_{h} (x),\rho)
$
for arbitrary $h \in\mathcal{F}_{d}$ and $\rho\in\mathbb{R}$.
But by assumption these vectors span $\mathbb{R}^{2n}\times\mathbb{R}$ so we can conclude
using Theorem~\ref{th:transv}.
\end{proof}

\medskip

Let us now return to our main aim: build families of vector fields spanning
$\mathbb{R}^{2n}$ in each point of $\Gamma^{c}$.
The neighborhoods of the diagonals $\Gamma$ are source of troubles so for the
moment let us restrict to the consideration of $n$-point configurations
belonging to an open set $G\subset\mathbb{T}^{2n}$ away from them.

Consider the map $\Phi: \mathcal{F}_{d}^{2 n M} \times G \times
(\mathbb{R}^{2n})^{M} \rightarrow(\mathbb{R}^{2 n}\times \mathbb{R})^{M}$ given by
\[
\Phi(F, x,U)  = ( (\sum_{i = 1}^{2 n} u_{1, i} A_{f_{1, i}} (x),|u_1|^2-1), \cdots, (\sum_{i =
1}^{2 n} u_{M, i} A_{f_{M, i}} (x),|u_M|^2-1)) .
\]
where $F = (f_{1, 1}, \ldots f_{1, 2 n}, \ldots, f_{M, 2 n}) \in\mathcal{F}%
_{d}^{2 n M}$ and $U = (u_{1}, \ldots, u_{M}) \in(
{\mathbb{R}^{2n}_0})^{M}$. Then
\[
D \Phi(F, x, U) : \mathbb{R}^{2 nMD} \times\mathbb{R}^{2 n} \times
\mathbb{R}^{2 n M} \rightarrow(\mathbb{R}^{2 n}\times\mathbb{R})^{M}
\]
The various components
of the Jacobian matrix   are given by (we denote by $\mathbb{I}_{a =
b}$
the indicator
function)
\[
(D_{u_{a, i}} \Phi(F, x, U))_{b} =\mathbb{I}_{a = b} (A_{f_{a, i}}
(x),2 u_{a,i}),
\]
\[
(D_{\hat{f}_{a, i} (q)} \Phi(F, x, U))_{b} =\mathbb{I}_{a = b} (u_{a, i}
D_{\hat{f}_{a, i} (q)} (f_{a, i} (x^{1}), \ldots, f_{a, i} (x^{n})),0)
\]
\[
=\mathbb{I}_{a = b} (u_{a, i} (q^{\bot} e^{i \langle q_{}, x^{1} \rangle}, \ldots,
q^{\bot} e^{i \langle q_{}, x^{n} \rangle}),0)
\]
\[
(D_{x^{i}} \Phi(F, x, U))_{a} =  \sum_{j = 1}^{2 n} (u_{a,
j}  (f_{a, j} (x^{1}), \ldots, D_{x^{i}} f_{a, j} (x^{i}), \ldots, f_{a, j}
(x^{n})),0)
\]
where $a, b = 1, \ldots, M$. The image of $D \Phi(F, x, U)$ contains then
vectors $v$ of the form
\[
v_{b} = \sum_{ a,i,q} \lambda_{a,i, q} (D_{\hat{f}_{a, i} (q)} \Phi(F, x, U))_{b}
=(\sum_{i} u_{b, i} (g_{b, i} (x^{1}), \ldots, g_{b, i} (x^{n})),0)
\]
with $a, b = 1, \ldots, M$, with arbitrary coefficients $\lambda_{a,
i, q}$ and where $g_{a, i} (x) = \sum_{q} \lambda_{a, i, q} q^{\bot}
e^{i \langle q_{}, x^{} \rangle} $ are  arbitrary vectors in
$\mathcal{F}^{d}$. Now note that for any $a = 1, \ldots, M$ the constraint $|u_{a}|^2=1$
imply that there exists $i = 1, \ldots, 2 n$ such that $u_{a, i}
\neq0$. This allows to conclude that in the image of $D \Phi(F, x,
U)$ belong all the vectors  $((A_{h_{1}} (x),\rho_1), \ldots,
(A_{h_{M}} (x),\rho_M))$ for an arbitrary family $\{h_{a}
\in\mathcal{F}_{d}\}_{a = 1, \ldots, M}$ and $\rho_1,..,\rho_M\in\mathbb{R}$. Now we use the assumption
that for any $x \in G$ we have vector fields $\sigma_{1}, \ldots,
\sigma_{2 n}$ such that $\{A_{\sigma_{i}} (x)\}_{i = 1, \ldots, 2
n}$ span all $\mathbb{R}^{2 n}$. This is enough to conclude that for
every $(F, x, U)$ we have $\operatorname{Im}  (D \Phi(F, x, U)) =
(\mathbb{R}^{2 n}\times\mathbb{R})^{M}$.

\medskip Now, by using Theorem~\ref{th:transv} we deduce that for
Lebesgue-almost every $F\in \mathcal{F}_{d}^{2 n M}$ the set of
configurations $x \in G$ and auxiliary vectors $U
\in({\mathbb{R}^{2n}})^{M}$ such that $\Phi(F, x, U) = 0$ is
empty. This in turn implies that for almost every realization of
Fourier coefficients the $2 n M$ vector fields $\{A_{f_{a, i}} \}_{a
= 1, \ldots, M, i = 1, \ldots, 2 n}$ span $\mathbb{R}^{2 n}$ in each
point $x \in G$ since for every $x\in G$ at least one of the
combinations
$
\sum_{i = 1}^{2 n} u_{1, i} A_{f_{1, i}} (x), \ldots, \sum_{i = 1}^{2 n}
u_{M, i} A_{f_{M, i}} (x)
$
is always different from zero for any possible choice of $u_{a, i}$. We just
proved that

\begin{theorem}
Assume that for every $x \in G$ there exists vector-fields $\sigma_{1},
\ldots,  \sigma_{2 n} \in\mathcal{F}_{d}$ such that $\operatorname{Span}
\{A_{\sigma_{i}} (x)\}_{i = 1, \ldots, 2 n}=\mathbb{R}^{2 n}$. Then for any $M
2 n$ and for Lebesgue almost every  realization of $2 n M$ vector fields
$\{f_{a, i} \in\mathbb{F}_{d} \}_{a = 1, \ldots, M, i = 1, \ldots, 2 n}$, the
family $\{A_{f_{a, i}} \}_{a = 1, \ldots, M, i = 1, \ldots, 2 n}$ spans
$\mathbb{R}^{2 n}$ in all the points $x \in G$.
\end{theorem}

Note that this theorem allows us to obtain a result valid in every point for a
generic set of vector fields form a construction of a set of vector fields
specific for each point, which is a lot easier to do.

For every $\delta> 0$ let us now define the open set $G_{\delta} =\{x
\in\mathbb{T}^{2 n} : \min_{i \neq j} |x^{i} - x^{j} | > \delta\}
\subset\mathbb{T}^{2 n}$ of points $\delta$-uniformly away from the diagonals.

A simple construction gives that for each $\delta>0$ there exist two smooth
divergence-free vector fields $g_{1}(x)$ and $g_{2}(x)$ with compact support
inside the ball $B(0,\delta/2)$ and such that $g_{1}(0)=(1,0)$ and
$g_{2}(0)=(0,1)$ (it is sufficient to use fields of the form $g\left(
x\right)  =\varphi( \vert x-x_{0}\vert ^{2}) (
x-x_{0})  ^{\perp}$ with suitable $x_{0}\in\mathbb{R}^{2}$ and smooth
scalar compact support function $\varphi$). In such a way, for any fixed point
$\hat{x}\in G_{\delta}$ we can obtain $2n$ vector fields $f_{1},\ldots,f_{2n}$
of the form
\[
f_{2k-1}(x)=g_{1}(x-\hat{x}_{k}),f_{2k}(x)=g_{2}(x-\hat{x}_{k}),k=1,\ldots,n
\]
such that $\{A_{f_{i}}(\hat{x})\}_{i=1,\ldots,2n}$ is the canonical basis of
$\mathbb{R}^{2n}$. A difficulty stems from the fact that these fields do not
necessarily belong to $\mathcal{F}_{d}$ for some $d$. We need then to
approximate the functions $g_{1}$ and $g_{2}$ by elements of $\mathcal{F}_{d}$
for $d$ large enough. Fix $\varepsilon>0$ small enough, by density of
trigonometric polynomials, there exists $d>0$ and $\widetilde{g_{1}%
},\widetilde{g_{2}}\in\mathcal{F}_{d}$ such that $\sup_{x\in\mathbb{T}^{2}%
}|g_{i}(x)-\tilde{g}_{i}(x)|<\varepsilon$. Note that the functions
\[
\tilde{f}_{2k-1}(x)=\tilde{g}_{1}(x-\hat{x}_{k}),\tilde{f}_{2k}(x)=\tilde
{g}_{2}(x-\hat{x}_{k}),k=1,\ldots,n
\]
belong to $\mathcal{F}_{d}$ for any $\hat{x}\in\mathbb{T}^{2}$ and that, for
example,
\[
|A_{\tilde{f}_{1}(\hat{x})}-(1,0,\ldots,0)|\leqslant C\varepsilon
\]
where the constant does not depend on the parameters of the problem. Then for
$\varepsilon$ small enough, the family $\{A_{\tilde{f}_{i}(\hat{x}%
)}\}_{i=1,\ldots,2n}$ spans all $\mathbb{R}^{2n}$. The value of $d$ depends
only on $\varepsilon$ and $\delta$ but not on $\hat{x}\in G_{\delta}$. This leads us to the following result.

\begin{lemma}
For each $\delta> 0$ there exists $d \geqslant1$ such that for every $x \in
G_{\delta}$ we can find $2 n$ vector fields $f_{1}, \ldots, f_{2 n}
\in\mathcal{F}_{d}$ with the property that $\operatorname{span}  \{A_{f_{i}
(x)} \}=\mathbb{R}^{2 n}$.
\end{lemma}

An easy implication is then

\begin{corollary}
For every $\delta> 0$ and $d > d_{0} (\delta)$, almost every realization of $2
n  M$ vector fields $\{f_{a, i} \in\mathcal{F}_{d} \}_{a = 1, \ldots, M, i =
1, \ldots, 2 n}$ is such that $\operatorname{span}  \{A_{f_{a, i}} (x)\}_{a =
1, \ldots, M, i = 1, \ldots, 2 n} =\mathbb{R}^{2 n}$ for all $x \in G_{\delta
}$.
\end{corollary}

By approximation of $C^{\infty}$ vector fields by elements in $\mathcal{F}%
_{d}$ we can conclude that also the set $Q_{\delta} \subset(C^{\infty
}(\mathbb{T}^{2};\mathbb{R}^{2}))^{2 n M}$ of $2 n M$ vector fields $\{f_{a,
i} \}_{a = 1, \ldots, M, i = 1, \ldots, 2 n}$ such that for all $x \in
G_{\delta}$ $\operatorname{span}  \{A_{f_{a, i}} (x)\}_{a = 1, \ldots, M, i =
1, \ldots, 2 n} =\mathbb{R}^{2 n}$ is dense in $(C^{\infty})^{2 n M}$.

Let us prove that $Q_{\delta}$ contains an open dense subset. For any compact
$K\subset\mathbb{T}^{2}$ define $Q_{K}$ as the subset of $(C^{\infty
}(\mathbb{T}^{2};\mathbb{R}^{2}))^{2 n M}$ which spans the full tangent space in
every point of $K$.

We first prove that the sets $Q_{K}$ are open: indeed assume that there exists
a sequence $(f^{(k)}_{i,a}) \in Q_{K}^{c}$ such that $f^{(k)}$ converge to a
point $f$ in $Q_{K}$. For each $f^{(k)}$ there exists a point $x^{(k)} \in K$
for which $\operatorname{span} (A_{f^{(k)}_{i,a}}(x^{(k)})) \neq
\mathbb{R}^{2n}$. By compactness of $K$ we can extract a subsequence, still
denoted by $(x^{k})_{k\ge1}$ which converges to $x\in K$. Then by uniform
convergence of $f^{(k)}$ to $f$ we deduce that we also have
$\operatorname{span} (A_{f_{i,a}}(x)) \neq\mathbb{R}^{2n}$ which is in
contradiction with the fact that $f\in Q_{K}$.

Then observe that for any $0< \delta^{\prime}< \delta$ there exists a compact
$K$ such that $G_{\delta} \subset K \subset G_{\delta^{\prime}}$ and then that
$Q_{\delta^{\prime}} \subset Q_{K} \subset Q_{\delta}$. The set $Q_{\delta
^{\prime}}$ is dense and contained in an open set $Q_{K}$ which proves that
the interior of $Q_{\delta}$ is both open and dense, that is a residual set
(or co-meagre).

At this point, by countable intersection, we get that $Q = \cap_{k} Q_{1 / k}$
is also residual and its elements are exactly the vector fields such that
$\operatorname{span}  \{A_{f_{a, i}} (x)\} =\mathbb{R}^{2 n}$ in every point
of $\Gamma^{c}$.
We have then proved Theorem~\ref{th:npoint-genericity}.

redpajama_set_name: RedPajamaArXiv


text:
Details for a publication entitled: Treasures of Cambridge (Illustrations)
Winged female figure, holding an arrow
Draughtsman: Mocetto, Girolamo (Attributed to)
Wooded landscape
Painter: Hobbema, Meindert
The Gust of Wind
Painter: Renoir, Pierre Auguste
William Pitt (1759-1806), Statesman and Prime Minister
Sculptor: Nollekens, Joseph
St Geminianus, St Michael and St Augustine, each with an Angel above
Painter: Martini, Simone (And workshop)
Bust of Charles II (1630-1685)
Sculptor: Bushnell, John
Sheet of studies: the draped figure of Christ; two studies of a left hand, one showing part of the forearm
Draughtsman: Michelangelo Buonarroti
Heneage Lloyd and his Sister, Lucy
Painter: Gainsborough, Thomas
Au Café
Painter: Degas, Edgar
The Fitzwilliam Museum (2023) "Publications details for Treasures of Cambridge (Illustrations)" Web page available at: https://data.fitzmuseum.cam.ac.uk/id/publication/publication-125 Accessed: 2023-02-02 16:05:00
{{cite web|url=https://data.fitzmuseum.cam.ac.uk/id/publication/publication-125|title=Publications details for Treasures of Cambridge (Illustrations)|author=The Fitzwilliam Museum|accessdate=2023-02-02 16:05:00|publisher=The University of Cambridge}}
https://data.fitzmuseum.cam.ac.uk/api/v1/publications/publication-125


text:
Terriers acquire Thomson from Notre Dame
Internet Photo.
The Terriers have acquired the rights to 96 defenceman Jordan Thomson from Notre Dame of the SJHL for Caleb Boman.
Thomson has four years, experience in the WHL and played on the Wheat Kings Championship team last season. The 5'11, 192 pound defenceman was traded to Brandon from the Swift Current Broncos and played in 46 games earning 12 points on the way to the Ed Chynoweth Cup.
Thomson chose to retire from the WHL but is considering playing in Portage this season.
Previous PostTerriers Hope for Good HarvestNext PostCJHL Announces Top 20 Pre-Season Rankings Terriers No.1


text:
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 */
export function loadScript(doc, url) {
  const script = /** @type {!HTMLScriptElement} */ (
    doc.createElement('script')
  );
  script.src = url;
  propagateNonce(doc, script);

  const promise = loadPromise(script).then(
    () => {
      doc.head.removeChild(script);
    },
    () => {}
  );
  doc.head.appendChild(script);
  return promise;
}

redpajama_set_name: RedPajamaGithub