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Government-made challenges facing industry sector - By Fahed Fanek, The Jordan Times
Prime Minister Hani Mulki will very soon, perhaps next week, meet with the leaders of the industrial sector.
The purpose is to look into the challenges and hurdles that are facing this vital sector and preventing its take-off as a leader of the economy.
This is of course a welcome step. It shows that the cries of the industrialists have finally reached the ears at the highest level of government to press officials to pay attention and discuss the subject in a way to reveal what is each party’s share of responsibility for this alarming situation.
The manufacturing industrial sector contributes around 17 per cent of the gross domestic product.
It provides jobs for more than 20 per cent of the country’s labour force. It produces the bulk of Jordan’s national exports; accordingly, it deserves full attention and looking at the challenges to industry from a wider view of the national economy.
Industrialists have a lot of issues to raise, but the bulk of the complaining will not be about actions that the government did not take to help the industry. It will be about actions the government took to the detriment of the industry.
What the government has to do, therefore, is not to support or subsidise industry; it is to put an end to policies and actions that have harmed industry and will continue to do so.
What I have in mind is Jordan’s entry into free trade agreements and sweeping exemptions on reciprocal bases with countries and trade blocs that local industry cannot match.
They do not allow fair competition. The end-result was a trade in one direction.
Suffice it to say that the EU exports to Jordan are currently equal to 35 times its imports from Jordan.
Europe is an agricultural area. They import phosphate and potash, but not from Jordan.
Would you believe that Jordan, with its huge deficit in its budget and dependence on foreign grants and loans, would agree to exempt all imports from Europe from customs duties based on the so-called partnership and reciprocal exemptions?
Such unwise policy has deprived the Treasury of hundreds of millions of dinars every year and has been instrumental in suffocating Jordan’s infant industry, which was born and has lived behind protection walls.
Who would believe that a country with an over 18 per cent unemployment rate would agree to give priory to Syrian labourers, not only by exempting them from all taxes and restrictions applied to all other nationalities, but also by securing a quota of jobs for starting with 15 per cent and rising gradually to 25 per cent of the workforce of Jordanian companies that hope to export to Europe?
Who would accept any formula of trade with countries that subsidise their exports, such as Turkey, or with countries that provide electricity, water, and fuel at nominal prices such as Arab Gulf states, which hardly import anything from Jordan beside vegetables and fruits?
Jordanian officials thought at the time that going too far in an open market policy would attract foreign investors to benefit from Jordan’s right to access the European market.
As a matter of fact, Jordanian industries are met in Europe with restrictions and hard rules of origin that are worse than what such foreign investors would face in Europe if they made it on their | 416,371 |
TITLE: Specific value of $\zeta(3/2)$?
QUESTION [3 upvotes]: Is anything known about the value of $$\zeta(3/2)=\sum_{n\geq 1}\frac{1}{n^{3/2}}?$$ It is a classical result that $\displaystyle \zeta(2)= \frac{\pi^2}{6}$ and $\zeta(3)$ has been shown to be irrational by Roger Apéry in 1979.
Do we even know if $$\zeta(3/2)=\sum_{n\geq 1}\frac{1}{n^{3/2}}$$ is an irrational number or not? Is it true that $$\zeta(3/2)=\sum_{n\geq 1}\frac{1}{n^{3/2}}=\frac{a}{b}\sum_{n\geq 1}\frac{(-1)^{k-1}}{n^{3/2}\binom{2n}{n}}?$$ Not sure if Apéry's proof can be adapted here since I haven't read it.
REPLY [1 votes]: By no means an answer but an extended comment. I should also preface I am an amateur mathematician with no formal education. Following Wikipedia's entry for the Riemann zeta function we have the following functional equation:
$$
\zeta(s)=2^{s}\pi^{s-1}\sin\left(\pi s/2\right)\Gamma(1-s)\zeta(1-s)\label{a}\tag{1}
$$
Suppose $s=3/2.$ Via direct substitutions I can rewrite LHS side of $\ref{a}$ explicitly as
$$
2^{3/2}\pi^{3/2-1}\sin\left(3\pi/4\right)\Gamma(1-3/2)\zeta(1-3/2);\label{b}\tag{2}
$$
which after regrouping and gathering like terms is equal to
$$
2^{3/2+1/2}\pi^{3/2-1+1/2}\zeta(-1/2);\label{c}\tag{3}
$$
which is reducible to
$$
4\pi\zeta(-1/2).\label{d}\tag{4}
$$
And so
$$
\zeta(3/2)=4\pi\zeta(-1/2), \label{e}\tag{5}
$$
Observe:
Case 1: Note $4\pi$ is an irrational number. If $\zeta(-1/2)$ is rational then RHS side of $\ref{e}$ is an irrational number and so to by equality is $\zeta(3/2).$
Case 2: Again note $4\pi$ is an irrational number. If $\zeta(3/2)$ is a rational number then the RHS of $\ref{e}$ is a rational number. If $\zeta(-1/2)$ is an rational number then the RHS of $\ref{e}$ is irrational; which is a contradiction. Subsequently $\zeta(-1/2)$ is an irrational number.
From cases 1 and 2 it seems at least one of $\zeta(3/2)$ or $\zeta(-1/2)$ is an irrational number. In particular they cannot both be rational. More strongly it would seem of the infinite number of pairs
$$
\{\zeta(j/2),\zeta(1-j/2)\}_{j\in\mathbb{N}}
$$
at least one number from each of the infinite pairs is irrational. | 131,561 |
Bombardier and Alstom in Morocco to fight mega transport contracts
Moroccan transport company connect millions of contracts with major European brands
French company signs contract for municipal waste millions of El Jadida
THINK wants money by Sylvana Simons to breach of contract
Renard tackles rumors: ' No salary increase received and will also not get it '
SNTL goes for 230 million dirham subsidized grain transport in Morocco
Two-thirds of the employees working without a contract in Morocco
Cameras in offices of notaries and lawyers to capture contracts in Morocco
Attijariwafa Bank allows American contractor for new headquarters Casablanca
Galatasaray rounds coming Belhanda, contract for 4 seasons
Exclusive interview: Ajax sign 17-year-old goalie Issam El Maach
Casablanca: M'dina Bus contract will not be extended
Municipality of Casablanca want to end contract with Sita El Beida
Exclusive interview: young FC Utrecht-Defender Ebie about his professional contract
Moroccan bond considering contract extension Renard to 2022
Lekjaa: Renard's ' contract with the League for a long time been extended to 2022 ' (video)
Vinci signs a contract of 284 million euros for the construction of a hydroelectric power plant in Morocco
' Renard refuses to extend contract, wait until after WORLD CUP ' | 152,474 |
The International Patrol.
In Vienna, the first district (which was also called the "inner city") was placed under quadripartite control, with the chairmanship changing every month. The fundamental thinking behind this four-power control was that the most important governmental authorities and administrative bodies were located in the inner city. If the inner city were controlled by only one of the Allied powers, that particular power could have exerted pressure on the Austrian government and public authorities.
In 1948, a reporter for The New Yorker wrote, "Vienna must be the most patrolled city on earth. Besides is the International Patrol. It was organized by the former American provost marshal in Vienna, [Colonel William] B. Yarborough." (1)
When duties began on 5 August 1945, the patrols consisted of three soldiers--one military police officer from each of the three Allied powers of the United States, Great Britain, and Russia. On 27 September 1945, French military police officers joined as fourth men. At first, the patrols used American jeeps as patrol vehicles but they proved too small for the four-man patrols; so larger command cars were then used. In later years, American and Russian sedans were used as patrol vehicles. The U.S. contingent was from Company C, 796th Military Police Battalion, and the British force consisted of members of the 105th Provost Company. Attempts to identify the Russian and French units met with no response from their respective embassies. The patrol members wore the uniforms of their own nations. In addition, each member wore a distinctive metal badge on his arm, midway between his shoulder and elbow. American, Russian, and French members wore their insignia on their left sleeves; British members wore theirs on their right sleeves.
Between 1300 and 0100 daily, five patrols were operating--one each in the American, British, Russian, and French sectors and a standby at the International Patrol Headquarters. Between 0100 and 0700, only three patrols were operating--one for the American and French sectors, one for the British and Russian sectors, and a standby. The patrols were constantly on the alert for traffic violations involving Allied vehicles and incidents involving personnel from any two different powers. Patrol members on the 1300-1900 shift ate their evening meals in the 796th Military Police Battalion mess. (2)
One narrative regarding the functions of the International Patrol comes from the British Royal Military Police:
There is a great temptation for each of us to feel that the problems of our own particular stations are, at least at times, unique in the field of provost experience; but when it is considered that the Royal Military Police in Vienna work day in and day out with the Americans, the French, and the Russians and that there is no hour of the day or night when you cannot find a [Royal Military Police] lance corporal sitting next to a Russian military policeman on patrol somewhere in the city, there is perhaps some justification. We conduct our business in four languages and have to get to know a fifth quadripartite, that peculiar hybrid in which Inter-Allied Command in Vienna, the Four Powers, decisions expressed in the form of protocol. (3)
A member of the 796th Military Police Battalion relates this story:
I was patrolling a street in the 10th Bezirk (Russian Zone), when suddenly the Russian member of the patrol noticed a vehicle ahead of us and told me to force the truck to the curb. Twice, I tried to force the vehicle to stop; but both times, the driver ignored the warnings and proceeded. Finally, the Russian member sitting next to me drew his pistol and fired at the truck, which continued on its course. After using his supply of ammunition, the Russian member borrowed the French member's pistol. Only after the Frenchman's ammunition had been exhausted did the truck halt. The driver was taken into custody, and the vehicle was driven to the Russian Kommandatura. The remarkable fact about this incident was that, although both pistols were fired, neither the driver nor the vehicle were seriously harmed. (4)
Another incident involving the Russian contingent was related by Ernest Holden of the Royal Military Police. He was on patrol when a Russian military policeman arrested a Russian man on the streets of Vienna. He wanted to take the man to the Russian Kommandatura; but the American driver refused, insisting that they take him to the International Patrol Headquarters. The Russian military police soldier drew his pistol, held it to the American driver's head, and told him to go to the Russian Kommandatura. Holden, in turn, drew his pistol, held it to the Russian's head, and told the American to do what standing orders said--go to the International Patrol Headquarters. So that's what they did, and the stand-off was resolved. Neither Holden nor the American Soldiers ever heard what happened to the Russian member of that patrol, but they never saw him on patrol again.
Corporal Edwin L. Luck, a patrolman from Amsterdam, New York, relates the following incident:
I received a message to proceed to the Astoria Hotel, a British hotel in the 1st Bezirk. When the other members of the patrol and I arrived, we learned that somebody had been dropping empty wine bottles from a window of the hotel onto the sidewalk on Karntnerstrasse, endangering the pedestrians. It was around midnight and extremely dark. At intervals, bottles continued to fall from an upper-story window but it was difficult to determine which window they were coming from. The British member of the patrol asked the assistance of the members, and a search of all the rooms was begun. After approximately an hour, two intoxicated men were found in a room on the top floor. When they were questioned, it was learned that they had been drinking and, after finishing a bottle, disposed of it by the easiest means--tossing it out of the window. The remarkable fact about this incident is not the amount of wine drunk by the men, but how none of the people passing below the window had been injured. (5)
Another excerpt from the 105th Journal states:
Lance Corporal Levi came close to death whilst on patrol. The patrol had stopped to check a Russian soldier who, taking exception as to how he was being spoken to, opened fire on the patrol, killing one patrolman and severely injuring another, the driver taking off. The gunman then approached [Lance Corporal] Levi (who could not drive), telling him to drive him to America. Levi drove instead to the International Patrol Headquarters; and seeing American [military police], the Russian exclaimed, "I'm in America," and surrendered. (6)
During the making of a film, The Third Man, which was set in Allied-occupied Vienna after World War II and starred Orson Wells and Trevor Howard, Howard was arrested by a British member of the patrol for impersonating an officer. The assistant director Guy Hamilton and the continuity assistant Angela Allen teamed up to describe what it was like to film in the Vienna sewers and to explain how a drunk Trevor Howard was arrested for impersonating a British officer while still in costume as the sober Major Calloway.
Due to an international agreement, the International Patrol ceased to exist on 14 September 1955, when all Allied armies left Austria.
Acknowledgement: I am deeply indebted to former Lance Corporal Brian Chammings, Royal Military Police, and Mr. Phil Holden, whose father was a British member of the patrol, for the documents and photographs they provided in researching this story. It was Holden's father, Ernest Holden, who detained actor Trevor Howard for impersonating an officer during the filming of the movie The Third Man.
Endnotes:
(1) Cameron et al., "A Reporter in Vienna," The New Yorker, 6 March 2948, p. 61, < /magazine/1948/03/06/cameron-lewis-laborde-gorodnistov>, accessed on 2 June 2015.
(2) "786th Military Police Battalion," <http:// usarmygermany.com/Sont_2.htm?http&&&usarmygermany .com/Units/USFA%20Units/USFA_796th%20MP%20Bn ,htm>, accessed on 8 June 2015.
(3) Major George Denis Pillitz, "Inter-Allied Command in Vienna," 105th Journal, Spring 2014.
(4) "796th Military Police Battalion: Vienna Command," <http:// usarmygermany.com/Sont_2.htm?http&&&usarmygermany .com/Units/USFA%20Units/USFA_796th%20MP%20Bn.htm>, accessed on 12 June 2015.
(5) Ibid.
(6) 105th Journal, Winter 2010, pp. 6-7.
Reference:
Carol Reed, director, The Third Man, British Lion Films, 2 September 1949.
By Master Sergeant Patrick V. Garland (Retired)
Master Sergeant Garland retired from the U.S. Army in 1974. During his military career, he served in military police units and criminal investigation detachments and laboratories. At the time of his retirement, Master Sergeant Garland was serving as a ballistics evidence specialist at the European Laboratory. He remained in this career field until retiring from civilian law enforcement in 1995. Photos Wanted
Military Police is always looking for good-quality, action photographs (no "grip and grins," please) to use on the outside covers. If you have photographs of Soldiers who are in the proper, current uniform and are participating in training events or operations or photographs of current, branch-related equipment that is being used during training or operations, please send them to us at <usarmy.leonardwood.mscoe.mbx.mdotmppb @mail.mil>.
Ensure that photographs depict proper safety and security procedures, and do not send copyrighted photographs. All photographs must be high-resolution; most photographs obtained from the Internet, made smaller for e-mailing, or saved from an electronic file such as a Microsoft[R] PowerPoint or Word document cannot be used for print. In addition, include a caption that describes the photograph and identifies the subject(s) and photographer (if known). Please see our photograph guide at < .wood. army.mil/mpbulletin/documents/PhotoGuide.pdf> for more detailed information.
MILITARY POLICE BRIGADE LEVEL AND ABOVE COMMANDS COMMANDER CSM/SGM CWO Mark Inch Timothy Fitzgerald Mark Inch Crystal Wallace John Welch Mark Spindler Richard Woodring Leroy Shamburger Mark Inch Timothy Fitzgerald Burton Francisco Jerome Wren Michael Hoban NA Phillip Churn Craig Owens Mary Hostetler Duane Miller Angelia Flournoy Eddie Jacobsen Winsome Laos Bryan Patridge David Tookmanian Erica Nelson Steven Raines Alexander Conyers Jeffrey Maddox Zane Jones James Breckinridge David Chase Jon Matthews Alex Reina Joseph Klostermann Peter Cross Joseph Menard Jr. Ross Guieb Bradley Cross Phillip Burton Jon Sawyer Malcom McMullen John Schiffli Richard Giles Abbe Mulholland Keith Nadig Andrew Lombardo Roger Hedgepeth Tara Wheadon Edgar Collins Ignatius Dolata Jr. Arthur Williams David Albaugh Thomas Denzler Clyde Wallace Celia Gallo David Heath Edwin Garris MILITARY POLICE BATTALION LEVEL COMMANDS Kevin Hanrahan Peter Harrington Anderson Wagner Gerald Mapp Chad Aldridge Billy Higgason Lawrence Stewart Mathew Walters Phillip Curran Larry Dewey Gordon Lawitzke Paul Bailey Christine Whitmer James Sanguins Joel Fitz Marcus Matthews Marcus Jackson Phillip Lenz Bryan Schoenhofer Jason Turner Kevin Pickrel Michelle Goyette Russell Erickson Chad Goyette Brian Flom Jeremy Willingham Daniel O'Brien Brian Carlson Lee Sodic Marc Hale Freddy Trejo Alexander Murray Kevin Rogers Michael Fowler Mark Duris Craig Maceri Scott Smilinich Steven Jackan Alpheus Haswell Robert Watras Darrell Masterson Mary Staab Aarion Franklin John Gobel Fowler L. Goodowens II Kenneth Niles Robert Wall Luis De La Cruz Jose Perez Haymet Llovet Francisco Ramos Norberto Flores II Roger Flores Dawn Bolyard James Summers John Dunn Gregory Derosier David Knudson William Allen Daniel Williams Jennifer Steed Victor Watson Erik Anderson Callie Leaver Larry Crowder Vacant Robert Paoletti Andraus Williams Paul Deal Boyd Dunbar Isaac Martinez Richard Yohn Timothy Starke Michael Rowan John Whitmire Nathan Deese Lance Shaffer Jonathan Stone Kenneth Dilg Ed Williams James Blake James Sartori Michael Treadwell Theodore Skibyak James Lake Robert Engle Timothy Winks Ben Adams James Rogelio Joseph Mitchell Charles Seifert Vacant Christine Borognoni Paul Shaw Richard Vanbuskirk Kyle Jenkins David Heflin Joseph Rigby Karen Connick Keith Magee Alexander Shaw Juan Mitchell Vance Kuhner Brett Goldstein Kelly Jones William Henderson William Rodgers Michael Robledo Steven Gavin Howard Anderson Vacant Michael Poll Victor Bakkila Ann Vega Manuel Ruiz Cheryl Clement Sheiita Taylor Eric Hunsberger Richard Cruickshank Susan Kusan Jason Litz Timothy Macdonald James Stillman Mauro Orcesi Caroline Horton Mark Hennessey Robert Arnold Jr. Lisa Piette-Edwards Jonathan Doyle Jeffrey Cereghino Jon Myers Clayton Sneed John Fivian Janet Harris Richard Millette Mark Bell Laura Steele Milton Hardy Kevin Smith Troy Gentry Christopher Wills Thomas Gray Rebecca Hazelett James Lea Matthew Gragg Michael Weatherholt Leevaine Williams Jr. John Vicars Karst Brandsma Darren Boruff James Eisenhart Robert Eichler Omar Lomas James Tyler Kenneth Powell Vacant Robert Mayo Stacy Garrity Rodney Ervin Jason Marquiss Scott Anderson Emma Thyen Shawn McLeod Mark Howard Todd Marchand Kenneth Richards Richard Weider Jeffrey Bergman Teresa Duncan Kirt Boston Bryan Lynch Mark McNeil Eric Vogt Lonnie Branum Jr. Scott Flint Sylvester Wegwu Cole Pierce Martin Eaves Dewey Haines David Astorga Lane Clooper Michael Thompson Keith Ford Gerald De Hoyos MILITARY POLICE BRIGADE LEVEL AND ABOVE COMMANDS COMMANDER UNIT LOCATION Mark Inch OPMG Alexandria, VA Mark Inch HQ USACIDC Quantico, VA Mark Spindler USAMPS Ft Leonard Wood, MO Mark Inch Army Alexandria, VA Corrections Cmd Burton Francisco 46th MP Cmd Lansing, Ml Michael Hoban USARC PM Ft Bragg, NC Phillip Churn 200th MP Cmd Ft Meade, MD Duane Miller 8th MP Bde Schofield Barracks, HI Eddie Jacobsen 11th MP Bde Los Alamitos, CA Bryan Patridge 14th MP Bde Ft Leonard Wood, MO Erica Nelson 15th MP Bde Ft Leavenworth, KS Alexander Conyers 16th MP Bde Ft Bragg, NC Zane Jones 18th MP Bde Sembach AB, Germany David Chase 42d MP Bde Ft Lewis, WA Alex Reina 43d MP Bde Warwick, Rl Peter Cross 49th MP Bde Fairfield, CA Ross Guieb 89th MP Bde Ft Hood, TX Phillip Burton 177th MP Bde Taylor, Ml Malcom McMullen 290th MP Bde Nashville, TN Richard Giles 300th MP Bde Inkster, Ml Keith Nadig 333d MP Bde Farmingdale, NY Roger Hedgepeth 3d MP Gp (C/D) Hunter Army Airfield, GA Ignatius Dolata Jr. 6th MP Gp (C/D) Joint Base Lewis- McChord, WA Thomas Denzler 701st MPGp (C/D) Quantico, VA David Heath Joint Detention Gp Guantanamo Bay, Cuba MILITARY POLICE BATTALION LEVEL COMMANDS Kevin Hanrahan 5th MP Bn (C/D) Kleber Kaserne, Germany Gerald Mapp 10th MP Bn (C/D) Ft Bragg, NC Lawrence Stewart 11th MP Bn (C/D) Ft Hood, TX Larry Dewey 19th MP Bn (C/D) Wheeler Army Airfield, HI Christine Whitmer 22d MP Bn (C/D) Joint Base Lewis- McChord, WA Marcus Matthews 33d MP Bn Bloomington, IL Phillip Lenz 40th MP Bn (C/D) Ft Leavenworth, KS Jason Turner 51st MP Bn Florence, SC Michelle Goyette 91st MP Bn Ft Drum, NY Chad Goyette 92d MP Bn Ft Leonard Wood, MO Jeremy Willingham 93d MP Bn Ft Bliss, TX Brian Carlson 94th MP Bn Yongsan, Korea Marc Hale 96th MP Bn (C/D) San Diego, CA Alexander Murray 97th MP Bn Ft Riley, KS Michael Fowler 102d MP Bn (C/D) Auburn, NY Craig Maceri 104th MP Bn Kingston, NY Steven Jackan 105th MP Bn (C/D) Asheville, NC Robert Watras 112th MP Bn Canton, MS Mary Staab 115th MP Bn Salisbury, MD John Gobel 117th MP Bn Athens, TN Kenneth Niles 118th MP Bn Warwick, RI Luis De La Cruz 124th MP Bn Hato Rey, Puerto Rico Haymet Llovet 125th MP Bn Ponce, Puerto Rico Norberto Flores II 136th MP Bn Tyler, TX Dawn Bolyard 151st MP Bn Dunbar, WV John Dunn 159th MP Bn (C/D) Terra Haute, IN William Allen 160th MP Bn (C/D) Tallahassee, FL Jennifer Steed 168th MP Bn Dyersburg, TN Erik Anderson 170th MP Bn Decatur, GA Larry Crowder 175th MP Bn Columbia, MO Robert Paoletti 185th MP Bn Pittsburg, CA Paul Deal 192d MP Bn (C/D) Niantic, CT Isaac Martinez 193d MP Bn (C/D) Denver, CO Timothy Starke 198th MP Bn Louisville, KY John Whitmire 203d MP Bn Athens, AL Lance Shaffer 205th MP Bn Poplar Bluff, MO Kenneth Dilg 210th MP Bn Taylor, Ml James Blake 211th MP Bn Lexington, MA Michael Treadwell 226th MP Bn Farmington, NM James Lake 231st MP Bn Prattville, AL Timothy Winks 304th MP Bn (C/D) Nashville, TN James Rogelio 310th MP Bn (C/D) Uniondale, NY Charles Seifert 317th MP Bn Tampa, FL Christine Borognoni 324th MP Bn (C/D) Fresno, CA 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[email protected]> or telephone (573) 563-7949. | 205,502 |
(1938-1993) including correspondence, specifications, biographical information, reminiscences, and photographs.
Papers, 1937-2001,.
Papers (1942-1945) of a U.S. Naval officer, USNA Class of 1941, consisting of Battle of Vella Gulf battle reports (1943), a history of the USS Lang (DD-399), USS Lang action reports (Feb. 1942-April 1944), naval communiques relating to USS Lang (1942-1944), and after-action reports for the battles of Vella Gulf, Guadalcanal, Wewak, (New Guinea), Morotai, Leyte Gulf, Okinawa, Lingayen Gulf, and other Pacific Ocean operations in which the....... (1917-1920) including correspondence, letters, contracts, comments on activities of war, etc.
Papers (1918-1957) including personal letters, correspondence, official naval orders, certificate of award and promotion, photographs, biographical sketches,...
Papers of Vice Admiral Frank H. Price, Jr., member of USNA Class of 1941, including correspondence, orders, a memoir, clippings, photographs, programs, and publications.
Records (1939-2013) of national and divisional offices of the U.S. Coast Guard auxiliary, including Flotilla 1301 records (1942-1945), consisting of correspondence, muster rolls, directives, minutes, services records, speeches, duty records, photographs, copies of the navigator and other publications, conference records, regulations, policy statements, training materials, histories, and scrap books. 718 boxes. 471 c.f. (c.f. rev. 8/21/2...
Papers of Naval officer, U.S. Naval Academy (USNA) Class of 1941, including autobiographical sketch, correspondence, global strategy conference file, orders, reports, flight records, speeches, registers, and data files. | 230,268 |
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TITLE: Statement about $(I-A)^{-1}$ matrices
QUESTION [5 upvotes]: Let $A \in \mathbb{R}^{n \times n}$ and let denote $I$ the $n \times n$ identitiy matrix.
Theorem. If $(I-A)$ is invertible and $(I-A)^{-1}$ is a nonnegative matrix and there is such a diagonal element in $(I-A)^{-1}$ which is less then $1$, then at least one of the elements in $A$ is negative.
Question. How could we prove this statement?
I've tried to use Perron–Frobenius theorem for $(I-A)^{-1}$, but after I've taken the eigenequation I don't know how to use the condition for the diagonal element.
I've also tried binomial inverse theorem and Woodbury matrix identity but they also didn't help me. Although I think using these tools lead us far away.
REPLY [3 votes]: Let
$$ B = (I-A)^{-1}, \qquad A = I-B^{-1}, $$
and let $B_{mm}<1$, $B\geq 0$, and let $e_m$ be the basis vector with $1$ in $m$-th position.
Then $Be_m = v$ has $v_m<1$ and $v\geq 0$, and $(Av)_m = v_m - 1 < 0$. | 30,328 |
East of England
Within the East of England region approximately 50,700 people work in the energy and utilities sector. This represents 11.8% of the total energy and utilities workforce in England. There are also 6,660 energy and utilities businesses based within the region.
24% of the energy and utility workforce in the East of England is female, compared with 43% of the region’s entire workforce. Across the industries:
- 30% of the power workforce is female
- 17% of the upstream gas workforce
- 20% of the waste management workforce
- 30% of the water workforce
97.7% of the regional workforce is White, with just 2.1% from other ethnic groups. The qualifications profile of the regional energy and utilities workforce is quite different from the UK average, with 8% less employees holding Level 3 or above qualifications but 11% more holding intermediate qualifications at Level 2.
25% of existing regional energy and utilities employers believe that at least some of their current workforce has skills gaps, including customer handing, problem solving and numeracy skills.
Source: Energy & Utility Skills Sector Skills Agreement Stage 1 2006 and Energy & Utility Skills Sector Skills Agreement Stages 1-3 Summary Report – East of England | 307,496 |
TITLE: Indefinite integral of $\frac{\sqrt{x}}{\sqrt{x}+1}$
QUESTION [2 upvotes]: For this I tried using the substitution technique, but it got me nowhere near the right answer.
What my notepad looks like:
$$f(x) = \dfrac{\sqrt{x}}{\sqrt{x}+1}$$
and
$$F(x) = \int f(x) =
\begin{cases}
\sqrt{x} &= t \\
dx &= 2t\cdot dt
\end{cases}
= \int\frac{t}{t+1}\cdot2t\cdot dt = \boxed{\dfrac{4t(t+1)-2t^2}{(t+1)^2}+c}$$
After swapping $t$ with $x$ again the answer is not even close; where did I go wrong?
Also as a follow up question:
How do I know when the proper situation is to use integration by substitution above integration by parts?
REPLY [2 votes]: In answer to the first question, the basic expression inside the OP's boxed formula, $4t(t+1)-2t^2\over(t+1)^2$, is the derivative of $2t^2\over t+1$, not its integral.
As for the second question, there's no hard and fast rule (at least none that I know of), but in this case using a substitution to get rid of the square root signs was a good idea.
REPLY [1 votes]: First:
$$\left(\frac{2t^2+4t}{(t+1)^2}\right)'=\frac{4(t+1)^2-4t(t+2)}{(t+1)^3}=\frac4{(t+1)^3}$$
and this doesn't look like the function you were trying to integrate: $\;\frac{2t^2}{t+1}\;$ !
Your function in the integral is :
$$\frac{2t^2}{t+1}=2t-\frac{2t}{t+1}=2t-2+\frac2{t+1}$$
and thus the integral in fact is
$$\int\left(2t-2+\frac2{t+1}\right)dt=t^2-2t+2\log|t+1|+C$$
and taking back the original variable according to your substitution:
$$t=\sqrt x\implies \int\frac{\sqrt x}{1+\sqrt x}dx=x-2\sqrt x+2\log(\sqrt x+1)+C$$ | 213,913 |
This page updated and corrected March 1, 2016.
The 2001 Anthrax Deception -- The Case for Domestic Conspiracy (Clarity Press, 2014) is a new book by Graeme MacQueen, a retired religion professor. Thoughtful people are sure to be profoundly affected by a reading of his deft analysis.
I came across his book while researching the article below. I have not so much focused on conspiracy as on examples of grotesque misrepresentations by federal lawyers and others in league with them. I wanted to look at concrete examples of the government casting events in a false light, but that doesn't mean I don't consider conspiracy as something the FBI seems to have been at pains to avoid investigating.
By PAUL CONANT
Conant is a former metropolitan New York newspaperman.
Permission is granted to reproduce this article in whole or in part, with or without attribution.
The Justice Department severelydistorted the facts it used to pin all blame for the 2001 anthrax attacks on a mentally ill scientist, a new review of federal documents shows.
In many instances in the federal case against Bruce E. Ivins, the meanings given to facts presented by federal authorities turn out to stray far from reality or to be highly ambiguous. The government bent the usual rules of evidence and logic wildly as it tried to make a case that the attack anthrax could only have come from a flask, numbered RMR-1029, "exclusively" controlled by Ivins.
A bit of background
The conclusions drawn from the scientific work performed by the FBI were strongly criticized by a National Academy of Sciences committee in 2011. The panel noted that though the FBI formally answered all questions, the bureau often rebuffed it with uninformative bureaucratic responses. The scientists added that, at the last minute when the bureau sensed the direction of the panel's upcoming report, the bureau suddenly reported that there was much classified information germane to the investigation and that a secret committee of White House appointees and scientists with high-level secrecy clearances had been quietly steering the FBI investigation.
The NAS panel decided against trying to review the national security data brought in at the last minute. The panel added that the FBI also suddenly introduced material supposedly pointing to potential al Qaeda involvement, but the panel decided that while authorities were welcome to seek more evidence in that matter, the rushed al Qaeda question was irrelevant to the panel's findings about the scientific methods used to directly link the attack anthrax to Ivins. The FBI had been led to the U.S. military early in the investigation because the attack spores showed that they came from the Ames strain, which was known to be used by the military in its biodefense work.
The problematic nature of the FBI's scientific work is buttressed by this writer's review of the investigative summary, which shows instances of trickery and serious distortion in its attempt to show that a pattern of circumstantial evidence strongly implied that Ivins, a federal anthrax scientist at the Army's Fort Detrick in Maryland, was solely responsible for the anthrax attacks.
Additionally, this writer's review of the psychiatric report written for the FBI, shows that its narrative portion differs greatly on an important incident from the Justice Department summary. None of the psychiatric review panelists was able to go on the record in response to queries from this writer. Other documents reviewed include transcripts of two FBI science briefings for the press and a number of released FBI reports.
The original Justice Department report was issued on Feb. 19, 2010, along with a group of FBI documents, in accord with a Freedom of Information Act request that was complied with at the formal closing of the anthrax case. However, it is apparent from the form of the summary that the document was intended for public consumption.
This analysis does not claim to exonerate Ivins. However, the idea that he could have acted alone has been vigorously challenged by his scientific colleagues working at the U.S. Army Medical Research Institute of Infectious Diseases (USAMRIID) at Fort Detrick.
The number of peculiarities with respect to the anthrax case is quite large, and no attempt has been made here to cover all bases.
This reporter notes that since he began researching this report in August 2014, the FBI has disabled its link to the Justice Department summary that appeared on the FBI's Amerithrax page.
This reporter has reviewed all testimony available to the public from depositions of Ivins's coworkers in a case brought in a Florida court against the Justice Department by Maureen Stevens, widow of the first anthrax fatality, Robert Stevens. The lawsuit was eventually settled after the Justice Department's criminal division forced the civil division to accept its claim that Ivins had had access to equipment for making the attack anthrax. The civil division had argued that, because there was no concrete evidence that the anthrax attacks had originated at Ivins's Army workplace, the U.S. government was not liable for a negligence damage claim.
The incidents cited below do not necessarily mean that the Justice Department and psychiatric reports are entirely inaccurate, but do show that one should beware any of their claims.
A slip of the zip
Item from the summary:
"On November 1, 2007, the Task Force executed these search warrants, which resulted in the recovery of numerous items of interest, including a large collection of letters that Dr. Ivins had sent to members of Congress and the news media over the previous 20 years -- including one sent to NBC News in 1987 at the same address for NBC used on the Brokaw letter."
The implication here is that it is odd that the NBC address on an anthrax letter matched an address used by Ivins decades previously. However, NBC has had the same address at Rockefeller Center since Rockefeller Center was built.
When checked on Sept. 4, 2014, the NBC Studios address, on the Rockefeller Center web site was
30 Rockefeller Plaza
(Entrance on 49th Street between Fifth & Sixth Avenues)
New York, NY 10012
The zip code is incorrect. The proper code is 10112.
(The page with that address has been taken down since Sept. 4 in an overhaul of the Rockefeller Center web site. Here is a URL to a partial copy of the deleted page: .)
However, if one had obtained the address via the NBC News web site "contact" page, the correct address was (as of Sept. 4, 2014) given:
NBC News, 30 Rockefeller Plaza, New York, N.Y. 10112.
As the deleted Rockefeller Center page said, NBC has been at Rockefeller Center since the center was built. "No major tenant has been here longer, occupied more space, or become so widely identified with Rockefeller Center than NBC" which opened its radio studios in 1933.
Nationwide postal zip codes date to the 1960s.
Aside from the conflicting zip codes, one other difference is that the attack letter uses "NBC TV," as opposed to "NBC News," in the address but this seems barely relevant.
So, the address that Ivins used in the early eighties would have been no different from the one the anthrax mailer used in 2001, though a number of correspondents in 2001 may have used the incorrect zip code given by NBC (yet it is not evident how long the wrong code had been up). But that is not very informative, as the attack mailer could easily have gone to the NBC News contact page to get the address.
If there were substantive differences, the summary does not disclose them.
So the insinuation is absurd. There is no strange coincidence between an old mailing address used by Ivins and the NBC anthrax letter.
The summary relates that in an Oct. 3, 2001 email from Ivins to a former coworker (Mara Linscott), Ivins talked about biowar scenarios but "the next day Ivins did not mention these more plausible possibilities" in an email to a CDC colleague, an email the summary portrayed as a sinister ploy.
A CDC website statement relates that on Oct. 4, 2001, the CDC "confirmed the first bioterrorism-related anthrax case identified in the United States in a resident of Palm Beach County, Florida." The CDC was initially highly uncertain as to whether a bioterror attack had occurred. Just because Ivins was aware of bioterror scenarios doesn't mean he would jump to the conclusion that the Florida case had resulted from bioterrorism.
Ivins's email to the CDC was fired off as soon as initial reports came in of Stevens's condition. But, he was writing in a professional capacity, and possibly was simply being cautious. The previous email had been sent in an informal vein to a friend.
The summary reprints the Oct. 3 email in full but paraphrases the scientist's email to the CDC selectively.
From the summary (numbering added):
"When Robert Stevens became the first victim of the anthrax attacks, Dr. Ivins sent an unexplainable [1] e-mail to a contact at the CDC on October 4, 2001, the day after [2] Stevens was diagnosed with inhalation anthrax. Dr. Ivins, one of the nation's foremost anthrax scientists, speculated that Mr. Stevens's infection could have been the result of Stevens drinking infected creek water. The proffered explanation was impossible because the anthrax had been inhaled [3]. Alternatively, he proposed to the CDC that Stevens could have contracted the disease from infected alpaca used in wool socks or a sweater [4]. Both a renowned microbiologist at another lab and a scientist at USAMRIID found these suggestions absurd. The microbiologist at the other lab described them as "laughable," and the USAMRIID scientist called them 'fishy, any reasonable scientist would say this doesn't make sense'." [5]
A footnote concedes that another anthrax researcher pointed out that everyone was wondering whether the microbe came from the environment, though no one had considered the idea of infection via ingestion of water.
Here is the Oct. 4 letter obtained from page 487 of Ivins's 2001 emails reproduced at ProPublica's Anthrax Files.
From: Ivins, Bruce E. Dr. USAMRIID
To: REDACTED
Subject: Florida case(?)
Date: Thursday, October 04, 2001 9:57:19 PM
Hi, REDACTED them and possibly inhaling them as an aerosol. Could this have happened? What if REDACTED, REDACTED
- Bruce
[1] After reading the email, would you call it unexplainable?
[2] The CDC says that it first confirmed that Stevens had inhalational anthrax on Oct. 4, the day Ivins wrote the email, not the day before. Hence, the dust had not yet settled and Ivins may well have been unsure the diagnosis was realistic.
[3] It's clear that he seems to distrust the inhalation diagnosis, though he is willing to entertain various possibilities -- possibilities he is thinking about in light of the news account he had read.
[4] The "alternative" explanation does not appear in the email and is not documented. However, cutaneous (infected skin lesion) anthrax is known as a disease of wool workers, and he may have wondered whether there was a slight, but not impossible, chance of transmission via infected wool clothing.
[5] It is apparent that when scientists were asked about Ivins's conjectures, they weren't privy to what he actually wrote. Even when it comes to the wild idea of anthrax contamination of drinking water, Ivins only throws out a question, which should be seen in context of his having heard of animals contracting anthrax at water holes, where they kicked up spores from the ground.
However, the Justice Department says, "This email to the CDC, fishing for information, is additional evidence of his guilty conscience."
Even if Ivins was fishing for information, is that really evidence of a guilty conscience? After all, his field was anthrax research.
No news is bad news
The summary bills another Ivins email as indicative of inside knowledge of the anthrax attacks.
On Sept. 26, 2001 Ivins wrote a former coworker:
"Of the people in my 'group'. You [REDACTED]."
The summary ominously notes that the public did not become aware of the first anthrax mailings until early October, about a week after this email. The email was sent six days after attack letters were postmarked in New Jersey.
The summary adds that in that same email, Ivins wrote: "Osama Bin Laden has just decreed death to all Jews and all Americans" -- language it held to be similar to the text of the anthrax letters postmarked two weeks later warning "DEATH TO AMERICA," "DEATH TO ISRAEL."
But, a point the summary omits is that there was much discussion on the internet and via other media about the possibility that bin Laden, or al Qaeda, possessed such weaponry. And, the public had been told of bin Laden's actions against Israel and his 1998 fatwa against Americans, saying they were permissible targets for attacks. In fact, the government's conspiracy theory concerning the 9/11 attacks hinges on bin Laden's death decree.
On Sept. 12, the atmosphere was set by former Defense Secretary William Cohen and CIA Director George Tenet.
A Sept. 12 report in the Guardian newspaper notes that Tenet told Americans that bin Laden had in 1998 served notice that any American was a legitimate target for attack. That same article asserts -- probably incorrectly -- that Sarin nerve gas was among "the many sinister components" of bin Laden's arsenal.
On the evening of Sept. 12, Cohen told CBS News that he fully expected there to be a full-scale deployment of biological and chemical "weapons of mass destruction" very soon. This reinforced his earlier statements as Clinton's Pentagon chief that a five-pound bag of anthrax bacteria in the hands of terrorists would likely cause the deaths of hundreds of thousands of Americans.
On Sept. 16, Defense Secretary Donald Rumsfeld told Fox News: "What they can do is use these asymmetrical threats of terrorism and chemical warfare and biological warfare and ballistic missiles and cruise missiles and cyber attacks."
On Sept. 19, a British TV station followed up another Telegraph report, and tied bin Laden, Sarin and anthrax together in one report.
A Sept 24 Time magazine article on terrorism via unconventional weapons discusses anthrax, Sarin and bin Laden, though the writer does not suggest bin Laden had access to such weapons.
On Sept. 26, Britain's Telegraph told of a bin Laden unit's plan to use Sarin nerve gas.
Though Ivins may not have seen the British reports, this type of information was picked up and circulated extensively on the internet and in American media.
On Sept. 26, columnist Maureen Dowd of the New York Times wrote, “Americans are now confronted with the specter of terrorists in crop dusters and hazardous-waste trucks spreading really terrifying, deadly toxins like plague, smallpox, blister agents, nerve gas and botulism." She added that women she knew "share information on which pharmacies still have Cipro, Zithromax and Doxycycline, all antibiotics that can be used for anthrax, the way they once traded tips on designer shoe bargains."
On Sept. 26, Bill Gertz of the Washington Times reports," Gertz wrote. "Sarin can be produced from the components used to make fertilizer and kills by disrupting the central nervous system. Anthrax is a highly lethal biological weapon that causes death after spores are ingested."
Gertz writes for the Washington Times, which still circulates in the Maryland suburbs of Washington, meaning that that story could well have been read by Ivins or a coworker.
It is true that none of these reports asserts that bin Laden definitely had Sarin and anthrax, but many casual listeners or readers would have assumed that bin Laden did have such capabilities. Even scientists can make rash assumptions when they are not focusing carefully.
Though the reports cited here mostly do not refer to bin Laden's death fatwa against Americans, it is safe to say various commentators had mentioned it. And it is quite plausible that Ivins heard a compressed version of Gertz's story from broadcast news or read a garbled account on the internet. Certainly there is nothing unlikely about someone rewriting the Gertz story and adding to it the bin Laden fatwas against Americans and Jews.
Another possibility is that Ivins overheard office scuttlebutt concerning the Gertz story or that he somehow became aware of a Pentagon threat analysis of al Qaeda's purported biowar potential. After all, he worked for the Pentagon.
Of course, it is possible Ivins's email was criminally anticipatory, but not only is there no evidence of that, there is plenty of evidence that such an awareness could have been picked up from news accounts.
Lies, damned lies and polygraphs
The summary relegated the fact that Ivins had passed a polygraph examination to a footnote.
."
Jeff Stein of the Washington Post wrote in February 2011 that the the FBI's case file contradicts the summary footnote.
Ivins “did not research anything about the test, to include ways to defeat its accuracy,” the FBI’s 2002 report on Ivins says.
“Likewise, he did not take any steps to defeat the tests [sic] accuracy or use countermeasures," the FBI report says. "In fact, IVINS stopped taking his anti-depression/anti-anxiety medication 48-72 hours before the polygraph, and he offered to provide blood and/or urine specimens at the time of the test to prove he was not medicated.”
Links to the report cited by Stein are now dead.
As Stein observes, "An obvious question might be whether, of the many other possible suspects who were eliminated, any were eliminated solely on the basis of polygraph examinations."
The polygraph method has been the subject of considerable controversy, but a National Academies of Science panel found that such examinations in the hands of skilled examiners are often effective, though far from perfect. Still, doubts remain about whether examiners know how to detect countermeasures and about the notion that psychiatric drugs are useful in that regard.
A few more points:
A heavily redacted Dec. 8, 2004, FBI report shows that, despite reservations, either Ivins or an associate agreed to submit to an FBI polygraph exam. Now, supposing the report refers to an Ivins associate, one still faces the question of why the FBI did not polygraph Ivins in 2004, when he had come under new scrutiny.
In April 2010, ProPublica reported that another Fort Detrick anthrax scientist, Henry S. Heine, said FBI agents gave him a polygraph exam and took statements from him several times between 2001 and 2003. Yet the summary implies that the bureau did not polygraph Ivins at all, but, rather, relied on Army polygraphers. So then, what stopped the bureau from relying on Army polygraphs of Heine?
So, aside from passing an Army polygraph, either Ivins also passed an FBI polygraph test -- with the government falsely implying that the FBI had not polygrapghed Ivins -- or the FBI was waved away from polygraphing him. As a Pentagon employee with a security clearance, Ivins's ability to refuse further polygraphing was limited, which was especially the case prior to his being officially considered a suspect.
In June 2002, the Hartford Courant noted that the FBI intended to interview and conduct polygraph tests of more than 200 former and current employees of Fort Detrick and the Army's Dugway Proving Grounds in Utah, where biowar anthrax tests had been carried out. An FBI source told the Courant that there were only about 25 people from Dugway on the list of those to be interviewed and tested, meaning most scientists to be scrutinized were from Fort Detrick.
Steven Hatfill, a former Fort Detrick scientist whom the FBI once considered a "person of interest" was among those polygraphed by the FBI.
So again, what really caused the FBI to seemingly give Ivins a waiver from polygraph testing?
It is hard to answer such questions because, as bioweapons expert Martin Furmanski observed concerning the released FBI files:
"Often the redactions are quite extensive, involving most or all of a paragraph. Generally, these carry the ‘personal privacy’ exemption notations, which seems unlikely. In some cases the redacted material can be reasonably surmised to be of scientific character from the context and the unredacted portions."
Furmanski, a Stanford University professor with degrees in pathology and microbiology, examined more than 2000 pages of FBI documents.
Despite his conclusion that Fort Detrick had possessed the equipment needed to make sufficient quantities of anthrax powder for the letter attacks, Furmanski noted, "There is a larger issue regarding the robustness of the material in the FBI FOIA documents. Although extensive, they are a selection of a much larger archive, estimated to be over 50,000 pages."
Furmanski, in his 2010 analysis for the Federation of American Scientists, said a fermenter at Ivins's work place was large enough to have produced in two batches enough anthrax for the attacks.
But in 2011, ProPublica and its partner news organizations reported that Gerard P. Andrews, a pathologist and Army officer who headed the bacteriology division where Ivins worked, described the division's fermenter as “indefinitely disabled,” with its motor removed. Assuming the Andrews statement is accurate, someone would not only have had to get the fermenter running, but to have removed and hidden the motor, at least twice. There is no indication that Ivins was skilled in such matters. So acceptance of the FBI theory tends toward the conclusion that Ivins very likely would have had confederates.
The FBI made no attempt to give a detailed explanation of how the powders were prepared, and so the public is left with unanswered questions about other fermenters at Fort Detrick and Ivins's access to them.
At any rate, in 2012 an NSA whistleblower, Russell Tice, revealed to Newsweek that if one has used trickery on a control question, then when the key questions are asked, the subject can daydream about something pleasant. It seems plausible that daydreaming might be easier to do if one is using psychotropic drugs.
It cannot easily be ruled out that Ivins had learned how to beat the test from associates in the Pentagon. Or, is it possible he was "passed" by confederates, which is another way of beating such tests?
One online report, quoting FindLaw.com, says that psychopaths and sociopaths (who lack empathy with other people) "may be excluded from polygraphs as the disorders reflect individuals who can control their emotional responses and do not exhibit a conscience."
A reader could get the impression -- though the summary is not explicit -- that Ivins's mental illness may have contributed to his ability to pass the polygraph test. In an acute paranoid schizophrenic episode, it is conceivable that an "alternate" personality could emerge which is capable of turning to murder. Later, when the "alter ego" has re-submerged, the "real Bruce Ivins" would sincerely believe that he had had nothing to do with the attacks.
For example, among the many personal emails he sent to his former coworkers Mara Linscott and Patricia Fellows, is this poem:
So now, please guess who
Is conversing with you.
Hickory dickory Doc!
Bruce and this other guy, sitting by some trees,
Exchanging personalities.
It’s like having two in one.
Actually it’s rather fun!
The summary adds:
"In the weeks that followed this e-mail, Dr. Ivins continued to discuss his 'terrifying' mental health issues, telling Former Colleague # 1 (Mara Linscott) in an e-mail on July 4, 2000: 'The thinking now by the psychiatrist and the counselor is that my symptoms may not be those of depression or bipolar disorder, they may be that of ‘Paranoid Personality Disorder'."
Ivins also wrote that he was seeking help, fearing the "terrible things some paranoid schizophrenics have done."
Under pressure from the Justice Department, the civil depositions of Linscott and Fellows were sealed, meaning there is no way at present to compare government claims with the recollections of the two women.
The psychiatric panel observed that "Ivins's writings referred, at times explicitly, to depression, paranoia, and delusional thoughts; described a sense of observing himself from the outside (depersonalization); talked and wrote about there being two Bruces (dissociation); described being harmed by the rejection of KKG [a sorority] members; and worried about becoming, and being, schizophrenic."
Yet, the psychiatric panel cited neither schizophrenia nor dissociative identity disorder (in which "alter egos" emerge) in its diagnosis, but found that Ivins lacked empathy (a problem associated with "psychopaths") and suffered from "personality disorder not otherwise specified, with narcissistic and antisocial features" which he medicated via drug and alcohol abuse.
The government may have had a problem with the "murder-by-alter-ego" idea, as such events are not well documented in the literature and the federal prosecutors may have been reluctant to set a precedent. Still, defense lawyers have tried to use the multiple personality defense in murder cases.
For example, lawyers for Richard Angelo, the "Angel of Death" nurse, fought to prove that Angelo suffered from dissociative identity disorder, which meant he would dissociate himself completely from the crimes he committed.
The lawyers fought to prove this theory by introducing polygraph exams which Angelo had passed during questioning about the murdered patients, it has been reported. The judge however, would not allow the polygraph evidence.
Considering the emphasis put on his mental stress and neurotic behavior before the anthrax attacks, one wonders how Ivins would not have panicked at the possibility he might flub his polygraph -- countermeasures or no -- and draw investigators down on his head. But if an alternate personality had been in control during key periods, Ivins may well have passed a polygraph test when that personality was submerged.
However, such a scenario has many difficulties, including a proposed ability to switch on the murderous "alter ego" during contacts with confederates.
Also, one should not automatically dismiss the possibility that Ivins's emails had been intercepted by security agents in 2000 and brought to the attention of politically powerful persons in early 2001 who were seeking war against Iraq and other Mideastern nations, as is documented in many places, including here:
[Another thought: .]
In such a scenario, Ivins would have been used as a witting or unwitting pawn, or perhaps as a potential "fall guy." How else does one account for the government notion that Ivins was guilty and also for his colleagues' assertion that Ivins could not have escaped notice while making the large amount of anthrax powders used in the mailings?
Shortly after the 9/11 attacks, Seymour M. Hersh, a highly regarded investigative reporter, wrote in the New Yorker magazine:
'."
What Went Wrong
The New Yorker, Oct. 8, 2001 issue
Also see,
From the psychiatric report, dubbed "Report of the Expert Behavioral Analysis Panel":
"The agent then asked Dr. Ivins whether that was all the anthrax he had. It was, Dr. Ivins said. The agent then specifically asked about the RMR-1029 flask.
"Dr. Ivins walked back into the cooler and returned with a standard, one-liter vessel, and labeled with a black Sharpie. The anthrax inside was in liquid form. Unlike the two samples he had readily volunteered, Dr. Ivins had held back the RMR-1029 — surrendering it only when specifically requested.
"Later, Dr. Ivins’ technicians reported that they had never seen the flask. He had been its sole custodian and presumably had kept it concealed in the cooler.
"Soon, anthrax from the seized RMR-1029 flask was tested with the more sophisticated assay techniques that scientists had been developing. The question was whether the anthrax from the flask would show the same genetic markers as the anthrax used in the mailings."
Why this incident is problematic is that the story is corroborated neither by the investigative summary nor pertinent FBI files.
The summary reads: "So, in April 2004, the flask containing RMR-1029, along with approximately 20 other samples of Ba [Bacillus anthracis]."
A federal affidavit for a 2007 search of Ivins's property discusses the April 2004 search. It says nothing about the alleged blatant evasion by Ivins.
"On December 12, 2003, an FBI Special Agent accompanied Dr. Ivins into Suite B3 at USAMRIID and identified additional Ames samples of Dr. Ivins's and others that had not been submitted as part of the above mentioned response. Dr. Ivins submitted slants [test tubes holding biological material] prepared from the newly identified samples to the FBIR [FBI repository] on April 7,. All of the samples were secured in the B3 walk-in cold room within a double-locked safe, and sealed with evidence tape until such time that they could be transported to the Navy Medical Research Center (NMRC), which was under contract by the FBI."
The affidavit, as of Oct. 26, 2014, is found at
There is no mention of Ivins attempting to withhold the critical flask from an FBI agent during the search of the hot suite.
ProPublica's Anthrax Files site has a document (pages 73 to 79 in "The FBI Bruce E. Ivins investigation") discussing the search conducted in early April 2004 which does not verify the psychiatric narrative's account.
ProPublica's research yielded nothing like the tale in the psychiatric report.
ProPublica:
[genetic markers].."
As discussed below, a Navy laboratory had measured the spore density in the seized anthrax, and obtained a count consistent with dilution. However, Ivins showed an FBI scientist that that the Navy measurement was unlikely.
On such an important matter as pronounced evasive behavior, the psychiatric report does not cite its source of evidence, whether that be an FBI document or an interview with the FBI agent who is said to have reported Ivins's suspicious behavior. In fact the redacted online report gives few if any citations for "facts" reported in its narrative.
The psychiatric report was published in 2011 by Research Strategies Network (RSN) under the leadership of Gregory B. Saathoff, a University of Virginia psychiatrist who has been an FBI behavioral consultant since 1996. The report's authors were given the mission of reviewing decades' worth of Ivins's psychiatric files and coming up with a behavioral analysis. The report was originally held under wraps, but eventually was released to the public.
Saathoff is currently RSN's president and treasurer.
Among RSN's reviewers were two American Red Cross officials with no record of mental health professionalism. Gerald M. DeFrancisco is head of Red Cross humanitarian services, who also sits on RSN's board, and Joseph C. White is a Red Cross senior vice president. Presumably, justification for their inclusion was the fact that Ivins had, late in his career, joined the Red Cross as a volunteer. Retention of these men demonstrates the political nature of the report.
Another reviewer is a fellow University of Virginia professor, Christopher P. Holstege MD, who shares with Saathoff management responsibilities at an outfit called the Critical Incident Analysis Group
Among luminaries on RSN's board are Edwin Meese III, President Ronald Reagan's attorney general, and Charles S. Robb, the former Virginia governor and U.S. senator. Meese also sits on the board of the critical incident group.
On Sept. 4, 2014 this reporter asked most RSN reviewers (some had no publicly available email addresses) these questions:
1. What did your contribution to the psychiatric panel entail?
2. Was there any thought of or opportunity for minority opinion?
3. Did you believe that the narrative portion of the report was reliable and gave you the necessary background for a psychiatric evaluation? Who prepared the narrative portion?
4. Did you personally subscribe to the belief that there was strong reason to conclude that Dr. Ivins had acted alone in carrying out the attacks?
5. Did you endorse all the findings and recommendations of the report?
No response was received from any of those emailed.
(For further information on RSN, please see Appendix A below.)
Princeton puzzles
"The letters were mailed from a mailbox in front of KKG in Princeton," the summary asserts, referring to offices of Kappa Kappa Gamma, a sorority with which Ivins was obsessed, at 20 Nassau Street.
Photo by Paul Conant
The storage box on the right dates from the 1960s. It is of a different style from the box on the right in the 2002 photo. In the 2005 photo, the box on the right is painted blue, though a careful look shows that is not identical to the other boxes. In the most recent photo, the right-hand box is a storage box that hasn't been painted in many years. Green-yellow boxes were phased out decades ago.
That's the FBI claim, based on the reported finding of spores at a mailbox a few steps from Nassau street -- a local name for Route 27 -- and Route 206. The mailbox site is at Nassau and Bank streets, across from University Place.
In 2002 there were three mailboxes at Nassau and Mercer, according to an observer, Richard M. Smith. Two were for receiving mail and one for storing mail temporarily. (Smith's URL has been deleted since this report appeared.)
There is now in November 2014 one standard mailbox and the storage box. So we can assume that the presumed spore-laden box was removed. It is not apparent whether the other standard mailbox was removed and replaced, though the Wall Street Journal in August 2002 said officials had removed "the box" for further tests. Attack letter postmarks had led to the swabbing of mailboxes in the Hamilton, N.J., postal region, with swabs tested for anthrax spores.
One can perhaps assume that the singular "the box" implies that the other standard mailbox at Nassau and Mercer tested negative. It is not evident whether the right-hand storage box was removed in 2002. Press accounts are foggy on whether the mailboxes flanking the tainted box were also removed.
The faded green storage box, which was manufactured in the 1960s, still stands at Mercer and Nassau.
This writer checked news reports from the period and found nothing about a decontamination operation at the busy intersection. One would think that once the site was definitively identified as anthrax-contaminated, the area around the tainted mailbox would have been decontaminated and, as a safety precaution, all other mailboxes on site removed. It seems unlikely that, had it been removed, the old storage box would have been returned to the spot.
Between 2005 and 2014, one box was removed and an ancient storage box was substituted.
In addition, nearby is a large shade tree, meaning that spores would have been protected from sunlight and could have posed a significant hazard for months or years. But, there seems to have been no concern about this possibility. It's possible that the local climate is inhospitable to anthrax spores. Yet, in light of the drastic anti-contaminant measures elsewhere, it seems odd that the public was allowed to stroll by the area around the tainted box.
Obviously, it is possible that a decontamination was conducted very quietly, or that authorities were simply negligent about public safety. However, it seems fair to wonder whether the letters were sent from the location claimed.
This suspicion is underpinned by Glenn Greenwald's report in Salon arguing that the FBI had been forced to change its story about the Princeton mailings after he and others spotted an absurd contradiction.
Greenwald challenged a Washington Post story that said, .
"Ivins."
Greenwald wrote, ."
The Post then ran a story based on federal sources that modified the timeline, having Ivins driving to Princeton on the evening of Sept. 17.
An editor might say that use of anonymous sources poses risks to a story's accuracy. And it is also possible a reporter misinterpreted what he or she was told.
But this mistake should be seen in light of a pattern of federal errors, deceptions and omissions as to its theory about Ivins.
Silicon alley
From the summary (with numbering added):
[1] .
[2] summary continues, '.”
This reporter's analysis follows:
[1] There are a number of means to weaponize anthrax spores: gene-splicing or possibly cultivation to increase resistance to antibiotics; microencapsulation to shield spores from the human immune system; purification, along with possible neutralization of electric charges to make spores go airborne easily; and addition of fused silica, which is commonly used to make powders less sticky and hence easier to aerosolize. When Ivins and his immediate supervisor, Lt. Col. Jeffrey J. Adamovicz, opened the plastic bag holding the Daschle powder, they were startled to see the spores fly out all over the place, according to Stevens case testimony.
Tests of the attack matter showed that it had no antibiotic resistance, indicating that advanced weaponization had not been used. However, virologist Jahrling was alarmed by the initial silicon signal and rushed to the White House to brief top officials.
[2] The Sandia analysis of attack powders sent to Sen. Patrick J. Leahy and the New York Post showed that fumed silica was not present in either sample, but that the spores tested positive for silicon.
According to NAS scientists, the evidence they saw convinced them that the amount of silicon detected in the Leahy powder could be completely accounted for by natural spore uptake from silicon in laboratory equipment. However, they pointed out that there was substantially more silicon detected in the Post powder than could be accounted for by silicon uptake in spore coats. This fact was obviously known to federal investigators when the summary was being prepared.
[For more on the silicon uptake matter, please see Appendix B below.]
The Post powder from the envelope postmarked Sept. 18 was a much more crudely prepared form of anthrax than the Leahy and Daschle powders sent two weeks later with an Oct. 9 postmark.
Asked at an FBI science briefing about the big discrepancy between silica levels in the attack powders, FBI scientist Vahid Majidi replied, "Well, the water in New Mexico has ten times more silicant in it then the water in some other states..."
The NAS panel would later observe: "The high levels of silicon found in the attack powders are extremely unusual."
A reporter at an FBI science briefing asked, "Would it be fair to say then that the silica and oxygen presence in these spores was, for want of a better term, accidental or not intentional or put there by God or something, but it just happened?"
An unnamed official responded: "Well, there are scientific reasons behind it. I mean, you know, Bacillus species often produce proteins that are -- whose sole purpose is to chellate metals and other minerals. And the theory behind it is that it makes the spore heartier. That if the spore mineralizes they're more -- so that's a scientific theory."
"But that's something the anthrax did, not man?"
"The understanding of that process is not well understood," the official replied.
Or, in other words, the science briefers have parried the issue about high silicon content in the Post letter without saying that its presence was regarded by microbiologists as very unusual.
A related question at that briefing: "Did you try and duplicate the process? And how close did you get to making something like, you know -- the finer preparation that appeared?"
Majidi replied, "We were able to get those spores minus the silicon signal."
At another briefing for science writers, a questioner asked whether there was any truth to news reports that the FBI had been unable to replicate the attack spores.
Majidi replied, "If I make soup at home at two different times, they are not going to taste the same. So the fact that we can't duplicate a single powder prep is not unusual in any realm." At this briefing, he kept mum about the inability to replicate the silicon signal.
The public has not been told what the FBI scientists did to reverse-engineer the attack anthrax, with Majidi having excused that opaqueness on national security grounds. But presumably the FBI's consultants used equipment comparable to what was available at Fort Detrick. Despite assuring the press that the silicon signal resulted from a "natural" process, the FBI scientists skirted around the anomalous Post result.
Ivins told the FBI that when he saw the purity of the Daschle powder. he immediately thought: "Fermenter!" He explained that fermenters tend to produce clean spore concentrations while matter grown on agar plates is much dirtier, or contaminated by other biological material.
Interestingly, the Post and NBC powders were quite crude. So the crude matter had silicon added, but silicon that would do no harm. One wonders if someone added silicon in an amateurish attempt to weaponize it, or whether someone added it to make it look as though semi-trained al Qaeda terrorists were responsible. Of course, one can come up with a scenario in which Ivins adds silicon in a devious ruse to fool investigators. According to the summary, Ivins's seemingly odd late hours in the lab began in August, weeks before Sept. 11 (and federal officials have never explicitly said that they believe Ivins began preparing attack powder before 9/11, though that is the logical conclusion of their scenario).
After Sept. 11, the summary shows, Ivins's lab hours zoomed up and fell rapidly once all anthrax attacks were completed.
Yet one wonders why Ivins didn't bother to add silicon to the new powder, and why he was no longer concerned with making the powder look crude (grown from agar plates, which would have been within al Qaeda's capability), but now went full-bore with a fermenter (assuming he somehow got his work place fermenter up and running).
Why the change in anthrax quality? It has been reported that at the time the Senate attack letters were mailed the Bush administration was high-pressuring Senate Democrats -- in particular Leahy and Daschle -- to get swiftly behind the emergency powers, or Patriot, bill,. Otherwise, the public is left to speculate that Ivins had already planned to blame al Qaeda when he was preparing the crude powder, but after 9/11 was electrified into throwing caution to the wind by dramatically ramping up his action.
The point here is that the FBI and Justice Department blurred the matter of high silicon levels in some attack powder by deliberately confusing that issue with the fact that other attack powder had routine levels of silicon.
A paper by biologists Martin E. Hugh-Jones, Barbara Hatch Rosenberg, and Stuart Jacobsen argues that the high level of silicon, along with iron and tin, detected in the attack spores suggests the possibility of microencapsulation of spores, which they say is a known method of weaponization. Rosenberg, an expert on biological weapons, has been a persistent critic of the FBI's anthrax probe.
However, if microencapsulation was used, one wonders why the cruder spore powder would contain encapsulated spores but not the purer powder sent later. (In this respect, it may be noted that the NAS panel cast a skeptical eye on other anomalies NAS scientists noticed in the spore test reports, as well as scorning the FBI's statistical analysis.)
The FBI's Amerithrax page has a link titled "Science Briefing on the Investigation," but only the opening remarks, which includes introduction of the FBI's blue-ribbon group of consultant scientists, is available. This reporter obtained transcripts of the two science briefings online after some effort, as at the time of writing they were not readily available via Google.
Aug. 18, 2008 science briefing for the general press
Aug. 18, 2008 briefing for science writers
or
Rubber sole
From the summary:
"Finally, when RMR-1029 was sent over for the aerosol challenges, it was frequently diluted substantially, usually 1,000-fold. Given the highly concentrated material used in the mailings, experts consulted have stated that it is extremely unlikely that such diluted material could have been used in the mailings."
This statement makes it seem as though the attack material would have come straight from the 1029 flask. But all scientists, including the FBI's scientists, were keenly aware the attack spores were grown from a seed batch of the 1029-type substrain of Ames. Scientists testified in the civil case that a 1029-type preparation could have been grown from a very low number of spores. Dilution would not have been an issue if the necessary time and equipment were available.
One must be cautious about the various Justice Department claims about dilutions, dilutions that could have been done deliberately in order to conceal the presence of the four genetic markers found in the attack spores. At one point, an FBI report says, a bureau microbiologist challenged Ivins on the Navy's finding that a 1029 sample its experts analyzed had on the order of 109 (or one billion) spores per unit milliliter. Ivins expressed skepticism and permitted the FBI scientist to stand at his side and replicate every part of Ivins's measurement process. Both Ivins and the FBI scientist obtained a 1010 (or 10 billion) measure.
From the behavioral analysis:
the summary:
."
In fact, Ivins only acknowledged being responsible for the flask, telling the FBI that the RMR-1029 flask had always been in building 1425, where he worked, but that vials of the material were held in building 1412 for use in animal experiments. However, after Ivins died, Adamovicz, his supervisor in 2001, produced a copy of an Army document showing that the RMR-1029 flask had been stored for a while in the late '90s in building 1412. Adamovicz told a court proceeding that the FBI "didn't want you to see" that receipt copy, which he testified he had found in his personal files.
There is nothing unusual about Ivins's apparent forgetfulness or misperception, and, in fact, Adamovicz's document establishes that various people other than Ivins had access to RMR-1029 prior to the attacks. Those with access or potential access included Steven Hatfill, who was reportedly suspected of possibly obtaining waste anthrax from containers that were awaiting sterilization in an autoclave. Louise Pitt, who ran the animal experiments, and various animal handlers and others had direct access.
Presumably, these experiments were still going on in 2001, though an interim lack of funding seems to have limited them in August and early September.
Note that the summary statements above convey the impression that Ivins had control of the 1029 anthrax fluid, as opposed to the beaker in which it was held, implying that the attack spores could only have been grown from the liquid in flask RMR-1029.
However, prodded by reporters, FBI scientist Majidi conceded that material identical to the RMR-1029 material had been located at Battelle Memorial Institute in Ohio. Battelle is a major CIA and defense contractor known to have worked on biological defense matters. Three biologists, citing the investigative summary, said the FBI made unwarranted assumptions that the attack anthrax could not have been made at Battelle while a buddy system was in force -- that is, the FBI assumed that only a single individual was involved -- and that Battelle and other laboratories were too far from New Jersey, where postmarks showed that the attack letters were mailed. The FBI reportedly checked commercial flight records and found nothing.
When a science writer asked where the 1029-type anthrax had been found, Majidi replied, "What we found was in RMR-1029, the repository, and then the laboratory, and the letters." Majidi was referring to the flask numbered RMR-1029, the FBI's repository of anthrax samples from numerous labs, Ivins's laboratory and the attack letters.
There is no mention of Battelle in that briefing.
Further, according to the three biologist critics, 10 laboratories showed samples that had one or more genetic markers. The fact that all four markers did not appear in these other samples does not mean none of them was a match, the trio of biologists wrote. False negatives are a rather common problem in biological experiments.
The scientists observed that among the eight laboratories that submitted a total of 63 samples with between one and three positive assay results were Dugway Proving Ground in Utah, the Naval Medical Research Center, Northern Arizona University, the Canadian Defense Research Establishment at Suffield, and a second sample from Battelle. Also, in December 2001, federal sources told the New York Times that Fort Detrick anthrax had been sent to the University of New Mexico.
"The submitters of the other three samples have not been revealed," the scientists wrote.
The three argued that the most likely sites of production of the attack anthrax are those that worked with dry spores: Battelle, Dugway, and Suffield, and their associated institutions and subcontractors.
"Battelle, for example, is well-known for its aerosol study capabilities and biodefense activities, for which dry spores are routinely needed," the three experts observed.
Ivins told the FBI that his institute never worked with dry spores because to do would pose an exceptional danger in the metropolitan area around Washington.
However, federal officials said that Ivins had had training on a lyophilizer (freeze dryer), which could have been used to dry anthrax. Ivins said he doubted use of that machine because it would damage the spore preparation too much for good quality powder. Ivins's colleagues argued that he couldn't have used the lyophilizer, which was not in a hot suite, undetected. One colleague, Patricia Worsham, testified that she would have thought non-vaccinated persons who worked in the unprotected area would have been sickened by anthrax.
Others testified that decontaminating the machine required a special mechanical process by trained technicians, and also that Ivins would have been unable to manhandle the machine into the hot suite in order to use it in a safe environment. But Paul Keim, the FBI consultant scientist whose genetic analysis pointed to flask RMR-1029, was skeptical of that argument, saying that microbiologists know how to clean up.
Though the Justice Department is unwilling to talk much about potential conspiracy, it seems as though FBI agents were not that naive. An anthrax scientist who worked at Battelle drew the FBI's interest in 2002 and quickly spiraled into alcoholism and death, relatives told the New York Times. Little is known about what caused Perry Mikesell to snap. He may have had remorse over recognition that his career choice was marked more by death than by life. Or, he may have felt shunned by fellow workers to the point that he was unable to cope. Or, Mikesell, who had worked alongside Ivins for years at Fort Detrick, may have felt remorse over complicity in treachery and anguish about the possibility of being tried for treason.
Whether Mikesell's anguish -- so similar to that experienced by Ivins as his end drew near -- was deserved or not, it seems apparent that the FBI field agents were unconvinced by the buddy system records at Battelle, which does much classified work. Nor were they convinced by commercial flight records.
However, no covert operation -- whether authorized or rogue -- via secret military flights could be officially considered -- though the attack anthrax is apparently derived from a military laboratory.
Hatfill, who was cleared by the Justice Department shortly before Ivins's death, had supervised construction of a mobile biowarfare lab while working for a major defense contractor, Science Applications International Corp. (SAIC).
Steven Hatfill vehemently denies involvement in anthrax terrorism. (A.P.)
The newspaper accounts do not say whether the FBI was able to keep the lab under eyeball surveillance at all times, or whether its examiners, as opposed to Army examiners, were permitted to swab it.
As noted above, the behavioral analysis says:"Later, Dr. Ivins’ technicians reported that they had never seen the flask. He had been its sole custodian and presumably had kept it concealed in the cooler."
It continues:
"When investigators pressed Dr. Ivins's two lab technicians to describe what RMR-1029 looked like. neither of them could do so. They were aware that Dr. Ivins had created a spore preparation called the 'Dugway Spores' -- which Dr. Ivins explained to the prosecution team was another name for RMR-1029.
"However, neither lab technician was aware that the 'Dugway Spores' were contained in two flasks, and neither knew what a flask containing the 'Dugway Spores' looked like."
The idea that Ivins had been deliberately concealing the flask is somewhat blunted by the recollection of the technicians that they knew about the special anthrax material and by the testimony of Ivins's boss, Adamovicz, who said that though he couldn't recall whether he had ever observed the flask, he certainly knew of its existence. Ivins also told the FBI that he didn't supply much of that anthrax to many researchers, though many wanted it.
From the behavioral analysis:
"Do the sealed psychiatric records support or refute the Department of Justice’s determination that Dr. Ivins was the sole mailer of the anthrax letters?"
The thrust of the report is yes. The report takes Ivins's mental health files and comes up with a psychological scenario that seems consistent with various clues, such as those in the attack letter addresses. The speculation is couched in scientific-sounding terms. And yet, it is quite interesting that patterns of evidence pointed to others, as well: Steven Hatfill, Perry Mikesell and Joseph Farchaus, all former Fort Detrick scientists.
A welter of intriguing associations surrounded Hatfill, prompting strong suspicions aired by Rosenberg and New York Times columnist Nicholas Kristof. Hatfill strenously denied having been involved in Rhodesia's germ warfare operations against rebels, though the Missourian returned from Africa in the 1990s and gravitated into Pentagon biological warfare work. His medical doctorate was valid but his PhD in microbiology was bogus. Nevertheless, after a two-year postdoctoral stint at the Centers for Disease Control, he went to Fort Detrick for another two-year post-doctorate as a virologist.
While there he became very friendly with the aging William Patrick, father of America's offensive biowarfare program, which had been officially scrapped by President Richard Nixon.
While at SAIC, Hatfill had commissioned a report from Patrick on risks presented by mailed anthrax that had been published in 1999. (Some observers have wondered whether the still-classified report was a "blueprint" for the 2001 mail attacks; Patrick had foreseen a low risk but had failed to consider the high-speed mail-sorting machinery that was said to have "milled" the powder into a purer form that made it highly dangerous.)
The FBI eventually ruled out Hatfill on grounds that his property and Patrick's basement, which contained a biological laboratory, were clean of anthrax.
The summary exonerates Hatfill thus:
bio-containment the Ames strain anthrax while at USAMRIID, he never had access to the particular spore-batch used in the mailings."
Now it is possible that the bureau was actually basing its decision to clear Hatfill on data that have been withheld, and so one should beware leaping to any conclusions in that respect.
However, consider these points:
The government is assuming Hatfill cannot be guilty because he could not have pulled off the attacks without inside help.
Interestingly, Ivins told the FBI that he had worked on a national security case concerning anthrax "from Iraq," and had deliberately mislabeled it in order to conceal its national security status. It is not necessarily ridiculous to wonder whether Ivins was a go-to person in what he thought was a national security matter. (If so, however, Ivins apparently went to his death with his lips sealed.)
The summary's contention that Hatfill had no access to the Ames 1029 substrain of attack anthrax is countered by testimony that Hatfill had worked largely in Building 1412 where the 1029 flask was stored for a time in the nineties and where vials of the 1029 substrain were used for animal experiments. Hatfill was at Fort Detrick until 1999, but the flask-1029 material was formulated sometime between 1997 and 1999.
Another scientist, Farchaus, lived within 15 minutes of Princeton, where the attack letters were apparently mailed. Additionally, he reportedly had a relative who lived very close to an elderly Connecticut woman who died from anthrax. To stretch a point, the locales of Kendall Park and Monmouth Junction, which have a return address and a zip code that relate to the mailings, are both roughly 15-minute drives from Princeton.
In the case of Mikesell, whom the summary doesn't mention at all, we don't know what the associations were that raised FBI suspicions. But apparently it was thought that some things didn't add up.
All this is to say that patterns of circumstantial evidence can be hung around the neck of any one of these men.
Tales from decrypt
The Justice Department, strongly assisted by the FBI psychiatric consultant team, made it appear that Ivins was strangely obsessed with codes and that this obsession is reflected in the code in the two Senate attack letters.
The emphasized A's and T's in the identical letters, the FBI found, relate to two of the letters used in DNA codons.
To buttress its case of a pattern of Ivins associations with DNA codons, the summary relates that on July 27, 2000, Ivins "forwarded" an email to former colleague no. 1 (Fellows), which began "Biopersonals: I have single-stranded too long! Lonely ATGCATG would like to pair up with congenial TACGTAG" along with a note "this is some cute humor for anyone who has ever had anything to do with biochemistry or molecular biology."
A number of coworkers have testified that Ivins enjoyed sending off funny emails, and the term "forwarded" implies the DNA joke was making the rounds among scientists. The joke is that in the replication process, a DNA string joins together with its inverse.
Perhaps the email would have been relevant had the joke DNA strings carried secret messages similar to the purported messages in two of the attack letters. But, following the FBI decoding procedure, the result for the first string is, "mhm" and "eee." For the inverse string, the result is "tv(stop)" and "ee(stop)" or perhaps "tvs" and "eep".
So how is the joke email relevant? It isn't. Not to worry. A footnote explains:
"This e-mail was notable not because of any particular meaning ascribed to those specific nucleic acids, but rather because it demonstrated Dr. Ivins’s familiarity with DNA, specifically As, Ts, Cs, and Gs."
And yet Ivins was a microbiologist who would have had at least some professional knowledge of DNA codons.
At any rate, the FBI code-crackers examined the "bolded letters" and discovered the left-to-right string
TTT AAT TAT – an apparent hidden message. The three-letter groups appeared to be codons, meaning that each sequence of three nucleic acids will code for a specific amino acid.
TTT = Phenylalanine (single-letter designator F)
AAT = Asparagine (single-letter designator N)
TAT = Tyrosine (single-letter designator Y)
The FBI, says the summary, gleaned two meanings from this analysis: "PAT" and "FNY."
"Pat" is short for his former technician, Patricia, with whom Ivins was allegedly obsessed, the summary says. Patricia Fellows left Fort Detrick to upgrade her education and further her career.
On the other hand, "Pat" is a common name. For example, there is another Fort Detrick anthrax scientist, Patricia Worsham. Then there is Patrick Leahy (though the coded mailings went to the media and not the Senate). Or what of the father of the U.S. bioweapons program, William Patrick?
As for, "FNY," this is taken as representative of Ivins's psychotic antipathy toward New York. But, no evidence of such an extreme hatred is produced. Rather, the summary mentions his dislike of New York City, which is a distaste shared by many Americans and the reason New York officials promoted the "I love N.Y." campaign. At another point, the summary portrays his fan loyalty, expressed in a joking manner, as a psychotic hatred of the New York Yankees.
The summary continues, "It was obviously impossible for the Task Force to determine with certainty whether either of these translations was correct." True, one can't even assign a statistical confidence interval. But, argues the Justice Department, "the key point is that there is a hidden message, not so much what that message is."
But if the FBI is unsure of the meaning, then the idea that the two messages refer to his young friend, who had gone on to pursue a medical degree, and New York City rests on air. In 2001, colleges around the nation were beginning to prepare students for the genetic engineering revolution sweeping industry and academia. Even non-biology majors were often given some awareness of DNA codons.
On the other hand, as officials assert, Ivins voluntarily told the FBI, after confiding in Fellows, that as a younger man he had broken into KKG sorority houses and stolen a decoder used for secret rituals and later the code book itself. Also, he liked to send gifts to women coworkers and let them guess who was sending them, which they usually did. The FBI sees this behavior as a desire to have the women "decode" his cryptic antics. As he was in his early thirties at the time of the sorority burglaries, such actions -- had they become known -- would have destroyed his career, underscoring the idea that the scientist suffered from a long-term mental illness.
A couple of other points:
The highlighted letters are the same as those found in Mohamed Atta's last name.
This reporter was able, by converting the T's and A's to the 1's and 0's of the binary number system and using a bit of numerology, to come up with a reading of "9/11."
But supposing the FBI's interpretation is partly correct, how do we know that the message wasn't intended simply as "FNY"? Why assume both messages were intended?
Also, why send letters overtly blaming Islamic extremists but "covertly" pointing to a "clever" scientist? The summary's theory is that Ivins was intrigued with the idea of wrapping one message inside another, as discussed in a book he owned.
Anyway, the summary relates, "Ivins showed a fascination with codes and also had an interest in secrets and hidden messages" and he "was also was familiar with biochemical codons." As a matter of fact, large numbers of people are enthusiastic about codes. An FBI website acknowledges this public interest and invites participation of code buffs.
The Justice Department goes to absurd lengths in its desire to persuade. Consider these words from the summary:
"Finally, Dr. Ivins’s own words demonstrated that he enjoyed playing detective and unlocking secrets. In an e-mail to Former Colleague #1 (Linscott) on June 26, 2000, he wrote: 'For me, it’s a real thrill to make a discovery, and know that I’ve just revealed something that no one else in the world ever knew before. I feel like a detective, and that which is unknown dares me to try to find out about it, to decipher its code, to understand it, to fit it into the puzzle or 'Big Picture'.”
Officials seem to be overlooking the fact that Ivins was a scientist, and this is the sort of thing scientists say.
The FBI reported that Ivins owned a 1992 copy of American Scientist that includes an article "The Linguistics of DNA," which discusses, "among other things, codons and hidden messages."
This reporter scanned the article by D.B. Searls and found that indeed there are a few graphs that contain codons. However, the article is a dense excursion on the DNA process as a form of information transfer. It is not concerned with secret messages or cryptography. The article is available at JSTOR. [Please see footnote on Searls background below.]
Another item owned by Ivins was a copy of the book Godel, Escher and Bach -- An Eternal Golden Braid (abbreviated GEB) by Douglas Hofstadter.e Morgan
Abel
Boole
Brouwer
Sierpinski
Weierstrass
In this passage, Achilles explains, “I believe it is supposed to be a Complete List of All Great Mathematicians. What I haven’t been able to figure out is why the letters running down the diagonal are so much bolder.” To which Tortoise replies that at "the bottom it says, ‘Subtract 1 from the diagonal, to find Bach in Leipzig'.”
If one decodes this list as directed, the answer is “Cantor.” To understand this message properly, one needs to know that Bach was the Cantor of Leipzig.
This passage is similar to the emphasized A's and T's in the anthrax attack letters, the summary writer believes. Except that the anthrax code seems to be far less clever than Hofstadter's.
By itself the coincidence is unremarkable. So what that scientist Ivins owned a book on scientific matters that happened to contain similarities to the anthrax letter code? And the paper by Searls seems barely relevant. Also, any American capable of launching the anthrax attacks is likely to have been familiar with DNA codons, and quite a few may have owned a copy of GEB, which achieved enduring popularity in the scientific community after its appearance in 1979.
But, the summary answers such doubts by relating that, soon after a search of his house, the harried scientist in November 2007 threw out GEB and the magazine containing the Searls article.
"The night he did so, Dr. Ivins behaved in the fashion of a nervous man, watching for the garbage truck, and then checking the garbage can to ensure that it was gone, and finally checking the bushes to see if he was being watched."
One may wonder when the video recording of this incident will be released.
The summary does not consider the possibility that, by this time, a panicky Ivins was worried that the FBI would "read something into" the book and the magazine. Another point: was Ivins really so naive as to not realize his garbage would be checked after pickup? If the written materials were so incriminating, why not wait until no one is home and burn them? Even so, it must be granted that anxious people make foolish mistakes.
At another point, the summary says, Ivins lent GEB to a female friend, recommending it enthusiastically. Upon learning sometime later that she hadn't read it, he asked for its return. This is meaningful, says the summary, because when asked by the FBI in January 2008 about books he had lent the woman, he did not mention GEB, "one of his favorites," that he had given her about a year earlier.
Granted, an alert FBI agent would see this as grounds for suspicion of evasiveness. On the other hand, human memory is notoriously fickle, especially after a year's lapse. And there remains the other possibility that an innocent man -- who as it happens suffered from episodes of clinical paranoia -- feared the government would want to read something in to his possession of GEB.
The summary adds, "Also reinforcing the importance of GEB to Dr. Ivins was the fact that he once sent an email to Janna Levin, complimenting her work, presumably referring to A Madman Dreams of Turing Machines, a book that discusses GEB and chronicles the lives of Godel, known as the world's greatest logician, and Alan Turing, known as an exceptional code-breaker."
A search of the novel's text via Amazon returns blanks for search terms "Escher," "Bach" and "Hofstadter."
The Justice Department chooses to emphasize Turing's code-breaking, but he was also a giant of mathematical logic, whose "Turing machine" was an intellectual exercise that, along with Godel's chief theorem, revolutionized both logic and mathematics.
In words paralleling those in the summary, Rachel Lieber, lead federal attorney in the Ivins case, told Frontline in 2011 that the “confluence of all these things taken together, that's the compelling evidence.”
“It's only when you take a step back and you look at all the evidence taken together can you realize this is the right person,” Lieber said.
In August 2008, soon after Ivins's suicide, U.S. Attorney Jeffrey Taylor, FBI Assistant Director Joseph Persichini and other officials assured the press that the pattern of circumstantial evidence against Ivins as the sole anthrax attacker was strong.
FOOTNOTE: Searls lists his expertise as computational and systems biology, pharmacoinformatics, macromolecular linguistics, data integration, philosophy of science
Searls, D. B., 1992. The linguistics of DNA. Am. Scient. 80: 579-591.
Appendix A
The psychiatric report lists Saathoff and DeFrancisco as chair and vice chair, respectively.
Others:
David Benedek, MD, psychiatry professor at the Uniformed Services University School of Medicine; Anita Everett, MD, psychiatrist with the Johns Hopkins University School of Medicine; Christopher P. Holstege, MD, a toxicologist at the University of Virginia School of Medicine; Sally C. Johnson, MD, psychiatry professor at the University of North Carolina-Chapel Hill; J. Steven Lamberti, MD, psychiatry professor at the University of Rochester Medical Center; and Ronald Schouten, MD, a psychiatrist specializing in legal matters with the Massachusetts General Hospital Harvard University School of Medicine.
J. Patrick Walsh is listed as a "special assistant and coordinator to the panel and its operations."
Appendix B
An FBI official explained that the possibility of natural silicon uptake into the spore coats had been discovered during a review of scientific literature. An old published paper led investigators to the widow of of one of the authors, A.P. Somlyo, who still had her husband's samples from the experiment on hand. Analysis of the Somlyo samples confirmed the presence of silicon in the spore coats, the official said.
The Somlyo article mentioned above says the silicon spike Somlyo and his coauthors detected was unlikely to have been solely from a contaminant, from the glass containers or from silicon material used in freeze drying. So, the online copy leaves us to conjecture that it came from the growth medium.
Stewart, M., A. P. Somlyo, A. V. Somlyo, H. Shuman, J. A. Lindsay, and W. G. Murrell. 1980. Distribution of calcium and other elements in cryosectioned Bacillus cereus T spores, determined by high-resolution scanning electron probe x-ray microanalysis. J. Bacteriol. 143:481-491. [PMC free article] [PubMed]
The online copy's description of method curtly says that the method is identical to that used by another research team as described in “Cytological and Chemical Structure of the Spore” by W.G. Murrell, D.F. Ohye and Rosalind A. Gordon. This reporter's copy of that article shows that no silicon was among chemicals tested, and there is no discussion of silicon uptake. However, the article says that a New Brunswick Scientific shaker was used. Current New Brunswick shakers come with a silicone mat.
One of Somlyo's coauthors was W.G. Murrell. He is also coauthor of another paper in which silicone antifoam is used in experimental preparations.
DRAFT 0: Nov. 3, 2014
DRAFT 1: Nov. 5, 2014
DRAFT 2: Nov. 7, 2014. Correction of a very minor inaccuracy.
DRAFT 3: Nov. 13, 2014. Very minor clarification.
DRAFT 4: March 1, 2016. Updated and corrected. | 76,100 |
\section{Topological toolkit}
The proofs of the following lemmas can be found in Chapter~1 of~\cite{Ribes2010}.
\begin{lemma}
\label{lem:surjections-between-cofiltered-diagrams}
Let $\tau\colon D_1 \to D_2$ be a natural transformation between cofiltered
diagrams (of the same index) in the category of compact Hausdorff
spaces. If each $\tau_i\colon D_1i \twoheadrightarrow D_2i$ is surjective,
so is the mediating map
\[
\Lim \tau \colon \Lim D_1 \to \Lim D_2.
\]
In particular, if $\tau_i\colon X \twoheadrightarrow Di$ is a cone of
surjections, then the mediating map $h = \lim\tau$ is surjective.
\end{lemma}
\begin{lemma}
\label{lem:dense-lemma}
Let $(X \xrightarrow{\rho_i} X_i)_i$ be a limit of a cofiltered diagram
in the category of topological spaces, and $(Y \xrightarrow{f_i} X_i)_i$ a
cone of that diagram consisting
of surjective
continuous functions. Then, the mediating map
\[
f\colon Y \to X
\]
is dense, i.e.\ the image~$f[Y]$ is a dense subset of~$X$.
\end{lemma}
\begin{lemma}
\label{prop:projection-is-surjective}
Let $D$ be a cofiltered diagram in the category of compact Hausdorff spaces.
If all $D(i \xrightarrow{f} j)$
are surjective, so is each projection $\rho_i\colon \Lim D \to Di$.
\end{lemma}
\section{Detailed proofs}
This appendix contains all proofs and additional details we omitted due to space restrictions.\\
\begin{defn}
An object $A$ in a category $\A$ is called \emph{finitely copresentable} if
the hom-functor $\A(\mathord{-},A): \A^{\op}\to\Set$ preserves filtered
colimits. Equivalently, given a cofiltered limit cone $\pi_i: B\to B_i$ ($i\in
I$) in $\A$, any morphism $f: B\to A$ factors essentially uniquely through
some $\pi_i$. The category $\A$ is \emph{locally finitely copresentable} if
(i) $\A$ is complete, (ii) $\A$ has only a set of finitely copresentable
objects up to isomorphism, and (iii) every object of $A$ is a cofiltered limit
of finitely copresentable objects.
\end{defn}
\begin{rem}\label{rem:lfcp}
For any small category $\A_0$ with finite limits the pro-completion $\A=\Pro{\A_0}$, i.e. the free completion of $\A_0$ under cofiltered limits, is locally finitely copresentable, and the full subcategory of finitely copresentable objects is equivalent to $\A_0$. See e.g. \cite[Theorem 6.23]{arv11}.
\end{rem}
\noindent\emph{Details for Remark \ref{re:hatD-is-procompletion}}. By \cite[Remark VI.2.4]{Johnstone1982} the category $\hatD$ is the pro-completion of $\D_f$. (The argument given there is for varieties of single-sorted algebras, but also applies to the ordered and many-sorted case.) Hence $\hatD$ is locally finitely copresentable and the objects of $\D_f$ are precisely the finitely copresentable objects of $\hatD$, see \Cref{rem:lfcp}. This implies (ii). For (i) see \cite[Proposition 1.22]{Adamek1994}.
\begin{lemma}\label{lem:hatdlimits}
$\hatD$ is closed under cofiltered limits in the category of topological
$\D$-algebras. In particular, the forgetful functor $V: \hatD\to \D$ preserves
cofiltered limits.
\end{lemma}
\begin{proof}
Let $\Top(\D)$ denote the category of topological $\D$-algebras. Since $\hat
J: \D_f \monoto \hatD$ is the pro-completion of $\D_f$ the inclusion functor
$I: \D_f\monoto \Top(\D)$ has an essentially unique extension $I': \hatD\to
\Top(\D)$ preserving cofiltered limits. The functor $I'$ maps an object $D\in
\hatD$ to the cofiltered limit $I'D$ of the diagram of all morphisms $f: D\to
D'$ with finite codomain. But since $D$ is profinite it follows $I'D \cong D$,
that is, $I'$ is just the inclusion.
That $V$ preserves cofiltered limits now follows from the fact that limits in $\Top(\D)$ are formed on the level of $\D$.
\end{proof}
\begin{rem}\label{rem:arrowcatclfp}
Since $\hatD$ is locally finitely copresentable so is the arrow category
$\hatD^\to$. Its finitely copresentable objects are precisely the morphisms
in $\D_f$, i.e. morphisms with finite domain and codomain, see
\cite[Corollary 1.54 and Example 1.55]{Adamek1994}. Hence every morphism
$f\colon A \to B$ in~$\hatD$ is a cofiltered limit (taken in $\hatD^\to$) of
morphisms in $\D_f$.
\end{rem}
\begin{proof}[Lemma \ref{lem:hatD-factorisation}]
Let $f\colon A \to B$ in~$\hatD$. Express $f$ as a cofiltered limit of
morphisms $f_i$ in $\D_f$ with the
limit cone $(a_i, b_i)\colon f \to f_i$, see \Cref{rem:arrowcatclfp}.
Factorise $f_i=m_i\o e_i$ into a quotient followed by a subalgebra in $\D$.
\[
\vcenter{
\xymatrix{
A \ar[rr]^f \ar[d]_{a_i} & & B \ar[d]^{b_i} \\
A_i \ar@{->>}[r]_{e_i} \ar `d[r]`[rr]_{f_i}[rr] & C_i
\ar@{ >->}[r]_{m_i} & B_i.
}
}
\]
Due to the diagonal fill-in property, the finite objects $C_i$ form a
cofiltered diagram with $(e_i)$ and $(m_i)$ natural transformations. Let $\left(c_i\colon C
\to C_i\right)$ be its limit in $\hatD$, and, $e \defeq \Lim e_i$, and $m \defeq \Lim m_i$ in $\hatD^\to$,
as shown in the diagram below:
\[
\vcenter{
\xymatrix{
A \ar`u[r]`r[rr]^{f}[rr] \ar[r]^{e} \ar[d]_{a_i} & C \ar[d]^{c_i}
\ar[r]^{m} & B
\ar[d]^{b_i} \\
A_i \ar`d[r]`r[rr]_{f_i}[rr] \ar@{->>}[r]^{e_i} & C_i
\ar@{ >->}[r]^{m_i} & B_i.
}
}
\]
The morphism $e$ is surjective by
\Cref{lem:surjections-between-cofiltered-diagrams}. To show that $m$ is
injective observe that $m(c) = m(c')$ implies
\[ m_i\o c_i(c) = b_i\o m(c) = b_i\o m(c') = m_i\o c_i(c') \]
thus $c=c'$ since $m_i$ is monic and the morphisms $c_i$ are jointly monic.
In the ordered case, to show that $m$ is order-reflecting use instead that
$m_i$ is order-reflecting and the morphisms $c_i$ are jointly
order-reflecting.
\end{proof}
\begin{proof}[\Cref{prop:profcomp}]
Since $V$ preserves cofiltered limits by Lemma \ref{lem:hatdlimits}, there is
a
unique mediating morphism $\eta_D: D \to V\hat D$ in $\D$ with $V\hat h \o
\eta_D = h$ for all $h: D\to D'$ with $D'\in\D_f$. We need to show that for
any $g: D\to VE$ with $E\in\hatD$ there is a unique $g^@: \hat D\to E$ with
$Vg^@\o \eta_D = g$.
Express $E$ as a cofiltered limit $p_i\colon E\ra E_i$, $i\in
I$, with $E_i\in\D_f$. Then the morphisms $\widehat{Vp_i\o g}: \hat D \to
E_i$ form a cone, so there is a unique $g^@\colon \hat{D}\ra E$
with $p_i\o g^@ = \widehat{Vp_i\o g}$ for all $i$. It follows that
\[ Vp_i \o Vg^@ \o \eta_D = V(\widehat{Vp_i\o g})\o \eta_D = Vp_i\o g. \]
Since $V$ preserves cofiltered limits the morphisms $Vp_i$ are jointly
monic, so $Vg^@\o \eta_D = g$. Moreover, since one can restrict the limit
cone defining $\hat D$ to surjective morphisms $h: D\epito D'$ (they form
an
initial subdiagram of $D\downarrow \D_f$), the morphism
$\eta_D$ is dense by \Cref{lem:dense-lemma} (where $D$ is viewed as a
discrete
topological space). This implies that the morphism $g^@$ is uniquely
determined by the equation $Vg^@ \o \eta_D = g$, as desired.
\[
\xymatrix{
D \ar[dr]_g\ar[r]^{\eta_D} \ar@/^2em/[rr]^f & V\hat{D} \ar[d]_{Vg^@}
\ar[r]^<<<<<{V\hat h} \ar[dr]^>>>>{V(\widehat{Vp_i\o g})} & VD' = D' \\
& VE \ar[r]_<<<<<{Vp_i} & VE_i = E_i
}
\]
We conclude that $V$ has a left adjoint whose action on objects is given by
$D\mapsto \hat D$, and on morphisms $h: D\to E$ by $h\mapsto (\eta_E\o h)^@$.
It remains to show that for $E$ finite we have $(\eta_E\o g)^@ = \hat g$. Indeed, in this case we have $\hat E = E$ and $\eta_E = \id$, and the limit cone $(p_i)$ can be chosen trivial (that is, $I=\{1\}$, $E_1=E$ and $p_1 = \id$). Then
\[ (g\eta_E)^@ = g^@ = p_1\o g^@ = \widehat{Vp_1\o g} = \hat g.\]
\end{proof}
\noindent\emph{Details for \Cref{re:factorisation_DT}.} We prove the claim that the factorisation system of $\D$ of lifts to $\D^\MT$. Letting
$h=m\o e$ be the canonical factorisation of $h$ in $\D$, diagonal fill-in
gives a unique $\gamma: TC\to C$ making the diagram below commute. One
easily verifies that $(C,\gamma)$ is a $\MT$-algebra, so $e:
(A,\alpha)\epito
(C,\gamma)$ and $m: (C,\gamma)\monoto (B,\beta)$ are $\MT$-homomorphisms.
\[
\xymatrix{
TA \ar@{->>}[r]^{Te} \ar@<-.1ex> `u[]`[rr]^{Th}[rr]
\ar[d]_{\alpha} & \ar[r]^{Tm} TC
\ar@{-->}[d]^{\gamma} & TB \ar[d]^{\beta} \\
A \ar@{->>}[r]_{e} \ar@<-.1ex> `d[]`[rr]_{h}[rr] & C \ar@{ >->}[r]_m &
B
}
\]
\begin{notation}\label{not:limitcone}
Recall from Remark \ref{rem:hattconst} that for $X\in\hatD$ the object $\hatt
X$
is the limit of the diagram
\[ (X \downarrow \hatJ U) \to \hatD,\quad (f: X\to \hatJ U(A,\alpha)) \mapsto
A. \]
We denote the limit projections by $f^* : \hatt X \to A$.
\end{notation}
\noindent\emph{Details for Remark \ref{rem:hattconst}.} We show that the
diagrams \eqref{eq:hat-eta} commute for all $h: (TD,\mu_D) \to (A,\alpha)$
with $(A,\alpha)\in \D^\MT_f$. As for the left hand diagram, recall the
definition of $\hateta = (\id_{\hatJ
U})^\dagger$
in the limit formula for right Kan extensions: for each $f\colon X \to
\hat{J} U(A,
\alpha)$, the collection of $f$'s forms a compatible cone for
the limit defining $\hatt X$, and $\hateta_X$ is the unique mediating
morphism satisfying the diagram
\[
\xymatrix{
X \ar[dr]_{f} \ar[r]^{\hateta_X} & \hatt X \ar[d]^{f^*} \\
& \hat{A}.
}
\]
For $X=\hat D$ and $f=\widehat{h\eta_D}$ we have $f^* = h^+$, which proves
the claim. Similarly for the right hand diagram.
\begin{proof}[\Cref{prop:characterisation-hatT}]
By \Cref{re:factorisation_DT},
we can factorise every morphism $h\colon (TD, \mu_D) \to (A, \alpha)$ as a
quotient followed by a subalgebra:
\[
\xymatrix{
TTD \ar@{->>}[r]^{Te} \ar[d]_{\mu_D} & \ar[r]^{Tm} TA_0
\ar@{-->}[d]^{\alpha_0} & TA \ar[d]^{\alpha} \\
TD \ar@{->>}[r]_{e} & A_0 \ar@{ >->}[r]_m & A.
}
\]
This shows the subdiagram of all quotients of $(TD, \mu_D)$ is initial in the
diagram defining~$\hatt \hat D$. Hence their limits are the same.
\end{proof}
\begin{proof}
By \Cref{prop:characterisation-hatT} we have the limit cone
$e^+: \hatt \hat D \to A$ in $\hatD$. That all $\br{e}$'s are surjective
follows by
\Cref{prop:projection-is-surjective}.
Further, the morphisms $\br{e}$ are $\hatT$-algebra homomorphisms and hence
$e^+:(\hatt
X, \hat\mu_X)\to (A, \alpha^+)$ is a cofiltered limit cone in
$\hatD^{\hatT}$
by~\eqref{eq:hat-eta},
,\Cref{rem:alphaplus} and the fact that limits of $\hatT$-algebras are
formed on the level of $\hatD$.
\end{proof}
\begin{proof}[\Cref{lem:hattcoflimsurj}]
(a) $\hatt$ preserves cofiltered limits: for the purpose of this proof it is
convient to express the limit defining $\hatt X$ for $X\in\hatD$, see
\Cref{not:limitcone}, by the end
formula
\[ \hatt X = \int_{(A, \alpha)\in \D_f^\MT} \widehat{\D}(X, \hat{A})
\pitchfork \hat{A}.\]
Note that the power functor $- \pitchfork B\colon \Set^\op \to \hatD$
preserves limits, and for $A$ finite the hom-functor $\hatD(-,
\hat{A})\colon
\hatD^\op
\to \Set$ preserves filtered colimits. Hence the composite $\hatD(-, \hat{A})
\pitchfork B\colon \hatD \to \hatD$ preserves cofiltered limits. Expressing
$\hatt$ as an end, it follows that $\hatt$ preserves
cofiltered limits:
\begin{align*}
\hatt (\Lim X_i)
= & \int_{(A, \alpha)} \widehat{\D}(\Lim X_i, \hat{A})
\pitchfork \hat{A} \\
= &\int_{(A, \alpha)} \Lim_i \left(\widehat{\D}(X_i, \hat{A}) \pitchfork
\hat{A} \right) \\
= &\Lim_i \left(\int_{(A, \alpha)} \widehat{\D}(X_i, \hat{A}) \pitchfork
\hat{A} \right) \\
= &\Lim_i (\hatt X_i).
\end{align*}
\noindent $\hatt$ preserves surjective morphisms:
let $f\colon X \to Y$
be a surjective morphism of~$\hatD$. We need to show that $\hatt f$ is
surjective.
\begin{enumerate}[(1)]
\item Suppose first that $X$ and $Y$ are finite. Then, $f =
\widehat{f_0}$ for
some
$f_0\colon X_0 \to Y_0$ in~$\D$. Since $T$ preserves
surjections by \Cref{asm:sec3}, the morphism $Tf_0 \colon TX_0 \to
TY_0$ is surjective, so every quotient $e$ of the free
$\MT$-algebra~$(TY_0, \mu_{Y_0})$ in~$\D^\MT$ yields a quotient
$\overline e
\defeq
e \o Tf_0$ of~$(TX_0, \mu_{X_0})$. Due
to~\Cref{prop:characterisation-hatT}, $\hatt f$ is the mediating morphism
\[
\xymatrix{
\hatt X \ar[r]^{\hatt f} \ar@{->>}[rd]_{\br{\overline e}} &
\hatt Y \ar@{->>}[d]^{\br{e}} \\
& \hatJ U(A, \alpha)
}
\]
for all quotients~$e$ of~$(TY_0, \mu_{Y_0})$. Each component
$\br{\overline e}$
is
surjective by \Cref{prop:projection-is-surjective}, so the mediating
morphism~$\hatt f$ is also surjective by
\Cref{lem:surjections-between-cofiltered-diagrams}.
\item Now let $X$ and $Y$ be arbitrary. By \Cref{rem:arrowcatclfp} the
morphism $f$ is the cofiltered limit in $\hatD^\to$ of all morphisms
$(x_i, y_i) \colon f \to f_i$ with $f_i\colon X_i \to Y_i$ in
$\D_f$. Let $f_i = e_i \o m_i$ be the canonical
factorisation of $f_i$ in $\hatD$.
\[
\xymatrix@-1em{
X \ar@{->>}[rr]^{f} \ar[dd]_{x_i} & & Y \ar[dd]^{y_i}
\ar@{-->}[ld]_{h_i} \\
& X_i' \ar@{ >->}[rd]_{m_i} \\
X_i \ar@{->>}[ru]^{e_i} \ar[rr]_{f_i} & & Y_i .
}
\]
There exists a unique morphism~$h_i \colon Y \to X_i'$ by the diagonal
fill-in property. It follows that surjections $(e_i)$ form an initial
cofiltered
subdiagram. Since $\hatt $ preserves cofiltered limits by (a),
$\hatt f$ is the limit of $\hatt e_i \colon \hatt X_i\to \hatt Y_i$.
However, each $\hatt e_i$ is surjective as proved in (1), so by
\Cref{lem:surjections-between-cofiltered-diagrams} it follows that
$\hatt f$ is surjective.\\
\noindent (c) We need to prove that finite $\hatT$-algebras are finitely copresentable in $\hatD^{\hatT}$. Consider the category~$\Alg(\hatt)$ of algebras for the
\emph{functor}~$\hatt$. In \cite[Lemma
3.2]{Adamek2004a} it is proved that for a cofinitary functor on a locally
finitely copresentable
category, every algebra $(A, \alpha)$ with $A$ finitely copresentable is
finitely copresentable in~$\Alg(\hatt)$. Now observe that $\hatD^{\hatT}$ is
a full subcategory of~$\Alg(\hatt)$ closed under
limits, $\hatD$ is
locally finitely copresentable, and finite objects in $\hatD$ are finitely
copresentable, see \Cref{re:hatD-is-procompletion}. Hence, finite
$\hatT$-algebras are finitely
copresentable in $\hatD^\MT$.
\end{enumerate}
\end{proof}
\begin{rem}\label{re:forgetful-monad-morphism}
Every right adjoint preserves right Kan extensions, so $(V\hatt,
V\epsilon)$
is the right Kan extension of $V\hatJ U$ along~$\hatJ U$. Moreover, there
is
a natural transformation $\gamma: TV\hat J U \to V\hat J U$ whose component
at $(A,\alpha)\in \D^\MT_f$ is given by $\alpha$ itself:
\[
\gamma_{(A, \alpha)} \defeq \alpha \colon TU(A, \alpha) \to U(A, \alpha).
\]
Therefore, by the universal property of $V\hatt$,
there is a unique natural transformation $\phi\colon TV \to V\hatt $ such
that~$\gamma = V \epsilon \o \varphi \hatJ U$. By the limit formula for
right Kan extensions and Remark
\ref{rem:hattconst}
the component
$\phi_{\hat X}$ for $X\in \D$ is the
unique mediating morphism making the following diagram commute for all $h:
(TX,\mu_X) \to (A,\alpha)$ with $(A,\alpha)\in\D^\MT_f$:
\begin{equation}\label{eq:phix}
\vcenter{
\xymatrix{
TV\hat X \ar[r]^{\phi_{\hat X}} \ar[d]_{TV(\widehat{h\eta_{X}})} &
V\hatt \hat X \ar[d]^{V\br{h}} \\
TA= TV\hat A \ar[r]_>>>>>>>>{\alpha} & **[r] V\hat A = A.
}
}
\end{equation}
In particular, putting $h=\alpha$ for $(A,\alpha)\in\D^\MT_f$, we get the
commutative triangle
\begin{equation}\label{eq:phia}
\vcenter{
\xymatrix{
TV A \ar[r]^{\phi_{ A}} \ar[dr]_\alpha &
V\hatt A \ar[d]^{V\br{\alpha}} \\
& A.
}
}
\end{equation}
\end{rem}
\begin{proposition} \label{prop:forgetful-functor}
\begin{enumerate}[(a)]
\item $\phi\colon TV \to V\hatt$ is a \emph{monad morphism~\cite{Street1972}},
i.e.\
the following diagrams commute:
\[
\vcenter{
\xymatrix@R-2em{
& TV \ar[dd]^{\phi} \\
V \ar[ru]^{\eta V} \ar[rd]_{V\hat \eta} & \\
& V\hatt
}
}
\quad\text{and}\quad
\vcenter{
\[email protected]{
TTV \ar[d]_{\mu V} \ar[r]^{T\phi} & TV\hatt \ar[r]^{\phi
\hatt} & V\hatt \hatt \ar[d]^{V\hat\mu} \\
TV \ar[rr]_{\phi} & & V\hatt
}
}
\]
\item For finite $X$ the morphism $\phi_X\colon TVX \to V\hatt X$ is dense.
\end{enumerate}
\end{proposition}
\begin{proof}
(a) The following pasting diagrams use the universality of $\epsilon$ and the
monad
laws.
\begin{itemize}
\item The preservation of unit $\phi \o \eta V = V\hat\eta$:
\begin{align*}
&
\vcenter{
\xymatrix{
\D^\MT_f \ar[d] \ar[r] \drtwocell<\omit>{\epsilon} &
\hatD \dtwocell{\hat\eta} \\
\D_f \ar[r] & \hatD \ar[r]_V & \D
}
}
\quad=\quad
\vcenter{
\xymatrix{
\D^\MT_f \ar[d] \ar[rr] \drrtwocell<\omit>{\id} & & \D
\ar[d]^{I} \\ \D_f \ar[rr] & & \D
}
}
\\
=
&
\vcenter{
\xymatrix{
\D^\MT_f \ar[d] \ar[rr] \drrtwocell<\omit>{\gamma} & & \D
\dtwocell{\eta} \\ \D_f \ar[rr] & & \D
}
}
\quad=\quad
\vcenter{
\xymatrix{
\D^\MT_f \ar[d] \ar[r] \drtwocell<\omit>{\epsilon} &
\hatD \ar[d] \ar[r]^V \drtwocell<\omit>{\phi}
& \D \dtwocell{\eta} \\
\D_f \ar[r] & \hatD \ar[r]_V & \D
}
}
\end{align*}
\item The preservation of multiplication $\phi \o \mu_V = V\hat\mu \o
\phi\hatt \o T\phi$:
\begin{align*}
&
\vcenter{
\xymatrix{
\D^\MT_f \ar[d]_{U_f} \ar[r] \drtwocell<\omit>{\epsilon} &
\hatD \ar[d]^{\widehat{T}} \ar[r]^V \drtwocell<\omit>{\phi}
& \D \dtwocell{\mu} \\
\D_f \ar[r] & \hatD \ar[r]_V & \D
}
}
=
\vcenter{
\xymatrix{
\D^\MT_f \ar[d] \ar[rr] \drrtwocell<\omit>{\gamma} &
& \D \dtwocell{\mu} \\
\D_f \ar[rr] & & \D
}
}
\\
= &
\vcenter{
\xymatrix@[email protected]{
\drtwocell<\omit>{\gamma} & \D \ar[d]^T \\
\D_f^\MT \ar[r] \drtwocell<\omit>{\gamma}
\ar@/^1pc/[ru]\ar@/_1pc/[rd] & \D \ar[d]^T \\
& \D
}
}
=
\vcenter{
\xymatrix@[email protected]{
\drtwocell<\omit>{\epsilon} & \D \dtwocell{\phi} \\
\D_f^\MT \ar[r] \drtwocell<\omit>{\epsilon}
\ar@/^1pc/[ru]\ar@/_1pc/[rd] & \D \dtwocell{\phi} \\
& \D
}
}
=
\vcenter{
\xymatrix@[email protected]{
\ddrtwocell<\omit>{\epsilon} & \D \dtwocell{\phi}
\ddlowertwocell{\mu} \\
\D_f^\MT
\ar@/^1pc/[ru]\ar@/_1pc/[rd] & \D \dtwocell{\phi} \\
& \D
}
}
\end{align*}
\end{itemize}
\noindent (b) If $X$ is finite we have for all $h: (TX,\mu_X) \to (A,\alpha)$
with $(A,\alpha)\in \D^\MT_f$:
\[ \alpha\o TV(\widehat{h\eta_X}) = \alpha \o Th\o T\eta_X = h\o \mu_X \o
T\eta_X = h \]
using that $h$ is a $\MT$-homomorphism and the unit law of the monad $\MT$.
In particular, if $h$ is surjective, so is the cone $\alpha\o
TV(\widehat{h\eta_X})$ in \eqref{eq:phiv}. By
\Cref{prop:characterisation-hatT} and Lemma \ref{lem:dense-lemma}
this implies that $\phi_X$ is dense.
\end{proof}
\begin{proof}[\Cref{prop:finite-algebras}]
(a) The maps $(A, \alpha) \mapsto ({A}, \br{\alpha})$ and $h
\mapsto \widehat{h}$ define a functor
\begin{equation}
\label{eq:comparison-functor}
K^\MT \colon \D_f^\MT \to \hatD_f^{\hatT}.
\end{equation}
To see this we only need to prove that for every $\MT$-homomorphism $h:
(A,\alpha)\to (B,\beta)$ is also a $\hatT$-homomorphism $(A,\alpha^+)\to
(B,\beta^+)$. Indeed, in the diagram below the right hand square commutes
when precomposed with $\hat\eta_A$, so it commutes by the universality of
$\hat\eta_A$.
\[
\xymatrix{
A \ar[r]^{\hat\eta_A} \ar[d]_h \ar@/^2em/[rr]^\id & \hatt A \ar[d]^{\hatt h}
\ar[r]^{\alpha^+} & A \ar[d]^h\\
B \ar[r]_{\hat\eta_B} \ar@/_2em/[rr]_\id & \hatt B \ar[r]_{\beta^+} & B
}
\]
\noindent(b) Conversely, the monad morphism $\phi: TV\to V\hatt$, see
\Cref{re:forgetful-monad-morphism} and
\Cref{prop:forgetful-functor}, induces a
functor
\[V^\MT \colon
\hatD^{\hatT}_f \to \D^\MT_f\]
mapping $(A, \alpha)$ to~$(VA, V\alpha \o \phi_A)$ and $h\colon (A,
\alpha) \to (B, \beta)$ to $Vh$.
\noindent (c) It remains to prove that $K^\MT$ and $V^\MT$ are mutually
inverse
functors. Clearly this holds on morphisms, since both functors are identity on
morphisms. As for objects, for every
finite $\MT$-algebra $(A, \alpha)$ we have
\begin{align*}
V^\MT K^\MT (A, \alpha)
& = V^\MT (\hat{A}, \br{\alpha})
&& \{\,\text{by definition}\,\} \\
& = (V\hat{A}, V\alpha^+ \o \phi_A)
&& \{\, \text{by definition}\,\} \\
& = (A, \alpha).
&& \{\, \text{by \eqref{eq:phia}}\,\}
\end{align*}
Conversely, let $(A, \alpha)$ be a finite $\hatT$-algebra
and $(A, \alpha') \defeq K^\MT V^\MT (A, \alpha)$. Applying~$V^\MT$ on both
sides, we have $V^\MT(A, \alpha') = V^\MT K^\MT V^\MT (A, \alpha) = V^\MT(A,
\alpha)$, using that $V^\MT K^\MT = \Id$ as proved above. That is,
$V\alpha'$ and
$V\alpha$ agree on the image
of~$\varphi_A$,
i.e.\ the diagram
\[
\xymatrix{
TVA \ar[r]^{\varphi_A} & V\hatt A \ar@<-1ex>[r]_{V\alpha}
\ar@<1ex>[r]^{V\alpha'} & VA
}
\]
commutes. By \Cref{prop:forgetful-functor}(b) the morphism $\phi_A$ is
dense, which implies $V\alpha' = V\alpha$ since $A$ is a
Hausdorff space. Since the forgetful
functor $V$ is faithful, we conclude $\alpha = \alpha'$.
\end{proof}
\begin{proof}[\Cref{lem:eqtovar}]
Let us first consider the unordered case. Since intersections of
pseudovarieties are pseudovarieties, we may assume that
our class is presented by a single equation $u=v$ over a finite set $X$.
\begin{enumerate}[(a)]
\item Closure under finite products: clearly the trivial one-element
$\MT$-algebra satisfies all profinite equations.
Let $(A_0,\alpha_0)$ and $(A_1,\alpha_1)$ be finite $\MT$-algebras satisfying
$u=v$ and let $p_i\colon (A_0\times A_1,\alpha)\to (A_i,\alpha_i)$ be their
product. For
any $\MT$-homomorphism $h\colon T\psi_X\to A_0\times A_1$ put $h_i = p_i \o
h$. Then
\[ p_i \o \br{h} (u) = \br{h_i} (u) = \br{h_i} (v) =
p_i \o\br{h} (v) \] so $\br{h} (u) = \br{h} (v) $ since
the projections $p_i$ are jointly monic. We conclude that
$(A_0\times A_1,\alpha)$ satisfies $u=v$.
\item Closure under subalgebras: let $(A,\alpha)$ be a finite $\MT$-algebra
satisfying $u=v$ and $m\colon (B,\beta)\monoto (A,\alpha)$ be a subalgebra.
Then
for any $\MT$-homomorphism $h\colon T\psi_X \to B$ we have
\[ m\o \br{h} (u) = \br{(m\o h)} (u) = \br{(m\o h)} (v) = m\o
\br{h} (v) \]
so $\br{h}(u) = \br{h} (v)$ since $m$ is monic. Hence $(B,\beta)$
satisfies
$u=v$.
\item Closure under quotients: let $(A,\alpha)$ be a finite $\MT$-algebra
satisfying $u=v$ and $e\colon
(A,\alpha)\epito (B,\beta)$ be a quotient. For any $h_0: \psi_X\to B$ choose a
morphism $h_0': \psi_X\to A$ with $h_0 = e\o h_0'$, cf. \Cref{rem:homtheorem}.
The corresponding $\MT$-homomorphisms $h: T\psi_X\to B$ and $h': T\psi_X\to A$
satisfy $h = e\o h'$, so
\[ \br{h} (u) = e\o \br{(h')} (u) = e\o \br{(h')}(v) =
\br{h} (v), \]
proving that $(B,\beta)$ satisfies $u=v$.
\end{enumerate}
The ordered case in analogous: replace profinite equations by inequations, in
(a) use that the projections $p_i$ are jointly order-reflecting, and in (b)
use that $m$ is order-reflecting.
\end{proof}
\begin{proof}[\Cref{lem:keylemma}]
Consider first the unordered case.
For any $\MT$-homomorphism $h\colon TX\to A$ with $(A,\alpha)\in \V$ we have
the
commutative triangle \eqref{eq:phiv}.
Since the projections $\brv{h}$ are jointly monic, it follows that
$\phi^\V_\hatX u = \phi^\V_\hatX v$ iff $\br{h} u = \br{h} v$
for all
$h$, i.e.\ iff every $(A,\alpha)\in \V$ satisfies $u=v$. For the ordered case
replace equations by inequations and use that the
projections $\brv{h}$ are jointly order-reflecting.
\end{proof}
\begin{proof}[Theorem \ref{cor:reitermanquasi}]
3$\Ra$1 requires a routine verification analogous to the proof of
\Cref{lem:eqtovar}, and 1$\Ra$2 is \Cref{thm:reiterman}. For 2$\Ra$3
consider first the unordered case. We may assume that $\V$ is presented by a
single profinite equation $u=v$ with $u,v$ elements of some $X\in\D_f$.
Express $X$ as a quotient $q\colon \Phi_Y\epito X$ for some finite set $Y$.
Let
$\{\,(u_i,v_i) : i \in I\,\}$ be the kernel of $\hat q\colon \hat{\Phi_Y}
\epito
X$ (consisting of all pairs $(u_i,v_i)\in \hat{\Phi_Y}\times\hat{\Phi_Y}$ with
$\hat q (u_i) = \hat q (v_i)$), and choose $u',v'\in \hat{\Phi_Y}$ with $\hat
q(u') = u$ and $\hat
q(v') = v$. We claim that a finite object $A\in\D_f$ satisfies the
profinite equation $u=v$ iff it satisfies the profinite implication
\begin{equation}
\bigwedge_{i\in I} u_i=v_i ~\Ra~ u'=v',\tag{\ref{eq:imp}}
\end{equation}
which proves that $\V$ is presented by that implication.
For the ``if'' direction suppose that $A$ satisfies \eqref{eq:imp}. For any morphism $h\colon X\ra A$ in $\D$ we have the commutative triangle below:
\[
\xymatrix{
\hat{\Phi_Y} \ar[r]^>>>>>>{\hat q} \ar[d]_{\widehat{hq}} & \hatX=X
\ar[dl]^{\hat{h}=h}\\
A &
}
\]
The morphism $\widehat{hq}$ merges $u_i,v_i$ for all $i$ since $\hat q$ does,
so by \eqref{eq:imp} it also
merges
$u',v'$. We conclude
\[ \hat{h}(u) = \hat{h}\hat q(u') = \widehat{hq}(u') = \widehat{hq}(v') =
\hat{h}\hat q(v') = \hat{h}(v),\]
so $A$ satisfies $u=v$.
For the ``only if'' direction, suppose $A$ satisfies $u=v$, and let $h\colon
\Phi_Y\ra A$ be a morphism with $\hat{h}(u_i)=\hat{h}(v_i)$ for all $i$. Since
$(u_i,v_i)$ are precisely the pairs merged by $\hat q$, the homomorphism
theorem (see \Cref{rem:homtheorem}) yields a morphism $h'\colon X \to A$
with $h'\o \hat q = \hat{h}$. Since
$A$ satisfies $u=v$, it follows that $h'(u)=h'(v)$ and hence
\[ \hat{h}(u') = h'\o \hat q (u') = h' (u) = h'(v) = h'\o \hat q (v') =
\hat{h}(v').\]
Hence $A$ satisfies the implication \eqref{eq:imp}.
The ordered case is analogous to the above argument, replacing
equations by inequations and using the homomorphism theorem for ordered
algebras in the ``only if'' direction.
\end{proof} | 105,032 |
TITLE: Probabilities of guessing correctly at least 3 of 4 randomly generated numbers
QUESTION [0 upvotes]: A system will randomly generate 4 numbers, each from 0-9. So there're 10,000 possible combinations.
Next I'll pick 4 numbers, each also from 0-9. What is the probability that at least 3 of my 4 numbers match the system-generated numbers (order doesn't matter)?
For example, if the system-generated combination is 0-0-1-2, eligible combinations are 0-1-0-2 (4 matches), 3-0-1-0 (3 matches), 2-1-0-8 (3 matches)...
REPLY [0 votes]: The required probability depends on how much there is matching among the picked numbers.
Say, if all picked numbers are different ($XYZT$), to get four matches the generated conbunations can be $XYZT$ or any permutation of this numbers. The probability to get four matches is
$$\dfrac{4!}{10^4}=\dfrac{24}{10^4}.$$
To get exactly three matches, one need to chose four ways to choose which numbers will coincide with generated: $XYZ$ or $XYT$ or $YZT$ or $XZT$. Say, it will be $XYZ$. Denote the last generated number by $U\neq T$.
The last number $U$ can either coincide with one of $XYZ$ or not. In the first case there is three ways to choose it and $\binom{4}{2}\cdot 2$ ways to rearrange $XXYZ$. In the second case there is six ways to choose it and $4!$ ways to rearrange $XYZU$.
Finally, the probability to get exactly three matches when the previously picked numbers are different one from another is
$$
\dfrac{4\cdot\left(3\cdot\binom{4}{2}\cdot 2+6\cdot 4!\right)}{10^4}=\dfrac{720}{10^4}.
$$
And the total probability to get at least 3 of 4 different numbers match the system-generated numbers is
$$\dfrac{24+720}{10^4}=\dfrac{744}{10^4}.$$
Similarly, you can find probability when two of four picked numbers are the same, then three, then four. The last case is simplest one: if picked numbers are $XXXX$, the probability of four matches is $1/10^4$ and the probability of three matches is $9\cdot 4/10^4$, totally
$$\dfrac{1+9\cdot 4}{10^4}=\dfrac{37}{10^4}.$$ | 74,776 |
TITLE: Does the set of all strings of letters form a group?
QUESTION [6 upvotes]: This is related to a course I'm taking in computer science theory.
Let $\sum$ be an alphabet. Then the set of all strings over $\sum$, denoted as $\sum^*$ has the operation of concatenation (adjoining two strings end to end). Clearly, concatenation is associative, $\sum^*$ is closed under concatenation, and the identity element is the empty string. I'm also taking a course in modern algebra, so I naturally ask can $\sum^*$ be formed into a group? Three of the four group axioms are satisfied.
REPLY [2 votes]: No, $\Sigma^*$ is not a group (unless $\Sigma = \emptyset $, in which case $\Sigma ^*$ is a trivial group with one element). The reason is that the only element having an inverse is the empty word. So if $a\in \Sigma$, then $a$ as an element of $\Sigma ^*$ does not have an inverse (rigorously, note that the length of words never decreases upon concatenation).
What $\Sigma ^*$ is is an example of a monoid. A monoid is an algebraic structure satisfying all axioms of a group except the requirement of identities. In fact, $\Sigma ^*$ is the free monoid on the set $\Sigma$. There is also the obvious notion of a commutative monoid, and a certain quotient of $\Sigma^*$, obtained by allowing elements to commute, gives rise to the free commutative monoid $\Sigma ^+$. | 169,661 |
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| 58,189 |
Childminder Jobs in Thompson Riverview Terrace
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There are currently no childminder jobs in Thompson Riverview Terrace matching these search criteria. | 91,596 |
TITLE: How to expand $\tan x$ in Taylor order to $o(x^6)$
QUESTION [10 upvotes]: I try to expand $\tan x$ in Taylor order to $o(x^6)$, but searching of all 6 derivative in zero (ex. $\tan'(0), \tan''(0)$ and e.t.c.) is very difficult and slow method.
Is there another way to solve the problem?
Any help would be greatly appreciated :)
REPLY [5 votes]: Here's the long division method suggested by coffemath.
\begin{align}
\sin(x)
= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots
&= \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} x^{(2n+1)}
&=& \sum_{n=0}^\infty a_{(2n+1)} x^{(2n+1)} \\
\cos(x)
= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots
&= \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!} x^{(2n)}
&=& \sum_{n=0}^\infty b_{(2n)} x^{(2n)} \\
\end{align}
\begin{align}
\tan(x) = \frac{\sin(x)}{\cos(x)}
\end{align}
\begin{align}
\sin(x) - \tan(x)\cos(x) &= 0 \\
\sin(x) - truncated(\tan(x))\cos(x) &= remainder\\
\left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots \right) -
\left( 0 \right)
\left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots \right)
&= \left( x - \frac{x^3}{6} + \frac{x^5}{120} - \dots \right) \\
\left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots \right) -
\left( x \right)
\left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots \right)
&= \left( \frac{x^3}{3} - \frac{x^5}{30} + \dots \right) \\
\left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots \right) -
\left( x + \frac{x^3}{3} \right)
\left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots \right)
&= \left( \frac{2x^5}{15} - \frac{4x^7}{315} + \dots \right) \\
\left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots \right) -
\left( x + \frac{x^3}{3} + \frac{2x^5}{15} \right)
\left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots \right)
&= \left( \frac{17x^7}{315} - \frac{29x^9}{5670} + \dots \right)
\end{align}
\begin{align}
\tan(x) = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \dots = \sum_{n=0}^{\infty}c_{(2n+1)}x^{(2n+1)}
\end{align}
Where the coefficients are given recursively as:
\begin{align}
a_{(2n+1)} &= \frac{(-1)^n}{(2n+1)!} \\
b_{(2n)} &= \frac{(-1)^n}{(2n)!} \\
c_{(2n+1)} &= a_{(2n+1)} - \sum_{k=0}^{n-1} b_{(2n-2k)} c_{(2k+1)}. \quad \blacksquare
\end{align}
This series converges with radius $\frac{\pi}{2}$ around $x=0$. | 209,413 |
Wellness Blooms During Artist’s Magical Nature Journey
You might say nature has been a life saver for artist Robyn Nola.
In her mid twenties, Nola learned she had Berger’s disease—a chronic, progressive disorder. The news was daunting for the health-conscious optimist.
Berger’s disease can be asymptomatic for years, so Nola didn’t precisely know when the debilitating protein antibodies began building up in her kidneys. And, with no family history of the disorder she and her doctor weren’t sure what caused it. After months of back pain, swelling and other issues, however, a renal biopsy led to a conclusive diagnosis. Immediately, Nola began treatment to relieve symptoms and to prevent or delay renal failure.
The months ahead were both physically and emotionally draining. So to help bolster her usually sunny outlook, Nola rekindled her love for photography. And when her husband Steve brought home a digital camera, Nola began photographing desert flowers in their yard.
“Those were the early days of digital photography,” Nola said. “And, immediately, I fell in love with the unlimited capabilities digital offered versus traditional film. I could take as many shots as I wanted and I would never run out of film! Being able to see immediate results from my shots was so satisfying. It really fed my creative side.”
Next, Nola began photographing flowers during the couple’s out-of-state trips. A summertime visit to the gardens within San Diego’s Balboa Park resulted in an photograph of a lotus flower that stunned her husband.
“Something was special about this photo, it had a magical quality. It was quite remarkable,” Steve said.
Nola said her love of nature photography seemed to take off from there. “My husband really encouraged me to share my gift with the world.”
Nola says that her love of nature photography just seemed to take off from there.
“It gives me a chance to become one with nature. It is a quiet time when I have the chance to capture beautiful, magical moments. Spending time in nature is a healing therapy for me.”
Always depicted in natural light, Nola’s vibrant, colorful photos are the result of an extremely detailed eye for capturing unusual combinations of composition and perspective. Nola’s art is captivating and distinctive. It’s no surprise that some of her growing base of devotees make stylistic comparisons with the raw brilliance of nature portrayed by American artist Georgia O’Keeffe (1887 – 1986).
As a result of Nola and Steve’s 2004 visit to Maui, Hawaii—a trip filled with adventures to secret gardens, waterfalls and awe-inspiring seascapes—there were plenty of diverse and exotic opportunities to develop her art.
“My creativity was sparked!” Nola said. “Every where you looked was a breathtaking vista or a beautiful flower just waiting to get captured into a magical moment.”
Four years later the couple returned to Hawaii; this time to the garden island of Kauai, which Nola calls “Heaven on Earth” because of its remote atmosphere and opportunities to become “one with nature.”
She realized that her photography—and the subsequent oneness with nature —had become a powerful and energizing healing therapy.
Upon returning to Arizona, Nola began transitioning her passion for the magic of nature photography into a full-time professional pursuit.
Little did she know, but there were other magical events apparently taking place at the same time.
Nearly a decade after Nola’s diagnosis, her nephrologist made an amazing discovery. Nola’s kidney function had returned to 100 percent normal. Robyn no longer needed to take medicine and could possibly live the rest of her life without ever needing a kidney transplant.
The chronic disorder, for which there is usually no cure, was gone.
Nola has absolutely no doubt about what led her back to vibrant health and wellness. “Focusing on the beauty in nature and practicing daily positive affirmations has really changed my life.” Nola said.
“I have now learned that this was a gift. I am now living my dream of capturing the beauty I see and sharing this with the world.” Nola said.
As her photography business progressed into a loving journey dedicated to capturing nature’s magic—Nola believes, “My body, including my kidneys, became balanced. Nature is filled with perfection and balance. My exposure to nature—my oneness with nature—manifested itself through a return to good health.”
Using her inherent gift to look at the familiar in a new way, Robyn Nola plans to continue her magical journey with a very personal affirmation. “Being in nature is the best medicine there is. Nature is my ultimate healer.” | 312,911 |
TITLE: What is the process of a photon transforming into an electron and positron?
QUESTION [0 upvotes]: How did they come into existence from a photon? Is it really understood how the process works? Is there even a process or is it just something fundamental?
REPLY [1 votes]: It is understood how the process works. In a nutshell, the energy of a photon is converted to the mass of an electron and positron as is given by Einstein's $E = mc^2$. This is spelled out as follows:
$γ → e^− + e^+$
There are two musts for this:
The photon must have a higher energy than the sum of the rest mass energies of a positron and electron for this to happen ($2 * 0.511 MeV = 1.022 MeV$).
The photon must be near a nucleus in order to satisfy conservation of energy and momentum.
The basic properties, energy transfer, and cross section can be specifically calculated. So, yes, I'd say the mathematics and process are fairly well defined.
More information can be found here.
Hope this helps! | 112,310 |
300 Union Plumbers Spent The Weekend Installing Water Filters For Flint Residents For Free
posted
Categories: News
Amazing collaborative work done by these plumbers. I'm sure the people of Flint are grateful for their dedication. All of this was prompted due to the inaction of state and federal officials, thousands of people in Flint have been exposed to unsafe levels of lead in their water. Now a group of union plumber are taking matters into their own hands.
On Saturday, 300 plumbers from unions across the country descended on Flint.
Residents of Flint, however, are still encouraged to use the filters. For most homes, they will work.
via ThinkProgress | 374,344 |
Applications for part-time or full-time employment opportunities are only accepted online.
For a listing of all open positions and to apply for a position go to the Homewood-Flossmoor Park District website by clicking.
Click the APPLY button to complete the online application.
CURRENT OPEN POSITIONS
Now hiring Fitness Center Attendants!
We are looking for outgoing individuals who enjoy helping others live happy, healthy lives! The ideal candidate will:
- Be friendly, hardworking and dependable.
- Enjoy working directly with the public.
- Possess a strong desire to provide members with excellent customer service.
- Be able to anticipate and solve problems before they escalate.
- Possess knowledge of basic anatomy.
- Be able to stand, bend, reach, and lift 45 lbs.
Apply at Homewood-Flossmoor Park District website by clicking. | 123,143 |
Ole Ernesto Valverde’s tactical input.
BASIC STATS
Goals – 3
Shots – 13
Shots on target – 6
Goals conceded – 0
Possession – 66%
Dribbles – 14
Tackles – 16
Lionel Messi has scored more Champions League goals against English clubs than any other player in #UCL history:— Squawka Football (@Squawka) April 16, 2019
9 vs. Arsenal
6 vs. Man City
4 vs. Man Utd
3 vs. Chelsea
2 vs. Spurs
Just waiting to add Liverpool now... pic.twitter.com/8EqgWgb1Hk
TACTICAL TALKING POINT
Busquets role
With Messi in the form that he was in on the night, it’s only natural that he would take all the plaudits. Whatever spotlight remains would undoubtedly be fixated on United’s glaring individual errors and general lack of quality. Spare a thought for Sergio Busquets though who almost single-handedly got Barca to settle down after a shaky start.
The experienced holding midfielder was made to sit a lot deeper after United’s early pressing and threat in the transition. He did that expertly, slotting in between the two centre-backs and helping Barca play their way out of danger. As the game wore on, that became a position of strength for the hosts. The Spaniard was critical to their build-up play, often leaving United’s forwards chasing shadows.
GOT RIGHT
Targeting United’s right side
As Barcelona increasingly gained control of the game, they started to target the right side of United’s defence, a ploy that was particularly noticeable after half-time. Especially after winning possession, their immediate passes were directed at finding either Coutinho or Jordi Alba. The duo threatened on a couple of occasions before linking up for Barca’s third.
VERDICT
Valverde did what has come to be expected of him. His 4-4-2 system is tailor-made for Messi to run the show and run the show he did. The tactician must be credited for the subtle tweaks he is known to make during games which help his side along their way. Has to be said that he benefited hugely from United’s own errors though, ultimately making things rather simple for him.
Rating 7/10 Ole Gunnar Solskjaer’s tactical input.
BASIC STATS
Goals – 0
Shots – 9
Shots on target – 3
Goals conceded – 3
Possession – 34%
Dribbles – 4
Tackles – 17
TACTICAL TALKING POINT
Diamond system
United needed to maintain a compact shape away to Barcelona but they also had to pose a threat in attack. Solskjaer had the right idea when he adopted a 4-4-2 diamond system and for the opening 10 minutes or so, Barcelona were second best.
The Catalans struggled to cope with the pace of Marcus Rashford and Anthony Martial up front while Scott McTominay and Fred had started well, giving United’s midfield plenty of mobility.
After Messi took control – aided by a self-destructive streak from United – Barca settled nicely and began to stretch the away side at will, exposing the narrow back-line.
3 - David de Gea has made three errors leading to opposition goals in all competitions this season - the only Premier League goalkeepers to have made more across all competitions in 2018-19 at club level are Asmir Begovic (5) and Jordan Pickford (4). Agonising. #FCBMUN pic.twitter.com/opnQREo1qK— OptaJoe (@OptaJoe) April 16, 2019
GOT RIGHT
Changes
Apart from his initial set-up, Solskjaer made the right call when he changed it entirely early in the second half after Coutinho’s strike put the game firmly beyond United’s reach. He went into damage-limitation mode, switching to a back five and under the circumstances, it was perfectly understandable.
The Norwegian didn’t get carried away by ludicrous hopes of a miraculous four-goal recovery. Even Sir Alex Ferguson was guilty of such misguided thoughts in the past. On one occasion it led to a 6-1 thumping in the Manchester derby at Old Trafford. Solskjaer at least spared United’s fans further humiliation.
GOT WRONG
Team selection
There was little wrong with Solskjaer’s set-up but perhaps he put a little too much faith in a couple of players, failing to appreciate their limitations. Ashley Young shouldn’t be a United player anymore, let alone starting the Champions League quarter-final away to Barcelona at left-back while sporting the captain’s armband. Diogo Dalot may have been a better option with Luke Shaw suspended.
But that still leaves Victor Lindelof playing at right-back. So there’s not too much for Solskjaer to work with. Four of United’s back five started in Basel when they were knocked out of the group stages in 2011. Eight years on, it’s farcical that the defensive personnel hasn’t been significantly improved. That speaks volumes of the task the United boss has on his hands.
2 GOALS in 3 minutes!!!! ⚽️⚽️🔥— Sport360° (@Sport360) April 16, 2019
THE G O A T 🐐#BARMUN #Messi pic.twitter.com/MZIbPXo4Ac
VERDICT
It’s hard to point fingers at Solskjaer, even after such a resounding defeat. You can’t fault the system he went with although there were one or two question marks over his team selection. Individual errors ultimately cost United. What else could Solskjaer have done? Have better players, maybe?
Rating 5/10 | 73,395 |
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It was a prayerful Lent, but I realized toward the end (as I always do), how much I need to grow in holiness and how much Our Lord loves me... deeply. Shaun & I made it through without breaking our TV fast (ok, we did on Sunday's but that was it) and praying practically every morning and night together, going to Mass more frequently at our new parish (which had a lovely evening mass and we were often actually able to go together) and trying our best to offer our days back to Him. I realized once again the many ways I need to love Jesus more, but find this year that I have tremendous gratitude for His unconditional love for me and patience as I slowly grow. I also find my motivation to grow spiritiually to be spurred more by love and less by "obligation", so that what I do is more ...heartfelt... even if I don't feel much. Shaun is steadily growing and always an inspiration to me. It's definitely a shift to realize that now I'm responsible for helping another person (not just myself) to get to heaven. THAT is something I need to work on. I feel like Shaun is the one waking me up to pray in the morning and happily pulling me along to Mass or say a rosary. He LOVES Jesus and Mary SO much (and I hear him tell them often)... *sigh* thank God for him, but I just hope I can become the wife he deserves for he's such a good man.
We had a beautiful Easter this year! Celebrated with Shaun's Mom and family at her home on the Saturday before and enjoyed watching all the little ones crawling into my lap and going a great Easter egg hunt prepared intricately by her (despite the rain).
On Easter morning it was so beautiful to come to the church all decorated with gorgeous lilies and hydrangeas with the cross covered in beautiful flowers.
MAY:
This brings us into not only the Easter Season, but the Lovely month of "May"... ie: "Mary's Month". I really enjoy this time of the year and teaching all my little ones about our "Queen of the May". It was beautiful to see our parish May Crowning done in such a simple humble way that I'm sure made Our Lady smile. Hopefully one day, our little one can participate in such an event
During the month of May we have also been able to see Shaun's nephew, Samuel recieve his First Holy Communion
A few days after Sam's 1st Communion, we headed back to Worscester for his nephew Martin's confirmation (for which Shaun was the sponsor). Here's a few photos:
FINANCES:
We've also been trying to take advantage of the low interest rates and refinance, so we've been busy with appointments, number crunching and appraisals. I'm NOT a "number" person, but... must say that I've learned a LOT and it's amazing how God gives you the sacramental grace (even in your weak areas) to help your family. Hopefully we'll be able to really pay our Mortgage down MUCH sooner without as much strain on us financially.
We're rounding up the year and coming to a close and with the last month rolling quickly by, I must say even with Spring Fever, I'm SO proud of where the kids I teach have come and feeling bittersweet about this being my last class. Thank God they were a GREAT one! 25-3 year olds every day leave me pretty tired at the end of it all, but have taught me SO much about getting ready to be a Mom myself.
THE HUBBY:
Working hard every day, taking time to go to basketball and softball games and practices, out for "father-daughter dinner dates" & help Cheyanne with homework he's out early and home late. He's also healing up since his surgery & goes to physical therapy at least twice a week. He seems to be getting better and better for sure (but slowly...) Keep up the prayers and I'm sure he'll be ready to hold our little girl when the time comes.
So... for those of you who haven't heard, Mother's Day weekend Shaun & I found out what we're having!!!
As you all know, I originally did not want to find out until the delivery, but Shaun was practically coming out of his skin.
On Wednesday the 4th of May, we went for our 18 week Ultrasound.
There the envelope stayed for the next few days.
I then dropped the envelope off to the baker who would be making Shaun's birthday cake & asked them to put pink frosting inside if it was a girl
and blue if the results said it was a boy.
On Saturday evening we all gathered for a birthday gathering of family members and some of our closest friends. It was a wonderful time with children everywhere and happy sounds flooding the house. When it came time to sing "Happy Birthday" and cut the cake everything stopped and everyone leaned in close with cameras and phones poised and ready...
It's a....GIRL!!!! We exclaimed as we saw the PINK frosting.
Shouts of happiness filled the air and cameras started clicking! My Mom had brought me Pink roses "from the baby" for Mother's Day and my mother-in-law had already bought a pink sleeper outfit (so cute) and brought it out right away.
What a beautiful time to find out while both celebrating Shaun's birthday AND Mother's day. We're SO excited. Though we would have been happy with either a boy OR a girl (or both) ...Shaun has a soft spot for little girls and it's so nice to be able to use her name. Shaun's daughter, Cheyanne seemed excited to know she's getting a little sister when we called her and immediately said, "Oh! NOW I can help Marijanna go shopping!!!"
YESTERDAY... I went for another Ultrasound (they couldn't get all the pictures and measurements they needed at the 18 week one) and got to see "Ava" putting on a show! Here are some pictures of the Ultrasound pictures.
Please pray for us during this happy time and know how much we appreciate all the support we have been recieving from all of you. You have made this time SO special. I feel like I'm bringing her into on big family with LOTS of auntie's and uncles to shower her with love. Can't wait to meet her!
A Baby Girl!!!
In honor of our Blessed Mother, "Ava Marie Fullen" will be her name...
(although if we have some great desire to change it... we still can!)
For a minute there I thought you weren't going to say what it was and keep us in agonizing suspense, but I'm glad you did! Girls are SO great! | 219,125 |
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\begin{document}
\title{\bf Lundberg-type inequalities
\\ for non-homogeneous
risk models
}
\renewcommand{\thefootnote}{}
\footnotetext{
CONTACT Junyi Guo [email protected]. School of Mathematical Sciences,
Nankai University, Tianjin, China.}
\author{\noindent{{Qianqian Zhou}$^{a}$, {Alexander Sakhanenko}$^{a,b}$ and {Junyi Guo}$^{a}$}\\
\small {\noindent{$^a$School of Mathematical Sciences,
Nankai University, Tianjin, China;}}\\
\small{ \noindent{$^b$Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue,
Novosibirsk, Russia}}}
\date{}
\maketitle
\vskip 0.5cm \noindent{\bf Abstract}\quad
In this paper, we investigate the ruin probabilities of non-homogeneous risk models. By employing martingale method, the Lundberg-type inequalities of ruin probabilities of non-homogeneous renewal risk models are obtained under weak assumptions. In addition, for the periodic and quasi-periodic risk models the adjustment coefficients of the Lundberg-type inequalities are obtained.
Finally, examples are presented to show that estimations obtained in this paper are more accurate and the ruin probability in non-homogeneous risk models may be fast decreasing which is impossible for the case of homogeneity.
\smallskip
\noindent {\bf Keywords:}\quad Non-homogeneous risk model; Martingale method; Ruin probability; Lundberg-type inequality
\smallskip
\noindent {\bf Mathematics Subject Classification (2010)}\quad 91B30; 60K10
\section{Introduction and some basic results}
In classical risk theory, the probability of ruin is a vital index of the robustness of an insurance company and is also a useful tool for risk management. Therefore, it has been studied by many authors and is still attracting the attention of many authors.
High probability of ruin means that the insurance company is not stable, and then the insurer should take corresponding measures to reduce risks.
Thus the research on ruin probability of risk model is a hot subject in risk theory.
The calculation of ruin probability is a classical problem in actuarial science. However, the exact value of ruin probability can only be calculated for exponential distribution or discrete distribution with finite values.
But the upper bounds of run probabilities of risk models can be obtained. The insurer may employ the upper bound of ruin probability to evaluate the stability of insurance company.
Let $R(t)$ be a risk reserve process, defined for all $t\ge 0,$ with non-random initial reserve $R(0) = u>0.$
The ultimate ruin probability $\psi(u)$ of the risk model is defined as
\begin{gather} \label{aa1}
\psi(u):={\bf P}\Big[\inf_{t\geq 0}R(t)<0\Big| R(0)=u\Big].
\end{gather}
After the classic works of Lundberg \cite{LF03} and Cram$\acute{e}$r \cite{C30, C55}, many research works, such as references
\cite{AA10, DC05, GH79, G91, RSST98}, have been devoted to studying ruin probabilities of homogeneous risk models.
For classical homogeneous risk models the most excellent result about the behavior of $\psi(u)$
is the Lundberg inequality, which states that under some appropriate assumptions (see \cite{C30, C55} and \cite{LF03} for more details)
\begin{gather} \label{aa9}
\psi(u)\leq e^{-Lu}\text{ for all } u\geq 0,
\end{gather}
The largest number $L$ in \eqref{aa9} is called the adjustment coefficient or Lundberg exponent.
In the homogeneous risk models, "identically distributed" assumption is imposed both on inter-occurrence times and claim sizes. However,
both factors are influenced by the economic environment and the usual assumptions on claim sizes and inter-occurrence times may be too restrictive for practical use. Thus relaxing the "identically distributed" assumption imposed on claim sizes or (and) inter-occurrence times is necessary, which yields non-homogeneous risk processes.
In Casta$\tilde{n}$er, et al. \cite{CCGLM13}, and Lef$\grave{e}$vre and Picard \cite{LP06}, the "identically distributed" assumption imposed on claim sizes is relaxed, while in Bernackait$\dot{e}$ and $\breve{S}$iaulys \cite{BJ17}, Ignator and Kaishev \cite{IK00} and Tuncel and Tank \cite{TT14}, the "identically distributed" assumption imposed on inter-occurrence times is also relaxed.
Without the condition of identical distribution on claim sizes and inter-occurrence times,
there are serious difficulties in evaluating the probability of ruin. Hence, the related papers generally investigate the recursive formulas for the finite time ruin probability under restrictive conditions. Such as in Bla$\check{z}$evi$\check{c}$ius, et al. \cite{BBE10}, the recursive formula of finite-time ruin probability of discrete time risk model with nonidentical distributed claims is obtained. In R$\check{a}$ducan, et al. \cite{RVZ15a}, the recursive formulae for the ruin probability at or before a certain claim arrival instant of the non-homogeneous risk model are obtained. In this risk model, the claim sizes are independent non-homogeneous Erlang distributed and independent of the inter-occurrence times, which are assumed to be i.i.d. random variables following an Erlang or a mixture of exponential distribution. R$\check{a}$ducan, et al. \cite{RVZ15b} extended those recursive formulae to a more general case when the inter-occurrence times are i.i.d. nonnegative random variables following an arbitrary distribution.
Our aim is to investigate non-homogeneous risk models and obtain Lundberg inequality for
the probability $\psi(u)$ under more general assumptions.
We consider a class of risk reserve processes, which need not to be homogeneous, with the following properties.
\medskip
\hspace*{0.5cm} (i) Process $R(t)$ may have positive jumps only at random or non-random times $T_1, T_2, \ldots$ such that
\begin{gather*}
\forall k=1, 2, \ldots,\ T_{k+1}>T_{k} >T_0:=0 \text{ and } T_n\rightarrow\infty\ a.s.
\end{gather*}
\hspace*{0.5cm} (ii) Process $R(t)$ is monotone on each time interval $[T_{k-1}, T_k), \ k=1,2,\ldots,$ and
$R(0)=u>0.$
\medskip
{\bf Model A.} \ Assume that the \emph{kth} claim $Z_k$ {occurs at} time $T_k$, i.e.,
\begin{gather} \label{aa5}
-Z_k:= R(T_k)-R(T_k-0)\le 0,\ \theta_k:=T_k-T_{k-1}>0,\ k=1,2,\ldots.
\end{gather}
Suppose that on each interval $[T_{k-1}, T_k)$ {the premium rate is} $p_k,$ i.e.,
\begin{gather}\label{aa6}
\forall t\in[T_{k-1}, T_k),\ R(t)-R(T_{k-1})=p_k(t-T_{k-1}), \ k=1,2,\ldots.
\end{gather}
Assume, also, that random vectors
\begin{gather*}\label{aa7}
(p_k, Z_k, \theta_k), \ k=1,2,\ldots
\end{gather*}
are mutually independent. Then conditions (i) and (ii) hold and random variables
\begin{gather*}
Y_k=R(T_{k-1})-R(T_k)=Z_k-X_k=Z_k-p_k \theta_k,\ k=1,2,\ldots
\end{gather*}
are also mutually independent. Here we denote by $X_k=R(T_k-0)-R(T_{k-1})=p_k \theta_k$ the total premium {collected by the insurer} over the time interval $[T_{k-1}, T_k)$.
Here we call model A is non-homogeneous renewal risk model and it is clear that model A is more general than the classical compound Poisson and renewal risk models introduced by Sparre Andersen \cite{AE57} in which the random vectors $(Z_k, \theta_k),\ k=1,2,\ldots$, are assumed to be i.i.d. and the premium rate $p_k\equiv p>0$ is fixed, positive and non-random.
\begin{thm}\label{thm1}
Assume that conditions (i) and (ii) hold and the random variables
\begin{gather} \label{aa3}
Y_k:=R(T_{k-1})- R(T_k),\ k=1,2,\ldots,
\end{gather}
are mutually independent.
Then for any $ u>0$ and $h\ge 0,$ the ruin probability $\psi(u)$ satisfies
\begin{gather} \label{aa4}
\psi(u)=
{\bf P}[\sup_{k\geq 1}S_k>u]\leq e^{-hu}\sup_{k\geq1}{\bf E}e^{hS_k}
=e^{-hu}\sup_{k\geq1}\prod_{j=1}^k{\bf E}e^{hY_j},
\end{gather}
where $S_k=Y_1+Y_2+\ldots+Y_k.$
Moreover, the Lundberg inequality (\ref{aa9}) holds with $L=L(S_\bullet),$
where
\begin{gather} \label{aa11}
L(S_\bullet):=\sup\{h\ge0: \sup_{k\ge 1}{\bf E}e^{hS_k}\le 1 \}.
\end{gather}
\end{thm}
Note that in Theorem \ref{thm1} the generalization of Lundberg inequality for non-homogeneous risk models is obtained. We also have the following assertion which follows immediately from Theorem \ref{thm1}.
\begin{cor} \label{CC1}
Under the assumptions of Theorem \ref{thm1}, for any $u>0,$ $\psi(u)$ satisfies that
\begin{gather} \label{aa12}
\psi(u)\leq \inf_{h\in[0, L(Y_\bullet)]}\{e^{-hu}{\bf E}e^{hY_1}\}\leq e^{-L(Y_\bullet) u}{\bf E}e^{L(Y_\bullet) Y_1}\leq e^{-L(Y_\bullet)u},
\end{gather}
with
${L(Y_\bullet)}:=\sup\{h\ge0:\sup_{j\geq 1}{\bf E}e^{hY_j}\leq 1\}\le L(S_\bullet).$
\end{cor}
Homogeneous renewal risk model is a special case of the model from Corollary \ref{CC1} when random variables $Y_1, Y_2, \ldots$ are i.i.d..
In this case from \eqref{aa12} we have the classical Lundberg inequality \eqref{aa9}
with the adjustment coefficient or Lundberg exponent $L=L(Y_1)$ given by
\begin{gather} \label{aa13}
L(Y_1):=\sup\{h\ge0: {\bf E}e^{hY_1}\leq 1\}.
\end{gather}
It can be seen that our inequality \eqref{aa12} is a little better than the Lundberg inequality in its classical form,
which can be found in \cite{AA10, G91,RSST98},
because
in our variant~\eqref{aa12} of Lundberg inequality we do not exclude
the cases of
${\bf E}e^{L(Y_1)Y_1}<1$ and/or ${\bf E}Y_1=-\infty.$
It follows from \eqref{aa13} that ${\bf E}e^{hY_1}>1$ for all $h>L(Y_1)$. So, for i.i.d. random variables $Y_1, Y_2, \ldots$ we have that the right hand side in \eqref{aa4} is $+\infty$
for all $h>L(Y_1)$.
Thus,
for classical renewal risk models the adjustment coefficient $L(Y_1)$ is the
natural boundary for possible values of the parameter $h$ in inequality \eqref{aa4} of Theorem \ref{thm1}. But for non-homogeneous risk models
the situation may be significantly different because
the optimal value $h=h(u)$ of the parameter $h$ in inequality \eqref{aa4} may be greater than $L(S_\bullet)$ and may tends to $+\infty$ as $u\to\infty$. Moreover, in Examples \ref{exa3} and \ref{exa4} below we present random variables $Y_1, Y_2, \ldots$
corresponding to risk models such that
\begin{gather} \label{aa15}
\psi(u)=o\big(e^{-Nu}\big) \text{ as } u\to\infty \text{ for all } N<\infty.
\end{gather}
Moreover, in Example \ref{exa3} we have:
\begin{gather} \label{aa16}
\forall u>0, \
\psi(2u)\leq
e^{-u^{3/2}}.
\end{gather}
So, very fast decreasing of ruin probabilities is possible in non-homogeneous cases.
Theorem \ref{thm1} allows us also to obtain a
generalization of the Lundberg inequality that for any $u>0$
\begin{gather} \label{aa17}
\psi(u)\leq Ce^{-Lu} \text{ with }\ C<\infty \text{ and } L>0.
\end{gather}
For example, for the periodic risk model with period $l$ Theorem \ref{thm1} yields immediately that
\begin{cor} \label{CC3}
Suppose that there exists an integer $l\geq 1$ such that for all $n=1,2, \ldots$ random variables $Y_{n+l}$ and $Y_n$
are identically distributed. Then under assumptions of Theorem \ref{thm1} inequality (\ref{aa4}) holds with
\begin{gather*}
\sup_{k\geq 1}{\bf E}e^{hS_k}=
\max_{1\le k\le l}{\bf E}e^{hS_k}\text{ for each } h\in[0,L(S_l)],
\end{gather*}
where
$$L(S_l):=\sup\big\{h\ge 0: {\bf E}e^{hS_{l}}\le 1 \big\}.$$
Moreover, for any $u>0$
$$\psi(u)\le \inf_{h\in [0, L(S_l)]}\{ e^{-hu}\max_{1\le k\le l}{\bf E}e^{hS_k} \}\le C_1 e^{-L(S_l)u}$$
with $C_1=\max_{1\le k\le l}{\bf E}e^{L(S_l)S_k}.$
\end{cor}
There are a few works in which the Lundberg inequalities of ruin probabilities of non-homogeneous models are studied.
In Andrulyt$\dot{e}$, et al. \cite{AE15}, Kievinait$\dot{e}$ \& $\check{S}$iaulys \cite{KD18}, and Kizinevi$\check{c}$ \& $\check{S}$iaulys \cite{EJ18}, the Lundberg inequalities of non-homogeneous renewal risk models were obtained. In their models, they all assume that claim sizes and inter-occurrence times are both independent but not necessarily identically distributed.
We'll show in Remarks \ref{Rem3+} and \ref{Rem3} that their results are special cases of our Theorem \ref{thm1}.
For example, it is not possible for them to obtain inequalities with properties \eqref{aa15} or \eqref{aa16}. In the following, a general non-homogeneous renewal risk model with interest rate is studied. Thus
the results of the present paper are complementary to the results in Andrulyt$\dot{e}$, et al. \cite{AE15}, Kievinait$\dot{e}$ \& $\check{S}$iaulys \cite{KD18}, and Kizinevi$\check{c}$ \& $\check{S}$iaulys \cite{EJ18}.
The rest of the paper is organized as follows.
In Section \ref{section2}, we investigate the ruin probabilities of non-homogeneous risk modes. The general risk model is studied then the special case of periodic and quasi-periodic risk model is studied. Finally, some remarks of our results are presented. Examples are given in Section 3 to show that our estimations are more accurate and the probability of ruin in the non-homogeneous risk model may be fast decreasing which is impossible in homogeneous case.
Almost all proofs are gathered in the last section.
Later on we regularly use the fact that expectations ${\bf E}e^{h S}\in(0, \infty]$ are everywhere defined for all random variables $S$ and all real number $h\ge 0$ but may take value of positive infinity.
By this reason all inequalities of the form ${\bf P}[A]\le \text{const} \cdot {\bf E}e^{h S}$ make sense even we omit, for brevity, the assumption ${\bf E}e^{h S}<\infty$.
Note also that for the probability $\psi(u)\le1$ the inequality $\psi(u)\le\psi^*(u)\le\infty$ means that $\psi(u)\le\min\{\psi^*(u),1\}\le1$.
Later on we will use an agreement that
\begin{gather*}
{ E}+\text{const}=\infty \text{ and } \text{const}/{E}=0 \text{ when } {E}=\infty.
\end{gather*}
All limits in this paper are taken with respect to $n\to\infty$ unless the contrary is specified.
And we use random variables $Y_1, Y_2, \ldots$
only when they are mutually independent.
\section{Ruin probability for non-homogeneous risk models}\label{section2}
\noindent{\bf 2.1. General risk model }
Now we consider a more general class of risk reserve processes $R(t)$ which, together with properties (i) and (ii), satisfy the following two assumptions.
\medskip
\hspace*{0.5cm} (iii) For some non-random $r_1, r_2,\ldots$
\begin{gather}\label{Riii}
\forall n\geq 1,\ R(T_k)\ge(1+\alpha_k)R(T_{k-1})-Y_k \text{ and } \alpha_k\ge r_k\ge0.
\end{gather}
\hspace*{0.5cm} (iv) Random variables $Y_k^*:=\frac{Y_k}{1+\alpha_k}, \ k=1,2,\ldots,$
are mutually independent.
\medskip
\medskip
{\bf Model B.} \
Instead of \eqref{aa6}
suppose that on each interval $[T_{k-1}, T_k)$ the surplus accumulates as follows
\begin{gather*}
R(t)=(1+\alpha_k(t))R(T_{k-1})+(1+\beta_k(t))p_k(t-T_{k-1}), \ k=1,2,\ldots,
\end{gather*}
where $(1+\alpha_k(t))R(T_{k-1})$ is the accumulated value of $R(T_{k-1})$ from $T_{k-1}$ to $t$ under interest rate $\alpha_k(t)$
and $(1+\beta_k(t))p_k(t-T_{k-1})$ is the accumulated value of premiums collected from $T_{k-1}$ to $t$ under premiun rate $p_k$ and interest rate $\beta_k(t)$.
Assume again that claims $Z_k$ arrive to an insurer only at time $T_k$, i.e.,~\eqref{aa5} holds.
Suppose now that processes $\alpha_k(t)\ge0$ and $\beta_k(t)\ge0$ are non-decreasing, and random vectors
\begin{gather*}
(p_k, Z_k, \theta_k, \alpha_k(T_k), \beta_k(T_k) ), \ k=1,2,\ldots,
\end{gather*}
are mutually independent.
Then conditions (i)-- (iv) hold and random variables
\begin{gather}\label{A1}
Y_k=(1+\alpha_k(T_k))R(T_{k-1})-R(T_k)=Z_k-X(T_k)=Z_k -(1+\beta_k(T_k))p_k \theta_k,\ k=1,2,\ldots,
\end{gather}
are also mutually independent.
\medskip
It is clear that the presented model is more general than model A.
For each $k=1, 2, \ldots$ introduce notations:
\begin{gather}\label{R+}
v_k:=\prod_{j=1}^{k}\frac{1}{1+r_j}\text{ and }
S_k^*:=\sum_{j=1}^k v_{j-1}Y_j^*,
\end{gather}
where $v_0:=1$.
\begin{thm}\label{thm2}
Under assumptions (i)--(iv), for any $u>0$ and any $h\ge 0$ the following inequality
\begin{gather}
\label{R6+}
\psi(u)\leq e^{-hu}\sup_{k\geq1}{\bf E}e^{hS_k^*}
=e^{-hu}\sup_{k\geq1}\prod_{j=1}^k{\bf E}e^{hv_{j-1}Y_j^*},
\end{gather}
holds.
In addition, the Lundberg inequality (\ref{aa9}) holds with $L=L(S_\bullet^*)$
where
\begin{gather}\label{A3}
L(S_\bullet^*):=\sup\Big\{h\ge0: \sup_{k\ge 1}{\bf E}e^{hS_k^*}\le 1 \Big\}.
\end{gather}
\end{thm}
\begin{rem}
As an analogue of Corollary \ref{CC1} for model A, it is easy to see that all the assertions of Corollary \ref{CC1} also hold for model B with
$$ L(Y_\bullet)=\underline{L}:=\sup\Big\{h\ge0:\sup_{j\geq 1}{\bf E}e^{hv_{j-1}Y_j^*}\leq 1\big\}\le L(S_\bullet^*).$$
\end{rem}
Theorem \ref{thm2} also yields the following result.
\begin{cor} \label{C2+}
Suppose that for each $k\ge 1$
\begin{gather*}
Y_k=b_k\xi_k\text{ and }b_kv_{k-1}\le1,
\end{gather*}
where random vectors $\{(\alpha_k,\xi_k),k=1,2,\ldots,\}$ are i.i.d..
In this case under assumptions (i)--(iii)
inequality \eqref{aa12} holds with
$$L(Y_\bullet)=L(Y_1^*):=\sup\{h\ge0: {\bf E}e^{hY_1^*}\leq 1\}.$$
In particular, for any $u>0$
\begin{gather} \label{c6+}
\psi(u)\leq e^{-\kappa u} \text{ if }
{\bf E}e^{\kappa Y_1^*}=1.
\end{gather}
\end{cor}
Thus, when $b_n=(1+r)^{n}$ and $v_n=(1+r)^{-n}<1$ we obtain two generalizations \eqref{aa12} with $L(Y_\bullet)=L(Y_1^*)$ and \eqref{c6+} of the famous Lundberg inequality for the case when $|Y_n|=(1+r)^{n}|\xi_n|\to\infty$ almost surely and with high speed.
\begin{rem} \label{Rem4}
Two special cases of inequality \eqref{c6+} are obtained in Corollaries 3.1 and 3.2 of Cai \cite{CJ02a} when
$$
Y_k=X_k-Z_k \text{ or }
Y_k=(1+\alpha_k)X_k-Z_k
$$
with $r_k\equiv0$.
Earlier a simpler case with
$$
\alpha_k=r\ge0\text{ and }
Y_k=(1+r)X_k-Z_k,
$$
is considered in Yang \cite{YH99}.
Underline that the mutual independence of non-negative random variables $X_k$, $Z_k$ and $\alpha_k$ is essential for the proofs in \cite{CJ02a, YH99}.
\end{rem}
\noindent{\bf 2.2. Periodic and quasi-periodic risk model }
Asmussen \& Rolski \cite{AR94} (see also Asmussen \& Albrecher \cite{AA10}) studied a kind of risk process which happens in a periodic environment. For the Lundberg-type inequality \eqref{aa17} they obtained that adjustment coefficient $L$ is the same as for the standard time-homogeneous Poisson risk process obtained by averaging the parameters over a period. In Corollary \ref{CC3} we have found the similar property for periodic risk models with period $l$ under assumptions of Theorem~\ref{thm1}. Now we present two more general results under conditions of Theorem \ref{thm2}.
\begin{cor}\label{corP7}
Suppose that there exist integers $l,m\geq 1$ and a real number $L^*\geq 0$ such that for any $n\ge m$
\begin{gather} \label{p1}
{\bf E}e^{L^*(S_{n+l}^*-S_{n}^*)}\le 1.
\end{gather}
Then under assumptions (i)--(iv), for any $h\in[0, L^*]$ inequality \eqref{R6+} holds with
\begin{gather} \label{p2}
\sup_{k\geq 1}{\bf E}e^{hS_k^*}
= \max_{1\leq k\leq l+m-1}{\bf E}e^{hS_k^*}.
\end{gather}
Moreover, for any $u>0$
\begin{gather}\label{A4}
\psi(u)\le \inf_{h\in [0, L^*]}\{e^{-hu}\max_{1\leq k\leq l+m-1}{\bf E}e^{hS_k^*} \}\le C_2 e^{-L^*u}
\end{gather}
with $C_2=\max_{1\leq k\leq l+m-1}{\bf E}e^{L^*S_k^*}.$
\end{cor}
\begin{thm} \label{thm3}
Suppose that there exist an integer $l\geq 1$ and a real number $q_l>0$ such that for all $n=1,2, \ldots$ random variables $Y_{n+l}^*$ and $q_lY_n^*$
are identically distributed.
Assume also that for each $n\ge 1$
\begin{gather} \label{p5}
q_l v_l\le1 \text{ and } r_{n+l}=r_n.
\end{gather}
And set
\begin{gather}\label{p6}
L(S_l^*):=\sup\big\{h\ge 0: {\bf E}e^{hS_{l}^*}\le 1 \big\}.
\end{gather}
Then under assumptions (i)--(iv), for any $h\in[0,L(S_l^*)]$ inequality \eqref{R6+} holds
with
\begin{gather} \label{p7}
\sup_{k\geq 1}{\bf E}e^{hS_k^*}\le
\max_{0\leq k< l}{\bf E}e^{hS_k^*}.
\end{gather}
And for any $u>0$ the following Lundberg-type inequality holds
\begin{gather}\label{A5}
\psi(u)\le \inf_{h\in [0, L(S_l^*)]}\{e^{-hu}\max_{0\le k<l}{\bf E}e^{hS^{*}_k} \}\le C_3 e^{-L(S_l^*)u}
\end{gather}
with $C_3=\max_{0\le k<l}{\bf E}e^{L(S_l^*)S^{*}_k}.$
Moreover, if $q_lv_l=1$ then \eqref{p2}
is also true with $m=1$ for all $h\in[0, L(S_l^*)]$.
\end{thm}
Note that the value in the right hand side of \eqref{p2}
may be less than 1. On the other hand, the right hand side of \eqref{p7}
may not be less than 1 since ${\bf E}e^{hS_0^*}=e^0=1.$
Thus, Corollary \ref{corP7}
may give sharper estimates than Theorem \ref{thm3}
because condition \eqref{p1} may be stronger than the assumptions in Theorem \ref{thm3}.
The models satisfying conditions of Corollary \ref{corP7} or Theorem \ref{thm3} will be called quasi-periodic.
For us it is essential that for $l>1$ all such risk models are automatically non-homogeneous.
Models satisfying assumptions of Theorem \ref{thm3} with $q_lv_l=1$ are naturally called periodic.
Periodic model from Corollary \ref{CC3} is a special case of Theorem \ref{thm3} with $q_l=v_l=1=m$ when $S_k^*=S_k$ and \eqref{p2} is true.
\noindent{\bf 2.3. General remarks. }
\begin{rem} \label{Rem5}
Theorem \ref{thm1} is an evident special case of Theorem \ref{thm2} when
$\alpha_k=r_k=0$ for all $k=1,2,\dots$. Hence, all corollaries from Theorem \ref{thm2}, which are presented below, may be considered also as corollaries from Theorem \ref{thm1}.
\end{rem}
\begin{rem} \label{Rem5}
Theorems \ref{thm1} and \ref{thm2} and all their corollaries remains valid also for non-homogeneous discrete-time risk models when claims arrive {at} non-random times $T_1,T_2,\dots$.
\end{rem}
\begin{rem} \label{Rem3+}
In \cite{AE15, KD18}, the authors use the trivial inequality
\begin{gather*}
\forall u>0, \ \forall h\ge0, \ {\bf P}[\sup_{k\geq 1}S_k>u]\leq\sum_{k=1}^\infty
{\bf P}[S_k>u]\leq e^{-hu}\sum_{k=1}^\infty {\bf E}e^{h S_k}
\end{gather*}
{instead of} our sharper estimate \eqref{aa4}. It is clear that the results in \cite{AE15, KD18} will be improved automatically by using our estimate \eqref{aa4}.
\end{rem}
\begin{rem} \label{Rem3}
In the proof of Theorem 4 in \cite{EJ18} it is shown (under several additional assumptions)
that for any $u>0$
\begin{gather} \label{c1-}
\psi(u)\leq \inf_{h\in[0, L(Y_\bullet)]}\{e^{-hu}\sup_{i\geq 1}{\bf E}e^{hY_i}\}.
\end{gather}
It is clear that the inequality \eqref{aa12} is better than \eqref{c1-}.
It also follows from Example \ref{exa3} below that our estimate \eqref{aa4} from Theorem \ref{thm1}
may be qualitatively better than the estimate \eqref{c1-}.
\end{rem}
\begin{rem} \label{Rem0}
Note that all independent random variables $\{Y_n\}$ (with all possible distributions) may appear
in Theorem \ref{thm1} since we everywhere may consider model A with values
\begin{gather*}
Z_n=Y_n^+:=\max\{Y_n,0\}, \ X_n=p_n\theta_n=Y_n^-:=\max\{-Y_n,0\},\ p_n=1.
\end{gather*}
For this reason, in Examples \ref{exa0}--\ref{exa4} below we do not construct risk models but only introduce random variables $\{Y_n\}$ that appear in risk models.
Similarly, all random variables $\{Y_n^*\}$ (with all possible distributions)
may appear in Theorem \ref{thm2}
with all possible {real numbers $r_n\ge 0$}. Indeed, we may consider model B with
\begin{gather*}
Z_n=(1+r_n)\max\{Y_n^*,0\}, \ X_n(T_n)=p_n\theta_n=\max\{-Y_n^*,0\},\ p_n=1,
\end{gather*}
and with $ \alpha_n(T_n)= \beta_n(T_n) =r_n\ge0$ for all $n=1,2,\ldots$.
\end{rem}
\section{Examples}
In this section, special examples are presented to show that our estimations are more accurate and the ruin probability in non-homogeneous risk model may be fast decreasing which is impossible in homogeneous case.
\begin{exa}\label{exa0}
Let $Y_1, Y_2, \ldots$ be independent normal
random variables with
\begin{gather} \label{p11}
Y_n \sim N(a_n, 1) \text{ and } a_n+a_{n+1}\le a_1+a_2=-1,\ n=1, 2, \ldots.
\end{gather}
It is easy to calculate that
\begin{gather} \label{p12}
{\bf E}e^{hY_n}=e^{ha_n+\frac{h^2}{2}} \text{ and } {\bf E}e^{hS_n}={\bf E}e^{h\sum_{i=1}^na_n+\frac{n}{2}h^2},\ n=1,2, \ldots.
\end{gather}
Hence,
$$L(S_2)=\sup\{h\ge 0: {\bf E}e^{hS_2}\le 1 \}=\sup\{h\ge 0:e^{-h+h^2}\le1 \}=1.$$
Thus, we have from \eqref{p11} and \eqref{p12} with $h=1$ that
\begin{gather} \label{p1+}
{\bf E}e^{S_{n+2}-S_{n}}=e^{a_{n+1}+a_{n+2}+1}\le e^0= 1 ,\ \forall n\ge 0.
\end{gather}
Comparing \eqref{p1} and \eqref{p1+} we obtain that random variables $\{Y_n\}$ from \eqref{p11}
satisfy all conditions of Corollary \ref{corP7}
with $l=2$, $m=1$ and $L^*=1$.
So, we have from \eqref{R6+} and \eqref{p2} with $h=L^*=1$ that for any $u>0$
\begin{gather} \label{p14}
\psi(u)\leq \max \{{\bf E}e^{S_1}, {\bf E}e^{ S_2}\}e^{-u}=e^{(a_1+1/2)^+-u},
\end{gather}
because
$$\max \{{\bf E}e^{S_1}, {\bf E}e^{ S_2}\}=\max \{e^{a_1+1/2}, 1\}=e^{\max \{a_1+1/2, 0\}}.$$
Inequality \eqref{p14} allows us to make several conclusions about risk models
with random variables $\{Y_n\}$ from \eqref{p11}. First, if $a_1\leq -1/2,$ we can obtain the Lundberg inequality~(\ref{aa9}) with $L=L^*=1.$ Second, if $a_1>-1/2,$ we can prove the generalization (\ref{aa17}) of the Lundberg inequality with $C=e^{a_1+1/2}>1.$
Third, in the case of $a_1\ge0$ we have
\begin{gather*}
\sup_{i\geq 1}{\bf E}e^{hY_i}\ge {\bf E}e^{hY_1}=e^{a_1h+h^2/2}\ge e^{h^2/2}\ge e^0=1,\ \forall h\ge 0.
\end{gather*}
Thus, it follows from \eqref{aa11} that in this case $L(Y_\bullet)=0$ and, hence, inequality \eqref{c1-} allows us to obtain only trivial estimate $\psi(u)\le1$.
So, (see Remark \ref{Rem3}) Theorem 4 from \cite{EJ18} does not work in this case, whereas, our results yield the estimate
(\ref{aa17}) with $C=e^{a_1+1/2}<\infty.$
\end{exa}
\begin{exa}(see \cite[Example 1]{KD18}) \label{exa1}
Suppose $Y_1, Y_2, \ldots$ are independent random variables such that:\\
\hspace*{0.5cm} $\bullet$ $Y_i$ are uniformly distributed on interval $[0,2]$ for $i\equiv 1 \mod 3;$\\
\hspace*{0.5cm} $\bullet$ $Y_i$ are uniformly distributed on interval $[-1,0]$ if $i\equiv 2 \mod 3;$\\
\hspace*{0.5cm} $\bullet$ $\overline{F}_{Y_i}(x)=1_{(\infty, -2)}(x)+e^{-x-2}1_{[-2, \infty)}(x)$ when $i\equiv 0 \mod 3.$
It is a periodic risk model with $l=3$ and it is easy to see that this model satisfies all the conditions of Corollary \ref{CC3}. We can calculate that
for all $0<h<1$
\begin{gather*}
{\bf E}e^{hY_2}=\frac{1-e^{-2h}}{2h}=e^{-2h}{\bf E}e^{hY_1}<1,\
{\bf E}e^{hY_3}= \frac{e^{-2h}}{1-h}.
\end{gather*}
Then for $h_0=2/3$
\begin{gather*}
{\bf E}e^{h_0S_3}=\frac{(1-e^{-2h_0})^2}{(2h_0)^2(1-h_0)}<1,
\ {\bf E}e^{h_0Y_1}=\frac{e^{2h_0}-1}{2h_0}<2.2.
\end{gather*}
Thus, we have from Corollary \ref{CC3} that
$$
\psi(u)\leq \max \{{\bf E}e^{S_1}, {\bf E}e^{ S_2}, {\bf E}e^{ S_3}\}e^{-h_0u}<2.2e^{-\frac{2}{3}u}.
$$
So, we have proved that
\begin{gather} \label{p20+}
\psi(u)\leq \psi^{*}_1(u):=\max \{1, 2.2e^{-\frac{2}{3}u}\}
\leq e^{(1-\frac{2}{3}u)^-},\ \forall u\geq 0.
\end{gather}
Remind that in \cite{KD18} the following bound is obtained
\begin{gather} \label{p20}
\psi(u)\leq \psi_1^{\star}(u):=\min \big\{1, 1502e^{-0.01269u} \big\},\ \forall u\geq 0.
\end{gather}
It is clear that estimate \eqref{p20+} is more accurate than \eqref{p20}. For example,
$$\psi^{*}_1(576)< e^{-381}<10^{-165} \text{ whereas } \psi^{\star}_1(576)=1.$$
\end{exa}
Here we are going to present examples of risk models with property (\ref{aa15})
which is impossible in homogeneous case.
\begin{exa} \label{exa3}
Consider again
independent random variables $Y_1, Y_2, \ldots$ with normal distributions from Example \ref{exa0}.
First, suppose that they are i.i.d. with $ N(-1/2,1)$.
In this case condition (\ref{p11}) holds and $L(Y_1):=\sup\{h\ge0: {\bf E}e^{hY_1}\leq 1\}=\sup\{h\ge0:e^{-h/2+h^2/2}\leq 1\}=1$.
Thus we have the Lundberg inequality (\ref{aa9}) with $L=1$ and it is the upper boundary for values $h$ which we use in inequality~(\ref{aa4}).
Suppose now that condition (\ref{p11}) takes the form
\begin{gather} \label{p11+}
Y_n \sim N(a_n, 1) \text{ with } a_n=(1-2n)/4,\ n=1, 2, \ldots.
\end{gather}
In this case $\sum_{i=1}^na_i=-n^2/4$ and we obtain from (\ref{p12}) that for any $h\ge 0$ and each $n=1,2, \ldots$
\begin{gather*} \label{p12+}
{\bf E}e^{hS_n}=e^{-hn^2/4+nh^2/2}=e^{h^3/4-h(n-h)^2/4}\le e^{h^3/4}.
\end{gather*}
Hence, we have from (\ref{aa4}) that for any $u>0$ and $h\ge 0$
\begin{gather}\label{p23}
\psi(u)\leq e^{-hu}\sup_{n\geq1}{\bf E}e^{hS_n}\le e^{-hu+h^3/4}.
\end{gather}
With $h=2\sqrt{u/3}$ we obtain from (\ref{p23}) that for any $u>0$
\begin{gather*} \label{p30}
\psi(u)\leq e^{-4(u/3)^{3/2}}= e^{-cu^{3/2}} \text{ where } c^2=16/27.
\end{gather*}
\end{exa}
So, we obtain an example of a risk model with property (\ref{aa16}) mentioned in the introduction.
\begin{exa} (see \cite[Example 2]{KD18}) \label{exa4}
Suppose that $Y_1, Y_2, \ldots$ are independent random variables with
$$
{\bf P}[Y_n=1]=\frac{1}{n+1}, \ {\bf P}[Y_n=-1]=1-{\bf P}[Y_n=1],\ n=1,2, \ldots.$$
In this case for all $h$
$$
{\bf E}e^{h Y_n}=e^{h}\frac{1}{n+1}+e^{-h}\Big(1-\frac{1}{n+1}\Big)=1+\frac{(1-e^{-h})(e^{h}-n)}{n+1}.
$$
So, ${\bf E}e^{h Y_n}\le1$ if and only if $n\ge e^{h}$. Hence, for each $h>0$
\begin{gather}\label{p25}
\sup_{n\geq1}{\bf E}e^{hS_n}={\bf E}e^{hS_m} \text{ iff } m+1\ge e^{h}\ge m.
\end{gather}
Thus, with $m$ from \eqref{p25}
$$
\sup_{n\geq1}{\bf E}e^{hS_n}={\bf E}e^{hS_m} <(e^{h})^m\le e^{he^{h}},
$$
and, using \eqref{aa4}, we obtain
\begin{gather}\label{p27}
\forall h,\ u>0, \
\psi(u)\leq e^{-hu}\sup_{n\geq1}{\bf E}e^{hS_n}\le e^{-hu+he^{h}}.
\end{gather}
With $h=\log(u/2)>0$ we have from (\ref{p27}) that
\begin{gather} \label{p28}
\forall u>2, \
\psi(u)\leq \psi^{*}_2(u):= e^{-(u/2)\log(u/2)}= \Big(\frac2u\Big)^{u/2}.
\end{gather}
So, we obtain another example of a risk model with property (\ref{aa15}).
Remind that in \cite{KD18} the following bound is given
\begin{gather}\label{p29}
\forall u\geq 0,\ \psi(u)\leq \psi^{\star}_2(u):= \min\big\{1, 178e^{-\frac{u}{20}} \big\}.
\end{gather}
It is clear that estimate \eqref{p28} is more accurate than \eqref{p29}. For example,
$$\psi^{*}_2(103)< 10^{-88}\text{ in \eqref{p28}, whereas } \psi_2^{\star}(103)=1 \text{ in \eqref{p29} }.$$
\end{exa}
\section{Proofs }
\subsection{Key Lemma}
Before proving our main results, we first introduce the following key lemma.
\begin{lem}\label{L1}
If random variables $X_1, X_2, \ldots$ are mutually independent, then for any real number $w$, any $h\ge 0$ and any $n\ge 1$
\begin{gather} \label{m1}
{\bf P}[\max_{1\leq k\leq n}W_k>w]\leq e^{-hw}\max_{1\leq k\leq n}{\bf E}e^{hW_k},
\end{gather}
where $W_k=X_1+X_2+\ldots+X_k.$ In addition, for any $w$ and any $h\ge 0$
\begin{gather} \label{m2}
{\bf P}[\sup_{k\geq 1}W_k>w]\leq e^{-hw}\sup_{k\geq 1}{\bf E}e^{h W_k}.
\end{gather}
\end{lem}
{\bf Proof.}
If $M_n(h):=\max_{1\leq k\leq n}{\bf E}e^{hW_k}=\infty$ then the inequality \eqref{m1} is obvious.
So, suppose that $0<M_n(h)<\infty$ and note that in this case the following sequence
\begin{gather*}
\mu_k=\frac{e^{hW_k}}{{\bf E}e^{hW_k}},\ k=0,1,2,\ldots, n, \text{ with }\mu_0=1,
\end{gather*}
is a martingale. This fact is evident, since for all $k\ge0$
\begin{gather*}
\mu_k=\mu_{k-1}\frac{e^{hX_k}}{{\bf E}e^{hX_k}} \text{ and }
{\bf E}\Big[ \frac{e^{hX_k}}{{\bf E}[e^{X_k}]} \Big| \mu_1, \ldots, \mu_{k-1}\Big]=1.
\end{gather*}
Thus, by maximal inequality for martingale
\begin{eqnarray*}
\forall x>0,\ {\bf P}\Big[\max_{1\leq k\leq n}\frac{e^{hW_k}}{{\bf E}e^{hW_k}}>x\Big]
= {\bf P}\Big[\max_{1\leq k\leq n}\mu_k>x\Big]\leq \frac{{\bf E}\mu_n}x=\frac1x.
\end{eqnarray*}
Hence, with $x=e^{hw}/M_n(h)$ we obtain that
\begin{eqnarray*}
{\bf P}\Big[\max_{1\leq k\leq n}W_k>w\Big]
&=&{\bf P}\Big[\max_{1\leq k\leq n}\frac{e^{hW_k}}{M_n(h)}> x=\frac{e^{hw}}{M_n(h)}\Big]\nonumber\\
&\leq& {\bf P}\Big[\max_{1\leq k\leq n}\frac{e^{hW_k}}{{\bf E}e^{hW_k}}>x\Big]
\leq \frac1x=e^{-hw}M_n(h).
\end{eqnarray*}
So, inequality \eqref{m1} is proved.
Note that
$\max_{1\leq k\leq n}W_n\uparrow \sup_{k\geq 1} W_k$. Hence
$${\bf P}\Big[\sup_{k\geq 1}W_k>w\Big]
=\lim_{n\rightarrow\infty}{\bf P}\Big[\max_{1\leq k\leq n}W_k>w\Big]
\leq e^{-hw}\sup_{n\ge1}M_n(h)
= e^{-hw}\sup_{k\geq 1}{\bf E}e^{h W_k}.$$
Thus, Lemma \ref{L1} is proved.
\subsection{ Proof of Theorem \ref{thm1}}
It follows from (i) and (ii) that
\begin{gather} \label{Ri}
\inf_{t\ge0}R(t)=\inf_{k\ge1}\inf_{t\in [T_{k-1}, T_k]}R(t)
=\inf_{k\ge1}\min\{R(T_{k-1}), R(T_k)\}=\inf_{k\ge0} R(T_k).
\end{gather}
Next, from the definition of $Y_k$ in \eqref{aa3} and using telescoping sum we have that
$$
R(T_k)= R(T_0)-\sum_{j=1}^kY_j=u-S_k,\ k=0,1,2,\ldots,
$$
with $S_k=Y_1+Y_2+\ldots +Y_k$ and $S_0=0$. Thus,
$\inf_{t\ge0}R(t)= u-\sup_{k\ge0}S_k$,
and hence, by the definition of $\psi(u)$ in \eqref{aa1} for any $u>0$ the following equality holds,
\begin{gather} \label{m12}
\psi(u)={\bf P}[u-\sup_{k\ge0}S_k<0]={\bf P}[\sup_{k\ge0}S_k>u].
\end{gather}
Since $S_0=0$, we have from \eqref{m12} that for any $u>0$
\begin{gather} \label{m13}
\psi(u)={\bf P}[\sup_{k\ge1}S_k>u].
\end{gather}
From \eqref{m13} and estimate \eqref{m2} of Lemma \ref{L1} we obtain the first inequality of \eqref{aa4}. And since $Y_1, Y_2, \ldots$ are independent, then all the assertions of (\ref{aa4}) are proved.
In addition, from (\ref{aa4}) and the definition of $L(S_\bullet)$ in (\ref{aa11}) we have that
\begin{eqnarray*}
\psi(u)&\le& \inf_{h\ge 0}\{e^{-hu}\sup_{k\ge 1}{\bf E}e^{hS_k} \}\\
&\le& e^{-L(S_\bullet)u}\sup_{k\ge 1}{\bf E}e^{L(S_\bullet) S_k}\\
&\le& e^{-L(S_\bullet)u},
\end{eqnarray*}
i.e., the Lundberg inequality (\ref{aa9}) holds with $L=-L(S_\bullet).$
Thus all the assertions are proved.
\subsection{ Proof of Theorem \ref{thm2}.}
For each $k=1, 2, \ldots,$ introduce notations:
\begin{gather} \label{c11}
v_k^*:=\prod_{j=1}^{k}\frac{1}{1+\alpha_j}\text{ and }
S_k^{**}:=\sum_{j=1}^k v_{j}^*Y_j=\sum_{j=1}^k v_{j-1}^*Y_j^*,
\end{gather}
with $v_0^*:=1$.
\begin{lem} \label{L-2}
If $v_0=v_0^*=1$, $S_0^{**}=S_0^*=0$ and $\alpha_k\ge r_k\ge0$ for all $k\ge0$, then for any $n\ge 0$
\begin{gather} \label{c12}
\max_{0\le k\le n}S_k^{**}\le \max_{0\le k\le n}S_k^*.
\end{gather}
\end{lem}
{\bf Proof.}
Since $\alpha_j\geq r_j\geq 0$ for $j=1,2, \ldots,$ we have that
\begin{gather} \label{c13}
\forall j\geq 1,\
c_{j}:=\prod_{i=1}^{j}\frac{1+r_i}{1+\alpha_i}
=c_{j-1}\frac{1+r_j}{1+\alpha_j}\ge c_{j-1}.
\end{gather}
Thus, real numbers $\{c_j\}$ have the following property
\begin{gather} \label{c14}
\forall j\geq 1,\
1=c_0\ge c_1\ge \ldots\ge c_{j-1} \ge c_j>0.
\end{gather}
Next, from \eqref{c11} and \eqref{c13} we have for any $j=1, 2, \ldots$ that
\begin{gather} \label{c15}
Y_j^* v^{*}_{j-1}
=Y^{*}_j \prod_{i=1}^{j-1} \frac{1}{1+r_i} \prod_{i=1}^{j-1}\frac{1+r_i}{1+\alpha_i}=Y^{*}_j v_{j-1} c_{j-1}=(S^{*}_j -S^{*}_{j-1}) c_{j-1}.
\end{gather}
Now, substituting \eqref{c15} into \eqref{c11} we obtain that for all $k\le n$
\begin{eqnarray*}
S^{**}_k&=&
\sum_{j=1}^k v_{j-1}^*Y_j^*=\sum_{j=1}^k
(S^{*}_j -S^{*}_{j-1}) c_{j-1}
=c_{k-1} S^{*}_k+\sum_{j=1}^{k-1}(c_{j-1}-c_j)S^{*}_j\nonumber\\
&\leq& c_{k-1} M_n^*+\sum_{j=1}^{k-1}(c_{j-1}-c_j)M_n^*=c_0 M_n^*=M_n^*:=\max_{1\leq j\leq n}S_j^*,
\end{eqnarray*}
where
we also use \eqref{c14}.
So, inequality \eqref{c12} is proved.
\medskip
{\bf Proof of Theorem \ref{thm2}.}
Multiplying \eqref{Riii} by $v_k^*$ we obtain for any $k\ge 1$ that
\begin{gather*}
v_k^*R(T_k)\ge v_{k-1}^*R(T_{k-1})-v_k^*Y_k=v_{k-1}^*R(T_{k-1})-v_{k-1}^{*}Y_k^{*}.
\end{gather*}
Hence, by induction for any $k\ge 1$
\begin{gather*}
v_k^*R(T_k)\ge v_0^* R(T_0)-\sum_{j=1}^k v_j^*Y_j=u-S_k^{**}.
\end{gather*}
This fact and Lemma \ref{L-2} imply for any $n\ge 0$ that
\begin{gather} \label{c18}
\min_{0\le k\le n}v_k^*R(T_k) \ge u-\max_{0\le k\le n}S_k^{**}\ge u-\max_{0\le k\le n}S_k^*.
\end{gather}
Inequality \eqref{c18} with $v_k^*>0$ implies that for any $u>0$ and each $n=0,1,2,\ldots$
\begin{gather*}
{\bf P}[ \min_{0\le k\le n}R(T_k)<0]={\bf P}[ \min_{0\le k\le n}v_k^*R(T_k)<0]
\le{\bf P}[u-\max_{0\le k\le n}S_k^*<0]={\bf P}[\max_{0\le k\le n}S_k^*>u].
\end{gather*}
Since $S^{*}_0=0$ and $\{S_k^*\}$ are
sums of independent random variables $\{v_{j-1}Y^{*}_j\},$ from Lemma \ref{L1} for any $u>0$ and each $ n=1,2,\ldots$ we have that
\begin{gather}\label{A2}
{\bf P}[ \min_{0\le k\le n}R(T_k)<0]\le{\bf P}[\max_{1\le k\le n}S_k^*>u]\le e^{-hu}\max_{1\le k\le n}{\bf E}e^{hS^{*}_k}.
\end{gather}
On the other hand, equality \eqref{Ri} again follows from (i) and (ii) for all $n\ge0$.
Then taking limit on both sides of \eqref{A2} as $n\to\infty$ we obtain the first inequality in \eqref{R6+}.
The equality in \eqref{R6+} directly comes from the definition of $S^{*}_k$ in (\ref{R+}) and the independence of $Y^*_j.$
The rest of proof is similar to Theorem \ref{thm1}.
\subsection{ Proof of Corollary \ref{corP7}}
For all $n\ge m+l,$ since $Y^*_1, Y^*_2, \ldots$ are independent then the random variables $\Delta^*_{n,l}:=S_{n}^* -S_{n-l}^*$ and $S_{n-l}^*$ are independent. Hence, by Jensen's inequality for any $h\in [0, L^*]$ we have from \eqref{p1} that
\begin{gather} \label{c21}
{\bf E}e^{h\Delta^*_{n,l}}\le \Big({\bf E}e^{L^*\Delta^*_{n,l}}\Big)^{h/L^*}\le1\text{ and }
{\bf E}e^{hS^*_n}={\bf E}e^{h\Delta^*_{n,l}}\cdot {\bf E}e^{hS^*_{n-l}}\le {\bf E}e^{hS^*_{n-l}}.
\end{gather}
Since \eqref{p1} holds for any $n\ge m,$ then we can do (\ref{c21}) again that
$${\bf E}e^{hS_{n-l}^*}\le {\bf E}e^{hS^*_{n-2l}}$$
if $n-l\ge m+l,$
otherwise ${\bf E}e^{hS^*_n}\le \max_{1\le k\le m+l-1}{\bf E}e^{hS^*_k}.$
Using induction with respect to $n$ it is not difficult to see that we can do (\ref{c21}) for $i$ times until $n-il<m+l,$ i.e., $n-il\le m+l-1,$ with
$${\bf E}e^{hS^*_n}\le {\bf E}e^{h S^*_{n-il}}.$$
Then we obtain that for any $n\ge m+l$
\begin{gather*}
{\bf E}e^{hS^*_n}\le \max_{1\le k\leq m+l-1}{\bf E}e^{hS^*_k}.
\end{gather*}
Hence
$$\max_{1\le k\leq m+l-1}{\bf E}e^{hS^*_k}\le\sup_{n\ge 1}{\bf E}e^{hS^*_n}\le \max_{1\le k\leq m+l-1}{\bf E}e^{hS^*_k}.$$
So, equality \eqref{p2} is proved.
From (\ref{p2}) and the results in Theorem \ref{thm2} we see that for any $h\in [0, L^*]$ and $u>0$
\begin{eqnarray*}
\psi(u)&\le& e^{-hu}\max_{1\le k\le l+m-1}{\bf E}e^{hS^*_k}\\
&\le& \inf_{h\in [0, L^*]}\{ e^{-hu}\max_{1\le k\le l+m-1}{\bf E}e^{hS^*_k} \}\\
&\le& e^{-L^*u}\max_{1\le k\le l+m-1}{\bf E}e^{L^* S^*_k}.
\end{eqnarray*}
Therefore, the result in (\ref{A4}) is proved with $C_2=\max_{1\leq k\leq l+m-1}{\bf E}e^{L^*S_k^*}.$
\subsection{ Proof of Theorem \ref{thm3}.}
It is clear that each integer $n\ge1$ may be represented in the following way
\begin{gather} \label{c24}
1 \le n=il+k \text{ for some } i\ge0 \text{ and some } 1\le k\le l.
\end{gather}
\begin{lem}\label{lem4}
Under assumptions of Theorem \ref{thm3}
for all $i=0,1,2, \ldots$ and $k=1,2, \ldots$, the following random variable
\begin{gather} \label{c26}
\Delta_{i,k}:=S^*_{il+k}-S^*_{il}=\sum_{j=1}^{k}v_{ik+j-1}Y_{ik+j}^*
\end{gather}
is identically distributed with $(q_lv_l)^iS^*_{k}$.
In particular, for any $ h\in[0,L(S_l^*)]$
\begin{gather} \label{c27}
\sup_{n\geq 1}{\bf E}e^{hS_n^*}\le
\max_{1\leq k\leq l}\sup_{i\geq 0}{\bf E}e^{h(q_lv_l)^iS_k^*},
\end{gather}
where $L(S_l^*)$ is defined in (\ref{p6})
\end{lem}
{\bf Proof. }
It follows from condition \eqref{p5} that
\begin{gather*}
v_{j-1}=\prod_{i=1}^{j-1}\frac{1}{1+r_i}
=\prod_{i=1}^{l}\frac{1}{1+r_i}\cdot\prod_{i=l+1}^{j-1}\frac{1}{1+r_i}
=v_{l}\cdot\prod_{i=1}^{j-l-1}\frac{1}{1+r_i}=v_{l}v_{j-l-1}.
\end{gather*}
Hence, for all $n=1,2, \ldots$, random variables $v_{n+l-1}Y_{n+l}^*$ and $(q_lv_l)v_{n-1}Y_n^*$
are identically distributed because $Y_{n+l}^*$ and $q_lY_n^*$
are identically distributed by the assumption of Theorem~\ref{thm3}.
For $n=il+k,$ using induction with respect to $i$ it is not difficult to obtain that
$v_{il+j-1}Y_{il+j}^*$ and $(q_lv_l)^iv_{j-1}Y_j^*$ are identically distributed
for all $i=0,1,2, \ldots$ and $j=1,2, \ldots$.
As a result, $\sum_{j=1}^k v_{il+j-1}Y^*_{il+j}$ are identically distributed with $\sum_{j=1}^k (q_l v_l)^iv_{j-1}Y^*_j=(q_l v_l)^i\sum_{j=1}^k v_{j-1}Y^*_k=(q_l v_l)^iS^*_k.$
So, using definition (\ref{c26}), we have the first assertion of the lemma that
$\Delta_{i,k}$ and $(q_lv_l)^iS^*_{k}$ are identically distributed.
Using telescoping sum, we note that $S^*_{il+k}=\sum_{m=0}^{i-1}\Delta_{m,l}+\Delta_{i,k}$. Hence
\begin{gather} \label{c29}
{\bf E}e^{hS^*_{il+k}}=\prod_{m=0}^{i-1}{\bf E}e^{h\Delta_{m,l}}\cdot {\bf E}e^{h\Delta_{i,k}}
=\prod_{m=0}^{i-1}{\bf E}e^{h(q_lv_l)^mS^*_{l}}\cdot {\bf E}e^{h(q_lv_l)^iS^*_{k}}.
\end{gather}
Since $0\le h(q_lv_l)^m\le h\le L(S_l^*)<\infty$, for any $h\in[0,L(S_l^*)]$ we have that
\begin{gather*} \label{c29+}
{\bf E}e^{h(q_lv_l)^mS^*_{l}}
\le \big({\bf E}e^{L(S_l^*)S^*_{l}}\big)^{h(q_lv_l)^m/L(S_l^*)}\le1
\end{gather*}
by definition of $L(S_l^*)$.
From this inequality and \eqref{c29}, for any $ h\in[0,L(S_l^*)]$ and each $i\ge 0$ and $k\ge 1$ we obtain
\begin{gather*} \label{c29++}
{\bf E}e^{hS^*_{il+k}}\le{\bf E}e^{h(q_lv_l)^iS^*_{k}}.
\end{gather*}
The latter yields (\ref{c27}) if only we remind that
each integer $n\ge1$ may be represented in the form (\ref{c24}).
Thus the lemma is proved.
\medskip
{\bf Proof of Theorem \ref{thm3}.}
If $q_lv_l=1,$ then for any $ h\in[0,L(S_l^*)]$ we have from \eqref{c27} that
$$
\sup_{k\geq 1}{\bf E}e^{hS_k^*}\le\max_{1\leq k\leq l}{\bf E}e^{hS_k^*}.
$$
So, the second assertion of Theorem \ref{thm3} is proved.
To prove the first one note that for $(q_lv_l)^i\le1$ and $1\le k\le l$ we have
\begin{gather} \label{c30}
{\bf E}e^{h(q_lv_l)^iS_k^*}\le \big({\bf E}e^{hS_k^*}\big)^{(q_lv_l)^i}
\le \max\{1,{\bf E}e^{hS_k^*}\}\le
\max_{0\leq k\leq l}{\bf E}e^{hS_k^*} =\max_{0\leq k< l}{\bf E}e^{hS_k^*} ,
\end{gather}
because $S_0^*=0$ and ${\bf E}e^{hS_0^*}=1$.
Inequality \eqref{c30} together with \eqref{c27} implies \eqref{p7} under assumptions of Theorem \ref{thm3}.
The rest of proof is similar to Corollary \ref{corP7}.
\bigskip
Remind that Corollary \ref{CC1} immediately follows from Theorem~\ref{thm1},
Corollary \ref{CC3} is a special case of Theorem~\ref{thm3},
whereas Corollary \ref{C2+} simply follows from Theorem~\ref{thm2}.
All examples and lemmas are proved after their statements.
Thus, all results of the paper are proved.
\bigskip
\noindent {\bf\large Acknowledgments} \quad
\noindent This work was supported by the National Natural Science
Foundation of China (Grant No. 11931018), Tianjin Natural Science Foundation
and the program of fundamental scientific researches of the SB RAS № I.1.3., project № 0314-2019-0008. | 148,349 |
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\begin{document}
\title{Combinatorial Proof of the Minimal Excludant Theorem}
\author{Cristina Ballantine
\\
\footnotesize Department of Mathematics and Computer Science\\
\footnotesize College of The Holy Cross\\
\footnotesize Worcester, MA 01610, USA \\
\footnotesize [email protected]
\and Mircea Merca
\\
\footnotesize Academy of Romanian Scientists\\
\footnotesize Ilfov 3, Sector 5, Bucharest, Romania\\
\footnotesize [email protected]
}
\date{}
\maketitle
\begin{abstract}
The minimal excludant of a partition $\l$, $\mex(\l)$, is the smallest positive integer that is not a part of $\l$. For a positive integer $n$, $ \sigma\, \mex(n)$ denotes the sum of the minimal excludants of all partitions of $n$. Recently, Andrews and Newman obtained a new combinatorial interpretations for $\sigma\, \mex(n)$. They showed, using generating functions, that $\sigma\, \mex(n)$ equals the number of partitions of $n$ into distinct parts using two colors.
In this paper, we provide a purely combinatorial proof of this result and new properties of the function $\sigma\, \mex(n)$. We generalize this combinatorial interpretation to $\sigma_r\, \mex(n)$, the sum of least $r$-gaps in all partitions of $n$. The least $r$-gap of a partition $\l$ is the smallest positive integer that does not appear at least $r$ times as a part of $\l$.
\\
\\
{\bf Keywords:} Minimal excludant, MEX, least gap in partition, partitions, $2$-color partitions
\\
\\
{\bf MSC 2010:} 11A63, 11P81, 05A19
\end{abstract}
\section{Introduction}
The minimal excludant or $\mex$-function on a set $S$ of positive integers is the least positive integer not in $S$. The history of this notion goes back to at least the 1930s when it was applied to combinatorial game theory \cite{S35, G39}.
Recently, Andrews and Newman \cite{AN19} considered the $\mex$-function applied to integer partitions.
They defined the minimal excludant of a partition $\l$, $\mex(\l)$, to be the smallest positive integer that is not a part of $\l$.
In addition, for each positive integer $n$, they defined
$$ \sigma\, \mex(n) := \sum_{\l\in \P(n)} \mex(\l),$$
where $\P(n)$ is the set of all partitions of $n$. Elsewhere in the literature, the minimal excludant of a partition $\l$ is referred to as the least gap or smallest gap of $\l$. An exact and asymptotic formula for $\sigma\, \mex(n)$ is given in \cite{GK06}. In \cite{BM19}, where $\mex(\l)$ is denoted by $g_1(\l)$ and $\sigma\, \mex(n)$ is denoted by $S_1(n)$, the authors study a generalization of $\sigma\, \mex(n)$ and its connection to polygonal numbers.
Let $\D_2(n)$ be the set of partitions of $n$ into distinct parts using two colors and let $D_2(n)=|\D_2(n)|$. For ease of notation, we denote the colors of the parts of partitions in $\D_2(n)$ by $0$ and $1$. In \cite{AN19}, the authors give two proofs of the following theorem.
\begin{theorem} \label{T1} Given an integer $n \geqslant 0$, we have $$\sigma\, \mathrm{mex}(n)=D_2(n).$$
\end{theorem} They also determine the parity of the $\smex$ function.
\begin{lemma} \label{L1} $\smex(n)$ is odd if and only if $n=j(3j\pm 1)$ for some non-negative integer $j$.
\end{lemma} We note that this parity result was also established in \cite[Corollary 1.6]{BM19}. Andrews and Newman write that ``it would be of great interest to have a bijective proof of Theorem \ref{T1}." They also ask for a combinatorial proof of Lemma \ref{L1}.
In section \ref{combAN} we provide these desired proofs.
In the proof of Theorem \ref{T1}, we make use of the fact that \begin{equation}\label{smex as sum} \sigma \, \mex(n)=\ds\sum_{k\geqslant 0} p(n-k(k+1)/2),\end{equation} where, as usual, $p(n)$ denotes the number of partitions of $n$.
A combinatorial proof of \eqref{smex as sum} is given in \cite[Theorem 1.1]{BM19} . The same argument is also described in the second proof of \cite[Theorem 1.1]{AN19}. We note the result proven in \cite{BM19} is a generalization of \eqref{smex as sum} to the sum of $r$-gaps in all partitions of $n$. The $r$-gap of a partition $\l$ is the least positive integer that does not appear $r$ times as a part of $\l$.
In \cite{Andrews12}, Andrews and Merca considered a restricted $\mex$-function and defined $M_k(n)$ to be the number of partitions of $n$ in which $k$ is the least positive integer that is not a part and there are more parts $>k$ than there are parts $<k$.
When $k=1$, $M_1(0)=0$ and, if $n>0$, $M_1(n)$ is the number of partitions of $n$ that do not contain $1$ as a part. Thus, if $n>0$, we have $M_1(n)=p(n)-p(n-1)$. Then, from $\eqref{smex as sum}$, we obtain \begin{equation}\label{T2k=1} \sigma\, \mathrm{mex}(n) -\sigma\, \mathrm{mex}(n-1) - \delta(n)
= \sum_{j=0}^\infty M_1\big(n-j(j+1)/2\big),\end{equation} where $\delta$ is the characteristic function of the set of triangular numbers, i.e.,
$$
\delta(n) = \begin{cases}
1,& \text{if $n=m(m+1)/2,\ m\in\mathbb{Z}$,}\\
0,& \text{otherwise.}
\end{cases}
$$
In section \ref{pfT2}, we prove the following generalization of \eqref{T2k=1}.
\begin{theorem}\label{T2}
Let $k$ be a positive integer. Given an integer $n\geqslant 0$, we have
\begin{align*}
& (-1)^{k-1} \left( \sum_{j=-(k-1)}^k (-1)^j \sigma\, \mathrm{mex}\big(n-j(3j-1)/2\big) - \delta(n) \right) \\
& = \sum_{j=0}^\infty M_k\big(n-j(j+1)/2\big).
\end{align*}
\end{theorem}
As a corollary of this theorem we obtain the following infinite family of linear inequalities involving $\sigma\, \mathrm{mex}$.
\begin{corollary}
Let $k$ be a positive integer. Given an integer $n\geqslant 0$, we have
\begin{align*}
& (-1)^{k-1} \left( \sum_{j=-(k-1)}^k (-1)^j \sigma\, \mathrm{mex}\big(n-j(3j-1)/2\big) - \delta(n) \right) \geqslant 0,
\end{align*}
with strict inequality if $n\geqslant k(3k+1)/2$.
\end{corollary}
In section \ref{pfT2} we also give a combinatorial interpretation for $$\ds \sum_{j=0}^\infty M_k\big(n-j(j+1)/2\big)$$ in terms of the number of partitions into distinct parts using three colors and satisfying certain conditions.
In sections \ref{op} and \ref{dist}, we introduce connections to certain subsets of overpartitions and partitions with distinct parts, respectively.
\section{Combinatorial Proofs of Theorem \ref{T1} and Lemma \ref{L1}} \label{combAN}
\subsection{Bijective Proof of Theorem \ref{T1}}\label{s1}
To prove the theorem, we adapt Sylvester's bijective proof of Jacobi's triple product identity \cite{S1882} which was later rediscovered by Wright \cite{W65}. For the interested reader, it is probably easier to follow Wright's short article for the description of the bijection.
Given a partition $\l$ in $\D_2(n)$, let $\l^{(j)}$, $j=0,1$, be the (uncolored) partition whose parts are the parts of color $j$ in $\l$. Then, $\l^{(1)}$ and $\l^{(2)}$ are partitions into distinct parts.
\begin{example} If $\l=4_1+3_0+3_1+2_0+1_0 \in \D_2(13)$, then $\l^{(0)}=3+2+1$ and $\l^{(1)}=4+3$.
\end{example}
Denote by $\eta(k)$ the staircase partition $\eta(k)=k+(k-1)+(k-2)+\cdots +3+2+1$. If $k=0$ we define $\eta(k)=\emptyset$.
For any partition $\l$ we denote by $\ell(\l)$ the number of parts in $\l$.
\begin{definition} Given diagram of left justified rows of boxes (not necessarily the Ferrers diagram of a partition), the \textit{staircase profile} of the diagram is a zig-zag line starting in the upper left corner of the diagram with a right step and continuing in alternating down and right steps until the end of a row of the diagram is reached.
\end{definition}
\begin{example} The staircase profile of the diagram $$\yng(3,5,8,4,4,2,2,1)$$ is
\begin{center}\tiny{\begin{tikzpicture}[inner sep=0in,outer sep=0in]\node (n) {\begin{varwidth}{0cm}{\ydiagram{3,5,8,4,4,2,2,1}
}\end{varwidth}};
\draw[ultra thick] (0,1.47)--(0.37,1.47)-- (0.37,1.1)--(.73,1.1)--(.73,.73)--(1.1,.73)--(1.1,.36)--(1.48,.36) ;
\end{tikzpicture}}\end{center}
\end{example}
Given a Ferrers diagram (with boxes of unit length) of a partition $\l$ into distinct parts, the \textit{shifted Ferrers diagram} of $\l$ is the diagram in which row $i$ is shifted $i-1$ units to the right.
We create a map $$\vp:\bigcup_{k \geqslant 0} \P(n-k(k+1)/2)\to\D_2(n)$$ as follows.
Start with $\l\in \P(n-k(k+1)/2)$ for some $k\geqslant 0$. Append a diagram with rows of lengths $1, 2, \ldots k$ (i.e., the Ferrers diagram of $\eta(k)$ rotated by $180^\circ$) the top of Ferrers diagram of $\l$. We obtain a diagram with $n$ boxes. Draw the staircase profile of the new diagram. Let $\a$ be the partition whose parts are the length of the columns to the left of the staircase profile and let $\b$ be the partition whose parts are the length of the rows to the right of the staircase profile. Then $\a$ and $\b$ are partitions with distinct parts. Moreover, $k\leqslant \ell(\a)- \ell(\b)\leqslant k+1$. Color the parts of $\a$ with color $k\!\!\pmod 2$ and the parts of $\b$ with color $k+1\!\!\pmod 2$. Then $\vp(\l)$ is defined as the 2-color partition of $n$ whose parts are the colored parts of $\a$ and $\b$.
Conversely, start with $\mu\in \D_2(n)$. Let $\ell_j(\m)$, $j=0,1$, be the number of parts of color $j$ in $\m$.
(i) If $\ell_0(\m)\geqslant \ell_1(\m)$, let $r=\ell_0(\m)-\ell_1(\m)$. Let $k=\ds r+\frac{(-1)^r-1}{2}$. Remove the top $k$ rows (i.e., the rotated Ferrers diagram of $\eta(k)$) from the conjugate of the shifted diagram of $\m^{(0)}$ and join the remaining diagram with the shifted digram of $\m^{(1)}$ so they align at the top. The obtained partition $\vp^{-1}(\m)$ belongs to $\P(n-k(k+1)/2)$.
(ii) If $\ell_1(\m)>\ell_0(\m)$, let $r=\ell_1(\m)-\ell_0(\m)$. Let $k=\ds r-\frac{(-1)^r+1}{2}$. Remove the top $k$ rows (i.e., the rotated Ferrers diagram of $\eta(k)$) from the conjugate of the shifted diagram of $\m^{(1)}$ and join the remaining diagram with the shifted digram of $\m^{(0)}$ so they align at the top. The obtained partition $\vp^{-1}(\m)$ belongs to $\P(n-k(k+1)/2)$.
\begin{example} \label{eg1} Let $n=38, k=3$, and let $\l=7+7+6+6+4+2$ be a partition of $n-k(k+1)/2=32$. We add the rotated Ferrers diagram of $\eta(3)$ to the top of the Ferrers diagram of $\l$ and draw the staircase profile. \begin{center}\tiny{\begin{tikzpicture}[inner sep=0in,outer sep=0in]\node (n) {\begin{varwidth}{0cm}{\ydiagram{1,2,3,7,7,6,6,4,2}
}\end{varwidth}};
\draw[ultra thick] (-0,1.67)--(0.37,1.67)-- (0.37,1.3)--(.73,1.3)--(.73,.93)--(1.1,.93)--(1.1,.56)--(1.48,.56)--(1.48,.21)--(1.85,.21)--(1.85,-0.16)--(2.22,-0.16) ;
\end{tikzpicture}}\end{center}
Then $\a=9+8+6+5+3+2$ and $\b=3+2$. Since $k$ is odd, we have $\vp(\l)=9_1+8_1+6_1+5_1+3_1+3_0+2_1+2_0\in \D_2(38)$.
Conversely, suppose $\mu=9_1+8_1+6_1+5_1+3_1+3_0+2_1+2_0\in \D_2(38)$. Then $\ell_0(\mu)=2$ and $\ell_1(\mu)=6$. We have $r=\ell_1(\m)-\ell_0(\m)=4$ and $k=3$. The diagrams of the conjugate of the shifted diagram of $\mu^{(1)}$ and the shifted diagram of $\mu^{(0)}$ are shown below. \begin{center} \begin{figure}[htbp]\begin{subfigure}{.7\textwidth}\centering \tiny{\begin{tikzpicture}[inner sep=0in,outer sep=0in]\node (n) {\begin{varwidth}{0cm}{\ydiagram{1,2,3,4,5, 6, 6, 4, 2}
}\end{varwidth}};
\end{tikzpicture}}\end{subfigure}
\begin{subfigure}{.5\textwidth}
\small{\begin{tikzpicture}[inner sep=0in,outer sep=0in]\node (n) {\begin{varwidth}{0cm}{\young(\e\e\e,:\e\e)
}\end{varwidth}};
\end{tikzpicture}}\end{subfigure}
\end{figure}\end{center}
Next, we remove the first $3$ rows from the conjugate of the shifted diagram of $\mu^{(1)}$ and join the remaining diagram and the shifted digram of $\m^{(0)}$ so they align at the top. We obtain $\vp^{-1}(\mu)=7+7+6+6+4+2\in \P(32)$.
\end{example}
\subsection{Combinatorial Proof of Lemma \ref{L1}}
To determine the parity of $\smex(n)$, we pair partitions in $\D_2(n)$ as follows. If $\m\in \D_2(n)$, we denote by $\tilde \m$ the partition in $\D_2(n)$ obtained by interchanging the colors of the part of $\m$. Then $\m\neq \tilde \m$ if and only if $\m^{(0)}\neq \m^{(1)}$. If $n$ is odd, $\m^{(0)}\neq \m^{(1)}$ for all $\m\in \D_2(n)$ and $\smex$ is even. If $n$ is even, $\smex \equiv q(n/2) \pmod 2$, where, as usual, $q(m)$ denotes the number of partitions of $m$ with distinct parts. Franklin's involution used to prove Euler's Pentagonal Number Theorem provides a pairing of partitions with distinct parts that shows that $q(m)$ is odd if and only if $m$ is a generalized pentagonal number. Thus, $\smex(n)$ is odd if and only if $n$ is twice a generalized pentagonal number.
\section{Proofs of Theorem \ref{T2}} \label{pfT2}
\begin{proof}[Analytic proof of Theorem \ref{T2}]
In \cite{Andrews12}, the authors considered Euler's pentagonal number theorem and proved the following truncated form:
\begin{equation} \label{TPNT}
\frac{(-1)^{k-1}}{(q;q)_\infty} \sum_{n=-(k-1)}^{k} (-1)^{j} q^{n(3n-1)/2}= (-1)^{k-1}+ \sum_{n=k}^\infty \frac{q^{{k\choose 2}+(k+1)n}}{(q;q)_n}
\begin{bmatrix}
n-1\\k-1
\end{bmatrix},
\end{equation}
where
$$(a;q)_n = \begin{cases}
1, & \text{if $n=0$,}\\
\prod\limits_{k=0}^{n-1} (1-aq^k), & \text{otherwise,}
\end{cases}
$$
and
$$
\begin{bmatrix}
n\\k
\end{bmatrix}
=
\begin{cases}
\dfrac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}, & \text{if $0\leqslant k\leqslant n$},\\
0, &\text{otherwise.}
\end{cases}
$$
Multiplying both sides of \eqref{TPNT} by
$$
\frac{(q^2;q^2)_\infty}{(q,q^2)_\infty} = \sum_{n=0}^\infty q^{n(n+1)/2},
$$
we obtain
\begin{align*}
& (-1)^{k-1} \left( \bigg( \sum_{n=0}^\infty \sigma\, \mathrm{mex}(n) q^n \bigg) \bigg( \sum_{n=-(k-1)}^{k} (-1)^{j} q^{n(3n-1)/2}\bigg) -\sum_{n=0}^\infty q^{n(n+1)/2}\right) \\
& = \left( \sum_{n=0}^\infty q^{n(n+1)/2} \right) \left( \sum_{n=0}^\infty M_k(n) q^n\right),
\end{align*}
where we have invoked the generating function for $ \sigma\, \mathrm{mex}(n)$ \cite{BM19, AN19},
$$
\sum_{n=0}^\infty \sigma\, \mathrm{mex}(n) q^n = \frac{(q^2;q^2)_\infty}{(q;q)_\infty(q;q^2)_\infty}
$$
and the generating function for $M_k(n)$ \cite{Andrews12},
$$
\sum_{n=0}^\infty M_k(n) q^n = \sum_{n=k}^\infty \frac{q^{{k\choose 2}+(k+1)n}}{(q;q)_n}
\begin{bmatrix}
n-1\\k-1
\end{bmatrix}.
$$
The proof follows easily considering Cauchy's multiplication of two power series.\medskip
\end{proof}
\begin{proof}[Combinatorial proof of Theorem \ref{T2}]
The statement of Theorem \ref{T2} is equivalent to identity \eqref{T2k=1} together with
\begin{align}
& \sigma \, \mex
\left(n-\frac{k(3k+1)}{2}\right)-\sigma \, \mex
\left(n-\frac{k(3k+5)}{2}-1\right)\nonumber \\
& \qquad\qquad = \sum_{j=0}^\infty \Big(M_k\big(n-j(j+1)/2\big) +M_{k+1}\big(n-j(j+1)/2\big)\Big).\label{T2kequiv}
\end{align}
Using \eqref{smex as sum}, identity \eqref{T2kequiv} becomes
\begin{align}
& \sum_{j=0}^\infty \Bigg(p
\bigg(n-\frac{j(j+1)}{2}-\frac{k(3k+1)}{2}\bigg)-p\bigg(n-\frac{j(j+1)}{2}-\frac{k(3k+5)}{2}-1\bigg)\Bigg)\nonumber \\
& \qquad\qquad = \sum_{j=0}^\infty \Big(M_k\big(n-j(j+1)/2\big) +M_{k+1}\big(n-j(j+1)/2\big)\Big). \label{T2kequiv1}
\end{align}
Identity \eqref{T2kequiv1} was proved combinatorially in \cite{Y15}. Together with the combinatorial proof of \eqref{smex as sum}, this gives a combinatorial proof of Theorem \ref{T2}.
\end{proof}
\medskip
Next, we give a combinatorial interpretation for $\ds \sum_{j=0}^\infty M_k\big(n-j(j+1)/2\big)$. First, we introduce some notation. Given an integer $r$, let $\rm{sign}(r)$ denote the sign of $r$, i.e.
$$\rm{sign}(r)=\begin{cases} 1 & \mbox{ if } r \geqslant 0\\ -1 & \mbox{ if } r <0.\end{cases}$$
For integers $k,n$ such that $k\geqslant 1$ and $n\geqslant 0$, we denote by $D^{(k)}_3(n)$ the number of partitions $\mu$ of $n$ into distinct parts using three colors, $0,1,2$, and satisfying the following conditions:
\begin{enumerate}
\item[(i)] $\mu$ has exactly $k$ parts of color $2$ and, if $k>1$, twice the smallest part of color $2$ is greater than largest part of color $2$.
\item[(ii)] Let $r=\ell_0(\mu)-\ell_1(\mu)$ be the signed difference in the number of parts colored $0$ and the number of parts colored $1$ in $\mu$. Let $j=\ds |r|-\frac{1}{2}+\rm{sign}(r)\frac{(-1)^r}{2}$.
The largest part of color $j \!\!\pmod 2$ must equal $j$ more that the smallest part of color $2$.
\end{enumerate}
Then, we have the following proposition.
\begin{proposition}\label{prop}
For integers $k,n$ such that $k\geqslant 1$ and $n\geqslant 0$, we have \begin{equation}\label{p1}\sum_{j=0}^\infty M_k\big(n-j(j+1)/2\big)=D^{(k)}_3(n).\end{equation}
\end{proposition}
\begin{proof}
Take a partition counted by $M_k\big(n-j(j+1)/2\big)$ and consider its Ferrers diagram. Remove the first $k$ columns and color the length of each of these columns with color $2$. To the remaining Ferrers diagram, add the rotated Ferrers diagram of a staircase $\eta(j)$ of height $j$ and perform the transformation in the combinatorial proof of Theorem \ref{T1}. It is now straight forward that this transformation is a bijection between the sets of partitions counted by the two sides of \eqref{p1}.\end{proof}
Combining Theorems \ref{T1} and \ref{T2}, and Proposition \ref{prop} we obtain the following corollary which, by the discussion above, has both analytic and combinatorial proofs.
\begin{corollary}
For integers $k,n$ such that $k\geqslant 1$ and $n\geqslant 0$, we have
$$(-1)^{\max(0,k-1)} \left( \sum_{j=-\max(0,k-1)}^k (-1)^j \sigma\, \mathrm{mex}\big(n-j(3j-1)/2\big) - \delta(n) \right) \\
= D^{(k)}_3(n).$$
\end{corollary}
Note that, if $k=0$, the statement of the corollary reduces to Theorem \ref{T1}.
\section{Connections with overpartitions} \label{op}
Overpartitions
are ordinary partitions with the added
condition that the first appearance
of any part may be overlined or not.
There are eight overpartitions of $3$:
$$
3, \overline{3}, 2+1, \overline{2}+1, 2+\overline{1}, \overline{2}+\overline{1}, 1+1+1, \overline{1}+1+1.
$$
In \cite{AM18}, the authors denoted by $\overline{M}_k(n)$ the number of overpartitions
of $n$ in which the first part larger than $k$ appears at least $k+1$ times.
For example, $\overline{M}_2(12)=16$,
and the partitions in question are
$4+4+4$,
$\overline{4}+4+4$,
$3+3+3+3$,
$\overline{3}+3+3+3$,
$3+3+3+2+1$,
$3+3+3+\overline{2}+1$,
$3+3+3+2+\overline{1}$,
$3+3+3+\overline{2}+\overline{1}$,
$\overline{3}+3+3+2+1$,
$\overline{3}+3+3+\overline{2}+1$,
$\overline{3}+3+3+2+\overline{1}$,
$\overline{3}+3+3+\overline{2}+\overline{1}$,
$3+3+3+1+1+1$,
$3+3+3+\overline{1}+1+1$,
$\overline{3}+3+3+1+1+1$,
$\overline{3}+3+3+\overline{1}+1+1$.\medskip
We have the following identity.
\begin{theorem}\label{T3}
For integers $k,n>0$, we have
\begin{align*}
& (-1)^k \left( \sigma\, \mathrm{mex}(n) + 2\sum_{j=1}^{k} (-1)^j \sigma\, \mathrm{mex}(n-j^2) - \delta'(n)\right) \\
& =\sum_{j=-\infty}^\infty (-1)^j \overline{M}_k\big(n-j(3j-1)\big),
\end{align*}
where
$$
\delta'(n) = \begin{cases}
(-1)^m,& \text{if $n=m(3m-1),\ m\in\mathbb{Z}$,}\\
0,& \text{otherwise.}
\end{cases}
$$
\end{theorem}
\begin{proof}
According to \cite[Theorem 7]{AM18}, we have
\begin{align}
& \frac{(-q;q)_{\infty}} {(q;q)_{\infty}} \left(1 + 2 \sum_{j=1}^{k} (-1)^j q^{j^2} \right) \label{eq:1.11} \\
& \qquad = 1+2 (-1)^k \frac{(-q;q)_k}{(q;q)_k} \sum_{j=0}^{\infty} \frac{q^{(k+1)(k+j+1)}(-q^{k+j+2};q)_{\infty}}{(1-q^{k+j+1})(q^{k+j+2};q)_{\infty}},\notag
\end{align}
where
$$\sum_{n=0}^\infty \overline{M}_k(n) q^n =
2 \frac{(-q;q)_k}{(q;q)_k} \sum_{j=0}^{\infty} \frac{q^{(k+1)(k+j+1)}(-q^{k+j+2};q)_{\infty}}{(1-q^{k+j+1})(q^{k+j+2};q)_{\infty}}.$$
Multiplying both sides of \eqref{eq:1.11} by
$$
(q^2,q^2)_\infty = \sum_{n=-\infty}^\infty (-1)^n q^{n(3n-1)},
$$
we obtain
\begin{align*}
& (-1)^k \left(\Bigg(\sum_{n=0}^\infty \sigma\, \mathrm{mex}(n) q^n \Bigg) \Bigg( 1 + 2 \sum_{j=1}^{k} (-1)^j q^{j^2}\Bigg) - \sum_{n=-\infty}^\infty (-1)^n q^{n(3n-1)} \right) \\
& = \left( \sum_{n=-\infty}^\infty (-1)^n q^{n(3n-1)} \right) \left( \sum_{n=0}^\infty \overline{M}_k(n) q^n \right)
\end{align*}
and the proof follows easily.
\end{proof}
Related to Theorem \ref{T3}, we remark that
there is a substantial amount of numerical evidence to conjecture the following inequality.
\begin{conjecture}\label{T4}
For $k,n>0$,
\begin{align*}
\sum_{j=-\infty}^\infty (-1)^j \overline{M}_k\big(n-j(3j-1)\big)\geqslant 0,
\end{align*}
with strict inequality if $n\geqslant (k+1)^2$.
\end{conjecture}
It would be very appealing to have a combinatorial interpretation for the sum in this conjecture.
\section{Connections with partitions into distinct parts} \label{dist}
Following the notation for the number of partitions of $n$ into distinct parts of two colors, we denote by $D_1(n)$ the number of partitions of $n$ into distinct parts. We prove the following identity.
\begin{theorem}\label{T5}
For any integer $n\geqslant 0$, we have
\begin{equation}\label{sumdist}
\sum_{j=0}^\infty (-1)^{j(j+1)/2} \sigma\, \mathrm{mex}\big(n-j(j+1)/2 \big)
= \sum_{j=0}^\infty D_1\left( \frac{n-j(j+1)/2}{2}\right),
\end{equation}
where $D_1(x)=0$ if $x$ is not a positive integer.
\end{theorem}
\begin{proof}
Considering the classical theta identity \cite[p. 23, eq. (2.2.13)]{Andrews98}
\begin{equation}\label{Eq:5}
\frac{(q^2;q^2)_\infty}{(-q;q^2)_\infty} = \sum_{n=0}^\infty (-q)^{n(n+1)/2},
\end{equation}
we can write
\begin{align*}
\left( \sum_{n=0}^\infty \sigma\, \mathrm{mex}(n) q^n \right) \left( \sum_{n=0}^\infty (-q)^{n(n+1)/2} \right)
& = \frac{(q^2;q^2)_\infty}{(q;q)_\infty (q;q^2)_\infty} \cdot \frac{(q^2;q^2)_\infty}{(-q;q^2)_\infty} \\
& = (-q^2;q^2)_\infty \cdot \frac{(q^2;q^2)_\infty}{(q;q^2)_\infty} \\
& = \left( \sum_{n=0}^\infty D_1(n) q^{2n} \right) \left( \sum_{n=0}^\infty q^{n(n+1)/2} \right)
\end{align*}
and the proof follows by equating the coefficients of $q^n$ in this identity.
\end{proof}
To obtain a combinatorial interpretation for the sum on the right hand side of \eqref{sumdist}, let $D_2^*(n)$ be the
number of partitions of $n$ with distinct parts using two colors such that: (i) parts of color $0$ form a gap-free partition (staircase) and (ii)
only even parts can have color $1$.
Then, we have the following identity of Watson type \cite{BM19a}.
\begin{proposition}
For $n\geqslant 0$,
$$\sum_{j=0}^\infty D_1\left( \frac{n-j(j+1)/2}{2}\right)= D_2^*(n).$$
\end{proposition}
\begin{proof}
To see this, let $\l$ be a partition counted by $\ds D_1\left(\frac{n-j(j+1)/2}{2}\right)$. Double the size of each part of $\l$ to obtain a partition $\mu$ of $n-j(j+1)/2$ whose parts are even and distinct. Color the parts of $\mu$ with color $1$ and add parts $1, 2, \ldots, j$ in color $0$ to obtain a partition counted by $D_2^*(n)$. This transformation is clearly reversible.
\end{proof}
In \cite{AM18}, the authors denoted by $MP_k(n)$
the number of partitions of $n$ in which the first part larger than $2k-1$ is odd and appears exactly $k$ times.
All other odd parts appear at most once. For example, $MP_2(19)=10$, and the partitions in question are
$9+9+1$,
$9+5+5$,
$8+5+5+1$,
$7+7+3+2$,
$7+7+2+2+1$,
$7+5+5+2$,
$6+5+5+3$,
$6+5+5+2+1$,
$5+5+3+2+2+2$,
$5+5+2+2+2+2+1$.
We remark the following truncated form of Theorem \ref{T5}.
\begin{theorem}\label{T6}
For integers $n,k>0$,
\begin{align*}
& (-1)^{k-1} \left( \sum_{j=0}^{2k-1} (-1)^{j(j+1)/2} \sigma\, \mathrm{mex}\big(n-j(j+1)/2 \big) -D^*_2(n) \right) \\
& = \sum_{j=0}^n MP_k(j) D^*_2(n-j).
\end{align*}
\end{theorem}
\begin{proof}
According to \cite[Theorem 9]{AM18}, we have
\begin{align}
& \frac{(-q;q^2)_\infty}{(q^2;q^2)_\infty} \sum_{j=0}^{2k-1}(-q)^{j(j+1)/2} \label{eq:1.13} \\
& \qquad = 1 + (-1)^{k-1} \frac{(-q;q^2)_k}{(q^2;q^2)_{k-1}}
\sum_{j=0}^{\infty} \frac{q^{k(2j+2k+1)}(-q^{2j+2k+3};q^2)_{\infty}}{(q^{2k+2j+2};q^2)_{\infty}},\notag
\end{align}
where
$$
\sum_{n=0}^\infty MP_k(n) q^n = \frac{(-q;q^2)_k}{(q^2;q^2)_{k-1}}
\sum_{j=0}^{\infty} \frac{q^{k(2j+2k+1)}(-q^{2j+2k+3};q^2)_{\infty}}{(q^{2k+2j+2};q^2)_{\infty}}.
$$
The proof follows easily by multiplying both sides of \eqref{eq:1.13} by
$$\frac{(q^2;q^2)_\infty}{(q;q)_\infty (q;q^2)_\infty} \cdot \frac{(q^2;q^2)_\infty}{(-q;q^2)_\infty}.$$
\end{proof}
A further interesting corollary of Theorem \ref{T6} relates to $\sigma\, \mathrm{mex}(n)$.
\begin{corollary}
For integers $n,k>0$,
\begin{align*}
(-1)^{k-1} \left( \sum_{j=0}^{2k-1} (-1)^{j(j+1)/2} \sigma\, \mathrm{mex}\big(n-j(j+1)/2 \big) -D^*_2(n) \right) \geqslant 0,
\end{align*}
with strict inequality if $n\geqslant k(2k+1)$.
\end{corollary}
On the other hand, the reciprocal of the infinite product in \eqref{Eq:5} is the generating function for $\mathrm{pod}(n)$, the number of partitions of $n$ in which odd parts are not repeated, i.e.,
\begin{equation}\label{gfpod}
\frac{(-q;q^2)_\infty}{(q^2;q^2)_\infty} = \sum_{n=0}^\infty \mathrm{pod}(n) q^n.
\end{equation}
The properties of the partition function $\mathrm{pod}(n)$ were studied in \cite{Hirschhorn} by Hirschhorn and Sellers. We easily deduce the following convolution identity.
\begin{corollary}
For $n\geqslant 0$,
$$
\sigma\, \mathrm{mex}(n) = \sum_{j=0}^n \mathrm{pod}(j) D^*_2(n-j).
$$
\end{corollary}
Finally, we remark that finding a combinatorial interpretation for
$$\sum_{j=0}^n MP_k(j) D^*_2(n-j) $$
would be very desirable.
\section{Concluding remarks}
The present work began with the search for a combinatorial proof of Theorem \ref{T1}. We were further able to prove several truncated series formulas involving the function $\sigma \, \mex$. In \cite{BM19}, we worked with the generalization of this function: the sum, $S_r(n)$, of $r$-gaps in all partitions of $n$. To keep notation uniform, we use $\smex_r(n)$ for $S_r(n)$. Recall that the $r$-gap of a partition $\l$ is the least positive integer that does not appear at least $r$ times as a part of $\l$. In \cite{BM19}, we proved combinatorially that \begin{equation}\smex_r(n)=\sum_{j \geqslant 0}p(n-rj(j+1)/2), \end{equation}
and we gave the generating function for $\sigma_r \, \mex(n)$, namely \begin{equation}\label{sr}\sum_{n\geq 0} \sigma_r \, \mex(n)q^n=\frac{(q^{2r};q^{2r})_\infty}{(q;q)_\infty (q^r;q^{2r})_\infty}.\end{equation}
Denote by $\widetilde D_2^{(r)}(n) $ the number of partitions $\l$ of $n$ using two colors, $0$ and $1$, such that:
\begin{enumerate}
\item[(i)] $\l^{(0)}$ is a partition into distinct parts divisible by $r$.
\item[(ii)] $\l^{(1)}$ is a partition with parts repeated at most $2r-1$ times.
\end{enumerate}
The following generalization of Theorem \ref{T1} is immediate from \eqref{sr}.
\begin{theorem} \label{TL}
Let $n,r$ be integers with $r>0$ and $n\geq 0$. Then $\sigma_r \, \mex(n)=\widetilde D_2^{(r)}(n).$ \end{theorem}
\begin{proof}[Combinatorial proof of Theorem \ref{TL}]
Let $\widetilde \D_2^{(r)}(n)$ be the set of partitions of $n$ counted by $\widetilde D_2^{(r)}(n)$ described above.
Let $\P_{r}(n)$ be the set of partitions of $n$ in which all parts are divisible by $r$. Let $\overline \P_{r}(n)$ be the set of partitions of $n$ in which all parts are not divisible by $r$. Finally, let $\Q_r(n)$ be the set of partitions of $n$ with parts repeated at most $r-1$ times.
Let $\psi$ denote Glaisher's bijection from $\overline \P_r(n)$ to $\Q_r(n)$.
We create a bijection $$\xi: \bigcup_{j \geqslant 0}\P(n-rj(j+1)/2)\to \widetilde \D_2^{(r)}(n). $$
Start with a partition $\l \in \P(n-rj(j+1)/2)$ for some $j \geqslant 0$. Let $\tilde \l$ be the partition consisting of the parts of $\l$ that are divisible by $r$ and $\bar \l$ be the partition consisting of the remaining parts of $\l$. Thus all parts of $\bar \l$ are not divisible by $r$.
Let $\tilde \l_{/r}$ be the partition obtained from $\tilde \l$ by dividing each part by $r$. To $\tilde \l_{/r}\in\P\left(\frac{n-|\bar \l|}{r}-\frac{j(j+1)}{2}\right)$ we apply the bijection $\vp$ from the combinatorial proof of Theorem \ref{T1} in section \ref{combAN}. (The appended rotated staircase is $\eta(j)$.) Then $\vp( \tilde \l_{/r})\in \D_2\left(\frac{n-|\bar\l|}{r}\right)$. In $\vp( \tilde \l_{/r})$, multiply each part of color $0$ by $r$ and repeat each part of color $1$ exactly $r$ times. These parts, together with the parts of $\psi(\bar \l)$ colored $1$, form the partition $\xi(\l)\in \widetilde \D_2^{(r)}(n)$.
\bigskip
Conversely, let $\mu\in \widetilde \D_2^{(r)}(n)$. Then $\mu^{(0)}$ is a partition with distinct parts all of which are multiples of $r$ and $\mu^{(1)}$ is a partition with parts repeated at most $2r-1$ times. We write $\mu^{(1)}$ as $\mu^{(1)}=\a^{(1)}\cup \b^{(1)}$, where all parts of $\a^{(1)}$ have multiplicity exactly $r$ and all parts of $\b^{(1)}$ have multiplicity at most $r-1$. Then $\psi^{-1}(\b^{(1)})$ has no part divisible by $r$. We have $|\mu^{(0)}|=rt_1$, $|\a^{(1)}|=rt_2$ and $|\b^{(1)}|=n-rt_1-rt_2$ for some non-negative integers $t_1$ and $t_2$.
Let $\mu^{(0)}_{/r}$ be the partition with parts colored $0$ obtained from $\mu^{(0)}$ by dividing each part by $r$. Then, $\mu^{(0)}_{/r}$ is a partition with distinct parts colored $0$. Let $\a^{(1)}_{\backslash r}$ be the partition with distinct parts colored $1$ with exactly the same set of parts as $\a^{(1)}$. We then apply $\vp^{-1}$ to $\mu^{(0)}_{/r}\cup \a^{(1)}_{\backslash r} \in \D_2(t_1+t_2)$ to obtain $\vp^{-1}(\mu^{(0)}_{/r}\cup \a^{(1)}_{\backslash r}) \in \P\left(t_1+t_2-\frac{j(j+1)}{2}\right)$ for some non-negative integer $j$. We multiply each part of $\vp^{-1}(\mu^{(0)}_{/r}\cup \a^{(1)}_{\backslash r})$ by $r$. These parts, together with the parts of $\psi^{-1}(\b^{(1)})$, form the partition $\xi^{-1}(\mu)\in \P(n-rj(j+1)/2)$.
\end{proof}
\begin{example} Let $n=167, r=3, j=3$ and consider $$\l=21+21+19+18+18+12+8+8+8+8+6+1+1 \in \P(167-3\cdot6)=\P(149).$$ Then, $$\tilde \l=21+21+18+18+12+6\in \P_3(96),$$ $$\bar \l=19+8+8+8+8+1+1\in \overline\P_3(53),$$ and $$\tilde \l_{/3}=7+7+6+6+4+2\in \P(32).$$ Applying Glaisher's bijection, we have $\psi(\bar \l)=24+19+8+1+1\in \Q_3(53)$. From Example \ref{eg1}, we have $\vp(\tilde \l_{/3})=9_1+8_1+6_1+5_1+3_1+3_0+2_1+2_0$. Now, we multiply parts of color $0$ by $3$, repeat each part of color $1$ three times, and include the parts of $\psi(\bar \l)$ with color $1$ to obtain \begin{align*}\xi(\l)=& 24_1+19_1+9_1+9_1+9_1+9_0+8_1+8_1+8_1+8_1+6_1+6_1+6_1+\\ &6_0+5_1+5_1+5_1+ 3_1+3_1+3_1+2_1+2_1+2_1+1_1+1_1\in \widetilde \D_2^{(3)}(167).\end{align*}
Conversely, let \begin{align*}\mu=& 24_1+19_1+9_1+9_1+9_1+9_0+8_1+8_1+8_1+8_1+6_1+6_1+6_1+\\ &6_0+5_1+5_1+5_1+ 3_1+3_1+3_1+2_1+2_1+2_1+1_1+1_1\in \widetilde \D_2^{(3)}(167).\end{align*}
Then, we have the following relevant partitions. \begin{align*}\mu^{(0)}=& 9_0+6_0\in \P_3(15)\\ \mu^{(1)}=& 24_1+19_1+9_1+9_1+9_1+8_1+8_1+8_1+8_1+6_1+6_1+6_1+\\ & 5_1+5_1+5_1+ 3_1+3_1+3_1+2_1+2_1+2_1+1_1+1_1\in \Q_6(152) \\\a^{(1)}=& 9_1+9_1+9_1+8_1+8_1+8_1+6_1+6_1+6_1+5_1+5_1+5_1+\\ & 3_1+3_1+3_1+2_1+2_1+2_1\in \P(99)\\ \b^{(1)}=& 24_1+19_1+8_1+1_1+1_1\in \Q_3(53)\\ \psi^{-1}(\b^{(1)})=&19_1+8_1+8_1+8_1+8_1+1_1+1_1\in \overline\P_3(53)\\ \mu^{(0)}_{/3}= & 3_0+2_0\\ \a^{(1)}_{\backslash 3}= & 9_1+8_1+6_1+5_1+3_1+2_1\\ \end{align*} Then $\mu^{(0)}_{/3}\cup\a^{(1)}_{\backslash 3} \in \D_2(38)$ and from Example \ref{eg1}, we have $j=3$ and $$\vp^{-1}(\mu^{(0)}_{/3}\cup\a^{(1)}_{\backslash 3})=7+7+6+6+4+2\in\P(32)= \P\left(38-\frac{3(3+1)}{2}\right).$$ Now we multiply all parts of $\vp^{-1}(\mu^{(0)}_{/3}\cup\a^{(1)}_{\backslash 3})$ by $3$ and include the parts of $\psi^{-1}(\b^{(1)})$ with the color removed to obtain $$\xi^{-1}(\mu)=21+21+19+18+18+12+8+8+8+8+6+1+1 \in \P(149)= \P\left(167-3\cdot 6\right).$$
\end{example} | 189,045 |
Not quite “sponge worthy” but… Hello World!
Posted by Maya on 03/19/08 in Uncategorized
I’ve been wanting to start blogging for too long and just kept putting it off. Every time I was about to set up a blog and start, the little voice in me would ask “is this really worthy of your first post?”
Sure enough, with every idea that didn’t cut it, expectations built up, till eventually, no post could measure up. No post was quite “sponge worthy”…
Today, on my morning Netvibes routine, I stumbled upon this post: Getting Writing Done: How to Stop Thinking About It and Write . Copyblogger, one of my favorite copy writing blogs, gave me the much needed kick in the butt and there you have it, my first post, the one I’ve been saving myself for, is about nothing. But it’s out there and it feels great!
tag this | permalink | trackback url
Darah | Mar 20, 2008 | Reply
Look at this, a comment on your first day!! hehe. I found you through copyblogger. Twas my first time there, I’ve gotten lost in reading about creative generalism and just creativity in general.
Anyhow I adore your first post title.
Top of the muffin, TO YOU!
Maya | Mar 22, 2008 | Reply
I’m really glad you found it “sponge worthy”. It put a smile on my face to read your comment. Thank you!
Or | Apr 8, 2008 | Reply
Hey, congratulations on your first blog! I am surprised you didn’t have one so far.
Found this on the web - check it out and think again before delving into the blogsphere…
Maya | Apr 9, 2008 | Reply
Thanks for the warning
Guy Ronen | Apr 9, 2008 | Reply
I can relate..
I also write a blog (well.. trying to keep posting with just a very little amount of success..)
and I felt more than once the feeling of too-much-perfectionism-to-even-start..
Every once in a while I get over that feeling + find the time, and I post something. Too little..
Anyway, cograts on the fist blog and even better - on getting this barier out of your system.
eran elhalal | Apr 26, 2008 | Reply
Ahot,
that is a properly structured blog!
looks great.
As thorough as always.
you da man!
eran | 248,922 |
\begin{document}
\begin{abstract}
Given a smooth manifold $M$ equipped with a properly and discontinuous smooth action of a discrete group $G$, the nerve $M_{\bullet}G$ is a simplicial manifold and its vector space of differential forms $\operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right)$ carry a $C_{\infty}$-algebra structure $m_{\bullet}$. We show that each $C_{\infty}$-algebra $1$-minimal model $g_{\bullet}\: : \: \left(W, {m'}_{\bullet} \right) \to \left( \operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right),m_{\bullet}\right) $ gives a flat connection $\nabla$ on a smooth trivial bundle $E$ on $M$ where the fiber is the Malcev Lie algebra of $\pi_{1}(M/G)$ and its monodromy representation is the Malcev completion of $\pi_{1}(M/G)$. This connection is unique in the sense that different $1$-models give isomorphic connections. In particular, the resulting connections are isomorphic to Chen's flat connection on $M/G$. If the action is holomorphic and $g_{\bullet}$ has holomorphic image (with logarithmic singularities), $(\nabla,E)$ is holomorphic and different $1$-models give (holomorphically) isomorphic connections (with logarithmic singularities).
These results are the equivariant and holomorphic version of Chen's theory of flat connections.
\end{abstract}
\maketitle
\section*{Introduction}
Let $H$ be an abstract group and let $\Bbbk$ be a field of characteristic zero. We denote by $\Bbbk\left[ H\right] $ the group ring. It is a Hopf algebra where the coproduct is given by $\Delta\left( g\right) :=g\otimes g$. Let $J$ be the kernel of the augmentation map $\Bbbk\left[ H\right]\to \Bbbk$ that sends each element of $H$ to $1$. The powers of $J$ (with respect to the multiplication) define a filtration $J^{i}$ such that the completion $\Bbbk\left[ H\right]^{\wedge }$ is a complete Hopf algebra, i.e a complete vector space such that the structure maps are continuous (see \cite{benoitfresse}). A group $H$ is said to be \emph{Malcev complete} if it is isomorphic to the group like elements of $\Bbbk\left[ G\right]^{\wedge }$ for some group $G$. For a group $H$ the primitive elements of $\Bbbk\left[ H\right]^{\wedge }$ are called the (complete) Malcev Lie algebra of $H$. The Malcev completion of a group $H$ (over $\Bbbk$) consists of a Malcev complete group $\widehat{H}$ and a universal group homomorphism $H\to \widehat{H}$. It can be constructed by taking the group like elements of $\Bbbk\left[ H\right]^{\wedge }$ and the adjunction between Hopf algebras and Malcev complete groups (see \cite{benoitfresse}). Let $M$ be a connected smooth manifold such that $H^{i}\left(M, \C \right) $ is finite dimensional for any $i>0$. In \cite{extensionChen} K.T. Chen gives a formula for the Malcev completion (over $\R$ or $\C$) of $H=\pi_{1}\left(M,p \right) $ for $p\in M$. In fact, he constructs a flat connection on a trivial bundle on $M$ such that its monodromy representation is the Malcev completion of $\pi_{1}\left(M,p \right) $. The monodromy representation can be explicitly calculated via iterated integrals. We denote by $A_{DR}^{\bullet}(M)$ the differential graded algebra of complex differential forms on $M$. In order to present the main theorem of \cite{extensionChen}, we need the following preliminaries: for any pronilpotent Lie algebra $\mathfrak{g}$, each
\[
C\in A_{DR}^{1}(M)\widehat{\otimes}\mathfrak{g}
\]
defines a connection $d-C$ on the trivial bundle $M\times\mathfrak{g}$, where the latter is considered to be equipped with the adjoint action. For a non-negatively graded vector space $V$ of finite type, we denote by $V_{+}$ its positively graded part and by $V[1]$ the shifted graded vector space such that $\left( V[1]\right)^{i}=V^{i+1}[1]=V^{i+1}$ for any $i$. Its graded dual is denoted by $V^{*}$. Let $\widehat{T}(V)$ be the complete free algebra on $V$. It is a Hopf algebra equipped with the shuffle coproduct. The primitive elements $\widehat{\mathbb{L}}(V)\subset\widehat{T}(V)$ are the complete free Lie algebra over $V$. Let $A\subseteq A_{DR}^{\bullet}(M) $ be a differential graded subalgebra equipped with a Hodge type decomposition i.e., a vector space decomposition
\[
A=W\oplus \mathcal{M}\oplus d\mathcal{M}
\]
where $W=\oplus_{p\geq 0}{W}^{p}$ is a graded vector subspace of closed elements such that the inclusion $W\hookrightarrow A$ is a quasi-isomorphism, and $\mathcal{M}$ is a graded vector subspace containing no exact elements except $0$. We assume that $A\hookrightarrow A_{DR}^{\bullet}(M)$ is a quasi-isomorphism, i.e. that $A$ is a model.
\begin{thmll}[\cite{extensionChen}]\label{Chenthm}
There exist
\begin{enumerate}
\item a Lie ideal $\mathcal{R}_{0}\subset \widehat{\mathbb{L}}\left( (W_{+}^{1}[1])^{*}\right) $, and
\item an element $C_{0}\in A^{1}\widehat{\otimes}\mathfrak{u}$, where $\mathfrak{u}=\widehat{\mathbb{L}}\left( (W_{+}^{1}[1])^{*}\right)/\mathcal{R}_{0}$,
\end{enumerate}
such that $\mathfrak{u}$ is the complete Malcev Lie algebra of $\pi_{1}\left(M,p \right)$ and $d-C_{0}$ defines a flat connection on the trivial bundle $M\times\mathfrak{u}$ whose monodromy represention at $p\in M$ is a group homomorphism
\[
\Theta\: : \: \pi_{1}\left(M,p \right)\to \widehat{T}\left( (W_{+}^{1}[1])^{*}\right)/\bar{\mathcal{R}}_{0}\subset \operatorname{End}\left(\mathfrak{u} \right)
\]
where $\bar{\mathcal{R}}_{0}\subset \widehat{T}\left( (W_{+}^{1}[1])^{*}\right)$ is the ideal generated by $\mathcal{R}_{0}$. Moreover, $\Theta\left(\pi_{1}\left(M,p \right) \right)\subset \exp(\mathfrak{u})$ and $
\Theta\: : \: \pi_{1}\left(M,p \right)\to\exp(\mathfrak{u})
$ is the Malcev completion of $\pi_{1}\left(M,p \right)$. The element $C_{0}$ depends uniquely on the choice of $A$, of the Hodge type decomposition and of basis for $W$.
\end{thmll}
The Lie ideal $\mathcal{R}_{0}$ is constructed in \cite{extensionChen} as the image of a differential $\delta^{*}$ on $\widehat{\mathbb{L}}\left( (W_{+}^{1}[1])^{*}\right)$ and the connection $C_{0}$ is constructed as a degree zero part of a degree one element $C\in A\widehat{\otimes} \widehat{\mathbb{L}}\left( (W_{+}[1])^{*}\right) $. The pair $(C,\delta^{*})$ satisfies some algebraic conditions and they are unique up to the choice of Hodge type decomposition.
Chen first proves the existence of an element $C$ and a differential $\delta^{*}$ that satisfy such algebraic conditions (see \cite{extensionChen}, Theorem 1.3.1). Secondly, he proves that they induce $C_{0}$ and $\mathcal{R}_{0}$ that satisfy Theorem \ref{Chenthm} (see \cite{extensionChen}, Theorem 2.1.1). The proof for the existence of $C$ and $\delta^{*}$ in \cite{extensionChen} is rather geometrical and involves iterated integrals. A purely algebraic proof can be obtained by homological perturbation theory (see for instance \cite{Algchentwist}) or equivalently the homotopy transfer theorem (see \cite{Huebsch}). In fact, $\delta^{*}$ corresponds to a $C_{\infty}$-structure on $W$ induced by $A$ along the inclusion $W\hookrightarrow A$ and $C$ corresponds to a $C_{\infty}$-quasi-isomorphism between $W$ and $A$. The connection obtained by Theorem \ref{Chenthm} is holomorphic if $A$ is a holomorphic model, i.e it contains only holomorphic forms. The celebrated
Knizhnik–Zamolodchikov connection can be constructed via Theorem \ref{Chenthm}, using a holomorphic model for the configuration space of points due to Arnold (see \cite{Arnold}).
In the present paper, we generalize Theorem \ref{Chenthm} as follows. Let $M$ be a smooth (complex or real) manifold equipped with a smooth properly discontinuous action of a discrete group $G$. The nerve gives a simplicial manifold $M_{\bullet}G$ called the action groupoid. In particular, $A_{DR}\left(M_{\bullet}G \right) $ is a simplicial commutative differential graded algebra. Let $\operatorname{Tot}_{N}^{\bullet}\left(A_{DR}\left(M_{\bullet}G \right)\right)$ be the normalized total complex. An element $w$ of degree $n$ in this complex can be written as $w=\sum_{p+q=n}w^{p,q}$, where $w^{p,q}$ is a set map
\[
w^{p,q}\: : \: G^{p}\to A_{DR}^{q}(M)
\]
such that $w^{p,q}(g_{1}, \dots, g_{n})=0$ if $g_{i}=e\in G$ for some $i$. The elements $w^{p,q}$ are called elements of bidegree $(p,q)$. In particular, $\operatorname{Tot}_{N}^{0,q}\left(A_{DR}\left(M_{\bullet}G \right)\right)=A_{DR}^{q}(M)$. An element $w$ is said to be holomorphic if each $w^{p,q}$ takes holomorphic values for any $(p,q)$. Since the action is properly discontinuous, we can visualize $A_{DR}^{\bullet}(M/G)\subset A_{DR}^{\bullet}(M)$ as the differential graded algebra of $G$-invariant differential forms. Getzler and Cheng (see \cite{Getz}) have shown that the normalized total complex $\operatorname{Tot}_{N}^{\bullet}\left(A_{DR}\left(M_{\bullet}G \right)\right)$ carries a natural unital $C_{\infty}$-structure $m_{\bullet}$
such that
\[
m_{1}=d,\quad m_{2}=\wedge,\quad m_{n}=0\text{ for }n\geq 3
\]
on $A_{DR}^{\bullet}(M/G)\subset\operatorname{Tot}_{N}^{0,\bullet}\left(A_{DR}\left(M_{\bullet}G \right)\right)$.
We denote the differential $m_{1}$ by $D$. In \cite{Dupont} Dupont has shown that $H^{\bullet}\left(\operatorname{Tot}_{N}^{\bullet}\left(A_{DR}\left(M_{\bullet}G \right)\right),D \right) $ corresponds to the singular cohomology of the fat geometric realization of $M_{\bullet}G$. Since the action is properly discontinuous, $H^{\bullet}\left( ||M_{\bullet} G||,\C\right)\cong H^{\bullet}\left( M/G,\C\right)$ and in particular the inclusion
\[
\left( A_{DR}(M/G),d, \wedge\right) \hookrightarrow\left( \operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right),m_{\bullet}\right)
\]
is a quasi-isomorphism of $C_{\infty}$-algebras. We assume that $M/G$ is connected and that $H^{i}\left(M/G, \C \right) $ is finite dimensional for any $i$. Let $B\subseteq \operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right)$ be a $C_{\infty}$-subalgebra. We assume that $\left( B,D\right) \hookrightarrow \left( \operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right),D\right) $ induces an isomorphism on the $0$-th and on $1$-th cohomology group and an injection on the $2$-th, i.e. $B$ is a $1$-model for $\left( \operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right),m_{\bullet}\right)$. A $1$-minimal model for $\operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right)$ consists of a $C_{\infty}$-morphism $g_{\bullet}\: : \: \left(W, {m'}_{\bullet} \right) \to \operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right)$ such that $m_{1}=0$ (i.e. $W$ is minimal) and
$g_{\bullet}$ induces an isomorphism on the $0$-th and on $1$-th cohomology group and an injection on the $2$-th one. Explicit examples of $1$-minimal models can be constructed via the homotopy transfer theorem (see \cite{Kadesh}, \cite{kontsoibel}, \cite{Markl} and \cite{Prelie}). In this paper, we introduce the category of $1-C_{\infty}$-algebras and $1-C_{\infty}$-morphisms. They are essentially ``$1$-truncated'' $C_{\infty}$-algebras, i.e. a graded vector space $A$ equipped with maps $\tilde{m}_{n}\: :\: \oplus_{i\leq n}\left( A^{\otimes n}\right)^{i}\to A$ that satisfy the ordinary relations for $C_{\infty}$-algebras only on $\oplus_{i\leq n-1}\left( A^{\otimes n}\right)^{i}$. There is a forgetful functor $\mathcal{F}$ from the category of $C_{\infty}$-algebras towards the category of $1-C_{\infty}$-algebras. The $1-C_{\infty}$-algebras have a well-defined notion of cohomology groups and morphisms. In particular, the second cohomology groups is generated by the higher Massey products between elements of lower degree. We introduce the notion of $1$-minimal models of $1-C_{\infty}$-algebras such that each ordinary $1$-minimal models gives a $1$-minimal model of $1-C_{\infty}$-algebras via $\mathcal{F}$.
A $1$-minimal model $g_{\bullet}\: : \: \left(W, {m'}_{\bullet} \right) \to \mathcal{F}\left(\operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right), m_{\bullet}\right)$ is said to be \emph{holomorphic } if the image of $g_{\bullet}$ contains only holomorphic elements. Let $\left( N,\mathcal{D}\right) $ be a smooth complex manifold with a normal crossing divisor $\mathcal{D}$ and $G$ be a group acting holomorphically on $N$ preserving $\mathcal{D}$. Assume that $M=N-\mathcal{D}$, then $g_{\bullet}$ is said to be \emph{holomorphic with logarithmic singularities} if the image of $g_{\bullet}$ contains only holomorphic elements with logarithmic singularities along $\mathcal{D}$. In the present paper, we prove the following. For a $p\in M$ we denote its class in $M/G$ by $\overline{p}$.
\begin{thmll}\label{Final1}
Let $M$ be a smooth complex manifold equipped with a properly discontinuous action of a discrete group $G$.
For each $1$-minimal model $g_{\bullet}\: : \: \left(W, {m'}_{\bullet} \right) \to \mathcal{F}\left(\operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right), m_{\bullet}\right) $ there exists
\begin{enumerate}
\item a Lie ideal $\mathcal{R}_{0}\subset \widehat{\mathbb{L}}\left( (W_{+}^{1}[1])^{*}\right) \subset \widehat{T}\left( (W_{+}^{1}[1])^{*}\right) $,
\item an element $r_{*}\left(\overline{C}\right) \in A_{DR}^{1}(M)\widehat{\otimes}\mathfrak{u}$, where $\mathfrak{u}=\widehat{\mathbb{L}}\left( (W_{+}^{1}[1])^{*}\right)/\mathcal{R}_{0}$, and
\item a fiber preserving smooth $G$-action on $M\times \mathfrak{u}$ such that $\mathfrak{u}$ is the complete Malcev Lie algebra of $\pi_{1}\left(M/G, {p} \right)$ and $d-r_{*}\left( \overline{C}\right)$ defines a flat connection on a trivial bundle $E:=\left( M\times \mathfrak{u}\right)/G $ on $M/G$ whose monodromy representation at $\overline{p}\in M/G$ is a group homomorphism
\[
\Theta\: : \: \pi_{1}\left(M/G,\overline{p} \right)\to \widehat{T}\left( (W_{+}^{1}[1])^{*}\right)/\bar{\mathcal{R}}_{0}\subset \operatorname{End}\left(\mathfrak{u} \right)
\]
where $\bar{\mathcal{R}}_{0}\subset \widehat{T}\left( (W_{+}^{1}[1])^{*}\right)$ is the ideal generated by $\mathcal{R}_{0}$. Moreover, for $\bar{p}\in M$ we have\\ $\Theta\left(\pi_{1}\left(M/G,p \right) \right)\subset \exp(\mathfrak{u})$ such that $
\Theta\: : \: \pi_{1}\left(M/G,\bar{p} \right)\to\exp(\mathfrak{u})
$ is the Malcev completion of $\pi_{1}\left(M/G,\bar{p} \right)$.
\item Assume that the action of $G$ on $M$ is holomorphic and $g_{\bullet}$ is holomorphic. Then $E$ is a holomorphic bundle on $M/G$ and $d-r_{*}\left(\overline{C}\right)$ defines a holomorphic flat connection on $E$.
\item Let $\left( N,\mathcal{D}\right) $ be a smooth complex manifold with a normal crossing divisor and $G$ be a group acting holomorphically on $N$ and that preserves $\mathcal{D}$. Assume that $M=N-\mathcal{D}$ and that $g_{\bullet}$ is holomorphic with logarithmic singularities. Then $E$ is a holomorphic bundle on $N/G$ and $d-r_{*}\left(\overline{C}\right)$ defines a holomorphic flat connection with logarithmic singularities on $E$.
\end{enumerate}
\end{thmll}
Theorem \ref{Chenthm} is special case of Theorem \ref{Final1}. Let $A\subset A_{DR}(M/G)$. The Hodge type decomposition $A=W\oplus \mathcal{M}\oplus d\mathcal{M}$ can be turned via the homotopy transfer theorem into a $1$-minimal model $g_{\bullet}\: : \: \left(W, {m'}_{\bullet} \right) \to A$ which is unique with respect some algebraic conditions relative to the decomposition. This $1$-minimal model corresponds via Theorem \ref{Final1} to a proof of \ref{Chenthm}. In this case, the group action on the bundle is trivial.\\
The strategy for the proof of Theorem \ref{Final1} is the following. The $1-C_{\infty}$-structure ${m'}_{\bullet}$ corresponds to a map $\delta'\: : \: \left( T^{c}(W_{+}[1])\right)^{0} \to \left( T^{c}(W_{+}[1])\right)^{1}$ and $\mathcal{R}_{0}$ is the Lie ideal generated by the image of ${\delta'}^{*}$. We construct a particular $L_{\infty}$-algebra called degree zero convolution Lie algebra and show that it contains some special Maurer-Cartan elements $\overline{C}$, each of which can be turned into a connection form $r_{*}\overline{C} \in A_{DR}(M)\widehat{\otimes}\left( \widehat{\mathbb{L}}\left( (W_{+}[1])^{*}\right)/\mathcal{R}_{0}\right) $ that fulfills the above conditions. In general a holomorphic $1$-minimal model with logarithmic singularities for $ \mathcal{F}\left( A_{DR}(M/G),d, \wedge \right)$ does not exist, because there are not enough closed holomorphic forms. In some cases, it is possible to find a holomorphic $1$-minimal model with logarithmic singularities for $ \mathcal{F}\left( \operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right),m_{\bullet}\right)$
This is the case for punctured Riemann surfaces of positive genus. In the present paper, we give a comparison between Theorem \ref{Chenthm} and Theorem \ref{Final1}, and we analyze the dependence on the choice of $1$-minimal model. Let $M$ be a smooth complex manifold and let $E_{1}$, $E_{2}$ be two smooth vector bundles on $M$. Let $\left(d-\alpha_{1},E_{1} \right) $, $\left(d-\alpha_{2},E_{2} \right) $ be two smooth connections. They are \emph{isomorphic} if there exists a bundle isomorphism $T\: : \: E_{1}\to E_{2}$ such that
\[
T\left( d-\alpha_{1}\right)T^{-1}=d-\alpha_{2}.
\]
\begin{thmll}\label{Final2}
Let $d-\alpha_{1}$ and $d-\alpha_{2}$ be two flat connections on two smooth vector bundles $E_{1}$ and $E_{2}$ on $M/G$ obtained by Theorem \ref{Final1} by choosing some $1$-minimal models.
\begin{enumerate}
\item The connections $\left( d-\alpha_{1},E_{1}\right) $ and $\left( d-\alpha_{2},E_{2}\right) $ are isomorphic.
\item Assume that $d-\alpha_{1}$ and $d-\alpha_{2}$ are constructed using holomorphic $1$-models. The above isomorphism is holomorphic.
\item Let $\left( N,\mathcal{D}\right) $ be a smooth complex manifold with a normal crossing divisor and $G$ be a group acting holomorphically on $N$ and that preserves $\mathcal{D}$. Assume that $M=N-\mathcal{D}$ and that $d-\alpha_{1}$ and $d-\alpha_{2}$ are constructed using holomorphic $1$-models with logarithmic singularities. The above isomorphism is holomorphic and it extends to an holomorphic isomorphism between bundles on $N/G$.
\end{enumerate}
In particular, the connections are (smoothly) isomorphic to the connection obtained in Theorem \ref{Chenthm} on $M/G$.
\end{thmll}
Theorem \ref{Final1} and \ref{Final2} are Theorem \ref{wehaveabundle} and Theorem \ref{wehaveabundleuniq}. In a forthcoming paper, we construct two holomorphic $1$-minimal models and we show that the KZB connection on the punctured torus and on the configuration space of points of the punctured torus presented in \cite{Damien} can be constructed explicitly using the homotopy transfer theorem and Theorem \ref{Final1} (see \cite{Sibilia1}). The methods can be used to construct holomorphic flat connection on punctured surfaces of arbitrary genera (compare with \cite{Enriquezconnection}). We mainly work with smooth complex manifolds but all the result contained here work for real manifolds as well.
\subsection*{Plan of the paper.}
In Section \ref{first} we discuss $1-C_{\infty}$-algebras and $L_{\infty}$-algebras and we construct the convolution $L_{\infty}$-algebra. This section is purely algebraic. In Section \ref{sectGetztler}, we recall a natural $C_{\infty}$-structure on the total complex of a cosimplicial commutative algebra due to Getzler and Cheng (\cite{Getz}). It gives a natural $C_{\infty}$-structure on the differential forms on a simplicial manifold and we analyze some properties of such a $C_{\infty}$-structure. In Section \ref{third}, given a simplicial manifold $M_{\bullet}$, we consider the convolution $L_{\infty}$-algebra constructed using such a $C_{\infty}$-structure. Certain Maurer-Cartan elements of this $L_{\infty}$-algebra corresponds to flat connections on a trivial bundle on $M_{0}$. In Section \ref{newadded}, we restrict our attention on action groupoids and we give a proof of Theorem \ref{Final1}, \ref{Final2}.
\begin{Ack}
This article is based on the first part of my PhD project. I wish to thank my two advisors Damien Calaque and Giovanni Felder for their guidance and constant support and Benjamin Enriquez for his interesting questions about my PhD thesis. I'm very much indebted to Daniel Robert-Nicoud for long discussion about Section \ref{sectGetztler}. I thanks the SNF for providing an essential financial support through the ProDoc module PDFMP2 137153 ``Gaudin subalgebras, Moduli Spaces and Integrable Systems" I also thank the support of the ANR SAT and of the Institut Universitaire de France.
\end{Ack}
\subsection*{Notation} Let $\Bbbk$ be a field of characteristic zero.
We work in the unital monoidal tensor category of graded vector spaces $(grVect,\otimes,\Bbbk, \tau)$ where the field $\Bbbk$ is considered as a graded vector space concentrated in degree $0$, the twisting map is given $\tau(v\otimes w):=(-1)^{|v||w|}w\otimes v$ and the tensor product is the ordinary graded tensor product. For a graded vector space $V^{\bullet}$, $V^{i}$ is called the homogeneous component of $V$, and for $v\in V^{i}$ we define its degree via $|v|:=i$. For a vector space $W:=\oplus_{i\in I} W_{i}$ we denote by ${pro}_{W_{i}}\: : \: W\to W_{i}$ the projection. For a graded vector space $V^{\bullet}$ we denote by $V[n]$ the $n$-shifted graded vector space, where $\left( V[n]\right) ^{i}=V^{n+i}$. For example $\Bbbk[n]$ is a graded vector space concentrated in degree $-n$ (its $-n$ homogeneous component is equal to $\Bbbk$, the other homogeneous component are all equal to zero). A (homogeneous) morphism of graded vector spaces $f\: : \: V^{\bullet}\to W^{\bullet}$ of degree $|f|:=r$ is a linear map such that $f(V^{i})\subseteq V^{i+r}$. Given two graded vector spaces $V$, $W$, then the set of morphisms of degree $n$ is denoted $\Hom^{n}_{gVect}\left(V,W \right)$. More generally the set of maps between $V$ and $W$ is again a graded vector space $\Hom^{\bullet}_{gVect}\left(V,W \right)$ for which the $i$-homogeneous elements are those of degree $i$.
The tensor product of homogeneous morphisms is defined through the \emph{Koszul convention}: given two morphisms of graded vector spaces $f\: : \: V^{\bullet}\to W^{\bullet}$ and $g\: : \: {V'}^{\bullet}\to {W'}^{\bullet}$ then its tensor product $f\otimes g\: : \: \left( V\otimes W\right)^{\bullet}\to \left( V'\otimes W'\right)^{\bullet}$ on the homogeneous elements is given by
\[
\left( f\otimes g\right) (v\otimes w ):=(-1)^{|g||v|}(f(v)\otimes g(w )).
\]
We denote by $s\: : \: V\to V[1]$, $s^{-1}\: : \: V[1]\to V$ the shifting morphisms that send $V^{n}$ to $V[1]^{n-1}=\Bbbk \otimes V^{n}=V^{n}$ resp. $V[1]^{n}=\Bbbk \otimes V^{n+1}=V^{n+1}$ to $V^{n+1}$. Those maps can be extended to a map $s^{n}\: : \: V\to V[n]$, (the identity map shifted by $n$).
Note that $s^{n}\in \Hom^{-n}\left(V, V[n] \right)$. A graded vector space is said to be of finite type if each homogeneous component is a finite vector space. A graded vector space $V^{\bullet}$ is said to be bounded below at $k$ if there is a $k$ such that $V^{l}=0$ for $l<k$. Analogously it is said to be bounded above at $k$ if there is a $k$ such that $V^{l}=0$ for $l>k$. For a non-negatively graded vector space we define the positively graded vector space
$W^{\bullet}_{+}$ as
\[
W_{+}^{0}:=0,\quad W_{+}^{i}:=W^{i}\text{ if }i\neq0
\]
Let $(V,d_{V})$ be a differential graded vector space, then $V^{\otimes n}$ is again a differential graded vector space with differential
\[
d_{V^{\otimes n}}(v_{1}\otimes\cdots\otimes v_{n}):=\pm\sum_{i=1}^{n}v_{1}\otimes\cdots\otimes d_{V}v_{i}\cdots\otimes v_{n},
\]
where the signs follow from the Koszul signs rule.
Let $(V,d_{V})$, $(W,d_{W})$ be differential graded vector spaces, then $\Hom^{\bullet}_{gVect}\left(V,W \right)$ is a differential graded vector space with differential
\[
\partial f:= d_{W}f-(-1)^{|f|}fd_{v}.
\]
\section{$A_{\infty}$, $C_{\infty}$-structures and homological pairs}\label{first}
In this section we construct a $L_{\infty}$-algebra (called convolution $L_{\infty}$-algebra) and we study their Maurer-Cartan elements.
\subsection{$A_{\infty}$, $C_{\infty}$, $L_{\infty}$-structures}\label{Intro A infty}
We introduce $A_{\infty}$, $C_{\infty}$, $L_{\infty}$-structures. Our reference is \cite{lodayVallette}. For a graded vector space $V$, let ${T^{c}}\left(V[1]\right)$ be the (graded) tensor coalgebra on shift of $V$. Let $NTS(V)\subset {T^{c}}\left(V[1]\right)$ be the subspace of non-trivial shuffles, i.e the vector space generated by $\mu'(a,b)$ such that $\mu'$ is the graded shuffle product and $a,b\notin \Bbbk\subset {T^{c}}\left(V[1]\right)$.
\begin{defi}\label{defAinfty}
Let $V^{\bullet}$ be a graded vector space. An \emph{$A_{\infty}$-algebra structure} on $V^{\bullet}$ is a coderivation $\delta\: : \: {T^{c}}\left(V[1]\right)\to {T^{c}}\left(V[1]\right)$ of degree $+1$ such that $\delta ^{2}=0$. A \emph{$C_{\infty}$-algebra structure} on $V^{\bullet}$ is an $A_{\infty}$-structure such that $\delta(NTS(V[1]))=0$.
\end{defi}
Each coderivation is uniquely determined by the maps of degree $1$
\[
\begin{tikzcd}
\delta_{n}\: : \:A[1]^{\otimes n}\arrow[hook]{r}&{T^{c}}\left(A[1]\right)\arrow{r}{\delta}&
{T^{c}}\left(A[1]\right)\arrow{r}{{pro}_{A[1]}} & A[1].\end{tikzcd}
\]
For $n>0$ we define maps $m_{n}\: : \: A^{\otimes n}\to A$ of degree $2-n$ via
\[
\begin{tikzcd}
A^{\otimes n}\arrow{r}{\left( s\right)^ {\otimes n}}& A[1]^{\otimes n}\arrow{r}{\delta_{n}}&A[1]\arrow{r}{s^{-1}}&A.
\end{tikzcd}
\]
The condition $\delta^{2}=0$ for the maps $m_{n}\: : \: A^{\otimes n}\to A$ implies $m_{1}^{2}=0$ and the relations
\begin{equation}\label{rel}
\sum_{\substack{p+q+r=n\\ k=p+1+r\\k,q>1}}(-1)^{p+qr}m_{k}\circ \left(\mathrm{Id}^{\otimes p}\otimes m_{q}\otimes \mathrm{Id}^{\otimes r} \right)=\partial m_{n}, \quad n>0.
\end{equation}
Conversely, starting from maps $m_{\bullet}:=\left\lbrace m_{n} \right\rbrace _{n\geq0}$ that satisfy the above relations, we get a sequence of maps $\delta_{n}\: : \:A[1]^{\otimes n}\to A[1]$, for $n\geq0$ defined via
\[
\begin{tikzcd}
\delta_{n}\: : \:\left( A[1]\right) ^{\otimes n}\arrow{r}{\left( s^{-1}\right)^ {\otimes n}}& A^{\otimes n}\arrow{r}{m_{n}}& A\arrow{r}{s}& A[1].
\end{tikzcd}
\]
In particular we have $m_{1}^2=0$. These maps can be viewed as the restriction of a coderivation $\delta$, which is a differential by \eqref{rel}. We denote an $A_{\infty}$- ( $C_{\infty}$- ) algebra $\left( {T^{c}}\left(V[1]\right),\delta\right)$ by $\left(A^{\bullet},m_{\bullet} \right) $ as well.
\begin{defi}
An $A_{\infty}$-algebra $\left(A,m_{\bullet} \right) $ is said to be \emph{unital} if there exists a $m_{1}$-closed element $1$ of degree zero such that $m_{2}(1,a)=1=m_{2}(a,1)$ and $m_{k}(a_{1}\dots,1,\dots, a_{k})=0$ for $k\geq 3$. $\left(A,m_{\bullet} \right) $ is said to be \emph{connected} if $A^{0}=\Bbbk1$ and $A$ is a non-negatively graded vector space.
Let $A^{\bullet}$, $B^{\bullet}$ be two $A_{\infty}$-algebras. A \emph{morphism between $A_{\infty}$-algebras} is a morphism of differential graded coalgebras
\[
F\: : \: \left({T^{c}}\left(A[1]\right),{\Delta}, \delta_{A}\right) \to \left( {T^{c}}\left(B[1]\right),{\Delta}, \delta_{B}\right).
\]
Each morphism is completely determined by the degree zero maps, i.e.
\[
\begin{tikzcd}
F_{n}\: : \:A[1]^{\otimes n}\arrow[hook]{r}& {T^{c}}\left(A[1]\right)\arrow{r}{F}&
{T^{c}}\left(B[1]\right)\arrow{r}{{pro}_{B[1]}} & B[1],\quad n>0.
\end{tikzcd}
\]
and $F_{0}(1):=F(1)=1$. $F$ is said to be a \emph{morphism of $C_{\infty}$-algebras} if $A$ and $B$ are $C_{\infty}$-algebras and $F_{n}(NTS(V[1])\cap V[1]^{\otimes n})=0$.
\end{defi}
We denote by $f_{\bullet}:=\left\lbrace f_{n} \right\rbrace_{n\in \N}$ the family of maps of degree $1-n$ given by
\[
\begin{tikzcd}
f_{n}\: : \: A^{\otimes n}\arrow{r}{\left( s\right)^{\otimes n}}& A[1]^{\otimes n}\arrow{r r}{{pro}_{B[1]}\circ F_{n}} & & B[1]\arrow{r}{s^{-1}}& B.
\end{tikzcd}
\]
Let $m_{n}^{A}$ be the degree $2-n$ maps obtained from $\delta_{A}$, and let $m_{n}^{B}$ be the ones from $\delta_{B}$. The condition $F\circ \delta_{A}=\delta_{B}\circ F$ implies the following equations $f_{1}m_{1}^{A}=m_{1}^{B}f_{1}$ and
\begin{align}\label{tricksign}
&\nonumber \sum_{\substack{p+q+r=n\\ k=p+1+r}}(-1)^{p+qr}f_{k}(Id^{\otimes p}\otimes m_{q}^{A}\otimes Id^{\otimes r})\\
&-\sum_{k\geq2,\,i_{1}+\dots +i_{k}=n}(-1)^{s}{m}_{k}^{B}\left(f_{i_1}\otimes\dots\otimes f_{i_k} \right)= \partial f_{n},\quad n> 1,
\end{align}
where $s=\left(k-1 \right) \left(i_{1}-1 \right) +\left(k-2 \right) \left(i_{2}-1 \right) +\dots+2\left(i_{k-2}-1 \right) +\left(i_{k-1}-1 \right) $. In particular $f_{1}$ is an ordinary cochain map. A family of maps $f_{\bullet}$ satisfying condition \eqref{tricksign} induces also a morphism of $A_{\infty}$-algebras $F\: : \: \left({T^{c}}\left(A[1]\right),{\Delta}, \delta_{A}\right) \to \left( {T^{c}}\left(B[1]\right),{\Delta}, \delta_{B}\right)$.
\begin{rmk}
We will adopt the following notation. We will use small dotted letters (e.g. $f_{\bullet}$) to denote morphisms of $A_{\infty}$, $C_{\infty}$-algebras. On the other hand, we will use capital letters (e.g. $F$) to denote morphism of $A_{\infty}$, $C_{\infty}$-algebras as morphism of quasi-free coalgebras.
\end{rmk}
\begin{defi}
A morphism between $A_{\infty}$ ( $C_{\infty}$ )-algebras is a \emph{quasi-isomorphism} if the cochain map
\[
f_{1}\: : \: \left(A, m_{1}^{A}\right)\to \left(B, m_{1}^{B}\right)
\]
is a quasi-isomorphism. A morphism between $A_{\infty}$ ( $C_{\infty}$ ) -algebras $F\: : \: {T^{c}}\left(A[1]\right)\to {T^{c}}\left(B[1]\right)$ is an isomorphism if $f_{1}$ is an isomorphism. A morphism $f_{\bullet}$ is called strict if $f_{n}=0$ for $n>1$.
\end{defi}
We denote by $\Omega(1)$ the free commutative graded algebra generated by $1,t_{0}$, $t_{1}$ of degree zero and by $dt_{0}$, $dt_{1}$ of degree $1$ such that
\[
t_{0}+t_{1}=1,\quad dt_{0}+dt_{1}=0.
\]
We put a differential $d$ on $\Omega(1)$ by $d1=0$ and $d(t_{j}):=dt_{j}$, for $ j=0,1$ such that $\Omega(1)$ is a differential free commutative graded algebra. Equivalently, we may define $\Omega(1)$ as the commutative free graded algebra generated by $1,t$ of degree $0$ and $dt$ in degree $1$. The differential here is given by $d(1):=0, d(t):=dt$. The two presentations are isomorphic via the map $t\mapsto t_{0}$. We denote by $i_{j}\: : \: \Omega(1)\to \Bbbk$ the dg algebra map sending $t_{j}$ to $1$ and $dt_{j}$ to $0$ for $j=0,1$.
\begin{lem}\label{tensorinfinity}
Let $f_{\bullet}\: : \: \left( A,m_{\bullet}^{A}\right)\to \left( B,m_{\bullet}^{B}\right) $ be a morphism of $A_{\infty}$-algebras and $g\: : \: \Omega(1)\to \Omega(1)$ be a morphism of differential graded algebras.
\begin{enumerate}
\item $\left( \Omega(1)\otimes A, m_{\bullet}^{ \Omega(1)\otimes A}\right) $ is an $A_{\infty}$-algebra via
\[
m_{n}^{ \Omega(1)\otimes A}(p_{1}\otimes a_{1}, \dots, p_{n}\otimes a_{n}):=\pm\left( p_{1} \cdots p_{n}\right) \otimes m_{n}^{A}( a_{1},\otimes ,a_{n}),
\]
where the sign $\pm$ follows by the signs rule. In particular $i_{j}\otimes Id\: : \: \Omega(1)\otimes A\to A $ is a well-defined strict $A_{\infty}$-morphism.
\item The map $(g\otimes f)_{\bullet}\: : \: \left( \Omega(1)\otimes A,m_{\bullet}^{ \Omega(1)\otimes A}\right)\to \left( \Omega(1)\otimes B,m_{\bullet}^{ \Omega(1)\otimes B}\right) $ defined by
\[
(g\otimes f)_{n}(p_{1}\otimes a_{1}, \dots, p_{n}\otimes a_{n}):= \pm g\left( p_{1} \cdots p_{n}\right) \otimes f_{n}( a_{1},\dots,a_{n}),
\]
where the signs $\pm$ follows by the signs rule, is a morphism of $A_{\infty}$-algebras. If $g=id$ we have $\left( i_{j}\otimes Id\right) (Id\otimes f)_{\bullet} =f_{\bullet}\left( i_{j}\otimes Id\right) $ for $j=0,1$.
\end{enumerate}
\end{lem}
\begin{proof}
Straightforward calculation.
\end{proof}
\begin{defi}\label{homotopy}
Let $f_{\bullet}, g_{\bullet}\: : \: A\to B$ be $A_{\infty}$-morphisms (resp. $C_{\infty}$-morphism). A \emph{homotopy} between $f_{\bullet}$ and $g_{\bullet}$ is an $A_{\infty} $ (resp. $C_{\infty}$) map $H_{\bullet}\: : \: A \to \Omega(1)\otimes B$ such that
\[
\left( i_{0}\otimes Id\right)H_{\bullet}=f_{\bullet}, \quad \left( i_{1}\otimes Id\right)H_{\bullet}=g_{\bullet},
\]
two morphisms are homotopy equivalent if there exists a finite sequence of homotopy maps connecting them.
\end{defi}
By Lemma \ref{tensorinfinity} we have the following. Let $f_{\bullet}, g_{\bullet}\: : \: A\to B$ be $A_{\infty}$-morphisms. Let $H_{\bullet}\: : \: A \to \Omega(1)\otimes B$ be a homotopy between $f_{\bullet}$ and $g_{\bullet}$, let $p^{1}_{\bullet}\: : \: A_{1}\to A$ and $p^{2}_{\bullet}\: : \: B\to A_{2}$ be $A_{\infty}$-maps. Then $H_{\bullet}(Id\otimes p^2)_{\bullet}$ is a homotopy between $f_{\bullet}p^{2}_{\bullet}$ and $g_{\bullet}p^{2}_{\bullet}$ and $(Id\otimes p^1)_{\bullet}H_{\bullet}$ is a homotopy between $p^{1}_{\bullet}f_{\bullet}$ and $p^{1}_{\bullet}g_{\bullet}$.\\
\begin{prop}\label{existence of a homotopical inverse}
Let $P_{\infty}$ be $A_{\infty}$ or $C_{\infty}$. Then any $P_{\infty}$-quasi-isomorphism $f_{\bullet}\: : \: A\to B$ has a $P_{\infty}$-homotopical inverse, i.e. there exists a $P_{\infty}$-map $g_{\bullet}\: : \: B\to A$ such that $g_{\bullet}\circ f_{\bullet}\cong Id_{A}$ and $f_{\bullet}\circ g_{\bullet}\cong Id_{B}$.
\end{prop}
\begin{proof}
This is Theorem 3.6 of \cite{BrunoHomotopy}. The $P_{\infty}$ objects enjoy this property because they are fibrant-cofibrant objects in a certain model category structure where $\Omega(1)\otimes \left(-\right) $ is a functorial cylinder object.
\end{proof}
\begin{defi}
Let $P_{\infty}$ be $A_{\infty}$ or $C_{\infty}$. We denote by $P_{\infty}-\operatorname{ALG}$ the category of bounded below $P_{\infty}$-algebras. We denote by $ P_{\infty}-\operatorname{ALG}_{\geq0}$ and $ P_{\infty}-\operatorname{ALG}_{>0}$ the two full subcategories whose objects are non-negatively graded $P_{\infty}$-algebras and positively graded $P_{\infty}$-algebras respectively.
\end{defi}
For a bounded below $P_{\infty}$-algebra $\left(A, m_{\bullet}^{A} \right)$ we denote by $\left(\overline{A}, m_{\bullet}^{A} \right)$ its positively graded $P_{\infty}$-subalgebra where $\overline{A}:=\oplus_{i>0} A^{i}$.
Let $A=\oplus_{i\leq 2} A^{i}$ be a bounded below graded vector space. A \emph{$1$-truncated $A_{\infty}$-algebra} or \emph{$1-A_{\infty}$-algebra} consists in a sequence of coderivations
\[
\begin{tikzcd}
\dots \arrow{r}{\delta^{A}} &\left( T^{c}\left(A[1] \right)\right)^{-2} \arrow{r}{\delta^{A}} & \left( T^{c}\left(A[1] \right)\right)^{-1} \arrow{r}{\delta^{A}} & \left( T^{c}\left(A[1] \right)\right)^{0}\arrow{r}{\delta^{A}} & \left( T^{c}\left(A[1] \right)\right)^{1}
\end{tikzcd}
\]
such that $\left(\delta^{A} \right)^{2}=0$. Equivalently, they corresponds to a sequences of degree $n-2$ maps $m_{n}\: :\: \oplus_{i\leq n}\left( A^{\otimes n}\right)^{i}\to A$ that satisfies \eqref{rel} on $\oplus_{i\leq n-1}\left( A^{\otimes n}\right)^{i}$. The structure is said to be $1-C_{\infty}$ if $\delta^{A}$ vanishes on non-trivial shuffles of degree smaller than $1$. A \emph{$1$-$A_{\infty}$-algebra morphism} consists in a graded coalgebra maps $$F\: : \: \oplus_{i\leq 1}\left( T^{c}\left(A[1] \right)\right)^{i} \to \oplus_{i\leq 1}\left( T^{c}\left(B[1] \right)\right)^{i} $$ such that $\left(F\otimes F \right) \Delta=\Delta F $ and $F\delta^{A}=\delta^{B} F$. Note that $F$ corresponds to a family degree $n-1$ maps $f_{n}\: :\: \oplus_{i\leq n+1}\left( A^{\otimes n}\right)^{i}\to A\to B$ that satisfies \eqref{tricksign} on $\oplus_{i\leq n}\left( A^{\otimes n}\right)^{i}$ for any $n>0$. We denote the $1-A_{\infty}$-morphism $F$ by ${f}_{\bullet}\: : \: \left(A, m_{\bullet}^{A} \right)\to \left(B, m_{\bullet}^{B} \right) $. If the two algebras are $1-C_{\infty}$ then $F$ is said to be $1-C_{\infty}$ if it vanishes on non-trivial shuffles of total degree smaller than $2$.
\begin{defi}
Let $P_{\infty}$ be $A_{\infty}$ or $C_{\infty}$. We denote by $1-P_{\infty}-\operatorname{ALG}$ the category of $1-P_{\infty}$-algebras and by $ P_{\infty}-\operatorname{ALG}_{\geq0}$ and $ P_{\infty}-\operatorname{ALG}_{>0}$ the two full subcategories whose objects are non-negatively graded $P_{\infty}$-algebras and positively graded $P_{\infty}$-algebras respectively.
\end{defi}
There is an obvious forgetful functor $\mathcal{F}\: : \: P_{\infty}-\operatorname{ALG}\to 1-P_{\infty}-\operatorname{ALG}$ which is left adjoint to the functor $\mathcal{E}\: : \: 1-P_{\infty}-\operatorname{ALG}\to P_{\infty}-\operatorname{ALG}$ obtained by extending the $1-P_{\infty}$ into a $P_{\infty}$-structure by $0$. We define the cohomology of a $1-P_{\infty}$ algebra $\left(A, m_{\bullet} \right)$ as the cohomology of the complex
\[
\begin{tikzcd}
\dots \arrow{r}{m_{1}} & A^{1}\arrow{r}{m_{1}} & A'\arrow{r}{0}& 0.
\end{tikzcd}
\]
where $A'\subset A^{2}$ is the subspace generated by $m_{n}(a_{1}, \dots a_{n})$ such that all the $a_{i}$ are $m_{1}$-closed. The cohomology of a $1-P_{\infty}$-algebra is again a a $1-P_{\infty}$-algebra and a $1-P_{\infty}$-morphism induces a $1-P_{\infty}$-morphism in cohomology. A $1-P_{\infty}$-morphism ${f}_{\bullet}\: : \: \left(A, m_{\bullet}^{A} \right)\to \left(B, m_{\bullet}^{B} \right) $ is an $1$-isomorphism if $f_{1}\: : \: A^{i}\to A^{i}$ is an isomorphism for $i\leq 1$ and $f_{1}\: : \: A^{2}\to A^{2}$ is an injection. ${f}_{\bullet}$ is a $1$-quasi-isomorphism if ${f}_{\bullet}$ induces a $1$-isomorphism in cohomology. Let $f_{\bullet}, g_{\bullet}\: : \: A\to B$ be $1$-$P_{\infty}$-morphisms. A \emph{homotopy} between $f_{\bullet}$ and $g_{\bullet}$ is a $1$-$P_{\infty} $ map $H_{\bullet}\: : \: A \to \Omega(1)\otimes B$ such that
\[
\left( i_{0}\otimes Id\right) H_{\bullet}=f_{\bullet}, \quad \left( i_{1}\otimes Id\right)_{\bullet}H_{\bullet}=g_{\bullet}.
\]
We conclude this subsection by defining $L_{\infty}$-structures. Let $S(V)$ be the symmetric algebra, then we denote by $S^{c}(V)$ the graded coalgebra given by the graded vector space $S(V)$ and the concatenation coproduct $\Delta$, i.e. $S^{c}(V)$ is the cocommutative cofree conilpotent coalgebra generated by $V$.
\begin{defi}
An \emph{$L_{\infty}$-structure} on a graded vector space $V$ is a coderivation $\delta\: : \: S^{c}(V[1])\to S^{c}(V[1])$ of degree $+1$ such that $\delta^{2}=0$.
\end{defi}
As for the $A_{\infty}$ case, the above definition is equivalent to family of maps $l_{n}\: : \: V^{\otimes n}\to V$ of degree $2-n$. These maps are skew symmetric and satisfy the relations
\[
\sum_{p+q=n+1,\,p,q>1 }\sum_{\sigma^{-1}\in Sh^{-1}(p,q)}\operatorname{sgn}(\sigma)(-1)^{(p-1)q}l_{p}\left(l_{q}\otimes Id^{\otimes \left( p-1\right) } \right)^{\sigma}=\partial l_{n},
\]
for $n\geq 1$.
The morphisms, quasi-isomorphisms and isomorphisms between $L_{\infty}$-algebras are defined in the same way, as for the $A_{\infty}$ case. \\
Given an associative algebra $A $, then it carries a Lie algebra structure where the bracket is obtained by anti-symmetrizing the product. The same is true between $A_{\infty}$ and $L_{\infty}$-algebras.
\begin{thm}\label{antilie}
Let $(V, m_{\bullet})$ be an $A_{\infty}$-algebra. The anti-symmetrized map $l_{n}\: : \: V^{\otimes}\to V$, given by
\[
l_{n}:=\sum_{\sigma\in S_{n}}\operatorname{sgn}(\sigma)m_{n}^{\sigma},
\]
define an $L_{\infty}$-algebra structure on $V$.
\end{thm}
\begin{defi}\label{Linftydef}
Given a $L_{\infty}$-algebra $\mathfrak{g}$ with structure maps $l_{\bullet}$.
\begin{enumerate}
\item An \emph{$L_{\infty}$-ideal} $I\subset \mathfrak{g}$ is a subgraded vector space such that $l_{k}(a_{1}, \dots ,a_{k})\in I$ if one of the $a_{i}$ lies in $I$ (in particular $\left( \mathfrak{g}/I,l_{\bullet}\right) $ is an $L_{\infty}$-algebra).
\item An $L_{\infty}$-algebra is said to be \emph{filtered} if it is equipped with a filtration $F^{\bullet}$ of $L_{\infty}$-ideals such that for $a_{i}\in F^{n_{i}}(\mathfrak{g})$, we have $l_{k}(a_{1}, \dots ,a_{k})\in F^{n_{1}+\dots +n_{k}}(\mathfrak{g})$.
\item A filtered $L_{\infty}$-algebra is said to be \emph{complete} if $\mathfrak{g}\cong \lim_{i}\mathfrak{g}/F^{i}(\mathfrak{g})$ as a graded vector space.
\item A \emph{Maurer-Cartan element} in a complete $L_{\infty}$-algebra $\mathfrak{g}$ is a $\alpha\in \mathfrak{g}^{1}$ such that
\begin{equation*}\label{infinitymaurercartan0}
\partial(\alpha)+\sum_{k\geq 2}^{\infty}\frac{l_{k} \left(\alpha,\dots, \alpha \right) }{k!}=0 .
\end{equation*}
We denote by $MC(\mathfrak{g})$ the set of Maurer-Cartan elements\footnote{ Notice that the above sum is well-defined in $\mathfrak{g}$ since it is complete.}.
\item Let $\mathfrak{g} $ be a complete $L_{\infty}$-algebra. Then $\Omega(1)\widehat{\otimes}\mathfrak{g}$ is again a complete $L_{\infty}$-algebra. An \emph{homotopy} between two Maurer-Cartan elements $\alpha_{0}, \alpha_{1}\in MC(\mathfrak{g})$ is a Maurer-Cartan element $\alpha(t)\in MC(\Omega(1)\widehat{\otimes}\mathfrak{g})$ such that $\alpha(0)=\alpha_{0}$ and $\alpha(1)=\alpha_{1}$. Two Maurer-Cartan elements are said to be \emph{homotopy equivalent} or \emph{ homotopic } if they are connected by a finite sequence of homotopies.
\end{enumerate}
\end{defi}
\begin{defi}
We denote by $\left( \mathcal{L}_{\infty}-\operatorname{ALG}\right)_{p}$ be the category whose objects are $L_{\infty}$-algebras and the arrows are defined as follows: an arrow $\mathfrak{g}\to \mathfrak{g}' $ is a set map $f\: :\: MC(\mathfrak{g})\to MC(\mathfrak{g})$.
\end{defi}
\subsection{Convolution $L_{\infty}$-algebras}
Let $C$ be a coalgebra and let $A$ be a differential graded algebra. The space of morphisms between graded vector spaces $\Hom^{\bullet}\left(C, A \right)$ is equipped with a differential graded Lie algebra structure called convolution algebra (see \cite{lodayVallette}, chapter 1).\\
Let $\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) $ be $A_{\infty}$-algebras and assume they are both bounded below.
Let $\delta$ be the codifferential of $T^{c}(V[1])$ and $\delta^{A}$ be the codifferential on $T^{c}(A[1])$. Consider $A^{\bullet}$ as a graded cochain vector space equipped with $m_{1}^{A}$ as differential. We have a differential graded vector space of morphisms between graded vector spaces
\[
\Hom^{\bullet}\left(T^{c}(V[1]), A \right).
\]
If $m_{n}^{A}=0$ for $n>2$, there is a one to one correspondence between coalgebras morphism $F\: : \:T^{c}(V[1])\to T^{c}(A[1])$ and twisting cochains. For general $A_{\infty}$-structure, there is a similar property. For each $F$, we associate a graded map $\alpha\in \Hom^{1}\left(T^{c}(V[1]), A \right)$ defined as
\begin{equation}\label{12}
\begin{tikzcd}
T^{c}(V[1]) \arrow{r}{F} & T^{c}(A[1])\arrow{r}{\text{proj}_{A[1]}} & A[1]\arrow{r}{s^{-1}}& A.
\end{tikzcd}
\end{equation}
with $\alpha(1)=0$. The condition $F\circ \delta|_{V[1]^{\otimes n}}= \delta^{A}\circ F|_{V[1]^{\otimes n}}$ reads
\[
\left( \alpha \circ \delta^{V}\right) _{V[1]^{\otimes n}}=\sum_{k\geq 1,\, i_{1}+\dots i_{k}=n}(-1)^{k+1}m_{k}^{A}\left(\alpha_{i_{1}}, \dots, \alpha_{i_{k}} \right)\circ \Delta^{k-1}
\]
where $\alpha_{i_{j}}:=\alpha|_{V[1]^{\otimes i_{j}}}$ and $\Delta^{k}$ is the iterated coproduct in the tensor coalgebra $T^{c}(V[1])$. It is an easy exercise to show that $$\tilde{m}^{A}_{k}:=(-1)^{k}m_{k}^{A}\: : \: A^{\otimes k}\to A$$ is again an $A_{\infty}$-structure on $A$. We conclude
\begin{equation}\label{newtwist}
\alpha \circ \delta=-\sum_{k\geq 1}\tilde{m}^{A}_{k}\left(\alpha , \dots, \alpha\right)\circ \Delta^{k-1}.
\end{equation}
The above equation can be interpreted as the $A_{\infty}$-version of the twisting cochain condition.
For $n>1$, we define the maps $M_{n}\: : \: \left( \Hom^{\bullet}\left(T^{c}(V[1]), A \right)\right)^{\otimes n}\to \Hom^{\bullet}\left(T^{c}(V[1]), A \right)$ via
\[
M_{n}(f_{1}, \dots, f_{n}):=\tilde{m}^{A}_{n}\left(f_{1}, \dots, f_{n}\right)\circ \Delta^{n-1},
\]
the map $M_{1}\: : \: \Hom^{\bullet}\left(T^{c}(V[1]), A \right)\to \Hom^{\bullet}\left(T^{c}(V[1]), A \right)$ as $M_{1}(f):=\tilde{m}^{A}_{1}(f)$ and the map $\partial\: : \: \Hom^{\bullet}\left(T^{c}(V[1]), A \right)\to \Hom^{\bullet}\left(T^{c}(V[1]), A \right)$ as
\[
\partial(f):=\tilde{m}^{A}_{1}f- (-1)^{|f|}f\circ \delta.
\]
We define $L_{V[1]^{*}}\left(A \right)$ be the set of morphisms $\Hom^{\bullet}(T^{c}(V[1]), A)$ whose kernel contains the set of non-trivial shuffles $NTS(V[1])$.
\begin{lem}\label{Algebrainfinitystruct}
Let $V$, $A$ as above.
\begin{enumerate}
\item $\left(M_{\bullet}, \Hom^{\bullet}\left(T^{c}(V[1]), A \right) \right)$ is an $A_{\infty}$-algebra,
\item $\left(\partial,\left\lbrace M_{n}\right\rbrace_{n\geq 2}, \Hom^{\bullet}\left(T^{c}(V[1]), A \right) \right)$ is an $A_{\infty}$-algebra,
\item We denote by $l_{\bullet}'$ the maps on $\Hom^{\bullet}\left(T^{c}(V[1]), A \right)^{\otimes n}$ induced by the anti-symmetrization of the maps $\left(\partial,\left\lbrace M_{n}\right\rbrace_{n\geq 2}\right) $ via Theorem \ref{antilie} and by $l_{\bullet}$ the maps induced by the anti-symmetrization of the maps $\left(\partial, M_{2}, M_{3}, \dots\right) $. Assume that $\delta\mu'(a,b)=0$ for any non-trivial shuffle $\mu'(a,b)\in T^{c}(V[1])$ and that the product $m_{2}^{A}$ is graded commutative (but not necessarily associative). Then we have two $L_{\infty}$-subalgebras
\[
\left(l_{\bullet}', L_{V[1]^{*}}(A)\right) \subset \left(l_{\bullet}', \Hom^{\bullet}\left(T^{c}(V[1]), A \right)\right) ,\quad \left(l_{\bullet}, L_{V[1]^{*}}(A)\right) \subset \left(l_{\bullet}, \Hom^{\bullet}\left(T^{c}(V[1]), A \right)\right) .
\]
\end{enumerate}
\end{lem}
For the proof see the Appendix, Section \ref{proof}.
\begin{defi}
Let $\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) $ be $A_{\infty}$-algebras.
\begin{enumerate}
\item We call $ \left( \partial,{\left\lbrace M_{n}\right\rbrace }_{n>1}\Hom^{\bullet}\left(T^{c}(V[1]), A \right) \right) $ the \emph{convolution $A_{\infty}$-algebra} \\associated to $\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) $.
\item We call $$
\operatorname{Conv}_{A_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right):= \left( {l'}_{\bullet},\Hom^{\bullet}\left(T^{c}(V[1]), A \right)\right)$$
the \emph{convolution $L_{\infty}$-algebra} associated to $\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) $.
\item Let $\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) $ be $C_{\infty}$-algebras. We call
\[
\operatorname{Conv}_{C_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right):= \left({l'}_{\bullet}, L_{V[1]^{*}}(A)\right) \subset \left(l_{\bullet}', \Hom^{\bullet}\left(T^{c}(V[1]), A \right)\right)
\]
the \emph{reduced convolution $L_{\infty}$-algebra } associated to $\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) $.
\end{enumerate}
\end{defi}
The next proposition is a consequence of the discussion due at the beginning of this subsection.
\begin{prop}\label{Cinftydictionary}
Let $\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) $ be $P_{\infty}$-algebras. There exists a one to one correspondence between
\begin{enumerate}
\item $P_{\infty}$-morphisms $f_{\bullet}\: : \: \left(V, m_{\bullet}^{V}\right) \to\left(A, m_{\bullet}^{A} \right) $.
\item Morphisms of differential graded quasi-free coalgebras $F\: : \:T^{c}(V[1])\to T^{c}(A[1])$ (resp. such that $F(NTS(V[1]))=0$).
\item Maurer-Cartan elements $\alpha\in \operatorname{Conv}_{P_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right)$ such that $\alpha(1)=0$.
\end{enumerate}
\end{prop}
\begin{defi}
Let $\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) $ be $P_{\infty}$-algebras. Assume that $V$ is positively graded. An element $\alpha\in\operatorname{Conv}^{1}_{P_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right)$ is a \emph{$1$-Maurer-Cartan element} if the corresponding tensor coalgebra morphism $F$ restricted on
\[
\left( T^{c}\left(V[1] \right)\right)^{0}\oplus \left( T^{c}\left(V[1] \right)\right)^{1}
\]
is a $1-P_{\infty}$-algebra morphism. We say that two $1$-Maurer Cartan elements are equal if they coincide on $\left( T^{c}\left(V[1] \right)\right)^{0}\oplus \left( T^{c}\left(V[1] \right)\right)^{1}$. We denote the sets of $1$-Maurer-Cartan elements with $MC_{1}\left( \operatorname{Conv}_{P_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right)\right)\right) $.
\end{defi}
\subsection{ Degree zero convolution $L_{\infty}$-algebras}
Let $(V, m_{\bullet}^{V})$ be a positively graded $1-A_{\infty}$-algebra and let $(A, m_{\bullet}^{A})$ be an $A_{\infty}$ algebra. Let $V[1]^{0}$ be the degree $0$ part of $V[1]$, we consider $\Hom^{\bullet}(T^{c}(V[1]^{0}) , A)\subset \Hom^{\bullet}(T^{c}(V[1]) , A)$ as the graded vector subspace of morphisms with support in $T^{c}(V[1]^{0})$. We define $L_{V[1]^{*}}^{0}\left(A \right) := L_{V[1]^{*}}\left(A \right)\cap \Hom^{\bullet}(T^{c}(V[1]^{0}) , A)$ and we denote $L_{V[1]^{*}}^{0}\left(\Bbbk\right)$ by $L_{V[1]^{*}}^{0}$. The restriction of the dual $\delta^{*}\: : \: \Hom^{\bullet}(\left( T^{c}(V[1])\right)^{1}\oplus T^{c}(V[1]^{0}) , A)\to\Hom^{\bullet}( T^{c}(V[1]^{0}) , A)$ vanishes on $\Hom^{\bullet}(T^{c}(V[1]^{0}) , A)$ if $V[1]$ is a non-negatively graded vector space (equivalently, if $V$ is positively graded).
\begin{cor}\label{Linftystructnotquot} Let $\left(M_{\bullet}, \Hom^{\bullet}\left(T^{c}(V[1]), A \right) \right)$ be as above.
\begin{enumerate}
\item $\left(M_{\bullet}, \Hom^{\bullet}\left(T^{c}(V[1]^{0}), A \right) \right)$ is an $A_{\infty}$-algebra,
\item Let $f_{1},\dots,f_{n}\in \Hom^{\bullet}\left(T^{c}(V[1]^{0}), A \right)$ and assume that there is a $g$ with $\delta^{*}g=f_{i}$ for some $i$, then $M_{n}(f_{1},\dots, f_{n})\in \operatorname{Im}\left(\delta^{*} \right) $ for $n>1$. In particular $$\left(M_{\bullet}, \Hom^{\bullet}\left(T^{c}(V[1]^{0}), A \right)/\operatorname{Im}(\delta^{*}) \right)$$ is an $A_{\infty}$-algebra.
\item Consider $\Hom^{\bullet}\left(T^{c}(V[1]^{0}), A \right)/{\operatorname{Im}(\delta^{*})}$ equipped with the $L_{\infty}$-structure $l_{\bullet}$ induced by the maps $M_{\bullet}$ via Theorem \ref{antilie}. Assume that $\delta\mu'(a,b)=0$ for any non-trivial shuffle $\mu'(a,b)\in T^{c}(V[1]^{0})$ and that the product $m_{2}^{A}$ is graded commutative. The subgraded vector space $ L_{V[1]^{*}}^{0}\left(A \right)/{\operatorname{Im}(\delta^{*})}$ equipped with $l_{\bullet}$ is a $L_{\infty}$-subalgebra of $$\left(l_{\bullet}, \Hom^{\bullet}\left(T^{c}(V[1]^{0}), A \right)/{\operatorname{Im}(\delta^{*})}\right).$$
\end{enumerate}
\end{cor}
For the proof see the Appendix, Section \ref{proof}.
\begin{defi}
We call $$
\operatorname{Conv}_{1-A_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right):= \left( {l}_{\bullet},\Hom^{\bullet}\left(T^{c}(V[1]^{0}), A \right)/{\operatorname{Im}(\delta^{*})}\right) $$ the \emph{degree zero convolution $L_{\infty}$-algebra } associated to $\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) $.\\ Assume that $\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) $ are $1-C_{\infty}$ and $C_{\infty}$ respectively. We call
$$
\operatorname{Conv}_{1-C_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right):= \left({l}_{\bullet}, L_{V[1]^{*}}(A)/{\operatorname{Im}(\delta^{*})}\right) \subset \left(l_{\bullet}, \Hom^{\bullet}\left(T^{c}(V[1]^{0}), A \right)/{\operatorname{Im}(\delta^{*})}\right)$$
the \emph{degree zero reduced convolution $L_{\infty}$-algebra} associated to $\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) $.
\end{defi}
\begin{prop}\label{Cinftydictionary1}
Let $\left(V, m_{\bullet}^{V}\right) $ be a positively graded $1-P_{\infty}$-algebra and a $P_{\infty}$-algebra respectively. There exists a one to one correspondence between the following.
\begin{enumerate}
\item $1-P_{\infty}$-morphism $f_{\bullet}\: : \: \left(V, m_{\bullet}^{V}\right) \to\mathcal{F}\left(B, m_{\bullet}^{B} \right) $.
\item The set of degree zero maps $$F=F^{0}\oplus F^{1}\: : \: \left( T^{c}\left(V[1] \right)\right)^{0}\oplus \left( \left( T^{c}\left(V[1] \right)\right)^{0}\right) \to \left( T^{c}\left(B[1] \right)\right)^{0}\oplus \left( \left( T^{c}\left(B[1] \right)\right)^{0}\right)$$ such that \begin{enumerate}
\item $\left(F\otimes F \right) \Delta=\Delta F $ and $F^1\delta=\delta F^0$, and
\item if $P_{\infty}=C_{\infty}$ $F \left(NTS\left(V[1] \right) \cap \left( \left( T^{c}\left(V[1] \right)\right)^{0}\oplus \left( T^{c}\left(V[1] \right)\right)^{0} \right) \right) =0$,
\end{enumerate}
modulo the following equivalence relation: $F,G$ are equivalent if the kernel of $F^{1}-G^{1}$ contains $ \delta\left( \left( T^{c}\left(V[1] \right)\right)^{0}\right)$.
\item Maurer-Cartan elements $\alpha\in \operatorname{Conv}_{1-P_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right)$ such that $\alpha(1)=0$.
\end{enumerate}
Let $\left(V', m_{\bullet}^{V'}\right)$ be a positively graded $P_{\infty}$-algebra such that $\mathcal{F}\left(V', m_{\bullet}^{V'}\right)=\left(V, m_{\bullet}^{V}\right)$. Then the above elements are in one to one correspondence with $1$-Maurer-Cartan elements\\ $\alpha\in \operatorname{Conv}_{P_{\infty}}\left(\left(V', m_{\bullet}^{V'}\right) ,\left(A, m_{\bullet}^{A} \right) \right)$ such that $\alpha(1)=0$.
\end{prop}
\begin{proof}
Let $F=F^{0}\oplus F^{1}$ be as in point 1. Then $F^{0}$ is a tensor coalgebra map and it is completely determined by an $\alpha \in \operatorname{Hom}^{1}\left( \left(T^{c}\left(V[1] \right)\right)^{0},A \right)= \operatorname{Hom}^{1} \left(T^{c}\left(V^{0}[1] \right),A \right) $. The condition $\delta^{A}F^{0}=F^{1}\delta^{V}$ correspond to the fact that $\alpha$ is a Maurer-Cartan element in the degree zero convolution algebra. Let $\alpha$ be a Maurer-Cartan element in the degree zero convolution algebra. In particular, $\alpha \in \operatorname{Hom}^{1} \left(T^{c}\left(V^{0}[1] \right),A \right) $ corresponds to a tensor coalgebra map $F^{0}\: : \: T^{c}\left(V^{0}[1] \right)\to T^{c}\left(A^{0}[1] \right)$ and the Maurer-Cartan equation implies that there exists a $F^{1}\: : \: \left( T^{c}\left(V[1] \right)\right)^{1}\to \left( T^{c}\left(A[1] \right)\right)^{1}$ such that $\delta^{A}F^{0}=F^{1}\delta^{V}$. This proves the equivalence between $1$ and $2$. The last step follows in a similar way.
\end{proof}
\begin{Warn}
For any Maurer-Cartan $\alpha$ in a convolution algebra (reduced, degre zero,.. ), we will assume $\alpha(1)=0$.
\end{Warn}
\subsection{Functoriality of convolution $L_{\infty}$-algebras}
Let $\left(V, m_{\bullet}^{V}\right)$ and $\left(W, m_{\bullet}^{W}\right)$ be positively graded $1-P_{\infty}$-algebras and let $\left(A, m_{\bullet}^{A} \right) $ be a $P_{\infty}$-algebra. A $1$-$P_{\infty}$-map $f_{\bullet}\: : \: \left(W, m_{\bullet}^{W}\right)\to \left(V, m_{\bullet}^{V} \right)$ corresponds via Proposition \ref{Cinftydictionary1} to certain morphism $F=F^{0}\oplus F^{1}$. In the same way, a Maurer-Cartan element $\alpha$ in $\operatorname{Conv}_{1-P_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right)$ can be written as ${F'}={F'}^{1}\oplus {F'}^{2}$. We denote with $f^{*}(\alpha)$ the Maurer-Cartan element that corresponds to the $1$-morphism $F{F'}={F}^{1}{F'}^{1}\oplus{F}^{2} {F'}^{2}$, this gives a well-defined map
$$f^{*}\: : \: \operatorname{Conv}_{1-P_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right)\to \operatorname{Conv}_{1-P_{\infty}}\left(\left(W, m_{\bullet}^{W}\right) ,\left(A, m_{\bullet}^{A} \right) \right).$$
Let $\left(V', m_{\bullet}^{V'}\right)$ and $\left(W', m_{\bullet}^{W'}\right)$ be $P_{\infty}$-algebras and let $g_{\bullet}\: : \: \left(V', m_{\bullet}^{V'}\right)\to\left(W', m_{\bullet}^{W'}\right) $ be a $P_{\infty}$-morphism. In the same way we have a map
$$g^{*}\: : \: \operatorname{Conv}_{P_{\infty}}\left(\left(V', m_{\bullet}^{V'}\right) ,\left(A, m_{\bullet}^{A} \right) \right)\to \operatorname{Conv}_{P_{\infty}}\left(\left(W', m_{\bullet}^{W'}\right) ,\left(A, m_{\bullet}^{A} \right) \right).$$
Assume that $V'$ is positively graded.
Consider $\pi:={p} \circ r$, where
\begin{equation}\label{restrquot}
r\: : \: \Hom^{\bullet}\left(T^{c}(V'[1]), A \right)\to \Hom^{\bullet}\left(T^{c}(V'[1]^{0}), A \right)
\end{equation}
is the restriction map and ${p}\: : \: \Hom^{\bullet}\left(T^{c}(V'[1]^{0}), A \right)\to \Hom^{\bullet}\left(T^{c}(V'[1]^{0}), A \right)/{\operatorname{Im}(\delta^{*})}$ is the quotient map. We have a well defined map
\begin{eqnarray}\label{notcompl}\pi\: : \: \operatorname{Conv}_{P_{\infty}}\left(\left(V', m_{\bullet}^{V'}\right) ,\left(A, m_{\bullet}^{A} \right) \right)\to \operatorname{Conv}_{1-P_{\infty}}\left(\left(V', m_{\bullet}^{V'}\right) ,\left(A, m_{\bullet}^{A} \right) \right).
\end{eqnarray}
\begin{prop}\label{pushforward}
Let $g_{\bullet}, f_{\bullet}$ as above and let $\left(A, m_{\bullet}^{A} \right) $ be a $P_{\infty}$-algebra. Assume that $V$ is positively graded.
\begin{enumerate}
\item The map
$$f^{*}\: : \: \operatorname{Conv}_{1-P_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right) \to\operatorname{Conv}_{1-P_{\infty}}\left(\left(W, m_{\bullet}^{W}\right) ,\left(A, m_{\bullet}^{A} \right) \right) $$
is a strict morphism of $L_{\infty}$-algebras.
\item The map
$$g^{*}\: : \: \operatorname{Conv}_{P_{\infty}}\left(\left(V', m_{\bullet}^{V'}\right) ,\left(A, m_{\bullet}^{A} \right) \right)\to \operatorname{Conv}_{P_{\infty}}\left(\left(W', m_{\bullet}^{W'}\right) ,\left(A, m_{\bullet}^{A} \right) \right) $$
is a strict morphism of $L_{\infty}$-algebras.
\item Assume that $\left(V', m_{\bullet}^{V'}\right)$, $\left(W', m_{\bullet}^{W'}\right)$ are positively graded $P_{\infty}$-algebras such that\\ $\mathcal{F}\left(V', m_{\bullet}^{V'}\right)=\left(V, m_{\bullet}^{V}\right)$ and $\mathcal{F}\left(W', m_{\bullet}^{W'}\right)=\left(W, m_{\bullet}^{W}\right)$. $f^{*}$ is a well defined map between set of $1$-Maurer-Cartan elements.
\item Assume that $\left(V', m_{\bullet}^{V'}\right)$, $\left(W', m_{\bullet}^{W'}\right)$ as in point 3. $\pi$ is a strict morphism of $L_{\infty}$-algebras. In particular, it sends $1$-Maurer-Cartan elements to Maurer-Cartan elements and $f^{*}\pi= \pi f^{*}$.
\end{enumerate}
\end{prop}
\begin{proof}
Direct verification.
\end{proof}
Let $\left(B, m_{\bullet}^{B} \right) $ be a $P_{\infty}$-algebra. For a $P_{\infty}$-map $h_{\bullet}\: : \: \left(A, m_{\bullet}^{A} \right)\to \left(B, m_{\bullet}^{B} \right)$ we have a well-defined map
$$h^{*}\: : \: \operatorname{Conv}_{1-P_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right)\to \operatorname{Conv}_{1-P_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(B, m_{\bullet}^{B} \right) \right)$$
defined as follows. Let $\alpha\in \operatorname{Conv}_{1-P_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right)$ and let $H$ and $F$ be the coalgebras morphism corresponding to $h_{\bullet}$ and $\alpha$ respectively. We consider the composition $F H$. It corresponds to a Maurer-Cartan element $h_{*}(\alpha)\in \operatorname{Conv}_{1-P_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(B, m_{\bullet}^{B} \right) \right)$ given by
\[
h_{*}(\alpha)|_{V[1]^{\otimes n}}=\sum _{l=1}^{n}\pm h_{l}\left(\sum_{i_{1}+\dots i_{l}=n} \alpha|_{V[1]^{\otimes i_{1}}}\otimes \cdots \otimes \alpha|_{V[1]^{\otimes i_{l}}} \right)
\]
where the signs are a consequence of the Koszul convention. In particular, if $h_{\bullet}$
is strict we have $h_{*}(\alpha)=h \alpha$.
\begin{prop}\label{pullback}\label{reductiontoquotient morphism}
Let $h_{\bullet}$ be as above.
\begin{enumerate}
\item The map
\[
h_{*}\: : \:MC\left( \operatorname{Conv}_{1-P_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right)\right) \to MC\left( \operatorname{Conv}_{1-P_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(B, m_{\bullet}^{B} \right) \right) \right)
\]
is well-defined.
\item The map
\[
h_{*}\: : \:MC\left( \operatorname{Conv}_{P_{\infty}}\left(\left(V', m_{\bullet}^{V'}\right) ,\left(A, m_{\bullet}^{A} \right) \right)\right) \to MC \left( \operatorname{Conv}_{P_{\infty}}\left(\left(V', m_{\bullet}^{V'}\right) ,\left(B, m_{\bullet}^{B} \right) \right) \right).
\]
\item Let $\overline{h}_{\bullet}\: : \: \mathcal{F}\left(A, m_{\bullet}^{A} \right)\to\mathcal{F} \left(B, m_{\bullet}^{B} \right)$ be a $1-P_{\infty}$-morphism. Then $\overline{h}_{*}$ is well-defined and the above statements are true as well if we replace ${h}_{*}$ by $\overline{h}_{*}$.
\item Assume that $\left(V', m_{\bullet}^{V'}\right)$ is a positively graded $P_{\infty}$-algebras such that $\mathcal{F}\left(V', m_{\bullet}^{V'}\right)=\left(V, m_{\bullet}^{V}\right)$. The map $h_{*}$ preserves $1$-Maurer-Cartan elements and $h_{*}\pi= \pi h_{*}$.
\end{enumerate}
\end{prop}
\begin{proof}
Direct verification.
\end{proof}
The propositions above and corollaries can be summarized as follows.
\begin{thm}\label{functorLinfty}
We have functors
\begin{align*}
&\operatorname{Conv}_{P_{\infty}}\: : \: \left( {P}_{\infty}-\operatorname{ALG}\right)^{op}\times {P}_{\infty}-\operatorname{ALG}\to \left( \mathcal{L}_{\infty}-\operatorname{ALG}\right)_{p},\\
&\operatorname{Conv}_{1-P_{\infty}}\: : \: \left( 1-P_{\infty}-\operatorname{ALG}_{>0}\right)^{op}\times {P}_{\infty}-\operatorname{ALG}\to \left( \mathcal{L}_{\infty}-\operatorname{ALG}\right)_{p}.
\end{align*}
\end{thm}
\subsection{Formal power series}
We want to express convolution $L_{\infty}$-algebras of the previous section in terms of formal power series, to do that we need to put some filtrations on $T(V)$.
We fix a finite type graded vector space $V$ which is bounded below at $k$. We define two filtrations.
Let $\epsilon\: : \: T(V)\to \Bbbk$ be the augmentation map, let $I:=\ker (\epsilon)$ be the augmentation ideal. The sequence given by the powers of $I$
\[
I^{0}:=T(V)\subset I\subset I^{2}\subset \dots
\]
is a filtration $I^{\bullet}$ on $T(V)$. We define the filtration $F^{\bullet}$ on $V$ as
$
F^{l}\left( V^{\bullet}\right):=\oplus_{j\geq k+l}V^{j}.
$
Let $\left\lbrace v^{l}_{i}\right\rbrace_{i\in J(l)} $ be a basis of $V^{l}$, hence $$\bigcup_{i\geq 0 } \bigcup_{j\in J(k+i) }v^{k+i}_{j}$$ is a basis of $V$. We denote such a basis by $\left\lbrace v_{i}\right\rbrace_{i\in J(l), l\geq 0}$. An element of $T(V)$ can be written in a unique way as a non commutative polynomial
\[
\displaystyle\sum_{p\geq 0}^{N} \displaystyle \sum_{(i_{1},\dots ,i_{p})}\lambda_{i_{1},\dots ,i_{p}} v_{i_{1}}\cdots v_{i_{p}},
\]
such that only finitely many $\lambda_{i_{1},\dots ,i_{p}} $ are different from $0$. The elements $f\in I^{i}\left( T(V)\right) \subset T(V)$ can be written in unique way as
\[
\displaystyle\sum_{p\geq i}^{N} \displaystyle \sum_{(i_{1},\dots ,i_{p})}\lambda_{i_{1},\dots ,i_{p}} v_{i_{1}}\cdots v_{i_{p}},
\]
such that only finitely many $\lambda_{i_{1},\dots ,i_{p}} $ are different from $0$. Let $A_{i}^{I}$ be the set of monomials $v_{i_{1}}\cdots v_{i_{p}}$
such that $p<i$, it forms a basis for $T(V)/I^{i}$. The tensor algebra $T\left(V/F^{j}\left(V \right)\right)$ may be considered as the graded algebra of non commutative polynomial $\Bbbk \left\langle \left\lbrace v_{i}\right\rbrace_{i\in J(l), l < j}\right\rangle $, where $\left\lbrace v_{i}\right\rbrace_{i\in J(l), l< j}$ is a basis of $V^{k}\oplus \dots V^{k+j-1}$. For each $j\geq 0$, let $G^{j}\left(T(V)\right)\subset T(V)$ be the subspace generated by
\[
\displaystyle\sum_{p\geq 0}^{N} \displaystyle \sum_{(i_{1},\dots ,i_{p})}\lambda_{i_{1},\dots ,i_{p}} v_{i_{1}}\cdots v_{i_{p}}\in \Bbbk \left\langle \left\lbrace v_{i}\right\rbrace_{i\in J(l), l\geq 0}\right\rangle,
\]
such that for each $v_{i_{1}}\cdots v_{i_{p}}$ there exists a $v_{i_{n}}\in V^{s}$, $s\geq j+k$. In particular
\[
G^{j}\left(T(V) \right)\oplus T\left(V/F^{j}\left(V \right)\right) =T(V).
\]
A basis $A^{G}_{j}$ of $T(V)/G^{j}\left(T(V) \right)= T\left(V/F^{j}\left(V \right)\right)$ is given by the monomials
$
v_{i_{1}}\cdots v_{i_{p}}
$
such that each $v_{i_{j}}\in V^{k}\oplus \dots V^{k+j-1}$. Let $$H_{i,j}:=\left( T(V)/I^{i}(T(V))\right)/G^{j}( T(V)),$$ we identify $H_{i,j}$ as the subspace of $T(V)$ generated by $A_{i,j}:=A^{G}_{j}\setminus A^{I}_{i}$. They form a diagram where all the maps are inclusion and two objects $H_{a,b}$, $H_{c,d}$ are connected by a map $H_{a,b}\hookrightarrow H_{c,d}$ if $a\leq b$ and $c\leq d$, in particular $\colim_{i,j} H_{i,j}=T(V)$. Let $W$ be a graded vector space equipped with the trivial filtration. We consider the tensor product of filtrations
\[
I^{i}(W\otimes T(V)):=W\otimes I^{i}(T(V)), \quad G^{j}(W\otimes T(V)):=W\otimes G^{j}(T(V)).
\]
Then $W\otimes T(V)\cong W\otimes \Bbbk \left\langle \left\lbrace v_{i}\right\rbrace_{i\in I(l), l\geq 0}\right\rangle$. In particular each element can be written
in unique way as
\begin{equation}\label{exampleform}
\displaystyle\sum_{p\geq 0}^{N} \sum_{q\geq 0}^{M}\sum_{v_{i_{1}}\cdots v_{i_{p}}\in A_{p,q}} w_{i_{1},\dots ,i_{p}}\otimes v_{i_{1}}\cdots v_{i_{p}}
\end{equation}
for some $N$, $M$. Let $W\widehat{\otimes}_{I} \Bbbk \left\langle \left\langle \left\lbrace v_{i}\right\rbrace_{i\in J(l), l\geq 0} \right\rangle \right\rangle$ be the completion of $W\otimes T(V)$ with respect to $I^{\bullet}$. It is the graded vector space of formal power series of the form $\eqref{exampleform}$ with $N=\infty$. On the other hand, let $W\widehat{\otimes}_{G} \Bbbk \left\langle \left\lbrace v_{i}\right\rbrace_{i\in J(l), l\geq 0}\right\rangle$ be the completion of $W\otimes T(V)$ with respect to $G^{\bullet}$, it is the formal graded vector space of formal power series $\eqref{exampleform}$ with $M=\infty$. We denote by $\mathcal{I}_{G}^{\bullet}$ the filtration obtained by the completion of $I^{\bullet}$ with respect to $G^{\bullet}$ on $W\widehat{\otimes}_{G} \Bbbk \left\langle \left\lbrace v_{i}\right\rbrace_{i\in J(l), l\geq 0}\right\rangle$. We denote by $\mathcal{G}_{I}^{\bullet}$ the filtration obtained by the completion of $G^{\bullet}$ with respect to $I^{\bullet}$ on $W\widehat{\otimes}_{I} \Bbbk \left\langle \left\lbrace v_{i}\right\rbrace_{i\in J(l), l\geq 0}\right\rangle$. The completion of $W\widehat{\otimes}_{G} \Bbbk \left\langle \left\lbrace v_{i}\right\rbrace_{i\in J(l), l\geq 0}\right\rangle$ with respect to $\mathcal{I}_{G}^{\bullet}$ coincides with the completion of $W\widehat{\otimes}_{I} \Bbbk \left\langle \left\lbrace v_{i}\right\rbrace_{i\in J(l), l\geq 0}\right\rangle$ with respect to $\mathcal{G}_{I}^{\bullet}$. We denote the obtained vector space by $ W\widehat{\otimes}\widehat{T}(V)$, it is the vector space of formal power series
\[
\displaystyle\sum_{p\geq l}^{\infty} \sum_{q\geq 0}^{\infty}\sum_{v_{i_{1}}\cdots v_{i_{p}}\in A_{p,q}} w_{i_{1},\dots ,i_{p}}\otimes v_{i_{1}}\cdots v_{i_{p}}
\]
We denote the induced filtrations on the completion by $\mathcal{I}$ and resp. $\mathcal{G}$. There exists a canonical isomorphism of complete graded vector spaces
\[
\Psi\: : \: \Hom\left(T^{c}\left(V[1] \right) , W \right) \to \widehat{T}\left(\left( V[1]\right)^{*} \right)\widehat{\otimes} W
\]
In particular $I^{\bullet}$ and $G^{\bullet}$ induce a filtration on the Lie algebra of primitive elements of $\mathbb{L}\left(\left( V[1]\right)^{*} \right)\subset T\left(\left( V[1]\right)^{*} \right)$ and $\Psi$ restricts to an isomorphism
\[
\Psi\: : \: L_{V[1]^{*}}\left(W \right)\to \widehat{\mathbb{L}}\left(\left( V[1]\right)^{*} \right)\widehat{\otimes} W.
\]
The filtration $\mathcal{I}$ on $\Hom^{\bullet}\left(T^{c}(V[1])^{\otimes n}, W \right)$ is given as follows: $\mathcal{I}^{i}$ is the graded vector subspace of morphisms in $\Hom^{\bullet}\left(T^{c}(V[1])^{\otimes n}, W \right)$ such that $f|_{I^{i}}\cong 0$. For each $n>0$, we consider $\Hom^{\bullet}\left(T^{c}(V[1]), A \right)^{\otimes n}$ equipped with the tensor product filtration $\mathcal{I}^{\otimes n}$.
Let $(A, m_{\bullet}^{A})$ and $(V',m_{\bullet}^{V'})$ be $P_{\infty}$-algebras. If the dual $\Hom\left( \delta, A\right) \: : \: \Hom^{\bullet}\left(T^{c}(V'[1]), A \right)\to \Hom^{\bullet}\left(T^{c}(V'[1]), A \right)$ preserves the filtration $\mathcal{I}$ then $$\left( \operatorname{Conv}_{P_{\infty}}\left(\left(V', m_{\bullet}^{V'}\right) ,\left(A, m_{\bullet}^{A} \right) \right), l_{\bullet}\right) $$ is a complete filtered $L_{\infty}$-algebra with respect to $\mathcal{I}^{\bullet}$ and the maps in \eqref{notcompl} are strict morphisms between complete filtered $L_{\infty}$-algebras.
For $V'$ of finite type, we denote the complete ideal generated by $\delta^{*}\: : \:\left( \widehat{T}\left( \left(V'[1]\right)^{*} \right)\right)^{1} \to\widehat{T}\left( \left(V'[1]^{0} \right)^{*} \right)$ by $\overline{\mathcal{R}}_{0}\subset \widehat{T}\left( \left(V'[1]^{0} \right)^{*} \right)$ and if $m_{\bullet}^{V'}$ is $C_{\infty}$, we denote by ${\mathcal{R}}_{0}\subset \widehat{\mathbb{L}}\left( \left(V'[1]^{0} \right)^{*} \right)$ the complete Lie ideal generated by $\delta^{*}$. Let $p,r$ and $\pi$ be as above. The next corollary is a standard exercise about filtrations.
\begin{cor}\label{maurercartandego0finitetype}Let $\left(A, m_{\bullet}^{A} \right) $ be a bounded below $A_{\infty}$-algebra and let $V$ be of finite type. The isomorphism $\Psi$ induces the following isomorphism between complete graded vector spaces.
\begin{enumerate}
\item If $\left(V', m_{\bullet}^{V'}\right) $ is a $A_{\infty}$-algebra$$
\operatorname{Conv}_{A_{\infty}}\left(\left(V', m_{\bullet}^{V'}\right) ,\left(A, m_{\bullet}^{A} \right) \right)\cong A\widehat{\otimes }\left( \widehat{T}(\left( V[1]\right)^{*}) \right).$$
\item If $\left(V', m_{\bullet}^{V'}\right) ,\left(A, m_{\bullet}^{A} \right) $ are $C_{\infty}$ we have $
\operatorname{Conv}_{C_{\infty}}\left(\left(V', m_{\bullet}^{V'}\right) ,\left(A, m_{\bullet}^{A} \right) \right)\cong A\widehat{\otimes }\left( \widehat{\mathbb{L}}(\left( V'[1]\right)^{*}) \right)$.
\item Assume that $\left( A, m_{\bullet}^{A}\right) $ is unital and $\left( V, m_{\bullet}^{V}\right) $ is $1-A_{\infty}$. Then
\[
\operatorname{Conv}_{1-A_{\infty}}\left(\left(V', m_{\bullet}^{V'}\right) ,\left(A, m_{\bullet}^{A} \right) \right)\cong A\widehat{\otimes }\left( \widehat{T}(\left( V[1]^{0}\right)^{*})/\overline{\mathcal{R}}_{0}\right) .
\]
\item Assume that $\left( A, m_{\bullet}^{A}\right) $ is unital and $C_{\infty}$ and $\left( V, m_{\bullet}^{V}\right) $ is $1-C_{\infty}$. Then
\[
\operatorname{Conv}_{1-C_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right) \cong A\widehat{\otimes }\left( \widehat{\mathbb{L}}(\left( V[1]^{0}\right)^{*})/{\mathcal{R}}_{0}\right).
\]
Moreover if $\delta^{*}$ preserves $\mathcal{I}^{\bullet}$, it is a morphism of complete $L_{\infty}$-algebras.
\item Assume that $\left(V', m_{\bullet}^{V'}\right)$ is a positively graded $P_{\infty}$-algebras such that $\mathcal{F}\left(V', m_{\bullet}^{V'}\right)=\left(V, m_{\bullet}^{V}\right)$. Let $C\in \operatorname{Conv}_{P_{\infty}}\left(\left(V', m_{\bullet}^{V'}\right) ,\left(A, m_{\bullet}^{A} \right) \right)$ be a Maurer-Cartan element. Then $\pi(C)$ is a Maurer-Cartan element in the $L_{\infty}$-algebra \\ $\operatorname{Conv}_{1-P_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right)$
\item Assume that $\left(V', m_{\bullet}^{V'}\right)$ is a positively graded $P_{\infty}$-algebras such that $\mathcal{F}\left(V', m_{\bullet}^{V'}\right)=\left(V, m_{\bullet}^{V}\right)$. Let $C^{1},C^{2}\in \operatorname{Conv}_{P_{\infty}}\left(\left(V', m_{\bullet}^{V'}\right) ,\left(A, m_{\bullet}^{A} \right) \right)$ be two $1$-Maurer-Cartan elements where $V'$ is positively graded. Let $f^{1}_{\bullet}, f^{2}_{\bullet}\: : \: \left(V', m_{\bullet}^{V'}\right)\to\left(A, m_{\bullet}^{A} \right) $ be the two corresponding $P_{\infty}$-maps and assume that they are $1$-homotopic. Then $\pi(C^{1}),\pi(C^{2}) $ are homotopic Maurer-Cartan elements in the $L_{\infty}$-algebra $\operatorname{Conv}_{1-P_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right)$.
\end{enumerate}
\end{cor}
Let $\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) $ be $P_{\infty}$-algebras, both assumed to be bounded below and that $V$ is of finite type. Then $m_{\bullet}^{V}$ corresponds to a codifferential $\delta$ on ${T}^{c}\left(V[1]\right)$.
\begin{defi}\label{defhomologicalpair}
Let $\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) $ be $P_{\infty}$-algebras. Let $C$ be a Maurer-Cartan element in $\operatorname{Conv}_{P_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right)$. We call $(C, \delta^{*})$ a \emph{homological pair}.
\end{defi}
Let $\left(V', m_{\bullet}^{V'}\right)$ be a positively graded $P_{\infty}$-algebra of finite type and let $\left(A, m_{\bullet}^{A} \right) $ be a unital $P_{\infty}$-algebra and let $\alpha^{0}, \alpha^{1}\in \operatorname{Conv}_{P_{\infty}}\left(\left(V', m_{\bullet}^{V'}\right) ,\left(A, m_{\bullet}^{A} \right) \right)$ be Maurer-Cartan elements. By Proposition \ref{Cinftydictionary}, there are two $P_{\infty}$-morphisms $f^{0}_{\bullet}, f^{1}_{\bullet}\: : \: \left(V', m_{\bullet}^{V'}\right)\to\left(A, m_{\bullet}^{A} \right) $ associated to them.
\begin{prop}\label{propequivhomot}
Let $\left(V', m_{\bullet}^{V'}\right)$ be a positively graded and of finite type $P_{\infty}$-algebra and let $\left(A, m_{\bullet}^{A} \right) $ be a unital $P_{\infty}$-algebra.
\begin{enumerate}
\item Let $\alpha^{0}, \alpha^{1}\in \operatorname{Conv}_{P_{\infty}}\left(\left(V', m_{\bullet}^{V'}\right) ,\left(A, m_{\bullet}^{A} \right) \right)$ be Maurer-Cartan elements.\\ Let $f^{0}_{\bullet}, f^{1}_{\bullet}\: : \: \left(V, m_{\bullet}^{V}\right)\to\left(A, m_{\bullet}^{A} \right) $ be the two corresponding $P_{\infty}$-map. Then if they are homotopic, so are $\alpha^{0}, \alpha^{1}$. Moreover the pullback and pushforward along homotopic maps gives homotopic Maurer-Cartan elements.
\item Let $\left(V,m_{\bullet}^{V} \right) $ be a $1-P_{\infty}$-algebra and let $\alpha^{0}, \alpha^{1}\in \operatorname{Conv}_{1-P_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right)$ be Maurer-Cartan elements. Let $f^{0}_{\bullet}, f^{1}_{\bullet}\: : \: \left(V, m_{\bullet}^{V}\right)\to\left(A, m_{\bullet}^{A} \right) $ be the two corresponding $P_{\infty}$-map viewed as $1$-morphism. Then if they are $1$-homotopic, it follows that $\alpha^{0}$ and $\alpha^{1}$ hare homotopic.
Moreover the pullback along $1$-homotopic maps and the pushforward along homotopic maps gives homotopic Maurer-Cartan elements.
\end{enumerate}
\end{prop}
\begin{proof}
We prove the first assertion. Let $H_{\bullet}$ be a homotopy between the two maps. We apply \eqref{Cinftydictionary} to $H_{\bullet}$. This gives a Maurer-Cartan element $$\alpha(t)\in \operatorname{Conv}_{A_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(\Omega(1)\otimes A, m_{\bullet}^{\Omega(1)\otimes A} \right)\right)\cong \Omega(1)\widehat{\otimes}\operatorname{Conv}_{A_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right),$$
where the last isomorphism is given by $\Psi$. On the left hand side we have the desired homotopy between $\alpha^{0}$ and $\alpha^{1}$ the sense of definition \eqref{Linftydef}. The other assertion follows similarly by using the map $\Psi$ as well and point 2 of Lemma \ref{tensorinfinity}.
\end{proof}
\subsection{ $1$-models and $1$-minimal models}\label{minimal model}
\begin{defi}
Let $\left(A,m^{A}_{\bullet}\right) $ be a non-negatively graded (unital) $P_{\infty}$-algebra. A $P_{\infty}$-sub algebra $\left(B,m^{B}_{\bullet}\right) $ is a $P_{\infty}$-algebra such that the inclusion is a strict morphism $i\: : \: B\hookrightarrow A$ of $P_{\infty}$-algebras. Hence $m_{\bullet}^{A}=m_{\bullet}^{B}$. Let $1\leq j\leq \infty$. A (unital) $P_{\infty}$-sub algebra $B$ is a $j$-model for $\left(A,m^{A}_{\bullet}\right) $ if
\begin{enumerate}
\item $i$ induces an isomorphism up to the $j$-th cohomology group and is injective on the $j+1$ cohomology group.
\item the inclusion $i^{l}\: : \: B^{l}\hookrightarrow A^{l}$ preserves non-exact elements for $0\leq l\leq j+1$.
\end{enumerate}
If $j=\infty$, we call $B$ a\emph{ model} for $\left(A,m^{A}_{\bullet}\right) $. A (unital) $P_{\infty}$-algebra $(W,m_{\bullet}^{W})$ is said to be minimal if $m_{1}^{W}=0$. A \emph{$j$-minimal model} of $\left(A,m^{A}_{\bullet}\right) $ consists in an (unital) morphism $g_{\bullet}\: : \: (W,m_{\bullet}^{W})\to \left(A,m^{A}_{\bullet}\right) $ such that $(W,m_{\bullet}^{W})$ is minimal and $g_{\bullet}$ induces an isomorphism up to the $j$-th cohomology group and is injective on the $j+1$ cohomology group. If $j=\infty$ we call $g_{\bullet}$ a \emph{minimal model}.
\end{defi}
\begin{defi}
Let $\left(A,m^{A}_{\bullet}\right) $ be a non-negatively graded (unital) $1-P_{\infty}$-algebra. A $1-P_{\infty}$-sub algebra $\left(B,m^{B}_{\bullet}\right) $ is a $1-P_{\infty}$-algebra such that the inclusion is a strict morphism $i\: : \: B\hookrightarrow A$ of $P_{\infty}$-algebras. A (unital) $P_{\infty}$-sub algebras $B$ is a $1$-model for $\left(A,m^{A}_{\bullet}\right) $ if $i$ is a quasi-isomorphism and the inclusion $i^{l}\: : \: B^{l}\hookrightarrow A^{l}$ preserves non-exact elements for $0\leq l\leq 2$. A (unital) non-negatively graded $1-P_{\infty}$-algebra $(V,m_{\bullet}^{V})$ is said to be minimal if $m_{1}^{V}=0$ and if $V^{2}$ is the vector space generated by $m_{n}(a_{1}, \dots, a_{n})$ where $a_{i}\in V^{1}$. Let $B$ be a positively graded $1-P_{\infty}$-algebra. A \emph{$1$-minimal model with coefficients in $B$} consists in an (unital) $1$-quasi-isomorphism $g_{\bullet}\: : \: (V,m_{\bullet}^{V})\to \mathcal{F}\left(B,m^{A}_{\bullet}\right) $ such that $(V,m_{\bullet}^{V})$ is minimal.
\end{defi}
Each $1$-minimal model for $P_{\infty}$-algebras $g_{\bullet}\: : \: \left(V', m_{\bullet}^{V'} \right) \to \left(A, m_{\bullet} \right) $ gives a $1$-minimal model $\mathcal{F}\left( {g'}_{\bullet}\right) $ for $\mathcal{F}\left(A, m_{\bullet} \right) $ via
\[
\begin{tikzcd}
{g'}_{\bullet}\: :\:\left(V, m_{\bullet}^{V'} \right) \arrow[r, hook] & \left(V', m_{\bullet}^{V'} \right)\arrow{r}{{g}_{\bullet}} & \left(A, m_{\bullet} \right)
\end{tikzcd}
\]
where $\left(V, m_{\bullet}^{V'} \right)\subset \left(V', m_{\bullet}^{V'} \right) $ is the $P_{\infty}$-subalgebra generated by $V^{0}$ and $V^{1}$.
\begin{rmk}
The minimal models and $1$-minimal models can be constructed explicit via the homotopy transfer theorem (see \cite{Kadesh}, \cite{kontsoibel},\cite{Markl} and \cite{Prelie}).
\end{rmk}
\begin{defi}
A non-negatively graded $P_{\infty}$-algebra (resp. $1-P_{\infty}$-algebra) has connected cohomology if its cohomology in degree zero has dimension $1$.
\end{defi}
Let $W$ be a non-negatively graded vector space. Let $m_{\bullet}^{W}$ be a $P_{\infty}$-structure on $W^{\bullet}$. For $n>1$, let $m_{n}^{W_{+}}$ be the maps given by the composition
\[
\begin{tikzcd}
W_{+}^{\otimes n}\arrow[hook]{r}& W^{\otimes n}\arrow{r}{m_{n}^{W}} & W_{+}.
\end{tikzcd}
\]
They are well-defined since $m_{n}^{W}$ are maps of degree $2-n$ and $(W_{+}^{\bullet}, m_{\bullet}^{W_{+}})$ is a $P_{\infty}$-algebra. Assume that $m_{1}^{W}=0$. Let $g_{\bullet}\: : \: (W, m^{W})\to (A, m^{A})$ a morphism of $P_{\infty}$-algebras. We denote by $g^{+}_{\bullet}$ the restriction of $g_{\bullet}$ to $\left(W^{\bullet}_{+}, m_{\bullet}^{W_{+}}\right)$. Then $g_{\bullet}^{+}\: : \: (W_{+}, m^{W_{+}})\to (A, m^{A})$ is a well-defined morphism of $P_{\infty}$-algebras. Assume that $(W, m^{W})$ is connected and that $ (A, m^{A})$ is unital. The map $g_{\bullet}\mapsto g_{\bullet}^{+}$ is a bijection.
Let $\left(A,m_{\bullet}\right) $ be a non-negatively graded unital $P_{\infty}$-algebra with connected cohomology. Let $g_{\bullet}^{1}\: : \:(W_{1}, m_{\bullet}^{1})\to \left(A,m_{\bullet}\right) $ be a $P_{\infty}$-algebra minimal model. By Proposition \ref{existence of a homotopical inverse} we get an inverse up to homotopy
\[
\begin{tikzcd}
g^{1}_{\bullet}\: : \: ({W_{1}}, m_{\bullet}^{1})\arrow[r, shift right, ""]&(A,m_{\bullet})\: : \: f^{1}_{\bullet} \arrow[l, shift right, ""].
\end{tikzcd}
\]
\begin{prop}\label{homotopyuniqueness}Let $g_{\bullet}^{1}$ be as above.\begin{enumerate}
\item Let $g_{\bullet}^{2}\: : \:(W_{2}, m_{\bullet}^{2})\to \left(A,m_{\bullet}\right) $ be $P_{\infty}$-algebra-morphism between non-negatively graded unital objects. Consider the diagram
\[
\begin{tikzcd}
({W_{1}}, m_{\bullet}^{1})\arrow[rrr, shift right, " g^{1}_{\bullet}"']&&&(A,m_{\bullet})
\arrow[lll, shift right, " f^{1}_{\bullet} "'] \\ \\ \\
({W_{2}}, m_{\bullet}^{2})\arrow[uuu, shift right, "k_{\bullet}"]\arrow[uuurrr, shift right, "g^{2}_{\bullet}"']
\end{tikzcd}
\]
where $k_{\bullet}:=f^{1}_{\bullet}g^{2}_{\bullet}$. For $j=1,2$ let $\alpha_{j}\in \operatorname{Conv}_{r}\left(\left( \left(W_{j}\right) _{+}, m_{\bullet}^{j}\right) ,\left(A, m_{\bullet}^{A} \right) \right)$ be the Maurer-Cartan elements corresponding to $g_{\bullet}^{j}$ via Proposition \ref{Cinftydictionary} in the reduced convolution algebra. Then
$\alpha_{2}$ is homotopic to $k^{*}\left( \alpha_{1}\right)$. Moreover, $k_{\bullet}$ is an isomorphism if $g_{\bullet}^{2}$ is a $1$-minimal model.
\item Let $g_{\bullet}^{2}\: : \:(W_{2}, m_{\bullet}^{2})\to \mathcal{F}\left(A,m_{\bullet}\right) $ be $1-P_{\infty}$-algebra-morphism between non-negatively graded unital objects. Consider the diagram
\[
\begin{tikzcd}
\mathcal{F}({W_{1}}, m_{\bullet}^{1})\arrow[rrr, shift right, "\mathcal{F}\left( g^{1}_{\bullet}\right) "']&&&\mathcal{F}(A,m_{\bullet})
\arrow[lll, shift right, " \mathcal{F}\left( f^{1}_{\bullet} \right) "'] \\ \\ \\
({W_{2}}, m_{\bullet}^{2})\arrow[uuu, shift right, "k_{\bullet}"]\arrow[uuurrr, shift right, "g^{2}_{\bullet}"']
\end{tikzcd}
\]
where $k_{\bullet}:=\mathcal{F}\left( f^{1}_{\bullet} \right)g^{2}_{\bullet}$. For $j=1,2$, let $\overline{\alpha}_{j}$ be the Maurer-Cartan elements corresponding to $g_{\bullet}^{j}$ via Proposition \ref{Cinftydictionary1} in the degree zero reduced convolution algebra. Then
$\overline{\alpha}_{2}$ is homotopic to $k^{*}\left( \overline{\alpha}_{1}\right)$. Moreover, $k_{\bullet}$ is a $1$-isomorphism if $g_{\bullet}^{2}$ is a $1$-minimal model.
\item Consider the situation at point 2 and assume that $g_{\bullet}^{2}$ is a $1$-minimal model. The $P_{\infty}$-map
\[
\mathcal{E}(k_{\bullet})\: : \: \mathcal{E}({W_{2}}, m_{\bullet}^{2})\to \mathcal{E}\mathcal{F}({W_{1}}, m_{\bullet}^{1})
\]
has a left-inverse ${k'}_{\bullet}$.
\end{enumerate}
In particular, $1$-minimal model for $1-P_{\infty}$-algebras are unique modulo $1$-isomorphisms.
\end{prop}
\begin{proof}
By Proposition \ref{existence of a homotopical inverse}, there exists a homotopy $H_{\bullet}$ between $\left( g^{1}\right)_{\bullet} \left( f^{1}\right)_{\bullet} $ and $Id_{A}$. Then $$g^{1}_{\bullet} k_{\bullet}=g^{1}_{\bullet} f^{1}_{\bullet}g^{2}_{\bullet}$$
is homotopic to $g^{2}_{\bullet}$ via $ H_{\bullet}g^{2}_{\bullet}$. We prove the second part. If $g_{\bullet}^{2}$ is a quasi-isomorphism then $k_{1}=f^{1}g^{2}\: : \: W_{2}\to W_{1}$ is a quasi-isomorphism. Since $m_{1}^{j}=0$ for $j=1,2$, it is an isomorphism. Part 2. follows analogously. We prove part 3. By construction we have $\left( \mathcal{E}(k)\right)_{1}\: : \: W^{i}\to W^{i}$ is an isomorphism for $i\neq 2$ and is injective for $i=2$. Then, there exists a left-inverse ${k'}\: : \: W\to W$. By using Theorem 10.4.2 in \cite{lodayVallette}, there exists an explicit $P_{\infty}$-map ${k'}_{\bullet}$ which extends ${k'}$.
\end{proof}
Consider the situation at point 2 in Proposition \ref{homotopyuniqueness}. We assume that $W_{i}$ are of finite type for $i=1,2$ and that $(W_{2},m_{\bullet}^{2})$ is minimal. By Proposition \ref{Cinftydictionary1}, $k_{\bullet}$ corresponds to a tensor coalgebra map $K=K^{0}\oplus K^{1}$ where
$$K\: : \: \left( T^{c}\left(\left( W_{2}\right)_{+}[1] \right)\right)^{0}\oplus \left( \left( T^{c}\left(\left( W_{2}\right)_{+}[1] \right)\right)^{1}\right) \to \left( T^{c}\left(\left( W_{1}\right)_{+}[1] \right)\right)^{0}\oplus \left( T^{c}\left(\left( W_{1}\right)_{+}[1] \right)\right)^{1}$$ such that
$\left(K\otimes K \right) \Delta=\Delta K $ and $K^0\delta^{1}=\delta^{1} K^1$. Its dual on degree-zero elements gives a map
$$
\left( K^{0}\right)^{*}\: : \:\widehat{T} \left(\left( W_{1}^{1}\right) _{+}[1]\right) ^{*}\to \widehat{T} \left(\left( W_{2}^{1}\right) _{+}[1]\right)^{*}
$$
which is a morphism of complete tensor algebras. These tensor algebras are in fact Hopf algebras, where the comultiplication is given by the shuffles coproduct.
We now assume that $k_{\bullet}$ is a $1-C_{\infty}$-morphism between $1-C_{\infty}$-algebras. It follows that $K$ vanishes on non-trivial shuffles, $K^{*}$ restricts to a morphism between complete free lie algebras
\begin{equation}
\left( K^{0}\right)^{*}\: : \:\widehat{\mathbb{L}} \left(\left( W_{1}^{1}\right) _{+}[1]\right) ^{*}\to \widehat{\mathbb{L}} \left(\left( W_{2}^{1}\right) _{+}[1]\right)^{*},
\end{equation}
and $\left( \delta^{i}\right)^{*}\: : \: \left( \widehat{T} \left(\left( W_{1}\right) _{+}[1]\right) ^{*}\right)^{-1}\to \widehat{T} \left(\left( W_{i}^{1}\right) _{+}[1]\right)^{*}$
restricts to a well-defined Lie algebra map $\left( \delta^{i}\right)^{*}\: : \: \left( \widehat{\mathbb{L}} \left(\left( W_{1}\right) _{+}[1]\right) ^{*}\right)^{-1}\to \widehat{\mathbb{L}} \left(\left( W_{i}^{1}\right) _{+}[1]\right)^{*}$ for $i=1,2$. We denote by ${\mathcal{R}}_{0}^{i}\subset \widehat{\mathbb{L}} \left(\left( W_{i}^{1}\right) _{+}[1]\right) ^{*}$ the complete Lie ideal in the complete tensor algebra generated by the image of $\left( \delta^{i}\right)^{*}$ for $i=1,2$. Let $\bar{\mathcal{R}}_{0}^{i}\subset \widehat{T} \left(\left( W_{i}^{1}\right) _{+}[1]\right) ^{*}$ be the ideals in the complete tensor algebra generated by ${\mathcal{R}}_{0}^{i}$ for $i=1,2$. Since $\delta^{i}$ vanishes on non-trivial shuffles we get that $\bar{\mathcal{R}}_{0}^{i}$ is a Hopf ideal for $i=1,2$ as well. Hence $
\left( K^{0}\right)^{*}\: : \:\widehat{T} \left(\left( W_{1}^{1}\right) _{+}[1]\right) ^{*}/\bar{\mathcal{R}}_{0}^{1}\to \widehat{T} \left(\left( W_{2}^{1}\right) _{+}[1]\right)^{*}/\bar{\mathcal{R}}_{0}^{2}
$ is a morphism of complete Hopf algebras and it restricts to a Lie algebra morphism on the primitive elements. By abuse of notation we denote its restriction by
\begin{equation}\label{mapKdual}
K^{*}:=\left( K^{0}\right) ^{*}\: : \:\widehat{\mathbb{L}} \left(\left( W_{1}^{1}\right) _{+}[1]\right) ^{*}/\mathcal{R}_{0}^{1}\to \widehat{\mathbb{L}} \left(\left( W_{2}^{1}\right) _{+}[1]\right)^{*}/\mathcal{R}_{0}^{2}.
\end{equation}
In particular, the pullback along $k_{\bullet}$
\[
k^{*}\: :\: \operatorname{Conv}_{1-P_{\infty}}\left(\left(V, m_{\bullet}^{V}\right) ,\left(A, m_{\bullet}^{A} \right) \right) \to\operatorname{Conv}_{1-P_{\infty}}\left(\left(W, m_{\bullet}^{W}\right) ,\left(A, m_{\bullet}^{A} \right) \right)
\]
corresponds under the identification of Corollary \ref{maurercartandego0finitetype} to
\[
\mathrm{Id}\widehat{\otimes }K^{*}\: : \: A\widehat{\otimes }\left( \widehat{\mathbb{L}} \left(\left( W_{1}^{1}\right) _{+}[1]\right) ^{*}/\mathcal{R}_{0}^{1} \right)\to A\widehat{\otimes }\left( \widehat{\mathbb{L}} \left(\left( W_{2}^{1}\right) _{+}[1]\right) ^{*}/\mathcal{R}_{0}^{2} \right)
\]
\begin{prop}\label{beh}
Consider the situation at point 2 in Proposition \ref{homotopyuniqueness} for $P_{\infty}=C_{\infty}$. Assume that $g_{\bullet}^{2}$ is a $1$-minimal model for $1-C_{\infty}$-algebras and that $W_{i}$ are of finite type for $i=1,2$. Then $
\left( K^{0}\right)^{*}\: : \:\widehat{T} \left(\left( W_{1}^{1}\right) _{+}[1]\right) ^{*}/\bar{\mathcal{R}}_{0}^{1}\to \widehat{T} \left(\left( W_{2}^{1}\right) _{+}[1]\right)^{*}/\bar{\mathcal{R}}_{0}^{2}
$
is an isomorphism of complete Hopf algebras. In particular its restriction on primitive elements $K^{*}$ is an isomorphism of complete Lie algebras. Moreover $K^{*}=\sum_{i=1}^{\infty}K_{i}^{*}$ where $K_{1}^{*}$ is induced by the dual of $k_{1}=f^{1}_{1}g^{2}_{2}\: : \: W_{2}\to W_{1}$.
\end{prop}
\begin{proof}
By Proposition \ref{homotopyuniqueness}, $K$ has a left inverse. Hence its dual has a right inverse which induces an isomorphism on the quotient.
\end{proof}
\section{ A $C_{\infty}$-structure on the total complex}\label{sectGetztler}
We give a very short introduction about the Dupont contraction and the results of \cite{Getz}. We introduce a $C_{\infty}$-structure that corresponds to the natural algebraic structure on the differential forms of a smooth complex simplicial manifold (see Theorem \eqref{functor}). This $C_{\infty}$-structure is in general hard to calculate. In the last we use a result of \cite{Getz} to present an almost complete formula on degree $1$-elements (see Theorems \ref{productthm>2} and \ref{productthm=2}). In this section we work on a field $\Bbbk$ of charactersitic zero.
\subsection{Cosimplicial commutative algebras}
We denote by $sSet$ the category of simplicial sets and by $\Delta\: : \:\boldsymbol{\Delta}\to sSet $ the Yoneda embedding.\\
For each $[n]\in \boldsymbol{\Delta}$ we define the $n$-gemetric simplex
\[
\Delta_{geo}[n]:=\left\lbrace(t_{0},t_{1}, \dots, t_{n})\in\R^{n}\: | \: t_{0}+t_{1}+\dots +t_{n}=1\right\rbrace.
\]
For each $[n]\in \boldsymbol{\Delta}, i=0,\dots, n+1$ we define the smooth maps $d^{i}\: : \: \Delta_{geo}[n]\to \Delta_{geo}[n+1]$
\[
d^{i}\left( t_{0},t_{1}, \dots, t_{n}\right)=\left( t_{0},t_{1}, \dots,t_{i-1},0,t_{i}, \dots t_{n}\right)
\]
and $s^{i}\: : \: \Delta_{geo}[n+1]\to\Delta_{geo}[n], i=0,\dots, n$ via
\[
s^{i}\left( t_{0},t_{1}, \dots, t_{n+1}\right)=\left( t_{0},t_{1}, \dots,t_{i-1},t_{i}+t_{i+1}, \dots, t_{n+1}\right).
\]
In particular, $\Delta[\bullet]_{geo}$ is a cosimplicial topological space. For each $[n]$ let $\Omega^{\bullet}(n)$ be the symmetric graded algebra (over $\Bbbk$) generated in degree $0$ by the variables $t_{0}, \dots , t_{n}$ and in degree $1$ by $dt_{0}, \dots , dt_{n}$ such that
\[
t_{0}+ \dots + t_{n}=1, \quad dt_{0}+\dots + dt_{n}=0.
\]
We equip $\Omega(n)$ with a differential $d\: : \: \Omega^{\bullet}(n)\to \Omega^{\bullet+1}(n)$ via $d\left( t_{i}\right) :=dt_{i}$. $\Omega(n)$ is the differential graded algebra of polynomial differential forms on $\Delta[n]_{geo}$. It follows that $\Omega(\bullet)$ is a simplicial commutative differential graded algebra, where the face maps $d_{i}$ are obtained via the pullback along $d^{i}$, and the codegenerancy maps are obtained via the pullback along $s^{i}$.\\
For a set $X$ we denote by $\Bbbk\left\langle X\right\rangle $ the module generated by $X$ and by $X^{\Bbbk}$ the module $\Hom_{Set}\left(X, \Bbbk \right)$. Thus for a simplicial set $X_{\bullet}$ we denote by $\Bbbk\left\langle X_{\bullet}\right\rangle $ the simplicial module $\Bbbk\left\langle X\right\rangle _{n}:=\Bbbk\left\langle X_{n}\right\rangle $ and by $X_{\bullet}^{\Bbbk}$ the cosimplicial module $\left( X^{\Bbbk}\right) _{n}:=X_{n}^{\Bbbk}$. Both of these constructions are functorials and $\left( -\right)^{\Bbbk}\: : \: sSet\to cMod$ is contravariant.\\
Let $\boldsymbol{\Delta}$ be the simplex category and let $\Delta\: : \:\boldsymbol{\Delta}\to sSet $ be the Yoneda embedding. Then $\Delta[\bullet]$ is a cosimplicial object in the category of simplicial sets and $C_{\bullet}:=\left( \Delta[\bullet]\right)^{\Bbbk}$ is a simplicial cosimplicial module. We get that $NC_{\bullet}$ is a simplicial differential graded module. Explicitly, for a fixed $n$ we have
\[
\left( NC_{n}\right)^{p}:=\begin{cases}
\Bbbk\left\langle \Hom_{Set}\left(\Delta[n]_{p}^{+}, \Bbbk \right) \right\rangle , \text{ if } & p\leq n,\\
0, \text{ if } & p>n
\end{cases}\]
where $\Delta[n]_{p}^{+}$ is the set of inclusions $[p]\hookrightarrow [n]$. A cosimplicial differential graded module is a cosimplicial object in the category $dgMod$ of differential graded modules. Explicitly, we denote this objects by $A^{\bullet, \bullet}$ where the first slot denote the cosimplicial degree and the second slot denotes the differential degree. It defines a functor $A^{\bullet, \bullet}\: : \: \boldsymbol{\Delta}\to dgMod$ and we get a bifunctor $NC_{\bullet}\otimes A^{\bullet, \bullet}\: : \: \boldsymbol{\Delta}^{op}\times \boldsymbol{\Delta}\to dgMod$. We consider the coend
\[
\int^{[n]\in\boldsymbol{\Delta}}NC_{n}\otimes A^{n, \bullet}\in dgMod.
\]
An element $v$ of degree $k$ in $\int^{[n]\in\boldsymbol{\Delta}}NC_{n}\otimes A^{n, \bullet}$ is a sequence $v:=(v_{n})_{n\in \N}$ where $v_{n}\in \left( NC_{n}\otimes A^{n, \bullet}\right)^{k}$ such that for any map $\theta\: : \: [n]\to [m]$ in $\boldsymbol{\Delta}$ we have
\[
\left( 1\otimes \theta^{*}\right)w_{n}=\left( \theta_{*}\otimes 1 \right)w_{m},
\]
where $\theta^{*}:=A^{\bullet, \bullet}(\theta)$, and $\theta_{*}:=NC_{\bullet}(\theta)$. Since
\[
\left( NC_{n}\otimes A^{n, \bullet}\right)^{k}=\oplus_{p+q=k}NC_{n}^{p}\otimes A^{n, q}
\]
we say that $v$ has bidegree $(p,q)$ if $v$ has degree $p+q$ and each $v_{n}\in NC_{n}^{p}\otimes B^{n, q}$ for each $n$. Let $\left( V,d_{V}\right) $ and $\left( W,d_{w}\right)$ be two cochain complexes, $\left( V\otimes W\right)^{\bullet}$ is again a cochain complex where the differential is
\[
d_{V\otimes W}\left( v\otimes w\right):=d_{V}(v)\otimes w+(-1)^{p}v\otimes d_{w}(w)
\]
for $v\otimes w\in V^{p}\otimes W^{q}$.
The differential on $\int^{[n]\in\boldsymbol{\Delta}}NC_{n}\otimes A^{n, \bullet}$ is defined via
\[
\left( dv\right)_{n}:= dv_{n}
\]
where $d$ is the induced differential on $\left( NC_{n}\otimes A^{n, \bullet}\right)^{\bullet}$. Consider the differential graded module (called Thom-Whitney normalization, see \cite{Getz})
\[
\operatorname{Tot}_{TW}\left(A\right):=\int^{[n]\in \boldsymbol{\Delta}} \Omega(n)\otimes A^{n,\bullet }\in dgMod.
\]
Explicitly, an element $v\in \operatorname{Tot}_{TW}\left(A\right)^{k}$ is a collection $v=\left( v_{n}\right)$ of $v_{n}\in \left( \Omega(n)\otimes A^{n,\bullet }\right)^{k}$ such that for any map $\theta\: : \: [n]\to [m]$ in $\boldsymbol{\Delta}$ we have
\[
\left( 1\otimes \theta_{*}\right)v_{n}=\left( \theta^{*}\otimes 1 \right)v_{m},
\]
where $\theta_{*}:=A^{\bullet, \bullet}(\theta)$, and $\theta^{*}:=\Omega(n)(\theta)$. Since
\[
\left(\Omega(n)\otimes A^{n, \bullet}\right)^{k}=\oplus_{p+q=k}\Omega^{p}(n)\otimes A^{n, q},
\]
we say that $v$ has bidegree $(p,q)$ if $v$ has degree $p+q$ and each $v_{n}$ is contained in $\Omega^{p}[n]\otimes A^{n, q}$. We denote by $\operatorname{Tot}_{TW}\left(A\right)^{p,q}$ the set of elements of bidegree $(p,q)$. If $A^{\bullet, \bullet}$ is a cosimplicial unital differential graded commutative algebra, then $\left(\operatorname{Tot}_{TW}\left(A\right), d_{\bullet,A},d_{\bullet,poly}\right)$ is a differential graded commutative algebra as well where the multiplication and the differential are
\[
(v\wedge w)_{n}:=(v)_{n}\wedge (w)_{n}, \quad (dv)_{n}:=d\left( v_{n}\right).
\]
\subsection{The Dupont retraction}\label{CinfiDerham}
We give a short summary of the results of \cite{Getz}, where a $C_{\infty}$-structure is induced on $\int^{[n]\in\boldsymbol{\Delta}}NC_{n}\otimes A^{n,\bullet }$ (and hence on $\operatorname{Tot}_{N}\left(A\right)$) from $\operatorname{Tot}_{TW}(A)$ via the homotopy transfer theorem.
\begin{thm}[\cite{Getz2},\cite{Dupont2}]\label{getzdup}
Let $\Omega(\bullet)$, $NC_{\bullet}$ be the two simplicial differential graded modules defined above. We denote the differential of $\Omega(\bullet)$ by $d_{\bullet, poly}$. There is a diagram between simplicial graded modules
\begin{equation}\label{simplicialdiag}
\begin{tikzcd}
E_{\bullet}\: : \: NC_{\bullet}\arrow[r, shift right, ""]&\Omega(\bullet)\: : \: \int_{\bullet}\arrow[l, shift right, ""]
\end{tikzcd}
\end{equation}
and a simplicial homotopy operator $s_{\bullet}\: : \: \Omega(\bullet)\to \Omega^{\bullet-1}(\bullet)$ between $\left( E_{\bullet}\int_{\bullet}\right) $ and the identity, i.e., $s_{\bullet}$ is a map between simplicial differential graded modules such that for each $n\geq 0$
\[
d_{n, poly}s_{n}+s_{n}d_{n, poly}=E_{n}\int_{n}-Id.
\]
In particular, for any $n$ we have $E_{n}$ is an injective quasi-isomorphism, $\int_{n}E_{n}=Id$ and \begin{align*}
\int_{n}s_{n}=s_{n}E_{n}=s_{n}^{2}=0.
\end{align*}
\end{thm}
\begin{proof}
The first statement is originally contained in \cite{Dupont2}. The second part of the theorem is proved in \cite{Getz2}.
\end{proof}
See Section \ref{sec2dimsimplex} for more details about the above maps. The homotopy transfer theorem (see \cite{Markl}) gives a $C_{\infty}$-algebra structure $m_{\bullet}^{[n]}$ on $NC_{n}$ induced by the above diagram. Let $A^{\bullet,\bullet}$ be a cosimplicial commutative algebra. The differential graded algebra $\Omega(n)\otimes A^{n, \bullet}$ is commutative as well. For any $n,m\geq 0$, we denote the $C_{\infty}$-structure induced along the diagram
\begin{equation}\label{simplicialdiagtens}
\begin{tikzcd}
E_{n}\otimes Id \: : \: NC_{n}^{\bullet}\otimes A^{ n,\bullet}\arrow[r, shift right, ""]& \Omega(n)\otimes A^{n, \bullet}\: : \: \int_{n}\otimes Id\arrow[l, shift right, ""]
\end{tikzcd}
\end{equation}
by $m_{\bullet}^{n,m}$. This structure depends only on $m_{\bullet}^{[n]}$. Let $w_{1}, \dots, w_{l}\in NC_{n}\otimes A^{n,\bullet }$ be such that $w_{i}=f_{i}\otimes a_{i}$ for $i=1, \dots, l$. Then
\begin{equation}\label{reduction}
m_{l}^{n,m}\left(w_{1}, \dots, w_{l} \right)=(-1)^{\sum_{i<j} |f_{i}||a_{j}|}m_{l}^{[n]}\left(f_{1}, \dots, f_{l} \right)\otimes \left( a_{1}\wedge \cdots \wedge a_{l}\right).
\end{equation}
The above structure defines a well-defined $C_{\infty}$-structure ${m}_{\bullet}$ on $\int^{[n]\in\boldsymbol{\Delta}}NC_{n}\otimes A^{n, \bullet}$. The maps ${m}_{\bullet}$ can be obtained in another way. We apply the coend functor on the simplicial diagram \eqref{simplicialdiagtens}. Since $s_{\bullet}$, $E_{\bullet}$, $\int_{\bullet}$ are all simplicial maps, they induce degree zero maps $E$, $\int$ between the coends such that
\begin{equation}\label{secondiag}
\begin{tikzcd}
E\: : \: \int^{[n]\in\boldsymbol{\Delta}}NC_{n}\otimes A^{n,\bullet }\arrow[r, shift right, ""]&\operatorname{Tot}_{TW}\left(A\right)\: : \: \int\arrow[l, shift right, ""]
\end{tikzcd}
\end{equation}
is a diagram that satisfies the same properties of the one in \eqref{simplicialdiag}. There is an unital $C_{\infty}$-structure induced on $\int^{[n]\in\boldsymbol{\Delta}}NC_{n}\otimes A^{n,\bullet }$ via the homotopy transfer theorem. By construction, this structure coincides with ${m}_{\bullet}$. Notice that the construction of $\int^{[n]\in\boldsymbol{\Delta}}NC_{n}\otimes A^{n, \bullet}$ gives a functor form the category of cosimplicial unital non-negatively graded commutative differential graded algebras ($\mathrm{cdgA}$ for short) toward the category of cochain complexes. We have a correspondence
\begin{equation}
\label{functor0}
A^{\bullet,\bullet}\mapsto \left( \int^{[n]\in\boldsymbol{\Delta}}NC_{n}\otimes A^{n, \bullet},m_{\bullet}\right).
\end{equation}
For a field $\Bbbk$ of characteristic zero we denote by $\left( C_{\infty}-\operatorname{Alg}\right)_{\Bbbk,str}$ the category of $C_{\infty}$-algebras on $\Bbbk$ and strict morphisms.
\begin{thm}\label{functorassoc0}
The correspondence \eqref{functor0} gives a functor $\mathrm{cdgA}\to \left( C_{\infty}-\operatorname{Alg}\right)_{\Bbbk,str}$.
\end{thm}
\begin{proof}
Let $f\: : \: A^{\bullet,\bullet}\to B^{\bullet,\bullet}$ be a morphism. The correspondence \eqref{functor0} is a functor toward the category of chain complexes. We denote its image by $f_{1}$. Since $f$ is a differential graded algebra map on each degree, the same argument of Lemma \ref{tensorinfinity} shows that $f_{1}$ induces a strict morphism of $C_{\infty}-$algebras.
\end{proof}
We give an explicit formula for $m_{\bullet}$. Fix a $n$ and a $p\leq n$. Notice that each inclusion $[p]\hookrightarrow [n]$ is equivalent to an ordered string $0\leq i_{0}<i_{i}<\dots <i_{p}\leq n$ contained in $\left\lbrace 0,1,\dots, n\right\rbrace $. For each string $0\leq i_{0}<i_{i}<\dots <i_{p}\leq n$, we denote the associated inclusion by $\sigma_{i_{0}, \dots i_{p}}\: : \:[p]\hookrightarrow [n]$, and we define the maps $\lambda_{i_{0}, \dots, i_{p}}\: : \: \Delta[n]_{p}^{+}\to \Bbbk$, via
\[
\lambda_{i_{0}, \dots, i_{p}}(\phi):=
\begin{cases}
1\text{ if }\sigma_{i_{0}, \dots i_{p}}=\phi,\\
0,\text{ otherwhise}.
\end{cases}
\]
Let $v_{1}, \dots, v_{n}\in \int^{[n]\in\boldsymbol{\Delta}}NC_{n}\otimes A^{n, \bullet}$ be elements of bidegree $\left(p_{i}, q_{i} \right)$. Then $m_{n}\left(v_{1}, \dots, v_{n} \right) $ is an element of bidegree $\left(\sum_{i}pi+2-n , \sum q_{i}\right) $. Let $l:=\sum_{i}pi+2-n$, then Lemma \ref{isompsi} implies that $m_{n}\left(v_{1}, \dots, v_{n} \right) $ is completely determined by $m_{n}\left(\left( v_{1}\right)_{l} , \dots, \left( v_{n}\right)_{l}\right)_{l}$. We write
$\left( v_{i}\right)_{p_{i}}=\lambda_{0, \dots, p_{i}}\otimes a_{i}\in NC_{p_{i}}^{p_{i}}\otimes A^{p_{i}, q_{i}}$ for all the $i$. We denote by $I$ the subsets $\left\lbrace i_{0}, \dots ,i_{p}\right\rbrace \subseteq \left\lbrace 0,\dots , l\right\rbrace $, for $I=\left\lbrace i_{0}, \dots ,i_{p}\right\rbrace $ we define $|I|:=p$ and we write $\lambda_{I}$ instead of $\lambda_{0, \dots, p}$. Each $I$ corresponds to an inclusion in $\boldsymbol{\Delta}$; we denote by $\sigma_{I}\: : \: [p]\to [l]$ the map induced by $I$. We have
\begin{align}
\nonumber m_{n}\left(v_{1}, \dots, v_{n} \right)_{l} &=m_{n}^{l,l}\left(\left( v_{1}\right)_{l} , \dots, \left( v_{n}\right)_{l}\right)_{l}\\\nonumber
& = m_{n}^{l,l}\left(\sum_{|I_{{1}}|=p_{1}}\lambda_{I_{1}}\otimes \left( \sigma_{I_{1}}\right)_{*}a_{1}, \dots, \sum_{|I_{n}|=p_{n}} \lambda_{I_{n}}\otimes \left( \sigma_{I_{n}}\right)_{*}a_{n}\right)\\
& =\sum_{|I_{1}|=p_{1},\dots ,|I_{n}|=p_{n} } (-1)^{\sum_{i<j} |p_{i}||q_{j}|}m_{n}^{[l]}\left(\lambda_{I_{1}}, \dots, \lambda_{I_{n}} \right)\otimes \left( \left( \sigma_{I_{1}}\right)_{*}a_{1}\wedge \cdots \wedge \left( \sigma_{I_{n}}\right) _{*}a_{n}\right).\label{formulafinale}
\end{align}
In particular, the above formula implies that if $v_{1}, \dots, v_{n}$ are all of degree $1$, then $m_{n}\left(v_{1}, \dots, v_{n} \right)$ would only depend on
\begin{itemize}
\item the restriction of $m_{\bullet}^{[2]}$ on the elements of degree $1$, if all the $v_{i}$ are of bidegree $(1,0)$;
\item $m_{\bullet}^{[0]}$, if all the $v_{i}$ are of bidegree $(0,1)$;
\item $m_{\bullet}^{[1]}$ in the other cases.
\end{itemize}
\begin{lem}\label{inclusiontrivial}
Consider $A^{0,\bullet}$ equipped with its differential graded algebra structure. There is a canonical inclusion $i\: :\:A^{0,\bullet}\hookrightarrow\operatorname{Tot}_{N}\left(A\right)$ which is a strict $C_{\infty}$-algebra map.
\end{lem}
\begin{proof}
The map $i$ is clearly a cochain map. The $m_{\bullet}^{[0]}$ is trivial, and by setting $l=0$ in \eqref{formulafinale} we obtain that $i$ is strict.
\end{proof}
The $m_{\bullet}^{[1]}$ is given in \cite{Getz}. We first set a convenient basis for $NC_{1}$. Notice that the maps $E_{n}$ are all injective. This allows us to interpret $NC_{1}$ as a submodule of $\Omega^{\bullet}(1)$. Recall that $\Omega^{\bullet}(1)$ is the free differential graded commutative algebra generated by the degree-zero variables $t_{0}$, $t_{1}$ modulo the relations
\[
t_{0}+t_{1}=1,\quad dt_{0}+dt_{1}=0.
\]
$NC_{1}^{0}$ is a two-dimensional vector space generated by $\lambda_{0}$ and $\lambda_{1}$ and $NC_{1}^{1}$ is one-dimensional generated by $\lambda_{0,1}$. We have
\begin{equation}\label{basisconv}
E_{1}\left( \lambda_{0}\right)=t_{0},\quad E_{1}\left( \lambda_{1}\right)=t_{1},\quad E_{1}\left( \lambda_{0,1}\right)=t_{0}dt_{1}-t_{1}dt_{0}.
\end{equation}
Let $t:=t_{0}$, hence $t_{1}=1-t$. Then $\Omega^{\bullet}(1)$ may be considered as the free differential graded commutative algebra generated by $t$ in degree zero and $NC_{1}$ is the subgraded module generated by $1,t,dt$. In particular $1$ is the unit of the $C_{\infty}$-structure.
\begin{prop}[\cite{Getz}]\label{Getzappl1}
The structure $m^{[1]}_{\bullet}$ on $NC_{1}$ is defined as follows:
\begin{enumerate}
\item $m_{2}^{[1]}(t,t)=t$,
\item $m_{n+1}^{[1]}(dt^{\otimes i}, t, dt^{\otimes n-i})=(-1)^{n-i}\binom{n}{i}m^{[1]}_{n+1}(t,dt,\dots ,dt)$,
\item $m_{n+1}^{[1]}(t,dt,\dots ,dt)=\frac{B_{n}}{n\text{!}}dt$, where $B_{n}$ are the second Bernoulli numbers,
\end{enumerate}
and all remaining products vanish.
\end{prop}
It remains to find a formula for $m_{n}^{[2]}|_{\left( NC_{2}^{1}\right)^{\otimes n}}$. For $n>2$ we are not aware of an explicit formula.
\begin{prop}\label{speriam} Consider $NC_{2}^{\bullet}$ equipped with $m_{\bullet}^{[2]}.$ We have $$m_{2}^{[2]}(\lambda_{01},\lambda_{02})=m_{2}^{[2]}(\lambda_{01},\lambda_{12})=m_{2}^{[2]}(\lambda_{02},\lambda_{12})=\frac{1}{6}\lambda_{012}.$$
\end{prop}
\begin{proof}
By explicit calculation. The details are given in Appendix \ref{sec2dimsimplex}.
\end{proof}
Consider $A^{\bullet, \bullet}$ as above. We denote by $N(A)^{\bullet, \bullet}$ its bigraded bidifferential module (see Appendix \ref{cosimplicial modules} for a definition) and by $\operatorname{Tot}_{N}\left(A\right)\in dgMod$ its associated total complex. It is well known that there is a natural isomorphism $\psi\: : \: \operatorname{Tot}_{N}\left(A\right)\to\int^{[n]\in\boldsymbol{\Delta}}NC_{n}\otimes A^{n, \bullet}$ of differential graded modules (see Lemma \ref{isompsi} for a proof). With an abuse of notation we denote again by $m_{\bullet}$ the $C_{\infty}$-structure induced on $\operatorname{Tot}_{N}\left(A\right)$ via the isomorphism $\psi$. We have the following.
\begin{thm}\label{functor}
The association \begin{equation}
A^{\bullet,\bullet}\mapsto \left( \operatorname{Tot}_{N}\left(A\right),m_{\bullet}\right)
\end{equation} is part of a functor $\mathrm{cdgA}\to \left( C_{\infty}-\operatorname{Alg}\right)_{\Bbbk,str}$.
\end{thm}
We give an explicit formula for $m_{\bullet}$ in $\operatorname{Tot}_{N}\left(A\right)$ for elements of degree $0,1$. We denote by $\tilde{\partial}\: : \: A^{\bullet,\bullet}\to A^{\bullet+1,\bullet}$ the differential given by the alternating sum of coface maps, in particular $\tilde{\partial}=d^0-d^1$ on $A^{0,\bullet}$.
\begin{thm}\label{productthm>2}Let $l>2$.
\begin{enumerate}
\item Let $a_{1}, \dots, a_{l}\in\operatorname{Tot}^{1}_{N}\left(A\right)$ and let $b_{i}\in A^{1,0}$, $c_{i}\in A^{0,1}$ be such that $a_{i}=b_{i}+c_{i}$ for every $i=1, \dots, l$. Then
\[
m_{l}\left( a_{1}, \dots, a_{l}\right) = \sum_{i=1}^{l}(-1)^{l-1}\binom{l-1}{i-1}\frac{B_{l-1}}{\left(l-1 \right) !}b_{1}\cdots \widehat{b}_{i}\cdots b_{l}\tilde{\partial}c_{i}+m_{l}\left(c_{1}, \dots, c_{n} \right) .
\]
\item Let $x\in A^{0,0}$. Then
\[
m_{l}\left(a_{1}, \dots,a_{i-1}, x, a_{i+1}, \dots, a_{l}\right)=(-1)^{i}\binom{l-1}{i-1}\frac{B_{l-1}}{\left(l-1 \right) !}b_{1}\cdots b_{l}\left( \tilde{\partial}x\right),
\]
and if we replace some $a_{i}$ by an element in $A^{0,0}$, the above expression vanishes.
\end{enumerate}
\end{thm}
\begin{proof}
For two subsets $\mathcal{B}, \mathcal{C}\subseteq \left\lbrace 1, \dots, l \right\rbrace $ such that $\mathcal{B}\cup\mathcal{C}=\left\lbrace 1, \dots, l \right\rbrace$, we denote by $m_{l}\left(b_{\mathcal{B}}, c_{\mathcal{C}}\right)$ the expression $m_{l}\left(y_{1}, \dots, y_{l} \right)$ such that $y_{i}=b_{i}$ for $i\in \mathcal{B}$ and $y_{i}=c_{i}$ for $i\in \mathcal{B}$. In particular, we have $\left| m_{l}\left(b_{\mathcal{B}}, c_{\mathcal{C}}\right)\right|= \left|\mathcal{B} \right|+2-l+\left| \mathcal{C}\right|$. It follows that $\left|\mathcal{B} \right| \geq l-2$ and hence $\left|\mathcal{C} \right| \leq 2$. We have
\begin{align*}
m_{l}\left( a_{1}, \dots, a_{l}\right) & = \sum_{\mathcal{B}, \mathcal{C}}m_{l}\left(b_{\mathcal{B}}, c_{\mathcal{C}}\right)\\& =
\sum_{\left| \mathcal{B}\right|=l-2 , \left| \mathcal{C}\right|=2}m_{l}\left(b_{\mathcal{B}}, c_{\mathcal{C}}\right)+ \sum_{\left| \mathcal{B}\right|=l-1 , \left| \mathcal{C}\right|=1}m_{l}\left(b_{\mathcal{B}}, c_{\mathcal{C}}\right)+ \sum_{\left| \mathcal{B}\right|=l , \left| \mathcal{C}\right|=0}m_{l}\left(b_{\mathcal{B}}, c_{\mathcal{C}}\right).\\&
\end{align*}
The first summand vanishes by \eqref{formulafinale}. We conclude
\begin{equation}\label{step1}
m_{l}\left( a_{1}, \dots, a_{l}\right) = \sum_{i=1}^{l}m_{l}\left(b_{1}, \dots, c_{i}, \dots, b_{l}\right)+m_{l}\left(b_{1}, \dots, b_{l}\right).
\end{equation}
We calculate explicitly $m_{l}\left(b_{1}, \dots, b_{l},c_{1}\right)\in \operatorname{Tot}^{2}_{N}\left(A\right)$. We work in $\int^{[n]\in\boldsymbol{\Delta}}NC_{n}\otimes A^{n,\bullet} $ and we use the isomorphism $\psi$ (see Appendix \ref{cosimplicial modules}). In particular, for $c\in \operatorname{Tot}^{0,1}_{N}\left(A\right)$ we have a $(0,1)$ element $\psi(b)\in \int^{[n]\in\boldsymbol{\Delta}}NC_{n}\otimes A^{\bullet,n} $. Its projection at $NC_{1}^{0}\otimes A^{1,1}\subset NC_{1}^{\bullet}\otimes A^{1,\bullet}$ is
\[
\lambda_{0}\otimes \sigma_{0}^{*}\left(c \right)+\lambda_{1}\otimes \sigma_{1}^{*}\left(c \right).
\]
Let $d^{0}, d^{1}$ be the coface maps of $A$. By definition, $\sigma_{0}$ corresponds to the coface map $d^{1}$ and $\sigma_{1}$ corresponds to the coface map $d^{0}$. Hence we can write $NC_{1}^{\bullet}\otimes A^{1, \bullet}$ as $t\otimes d^{1}c +(1-t)\otimes d^{0} c$ (see \eqref{basisconv}). Similarly, an element $b$ of bidegree $(0,1)$ can be written as $-dt\otimes b \in NC_{1}^{1}\otimes A^{1,0}$. We have
\begin{align*}
\nonumber m_{l}^{1,1}\left(\phi(b)_{1}, \dots, \phi(b)_{l-1}, \phi(c)_{l}\right)_{1}
&\nonumber = m_{l+1}^{1,1}\left(-dt\otimes b_{1}, \dots,-dt\otimes b_{l-1}, t\otimes d^{1}c_{l} +(1-t)\otimes d^{0} c_{l}\right)\\
&=(-1)^lm_{l+1}^{[1]}\left(dt, \dots,dt,t\right)\otimes b_{1}\cdots b_{l-1}\left( d^{1}c_{l}-d^{0}c_{l}\right) \\
& =(-1)^l\frac{B_{l-1}}{\left(l-1 \right) !}dt\otimes\left( b_{1}\cdots b_{l-1}\right) \tilde{\partial}c_{l}\in NC_{1}^{1}\otimes A^{1,1}
\end{align*}
where $ b_{1}\cdots b_{l-1}$ has to be understood as a multiplication inside $A^{1,0}$. The above expression defines an element of bidegree $(1,1)$ inside $\int^{[n]\in\boldsymbol{\Delta}}NC_{n}\otimes A^{\bullet,n} $. By applying $\psi^{1}$ we get
\[
m_{l}\left(b_{1}, \dots, b_{l-1},c_{l}\right)=(-1)^{l-1}\frac{B_{l-1}}{\left(l-1 \right) !}b_{1}\cdots b_{l-1}\tilde{\partial}c_{l}.
\]
Thanks to point 2 in Proposition \ref{Getzappl1}, we have
\[
m_{l}\left(b_{1}, \dots,b_{i-1}, c_{i}, b_{i+1}, \dots, b_{l}\right)=(-1)^{l-1}\binom{l-1}{i-1}\frac{B_{l-1}}{\left(l-1 \right) !}b_{1}\cdots b_{l}\tilde{\partial}c_{i}.
\]
By \eqref{step1}, we have
\begin{equation}\label{step2}
m_{l}\left( a_{1}, \dots, a_{l}\right) = \sum_{i=1}^{l}(-1)^{l-1}\binom{l-1}{i-1}\frac{B_{l-1}}{\left(l-1 \right) !}b_{1}\cdots \widehat{b}_{i}\cdots b_{l}\tilde{\partial}c_{l}+m_{l}\left(b_{1}, \dots, b_{l}\right).
\end{equation}
Now let $x\in A^{0,0}$. Then, the above computations give
\begin{align}
\nonumber m_{l}\left(a_{1}, \dots,a_{i-1}, x, a_{i+1}, \dots, a_{l}\right)& =m_{l}\left(b_{1}, \dots,b_{i-1}, x, b_{i+1}, \dots, b_{l}\right)\\
\nonumber& =(-1)^{l-1}(-1)^{l-i-1}\binom{l-1}{i-1}\frac{B_{l-1}}{\left(l-1 \right) !}b_{1}\cdots b_{l}\left( \tilde{\partial}x\right) \\
\label{step2unit}& =(-1)^{i}\binom{l-1}{i-1}\frac{B_{l-1}}{\left(l-1 \right) !}b_{1}\cdots b_{l}\left( \tilde{\partial}x\right).
\end{align}
\end{proof}
The following corollary follows directly from the previous proof.
\begin{cor}\label{corresiduemap}
Let $n>2$ and and let $a_{1}, \dots, a_{n}\in \operatorname{Tot}^{1}_{N}\left(A\right)$. We have
\[
m_{n}\left( a_{1}, \dots, a_{n}\right)=d_{1}+d_{2}+d_{3},
\]
where $d_{1}\in \operatorname{Tot}^{0,2}_{N}\left(A\right)$, $d_{2}\in \operatorname{Tot}^{1,1}_{N}\left(A\right)$ and $d_{3}\in\operatorname{Tot}^{0,2}_{N}\left(A\right)$. Then $d_{1}=0$ and $d_{3}=m_{n}\left( b_{1}, \dots, b_{n}\right) $.
\end{cor}
\begin{thm}\label{productthm=2}
Let $a_{1},a_{2}\in\operatorname{Tot}^{1}_{N}\left(A\right)$ and let $b_{i}\in \operatorname{Tot}^{1,0}_{N}\left(A\right)$, $c_{i}\in \operatorname{Tot}^{0,1}_{N}\left(A\right)$ such that $a_{i}=b_{i}+c_{i}$ for $i=1,2$. Then
\begin{align*}
m_{2}\left( a_{1},a_{2}\right)& = m_{2}\left( c_{1},c_{2}\right)+ m_{2}\left( b_{1},c_{2}\right)+m_{2}\left( c_{1},b_{2}\right)+m_{2}\left( b_{1},b_{2}\right),
\end{align*}
where
\begin{enumerate}
\item $m_{2}(c_{1}, c_{2})=c_{2}c_{2}\in A^{0,2},$
\item $m_{2}\left(b_{1},c_{2} \right)= -\frac{1}{2}b_{1}\tilde{\partial}c_{2}+b_{1}d^{0}c_{2}$,
\item $m_{2}\left(b_{1},b_{2} \right)= \frac{1}{6}\left(- \left(d^0b_{1}\left(d^1b_{2}+d^2b_{2} \right) \right)
+ \left(d^1b_{1}\left(d^0b_{2}-d^2b_{2} \right) \right)
+ \left(d^2b_{1}\left(d^0b_{2}+d^1b_{2} \right) \right)\right).$
\end{enumerate}
Let $x,y\in \operatorname{Tot}^{0}_{N}\left(A\right)$. Then, $m_{2}(c_{1},x)=c_{1}x\in \operatorname{Tot}^{0,1}_{N}\left(A\right)$ , $m_{2}(x,y)=xy\in A^{0,1}$, and $m_{2}\left( b_{1},x\right)=-\frac{1}{2}b_{1}\tilde{\partial}x+b_{1}d^{0}x\in\operatorname{Tot}^{1,0}_{N}\left(A\right)$.
\end{thm}
\begin{proof}
The second and last terms of $m_{2}(a_{1},a_{2})$ can be calculated by a computation similar to the proof above. We have
\begin{align*}
\nonumber m_{2}^{1,1}\left(\psi(b)_{1}, \psi(c)_{2}\right)_{1}
&\nonumber = m_{2}^{1,1}\left(-dt\otimes b_{1}, t\otimes d^{1}c_{2} +(1-t)\otimes d^{0} c_{2}\right)\\
&=-m_{2}^{[1]}\left(dt,t\right)\otimes b_{1}c_{2}+m_{2}^{[1]}\left(dt,1\right)\otimes \left(- b_{1}d^{0}c_{2}\right) \\
& +m_{2}^{[1]}(dt,t)\otimes \left(b_{1}d^{0}c_{2} \right) \\
& =dt\otimes \left( B_{1}b_{1}\tilde{\partial}c_{2}-b_{1}d^{0}c_{2}\right),
\end{align*}
and thus $m_{2}\left(b_{1},c_{2} \right)= -B_{1}b_{1}\tilde{\partial}c_{2}+b_{1}d^{0}c_{2}$. It remains to add $m_{2}\left(b_{1},b_{2}\right)$ where $B_{1}=\frac{1}{2}$. The expression for $m_{2}\left(b_{1},x\right)$ can be computed in the same way.
By Appendix \ref{sec2dimsimplex}, $NC_{2}$ is the graded vector space generated by $\lambda_{0}, \lambda_{1}, \lambda_{2}$ in degree $0$, $\lambda_{12}, \lambda_{02}, \lambda_{01}$ in degree $1$ and $\lambda_{012}$ in degree 2. In particular, for $b\in \operatorname{Tot}^{1,0}_{N}\left(A\right)$ we have a $(1,0)$ element $\psi(b)\in \int^{[n]\in\boldsymbol{\Delta}}NC_{n}\otimes A^{\bullet,n} $. Its projection at $ NC_{2}^{\bullet}\otimes A^{2,\bullet}$ is
\[
\lambda_{12}\otimes d^{0}\left(b \right)+\lambda_{02}\otimes d^{1}\left(b \right)+\lambda_{01}\otimes d^{2}\left(b \right).
\]
By Proposition \ref{speriam} we have
\begin{align*}
& m_{2}^{2,2}\left(\psi\left( b_{1}\right) ,\psi\left( b_{2}\right)\right)_{2}=\\
& =m_{2}^{2,2}\left(\lambda_{12}\otimes d^{0}\left(b_{1} \right)+\lambda_{02}\otimes d^{1}\left(b_{1} \right)+\lambda_{01}\otimes d^{2}\left(b_{1} \right) ,\lambda_{12}\otimes d^{0}\left(b_{2} \right)+\lambda_{02}\otimes d^{1}\left(b_{2} \right)+\lambda_{01}\otimes d^{2}\left(b_{2} \right)\right)_{2}\\
&=\frac{1}{6}\lambda_{012}\otimes \left( -\left(d^0b_{1}\left(d^1b_{2}+d^2b_{2} \right) \right)
+ \left(d^1b_{1}\left(d^0b_{2}-d^2b_{2} \right) \right)
+ \left(d^2b_{1}\left(d^0b_{2}+d^1b_{2} \right) \right)\right). \\
\end{align*}
Then
\[
m_{2}\left(b_{1},b_{2} \right)= \frac{1}{6}\left( -\left(d^0b_{1}\left(d^1b_{2}+d^2b_{2} \right) \right)
+ \left(d^1b_{1}\left(d^0b_{2}-d^2b_{2} \right) \right)
+ \left(d^2b_{1}\left(d^0b_{2}+d^1b_{2} \right) \right)\right).
\]
\end{proof}
\section{Geometric connections}\label{third}
We put the $C_{\infty}$-structure of Section \ref{sectGetztler} on the cosimplicial module of differential forms on a simplicial manifold $M_{\bullet}$. We show that Maurer-Cartan elements induce a flat connection form on $M_{0}$ (Corollary \ref{corpronilpot}).
\subsection{Simplicial De Rham theory}\label{sectionsimplicial De Rham Theory}
We recall some basic notions about simplicial manifolds. Our main reference is \cite{Dupont} and \cite{Dupont2}. By complex manifold we mean a smooth complex manifold. Let $\mathrm{Diff}_{\C}$ be the category of complex manifold. We denote with $\mathrm{dgA}$ the category of complex commutative non-negatively graded differential graded algebras.
\begin{defi}
A \emph{simplicial manifold} $M_{\bullet}$ is a simplicial object in $\mathrm{Diff}_{\C}$.
\end{defi}
Each smooth complex manifold $M$ can be viewed as a constant simplicial manifold $M_{\bullet}$, where $M_{n}:=M$ and all the degenerancy and face maps are equal to the identity. Let $G$ be a Lie group, and let $M$ be a manifold equipped with a left smooth (or holomorphic) $G$-action. We define the simplicial manifold $M_{\bullet}G$ as follows: $M_{n}G=M\times G^{n}$ and
The face maps $d^{i}\: : \: M_{n}G\to M_{n-1}G$ for $i=0,1,\dots n$ are
\[
d^{i}(x,g_{1}, \dots , g_{n}):=\begin{cases}
& (g_{1}x, g_{2}, \dots , g_{n}),\text{ if }i=0,\\
& (x,g_{1}, \dots , g_{i}g_{i+1}, \dots ,g_{n}),\text{ if }1<i<n\\
& (x,g_{1}, \dots , g_{n-1}),\text{ if }i=n.
\end{cases}
\]
The degenerancy maps $s^{i}\: : \: M_{n}G\to M_{n+1}G $ are defined via
\[
s^{i}(x,g_{1}, \dots , g_{n}):=\left(x,g_{1}, \dots, g_{i}, e, g_{i+1}, \dots , g_{n} \right)
\]
for $i=1, \dots ,n$.
\begin{defi}
We call the simplicial manifold $M_{\bullet}G$ the action groupoid.
\end{defi}
In particular the geometric realization of $M_{\bullet}G$ is weakly equivalent to the Borel construction $EG\times_{G}M$ and if the action of $G$ is free, the projection $\pi\: : \: EG\times_{G}M\to M/G$ is a homotopy equivalence. Let $A_{DR}\: : \: \mathrm{Diff}_{\C}\to \mathrm{dgA}$ be the smooth complex De Rham functor, i.e., $A_{DR}\left( M\right) $ is the differential graded algebra of smooth complex valued differential forms on $M$. $A_{DR}$ is contravariant and $A_{DR}\left(M_{\bullet} \right)$ is a cosimplicial complex commutative differential graded algebra for any simplicial manifold $M_{\bullet}$. As explained in the previous section, $\operatorname{Tot}_{TW}\left( A_{DR}\left(M_{\bullet} \right)\right) $ is a differential graded commutative algebra over $\C$. We obtain a contravariant functor
\[
\operatorname{Tot}_{TW}\left( A_{DR}\left(- \right)\right)\: :\: \mathrm{Diff}^{\boldsymbol{\Delta}^{op}}_{\C}\to \mathrm{dgA}.
\]
A smooth map between simplicial manifolds $f_{\bullet}\: : \: M_{\bullet}\to N_{\bullet}$ induces a morphism of differential graded algebras via $\left( f^{*}(w)\right)_{n}:=f^{*}_{n}(w_{n})$. If $M_{\bullet}$ is a constant simplicial manifold then $\operatorname{Tot}_{TW}\left( A_{DR}\left(M_{\bullet} \right)\right)$ is naturally isomorphic to $A_{DR}(M_{\bullet})$. Given a simplicial manifold $M_{\bullet}$ and consider the cosimplicial complex commutative differential graded algebra $A_{DR}\left(M_{\bullet} \right)$, by Theorem \ref{functor}, we have a functor
\[
\operatorname{Tot}_{N}\left( A_{DR}\left(- \right)\right)\: :\: \mathrm{Diff}_\C^{\boldsymbol{\Delta}^{op}}\to \left( C_{\infty}-\operatorname{Alg}\right)_{\C,str}.
\]
There is a De Rham theorem for simplicial manifolds. This is very useful because it allows us to determine the cohomology of $\operatorname{Tot}_{N}\left( A_{DR}\left(M_{\bullet} \right)\right)$. Let $\mathrm{Top}$ be the category of topological spaces. Let $T_{\bullet}\in \mathrm{Top}^{\boldsymbol{\Delta}^{op}}$ be a simplicial topological space.
Let $\boldsymbol{\Delta}_{+}$ be the subcategory of $\boldsymbol{\Delta}$ with the same objects but only injective maps. By restriction, we have a functor $T_{\bullet}\: : \: \boldsymbol{\Delta}_{+}^{op}\to \mathrm{Top}$. \\
\begin{defi}
We call the topological spaces
\[
\left| \left| T_{\bullet} \right| \right|:=\int^{[n]\in \boldsymbol{\Delta}_{+}}T_{n}\times \Delta_{geo}[n]\text{ and }\left| T_{\bullet} \right| :=\int^{[n]\in \boldsymbol{\Delta}}T_{n}\times \Delta_{geo}[n]
\]
the \emph{fat realization} and the \emph{geometric realization} of $T_{\bullet}$ respectively.
\end{defi}
\begin{rmk}
The natural quotient map $||T_{\bullet}||\to |T_{\bullet}|$ is not a weak equivalence in general. However, if the simplicial topological space is ``good'' (see the appendix of \cite{Segal}), then it is a weak-equivalence. In particular, for a simplicial manifold $M_{\bullet}$, this is true when its degeneracy $s^{i}$ maps are embeddings. We call this class of simplicial manifolds \emph{good} simplicial manifolds.
\end{rmk}
\begin{prop}[\cite{Dupont}]\label{Derhamthmsimpl}
Let $M_\bullet$ be a simplicial manifold, we have a sequence of multiplicative isomorphisms:
\[
H^{\bullet}\left(\operatorname{Tot}_{TW}\left( A_{DR}\left(M_{\bullet} \right)\right) \right)\cong H^{\bullet}\left(\operatorname{Tot}_{N}\left( A_{DR}\left(M_{\bullet} \right)\right)\right) \cong H^{\bullet}\left(||M_{\bullet}||, \C \right).
\]
\end{prop}
We introduce the complex of smooth logarithmic differential forms. Let $M$ be a complex manifold.
\begin{defi}
A normal crossing divisor $D\subset M$ is given by $\cup_{i} D_{i}$, where each $D_{i}$ is a non-singular divisor (a codimension $1$ object) and for each $p\in D_{i_{1}}\cap \dots \cap D_{i_{l}}$ there exist local coordinates $z=\left( z_{1}, \dots, z_{n}\right)\: : \: \C^{n}\to U\subset M$ near $p$ such that $U\cap D_{i_{1}}\cap \dots \cap D_{i_{l}}$ is given by the equation $\prod^{q}_{i=1}z_{j_{i}}=0$.
\end{defi}
A differential form $w$ in $A_{DR}^{\bullet}(\log \left( D\right) )$ is a smooth complex valued differential form on $M-D$ whose extension on $M$ admits some singularites along $D$ of degree $1$. More precisely it can be viewed as a non-smooth complex differential form on $M$ such that
\begin{enumerate}
\item for any given holomorphic coordinates $z=\left( z_{1}, \dots, z_{n}\right)\: : \: \C^{n}\to U\subset M$ such that $U\cap D=\emptyset$, $w$ can be written as an ordinary smooth complex valued differential forms
\[
\sum f(z_{1}, \dots z_{n})d{z_{i_{1}}\wedge\dots \wedge dz{i_{p}}}\wedge d{\overline{z}_{j_{1}}\wedge\dots \wedge d\overline{z}{j_{q}}}
\]
where $f$ is a smooth function over $U$.
\item For any given holomorphic coordinates $z=\left( z_{1}, \dots, z_{n}\right)\: : \: \C^{n}\to U\subset M$ near $p\in D$, such that $U\cap D_{i_{1}}\cap \dots \cap D_{i_{l}}$ is given by the equation $\prod^{q}_{i=1}z_{j_{i}}=0$; $w$ can be written as
\[
\sum w_{J}\wedge \frac{dz_{j_{1}}}{z_{j_{1}}}\wedge\cdots \wedge\frac{dz_{j_{l}}}{z_{j_{l}}}
\]
for $j_{i}\in \left\lbrace 1, \dots, q \right\rbrace $ and where $w_{J}$ is a smooth complex valued differential forms on $U$.
\end{enumerate}
The graded vector space $A_{DR}^{\bullet}(\log \left( D\right))$ equipped with the differential $d$ and wedge product $\wedge$ is a commutative differential graded algebra. Its cohomology gives the cohomology of $M-{D}$.
\medskip
\begin{prop}[\cite{Deligne2}]\label{delignholom}
The inclusion $M-D\hookrightarrow M$ induces a map
\[
A_{DR}^{\bullet}(\log \left( D\right) )\to A_{DR}^{\bullet}(M-D )
\]
which is a quasi-isomorphism of differential graded algebras.
\end{prop}
\begin{defi}
Let $\mathrm{Diff}_{Div}$ be the category of complex manifolds equipped with a normal crossing divisors, i.e the objects are pairs $(M,\mathcal{D})$ where $\mathcal{D}$ is a normal crossing divisor of $M$ and the maps are holomorphic maps $f\: : \: \left(M,\mathcal{D} \right)\to \left(N,{\mathcal{D}'}\right) $ such that
\[
f^{-1}(\mathcal{D}')\subset {\mathcal{D}}.
\]
A \emph{simplicial manifold $M_{\bullet}$ with a simplicial normal crossing divisor $\mathcal{D}_{\bullet}$ }is a simplicial objects in $\mathrm{Diff}_{Div}$.
\end{defi}
So given a simplicial manifold with divisor $(M_{\bullet}, \mathcal{D}_{\bullet})$, then $\left( M-D\right)_{\bullet}$ defines by $\left( M-\mathcal{D}\right)_{n}=M_{n}-\mathcal{D}_{n}$ is again a simplicial manifold and $A_{DR}\left(\left( M-\mathcal{D}\right)_{\bullet}\right)$ and $A_{DR}\left( \log\left( \mathcal{D}_{\bullet}\right) \right)$ are cosimplicial commutative dg algebras.
\medskip
\begin{prop}\label{simplicialcrossdiv}
The inclusion $\left( M-\mathcal{D}\right)_{\bullet} \hookrightarrow M_{\bullet}$ induces a map between chain complexes
\[
\operatorname{Tot}_{N}\left(A_{DR}(\log \left( \mathcal{D}_{\bullet}\right) ) \right)\to\operatorname{Tot}_{N}\left(A_{DR}\left(\left( M-\mathcal{D}\right)_{\bullet}\right) \right)
\]
which is a quasi-isomorphism.
\end{prop}
\begin{proof}
This is a direct consequence of Proposition \eqref{delignholom} and Lemma below. For a double complex $\left( A^{\bullet,\bullet},d,d'\right), $ we define a new double complex $\left(\left( E^{A} _{1}\right)^{\bullet,\bullet},d_{1}, {d'}_{1}\right) $, where $\left( E^{A}_{1}\right)^{p,q}:=H^{p}\left(A^{\bullet,q},d \right)$, $d_{1}=d$ and ${d'}_{1}=0$.
\begin{lem}[\cite{Dupont}, Lemma 1.19]\label{spectralcomplexes}
Let $f\: : \: A^{\bullet, \bullet}\to B^{\bullet,\bullet}$ be a morphism of double complex. Assume that $A^{p,q}=B^{p,q}=0$ if $p$ or $q$ are negative and that $f\: : \: E^{A}_{1}\to E^{B}_{1}$ is an isomorphism between cochain complexes. Then $f$ induces an isomorphism in the cohomology of the total complex.
\end{lem}
\end{proof}
\subsection{Restriction to ordinary flat connections on $M_{0}$}\label{restr}
Let $M_{\bullet}$ be a simplicial manifold with connected cohomology. Then $A_{DR}(M_{\bullet})$ is a cosimplicial unital commutative differential graded algebra. By Theorem \ref{functor} $\operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet})\right)$ is an unital $C_{\infty}$-algebras. We denote its structure by ${m}_{\bullet}$ where ${m}_{1}=D$ such that $D$ is the differential on $\operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet})\right)$ defined on elements of bidegree $(p,q)$ by
\[
D(a)=\tilde{\partial} a+(-1)^{p}da,
\]
where $\tilde{\partial}$ is differential obtained by the alternating sum of the pullback of the cofaces maps of the simplicial manifold. We denote the unit by $1$. It corresponds to the constant function at $1$ inside $A_{DR}^{0}(M_{0})$.
\begin{defi}\label{reducedhomologicalparifed}
Let $B\subseteq \left( \operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet})\right), m_{\bullet}\right)$ be a $C_{\infty}$-subalgebra. Let $W$ be a non-negatively graded vector space of finite type such that $W^{0}=\C$.
\begin{enumerate}
\item Let $\left( W, m_{\bullet}^{W}\right)$ be a unital minimal $C_{\infty}$-algebra. Let $\delta$ be the corresponding differential induced on $T^{c}\left(W_{+}[1] \right) $. Let $g_{\bullet}\: : \:\left( W, m_{\bullet}^{W}\right)\to B$ be a unital $C_{\infty}$-map and let $C$ be the corresponding Maurer-Cartan element in $C\in \operatorname{Conv}_{C_{\infty}}\left(\left(W_{+}, m_{\bullet}^{W_{+}}\right) ,\left(B, m_{\bullet} \right) \right)$
We call $(C, {\delta}^{*})$ the \emph{reduced homological pair associated to $g_{\bullet}$} or simply\emph{ reduced homological pair with coefficients in $B$}. If $g_{\bullet}\: : \:\left( W, m_{\bullet}^{W}\right)\to \left( B, m_{\bullet}\right)$ is a minimal model and $B$ is a model, we call $(C, {\delta}^{*})$ \emph{a good reduced homological pair associated to $g_{\bullet}$ } or simply \emph{ good reduced homological pair with coefficients in $B$}.
\item Let $\left( W, m_{\bullet}^{W}\right)$ be a unital minimal $1-C_{\infty}$-algebra. Let $\delta$ be the corresponding differential induced on $\left( T^{c}\left(W_{+}[1] \right)\right)^{0}$.
Let $f_{\bullet}\: : \:\left( W, m_{\bullet}^{W}\right)\to \mathcal{F}\left(B, m_{\bullet} \right)$ be a unital $1$-$C_{\infty}$-map and let $\overline{C}$ be the corresponding Maurer-Cartan element in $\operatorname{Conv}_{1-C_{\infty}}\left(\left(W_{+}, m_{\bullet}^{W_{+}}\right) ,\left(B, m_{\bullet} \right) \right)$. We call it the \emph{ degree zero geometric connection associated to $f_{\bullet}$} or simply \emph{ degree zero geometric connection with coefficients in $B$}. If $f_{\bullet}\: : \:\left( W, m_{\bullet}^{W}\right)\to \mathcal{F}\left(B, m_{\bullet} \right)$ is a $1$-minimal model and $B$ is a $1$-model, we call $\overline{C}$ \emph{a good degree zero geometric connection with coefficients in $B$}.
A $1$-minimal model $f_{\bullet}\: : \:\left( W, m_{\bullet}^{W}\right)\to \mathcal{F}\left(B, m_{\bullet} \right)$ is said to be holomorphic (with logarithmic singularities) if the image of $f_{n}$ contains only holomorphic elements (with logarithmic singularities). If $f_{\bullet}\: : \:\left( W, m_{\bullet}^{W}\right)\to \mathcal{F}\left(B, m_{\bullet} \right)$ is holomorphic (with logarithmic singularities) we say that $\overline{C}$ is holomorphic (with logarithmic singularities).
\end{enumerate}
\end{defi}
\begin{rmk}\label{masseyprod}
Let $M_{\bullet}$ be a simplicial manifold with connected and finite type cohomology. Let $f_{\bullet}\: : \:\left( W, m_{\bullet}^{W}\right)\to \mathcal{F}\left( \operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet})\right), m_{\bullet}\right)$ be a $1$-minimal model. By Proposition \ref{homotopyuniqueness}, the $1-C_{\infty}$ structure $m_{\bullet}^{W}$ is unique up to isomorphism. This products can explicit determined via the homotopy transfer theorem.
\end{rmk}
We fix a $C_{\infty}$-subalgebra $B\subset \left( \operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet})\right), m_{\bullet}\right)$ with connected and finite type cohomology and let $\overline{C}$ be the degree zero geometric connection associated to a $f_{\bullet}\: : \:\left( W, {m'}_{\bullet}\right)\to \mathcal{F}\left(B, m_{\bullet} \right)$. Let $
w_{1}, \dots, w_{i}, \dots$
be a basis of $W_{+}$. Let $X_{1}, \dots X_{n}, \dots$ be the basis of $\left( W[1]\right)^{*}$ dual to $ s\left( w_1 \right) ,\dots s\left( w_{n}\right) , \dots\in W_{+}[1]$. Let $\mathcal{R}_{0}\subset \widehat{\mathbb{L}}\left( \left( W_{+}^{1}[1]\right)^{*} \right)$ be the completed Lie ideal generated by ${\delta'}^{*}$. Hence $\overline{C}$ can be written as
\begin{equation}\label{formulaconnection}
\overline{C}=\sum{v_{i}X_i}+\sum{v_{ij}X_{i}X_{j}}+\dots+\sum{v_{i_{1}\dots i_{r}}X_{i_{1}}\dots X_{i_{r}}}+\dots \in B^{1} \hat{\otimes} \left( \widehat{\mathbb{L}}(\left( W^{1}_{+}[1]\right)^{*})/\mathcal{R}_{0}\right) .
\end{equation}
We show how to construct a flat connection starting from $\overline{C}$. Consider the projection $${r}\: : \:B\to B^{0,1}\subset A_{DR}\left(M_{0} \right) $$ that sends forms of bidegree $(p,q)$ to $0$ if $p\neq 0$ and preserves forms of bidegree $(0,q)$. Consider $A_{DR}\left(M_{0} \right)$ equipped with its differential graded algebra structure. Then
\[
\operatorname{Conv}_{1-C_{\infty}}\left( \left( W_{+}, {m'}_{\bullet}\right) , A_{DR}\left(M_{0} \right) \right)
\]
is a differential graded Lie algebra. In particular, it is a ordinary convolution Lie algebra (compare with \cite{lodayVallette}).
\begin{prop}\label{pres}
The pushforward along $r$ induces a map
\[
r_{*}:=r\widehat{\otimes}\mathrm{Id}\: : \: B \hat{\otimes} \left( \widehat{\mathbb{L}}(\left( W^{1}_{+}[1]\right)^{*})/\mathcal{R}_{0}\right) \to A_{DR}\left(M_{0} \right) \hat{\otimes} \left( \widehat{\mathbb{L}}(\left( W^{1}_{+}[1]\right)^{*})/\mathcal{R}_{0}\right)
\]
which is in $\left( L_{\infty}-ALG\right)_{p}$.
\end{prop}
\begin{proof}
By Corollary \ref{corresiduemap}, we have
\[
r{m}_{n}(a_{1}, \dots, a_{n})={m}_{n}(ra_{1}, \dots, ra_{n})=\begin{cases}
d ra_{1}, & \text{ if }n=1,\\
{m}_{2}(ra_{1}, ra_{2}), & \text{ if }n=2,\\
0, & \text{ otherwise }.
\end{cases}
\]
for $a_{1}, \dots, a_{n}\in B^{1}$. Let $\overline{C}\in B \hat{\otimes} \left( \widehat{\mathbb{L}}(\left( W^{1}_{+}[1]\right)^{*})/\mathcal{R}_{0}\right) $ be a Maurer-Cartan element and let
$${C}'\in\left(A_{DR}^{1}(M_{0})\right)\widehat{\otimes} \widehat{\mathbb{L}} \left( \left( W^{1}_{+}[1]\right)^{*}\right), \quad \text{ resp. } \tilde{C}\in\left(A_{DR}^{0}(M_{1})\right)\widehat{\otimes} \widehat{\mathbb{L}} \left( \left( W^{1}_{+}[1]\right)^{*}\right)$$ be such that $\overline{C}={C}'+\tilde{C}$. In particular, ${r_{*}}C=C'$ and
\begin{align*}
0& =r_{*}\left( D C+\sum_{k>1}\frac{{l}_{k}\left(C,\dots,C\right)}{k \text{!}}\right) \\
& = \left( d r_{*}C+\sum_{k>1}\frac{{l}_{k}\left(r_{*}C,\dots,r_{*}C\right)}{k \text{!}}\right)\\
& = \left( d r_{*}C+\frac{{l}_{2}\left(r_{*}C,r_{*}C\right)}{2}\right),\\
\end{align*}
i.e., $r_{*}C$ is a Maurer-Cartan element in $\operatorname{Conv}_{1-C_{\infty}}\left( \left( W_{+}, {m'}_{\bullet}\right) , A_{DR}\left(M_{0} \right) \right).$
\end{proof}
\begin{rmk}\label{mapr}
Let $M_{\bullet}=M_{\bullet}G$ for some complex manifold $M$ and discrete group $G$. The morphism of simplicial manifolds $ M_{\bullet}\left\lbrace e\right\rbrace \to M_{\bullet}G$ given by the inclusion gives the map
\begin{equation}
r\: : \: \operatorname{Tot}_{N}\left(A_{DR}\left( M_{\bullet}G\right) \right)\to A_{DR}\left( M\right)
\end{equation}
which is a strict morphism of $C_{\infty}$-algebras.
\end{rmk}
A Lie algebra $\mathfrak{u}$ is said to be \emph{pronilpotent} if is the projective limit of finite dimensional Lie algebras
\[
\mathfrak{u}\cong \varprojlim_{i}\left( \mathfrak{u}/I^{i}\right),
\]
where $I^{\bullet}$ is defined via $I^{0}=\mathfrak{u}$, $I^{i}:=\left[I^{i-1}, \mathfrak{u}\right]$ for $i\geq 1$.
\begin{cor}\label{corpronilpot}
Let $\mathfrak{u}':=\widehat{\mathbb{L}} \left( \left( W^{1}_{+}[1]\right)^{*}\right)/\mathcal{R}_{0}$.
\begin{enumerate}
\item We have $
\operatorname{Conv}_{1-C_{\infty}}\left( \left( W_{+}, {m'}_{\bullet}\right) , \left( A_{DR}\left(M_{0} \right)\right) \right)=\left( {l_{\bullet}},A_{DR}(M_{0})\widehat{\otimes} \mathfrak{u}'\right).
$\\
such that
$${l_{1}}=-d,\quad {l_{2}}=[-,-] ,\quad{l_{n}}=0,\text{ for }n>2,$$ where $[-,-]$ is the obvious Lie bracket on $A_{DR}^{\bullet}(M_{0})\widehat{\otimes} \mathfrak{u}'$. In particular $\mathfrak{u}'$ and $A_{DR}^{\bullet}(M_{0})\widehat{\otimes} \mathfrak{u}'$ are complete Lie algebras and $ \mathfrak{u}'$ is pronilpotent.
\end{enumerate}
\end{cor}
\begin{proof}
${\delta'}^{*}$ preserves the filtration given by the power of the augmented ideal $I$ in ${\mathbb{L}} \left( \left( W^{1}_{+}[1]\right)^{*}\right)$, since it satisfies the Leibniz rule and ${\delta'}^{*}I\subset I^{2}$ by ${m'}_{1}=0$. Hence $ \mathfrak{u}'$ can be written as a projective limit of finite dimensional nilpotent Lie algebras and the lie algebras $A_{DR}^{\bullet}(M_{0})\widehat{\otimes} \mathfrak{u}'$, $ \mathfrak{u}'$ are complete.
\end{proof}
For a Lie algebra $\mathfrak{u}'$ we define the adjoint action $\operatorname{ad}\: : \: \mathfrak{u}'\to \operatorname{End}\left(\mathfrak{u}' \right)$ via $\operatorname{ad}_{v}(w):=\left[v,w \right] $.
\begin{thm}\label{flatconnection on M0}
Let $M_{\bullet}$, $B$, $\overline{C}$ and $\mathfrak{u}'$ be as above. Consider the adjoint action $\operatorname{ad}$ of $\mathfrak{u}'$ on itself. Then $d-r_{*}\overline{C}$ defines a flat connection on the trivial bundle over $M_{0}$ with fiber $\mathfrak{u}'$, where the latter is considered to be equipped with the adjoint action. The connection is holomorphic (with logarithmic singularities) if so is $\overline{C}$.
\end{thm}
\begin{proof}
Since $r_{*}\overline{C}$ is a Maurer-Cartan element it defines a flat connection.
\end{proof}
\begin{rmk}\label{Chenhomologicalpair}
In \cite[Theorem 1.3.1]{extensionChen} Chen constructs a good homological pair $\left(C, \delta^{*} \right)$ on a complex manifold $M$ such that $d-\pi(C)$ is a flat connection on $M\times\mathfrak{u}$ and $d-\pi(C)$ satisfies Theorem \ref{Chenthm} (see \cite[Theorem 2.1.1]{extensionChen}). By definition, such a homological pair corresponds to a minimal model $g_{\bullet}\: : \:\left( W, m_{\bullet}^{W}\right)\to \left( A_{DR}(M), d, \wedge\right)$. In \cite{Huebsch}, it is shown that $g_{\bullet}$ is constructed via the homotopy transfer theorem. In \cite{Prelie}, it is shown that $g_{\bullet}$ has a quasi-inverse $f_{\bullet}$ (explicitly constructed) such that $f_{\bullet}g_{\bullet}=\mathrm{Id}_{W}$.
\end{rmk}
\begin{defi}
Let $\overline{C}$ be a good degree zero geometric connection. We call $\mathfrak{u}=\widehat{\mathbb{L}} \left( \left( W^{1}_{+}[1]\right)^{*}\right)/\mathcal{R}_{0}$ the \emph{fiber Lie algebra of the simplicial manifold} $M_{\bullet}$.
\end{defi}
The next proposition shows that the fiber Lie algebra is well-defined.
\begin{prop}\label{fibermalcev} Let $M_{\bullet}$ and $N_{\bullet}$ be simplicial manifolds with finite type connected cohomology. Assume that there is a smooth map $p_{\bullet}\: : \: \:N_{\bullet}\to M_{\bullet}$ that induces a quasi-isomorphism in cohomology.
\begin{enumerate}
\item The fiber Lie algebra of $M_{\bullet}$ is isomorphic to the fiber Lie algebra of $N_{\bullet}$.
\item Let $G$ be a discrete group acting properly and discontinuously on a complex manifold $M$. In this case the quotient $M/G$ is again a complex manifold. Consider the action groupoid $M_{\bullet}G$. Assume that the cohomology of $M/G$ is of finite type. The fiber Lie algebra of $M_{\bullet }G$ is the Malcev Lie algebra of $\pi_{1}\left( M/G\right) $.
\end{enumerate}
In particular the fiber Lie algebra is independent (up to isomorphism) by the choice of good degree zero geometric connection.
\end{prop}
\begin{proof}
We prove 1. We fix a good reduced homological pair $(C, {\delta}^{*})$ with coefficients in $\operatorname{Tot}_{N}\left( A_{DR}\left(M_{\bullet } \right)\right) $ and a good reduced homological pair $(C', {\delta'}^{*})$ with coefficients in $\operatorname{Tot}_{N}\left( A_{DR}\left(N_{\bullet} \right)\right) $. In particular,
\[
p^{*}\: : \:\operatorname{Tot}_{N}\left( A_{DR}\left(N_{\bullet } \right)\right) \to \operatorname{Tot}_{N}\left( A_{DR}\left(M_{\bullet } \right)\right)
\]
is a quasi-isomorphism and a strict $C_{\infty}$-map.
We get a diagram
\[
\begin{tikzcd}
\operatorname{Tot}_{N}\left( A_{DR}\left(M_{\bullet } \right)\right)\arrow[r, "p^{*}"]\arrow[d, , "{g}_{\bullet}"]
& \operatorname{Tot}_{N}\left( A_{DR}\left(N_{\bullet } \right)\right)\arrow[d, , "{g'}_{\bullet}"]\\(W ,m_{\bullet}^{W})\arrow[u, shift right, "f_{\bullet}"]& ( W' ,m_{\bullet}^{W'})\arrow[u, shift right, "{f'}_{\bullet}"].
\end{tikzcd}
\]
of $C_{\infty}$-quasi isomorphism. In particular the map $(g'pf)_{\bullet}$ is an isomorphism. It follows that the two fiber Lie algebras are isomorphic. Let $\overline{C}$ be a good degree zero geometric connection with coefficients in $\operatorname{Tot}_{N}\left( A_{DR}\left(M_{\bullet } \right)\right)$. Its fiber Lie algebra is isomorphic to the fiber Lie algebra of $\pi(C)$ via the isomorphism $K^{*}$ as constructed in \eqref{mapKdual}. Let $\overline{C}'$ be a good degree zero geometric connection with coefficients in $\operatorname{Tot}_{N}\left( A_{DR}\left(N_{\bullet } \right)\right)$, by the same reasoning we get that its fiber Lie algebra is isomorphic to the one of $\overline{C}$. We prove 2. We set $N_{\bullet}=M_{\bullet }G$ and $M_{\bullet}$ is the constant simplicial manifold $M/G$. There is a quasi-isomorphism
\[
i\: : \:A_{DR}\left(M/G \right) \to \operatorname{Tot}_{N}\left( A_{DR}\left(M_{\bullet } G\right)\right).
\]
We consider the good homological pair $\left(C, \delta^{*} \right)$ on constructed by Chen as in Remark \ref{Chenhomologicalpair}. Then it defines a flat connection on $M/G$ whose fiber is the Malcev Lie algebra of $\pi_{1}\left( M/G\right) $. The statement follows from 1.
\end{proof}
Let $(N,\mathcal{D})$ be a complex manifold with a normal crossing divisor. Let $G$ be a discrete group acting smoothly on $N$ and that preserves $\mathcal{D}$. Then $(N-\mathcal{D})_{\bullet}G$ is a complex manifold equipped with a simplicial normal crossing divisor.
\begin{cor}[\cite{Morgan}]\label{Morgancor}
Let $(N,\mathcal{D})$ and $G$ be as above. Assume that $N/G$ is a connected projective non-singular variety with a normal crossing divisor $\mathcal{D}/G$. Then the fiber Lie algebra $\mathfrak{u}$ of $(N-\mathcal{D})_{\bullet}G$ is given as follows: there exists a graded vector space $V=V^{1}\oplus V^{2}$ concentrated in degree $1$ and $2$ and a completed homogeneous ideal $\mathcal{J}\subset\widehat{\mathbb{L}\left(V\right)}$ where the degree of the generators is $2$, $3$ and $4$ such that
\[
\mathfrak{u}= \widehat{\mathbb{L}\left(V\right)}/\mathcal{J}
\]
\end{cor}
\begin{proof}
This is Corollary 10.3 in \cite{Morgan}.
\end{proof}
\section{Connections and bundles}\label{newadded}
Let $M$ be a complex manifold. Let $M_{\bullet}G$ be an action groupoid where $G$ is discrete and it acts properly and discontinuosly on $M$. Assume that the chomology of $M/G$ is connected and of finite type. The results of the previous section allow the construction of the desired flat connections on $M/G$.
\subsection{Gauge equivalences}\label{gauge equiv}
The next lemma is a standard exercise about filtered vector spaces.
\begin{lem}\label{lemmapronilp}
Let $\mathfrak{u}$ be a pronilpotent graded Lie algebra concentrated in degree $0$ and let $\left( A,d,\wedge\right) $ be a non-negatively graded differential graded algebra.
\begin{enumerate}
\item The vector space $A\widehat{\otimes }\mathfrak{u}$ is a differential graded Lie algebra where the differential is given by the tensor product and the brackets are defined via
\[
[a\otimes v, b\otimes w]:=\pm \left( a\wedge b\right) \otimes [v,w],
\]
where the signs follow from the signs rule.
\item The Lie algebra $A^{0}\widehat{\otimes }\mathfrak{u}$ is complete with respect to the filtration induced by $I$.
\item Let $\Omega(1)$ be the differential graded algebra of polynomials forms on the interval $[0,1]$. Consider the differential graded Lie algebra $\Omega(1)\widehat{\otimes }\left( A\widehat{\otimes }\mathfrak{u}\right) $ obtained as in point 1. Then, there is a canonical isomorphism
\[
\Omega(1)\widehat{\otimes }\left( A\widehat{\otimes }\mathfrak{u}\right) \cong \left(\Omega(1)\otimes A\right)\widehat{\otimes }\mathfrak{u}.
\]
\end{enumerate}
\end{lem}
Let $\mathfrak{u}$ be a pronilpotent differential graded Lie algebra. The pronilpotency guarantees that the (complete) universal enveloping algebra $\mathbb{U}\left( \mathfrak{u}^{0}\right) $ is a complete Hopf algebra and its group-like elements can be visualized as $U:=\exp\left(\mathfrak{u}^{0} \right) $. Assume that $\mathfrak{u}$ is concentrated in degree zero and that it is equipped with the trivial differential. Let $A$ be a differential graded commutative algebra an assume that it is non-negatively graded. We consider the complete differential graded Lie algebra $A\widehat{\otimes}\mathfrak{u}$. The universal enveloping algebra of $\mathfrak{u}$ is equipped with the filtration induced by $I^{\bullet}$. In particular $U$ is complete with respect to such a filtration and the exponential map can be extended to
\[
Id\widehat{\otimes }\exp\: : \: A^{0}\widehat{\otimes}\mathfrak{u}\to A^{0}\widehat{\otimes}U\subset A^{0}\widehat{\otimes}\mathbb{U}\left( \mathfrak{u}^{0}\right)
\]
By abuse of notation we denote the above map again by $\exp$. The complete tensor product gives to $ A^{0}\widehat{\otimes}\mathbb{U}\left( \mathfrak{u}^{0}\right) $ the structure of an associative algebra. The image of $Id\widehat{\otimes }\exp$ is again a group where the inverse of $e^{h}$ is given by $e^{-h}$ for a $u\in A^{0}\widehat{\otimes}\mathfrak{u}$. This group acts on the set of Maurer-Cartan elements $MC\left(A^{0}\widehat{\otimes}\mathfrak{u} \right) $ via
\[
e^{h}(\alpha):=e^{\operatorname{Ad}_{h}}(\alpha)+\frac{1-e^{\operatorname{Ad}_{h}}}{\operatorname{Ad}_{h}}(dh).
\]
We call this action the \emph{gauge-action} and the above group the \emph{gauge group}. If $A=A_{DR}(M)$ for some complex manifold $M$, then $d-\alpha$ defines a flat connection on $M\times \mathfrak{u}$ and the action of the gauge group can be written as $e^{h}(\alpha)=e^{h}(d-\alpha)e^{-h}$.
\begin{defi}
Two Maurer-Cartan elements $\alpha_{0},\alpha_{1}$ are said to be \emph{gauge equivalent} in $A\widehat{\otimes}\mathfrak{h}$ if there is a $h$ in $A^{0}\widehat{\otimes}\mathfrak{u}$ such that $e^{h}(\alpha_{0})=\alpha_{1}$.
\end{defi}
\begin{prop}\label{equivMaurer}
Let $A\widehat{\otimes }\mathfrak{h}$ be as above. Two Maurer-Cartan elements are gauge equivalent if and only if they are homotopy equivalent in the sense of Definition \ref{Linftydef}.
\end{prop}
\begin{proof}
We have that $A\widehat{\otimes }\mathfrak{h}$ is a complete Lie algebra with respect to $I^{\bullet}$. The result follows from \cite[Section 2.3]{Yalin}.
\end{proof}
\subsection{Factors of automorphy}\label{factorofautomorphy}
We show how to construct a flat connection on the quotient. We use the same point of view of \cite{Hain}. Let $X$ be a set and $G$ a group acting on it from the left. Let $V$ be a vector space. A \emph{factor of automorphy} is a map $F\: : \:G\times X\to \operatorname{Aut}(V) $ such that $g\: : \: X\times V\to X\times V$ defined by
$
(x,v)\mapsto (gx,F_{g}(x)v)
$
gives a group action of $G$ on $X\times V$. This is equivalent to
\[
F_{gh}(x)(v)=F_{g}(hx)F_{h}(x).
\]
Let $M$ be a complex manifold equipped with a smooth and properly discontinuous action of a discrete group $G$ (from the left). Let $V$ be a vector space and let $F\: : \:G\times M\to \operatorname{Aut}(V) $ be a factor of automorphy. Then $F$ induces a $G$-action on $M\times V$. In particular, the quotient
\[
\left( M\times V\right) /G
\]
is a vector bundle on $M/G$ where the sections satisfy $s(gx)=F_{g}(x)s(x).$ We denote by $E_{F}$ the vector bundle induced by the factor of automorphy $F$.
\begin{prop}\label{welldefined}
Let $M$ be a complex manifold equipped with a smooth action of a group $G$. Let $V$ be a complete vector space and let $F\: : \:G\times M\to \operatorname{Aut}(V) $ be a factor of automorphy. Let $\alpha\in A^{1}_{DR}(M)\widehat{\otimes }\operatorname{End}(V)$ be a $1$-form with vaules in $\operatorname{End}(V)$.
\begin{enumerate}
\item The connection $d-\alpha$ induces well-defined connection on $E_{F}$ if
\[
d-g^{*}\alpha=F_{g}\left(d-\alpha \right)F^{-1}_{g}
\]
for any $g\in G$.
\item For a smooth path $\gamma\: : \: [0,1]\to M$ we denote with $T(\gamma)$ the parallel transport along $\alpha$ of $d-\alpha$ cosnidered as a connection over $M\times V$. Assume that $\alpha$ is well-defined on $E_{F}$, Then
\[
T(g\gamma)={F}_{g}(\gamma(0))T(\gamma){F}_{g}(\gamma(1))^{-1}.
\]
\end{enumerate}
\end{prop}
\begin{proof}
The two statements are in \cite{Hain}( Proposition 5.1 and Proposition 5.7 respectively).
\end{proof}
Let $M$, $G$ be as above. We fix a pronilpotent Lie algebra $\mathfrak{u}$ concentrated in degree zero and we set $U=\exp(\mathfrak{u})$. Let $\alpha\in A^{1}_{DR}(M)\widehat{\otimes }\operatorname{End}(\mathfrak{u})$ be a well-defined connection form on a $E_{F}$. We fix a $p\in M$ and we denote its class in $M/G$ by $\overline{p}$. By covering space theory, this choice induces a group homomorphism
\[
\rho\: : \: \pi_{1}\left(M/G,\overline{p} \right) \to G,
\]
which is constructed via the homotopy lifting property. For a path starting at $\overline{p}$, we denote its (unique) lift starting at $p$ with $c_{\gamma}$. In particular, $c_{\gamma}(1)=\rho(\gamma)p$. Given two loops $\gamma_{1}, \gamma_{2}$ starting at $\overline{p}$ on $M/G$, we have $c_{\gamma_{1}\cdot \gamma_{2}}=c_{\gamma_{1}}\cdot\left(\rho(\gamma_{1}) \gamma_{2}\right) $ (see \cite{Hain}, Lemma 5.8).
\begin{prop}\label{holonomyrep}
Let $U=\exp(\mathfrak{u})$ and let $p$ be as above. The parallel transport $T$ of $\alpha$ on $M\times \mathfrak{u}$ induces a group homomorphism
\[
\Theta_{0}\: : \: \pi_{1}\left(M/G,\overline{p} \right) \to U
\]
given by $\Theta_{0}\left(\gamma \right):=T(c_{\gamma})F_{\rho(\gamma)}(p)$.
\end{prop}
\begin{proof}
See Proposition 5.9 in \cite{Hain}.
\end{proof}
We call $\Theta_{0}$ the\emph{ holonomy representation} of $\alpha$ on $M/G$. Consider the action groupoid $M_{\bullet}G$ and assume that the cohomology of $M/G$ is connected and of finite type. Let $$g_{\bullet}\: : \: \left(W, m_{\bullet}^{W} \right)\to \mathcal{F}\left( \operatorname{Tot}_{N}^{\bullet}\left(A_{DR}(M_{\bullet}G)\right) ,m_{\bullet} \right)$$
be a $1$-minimal model and let $\overline{C}$ be its associated good degree zero geometric connection. By the results of the previous section, $\overline{C}$ induces a flat connection form $r^{*}\overline{C}\in A_{DR}^{1}(M)\widehat{\otimes}\mathfrak{u}$ such that $d-r^{*}\overline{C}$ is a flat connection on the trivial bundle on $M$ with fiber $\mathfrak{u}$. By repeating the same reasoning for another $1$-minimal model $${g'}_{\bullet}\: : \: \left(W', m_{\bullet}^{W'} \right)\to\mathcal{F}\left( \left(A_{DR}(M/G)\right) ,m_{\bullet} \right)\subset \mathcal{F}\left( \operatorname{Tot}_{N}^{\bullet}\left(A_{DR}(M_{\bullet}G)\right) ,m_{\bullet} \right)$$
we get a good degree zero geometric connection $\overline{C}'$ and a flat connection form $r^{*}\overline{C}'\in A_{DR}^{1}(M)\widehat{\otimes}\mathfrak{u}'$ such that $d-r^{*}\overline{C}'$ is a flat connection on the trivial bundle on $M$ with fiber Lie algebra $\mathfrak{u}'$.
Let ${g''}_{\bullet}\: : \: (W'',m_{\bullet}^{W''})\to \left( \operatorname{Tot}_{N}^{\bullet}\left(A_{DR}(M_{\bullet}G)\right) ,m_{\bullet} \right)$ be a $C_{\infty}$-algebra minimal model. Assume that there exist a quasi-isomorphism $f_{\bullet}\: : \: \left(A_{DR}\left( M_{\bullet}G\right) ,m_{\bullet} \right)\to(W'',m_{\bullet}^{W''})$ such that $f_{\bullet}\circ g''_{\bullet}=\mathrm{Id}_{W}$. Let $\overline{C''}$ be its associated good degree zero geometric connection. By the results of the previous section, $\overline{C''}$ induces a flat connection form $r^{*}\overline{C''}\in A_{DR}^{1}(M)\widehat{\otimes}\mathfrak{u}$ such that $d-r^{*}\overline{C''}$ is a flat connection on the trivial bundle on $M$ with fiber $\mathfrak{u''}$.
\begin{prop}\label{gaugeuniqueness}
Let $g_{\bullet}$ and ${g'}_{\bullet}$ be as above.
\begin{enumerate}
\item There exists a morphism of Lie algebras $K^{*}\: : \: \mathfrak{u}'\to \mathfrak{u}$
such that $$r_{*}k^{*}\overline{C}',r_{*}\overline{C}\in A_{DR}(M)\widehat{\otimes } \mathfrak{u}$$ are gauge equivalent. Assume that $M_{\bullet}G=(N-\mathcal{D})_{\bullet}G$, where $\left( N,\mathcal{D}\right) $ is a complex manifold with a normal crossing divisors and $G$ is a group acting holomorphically on $N$ and that preserves $\mathcal{D}$. If ${g'}_{\bullet}$ and $g_{\bullet}$ are holomorphic with logarithmic singularities, the gauge $e^{h'}$ is in $ \exp\left(A^{0}_{DR}(M)\widehat{\otimes }\mathfrak{u}\right)$.
\item There exists a morphism of Lie algebras ${K''}^{*}\: : \: \mathfrak{u}''\to \mathfrak{u}$
such that $$r_{*}{k''}^{*}\overline{C}'',r_{*}\overline{C}\in A_{DR}(M)\widehat{\otimes } \mathfrak{u}$$ are gauge equivalent. Moreover the morphism depends only by $\mathcal{F}\left( f_{\bullet}\right) $ and $g_{\bullet}$. Assume that $M_{\bullet}G=(N-\mathcal{D})_{\bullet}G$, where $\left( N,\mathcal{D}\right) $ is a complex manifold with a normal crossing divisors and $G$ is a group acting holomorphically on $N$ and that preserves $\mathcal{D}$.If ${g''}_{\bullet}$ and $g_{\bullet}$ are holomorphic with logarithmic singularities, the gauge $e^{h''}$ is in $ \exp\left(A^{0}_{DR}(M)\widehat{\otimes }\mathfrak{u}\right)$.
\end{enumerate}
\end{prop}
\begin{proof}
The proof of 1. is an application of Proposition \ref{homotopyuniqueness}, Proposition \ref{pres} and Proposition \ref{equivMaurer}. The second part follows by the definition of $A_{DR}^{0}(\log D)$. We prove 2. We have a diagram of $1-C_{\infty}$-algebras
\[
\begin{tikzcd}
\mathcal{F}({W''}, m_{\bullet}^{W''})\arrow[rrr, shift right, "\mathcal{F}\left( {g''}_{\bullet}\right) "']&&&\mathcal{F}\left( \operatorname{Tot}_{N}^{\bullet}\left(A_{DR}(M_{\bullet}G)\right) ,m_{\bullet} \right)
\arrow[lll, shift right, " \mathcal{F}\left( f_{\bullet} \right) "'] \\ \\ \\
({W}, m_{\bullet}^{W})\arrow[uuu, shift right, "{k''}_{\bullet}"]\arrow[uuurrr, shift right, "g_{\bullet}"']
\end{tikzcd}
\]
where ${k''}_{\bullet}=\mathcal{F}(f_{\bullet})g_{\bullet}$. Notice that ${g''}_{\bullet}f_{\bullet}$ is the identity in cohomology since
\[
[{g''}_{\bullet}]^{-1}=[{f}_{\bullet}][{g''}_{\bullet}][{g''}_{\bullet}]^{-1}=[{f}_{\bullet}].
\]
Then there exist an inverse $t_{\bullet}\: : \: \left( \operatorname{Tot}_{N}^{\bullet}\left(A_{DR}(M_{\bullet}G)\right) ,m_{\bullet} \right)\to \left( \operatorname{Tot}_{N}^{\bullet}\left(A_{DR}(M_{\bullet}G)\right) ,m_{\bullet} \right)$ such that $t_{\bullet}{g''}_{\bullet}f_{\bullet}$ is homotopic to the identity via an homotopy $H_{\bullet}$. In particular
${g''}_{\bullet}f_{\bullet}$ is homotopic to $t_{\bullet}{g''}_{\bullet}$ via $H_{\bullet}{g''}_{\bullet}f_{\bullet}$.
Let $\overline{C}_{1}$, $\overline{C}_{2}$ be the Maurer-Cartan elements corresponding to $\mathcal{F}({g''}_{\bullet}f_{\bullet})g_{\bullet}$ and $\mathcal{F}(t_{\bullet}{g''}_{\bullet}f_{\bullet})g_{\bullet}$. By Proposition \ref{homotopyuniqueness}, Proposition \ref{pres} and Proposition \ref{equivMaurer} we get that $r_{*}\overline{C}_{1}$ is gauge equivalent to $r_*\overline{C}_{2}$ which is gauge equivalent to $r_{*}\overline{C}$. By Baker-Campbell-Hausdorff formula $BCH(-,-)$ on $\mathfrak{u}$ we conclude that $r_{*}\overline{C}_{1}$ is gauge equivalent to $r_{*}\overline{C}$. We have $r_{*}\overline{C}_{1}=r_{*}\left(\mathrm{Id}\otimes {K''}^{*} \right) \overline{C}''=r_{*}{k''}^{*}\overline{C}''$ where ${K''}^{*}$ is the isomorphism of Lie algebras constructed in Proposition \ref{beh} induced by $k_{\bullet}=\mathcal{F}(f_{\bullet})g_{\bullet}$. This conclude the proof.
\end{proof}
\begin{rmk}\label{explicitcalculation} If ${g''}_{\bullet}$ is a minimal model constructed via the homotopy transfer theorem (for instance the $1$-minimal model constructed in Remark \ref{Chenhomologicalpair}). In \cite{Prelie}, Theorem 5 there is an explicit formula for $ f_{\bullet}$.
\end{rmk}
\begin{prop}\label{fiber}
Let $M$, $G$ be as above. We fix a pronilpotent Lie algebra $\mathfrak{u}$ concentrated in degree zero. Let $\alpha'$ be a Maurer-Cartan elements in $A_{DR}(M)\widehat{\otimes} \mathfrak{u}$ such that it defines a well-defined flat connection on the bundle $E_{F'}$ with fiber $\mathfrak{u}'$ and factor of automorphy ${F'}_{g}(p)=Id$. Let $\alpha$ be a Maurer-Cartan element in $A_{DR}(M)\widehat{\otimes} \mathfrak{u}$. Assume that it is gauge equivalent to $\alpha'$ via some $h\in A^{0}_{DR}(M)\widehat{\otimes} \mathfrak{u}$. Then $\alpha$ is a well-defined connection form on the bundle $E_{F}$, where $F$ is given by ${F}_{g}(p):=e^{-h(gp)}e^{h(p)}$. Assume that there exists a finite dimensional positively graded vector space $V$ and a ideal $\mathcal{J}\subset \mathbb{L}\left(V \right)$ such that $\mathfrak{u}=\mathbb{L}\left(V \right)/\mathcal{J}$. Then if $\alpha$ and the group action are holomorphic, so is ${F}_{g}(p)$.
\end{prop}
\begin{proof}
${F}_{g}(p)$ is clearly a factor of automorphy. We have
\begin{align*}
e^{-g^{*}h}e^{h}\left( d-\alpha\right) e^{-h}e^{g^{*}h}& = e^{-g^{*}h}\left( d-\alpha'\right) e^{g^{*}h} \\
&=e^{-g^{*}h}\left( d-g^{*}\alpha'\right) e^{g^{*}h} \\
&=d-g^{*}\alpha.
\end{align*}
We prove the second part. By Baker-Campbell-Hausdorff formula $BCH(-,-)$ on $\mathfrak{u}$ we can define $k(p):= BCH(-g^{*}h(p),h(p))$. In particular, $k\in A^{0}_{DR}(M)\widehat{\otimes} \mathfrak{u}$.
We assume that $\alpha$ is holomorphic. Since
\begin{align*}
d-g^{*}\alpha & =e^{-g^{*}h}e^{h}\left( d-\alpha\right) e^{-h}e^{g^{*}h}\\
& = e^{k}\left( d-\alpha\right) e^{k} \\
&=e^{\operatorname{Ad}_{k}}(\alpha)+\frac{1-e^{\operatorname{Ad}_{k}}}{\operatorname{Ad}_{k}}(dk) \\
\end{align*}
and $dk=\sum_{i}\left( \frac{\partial}{\partial z_{i}}kd{z}_{i}+\frac{\partial}{\partial \overline{z}_{i}}kd\overline{z}_{i}\right)$, we set $\overline{\partial}k:=\sum_{i}\left( \frac{\partial}{\partial \overline{z}_{i}}kd\overline{z}_{i}\right)$. Let $\left\lbrace \mathcal{I}^{i}\right\rbrace $ be the filtration induced by the degree on $\widehat{\mathbb{L}}\left(V \right)$. Let $\underline{k}\in A^{0}_{DR}(M)\widehat{\otimes} \widehat{\mathbb{L}}\left(V \right)$ be a preimage of $k$ under the projection $A^{0}_{DR}(M)\widehat{\otimes}\widehat{\mathbb{L}}\left(V \right)\to \mathfrak{u}A^{0}_{DR}(M)\widehat{\otimes} \mathfrak{u}$. By above, we have $\frac{1-e^{\operatorname{Ad}_{\underline{k}}}}{\operatorname{Ad}_{\underline{k}}}\overline{\partial}\underline{k}\in \mathcal{J}$. The map $\frac{1-e^{\operatorname{Ad}_{\underline{k}}}}{\operatorname{Ad}_{\underline{k}}}\overline{\partial}$ defines an automorphisms on $A^{0}_{DR}(M)\widehat{\otimes}\widehat{\mathbb{L}}\left(V \right)$ and its inverses preserves $\mathcal{J}$. In particular, $\overline{\partial}\underline{k}\in \mathcal{J}$. We write $\underline{k}=\sum_{i=1}^{\infty}\underline{k}_{i}$ where each $\underline{k}_{i}$ is has homogeneous degree in $\widehat{\mathbb{L}}\left(V \right)$. Then $\overline{\partial}\underline{k}_{1}=\mathcal{J}\cap\mathcal{I}^{1}$ and
\[
\overline{\partial}\underline{k}_{1}=k_{1}^{\operatorname{hol}}+{k'}_{1}
\]
where ${k'}_{1}\in A^{0}_{DR}(M)\widehat{\otimes} \mathcal{J}$ and $k_{1}^{\operatorname{hol}}\in \widehat{\mathbb{L}}\left(V \right)$ is a primitive element whose coefficients are holomorphic functions. We set $\underline{k}^{\operatorname{hol},1}:=k_{1}^{\operatorname{hol}}+\underline{k}_{2}+\dots$. Continuing in this way we get a
$\underline{k}^{\operatorname{hol}}:= \underline{k}^{\operatorname{hol},\infty}\in A^{0}_{DR}(M)\widehat{\otimes}\widehat{\mathbb{L}}\left(V \right)$ whose coefficients are holomorphic functions. Let $k^{\operatorname{hol}}$ be its projection on $A^{0}_{DR}(M)\widehat{\otimes}\mathfrak{u}$. The gauge action of $k^{\operatorname{hol}}$ corresponds to the gauge action of $k$ and hence
\[
{F}_{g}(p):=e^{-h(gp)}e^{h(p)}=e^{k(p)}=e^{k^{\operatorname{hol}}(p)}\in \operatorname{Aut}\left( \mathfrak{u}\right)
\]
\end{proof}
Notice that there is canonical bundle isomorphism $E_{F'}\to E_{F}$ defined by
\begin{equation}\label{bundlemap}
(p,v)\mapsto (p,e^{{-h(gp)}}e^{{h(p)}}v).
\end{equation}
\begin{defi}
Let $M$ be a complex manifold and let $E$, $E'$ be two smooth vector bundles on $M$. Let $\left(d-\alpha,E \right) $, $\left(d-\alpha',E' \right) $ be two smooth connections. They are \emph{isomorphic} if there exists a (holomorphic) bundle isomorphism $T\: : \: E\to E'$ such that
\[
T\left( d-\alpha\right)T^{-1}=d-\alpha'.
\]
Assume that $E$, $E'$ are bundles whose fiber is a pronilpotent Lie algebra $\mathfrak{u}$ and that $\alpha,\alpha'\in A^{1}_{DR}(M)\widehat{\otimes }\mathfrak{u}$ (we consider to be equipped with the adjoint action). An isomorphism $T\left(d-\alpha,E \right) \to\left(d-\alpha',E' \right) $ is said to be \emph{induced by a (holomorphic) gauge} if there exists a (holomorphic) $h\in A^{0}_{DR}(M)\widehat{\otimes }\mathfrak{u} $ such that $T=e^{h}$.
\end{defi}
Let ${g'}_{\bullet}\: : \: \left(W', m_{\bullet}^{W'} \right) \to A_{DR}(M/G)$ be the minimal model (in the sense of $C_{\infty}$-algebras) obtained by the good homological pair $(C', { \delta'}^{*})$ and constructed by Chen as in Remark \ref{Chenhomologicalpair}. In particular it has a quasi-inverse $f_{\bullet}$ such that $f_{\bullet}{g'}_{\bullet}=\mathrm{Id}_{W}$. Let $\pi(C')$ be the good degree zero geometric connection associated to ${g'}_{\bullet}$ and let $\mathfrak{u}'$ be its fiber Lie algebra. Let $f_{\bullet}\: : \: \left(W, m_{\bullet}^{W} \right)\to \mathcal{F}\left( \operatorname{Tot}_{N}^{\bullet}\left(A_{DR}(M_{\bullet}G)\right) ,m_{\bullet} \right)$
be a $1$-minimal model and let $\overline{C}$ be its associated good degree zero geometric connection with fiber Lie algebra $\mathfrak{u}$. Consider the situation of Proposition \ref{gaugeuniqueness}, then there is an isomorphism of Lie algebra $K^{*}\: : \: \mathfrak{u}'\to \mathfrak{u}$, a Maurer-Cartan element $C\in A_{DR}(M)\widehat{ \otimes}\mathfrak{u}$ and a gauge $h\in A^{0}_{DR}(M)\widehat{\otimes }\mathfrak{u}$ such that $e^{h}\left(d- r_{*}\overline{C}\right)e^{-h}= k^{*}\pi{C'}$.
\begin{thm}\label{wehaveabundle}
Let $\pi(C')$ be the good degree zero geometric connection associated to $g_{\bullet}$ and let $\mathfrak{u}'$ be its fiber Lie algebra. Assume that there is a triple $(\tilde{C},K*, h)$ consisting in an isomorphism of Lie algebra $K^{*}\: : \: \mathfrak{u}'\to \mathfrak{u}$, a Maurer-Cartan element $\tilde{C}\in A_{DR}(M)\widehat{ \otimes}\mathfrak{u}$ and a gauge $h\in A^{0}_{DR}(M)\widehat{\otimes }\mathfrak{u}$ such that $e^{h}\left(d- \tilde{C}\right)e^{-h}= k^{*}\pi{C'}$ as Maurer-Cartan elements in $A_{DR}(M)\widehat{ \otimes}\mathfrak{u}$.
\begin{enumerate}
\item $h$ induces a smooth factor of automorphy $F_{g}(p):=e^{-h(gp)}e^{h(p)}$ such that $d- \tilde{C}$ is a well-defined flat connection on $M/G$ on the bundle $E_{F}$ where the fiber corresponds to the Malcev completion of $\pi_{1}(M/G)$.
\item Assume that $\tilde{C}$ has holomorphic coefficients and the group action on $M$ is holomorphic. Then $h$ and $F$ are holomorphic and $d- \tilde{C}$ is a well-defined holomorphic flat connection on $M/G$ on the holomorphic bundle $E_{F}$.
\item Assume that $M_{\bullet}G=(N-\mathcal{D})_{\bullet}G$, where $\left( N,\mathcal{D}\right) $ is a complex manifold with a normal crossing divisors and $G$ be a group acting holomorphically on $N$ and that preserves $\mathcal{D}$. Assume that $\tilde{C}$ is holomorphic with logarithmic singularities. Then $F\: : \:N\times \mathfrak{u}\to N\times \mathfrak{u}$ is holomorphic and $d- \tilde{C}$ is a well-defined holomorphic flat connection with logarithmic singularities on $M/G$ on the holomorphic bundle $E_{F}$.
\end{enumerate}
In particular, any $1$-minimal model ${g}_{\bullet}$ for $\mathcal{F}\left( \operatorname{Tot}_{N}^{\bullet}\left(A_{DR}(M_{\bullet}G)\right) ,m_{\bullet}\right) $ produces a triple $(\tilde{C},K^*, h)$ where $\tilde{C}=r_{*}\overline{C}$, where $\overline{C}$ is the geometric degree zero connection associated to $g_{\bullet}$ and $K^{*}$ is completely determined by $\mathcal{F}(f_{\bullet})g_{\bullet}$.
\end{thm}
\begin{proof}
By Proposition \ref{fiber} and Proposition \ref{fibermalcev} and we get a flat connection $\left( d- r_{*}\tilde{C}, E_{F}\right) $ whose fiber $\mathfrak{u}$ is the Malcev Lie algebra of $\pi_{1}(M/G)$ and $\left( d-k^{*}\pi{C'}, M\times \mathfrak{u}\right)$ is isomorphic via a gauge to $\left( d- r_{*}\tilde{C}, E_{F}\right) $. We calculate its monodromy representation. We consider $d-k^{*}\pi{C'}$ as flat connection on $M\times \mathfrak{u}$. Its parallel transport $T^{K}$ is defined via iterated integrals (see \cite{IteratedChen}) and gives a map from the path space $PM$ of $M$ to $\widehat{T}\left( {W'}^{1}_{+} [1]\right) ^{*}/{\bar{\mathcal{R}}'}_{0}$, i.e the complete universal enveloping algebra of $\mathfrak{u}'$. In particular, we have $T^{K}(\gamma)=K^{*}T(\gamma)$ where $\gamma\in PM$ and $T'$ is the parallel transport of $d-\pi{C'}$ considered as a flat connection on $M\times \mathfrak{u}$. Let $T$ be the parallel transport of $d-r_{*}\tilde{C}$ considered as a flat connection on $M\times \mathfrak{u}$. Since they are gauge equivalent via $h$ we have
\[
T(\gamma)=e^{-h(\gamma(0))} T^{K}(\gamma)e^{h(\gamma(1))}\in \widehat{T}\left( {W}^{1}_{+}[1]\right) ^{*}/{\bar{\mathcal{R}}}_{0}
\]
for a path $\gamma\: : \: [0,1]\to M$. We fix a $p\in M$ and we choice a representative $\overline{p}$ for its class in $M/G$. We denote with $ \rho\: : \: \pi_{1}\left(M/G,\overline{p} \right) \to G$ its induce group homomorphism. Let $\gamma$ be a loop based at $\overline{p}$ and let $c_{\gamma}$ be its unique lift starting at $p$. We denote with ${\Theta}_{0}$ the holonomy representation of $d-r_{*}\tilde{C}$ and by ${\Theta'}_{0}$ the holonomy representation of $d-\pi{C'}$, then
\begin{align*}
{\Theta'}_{0}(\gamma) & = T^{K}(c_{\gamma})e^{-h(\rho(\gamma)p)}e^{h(p)}\\
& = e^{-h(c_{\gamma}(0))}T(c_{\gamma})e^{h(c_{\gamma}(1))}e^{-h(\rho(\gamma)p)}e^{h(p)}\\
& = e^{-h(p)}T(c_{\gamma})e^{h(p)}\\
& = e^{-h(p)}{\Theta}_{0}(\gamma)e^{h(p)}.
\end{align*}
Let $U=\exp\left(\mathfrak{u}\right) $. Since $K^{*}$ is an Hopf algebra isomorphism, it preserves group-like elements. The above calculation shows that the monodromy representation $\Theta_{0}\: : \: \pi_{1}\left(M/G,\overline{p} \right) \to U$ is conjugate with $K^{*}{\Theta'}_{0}$, in particular it corresponds to the Malcev completion of the fundamental group. The second statement follows by Proposition \eqref{fiber}. We prove the last statement. By Proposition \eqref{fiber} we have a well-defined map $F_{g}(p)=e^{-h(gp)}e^{h(p)}$ where $h$ is holomorphic on $N-\mathcal{D}$. By Proposition \ref{gaugeuniqueness}, we have that $h$ is smooth on $N$. Hence by Cauchy's formula in several complex variables we conclude that $h$ is holomorphic on $N$ and then $F_{g}(p)=e^{-h(gp)}e^{h(p)}$ is the factor of automorphy of a holomorphic bundle on $N/G$. The final part follows directly by Proposition \ref{gaugeuniqueness}.
\end{proof}
\begin{rmk}\label{FinalRMK0}
Given a $1$-minimal model ${g}_{\bullet}$ for $\mathcal{F}\left( \operatorname{Tot}_{N}^{\bullet}\left(A_{DR}(M_{\bullet}G)\right) ,m_{\bullet}\right) $ and let $\tilde{C}$ be its degree zero geometric connection. Then in general there may be different choices for $K^{*}$ and $h$.
\end{rmk}
Let $M$, $G$ be as above and let $\mathfrak{u}$ be a pronilpotent Lie algebra concentrated in degree zero. Let $\alpha\in A^{1}_{DR}(M)\widehat{\otimes} \mathfrak{u}$ such that $d-\alpha$ be a well-defined connection on the bundle $E_{F}$, with fiber $\mathfrak{u}$. Let $K^{*}\: : \: \mathfrak{u}\to \mathfrak{u}'$ be a Lie algebra isomorphism. Then $d-\left(\mathrm{Id}\widehat{\otimes }K\right)\alpha=d-k^{*}\alpha$ is a well-defined connection on the bundle $E_{K^{*}F}$ with fiber $\mathfrak{u}'$.
\begin{thm}\label{wehaveabundleuniq}
Let $\tilde{C}_{1}$ and $\tilde{C}_{2}$ be two Maurer-Cartan elements satisfying the conditions of Theorem \ref{wehaveabundle} with factor of automorphy $F_{1}$ and $F_{2}$. Let $\mathfrak{u}_{1}$ and $\mathfrak{u}_{2}$ be the fiber Lie algebra of $\tilde{C}_{1}$ and $\tilde{C}_{2}$ respectively.
\begin{enumerate}
\item There exists a Lie algebra automorphism $K^{*}\: : \: \mathfrak{u}_{1}\to \mathfrak{u}_{2}$ such that $\left( d- k^{*} \tilde{C}_{2}, E_{K^{*}F_{2}}\right) $ is isomorphic via a gauge to $\left( d- \tilde{C}_{1}, E_{F_{1}}\right)$ as a flat connection on $M/G$.
\item Assume that $\tilde{C}_{1}$ and $\tilde{C}_{2}$ have holomorphic coefficients. The isomorphism is induced via a holomorphic gauge $e^{h}$, where $h\in A_{DR}(M)\widehat{ \otimes}\mathfrak{u}$ is a function with holomorphic coefficients.
\item Assume that $M_{\bullet}G=(N-\mathcal{D})_{\bullet}G$, where $\left( N,\mathcal{D}\right) $ is a complex manifold with a normal crossing divisors and $G$ is a group acting holomorphically on $N$ and that preserves $\mathcal{D}$. Assume that $\tilde{C}_{1}$ and $\tilde{C}_{2}$ have holomorphic coefficients with logarithmic singularities. The isomorphism is induced via a holomorphic gauge $e^{h}$, where $h\in A_{DR}(N)\widehat{ \otimes}\mathfrak{u}$ is a function with holomorphic coefficients.
\end{enumerate}
\end{thm}
\begin{proof}
Let $(C', { \delta'}^{*})$ be the good homological pair constructed by Chen as in Remark \ref{Chenhomologicalpair}. By Theorem \ref{wehaveabundle}, we have the following.
\begin{enumerate}
\item There is a Lie algebra isomorphism $\left( L'\right)^{*}\: : \: \mathfrak{u}'\to \mathfrak{u}_{1}$ such that $\left( d-(l')^{*}\pi{C'}, M\times \mathfrak{u}_{1}\right) $ and $\left( d- r_{*}\tilde{C}_{1}, E_{F_{1}}\right)$ are isomorphic via a gauge.
\item There is a Lie algebra isomorphism $\left( L\right)^{*}\: : \: \mathfrak{u}'\to \mathfrak{u}_{2}$ such that $\left( d-l^{*}\pi{C'}, M\times \mathfrak{u}_{2}\right) $ and $\left( d- r_{*}\tilde{C}_{2}, E_{F_{2}}\right)$ are isomoprhic via a gauge.
\end{enumerate}
An explicit calculation shows that $$\left(d-\left({l'}^{-1}l \right)^{*} \tilde{C}_{2} , E_{\left({L'}^{-1}L \right)^{*}F_{2}} \right)\text{ and }\left(d- r_{*}\tilde{C}_{1} , E_{F_{1}} \right)$$ are isomorphic via a gauge to $\left(d- r_{*}l^{*} \tilde{C}' , M\times \mathfrak{u}' \right)$. This prove point 1. We prove point 2. Assume that $\tilde{C}_{1}$ and $\tilde{C}_{2}$ have holomorphic coefficients. Then so are $\tilde{C}_{1} $ and $\left({l'}^{-1}l \right)^{*} \tilde{C}_{2}$. These two connection forms are gauge equivalent as Maurer-Cartan elements in $A^{0}_{DR}(M)\widehat{\otimes }\mathfrak{u}_{1}$ for a certain $h\in A^{0}_{DR}(M)\widehat{\otimes }\mathfrak{u}_{1}$. The same argument used in Proposition \ref{fiber} shows that the gauge is holomorphic. Point 3 follows by point 2 and Cauchy's formulas for holomorphic functions in several variables.
\end{proof}
\subsection{Formality of the fundamental group and Hopf algebra isomorphisms}
In \cite{Sibilia1}, we show that the KZB connection presented in \cite{Damien} on the configuration space of points on the punctured torus can be constructed via Theorem \ref{wehaveabundle}. More precisely, we construct a
a $1$-minimal model via the homotopy transfer theorem which gives degree zero geometric connection $\overline{C}$ such that $d-r_{*}\overline{C}$ is the KZB connection on the configuration space of points on the punctured torus. The monodromy of this connection is used in \cite{Damien} to shows the formality of the pure braids group on the torus.
This fact can be proved via Chen's theory. Let $\mathfrak{u}$ be a pronilpotent Lie algebra concentrated in degree zero. We denote its associated graded (with respect to the filtration $I^{\bullet}$) by $\operatorname{gr}\left( \mathfrak{u}\right)=\oplus_{i\geq 0}I^{i}/ I^{i+1}$ and the complete associated graded by $\widehat{\operatorname{gr}}\left( \mathfrak{u}\right)=\widehat{\oplus}_{i\geq 0}I^{i}/ I^{i+1}$. Notice that they are both filtered.
\begin{defi}[\cite{Damien}]
The Lie algebra $\mathfrak{u}$ is said to be \emph{formal} if there is an isomorphism of filtered Lie algebras $\mathfrak{u}\to \widehat{\operatorname{gr}}\left( \mathfrak{u}\right)$ whose associated graded is the identity. The group $U=\exp(\mathfrak{u})$ is \emph{formal} if so is $\mathfrak{u}$. A group $G$ is \emph{formal} if so is its Malcev completion.
\end{defi}
In particular, if there exists a positively graded Lie algebra $\mathfrak{t}$ and an isomorphism of filtered Lie algebras $\mathfrak{u}\to \widehat{ \mathfrak{t}}$ (called \emph{formality isomorphism}), then $\mathfrak{u}$ is formal and the map induced via its associated graded is an isomorhism of graded Lie algebra $\operatorname{gr}\left( \mathfrak{u}\right)\to \mathfrak{t}$.\\
Let $M$, $G$ be as in Theorem \ref{wehaveabundle}. We fix a $p\in M$ and we choice a representative $\overline{p}$ for its class in $M/G$. Let $\overline{C}$ be a good degree zero geometric connection.
The monodromy representation of $d-r_{*}\overline{C}$ gives a group homomorphism
\[
\Theta_{0}\: : \: \pi_{1}\left(M/G,\overline{p} \right) \to U\subset \widehat{T}\left( W_{+}^{1}[1]\right). ^{*}/\bar{\mathcal{R}}_{0}
\]
We consider the Hopf algebra $\C\left[ \pi_{1}\left(M/G,\overline{p} \right) \right]$. The above map can be extended into an Hopf-algebra morphism
\begin{equation}\label{Malcevcompletionintro}
\Theta_{0}\: : \: \C\left[ \pi_{1}\left(M/G,\overline{p} \right) \right] \to H^{0}\left( {T}\left(W[1]^{*}\right), \delta^{*}\right)
\end{equation}
Let $J$ be the kernel of the augmentation map $\C\left[ \pi_{1}\left(M/G,\overline{p} \right) \right]\to \C$ that sends each element of $\pi_{1}(M,p)$ to $1$. The powers of $J$ (with respect to the multiplication) define a filtration $J^{i}$ and the completion $\C\left[ \pi_{1}\left(M/G,\overline{p} \right) \right]^{\wedge }$ is a complete Hopf algebra. Theorem 2.1.1 in \cite{extensionChen} and Theorem \ref{wehaveabundle} implies the following\footnote{ Theorem 2.1.1 is not written in this way, but this fact is implies by the last isomorphism at page 209 in loc. cit.}: the map $\Theta_{0}$ preserves $J^{i}$ and it is the $J$-adic completion of $\C\left[\pi_{1}\left(M/G,\overline{p} \right) \right]$ as a Hopf algebra. Hence by looking at the group-like elements we conclude that $\Theta_{0}$ gives the Malcev completion of $\pi_{1}\left(M/G,\overline{p} \right) $. The following is immediate.
\begin{prop}\label{formality}
Assume that ${\mathcal{R}}_{0}$ is homogeneous, then $\pi\left(M/G, \overline{p} \right)$ is formal.
\end{prop}
In particular Chen'theory gives an explicit formality isomorphism by taking the $\log$ of the monodromy representation $\Theta_{0}$. The formality of the fundamental group of $M=\left( N-\mathcal{D}\right)/G$ where $\mathcal{D}$ is a normal crossing divisor in a smooth complex algebraic variety $N$ and $G$ is a discrete group acting smoothly on $N$ and preserving $\mathcal{D}$ follows from Corollary \ref{Morgancor} and this shows the formality of the braid group on the torus, which corresponds to the fundamental group of the configuration space of points of the punctured torus. Let $M$ be a complex manifold equipped with a smooth properly discontinuous action of a group $G$, we assume that the quotient is connected an that the cohomology is of finite type. Let $f_{\bullet}\: : \:\left( W, m_{\bullet}^{W}\right)\to \mathcal{F}\left( \operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right), m_{\bullet}\right)$ be a $1$-minimal model. The homogeneity of ${\mathcal{R}}_{0}$ can translated in terms of $m^{W}_{\bullet}$. Let $V_{n}\subset W^{2}$ be the subspace generated by $m_{n}^{W}(w_{1}, \dots,w_{n} )$ where $w_{i}\in W$ for $i=1, \dots, n$, then ${\mathcal{R}}_{0}$ has homogeneous generators if $\bigcap_{i=1}^{n}(V_{i})=0$.
\begin{rmk}
There is a topological interpretation of the $m^{W}_{n}$. Consider the singular cohomology $C^{\bullet}\left( M/G\right)$, in particular the cup products gives a differential graded algebra structure on $C^{\bullet}\left( M/G\right)$. In \cite{Camillo}, it is constructed an $A_{\infty}$-quasi-isomorphism between $C^{\bullet}\left( M/G\right)$ and $A_{DR}(M/G)$. Since $\left( A_{DR}(M/G),d,\wedge\right) \subset \left( \operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right), m_{\bullet}\right)$ is a strict $C_{\infty}$-quasi-isomorphism, it follows that $m^{W}_{n}$ corresponds to the higher Massey products in singular cohomology of $M/G$.
\end{rmk}
In a forthcoming paper, we analyze the case where the action of $G$ is not properly discontinuous and we give sufficient condition to have a rational Malcev completion.
\section{Appendix}
\subsection{Proof of Lemma \ref{Algebrainfinitystruct} and corollary \eqref{Linftystructnotquot}}\label{proof}
We start with the proof of Lemma \ref{Algebrainfinitystruct}.
\begin{proof}
Clearly $\partial^{2}=0$ and $M_{1}^{2}=0$. The relations \eqref{rel} for $n>1$ are equivalent to
\begin{equation}\label{rel2'}
\sum_{\substack{p+q+r=n\\
p+1+r>1, q>1}}(-1)^{p+qr}m_{p+1+r}\circ \left(Id^{\otimes p}\otimes m_{q}\otimes Id^{\otimes r} \right)= D\circ m_{n},
\end{equation}
where $D\circ m_{n}:=m_{1}\circ m_{n}+(-1)^{n+1}m_{n}\circ \left(\sum_{i=1}^{n-1} Id^{\otimes i}\otimes m_{1}\otimes Id^{\otimes n-1-i} \right)$. Chose homegeneous elements $f_{1}, \dots, f_{n}$. The expression \eqref{rel2'} in our case is
\begin{align*}
&\sum_{\substack{p+q+r=n\\
p+1+r>1, q>1}}(-1)^{p+qr}M_{p+1+r}\circ \left(Id^{\otimes p}\otimes M_{q}\otimes Id^{\otimes r} \right)(f_{1}\otimes \cdots\otimes f_{n}) =\\
& \sum_{\substack{p+q+r=n\\
p+1+r>1, q>1}}(-1)^{p+qr}\tilde{m}^{A}_{p+1+r} \left( \left(Id^{\otimes p}\otimes M_{q}\otimes Id^{\otimes r} \right)\circ (f_{1}\otimes \cdots\otimes f_{n})\right)\circ \Delta^{p+r} =\\
&\pm \sum_{\substack{p+q+r=n\\
p+1+r>1, q>1}}(-1)^{p+qr}\tilde{m}^{A}_{p+1+r} \left(f_{1}\otimes \cdots \otimes f_{p}\otimes \left( \tilde{m}^{A}_{q}\left(f_{p+1},\dots ,f_{p+q} \right)\circ \Delta^{q-1} \right) \otimes f_{p+q+1}\otimes \cdots \otimes f_{n} \right)\circ \Delta^{p+r} =\\
&\pm \sum_{\substack{p+q+r=n\\
p+1+r>1, q>1}}(-1)^{p+qr}\tilde{m}^{A}_{p+1+r} \left(f_{1}\otimes \cdots \otimes f_{p}\otimes \tilde{m}^{A}_{q}\left(f_{p+1},\dots ,f_{p+q} \right)\otimes f_{p+q+1}\otimes \cdots \otimes f_{n} \right)\circ \Delta^{n-1} =\\
& \sum_{\substack{p+q+r=n\\
p+1+r>1, q>1}}(-1)^{p+qr}\tilde{m}^{A}_{p+1+r} \left(Id^{\otimes p}\otimes \tilde{m}^{A}_{q}\otimes Id^{\otimes r} \right)\circ (f_{1}\otimes \cdots\otimes f_{n}) \circ \Delta^{n-1}.\\
\end{align*}
Set $m_{1}=\partial$, we have
\begin{align*}
\left( \partial\circ M_{n}+(-1)^{n+1}M_{n}\circ \left(\sum_{i=1}^{n-1} Id^{\otimes i}\otimes \partial\otimes Id^{\otimes n-1-i} \right)\right)(f_{1}\otimes \cdots\otimes f_{n}). \\
\end{align*}
The first summand is
\begin{align*}
& \partial\circ M_{n}(f_{1}\otimes \cdots\otimes f_{n})\\
&= \tilde{m}^{A}_{1}\tilde{m}^{A}_{n}(f_{1}\otimes \cdots\otimes f_{n})\circ\Delta^{n-1}+ (-1)^{|M_{n}(f_{1}\otimes \cdots\otimes f_{n})|+1}\tilde{m}^{A}_{n}(f_{1}\otimes \cdots\otimes f_{n})\circ\Delta^{n-1}\circ\delta,
\end{align*}
the second is
\begin{align*}
& (-1)^{n+1}M_{n}\circ \left(\sum_{i=1}^{n-1} Id^{\otimes i}\otimes \partial\otimes Id^{\otimes n-1-i} \right)(f_{1}\otimes \cdots\otimes f_{n})=\\
& (-1)^{n+1+|f_{1}|+\dots +|f_{p}|}\sum_{i=1}^{n-1}M_{n}\left( f_{1}\otimes \cdots \otimes f_{p}\otimes\left( \tilde{m}^{A}_{1} f_{p+1}+(-1)^{|f_{p+1}|+1}f_{p+1}\circ \delta \right) \otimes f_{p+2}\otimes \cdots \otimes f_{n} \right)=\\
& (-1)^{n+1}\sum_{i=1}^{n-1}\tilde{m}^{A}_{n}\left( Id^{\otimes p}\otimes \tilde{m}^{A}_{1} \otimes Id^{\otimes n-p-1}\right)(f_{1}\otimes \cdots\otimes f_{n})\circ \Delta^{n-1}+\\
&(-1)^{n+|f_{1}|+\dots +|f_{n}|} \tilde{m}^{A}_{n}(f_{1}\otimes \cdots\otimes f_{n})\circ \left(\sum_{i=1}^{n-1}Id^{\otimes p}\otimes \delta \otimes Id^{\otimes r} \right)\circ \Delta^{n-1} =\\
& (-1)^{n+1}\sum_{i=1}^{n-1}\tilde{m}^{A}_{n}\left( Id^{\otimes p}\otimes \tilde{m}^{A}_{1} \otimes Id^{\otimes n-p-1}\right)(f_{1}\otimes \cdots\otimes f_{n})\circ \Delta^{n-1}+\\
&(-1)^{n+|f_{1}|+\dots +|f_{n}|} \tilde{m}^{A}_{n}(f_{1}\otimes \cdots\otimes f_{n})\circ \Delta^{n-1}\circ \delta.
\end{align*}
Since $\tilde{m}_{\bullet}^{A}$ is an $A_{\infty}$-structure
\begin{align*}
& \sum_{\substack{p+q+r=n\\
p+1+r>1, q>1}}(-1)^{p+qr}\tilde{m}^{A}_{p+1+r} \left(Id^{\otimes p}\otimes \tilde{m}^{A}_{q}\otimes Id^{\otimes r} \right)\circ (f_{1}\otimes \cdots\otimes f_{n}) \circ \Delta^{n-1}=\\
& \tilde{m}^{A}_{1}\tilde{m}^{A}_{n}(f_{1}\otimes \cdots\otimes f_{n})\circ\Delta^{n-1}+(-1)^{n+1}\sum_{i=1}^{n-1}\tilde{m}^{A}_{n}\left( Id^{\otimes p}\otimes \tilde{m}^{A}_{1} \otimes Id^{\otimes n-p-1}\right)(f_{1}\otimes \cdots\otimes f_{n})\circ \Delta^{n-1}.
\end{align*}
This shows that $\left( \partial, M_{2}, M_{3},\dots\right) $ is an $A_{\infty}$-structure. The second summand is canceled by the second summand of the first summand and this shows that $\left( M_{\bullet}\right) $ is an $A_{\infty}$-structure. We prove the third statement. By direct calculation $n=1,2$, we have that $l_{n}$ and $l_{n}'$ are well-defined on
$L_{V[1]^{*}}\left(A \right)$. Let $n>2$, and
$f_{1}, \dots, f_{n}\in L_{V[1]^{*}}(A)$ be homogeneous elements. Let $\mu'(a,b)$ be a non trivial shuffle in $T^{c}(V[1])\otimes T^{c}(V[1])$. For any $n$ we have
\begin{align*}
&\tilde{m}_{n}^{A}\left(f_{1}\otimes \cdots\otimes f_{n} \right)\circ \Delta^{n-1}\circ\mu'(a,b) =\\
& = \tilde{m}_{n}^{A}\left(f_{1}\otimes \cdots\otimes f_{n} \right)\big(\sum_{i\neq j}^{n} 1\otimes \cdots 1\otimes a\otimes 1\cdots 1\otimes b\otimes 1 \cdots \otimes 1\\
&+ (-1)^{|a||b|}\sum_{i\neq j}^{n} 1\otimes \cdots 1\otimes b\otimes 1\cdots 1\otimes a \otimes 1 \cdots \otimes 1\big)\\
& = 0
\end{align*}
since $f_{i}(1)=0$ for each $i$.
\end{proof}
Proof of corollary \eqref{Linftystructnotquot}
\begin{proof}
The first claim is immediate. We prove the second statement. Let $f_{1},\dots,f_{n}\in \Hom^{\bullet}\left(T^{c}(V[1]^{0}), A \right)$ and assume that there is a $g$ with $\delta^{*}g=f_{i}$ for some $i$. Then
\begin{align*}
&\tilde{m}_{n}^{A}\left(f_{1}\cdots \otimes\delta^{*}g\otimes \cdots\otimes f_{n} \right)\circ \Delta^{n-1} =\\
& =\tilde{m}_{n}^{A}\left(f_{1}\cdots \otimes g\otimes \cdots\otimes f_{n}\right)\circ (Id\otimes\cdots \otimes \delta\otimes\cdots\otimes Id)\circ \Delta^{n-1} \\
&= \tilde{m}_{n}^{A}\left(f_{1}\cdots \otimes g\otimes \cdots\otimes f_{n}\right)\circ \left( \sum_{i=1}^{n}Id\otimes\cdots \otimes \delta\otimes\cdots\otimes Id\right) \circ \Delta^{n-1}\\
\end{align*}
since $\delta^{*}f_{i}=0$ for any $i$. Then
\begin{align*}
& \tilde{m}_{n}^{A}\left(f_{1}\cdots \otimes g\otimes \cdots\otimes f_{n}\right)\circ \left( \sum_{i=1}^{n}Id\otimes\cdots \otimes \delta\otimes\cdots\otimes Id\right) \circ \Delta^{n-1}=\\
&=\tilde{m}_{n}^{A}\left(f_{1}\cdots \otimes g\otimes \cdots\otimes f_{n}\right)\circ\Delta^{n-1}\circ \delta\\
&=\delta^{*}\circ\tilde{m}_{n}^{A}\left(f_{1}\cdots \otimes g\otimes \cdots\otimes f_{n}\right)\circ\Delta^{n-1}.\\
\end{align*}
The third assertion is straightforward.
\end{proof}
\subsection{Conormalized graded module}\label{cosimplicial modules}
The goal of this subsection is to construct an isomorphism $\psi\: : \: \operatorname{Tot}_{N}\left(A\right)\to \int^{[n]\in\boldsymbol{\Delta}}NC_{n}\otimes A^{n,\bullet }$ between differential modules. Fix a field $\Bbbk$ of charactersitic zero. Let $B^{\bullet}$ be a cosimplicial module. The \emph{conormalized graded module} $N(B)^{\bullet}$ is a (cochain) graded modules defined as follows,
\[
N(B)^{p}:=\begin{cases}
B^{0}, & \text{ if }p=0,\\
\bigcap_{i=0}^{p}\operatorname{Ker}\left(s^{i}_{p-1}\: : \: B^{p}\to B^{p-1} \right) , & \text{ otherwise}.
\end{cases}
\]
where $s^{p}_{i}$ for $i=0,\dots p-1$ are the codegenerancy maps.
The differential $\partial\: : \: N(B)^{p}\to N(B)^{p+1}$ is given by the alternating sum of the coface maps
\[
\partial=\sum_{i=0}^{p}(-1)^{i}d^{i}.
\]
In particular $N^{\bullet}$ is a functor from the category of cosimplicial modules $cMod$ toward the category of (cochain) differential graded modules $dgMod$.\\
A cosimplicial differential graded module is a cosimplicial objects in $A^{\bullet, \bullet}\in dgMod$ where the first slot denotes the cosimplicial degree and the second slot denotes the differential degree. It can be visualized as a sequence of cosimplicial modules
\[
\begin{tikzcd}
A^{\bullet,0 }\arrow{r}{d}& A^{ \bullet,1}\arrow{r}{d}& A^{ \bullet,2}\arrow{r}{d}&\dots
\end{tikzcd}
\]
If we apply the functor $N$ we turn the cosimplicial structure of each terms into a differential graded structures
\[\begin{tikzcd}
N\left( A^{\bullet,0 }\right)^{\bullet}\arrow{r}{d}& N\left( A^{ \bullet,1}\right)^{\bullet}\arrow{r}{d}& N\left( A^{ \bullet,2}\right)^{\bullet}\arrow{r}{d}&\dots
\end{tikzcd}\]
moreover since each cosimplicial maps commutes with the differentials, we get a bicomplex $\left(N(A), d, \partial \right) $, where $N(A)^{p,q}:=N(A^{p,\bullet})^{q}$. We define $\operatorname{Tot}_{N}\left(A\right)\in dgMod$ as the total complex associated to the bicomplex above. Explicitly an element $a\in \operatorname{Tot}\left(N(A) \right)^{k} $ is a collection
\[
(a_{0}, \dots , a_{k})\in A^{k,0}\oplus A^{1,k}\oplus
\dots A^{0,k}
\]
such that each $a_{i}$ is contained in the kernel of some codegenrancy map. The following is well-known.
\begin{lem}\label{isompsi}
Let $A^{\bullet,\bullet}$ be a cosimplicial differential graded module.
\begin{enumerate}
\item Let $v$ be an elements of bidegree $(p,q)$. Then each $v_{n}\in NC_{n}^{p}\otimes A^{n,q}$ is equal to zero for $p>n$.
\item $NC_{p}^{p}$ is a one dimensional module.
\item An element $v$ with bidegree $(p,q)$ is completely determined by
\[
v_{p}\in NC_{p}^{p}\otimes A^{p,q}.
\]
\item There is an isomorphism between differential modules $\psi\: : \: \operatorname{Tot}_{N}\left(A\right)\to \int^{[n]\in\boldsymbol{\Delta}}NC_{n}\otimes A^{n,\bullet }$ such that for $v$ with bidegree $(p,q)$ we have
\[
\psi(b)_{p}=b\in NC_{p}^{p}\otimes A^{p,q}
\]
\end{enumerate}
\end{lem}
\begin{proof}
The first two points are immediate.
Fix a $n$ and a $p\leq n$. Notice that each inclusion $[p]\hookrightarrow [n]$ is equivalent to an ordered string $0\leq i_{0}<i_{i}<\dots <i_{p}\leq n$ contained in $\left\lbrace 0,1,\dots, n\right\rbrace $. For each string $0\leq i_{0}<i_{i}<\dots <i_{p}\leq n$ we denote the associated inclusion by $\sigma_{i_{0}, \dots i_{p}}\: : \:[p]\hookrightarrow [n]$, and we define the maps $\lambda_{i_{0}, \dots, i_{p}}\: : \: \Delta[n]_{p}^{+}\to \Bbbk$, via
\[
\lambda_{i_{0}, \dots, i_{p}}(\phi):=
\begin{cases}
1\text{ if }\sigma_{i_{0}, \dots i_{p}}=\phi,\\
0,\text{ otherwhise}
\end{cases}
\]
Clearly $\left\lbrace \lambda_{i_{0}, \dots i_{p}}\right\rbrace _{0\leq i_{0}<i_{i}<\dots <i_{p}\leq n}$ is a basis of $NC_{n}^{p}$. It turns out that $v_{n}$ can be written as
\[
v_{n}=\sum_{\theta \in \Delta[n]_{p}^{+}}\lambda_{i_{0}, \dots, i_{p}}\otimes b^{i_{0}, \dots, i_{p}}
\]
for some $b^{i_{0}, \dots i_{p}}\in B^{n,q}$. Let $v_{p}\in NC_{p}^{p}\otimes A^{p,q}$. Since $NC_{p}^{p}$ is one dimensional we write $v_{p}=\lambda_{0, \dots, p}\otimes b$, for some $b\in A^{p,q}$. We shows that $b^{i_{0}, \dots i_{p}}$ is completely determined by $b$. Fix a $0\leq i_{0}<i_{i}<\dots <i_{p}\leq n$. Then the above relations read as follows
\[
\left( 1\otimes\sigma_{i_{0}, \dots i_{p}}^{*}\right)v_{p}=\left({\sigma_{i_{0}, \dots i_{p}}}_{*}\otimes 1 \right)v_{n}.
\]
In particular the map ${\sigma_{i_{0}, \dots i_{p}}}_{*}\: : \: NC_{n}^{p}\to NC_{p}^{p}$ is the linear map that sends $\lambda_{i_{0}, \dots i_{p}}$ to $\lambda_{{0}, \dots, {p}}$ and $\lambda_{j_{0}, \dots j_{p}}$ to $0$ for $\left( j_{0}, \dots j_{p}\right) \neq \left( i_{0}, \dots i_{p}\right) $.
Then
\begin{align*}
\lambda_{{0}, \dots ,{p}}\otimes \sigma_{i_{0}, \dots i_{p}}^{*}(b)=\left( 1\otimes\sigma_{i_{0}, \dots i_{p}}^{*}\right)v_{p}=\left( {\sigma_{i_{0}, \dots i_{p}}}_{*}\otimes 1 \right)v_{n}=\lambda_{{0}, \dots ,{p}}\otimes b^{i_{0}, \dots i_{p}}.
\end{align*}
Since each degree $k$ elements can be written uniquely as a sum of elements of bidegree $(p,q)$ with $p+q=k$ we get a natural isomorphism
$\psi\: : \: \operatorname{Tot}_{N}\left(A\right)\to \int^{[n]\in\boldsymbol{\Delta}}NC_{n}\otimes A^{n,\bullet }$ of differential graded modules.
\end{proof}
\subsection{The $C_{\infty}$-structure on the 2 dimensional simplex}\label{sec2dimsimplex}
This diagram originally defined in \cite{Dupont2} is intensively studied in \cite{Getz2}. We define two maps between simplicial differential graded module (see \cite{Dupont2})
\[
\begin{tikzcd}
E_{\bullet}\: : \: NC_{\bullet}^{\bullet}\arrow[r, shift right, ""]&\Omega^{\bullet}(\bullet)\: : \: \int_{\bullet}.\arrow[l, shift right, ""]
\end{tikzcd}
\]
\begin{enumerate}
\item Fix a $[n]\in \boldsymbol{\Delta}$. For each $p\leq n$ we define $\int_{n} \: : \: \Omega^{p}(n)\to NC_{n}^{p}$ via
\[
\int_{n}(w)(\sigma_{i_{0}, \dots, i_{p}}):= \int_{\Delta_{geo}[p]}\sigma_{i_{0}, \dots, i_{p}}^{*}(w)
\]
i.e, we pulled back $w$ along the smooth inclusion $\sigma_{i_{0}, \dots, i_{p}}\: : \: \Delta[p]_{geo}\to \Delta[n]_{geo}$ and we integrate along the geometric standard $p$-simplex. $\int \: : \: \Omega^{\bullet}(n)\to NC_{n}$ is indeed a map between graded modules. The Stokes'theorem implies
\[
\int_{n}(dw)(\sigma_{i_{0}, \dots, i_{p}})=\sum_{j=0}^{p}(-1)^{j}\int_{n}(w)(\sigma_{i_{0}, \dots, \hat{i_{j}}, \dots i_{p}}),
\]
i.e $\int_{n} \: : \: \Omega^{\bullet}(n)\to NC_{n}$ is a map between differential graded modules. Moreover, the above construction is compatibAle with the simplicial structure, i.e $\int_{\bullet} \: : \: \Omega^{\bullet}(\bullet)\to NC_{\bullet}^{*}$ is a map between simplicial differential graded modules.
\item We define the quasi inverse of $\int_{\bullet}$. We fix a $[n]\in \boldsymbol{\Delta}$. For each string $0\leq<i_{0}<i_{1}<\dots <i_{p}\leq n$ we define the \emph{Whitney elementary form} $\omega_{i_{0}, \dots, i_{p}}\in \Omega^{p}(n)$ via
\[
\omega_{i_{0}, \dots , i_{p}}:=k!\sum_{j=0}^{p}(-1)^{j}t_{i_{j}}dt_{i_{0}}\wedge \dots \wedge \hat{dt_{i_{j}}}\wedge \dots \wedge dt_{i_{p}}
\]
We define $E_{n}\: : \: NC_{n}^{p}\hookrightarrow \Omega^{p}(n)$ via
\[
E_{n}(\lambda):=\sum_{0\leq i_{0}< \dots < i_{p}\leq n} \lambda(\sigma_{i_{0}, \dots , i_{p}})\omega_{i_{0}, \dots , i_{p}}
\]
The above map defines a map between differential graded modules $E_{n}\: : \: NC_{n}\hookrightarrow \Omega^{\bullet}(n)$. Since the construction is compatible with the simplicial maps we get a map between simplicial differential graded modules $E_{\bullet}\: : \: NC_{\bullet}^{\bullet}\hookrightarrow \Omega^{\bullet}(\bullet)$. Moreover since
\[
\int_{\Delta[n]_{geo}}t_{1}^{a_{1}}\cdots t_{n}^{a_{n}}dt_{1}\wedge dt_{n}:=\frac{a_{1}!\cdots a_{n}!}{\left( a_{1}+\dots a_{n}+n\right)!}
\]
an easy computation demonstrates that
\[
\left( \int_{\bullet}\right) \circ E_{\bullet}=Id_{ NC_{\bullet}^{\bullet}}.
\]
\end{enumerate}
It remains to construct an explicit simplicial homotopy between $ E_{\bullet}\circ\left( \int_{\bullet}\right)$ and $\mathrm{Id}_{ \Omega^{\bullet}(\bullet)}$. We recall the construction of \cite{Dupont2} (see also \cite{Getz2}). Fix a $n$, for $0\leq i\leq n$ we define the map $\phi_{i}\: : \: [0,1]\times \Delta_{geo}[n]\to \Delta_{geo}[n]$ via
\[
\phi_{i}\left(u, t_{0}, \dots, t_{n}\right):=\left(t_{0}+(1-u)\delta_{i0}, \dots,t_{0}+(1-u)\delta_{in} \right)
\]
Let $\pi\: : \: [0,1]\times \Delta_{geo}[n]\to \Delta_{geo}[n]$ be the projection at the second coordinate and let $\pi_{*}\: : \: \Omega^{\bullet}(n)\to \Omega^{\bullet-1}(n)$ be the integration along the fiber. We define $h^{i}_{n}\: : \: \Omega^{\bullet}(n)\to \Omega^{\bullet-1}(n)$ via
\[
h_{n}^{i}(w):=\pi_{*}\circ\phi^{*}_{i}(w)
\]
We define $s_{n}\: : \: \Omega^{p}(n)\to \Omega^{p-1}(n)$ as follows:
\[
s_{n}w:=\sum_{j=0}^{p-1} \sum_{0\leq i_{0}<\dots<i_{j}\leq n}\omega_{i_{0}\dots i_{j}}\wedge h^{i_{j}}_{n}\dots h^{i_{0}}_{n}(w)
\]
We start with the proof of Proposition \ref{speriam}.
\begin{proof}
Recall that $\Omega^{\bullet}(2)$ is the free differential graded commutative algebra generated by the degree zero variables $t_{0}$, $t_{1}$ and $t_{2}$ modulo the relations
\[
t_{0}+t_{1}+t_{2}=1,\quad dt_{0}+dt_{1}+dt_{2}=0.
\]
On the other hand $NC_{2}^{0}$ is the vector space generated by $\lambda_{0}$, $\lambda_{1}$ and $\lambda_{2}$, $NC_{2}^{1}$ is generated by $\lambda_{01},\lambda_{02} $ and $\lambda_{12}$, and $NC_{2}^{2}$ is the one dimensional vector space generated by $\lambda_{0012}$. We have
\begin{enumerate}
\item $E_{2}\left( \lambda_{0}\right)=t_{0}$, $E_{2}\left( \lambda_{1}\right)=t_{1}$, and $E_{2}\left( \lambda_{2}\right)=t_{2}$.
\item $E_{2}\left( \lambda_{01}\right)=t_{0}dt_{1}-t_{1}dt_{0}$, $E_{2}\left( \lambda_{02}\right)=t_{0}dt_{2}-t_{2}dt_{0}$, and $E_{2}\left( \lambda_{12}\right)=t_{1}dt_{2}-t_{2}dt_{1}$.
\item $E_{2}\left( \lambda_{012}\right)=2t_{0}dt_{1}dt_{2}-2t_{1}dt_{0}dt_{2}+2t_{2}dt_{0}dt_{1}$.
\end{enumerate}
We prove a). For degree reason we have $m_{2}(\lambda_{01},\lambda_{02})=\mu_{01/02}\lambda_{012}$. By the homotopy transfer theorem (see \cite{kontsoibel}) we have
\begin{align*}
\mu_{01/02} &=\left( \int_{\Delta[2]}E_{2}\left( \lambda_{01}\right)E_{2}\left( \lambda_{02}\right)\right) \\
& = \int_{\Delta[2]} t_0 (t_0dt_1dt_2 -t_1 dt_0dt_2 +t_2 dt_0dt1 )\\
& =\frac{1}{6}
\end{align*}
and analogously the other cases.
\end{proof} | 63,796 |
How to tie a tie properly - Windsor Knot
The Low Down on Tie Tying
Tying a tie knot can be a frustrating time. Believe me you are not alone. Unless you use a tie on almost a daily basis you will probably forget every single time on how to tie a tie.
I am one of those people that only wears a tie on special occasions and every time I need it I really have to think hard to remember how to tie it.
I think this scenario is more common than not. Some men keep their ties tied up and just loosen them enough to get it around their head when taking it off and leave it that way until the next time they need it.
I don’t like doing that myself because I think the knot looks better if you tie it fresh for each use. I think it looks better when you tie a new knot each time.
One thing you must know when wanting to tie a tie is what kind of a knot are you going to use. Yes there are many different knots and ways to get those knots.
There are a few different types of tie knots that are quite common.
There is the half Windsor knot which is my favourite and probably the easiest to do, the Windsor knot, the four in hand knot and probably other knots but those are the most common and main stream to use.
The Differences in Tying a Tie
Each knot looks somewhat similar when tied up but there are slight differences in each.
Before we get started with learning how to tie a tie using the different knot styles an important part about neck tie tying to know is how long or where the end of the tie should end up once you are finished.
The bottom of your neck tie should just touch the bottom of your belt buckle. The type of knot you choose to use will affect the length of your tie, you will have to experiment and see which one works best for you.
If you can’t get it to that spot you should at least have the bottom of the tie going just past the top of your pants. Sometimes when you have a belly you might have to accept that your tie will be sitting a bit higher because it is being pushed out by your belly. Not a problem really. Just different builds between tie wearers.
If the bottom of your tie is above your pant line it just won’t look as good as it could. Anyone that knows anything about fashion will notice your short tie and point at you and start laughing out loud.
Ok maybe that won’t happen but I have you thinking now.
Another thing you might want to consider is putting a dimple in the knot of your tie. It just adds to the whole look making you look stylish and professional.
How To Tie The Half Windsor Knot
The half Windsor is a version of the Windsor knot. You don’t use as much tie for the knot so that makes this a good one to use if you need your tie to be longer.
Honestly the Half Windsor is a good knot to use for most if not all shirt styles. It is probably the easiest knot to tie and the most commonly used.
So I am going to show you how to tie a Half Windsor because I think it is the only knot you will ever need. I say this from experience.
There is a diagram and video at the bottom of my instructions to help you with getting your tie tied properly.
Step 1 in tying a half windsor knot
Take your tie and drape it over the back of your neck and hanging on either side of your chest. The wider part of your tie should be hanging on the right side of your chest about a foot lower than the narrow side.
Step 2
Take the wide part of your tie and cross it over the narrow part of your tie and then turn it back under the narrow end.
Bring the wide end up and put it through the front of the loop around your neck pulling it through and down.
A few different Tie knots
Step 3 & 4
Step 3
Taking the wide end and wrapping it around the front of the narrow end from the left side to the right.
Take the wide end and pull it up through the bottom of the loop and then down and through the loose knot.
step 4
Tighten up the knot slowly snugging it up to your neck. Make sure not to over tighten.
Hopefully your tie is laying at the correct length. If it is too short then you need to start over and this time start with the wider side a bit lower than a foot and see if that works better.
If the tie was too long start with the wide side not as far down as the last time.
Hope that helps. Good luck.
Great Hub and very informative. I agree that it makes such a huge difference in how a tie is tied. Your article is a perfect step-by-step guide.
Grant's World, this is a very informative hub. I do not know how many men I see on a daily basics with improperly tied ties or crooked ties around their neck. These guys all need to read this hub. Fortunately, I learned to tie my ties from a section in my Boy Scout manual years ago.
Instructions for tying a Half Windsor knot
| 324,146 |
Brandt’s Woodwind Quintet List – E
Eastman, Donna Kelly (1945- )
Sir Gawain and the Green Knight (1992)
Available from the composer: 6812 Dina Leigh Ct., Springfield, VA 22153 USA. See also: .
Based on the composer‘s opera of the same name. Duration: 14:00. Thesis (D.M.A.)—University of Maryland at College Park, 1992.
Eben, Petr (1929-2007)
Dechovy kvintet, Op. 34 (Woodwind Quintet) (1965)
Praha: Edition Supraphon, 1967 /
Toccata Classics
Duration: 12:50. Eben uses Czech folklore and some medieval musical elements. Movements: I. Monologo; II. Dialogo I; III. Coro I; IV. Dialogo II; V. Coro II. See also: and .
also:
Nocni hodiny (Night Hours) (1975) for woodwind quintet and orchestra
Artia
Panton
Scored for woodwind quintet, tuba, string orchestra, piano, percussion
with children‘s choir, ad lib. Recorded on Panton 8110 0037. Duration: 25:00.
Konzert “Hommage à Antonin Rejcha” (1975) for woodwind quintet and orchestra
Ebenhöh, Horst(1930- )
Divertipentephonien, für Bläserquintett, Op. 70/1 (1987)
See: . Duration: 10:00. Movements: Allegro; Moderato; Con moto; Poco adagio; Poco presto e staccato; poi legato; Lento; Allegro.
Eberhard, Dennis J. (1943- )
Paraphrases (1968)
Chicago, IL: Media Press, 2011
Manuscript: University of Illinois.”
Echeverria, Jesus
Cuatro Movimientos (2002)
Piles Editorial de Musica
Duration: 12:00. See also: .
Echevarria, Victorino (1898-1958)
Quinteto en Re menor
Manuscript: Archivo Sinfonico de la S.G.A.E., Madrid
Duration: 17:00.
Suite de Camara
Manuscript: Archivo Sinfonico de la S.G.A.E., Madrid
Eckartz, Hubert (1903-1962)
Bläserquintett
Recklinghausen: Iris, 1960‘s?
Eckert, Michael (1950- )
Wind quintet (2006)
Duration: 13:25.
Eckhardt-Gramatté, Sophie-Carmen (1899-1974)
Woodwind Quintet (1962-63)
Winnipeg: Estate S. C. Eckhardt-Grammatte, 1980 /
Toronto: Canadian Music Centre, 1963
Duration: 23:00. Movements: Entrada: Lento ma non troppo; Allegretto; Lento ma non troppo; Gioviale e sprituoso. Commissioned by the CBC. First performed by the Toronto Woodwind Quintet in 1963. Some sources list her birth date as 1902. See also:
also:
Nonet for woodwind quintet and string quartet (violin, viola, cello, double bass) (1966)
Commissioned by the University of Saskatchewan Regina Campus. Duration: 25:00. Movements: I. Vivo; II. Molto andante ma non adagio; III. Kaleidoscope.
Edel, Yitzhak (1896-1973)
Woodwind Quintet
Tel-Aviv: Israel Music Institute
Eder, Eguzki
Septet for Winds, Op. 55 for flute, oboe, clarinet, bassoon, 2 horns, trumpet
Eder, Helmut (1916-2005)
Quintet, Op 25 (1958)
Vienna: Doblinger, 1958
Duration: 10:15. 3 movements. Recorded by the Vienna Wind Quintet on Deutsche Grammophon.
Septuagesima instrumentalis (Zweites Bläserquintett), Op. 51 (1969) (Second Wind Quintet)
Wien: Verlag Doblinger 1970
Duration: 14:00. 4 movements. Recorded by the Vienna Wind Quintet.
Begegnung, Drittes Bläserquintett (Third Wind Quintet), Op. 91
Vienna: Doblinger, 1988
Duration: 17:00. Recorded on Sony Music Entertainment Austria GmbH
also:
Septett für Bläser, Hommage a Johannes Kepler, Op. 55 (woodwind quintet, 2nd horn, trumpet) (1970)
Wein: Verlag Doblinger
Duration: 13:00.
Ottetto breve, Op. 33 (flute/piccolo, oboe, clarinet, bassoon, string quartet)
Wein: Verlag Doblinger
Duration: 11:30.
Eder de Lastra, Erich (1933- )
Bläserquintet
Wien: Doblinger, 1958
Duration: 10:00.
Edlund, Mikael (1950- )
Music for Double Wind Quintet (Musik för dubbel blåskvintett)(1983-4)
Stockholm: Nordiska Musikforlaget and Wilhelm Hansen, 1984
Duration: 16:00. Although the WorldCat database gives the above publishers, this work doesn’t appear to be listed in any current catalogs. It is held by the Deutsche Nationalbibliothek in Leipzig, the Danish National Library and the The Royal Library of the Copenhagen University Library, if you want to find a copy.
Edwards, Paul
arrangements
Beautiful Savior
Integra Music, 23 Music Square East, Nashville, TN 37203 USA /
River Song Productions
Alternate parts for alto sax and bass clarinet.
Come Thou Fount
River Song Productions
(Extra bass clarinet part in lieu of bassoon)
God Rest Ye Merry Gentlemen
Integra Music, 23 Music Square East, Nashville, TN 37203 USA (out of business?)/
Norwalk, Calif.: River Song Productions , Anderson, IN: Distributed by Intrada /
Deckerville, MI: David E. Smith Publications
Music Group, 1998.
Substitute parts of clarinet for oboe; and bass clarinet for bassoon.
My Jesus I Love Thee
Integra Music, 23 Music Square East, Nashville, TN 37203 USA /
River Song Productions
Oh Come, All ye Faithful
River Song Productions
What Wondrous Love is This
Integra Music, 23 Music Square East, Nashville, TN 37203 USA
Edwards, Ross (1943- )
Wind Quintet No. 1 (1963)
See also: and
Wind Quintet No. 2 (1965)
Both works possibly published by London: Universal Editions, or in Sydney, Australia.
Duration: 8:30.
Maninya III
Universal Edition, [1996]
Duration: 10:00. Work was revised in 1985 and renamed Incantations.
Incantations (1985 / 2006)
Grosvenor Place, NSW: Australian Music Centre, 2006
Duration: 12:30; in 3 movements. The work was previously titled “Maninya III,” revised in 2006; and re-named “Incantations.”
The Laughing Moon, five bagatelles for wind quintet (2012)
Grosvenor Place, NSW: Australian Music Centre, 2006
Duration: 13:00. Movements: Laughing dance; Interlude with imaginary birds; Ecstatic dance; Moon song; Clapping dance.
also:
Sonata for Wind Quintet, Violin, Viola, Cello, & Harp (1967)
Eerola, Lasse (1945-2000)
Three Pieces for Wind Quintet (1994)
Helsinki: Suomalaisen musilikin tiedotuskeskus – Finnish Music Information Centre
Sorbus Suite (Sorbus-sarja) (1996)
Helsinki: Music Finland
Duration: 20:00.
Viisi väriä (Five Colours) (1998) for wind quintet
Helsinki: Music Finland
Duration: 10:00.
Neljä väriä puhallinkvintetille (Four colours for wind quintet) (1999)
Savonlinna, Finland: Modus Musiikki, 2000
Movements: Fustikki; Gambiiri; Gretti; Henna.
also:
“Harmio” (Berteroa incana) for wind sextet (instrumentation not listed)
Effinger, Cecil (1914-1990)
Quintet (1947)
Manuscript (USA)
Egge, Klaus (1906-1979)
Quintet for Winds No.1, Op. 13 (1939)
Drammen, Norway: H. Lyche, 1969, Sole agents, C.F. Peters
Duration: 12:00.
Quintet for Winds No 2, Op. 34 (1976)
Oslo: NB noter (Contemporary Norwegian sheet music)
See Norwegian Music Information Centre. Duration: 24:00.
Egidi, Arthur (1859-1943)
Quintett in B-flat major, Op. 18
Verlag für Musikalische Kultur und Wissenschaft, 1937
Egilsson, Arni
Blær (Breeze) (1997)
Port Hueneme, Calif.: Arnaeus Music, 1997
Egk, Werner (1901-1983)
Five Pieces (Fünf Stücke)
Mainz: Schott, 1974 /
San Antonio: Southern Music Co. /
Edition Musicus
Flute doubles on piccolo, oboe doubles on oboe d’amore and English horn, clarinet doubles on basset horn. If oboe d’amore and basset horn are not available, the players can substitute oboe and clarinet respectively, but piccolo and English horn are required. Duration: 15:00. Movements: Monolog; Choral; Mobile; Dialog; Finale.
Arthur Cohn in “The Literature of Chamber Music” notes that there is a great variety and sound between the different movements, calling it “colorful music” … “far from the usual patter and lighter affectations of so many wind quintets.” “…a welcome addition to the literature of the medium.”
also:
Divertissement for ten winds (1973)
Mainz: Schott Music
For 1 flute (doubling piccolo), 2 oboes, 2 clarinet (2nd doubles bass clarinet), 2 bassoons (2nd doubles contrabassoon), 2 horns, 1 trumpet. 3 movements: Contredanse (Allegro molto); Air (Lento); Rondeau (Molto allegro). Duration: 10:00.
Die Zaubergeige Overture (1980) for ten winds
Mainz: Schott Music
For 1 flute (doubling piccolo), 2 oboes (2nddoubles English horn), 2 clarinet, 2 bassoons (2nd doubles contrabassoon), 2 horns, 1 trumpet. This is an arrangement of the orchestral version of the overture by the composer.
Ehle, Robert C. (1939- )
North American Woodwind Quintet, Op. 59 (1980)
Austin, Texas: Jomar Press, 1999
Duration: 7:30. See also:
Ehmann, Wilhelm (1904-1989)
Bläser-intraden zum Wochenlied.
Kassel: Baerenreiter, 1957
Ehrlich, Abel (1915-2003)
4 Woodwind Quintets (1966, 1969, 1970, 1970)
possibly Tel Aviv: Israel Music Institute
Woodwind Quintet 28.2.91 (1991)
Tel Aviv: Israel Music Center
Duration: 8:00.
also:
Many Wonders (1977) for woodwind quintet and soprano
Einaudi, Ludovico (1955- )
Ai Margini dell’Aria
Milano: Ricordi, 1988 1982
Duration: 9:00.
Einbond, Aaron (1978- )
Arie (2004)
New York: American Music Center
Duration: 8:00.
Einem, Gottfried von (1918-1996)
Quintet, Op. 46 (1975)
London: Boosey & Hawkes, 1978
Duration: 13:30. First performed by the Vienna Wind Quintet. See also:
Eisler, Hanns (1898-1962)
Divertimento, Op. 4 (1923)
Wien: Universal Edition, 1977
2 movements.
Arthur Cohn in “The Literature of Chamber Music” states, “The spirit of Schoenberg is very strong” in this work.
Eisma, Will (1929- )
Fontemara (1965)
Amsterdam: Donemus, 1966
Duration: 12:00.
Quintett (1955)
Amsterdam: Donemus, 1960
Faomar, la poursuite for wind quintet (1999)
Amsterdam: Donemus, 1999
Eitler, Esteban (1913-1960)
Quintet (1945)
Manuscript
An Austrian (Tirol) born composer, he left in 1936 for Buenos Aires, then moved to Santiago, Chile, in 1945, the year of this work. Possibly printed in Argentina or Chile.
Eklund, Hans (1927-1999)
Improvisata (1958)
Stockholm: STIMS Informationscentral for Svensk Musik
Duration: 3:00.
Sommarparafras (1968)
Stockholm: STIMS Informationscentral for Svensk Musik
Duration: 9:00.
El-Dabh, Halim and Charles J. Coven (1921- )
Belly Dance Classic for percussion and woodwind quintet
Kent, OH: Halim El-Dabh Music, 2010
Percussion includes maracas, tambourine or zills, 2 djembes, 3 or 4 derabuccas and double bass.
According to “The music should be viewed as a loose blue print; thus, performers are encouraged to improvise. ‘The most beautiful things are found in the spur of the moment’, Coven attests to. El-Dabh says that “the music remains true to the Arabic melismatic passages.”
Elder, Daniel (1986- )
Due Miniature (2014)
Medina, NY: Imagine Music
See also:.
Elgar, Edward William (1857-1934)
Music for Woodwind Quintet, Op. 6 (for 2 flutes, oboe, clarinet and bassoon or cello) (1878-1881).
22 pieces have been, somewhat haphazardly, included under this opus including one which is 4 movements long. They were unpublished until 1976, edited by Richard McNichol who divided the pieces into the 7 volumes below. Written for his own use, Elgar played bassoon AND cello on these pieces. Taken together, they are a substantial portion of Elgar’s early work which only recently came to light. See Arthur Cohn, “The Literature of Chamber Music” for much more info on these volumes.
These works were recorded by the Athena Ensemble in their original instrumentation in 1978, released in 1992 by Chandos (CHAN 6553). Very brief sound samples are available at . The site also features a bit of a condescending review by James Leonard. (I wish I knew which Mozart wind quintets he believes Elgar modeled his pieces after, since Mozart didn’t actually write any, except the piano quintet.) More recently, several of these recordings have found their way to YouTube at .
Below, first see the original 7 volumes (with timings), then newer printed versions of these works (for both the original instrumentation and arranged for standard woodwind quintet).
Vol. I: Six Promenades (1878) (14:00)
Croydon: Belwin Mills, 1977
The Belwin Mills publications appear to be just for the original scoring, without any arrangement for the standard woodwind quintet instrumentation. See below for the Trevco publications with both.
Vol. II: Harmony Music No. 1 (4:02), Harmony Music No. 2 (10:18) (1879)
Croydon: Belwin Mills, 1977
Both are single movement works.
Vol. III: Harmony Music 3 & 4 (1879)
Croydon: Belwin Mills, 1977
No. 3 is incomplete (3:17), No. 4 “The Farmyard” (12:00), a longer sonata form.
Vol. IV: Harmony Music 5 (1879)
Croydon: Belwin Mills, 1977
Movements: The mission (11:40); Menuetto and trio (5:34); Noah‘s ark (4:15); Finale (5:46).
Vol. V: Five Intermezzos (1879)
Croydon: Belwin Mills, 1977
Elgar reportedly liked this set the best out of the entire collection.
Movements: No. 1 Nancy (1:30); No. 2 Mrs. and Miss Howell (1:23); No. 3 (1:10); No. 4 (2:05); No. 5 (1:30).
Vol. VI: Four Dances (1879) (12:30).
Croydon: Belwin Mills, 1977
Vol. VII: Adagio cantabile (“Mrs. Winslow‘s Soothing Syrup”) (3:42) and Andante con variazione (“Evesham Andante”) (5:00) (1879)
Croydon: Belwin Mills, 1977
Below are “modern” versions which include arrangements for standard woodwind quintet:
Six Promenades (see Volume I above)
TrevCo Music Publishing
Includes parts for the original scoring (2 flutes, oboe, clarinet and bassoon) and an arrangement for modern woodwind quintet by Trevor Cramer.
Harmony Music No. 1 (see Volume II of the list above)
TrevCo Music Publishing
Includes parts for the original scoring (2 flutes, oboe, clarinet and bassoon) and an arrangement for modern woodwind quintet by Trevor Cramer.
Harmony Music No. 2 (see Volume II of the list above.)
Trevco Music Publishing
Includes parts for the original scoring (2 flutes, oboe, clarinet and bassoon) and an arrangement for modern woodwind quintet by Trevor Cramer.
Five Intermezzos (see Volume 5 of the list above)
TrevCo Music Publishing
Includes parts for the original scoring (2 flutes, oboe, clarinet and bassoon) and an arrangement for modern woodwind quintet by Trevor Cramer.
Four Dances (see Vol. 6 of the list above)
TrevCo Music Publishing
Includes parts for the original scoring (2 flutes, oboe, clarinet and bassoon) and an arrangement for modern woodwind quintet by Trevor Cramer.
Adagio Cantabile “Mrs. Winslow’s Soothing Syrup” (see Vol. 7 of the list above)
Trevco Music Publishing
Includes parts for the original scoring (2 flutes, oboe, clarinet and bassoon) and an arrangement for modern woodwind quintet by Trevor Cramer.
Andante con variazione (“Evesham Andante”) (See Vol. 7 of the list above)
Trevco Music Publishing
Includes parts for the original scoring (2 flutes, oboe, clarinet and bassoon) and an arrangement for modern woodwind quintet by Trevor Cramer.
More versions of these works by Trevor Cramer appear to be likely in the future.
Elgar arrangements:
“Trio” from Pomp and Circumstance (arr. Earl C. North)
Richmond, VA: International Opus
The famous graduation tune, with optional endings for longer processions.
Pomp and Circumstance, No. 1 (arr. Holcombe and Bill Holcombe, Jr.)
Chesapeake, VA: Bill Holcombe‘s Musicians Pub‘s ()
“Nimrod” Variation 9 from The Enigma Variations, Op. 36 (arr. Jeanie Murrow)
Salem, Ct.: Cimarron Music Press, 2007. cimarronmusic.com
The only variation of the Enigma Variations frequently performed on its own, this and Barber’s Adagio for Strings are often performed as memorial works. Honestly, it‘s hard to see how a quintet could match the dramatic impact if used for this purpose and, if taken as slow as usually done with strings, there might be an endurance issue, too. (I’m willing to be proved wrong, though.) [Andrew Brandt]
“Nimrod” Variation 9 from The Enigma Variations, Op. 36 (arr. Ray Thompson)
Published by the arranger. See Boston, MA.: Ione Press
“Nimrod” from The Enigma Variations, Op. 36 (arr. Willy Hautvast)
Netherlands: Mansarda-Sintra, 2010
Chanson de Matin (arr. Ray Thompson, 2010)
Ray Thompson Music:
Duration: 3:00.
Chanson de Matin (arr. Guy de Cheyron)
Accolade; see Trevco Music
Chanson de matin, Op. 15, No. 2(arr. unlisted)
Hop Vine Music; see Trevco Music
Salut d’amour, Op. 12 (arr. Ray Thompson)
Ray Thompson Music:
Duration: 2:00. Transposed from the original E major to E-flat major.
Salut d’Amour (arr. George Trinkaus)
Andraud (now Southern Music)
Serenade for Strings, first movement “Allegro” (arr. Ray Thompson)
Ray Thompson Music:
Duration: 3:00. Includes parts for A and B-flat clarinet.
“Serious Doll” from the Nursery Suite (1931) (arr. Michael Axtell) for double woodwind quintet
Three Miniatures (arr. Newhill)
Available from the arranger. See also June Emerson
One of Elgar’s last works, although based on early sketches. The original orchestra version was written for and dedicated to the young Princess Margaret Rose (performed for her when she was aged four). A 7-movement work, it was later also staged as a ballet.
The Wand of Youth (arr. John Gibson)
JB Linear; see also
3 pieces from Suite No. 2: Moths and Butterflies; The Tame Bear; The Wild Bears.
also:
Serenade for Strings, Op. 20 for double woodwind quintet plus bass (arr. Geoffrey Emerson)
Yorkshire: The Red Hedgehog (45 Falsgrave Rd., Scarborough, Yorkshire, England)
Originally for strings.
The Severn Suite for 2 flutes, 2 oboes, 2 clarinets, 2 bassoons, 4 horns, bass (arr. Geoffrey Emerson)
Yorkshire: The Red Hedgehog (45 Falsgrave Rd., Scarborough, Yorkshire, England)
4 movements, “including a fine toccata and a lovely slow fugue.”
Sonata in G for 2 flutes (piccolo), 2 oboes, 2 clarinets, 2 bassoons, 4 horns, bass (arr. Geoffrey Emerson)
Originally for organ. Mr. Emerson calls this “A major work, built like a symphony — sounds really powerful on wind[s].”
Eliasson, Anders (1947- )
Picknick (Picnic) (1972)
Stockholm: STIMS Informationscentral for Svensk Musik
Duration: 5:00.
La Fievre (1978)
Danderyd, Sweden: Warner/Chappell Music Scandinavia AB
Duration: 8:30. Recorded: Caprice CAP 1271.
Elkana, Amos (1967- )
Ru’akh Quintet (1995)
Tel Aviv: Israel Music Institute
Duration: 10:00.
According to the composer, “Ru’akh in Hebrew literally means ‘wind’. Ru’akh Quintet was composed for the New Israeli Woodwind Quintet… I wanted the piece to feature the individual character of each of the members of the quintet. The piece is built of six sections, five of which are instrumental solos, one for each member of the group, while the others are accompanying. The remaining section acts as a middle section where all are playing together on an equal basis.”
El-Khoury, Bechara (1957- )
Quintette, Op. 46, Variations sans theme
Paris: M. Eschig, 1998
Ellerby, Martin (1957- )
Four Miniatures (1980)
London: Nova Music, 1984
See also
Also:
also:
Divertimento for Wind Dectet (1999) for double woodwind quintet
London: Studio Music, 2001
First flute doubles on piccolo; second oboe doubles English horn; second clarinet doubles bass clarinet. 7 movements: 1. Aubade; 2. Toccatina; 3. Eine kleine Wiener-Waltzer; 4. Notturno; 5. Scherzettino; 6. Siesta; 7. Doctor Murphy – His Burlesque. See also: Catherine Gerhart’s Annotated Bibliography of Double Wind Quintet Music,
Elliot, Willard (1926-2000)
Two Sketches
Manuscript, North Texas State University, Denton, TX /
Evanston, IL: Bruyere Music Publishers, 1986, now available from Trevco Music Publishing
Quintet
Manuscript
Ellis, David (1933- )
Quintet, Op. 17 (1956)
British Music Information Center,
Duration: 12:00. See also. Flute doubles on piccolo.
Elmer, Roderick (1947- )
Suite of Folk-tunes from Transylvania and Banat (1998)
Bradfield, Berkshire, England: Rosewood Publications, 1998
See also June Emerson Wind Music
Movements: Fair the maid, sweet and clean; Mummy’s girl; Good evening to thee, good woman; Down in the valley by the mill; Folk Dance; The Old Witch; He who crosses thickets green; Birdie, my little bird; May wet hay go up in flames; A Serbian circle dance.
also:.”
See also Elmer’s Sextet for piano and winds.
Emborg, Jens Laurson (1876-1957)
Quintet, Op. 74
Copenhagen: Edition Dania, 1937
This work]
Emilsen, Per-Anders (1964- )
Seven Pieces for Wind Quintet (1992)
Oslo: NB noter (Contemporary Norwegian sheet music)
See Norwegian Music Information Centre
Emilsson, Anders (1963- )
Blaskvintett ‘En vallare-lat’ (1987)
Stockholm: Svensk Music
Duration: 10:00.
Emmert, Frantisek Gregor (1940- )
Dechovy kvintet (1978)
Prague: Ceske hudebni fond, 1981. Rental.
Duration: 8:00.
Double Quintet (Dvojkvintet pro dvě dechová kvinteta) (1981)
Duration: 9:00.
Emmett, Dan (Daniel Decatur Emmett)
Dixie (arr. Greg Bartholomew)
Burke & Bagley; see also
Duration: 1:30.
Encinar, Jose Ramon (1954- )
Quinteto No 3
Madrid: Editorial Alpuerto, 1977
End, Jack (1918-1986)
Fantasy for wind quintet
Athens, Ohio, USA: Accura Music, 1982
Memo to a Woodwind Quintet
Manuscript: Eastman School of Music, Sibley Music Library, Rochester, NY
Born in Rochester, New York, he was long affiliated with the Eastman School of Music, first as a student, then later as faculty and staff.
Enescu, George (or, in France, Georges Enesco) (1881-1955)
Romanian Rhapsody No. 1 (arr. John Gibson)
JB Linear; See also
Dixtuor in D Major
Scored for 1 oboe and 1 English horn and pairs of flute, clarinet, bassoon and horn. Three movements: Doucement mouvementé; Modérément (D minor); Allègrement, mais pas trop vif. Score and parts are available to download from the International Music Score Library Project for those countries where the copyright has expired. There are numerous recordings of this work available, commercially and online. See also:
Engel, Paul (1949- )
Messanza, Quodlibet für Bläserquintett (1983)
Wein: Doblinger
Duration: 18:00. See also:
also:
Variationen uber ein australisched Volkslied (1967) for flute, oboe, clarinet, bassoon
Published by the composer
Choral, Musik für Holzbläserquartett und Cembalo mit einer kleinen Trommel ad lib (1972) for flute, oboe, clarinet, bassoon, harpsichord, optional snare drum
Published by the composer
Duration: 13:00.
See also Engel’s Sextet for winds and piano.
Engela, David (probably Dawid Sofius Engela, 1931-1967, South African composer)
Divertimento
Manuscript
Engelbrecht, Richard (1907 – 2001)
Wind Quintet (1961)
Manuscript
Duration: 25:50.
Englert, Giuseppe Giorgio (1927-2007)
Musica da Camera – Rime Serie, Op. 5 (1958-1961)
Manuscript
Duration: 11:00.
English, George Selwyn (1912-1980)
Quintet for wind instruments
London: Chappell 1973 1972
Written for the New Sydney Wind Quintet.
also:
First Waratah for narrator, wind quintet and percussion (1972)
Englund, (Sven) Einar (1916-1999)
Kvintett (1989)
Helsinki: Suomalaisen musiikin tiedotuskeskus, 1989
Duration: 17:30. Commissioned by the Visby Music Foundation.
See Englund’s piece for wind + string quintets and piano.
Enriquez-Salazar, Manuel (1926-1994)
Pentamusica (1963)
I do not know the publisher, but it is recorded by the Quintetto de Alientos de la Ciudad de Mexico on Spartacus 21018 (A mexican label), “Nueva Musica Mexicana.”
Eppert, Carl (1882-1961)
A Little Symphony for woodwind quintet, Op. 52
Uses alto clarinet instead of horn. 3 movements.
Suite No. 2 for Modern Woodwind Quintet, Op. 57 (1935)
Manuscript
Duration: 16:00.
Original Theme and 12 Variations, Op. 63 (1935)
Manuscript
Duration: 23:00.
Suite Pastoral for Modern Woodwind Quintet, Op. 64 (1936)
Manuscript
Duration: 24:00.
Epstein, Alvin (1926- )
Quintet
New York: Seesaw Music
Probably not the actor also named Alvin Epstein.
Epstein, Douglas
Commitments (1974)
Premiered by the Dorian Wind Quintet in 1975.
Epstein, Paul (1938- )
Variations for Wind Quintet (1991)
New York: Seesaw Music
Duration: 13:00. See also: .
Erb, Donald (1927-2008)
The Last Quintet (1978)
Bryn Mawr, Pa.: Merion Music, Theodore Presser, sole selling agent, 1984
Duration: 8:00. For quintet and electronic tape. Recorded: Redwood ES 28.
Ercolano, Tommaso
Trittico Carnascialesco
Bologna: Bongiovanni, 1992
Duration: 10:10. Movements: Stenterello; Pulcinella; Arlecchino.
Erdlen, Hermann Christian Georg (1893-1972)
Kleine Variationen uber ein Fruhlingslied (Little variations over
a spring-song), Op. 27, No. 1
Leipzig: W. Zimmermann, 1932
Duration: 4:30.
Erdman, Helmut W. (1947- )
Miniaturen (1977)
Rimsting: Keturi Musikverlag, 2006
Erichsen, Poul Allin (1910-1970)
2 ironische Stucke
Manuscript: Danischer Rundfunk, Copenhagen
Duration: 8:00.
Erickson, Elaine (1941- )
Two Pieces for Woodwind Quintet (1979)
New York: American Music Center
Des Moines, Iowa-based composer.
Erickson, Frank (1923-1996)
Woodwind Quintet No. 2 (1951)
Reproduced from holograph dated 5/6/51.
Composer is a member of ASCAP. Also wrote over 100 pieces for band.
Ericsson, Jan (1943- )
Blaskvintett
Manuscript: Swedish Music Information Center, Stockholm
Duration: 11:00.
Ernst, David (1945- )
Shapes: to the New Metropolitan Woodwind Quintet (1973)
Manuscript Publications, Pendleton, Or. (228 S.W. 28th Dr. #14)
ASCAP.
Ernst, Leb (? – 2013)
The “Albert” Quintet (1996)
Tel Aviv: Israel Music Center
Eröd, Iván (1936- )
Quintetto Ungherese (Hungarian Quintet), Bläserquintett, Op. 58 (1990)
Wien: Doblinger, 1992
Duration: 15:00. Dedicated to the Ensemble Wien-Berlin. See:
also:
Capriccio, Op. 23 for double woodwind quintet
Washington, DC: Library of Congress M957 .E76C3 1980 /
Wein: Doblinger
First flute doubles on piccolo. 1 movement: Allegro vivace. Duration: 7:00. First performed by the Niederosterreichisches Bläserquintett and the Bläserquintett der Jeunesse musicale Budapest.
Schnappschusse, fünf Portraits für Flote und Bläseroktett, Op. 52 for flute, 2 oboes, 2 clarinets, 2 bassoons, 2 horns.
Wein: Doblinger
Eroles, Carles M. (1957- )
Erebo
Barcelona: Clivis, 2002
See bio at: Associació Catalana de Compositors, .
Errante, Belisario Anthony (1920-2003?)
Schizzo Moderno
Pro Art Publications (see Belwin-Mills)
Can also be performed by flute, 2 clarinets, alto clarinet and bass clarinet. Composer is a member of ASCAP.
Ertamo, Sampsa (1973- )
Flirt (2006)
Helsinki: Music Finland
Duration: 6:00.
Ervin, Karen (also Karen Ervin-Pershing)(1943-2004)
Tracks
New York: Seesaw Music Corp., 1976
Duration: 12:00.
Escaich, Thierry (1965- )
Trois Instants fugitifs (1994)
Paris: Gérard Billaudot
Duration: 10:00. Recorded by Quintette Aquilon. See.
Also see Escaich’s Mechanic Song for woodwind quintet and piano.
Escher, Rudolf George (1912-1980)
Bläserquintett (1967)
Amsterdam: Donemus, 1967 1970
Doublings: flute/alto flute, oboe d’amore, clarinet/bass clarinet, bassoon, horn. Duration: 10:20. The New Grove’s Dictionary gives almost an entire paragraph to this work in Vol. 6, p. 242. Cohn also describes it in Vol. 2, p. 802 of his work, “Tonal but highly chromatic with frequent tempo and metric changes (often metric modulations), yet colorful and rhapsodic.”
also:
Sinfonia for wind quintet, string quintet and bass (1976)
Escobar, Luis Antonio (1925-1993)
Quinteto “La Curaba” (1959)
Manuscript
Escudero, Francisco (1913-2002)
Gnosis
Barcelona: Trito: Eresbil, 2004, 2002
Duration: 35:00. Movements: Aqueos; Nocturno en el Valle Lidio; Jonio; Eólide; División de los Locrios. Preface in Basque, English, Portuguese and Spanish.
Eshpai, IAkov (or Jacov) Andreevich (1890-1963)
Mariiskie melodii
Moskva (Moscow): Gos. Muzykal’noe Izd-vo, 1931
(Also try Interlibrary loan: U. Rochester)
Uses English horn instead of French horn! Composer is father to Andrei Eshpai and grandfather to filmmaker Andrei Andreyevich Eshpai.
Esley, Evgeny(1981- )
Divertimento for double woodwind quintet (2007)
Saint Petersburg: Compozitor Pub. House
4 parts or movements. Duration: 9:20. You can contact the publisher via mariapetrenko [at] gmail[dot]com.
Esnaola, Pedro (1808-1878)
La Delicada, Minué (arr. Mariano Drago)
Manuscript
Esnaola was an Argentine composer.
There is a truly awful (albeit, amateur) recording of this work on many of the free MP3 sites on the web. Your quintet can do better, if you can find the music. [Andrew Brandt]
Essex, Kenneth (1915-1955)
Wind Quintet (1941)
London: Hinrichsen, 1949 /
Peters, 1967
Duration: 13:00. Kenneth Essex appears to be a pseudonym for Rufus Isaacs.
Ethridge, Jean (1943- )
Three Pieces for Woodwind Quintet (1969-1979)
Toronto: Canadian Music Center
Duration: 7:00. Movements: I. Scherzo; II. Interlude; III. Fugue.
Etkin, Mariano (1943- )
Aleatorisches Bläserquintett (1961)
Composer is from Argentina.
Etler, Alvin Derald (1913-1973)
Quintet No. 1 (1955)
New York: Associated Music Publishers 1960, 1962
Duration: 15:30.
Barbera Secrist-Schmedes, in Wind Chamber Music, describes this work as “contrapuntal in style and similar to Fine‘s Partita in that the focus is on movement of parts rather than on instrumental color. Etler is economical–many times only two voices sounding.”
Arthur Cohn in “The Literature of Chamber Music” informs us that Etler was “a first rate oboist,” and that this work‘s “sonorities are a constant joy of fresh discovery…”
Quintet No. 2 (1957)
New York: Associated Music Publishers, 1960
Dexter, Michigan: TrevCo-Varner Music 2014?
Duration: 16:00. Recorded by the Sierra Wind Quintet.
Very challenging 4-movement work, using serial techniques. Has some lovely bassoon solos (bassoonist needs to read treble clef for third movement). Haven’t performed the work but would like to. [Andrew Brandt]
Arthur Cohn calls this “Sterner, stronger soundstuff” than Etler‘s first quintet, and “no nonsense.”
also:
Concerto for wind quintet and orchestra (1960)
New York: Associated Music Publishers
Recorded by the Louisville Orchestra in 1965.
Concerto for Violin and Wind Quintet (1958)
New York: G. Schirmer
For flute, oboe, clarinet, bassoon, horn and violin. Duration: 20:00.
Etti, Karl (1912-1996)
Bläserserenade für fünf Solisten (1967)
Manuscript
Duration: 15:00.
Variationen und Fuge
Wien: Doblinger
Es war amål an Åbend spåt (Austrian folksong – arranged)
Wien: Doblinger
Euba, Akin (1935- )
Woodwind Quintet (1967)
Manuscript
Composer‘s address: P. O. Box 27, University of Lagos Post Office, Akoka, Lagos, Nigeria.
Thoroughly trained in western composition, his works combine Western and African musical philosophy and procedures.
Evans, Robert Bruce (1933-2005)
Prelude & Fugue
Toronto: Berandol Music /
Don Mills, Ontario: BMI Canada, 1967
Evensen, Bernt Kasberg (1944- )
Quinta essensia (1981)
Oslo: NB noter (Contemporary Norwegian sheet music)
See Norwegian Music Information Centre
Van Walsum Suite: Six Movements for Wind Quintet (1971)
Oslo: NB noter (Contemporary Norwegian sheet music)
Everett-Salicco, Betty Lou (1925- )
Quintet (1970)
Everson, Dana F.
arrangements
Blessed Assurance
Deckerville, MI: David E. Smith Publications
Brethren We have Met to Worship
Deckerville, MI: David E. Smith Publications
Christmas Collection
Deckerville, MI: David E. Smith Publications
Christmas for WW
Deckerville, MI: David E. Smith Publications
Come Thou Fount
Deckerville, MI: David E. Smith Publications
Count Your Blessings
Deckerville, MI: David E. Smith Publications
Guide Me, Oh Great Jehovah
Deckerville, MI: David E. Smith Publications
I Will Sing of the Mercies
Deckerville, MI: David E. Smith Publications
Only a Sinner
Deckerville, MI: David E. Smith Publications
To God be the Glory
Deckerville, MI: David E. Smith Publications
Optional extra parts: also sax for horn, bass clarinet for bassoon.
We’re Marching to Zion
Deckerville, MI: David E. Smith Publications
Optional extra parts: alto sax for horn, bass clarinet for bassoon.
Ewazen, Eric (1954- )
Roaring Fork
San Antonio, Tex.: Southern Music Co., 1997
Movements: Whitewater Rapids (Maroon Creek); Columbines (Snowmass Lake); At the Summit (Buckskin Pass). Recently recorded by members of the U.S. Air Force‘s Heartland of America Band on an album entitled “Brass and Woodwind Quintets.” Recording can be ordered from the Director of Operations, 109 Washington Sq. STE 111, Offutt AFB, NE 68113-2126. Phone: 402/294-6046. Also recorded by the Borealis Wind Quintet.
Cumberland Suite
King of Prussia, PA: Theodore Presser Co., 2012
Duration: 18:00. 4 movements of “Baroque spirit.”
also:
Cascadian Concerto for woodwind quintet and orchestra
San Antonio, TX: Southern Music Company, 2009
This edition is for woodwind quintet and piano. Orchestral version premiered June 20, 2003 at Town Hall Seattle with the Cascadian Wind Quintet and the Lake Union Civic Orchestra conducted by Christophe Chagnard.
Exton, John (1932- )
Quintet No. 1
Composer is Australian.
Eyerly, Scott
Birch Music
Eyken, Ernest van der (1913-2010)
Refereynen ende liedekens (Refrains and Songs) (2002)
Bruxelles: CeBeDeM, 2003
Duration: 14:00.
also:
Concerto per Otto Strumenti a Vento (1999) for 1 flute, 2 oboes, 2 clarinets, 2 bassoons, 1 horn
Bruxelles: Le Centre Belge de Documentation Musicale (CeBeDeM)
Duration: 23:00.
Eyser, Eberhard Friedrich (1932- )
Blaskvintett (1970)
Stockholm: Svensk Musik
For flute, E-flat clarinet doubling on bass clarinet, contrabassoon and horn (presumably also with oboe).
Duration: 12:00.
Ezell, Helen Ingle (1903-1985)
Quintet (1959) | 298,487 |
Mountain Memories
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Price: $175 - $648: $175 - $648
Bedrooms: 3 Bathrooms: 3.5 Sleeps: 12
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Mountain Memories has it all! Magnificent views in private setting offering luxury accommodations while close to all the Pigeon Forge and Gatlinburg activities. Start creating your special memories for your next vacation to the Smoky Mountains by staying at Mountain Memories...a cabin you will always remember!
- Hot Tub
- Washer & Dryer
- Pool Table
- Type: Cabin
- Location: Wears Valley
- king
- Air Conditioning
- Refrigerator
- Coffee Maker
- Microwave
- Toaster
- Stove
- Dishwasher
- Jacuzzi Tub
- Deck
- Linens Provided
- Towels Provided
- Dishes & Utensils
- Oven
- children welcome
- pets not allowed
- Parking | 146,851 |
TITLE: Constant terms of asymptotic expansions of smoothed sums of all prime numbers
QUESTION [2 upvotes]: Consider the series
$$\sum_{n=1}^\infty n e^{-n \varepsilon}$$
For $\varepsilon \leq 0$, it diverges. For $\varepsilon > 0$, it converges and equals
$$\frac{e^\varepsilon}{(e^\varepsilon - 1)^2}$$
which has the asymptotic expansion
$$\frac{1}{\varepsilon^2} - \frac{1}{12} + \frac{\varepsilon^2}{240} - \frac{\varepsilon^4}{6048} + O(\varepsilon^6)$$
which has a constant term of $-\frac{1}{12}$.
Let $p_n$ denote the $n$th prime number. $p_n \sim n \log n$ by the prime number theorem. Let
\begin{align}
f_1(\varepsilon) &= \sum_{n=1}^\infty p_n e^{-n \varepsilon} \\
f_2(\varepsilon) &= \sum_{n=1}^\infty p_n e^{-p_n \varepsilon}.
\end{align}
These series converge for all $\varepsilon > 0$. Do they have asymptotic expansions in $\varepsilon$? If so, is it possible to extract their constant terms?
[Related question on MO. For more on smoothed sums, see reference 1.]
References:
Terence Tao. The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation. 2010-04-10.
REPLY [2 votes]: Heuristically, using the prime number theorem,
\begin{align*}
& f_2 (\varepsilon ) = \sum\limits_{n = 1}^\infty {p_n \,\mathrm{e}^{ - p_n \varepsilon } } = \int_2^{ + \infty } {t\,\mathrm{e}^{ - t\varepsilon } \mathrm{d}\pi (t)} \sim \int_2^{ + \infty } {\frac{{t\,\mathrm{e}^{ - t\varepsilon } }}{{\log (t)}}\mathrm{d}t}
\\ & = \frac{1}{{\varepsilon ^2 \log (1/\varepsilon )}}\int_{2\varepsilon} ^{ + \infty } {\frac{{s\,\mathrm{e}^{ - s} }}{{1 + \log (s)/\log (1/\varepsilon )}}\mathrm{d}s}
\\ &
\sim \frac{1}{{\varepsilon ^2 \log (1/\varepsilon )}}\sum\limits_{n = 0}^\infty {\frac{{( - 1)^n }}{{\log ^n (1/\varepsilon )}}\int_{2\varepsilon} ^{ + \infty } {s\,\mathrm{e}^{ - s} \log ^n (s)\,\mathrm{d}s} }
\\ &
\sim \frac{1}{{\varepsilon ^2 \log (1/\varepsilon )}}\sum\limits_{n = 0}^\infty {\frac{{( - 1)^n }}{{\log ^n (1/\varepsilon )}}\int_0^{ + \infty } {s\,\mathrm{e}^{ - s} \log ^n (s)\,\mathrm{d}s} }
\\ &
= \frac{1}{{\varepsilon ^2 \log (1/\varepsilon )}}\left( {1 + \frac{{\gamma - 1}}{{\log (1/\varepsilon )}} + \frac{{\frac{{\pi ^2 }}{6} + \gamma ^2 - 2\gamma }}{{\log ^2 (1/\varepsilon )}} + \frac{{\frac{{\pi ^2 }}{2}(\gamma - 1) + \gamma ^2 (\gamma - 3) + 2\zeta (3)}}{{\log ^3 (1/\varepsilon )}} + \ldots } \right)
\end{align*}
as $\varepsilon \to 0^+$. The error should be exponentially small compared to any of the terms in the series. Here $\pi(t)$ is the prime counting function, $\gamma$ is the Euler–Mascheroni constant, and $\zeta(3)$ is Apéry's constant. For a numerical example consider $\varepsilon =5 \cdot 10^{-5}$. With this value, $f_2(\varepsilon)\approx 1.8512832 \cdot 10^{10}$, whereas the approximation, with the terms neglected after order $\log ^{-3} (1/\varepsilon )$, gives $\approx 1.8524638\cdot 10^{10}$.
Note that the coefficients in the asymptotic expansion may be obtained via the exponential generating function
$$
\exp\! \bigg( {(\gamma - 1)z + \sum\limits_{n = 2}^\infty {\frac{{\zeta (n) - 1}}{n}z^n } } \bigg) = 1 + \frac{{\gamma - 1}}{{1!}}z + \frac{{\frac{{\pi ^2 }}{6} + \gamma ^2 - 2\gamma }}{{2!}}z^2 + \ldots ,
$$
with $|z|<2$. This relation may be proved using the observation $
\int_0^{ + \infty }s\,\mathrm{e}^{ - s} \log ^n (s)\,\mathrm{d}s = \Gamma ^{(n)} (2)
$. The latter also leads to the simple asymptotic approximation
$$
f_2 (\varepsilon ) \sim \int_0^1 {\frac{{\Gamma (1 + s)}}{{\varepsilon ^{1 + s} }}{\rm d}s}
$$
as $\varepsilon \to 0^+$.
Addendum. I shall show that the error coming from the first approximating step can be absorbed into any of the error terms of the final asymptotic expansion. We use the prime number theorem in the form
$$
\pi (t) = \int_2^t {\frac{{{\rm d}s}}{{\log (s)}}} + R(t),\quad R(t) = \mathcal{O}\!\left( {\frac{t}{{\log ^{N + 2} (t)}}} \right)
$$
where $N$ is an arbitrary fixed positive integer. Then
\begin{align*}
f_2 (\varepsilon ) & = \int_2^{ + \infty } {\frac{{t\,\mathrm{e}^{ - t\varepsilon } }}{{\log (t)}}{\rm d}t} + \int_2^{ + \infty } {t\,\mathrm{e}^{ - t\varepsilon } {\rm d}R(t)} \\ & = \int_2^{ + \infty } {\frac{{t\,\mathrm{e}^{ - t\varepsilon } }}{{\log (t)}}{\rm d}t} + \int_2^{ + \infty } {(\varepsilon t - 1)\,\mathrm{e}^{ - t\varepsilon } R(t)\,{\rm d}t} + \mathcal{O}(1).
\end{align*}
But
\begin{align*}
\int_2^{ + \infty } {(\varepsilon t - 1)\,\mathrm{e}^{ - t\varepsilon } R(t){\rm d}t} & = \mathcal{O}(1)\int_2^{ + \infty } {\frac{{t(\varepsilon t - 1)\,\mathrm{e}^{ - t\varepsilon } }}{{\log ^{N + 2} (t)}}{\rm d}t}
\\ & = \mathcal{O}(1)\frac{1}{{\varepsilon ^2 \log ^{N + 2} (1/\varepsilon )}}\int_{2\varepsilon}^{ + \infty } {\frac{{s(s - 1)\,\mathrm{e}^{ - s} }}{{(1 + \log (s)/\log (1/\varepsilon ))^{N + 2} }}{\rm d}s}
\\ & = \frac{1}{{\varepsilon ^2 \log ^2 (1/\varepsilon )}}\mathcal{O}\!\left( {\frac{1}{{\log ^N (1/\varepsilon )}}} \right)
\end{align*}
and the claim follows. | 31,103 |
Discussion
Do you use coupons at restaurants?
I'm sure this was asked before before but I'm curious how people feel about this today...throughout the year retailers, shopkeepers and restaurants offer them...how come then, some people find using coupons in a restaurant a negative and not so in a department store or such? How do you chowhounders feel about coupons offered by restaurants? (Does offering a coupon to dine at a restaurant give off a negative connotation?)
I've used them on occasion. My feeling is that most of the restaurants I see coupons for are not restaurants I would normally go to and even a coupon for 50% off does not entice me. Remember, this is food you are ingesting and even a discount is not going to get me to eat at a restaurant I do not find appetizing. Also, I find that a lot of restaurants that advertise with coupons come off as being desperate for business.
As for department stores, coupons are part of the sales strategy and it is a whole different game.
- re: ttoommyy
Good point.
I see much the same. It is like the book of coupons, that get sold - I would never dine at anything listed in most.
However, we do get various cards, and coupons, and some ARE for restaurants, that we might dine with - though not always. So very much depends.
It's almost like the coupons in the various tourist magazines, such as "Where Magazine." I have never used one of those, and likely would never use one. I would also likely never dine at any restaurant, that had a coupon in such a magazine. Just not likely to be my "style."
Hunt
Coupons in the restaurant business are rarely associated with successful high-end places, so their use tends cheapen the image of the restaurant as well as the patrons who use them. Then there is the issue of the kind of crowd they tend to attract. I once took a class for a guy who used to own a restaurant that used coupons. He and his maitre'd developed a game in which they would take a quick glance at a guest entering the restaurant and then would try to predict whether that guest would turn out to be a "coupon." He said they got really good at it. Eventually, he abandoned coupons altogether.
There are other ways to give discounts that are less controversial, such as happy-hour menus.
- re: nocharge
I agree, most of the coupons I've seen for restaurants are more take out or quick sit down places like Chipotle or chains like Outback and Olive Garden. A few higher-end places here offer discounts via Groupon but that usually does not include the highest-end of places and more so places in the low-to-mid tier and even then they are usually either on the brink of closing or new and trying to brew up business. That being said, I have used coupons at Chipotle and places like that before although I usually forget as the cost of the total meal doesn't really trigger me to search for some discount and I usually end up paying it and remember in the car that I could have saved $2 off a $12 meal. I've never used a Groupon either although I might. Though, I've always wondered the most polite way to announce that you are using a coupon - at the beginning, when the bill comes?
- re: fldhkybnva
- re: fldhkybnva
Maybe Seattle is just a different kind of restaurant market from where some of you live. I see some coupons (in newspapers, Groupon and their clones, the Entertainment Book, Val-Pak, etc.) for mediocre chains, but I see a lot more here for one-off neighborhood restaurants, many of them "ethnic." They usually aren't elegant or trendy places, but some of them have some damn good food.
Or maybe we're just living at a different price point, and you would see the kind of places I frequent (some of which have become my favorites after a coupon first got me in the door) as too "low-tier" for your tastes. To each their own.
I just came off of almost a year of being unemployed, and the judicious use of coupons made the difference between eating at home without a break for months on end, and being able to get out once in a while. Even now that I'm working again, I wish my budget allowed for fine dining more often--but even if it did, I wouldn't give up on my favorite dives and neighborhood regulars. And I don't anticipate ever being so well-off that I'll be able to sneer at a coupon for a place I've been wanting to try!
If a place offers a coupon or any other form of reduced price offer, I'd doubt whether they would then regard it as a negative when you use it.
I don't think I've paid full price at my nearest pizza place in the last couple of years (if not longer)
- re: Harters
I can easily imagine a restaurant owner being eager to reap the benefits of offering a coupon (more customers through the door), but less than enthusiastic when it comes to actually honoring it. It's just human nature to be self-centered, greedy, and short-sighted… Or maybe the coupon was a big miscalculation and is really causing the restaurant financial difficulty. It's good business practice to hide these sentiments from the customer, but not all owners/staff manage to do so.
- re: DeppityDawg
- re: DeppityDawg
I just do NOT understand this. Remind me...who was it that decided to offer the coupon in the first place? Oh, yeah...it was the restaurant's management! And now they're going to call me "riff raff" or "cheapskate" and act like they're doing me a favor to accept it?
If they're going to give me attitude for taking them up on the deal they put out there, this is an establishment that's definitely not going to get any more business from me. They would do better not to have the new customers that a coupon might bring in, than to bring in one-time customers, piss them off, and have them leave and tell all their friends about their unpleasant experience.
- re: MsMaryMc
- re: ttoommyy
Could be. But you know...into everybody's job, some bulls*it from their bosses must fall. If customers who use coupons bother the in-house staff so very much, they should take it up with their owner--and if that doesn't work, perhaps consider finding a job somewhere that better meets their standards. Because I'm pretty sure that being obviously snotty and disdainful of customers is NOT in their job description!
- re: MsMaryMc
I think that in the early days of Groupon, there were many businesses that did deals that came back to haunt them because they didn't properly understand the ramifications of the format and were overly optimistic about how the deal would result in repeat customers paying full price. That may have led to frustration on part of some business owners, but it's obviously not an excuse for any form of rudeness. If offering discount coupons doesn't draw the kind of crowd the restaurant desires, it shouldn't be offering them.
- re: DeppityDawg
" Or maybe the coupon was a big miscalculation and is really causing the restaurant financial difficulty. It's good business practice to hide these sentiments from the customer, but not all owners/staff manage to do so."
The last coupon we used was an Amazon Local coupon for an Italian deli (Sorriso's in Astoria, Queens - they are fantastic, make the most amazing homemade soppresata, mozzarella, pasta and pasta sauces, etc. - highest possible recommendation from me, for any of you in NYC). We were pretty surprised to see that they would offer a deal like this since they are a longtime established business, very busy, a line at the counter at any time of the day or week when you go in. But since we shop there anyway we jumped on the chance and bought the maximum number of coupons, used two of them right away, and promptly forgot about the other two until last week right before they were set to expire. Apparently a lot of other people also ran into redeem their coupons before the end of the month - because when we brought our order up to the counter and presented the coupon, the counter guy's face kind of fell and he said, "you know, we don't actually come out ahead on these things unless you buy MORE than the coupon amount." Mr Rat being the kind of impulse shopper he is, we ended up having $80 worth of stuff on a $30 dollar coupon, so the guy perked right up. And to be fair, he was not somebody I'd seen there before and was definitely not one of the owners/managers - but still it was funny that he'd come right out and say what we all know to be true, but shouldn't say to a customer. Not a politic sales style, to be sure :)
- re: DeppityDawg | 155,799 |
\begin{document}
\title{Coorbit description and atomic decomposition of Besov spaces }
\author{Jens G. Christensen}
\address{Tufts University, Department of Mathematics,
503 Boston Avenue, Medford, MA 02155}
\thanks{The research of J. G. Christensen was partially supported by
NSF grant DMS-0801010, and ONR grants NAVY.N0001409103, NAVY.N000140910324}
\email{[email protected]}
\author{Azita Mayeli}
\address{Department of Mathematics and Computer Sciences, City University of New York (CUNY), Queensborough College,
222-05 56th Avenue Bayside, NY 11364}\thanks{The research of A. Mayeli was partially supported by
NSF grant DMS-0801010}
\email{[email protected]}
\author{Gestur {\'O}lafsson}
\address{Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803}
\email{[email protected]}
\thanks{The research of G. {\'O}lafsson was supported by DMS-0801010 and DMS-1101337}
\keywords{Coorbit spaces; Sampling; Stratified Lie group; Besov spaces; Sub-Laplacian;}
\begin{abstract} Function spaces are central topic in analysis. Often those spaces and related analysis involves symmetries in form of an action of a Lie group. Coorbit theory as introduced by Feichtinger and Gr\"ochenig and then later extended in \cite{Christensen2011} gives a
unified method to construct Banach spaces of functions based on
representations of Lie groups. In this article we identify the
homogeneous Besov spaces on stratified Lie groups introduced in
\cite{FuMa} as coorbit spaces in the sense of \cite{Christensen2011}
and use this to derive atomic decompositions for the Besov spaces.
\end{abstract}
\maketitle
\section*{Introduction}
\noindent
Banach spaces of functions are central topics in analysis on $\RR^n$. Those spaces are usually translation invariant, a property that can be expressed as invariance under the Lie group $\RR^n$ acting on itself. More generally, one considers Banach spaces defined on a manifold $X$ and invariant under an action of a Lie group $G$. This simple idea combines two central topics in mathematics: Harmonic analysis on manifolds and representation theory of Lie groups. This idea was exemplified in the fundamental
construction of Feichtinger and Gr\"ochenig \cite{Feichtinger1988,Feichtinger1989a,Feichtinger1989b} where the authors proposed
a unified way to construct function spaces based on representation theory. This method was generalized in \cite{Christensen2009} by removing
some assumptions on the representation
from the work of Feichtinger and Gr\"ochenig, and
a method for atomic decompositions of these spaces was developed in
\cite{Christensen2010}. Similar techniques for decomposition
can be found in for example \cite{Grochenig1991} and \cite{Fornasier2005},
but the use of differentiable representations in \cite{Christensen2010}
makes those results particularly convenient for our purposes.
A different way to introduce function spaces is to use smoothness
conditions. As
important examples of that are the well known
Besov and and Triebel-Lizorkin-type spaces. Often those
spaces are described using Littlewood-Paley methods \cite{FJW}, but they
can also be described using wavelet type decompositions,
see \cite{HL} and the reference therein. Stratified Lie groups are
natural objects for extending this line of study, since they
come with natural dilations and a sub-Laplacian satisfying
the H\"ormander conditions.
The inhomogeneous Besov spaces on stratified Lie groups was introduced by Saka \cite{Saka79}. A characterization of inhomogeneous Besov groups in terms of Littlewood-Paley-decomposition was shown for all groups of polynomial growth in \cite{FMV}. The construction of the homogeneous Besov spaces was extended to stratified
Lie groups in \cite{FuMa}
using a discrete collection
of smooth band-limited wavelets with vanishing moments
of all orders. These wavelets were introduced in \cite{gm1}.
In \cite{FuMa}, it was also shown that the definition is equivalent with a definition in terms of the heat semigroup
$\{e^{-t\sL}\}_{t>0}$ for $\sL$, the sub-Laplacian operator for the group $G$.
In this paper we extend this result and show that the definition of Besov spaces also coincides with a definition based on
a family of
operators $\{\widehat\varphi(t\sL)\}_{t>0}$ where $\widehat\varphi\in
C_c^\infty(\RR^+)$ (Theorem \ref{norm-equi-general}).
We further use a representation theory approach
to show that the homogeneous
Besov spaces on stratified Lie groups in \cite{FuMa} are coorbit spaces in the sense of
\cite{Christensen2009,Christensen2011} and to use the results
of \cite{Christensen2010} to obtain atomic decompositions and frames
for those spaces.
The article is organized as follows. In Section \ref{Se-Coorbit} we recall the definition of coorbit spaces and their atomic decomposition as needed for the theory of Besov spaces. The details and proofs can be found in \cite{Christensen2009,Christensen2010,Christensen2011}. We start Section \ref{besov-spaces} by recalling the basic definitions for stratified Lie groups. In particular we introduce the sub-Laplacian and
its spectral theory. This enables us to formulate the needed results from \cite{FuMa}. In particular we recall in Theorem
\ref{equivalency-with-heat-kernel} some properties of the homogeneous Besov spaces and their descriptions in terms of the heat kernel:
For any
$k\in \NN$, $|s|<2k$, and any $f\in \cSZd$
\begin{align*}
\| f \|_{{\dot B}_{p,q}^s} \asymp
\left( \int_0^\infty t^{-sq/2 } \|(t\sL)^ke^{-t\sL}f \|_p^q \frac{dt}{t}
\right)^{1/q}
\end{align*}
where $\mathcal{L}$ is the sub-Laplacian (\cite{FuMa}, Theorem 4.4).
This description is not well adapted to the coorbit theory. For that
we need to replace the heat kernel by
a smooth bandlimited function $u$. This is achieved in
Theorem \ref{norm-equi-general}, which is the fundamental step in describing the
Besov spaces as coorbit spaces and their atomic decomposition in Theorem \ref{analyze vector}:
Let $1\leq p,q\leq \infty$ and $s\in\RR$. Let
$\widehat u\in \cS(\RR^+)$ be compactly
supported and satisfying some extra conditions. If $u$
denotes the distribution kernel of $\widehat u(\sL)$,
then $u$ is an analyzing vector and up to norm equivalence
$$B_{p,q}^{Q-2s/q}(G)=\mathrm{Co}_{\cSZ}^u L^{p,q}_s\, .$$
Here $Q$ is the homogeneous degree of the stratified Lie group $G$.
\section{Coorbit spaces and their atomic decomposition}\label{Se-Coorbit}
\noindent
In this section we introduce the notion of coorbit spaces based on \cite{Christensen2009,Christensen2011}.
We then discuss the
discretization from \cite{Christensen2010}
of reproducing kernel Banach spaces and apply it to coorbit spaces.
\subsection{Construction}
The starting point is a continuous representation $\pi$ of
a locally compact group $G$ on a Fr\'echet space $S$. In most applications
the space $S$ has a natural candidate.
Let $S^\cdual$ be the
conjugate linear dual equipped with the weak* topology and
assume that $S$ is continuously embedded and weak*
dense in $S^\cdual$. The conjugate linear
dual pairing of elements $v\in S$ and $\phi\in S^\cdual$ will be denoted
by $\dup{\phi}{v}$.
In particular
$\dup{u}{v}\in \mathbb{C}$ is well defined for $u,v\in S$ and if
$\dup{u}{v}=0$ for all $v\in S$ then $u=0$.
A vector $v\in S$ is called \emph{cyclic} if
$\dup{\phi}{\pi(x)v}=0$ for all $x\in G$ means that
$\phi=0$ in $S^*$, and if such a vector
exists $\pi$ is called a \emph{cyclic representation}.
Define the contragradient
representation $(\pi^\cdual,S^\cdual)$ by
\begin{equation*}
\dup{\pi^\cdual(x)\phi}{v}:=\dup{\phi}{\pi(x^{-1})v}.
\end{equation*}
Then $\pi^*$ is a continuous representation of $G$ on $S^\cdual$.
For a fixed vector
$u\in S$ the \emph{wavelet transform} $W_u:S^*\to C(G)$
\begin{equation*}
W_u(\phi)(x) := \dup{\phi}{\pi(x)u} = \dup{\pi^*(x^{-1})\phi}{u}.
\end{equation*}
is a linear mapping. Note that $W_u$ is injective if and only if $u$ is cyclic.
Denote by $\ell_x$ and $r_x$ the left and right
translations on functions on $G$ given by
\begin{equation*}
\ell_x f(y) := f(x^{-1}y)
\qquad\text{and}\qquad
r_xf(y) := f(yx).
\end{equation*}
A Banach space of functions $B$ is called left
invariant if $f\in B$ implies that $\ell_x f\in B$ for
all $x\in G$ and there is a constant
$C_x$ such that $\| \ell_x f\|_B\leq C_x \| f\|_B$
for all $f\in B$. Define right invariance similarly.
We will always assume that for each non-empty compact set $U$ there is a
constant $C_U$ such that for all $f\in B$
\begin{equation*}
\sup_{y\in U} \| \ell_yf \|_B \leq C_U \| f\|_B
\qquad\text{and}\qquad
\sup_{y\in U} \| r_yf \|_B \leq C_U \| f\|_B.
\end{equation*}
If $x\mapsto \ell_xf$ or $x\mapsto r_xf$ are continuous from $G$ to $B$
for all $f\in B$, then we say that left or
right translation are continuous respectively.
Fix a Haar measure $\mu$ on $G$. We will only work with spaces $B$ of functions on $G$
for which convergence in $B$ implies convergence (locally) in left
Haar measure on $G$. Examples of such spaces are all spaces continuously included in some $L^p(G)$ for $1\leq p \leq \infty$.
When $f,g$ are measurable functions on $G$ for which
the product $f(x)g(x^{-1}y)$ is integrable for all $y\in G$
we define the convolution $f*g$ as
\begin{equation*}
f*g(y) := \int_G f(x)g(x^{-1}y)\,d\mu(x).
\end{equation*}
For a function $f$ on $G$ we define ${f}^\vee(x) := f(x^{-1})$ and
$f^* := \overline{{f}^\vee}$. We will frequently use that for
$f \in {\rm L}^1(G)$, the adjoint of the convolution
operator $g \mapsto g \ast f$ is provided by $h \mapsto h \ast f^*$.
For a given representation $(\pi,S)$ and a vector $u\in S$
define the space
$$B_u := \{ f\in B\, |\, f=f*W_u(u) \}$$
with norm inherited from $B$.
Furthermore define the space
$$\Co_S^u B := \{ \phi\in S^*\, |\, W_u(\phi)\in B \}$$
with norm $\| \phi \| = \| W_u(\phi)\|_B$.
In \cite{Christensen2011} minimal conditions were listed to ensure
that a $B_u$ and $\Co_S^u B$ are isometrically
isomorphic Banach spaces. In that case, the space $\Co_S^u B$ is called
the \emph{coorbit space of $B$
with respect to $u$ and $(\pi,S)$}.
In this paper we
only need the following result which can be
found in \cite{Christensen2011}.
\begin{theorem}\label{thm:coorbitsduality}
Let $\pi$ be a cyclic representation of a group $G$ on a
Fr\'echet space $S$ which is continuously included in
its conjugate dual $S^*$. Fix a cyclic vector $u\in S$
and assume that
$W_u(\phi )\ast W_u(u)=W_u(\phi )$ for all $\phi \in S^*$.
If for the Banach function space $B$ the mapping
\begin{equation}\label{eq-doubleCont}
B\times S\to \mathbb{C}\, , \quad (f,v)\mapsto \int_G f(x) W_v(u)^\vee (x)\, d\mu (x)
\end{equation}
is well defined and continuous, then
\begin{enumerate}
\item The space $\Co_S^u B$ is a $\pi^\cdual$-invariant Banach
space. \label{prop2}
\item The space $B_u$ is a left invariant reproducing kernel
Banach subspace of $B$.
\item $W_u:\Co_S^u B\to B_u$ is an isometric isomorphism which
intertwines $\pi^\cdual$ and left translation. \label{prop3}
\item If left translation is continuous on $B,$ then
$\pi^*$ acts continuously on $\Co_S^u B$.\label{prop4}
\item $\Co_S^u B = \{ \pi^\cdual(f)u \mid f\in
B_u\}$. \label{prop5}
\end{enumerate}
\end{theorem}
A vector $u$ satisfying the requirements of the theorem
is called an \emph{analyzing vector}.
\begin{remark} Note that the condition (\ref{eq-doubleCont}) implies,
in particular, that for a fixed $u\in S$ we have
$\pi^*(f)\in S^*$ for all $f\in B$. Also, if $S$ is dense in a Hilbert space $H$ and $\pi$ extends to an unitary
representation, also denoted by $\pi$, of $G$ on $H$ such that for $u,v\in S$ (or $H$) $\dup{u}{v}=(u,v)$, then (\ref{eq-doubleCont})
says that
$$(f,v)\mapsto (f,W_u(v))$$
is continuous, where the $(\cdot,\cdot )$ on the right refer to the K\"othe
dual pairing on $B$.
\end{remark}
\subsection{Atomic decompositions and frames}
The previous section sets up a correspondence between
a space of distributions and a reproducing kernel
Banach space. We now investigate the discretization
operators introduced in \cite{Feichtinger1989a}
and \cite{Grochenig1991}, but we do so without assuming integrability
of the reproducing kernel.
The reproducing kernel Banach space $B_u$ has
reproducing kernel $W_u(u)(y^{-1}x)$ and
is continuously included in an ambient Banach space $B$.
From now on we will
assume that the space $B$ is solid, which means that
if $g\in B$ and $|f (x) | \leq |g(x)|$ $\mu$-almost everywhere then $f \in B$
and $\| f\|_B \leq \| g\|_B$.
Frames and atomic decompositions for Banach spaces were first introduced
in \cite{Grochenig1991} in the following manner.
\begin{definition}
\label{def:1}
Let $B$ be a Banach space and $B^\#$ a Banach sequence
space with index set $I$. If for $\lambda_i\in B^*$ and $\phi_i \in B$
we have
\begin{enumerate}
\item $\{\lambda_i(f) \}_{i\in I} \in B^\#$ for all $f\in B$;
\item the norms
$\| \lambda_i(f)\|_{B^\#}$ and $\| f\|_B$ are equivalent;
\item $f$ can be written as $f = \sum_{i\in I} \lambda_i(f) \phi_i$;
\end{enumerate}
then $\{(\lambda_i,\phi_i) \}_{i\in I}$ is an atomic decomposition
of $B$ with respect to $B^\#$.
\end{definition}
More generally a Banach frame for a Banach space can be defined as:
\begin{definition}
\label{def:3}
Let $B$ be a Banach space and $B^\#$ a Banach sequence
space with index set $I$. If for $\lambda_i\in B^*$
we have
\begin{enumerate}
\item $\{\lambda_i(f) \}_{i\in I} \in B^\#$ for all $f\in B$,
\item the norms
$\| \lambda_i(f)\|_{B^\#}$ and $\| f\|_B$ are equivalent,
\item there is a bounded reconstruction
operator $T:B^\# \to B$
such that $T(\{\lambda_i(f) \}_{i\in I}) = f$,
\end{enumerate}
then $\{ \lambda_i \}_{i\in I}$ is an Banach frame for
$B$ with respect to $B^\#$.
\end{definition}
In the sequel we will often suppress the use of the index set $I$.
The Banach sequence spaces we will use were introduced in
\cite{Feichtinger1989a} and they are constructed from
a solid Banach function space $B$ as described below.
For a compact neighbourhood $U$ of the identity we call
the sequence $\{ x_i\}_{i\in I}$ $U$-relatively separated
if $G\subseteq \bigcup_i x_iU$ and there exists $N\in \mathbb{N}$ such that
\begin{equation*}
\sup_i (\# \{ j \mid x_iU\cap x_jU \neq \emptyset \}) \leq N.
\end{equation*}
For a $U$-relatively separated sequence $X=\{ x_i\}_{i\in I}$
define the space $B^\#(X)$ to be the collection
of sequences $\{ \lambda_i\}_{i\in I}$ for which
the pointwise sum
\begin{equation*}
\sum_{i\in I} |\lambda_i| 1_{x_iU}(x)
\end{equation*}
defines a function in $B$.
If the compactly supported continuous functions are dense in $B$ then
this sum also converges in norm (for an argument see \cite{Rauhut2005}
Lemma 4.3.1).
Equipped with the norm
$$\| \{\lambda_i \}\|_{B^\#} := \|
\sum_{i\in I} |\lambda_i| 1_{x_iU} \|_B$$
the space $B^\#(X)$ is a solid
Banach sequence space, i.e., if $\tau=\{\tau_i\}$ and $\lambda=\{\lambda_i\}$ are sequences such that
$\lambda \in B^\# (X)$ and for all $i\in I$ we have $|\tau_i |\le |\lambda_i|$, then
$\tau \in B^\#(X)$ and $\|\tau \|\le \|\lambda\|$. In the case were $B=L^p(G)$ we have $B^\#(X)=\ell^p(I)$.
For fixed $X=\{ x_i\}_{i\in I}$ the space $B^\#(X)$ only depends
on the compact neighborhood $U$ up to norm equivalence.
Further, if $X=\{ x_i \}_{i\in I}$ and $Y=\{ y_i \}_{i\in I}$ are
two $U$-relatively separated sequences
with same index set such that $x_i^{-1}y_i \in V$
for some compact set $V$, then $B^\#(X)=B^\#(Y)$ with equivalent norms.
For these properties consult Lemma 3.5 in \cite{Feichtinger1989a}.
For a given compact neighbourhood $U$ of the identity
a sequence of non-negative functions $\{ \psi_i\}$ is called a
\emph{bounded uniform partition of unity} subordinate to $U$ (or $U$-BUPU), if
there is a $U$-relatively separated
sequence $\{ x_i\}$, such that
$\supp(\psi_i)\subseteq x_i U$ and for all $x\in G$ we have
$\sum_i \psi_i (x) =1$. Note that for a given $x\in G$ this sum is over
finite indices.
\subsection{Discretization operators for
reproducing kernel Banach spaces on Lie groups}
Let $G$ be Lie group with Lie algebra $\mathfrak{g}$
and exponential map $\exp:\mathfrak{g}\to G$. Then for
$X\in\mathfrak{g}$ we define the right and left
differential operators (if the limits exist)
\begin{equation*}
R(X)f(x) := \lim_{t\to 0} \frac{f(x\exp(tX))-f(x)}{t}
\qquad\text{and}\qquad
L(X)f(x) := \lim_{t\to 0} \frac{f(\exp(-tX)x)-f(x)}{t}.
\end{equation*}
We note that $R([X,Y])=R(X)R(Y)-R(Y)R(X)$ and $L([X,Y])=L(X)L(Y)-L(Y)L(X)$.
Fix a basis $\{ X_i\}_{i=1}^{\mathrm{dim}(G)}$
for $\mathfrak{g}$. For a multi index $\alpha$ of
length $|\alpha| =k$
with entries between $1$ and $\mathrm{dim}(G)$
we introduce the operator $R^\alpha$ of subsequent right differentiations
\begin{equation*}
R^{\alpha} f
:= R(X_{\alpha(k)}) R(X_{\alpha(k-1)}) \cdots R(X_{\alpha(1)}) f.
\end{equation*}
Similarly we introduce the operator $L^\alpha$ of subsequent
left differentiations
\begin{equation*}
L^{\alpha} f
:= L(X_{\alpha(k)}) L(X_{\alpha(k-1)}) \cdots L(X_{\alpha(1)}) f.
\end{equation*}
We call the function $f$ right (respectively left) \emph{differentiable of order $n$},
if for every $x$ and all $|\alpha|\leq n$ the derivatives
$R^\alpha f(x)$ (respectively $L^\alpha f(x)$) exist.
In the remainder of this section we will assume that
\begin{enumerate}
\item $B$ is a solid left- and right-invariant Banach function
space on which right translation
is continuous.
\item The kernel $\Phi$ is smooth and
satisfies $\Phi*\Phi = \Phi$.
\item The mappings $f\mapsto f*|L^\alpha \Phi|$ and
$f\mapsto f*|R^\alpha \Phi|$ are continuous on $B$ for
all $\alpha$ with $|\alpha |\leq \dim(G)$.
\end{enumerate}
By setting $\alpha=0$
we see that convolution with
$\Phi$ is a continuous projection from $B$ onto
the reproducing kernel Banach space
\begin{equation*}
B_\Phi := \{ f\in B \mid f*\Phi =f \}.
\end{equation*}
We now describe sampling theorems and atomic decompositions
for the space $B_\Phi$ based on the smoothness of the (not necessarily
integrable) reproducing kernel $\Phi$.
The results can be found in
\cite{Christensen2010} and build
on techniques from
\cite{Feichtinger1988,Feichtinger1989a,Feichtinger1989b,Grochenig1991},
where it should be mentioned that integrability is required.
For integrable kernels not parametrized by groups
see also \cite{Fornasier2005} and \cite{Dahlke08}.
\begin{lemma}
The mapping
\begin{equation*}
B^\# \ni \{ \lambda_i \}
\mapsto \sum_i \lambda_i \ell_{x_i}\Phi \in B_\Phi
\end{equation*}
is continuous.
The convergence of the sums above is pointwise, and
if $C_c(G)$ is dense in $B$
the convergence is also in norm.
\end{lemma}
From now on we let $U_\epsilon$ denote the set
\begin{equation*}
U_\epsilon := \left\{
\exp(t_1X_1)\cdots\exp(t_nX_n)
\Mid -\epsilon \leq t_k \leq \epsilon ,1\leq k\leq n
\right\}.
\end{equation*}
\begin{theorem}\label{thm:8}
With the assumptions on $B$ and $\Phi$
listed above we can choose $\epsilon$ small enough
that for any $U_\epsilon$-BUPU $\{ \psi_i\}$
the following three operators are all invertible on $B_\Phi$.
\begin{align*}
T_1f &:= \sum_i f(x_i)(\psi_i*\Phi) \\
T_2f &:= \sum_i \lambda_i(f) \ell_{x_i}\Phi\quad
\text{with $\lambda_i(f) = \textstyle\int f(x) \psi_i(x)\, dx$} \\
T_3f &:= \sum_i c_i f(x_i) \ell_{x_i}\Phi
\quad\text{ with $c_i = \textstyle\int \psi_i(x)\,dx$}.
\end{align*}
The convergence of the sums above is pointwise and,
if $C_c(G)$ is dense in $B$,
the convergence is also in norm.
Furthermore
both $\{ \lambda_i(T^{-1}_2 f),\ell_{x_i}\Phi\}$
and
$\{ c_iT_3^{-1}f(x_i),\ell_{x_i}\Phi\}$
are atomic decompositions of $B_\Phi$ and
$\{c_i\ell_{x_i}\Phi \}$ is a frame.
\end{theorem}
\subsection{Decomposition of coorbits using smoothness}
To obtain frames and atomic decompositions via Theorem~\ref{thm:8}
we will need to introduce differentiability with respect to $\pi$ and
$\pi^*$.
A vector $u\in S$ is called $\pi$-weakly differentiable in the direction
$X\in\mathfrak{g}$ if there is a vector, denoted by $\pi(X)u$, in $S$
such that for all $\phi\in S^*$
\begin{equation*}
\dup{\phi}{\pi(X)u}=
\frac{d}{dt}\Big|_{t=0}\dup{\phi}{\pi(e^{tX})u}.
\end{equation*}
For a fixed basis $\{X_i \}\subseteq \mathfrak{g}$
and for a multi-index $\alpha$ define
$\pi(X^\alpha)u$ (when it makes sense) by
\begin{equation*}
\dup{\phi}{\pi(X^\alpha)u}:=
\dup{\phi}{\pi(X_{\alpha(k)}) \pi(X_{\alpha(k-1)}) \cdots \pi(X_{\alpha(1)})u}.
\end{equation*}
We note that if $\Phi(x) = W_u(\phi)(x)$
and $u$ is $\pi$-weakly differentiable,
then $R^\alpha\Phi (x) = W_{\pi(X^\alpha)u}(u)(x)$. We have similar
notion for the representation $(\pi^*,S^*)$.
A vector $\phi\in S^*$ is called $\pi^*$-weakly differentiable
in the direction
$X\in\mathfrak{g}$ if there is a
vector denoted $\pi^*(X)\phi \in S^*$
such that for all $v\in S$
\begin{equation*}
\dup{\pi^*(X)\phi}{v}=
\frac{d}{dt}\Big|_{t=0}\dup{\pi^*(e^{tX})\phi}{v}.
\end{equation*}
For a multi-index $\alpha$ define
$\pi^*(X^\alpha)\phi$ by (when it makes sense)
\begin{equation*}
\pi^*(X^\alpha)\phi =
\pi^*(X_{\alpha(k)}) \pi^*(X_{\alpha(k-1)}) \cdots \pi^*(X_{\alpha(1)})\phi
\end{equation*}
We say that $v\in S$, respectively $\phi\in S^*$, is
$k$-times differentiable if $\pi (X^\alpha )v$, respectively
$\pi^*(X^\alpha)\phi$, exists for all $\alpha$ with $|\alpha|\le k$.
We note that if $\Phi(x) = W_u(\phi)(x)$ and $\phi$ is $\pi^*$-weakly
differentiable, then $L^\alpha\Phi (x) = W_{u}(\pi^*(X^\alpha) \phi)(x)$.
The following result provides atomic decompositions and frames
for coorbit spaces.
\begin{theorem}
\label{thm:7}
Let $B$ be a solid and left and right invariant
Banach function space for which right translations are continuous.
Assume that $u \in S\subseteq S^*$ is both
$\pi$- and $\pi^*$-weakly differentiable and
satisfies the requirements in
Theorem~\ref{thm:coorbitsduality}.
Furthermore assume the mappings
\[f \mapsto f*W_{\pi(X^\alpha)u}(u)\text{ and } f\mapsto f*W_{u}(\pi^*(X^\alpha)u)\]
are continuous on $B$
for $|\alpha|\leq \dim(G)$.
Then we can choose $\epsilon$ small enough that for any
$U_\epsilon$-relatively separated set $\{ x_i\}$
the family
$\{ \pi(x_i)u\}$ is a frame for $\Co_S^u B$.
Furthermore the families $\{ \lambda_i\circ T_2^{-1}\circ W_u,\pi(x_i)u \}$
and $\{ c_i T_3^{-1}\circ W_u,\pi(x_i)u \}$ both
form atomic decompositions for $\Co_S^u B$.
In particular $\phi\in \Co_S^u B$ can be reconstructed by
\begin{align*}
\phi &=
W_u^{-1 }T_1^{-1} \left( \sum_i W_u(\phi)(x_i)\psi_i*W_u(u)
\right) \\
\phi &= \sum_i \lambda_i(T^{-1}_2 W_u(\phi)) \pi(x_i)u \\
\phi &= \sum_i c_i (T^{-1}_3W_u(\phi))(x_i) \pi(x_i)u
\end{align*}
The convergence of the last two sums is in $S^*,$
and with convergence in $\Co_S^u B$ if
$C_c(G)$ is dense in $B$.
Here $\{\psi_i \}$ is any $U_\epsilon$-BUPU for which
$\supp(\psi_i)\subseteq x_iU_\epsilon$.
$T_1,T_2,T_3$ and $c_i$ and $\lambda_i$ are defined as in
Theorem~\ref{thm:8}.
\end{theorem}
\section{Besov spaces on stratified Lie Groups}\label{besov-spaces}
\noindent
In this section we introduce the notion of \emph{stratified Lie groups}. We collect some needed
fundamental work by Folland, Hulanicki and Stein \cite{Folland75,FollandStein82,Hulanicki84}. We then
introduce \emph{homogeneous Besov spaces} for stratified Lie groups following \cite{FuMa}. Our main result in this section
is Theorem \ref{norm-equi-general} which is crucial for coorbit description and atomic decomposition of the Besov spaces.
\subsection{Stratified Lie groups}\label{preliminary-and-notations}
A connected and simply connected Lie group ${G}$
is called stratified
if its Lie algebra $\mathfrak{g}$ decomposes as a direct sum
\begin{equation*}
\mathfrak{g}= V_1\oplus \cdots \oplus V_m=\bigoplus_{j=1}^m V_j\quad \text{ such that } \quad [V_1,V_k] =V_{k+1} \text{ for }1\leq k\leq m
\end{equation*}
where we set
$V_{m+1}=\{0\}$, see \cite{Folland75}. Note that induction shows that
\begin{equation}\label{eq-ViVj}
[V_i,V_j]\subseteq V_{i+j} \text{ for } i,j\ge 1\, .
\end{equation}
Thus $\fg$ is step $m$ nilpotent.
{}From now on $\fg$ is assumed stratified with fixed stratification $(V_1,\ldots ,V_m)$.
We identify $G$ with $\mathfrak{g}$ through the exponential map. The multiplication is then given by the
Cambell-Hausdorff formula. Examples of stratified Lie groups are Euclidean spaces $\RR^n$, the
Heisenberg group $\HH^n$, and the upper triangular matrices with ones
on the diagonal.
For $r>0$ define the $\delta_r : \fg\to \fg$ by
$\delta_r (X)=r^k X$ if $x\in V_k$. Equation (\ref{eq-ViVj}) shows that $\delta_r$ is an automorphism of $\fg$. The family $\{\delta_r\}_{r> 0}$ is called the canonical family of dilations for $\mathfrak g$. It gives rise to dilations $\gamma_r : G\to G$ given by
\begin{equation*}
\gamma_r(x)=\exp \circ \delta_r \circ \log (x)
\end{equation*}
with $\log =\exp^{-1}$.
We note that $\gamma_r$ is a homomorphism, and
following \cite{Folland75} we often write $rx$ for $\gamma_r(x)$.
The Lebesgue measure on $\fg$ defines a left and right Haar measure $\mu_G$ on $G$.
We will often write $dx$ instead of $d\mu_G(x)$.
Let the number $Q:= \sum_1^m j(\dim V_j)$ be
the \emph{homogeneous degree} of $G$, then we have
\begin{equation*}
\int_G f(rx)\,dx = r^{-Q} \int_G f(x)\,dx.
\end{equation*}
For example, the homogenous degrees for $G=\RR^n$ and $\HH^n$ are
$Q=n$ and $Q=2n+2$, respectively.
A homogeneous norm on $G$ is a map $|\cdot |: \ G\rightarrow [0,\infty)$, $x\mapsto |x|$,
which is continuous and smooth, except at zero, and satisfies
the following:
\begin{enumerate}
\item $|x|=0$ if and only if $x=0$,
\item $|rx|=r |x|$ for all $r>0$, and $x\in G$,
\item $|x|=|x^{-1}|$.
\end{enumerate}
Homogenous norms always exist on stratified Lie groups
\cite{Folland75,FollandStein82}.
For example, on
the Heisenberg group $\HH^n$ with the underlying manifold $\RR^{2n}\times \RR$, a homogenous norm is given by
\begin{align}
\notag |(x,t)|=(\sum x_j^4+t^2)^{\frac{1}{4}}
\end{align}
It follows by \cite{Folland75}, Lemma 1.2, that
the balls $\{x\in G\mid |x|\le R\}$ are compact and by
\cite{Folland75}, Lemma 1.4,
there exists a
constant $C>0$ such that $|xy| \le C(|x|+|y|)$ holds for all $x, y\in G$.
\subsection{The sub-Laplacian on stratified Lie groups}
Let $n=\dim G$, $n_k=\dim V_k$ and let $X_1,\ldots ,X_n$ be a basis
for $\fg$ as before.
We require that
$X_j\in V_k$ for $n_1+\ldots +n_{k-1}+1\le j\le n_1+\ldots +n_k$. In particular, $X_1,\ldots ,X_{n_1}$ is a basis for
$V_1$. Thus, the left invariant differential operators $R(X_1),\ldots ,R(X_{n_1})$ satisfy the H\"ormander condition
\cite{H67} or \cite{Folland75}. We will use coordinates $(x_1,\ldots ,x_n)$ on $\fg$ with respect to this basis. Note that those
coordinates also form a global chart for $G$. In particular, the space $\cP$ of polynomial functions on $G$ is just the space
$\mathbb{C}[x_1,\ldots ,x_n]$ of polynomials in the coordinates $(x_1,\ldots ,x_n)$.
Denote by $\mathbb{D}(G)$ the algebra of left invariant differential operators on $G$. Recall that $X\mapsto R(X)$ defines a Lie algebra homomorphism into $\mathbb{D}(G)$ and $R(\fg )$ generates $\mathbb{D}(G)$. Note that $\mathbb{D}(G)$ is contained in the Weyl algebra
$\mathbb{C}[x_1,\ldots ,x_n,\partial_1,\ldots ,\partial_n]$ of differential operators on $\fg$ with polynomial coefficients.
Denote by
$\cSg$ the usual space of Schwartz
functions on $\fg$ and set $\cSG : = \cSg$. Thus, the
Schwartz functions on $G$
are the smooth functions on $G$ for which
\[
|f|_N :=
\sup_{|\alpha | \le N, x \in G} (1+|x|)^{N} |R^\alpha f(x)| < \infty\, \quad \text{ for all } N\in \mathbb{N}\, .\]
The topology on $\cSG$ is defined by the family of seminorms $\{|\, \cdot \, |_N \mid N\in\mathbb{N}\}$, see \cite{FollandStein82}, Section D, Page 35.
This topology does not depended on
the stratification $(V_1,\ldots ,V_m)$.
According to our previous notation, the conjugate linear dual of $\cSG$ will be denoted by
$\cSd$ and the conjugate dual pairing is
denoted by $\dup{\cdot}{\cdot}$.
Define the space $\cSZ$ of Schwartz functions with vanishing moments by
\begin{align}\notag \cSZ:=\left\{ f\in \cSG \,\left|\,
\int_G x^\alpha f(x)d\mu (x)=0~\text{ for all multi-indices}~ \alpha\in
\NN^n\right.\right\}.
\end{align}
The space $\cSZ$ is a closed subspace of $\cSG$ with the relative
topology, and it forms an ideal under convolution:
$\cSG \ast \cSZ \subset \cSZ$.
The inclusion $\cSZ \hookrightarrow \cSG$ induces a continuous linear map $\cSd \to \cSZd$ with kernel
$\cP$, and thus $\cSZd\simeq \cSd/\cP $ in a canonical way.
For the proof of these facts we refer to \cite{FuMa}.
The sub-Laplacian operator on $G$ is
defined by
\begin{equation*}
\sL:=-\sum_{i=1}^{n_1} R(X_i)^2\in \mathbb{D}(G)\, .
\end{equation*}
Note that $\sL$ depends on the space $V_1$ in the given stratification. As $R(X_1),\ldots ,R(X_{n_1})$ satisfies the H\"ormander condition it follows that
$\sL$ is hypoelliptic. Also, $\sL$ is formally self-adjoint and: for
any $f,g\in C_c^\infty(G)$, $\langle \sL f, g\rangle= \langle f,
\sL g\rangle$ which follows by partial integration.
The closure of $\sL$ is also denoted by $\sL$ and
has domain $\mathcal{D} = \{u \in L^2(G)\mid \sL u \in L^2(G)\}$,
where we use $\sL u$ in the sense of distributions. For more information we refer to \cite{gm1,Folland75,FollandStein82}.
A linear differential operator $D$ on $G$ is called
homogenous of degree $p$ if $D (f\circ \gamma_a)= a^p (D f)\circ
\gamma_a$ for any $f$. If $X\in V_j$,
then $R(X)$ is homogeneous of
degree $j$. It follows that $\sL^k$ is homogenous of degree $2k$.
For the spectral theory for unbounded operators we refer to \cite{Rudin}, in particular p. 356--370.
Since the closure of $\sL$ is self-adjoint and positive it follows that $\sL$ has spectral resolution
\begin{align}
\notag \sL= \int_0^\infty \lambda \, dP_\lambda,
\end{align}
where $dP_\lambda$ is the corresponding projection valued measure.
For $\widehat \phi \in L^\infty(\RR^+)$
define the
commutative integral operator on $L^2(G)$ by
\begin{align*}
\widehat \phi (\sL):=\int_0^\infty
\widehat \phi (\lambda)\, dP_\lambda.
\end{align*}
The operator norm is
$\|\widehat \phi (\sL)\| = \| \widehat \phi \|_\infty$, and
the correspondence $\widehat\phi \leftrightarrow \widehat\phi (\sL)$ satisfies
\begin{enumerate}
\item $(\widehat \phi \widehat \psi )(\sL)=\widehat \phi (\sL)\widehat \psi (\sL)$;
\item $\widehat \phi (\sL)^* =\overline{\widehat{\phi}}(\sL)$.
\end{enumerate}
\begin{theorem}[\cite{Hulanicki84}]\label{th-Hulan}
Let $\widehat{\phi}\in \cS(\RR^+)$. Then there exists $\phi \in \cSG$ such that
\begin{align}\label{eq-Hulan}
\widehat \phi (\sL)f = f \ast \phi \quad \text{ for all }
\; f\in \cSG~.
\end{align}
\end{theorem}
The function $\phi$ is called the
\emph{distribution kernel} for $\widehat{\phi}$,
and from now on, if $\widehat{\phi}\in \cS (\RR^+)$
then $\phi\in\cSG$ will always denote the distribution kernel for
$\widehat{\phi}(\sL)$.
The following two lemmas are simple applications of Hulanicki's theorem.
\begin{lemma}\label{le_star}
Let $\widehat{\psi}\in\cS (\RR^+)$ with distribution kernel $\psi\in\cSG$. Then
the distribution kernel for $\overline{\widehat{\psi}}$ is $\psi^*$
where $\psi^*(x) := \overline{\psi(x^{-1})}.$
\end{lemma}
\begin{proof} Let $f,g\in \cSG$ and recall that $\overline{\widehat{\psi}}(\sL)=\widehat{\psi}(\sL)^*$. The claim follows now because
\[(f,\widehat{\psi}(\sL)^*g)=(\widehat{\psi}(\sL)f,g)=(f*\psi, g)=(f,g*\psi^*)\, .\qedhere\]
\end{proof}
\begin{lemma}\label{le-LkpZero} Let $p$ be a polynomial function on $G$. Then there exists $k\in\NN$ such that $\sL^kp=0$.
\end{lemma}
\begin{proof} Let $P\in \mathbb{C}[\fg]$ be such that $p(\exp (X))=P(X)$. Define
\[q_X(t_1,\ldots ,t_{n_1}):= \sum_{j=1}^{n_1} P(\log (\exp (X)\exp (t_jX_j)))\, .\]
As $G$ is nilpotent, then there exists $m$ such that $q_X$ is a polynomial of degree $\le m$ for all $X\in \fg$. But then
\[\sL^kp (\exp X)
=
\left(
- \sum_{j=1}^{n_1}\dfrac{\partial^2}{\partial t_j^2}
\right)^k
q_X(0)=0\]
for all $k> m/2$ and all $X\in\fg$.
\end{proof}
If $\widehat{\eta}\in \cS (\RR^+)$ and $\eta$ is the corresponding
kernel function, then we say that $\eta$ is \emph{band-limited} if $\widehat{\eta}$ is compactly supported. In
that case
there exist $\epsilon, \delta>0$ such that $\supp (\widehat{\eta})\subseteq [\epsilon, \delta]$.
\begin{lemma}\label{all-vanishing-moments} Let $\widehat \phi \in
\cS (\RR^+)$ with $\widehat \phi \equiv 0$ in some neighborhood of zero and let $\epsilon,\delta>0$. Then
the distribution kernel $\phi $ of $\widehat \phi (\sL)$ lies in $\cSZ$,
and for each $m\in\NN$ there
exists $\widehat \phi_m\in \cS (\RR^+)$ such that $\phi =\sL^m\phi_m$. Furthermore, $\supp (\widehat \phi_m) \subseteq [\epsilon ,\delta]$ if and only if $\supp (\widehat \phi ) \subseteq
[\epsilon ,\delta ]$.
\end{lemma}
\begin{proof} Define $\widehat{\phi}_m (s) := s^{-m}\widehat{\phi}(s)$. Then $\widehat{\phi}(\sL)=\sL^m\widehat\phi_m(\sL)$ and
by Theorem \ref{th-Hulan} and our assumption
on $\phi$
we have $\phi_m\in \cSG$ for all $m\in\NN$ and $\phi=\sL^m \phi_m$. Let $p$ be any polynomial on $G$ of degree $k\leq 2m-1$. Then
$\sL^mp=0$ and we have
\[
\int_G p(x) \phi (x)\, d\mu (x) = \int_G p(x)\sL^m\phi_m(x)\, d\mu (x)=\int_G \sL^mp(x)\phi_m(x)\, d\mu (x)=0\]
Thus $\phi $ has vanishing moments of arbitrary order. The last statement is obvious.
\end{proof}
If $\widehat{\varphi}\in \cS(\RR^+)$ and
$\varphi$ has vanishing moments of all orders, then
$p\ast \varphi=0$ for all $p\in \cP$. Hence
$(f+\cP)\mapsto f*\varphi$ is a well defined linear map on
$\cSZd$.
\subsection{Besov spaces}
To define Besov spaces on an abstract stratified group $G$ we begin as
follows: Pick a multiplier $\widehat\psi\in C_c^\infty(\RR^+)$ with
support in $[2^{-2}, 2^{2}]$. Denote by $\psi$ the distribution
kernel of the operator $\widehat\psi(\sL)$ as before.
Define
$\widehat\psi_j(\lambda ) := \widehat\psi(2^{-2j}\lambda)$. We assume that
\begin{align}\label{unity-partition}
\sum_{j\in \ZZ}
|\widehat\psi_j(\lambda)|^2=1~~~ a.e ~~ \lambda \in \RR^+ ~.
\end{align}
\begin{lemma}\label{eq-psij}
For $\widehat{\psi}\in\cS(\RR^+)$ and $r>0$. The distribution kernel $\psi^r$ corresponding
to the operator $\widehat{\psi}(r\sL)$ is $\psi^r(x)= r^{-Q/2}\psi (r^{-1/2}x)$.
In particular, let $\psi_j\in \cS(G)$
denote the distribution kernel of the
operator $\widehat\psi_j(\sL) $. Then $
\psi_j (x)= 2^{jQ}\psi(2^jx)$.
\end{lemma}
This follows easily from the homogeneity of $\sL$. The result is also
true for bounded $\widehat\psi$ on $\RR^+$. For a proof see
\cite{FollandStein82} Lemma 6.29.
Equation (\ref{unity-partition}) implies that
\begin{align*}
\sum_{j\in \ZZ}
\widehat\psi_j(\sL)^\ast \circ \widehat\psi_j(\sL)=I~,
\end{align*}
where the sum converges in the strong $L^2$-operator topology.
By (\ref{eq-Hulan}) we therefore get the following Calder\'{o}n decomposition for all $g\in
L^2(G)$:
\begin{align}\label{decomp-in-L2}
g=\sum_{j\in \ZZ} g\ast \psi_j \ast
\psi_j^*
\end{align} where the series converges unconditionally in
$L^2$-norm.
For $g\in \cS_0(G)$,
the expansion (\ref{decomp-in-L2}) also converges in $\cSZ$ in the Schwartz space topology (\cite{FuMa}, Lemma 3.7). Therefore by duality, for $\cSZd$ in the weak $^*$ topology.
Assume that (\ref{unity-partition}) holds,
and define the \emph{homogeneous Besov space}
${\dot B}_{p,q}^{s}(G):={\dot B}_{p,q}^{s}$ for $s\in \RR$, $1\leq p \le \infty$
and $1 \le q < \infty$,
to be the set of all $f\in \cSZd$ for
which
\begin{align}\label{norm-besov-space} \|f\|_{{\dot B}_{p,q}^{s}}:=
\left( \sum_{j\in \ZZ} 2^{jsq} \| f \ast \psi_j \|_p^q \right)^{1/q}<
\infty~,
\end{align} with
the natural convention for when $p=\infty$ or $q=\infty$. (Note that the convolution with a distribution is understood in the weak sense.)
Here we collect some of properties of the homogeneous Besov spaces
in the following lemma. For proofs we refer to \cite{FuMa}:
\begin{lemma}\label{equivalency-with-heat-kernel} The following holds:
\begin{enumerate}
\item The Besov spaces ${\dot B}_{p,q}^{s}$ with norm $ \|\cdot\|_{{\dot B}_{p,q}^{s}}$ are Banach spaces.
\item For all $1 \le p,q \le \infty$ and all $s \in \RR$, one has
continuous inclusion maps $\cSZ \hookrightarrow {\dot
B}_{p,q}^s \hookrightarrow \cSZd$. And for any $1\leq p,q<\infty$ and $k\in \NN_0$, $\sL^k\cSZ$ is dense in
${\dot B}_{p,q}^s$.
\item For any given $1 \le p,q < \infty$ the
decomposition (\ref{decomp-in-L2}) converges for all
$g \in {\dot B}_{p,q}^s$ in the Besov space norm.
\item For any
$k\in \NN$, $|s|<2k$, and any $f\in \cSZd$
\begin{align}
\label{asympto_rel} \| f \|_{{\dot B}_{p,q}^s} \asymp
\left( \int_0^\infty t^{-sq/2 } \|(t\sL)^ke^{-t\sL}f \|_p^q \frac{dt}{t}
\right)^{1/q}~.
\end{align}
\end{enumerate}
\end{lemma}
We note that the corresponding kernel function to
$\widehat \psi (\lambda) =(\lambda )^k e^{-\lambda}$ is \textbf{not} in $\cSZ$ so we can not use (\ref{asympto_rel})
for our purposes. Our aim is therefore to show that we can replace this function with a bandlimited function.
\subsection{Equivalent norms on Besov spaces}
The chief result of this section
is stated in Theorem \ref{norm-equi-general} where the Besov norm (\ref{norm-besov-space}) is described in terms of
band-limited kernel functions with vanishing moments of all orders.
In the sequel, we will use the notation ``$\preceq$'' to mean ``$\leq$''
up to a positive constant.
\begin{lemma}\label{exchang-dilation-parameters} Let $
\widehat\varphi,~ \widehat\psi \in \cS(\RR^+)$. Then for any $r, s>0$ and $f\in L^2(G)$
\begin{align*} \widehat\varphi(r\sL)\widehat\psi(s\sL)f=
\widehat\varphi(rs^{-1}\sL)\widehat\psi(\sL)f= \widehat\varphi(\sL)\widehat\psi(sr^{-1}\sL)f ~.
\end{align*}
\end{lemma}
\begin{proof} This follows easily from Lemma \ref{eq-psij} and the equation (\ref{eq-Hulan}).\end{proof}
\begin{lemma}\label{exponential estimation theorem} Let $\widehat
\varphi, \widehat\psi\in C_c^\infty(\RR^+)$ and $m\in\ZZ$. Then there exists a constant
$c=c(m, \varphi, \psi)$ such that
$$
\| \widehat\varphi(r\sL) \psi \|_1 \leq c r^m\quad \forall ~r>0\, .
$$
\end{lemma}
\begin{proof} By Lemma \ref{all-vanishing-moments} we have $\psi=\sL^m \psi_m$ for some
$\psi_m\in \cSZ$. Denote the distribution kernel of $\widehat{\varphi}(r\sL)$ by
$\varphi^r\in \cSZ$. Then by the equation (\ref{eq-Hulan})
\begin{equation*}
\| \widehat\varphi (r \sL)
\psi\|_1= \| \widehat\varphi(r \sL) \sL^m\psi_m\|_1 = \| (\sL^m\psi_m)\ast
\varphi^r \|_1 = \|\psi_m\ast (\sL^m \varphi^r)\|_1\, .
\end{equation*}
Now, Lemma \ref{eq-psij}, the homogeneity of $\sL^m$, and Young's inequality for $p=q=1$
implies
\[ \| \widehat\varphi (r \sL)
\psi\|_1\leq \| \sL^m \varphi^r \|_1~\|\psi_m
\|_1\leq r^{m} \| \sL^{m} \varphi \|_1~
\|\psi_m \|_1= c \ r^{m}\]
with $c= \| \sL^{m}\varphi\|_1~ \|\psi_m
\|_1$.
\end{proof}
The following is the main result of this section.
\begin{theorem}\label{norm-equi-general} Let $ \widehat\varphi\in \cS(\RR^+)$ with compact support. Then
for any $f\in \cSZd$
\begin{align}\label{asympto-relation} \| f \|_{{\dot B}_{p,q}^s}
\asymp \left( \int_0^\infty t^{-sq/2 } \|\widehat\varphi(t\sL)f \|_p^q
\frac{dt}{t} \right)^{1/q}~.
\end{align}
\end{theorem}
\begin{proof} Fix $1\leq p\leq \infty$. We first show that ``$\succeq$" holds. Let
$0\not = f\in \cS_0^*(G)$
and assume that the right hand side in (\ref{asympto-relation}) is finite.
We can rewrite the integral
as follows.
\begin{align*}
\int_0^\infty t^{-sq/2 }
\|\widehat\varphi(t\sL)f \|_p^q \frac{dt}{t} = \int_1^4 \sum_{l\in \ZZ}
(2^{2l}t)^{-sq/2} \|\widehat\varphi(2^{2l}t\sL)f\|_p^q ~\frac{dt}{t} ~.
\end{align*}
As $\widehat\varphi(2^{2l}t\sL)f$ is in $\cS_0^*(G)$ and that
\eqref{decomp-in-L2} converges in $\cS_0^*(G)$ in weak $^*$ topology, we can write
\begin{equation}\label{Young-Ineq-1}
\|\widehat\varphi(2^{2l}t\sL)f\|_p=\left\| \sum_{j\in\ZZ}
\widehat\varphi(2^{2l}t\sL)f\ast \psi_j \ast \psi_j^\ast \right\|_p \leq
\sum_{j\in\ZZ} \| \widehat\varphi(2^{2l}t\sL) f\ast \psi_j \ast \psi_j^\ast \|_p~.
\end{equation}
By the left invariant property of the operator $\sL$ and hence $\widehat\varphi(2^{2l}t\sL)$, along with an application of Young's inequality
for $p\geq 1$ and Lemma \ref{exchang-dilation-parameters} we get
\begin{align}\notag \sum_j \| \widehat\varphi(2^{2l}t\sL) f\ast \psi_j
\ast \psi_j^\ast \|_p \leq \sum_j \| \widehat\varphi(2^{2l}t\sL)
\psi_j^\ast\|_1~ \|f \ast \psi_j \|_p= \sum_j \|
\widehat\varphi(2^{2(l+j)}t\sL) \psi^\ast\|_1~ \|f\ast \psi_j \|_p ~.
\end{align}
By interfering the preceding in (\ref{Young-Ineq-1}) and
then multiplying both sides by $2^{-ls}$ we obtain the following.
\begin{align}\label{last}
2^{-ls} \| \widehat\varphi(2^{2l}t\sL)f\|_p
& \leq \sum_j 2^{-ls} \| \widehat\varphi(2^{2(l+j)}t\sL) \psi_j^\ast\|_1 \|f\ast
\psi_j \|_p\\
&= \sum_j \left( 2^{-(l+j)s} \|
\widehat\varphi(2^{2(l+j)}t\sL) \psi^\ast\|_1\right) \left(2^{js} \|f\ast \psi_j \|_p\right)~.\nonumber
\end{align}
To simplify the notations above, let $A_j^t:= 2^{js}
\| \widehat\varphi(2^{-2j}t\sL) \psi^\ast\|_1$ and $B_j:= 2^{js} \|f\ast
\psi_j \|_p$.
Using new definitions and the definition of convolution on $\ZZ$,
the summation in the inequality (\ref{last}) is a convolution on $\ZZ$. Therefore the inequality can be rewritten as
\begin{align}\notag
2^{-ls} \| \widehat\varphi(2^{2l}t\sL)f\|_p &\leq \{B_j\}_j\ast \{A_j^t\}_j(-l).
\end{align}
Taking the sum on $l\in \ZZ$ above and then again applying Young's inequality
yields
\begin{align}\label{double-star}
\sum_j 2^{-lsq} \|
\widehat\varphi(2^{2l}t\sL)f\|_p^q &\leq \|\{B_j\}\|_q^q \|
\{A_j^t\}\|_1^q~.
\end{align}
Integrating both sides of (\ref{double-star}) over $t$ in $[1,4]$, substituting $ \{A_j^t\}$ back, and that $\|\{B_j\}\|_q^q= \|f\|_{B_{p,q}^s}^q$, implies
\begin{align}\label{integral}
\int_0^\infty t^{-sq/2} \|
\widehat\varphi(t\sL)f\|_p^q ~\frac{dt}{t}
\leq \|f\|_{B_{p,q}^s}^q \int_1^4
\left(\sum_j 2^{js}\| \widehat\varphi(2^{-2j}t\sL)\psi^*\|_1\right)^q ~\frac{dt}{t}
\end{align}
Therefore to complete the proof it remains to show that the sum in the preceding relation is finite.
Without loss of generality, we shall prove this only for negative
$j$'s. For positive $j$ the assertion follows with a similar argument.
Let $m\in \ZZ$ with $m<s/2$. By Lemma \ref{exponential estimation theorem} we have
$$\| \widehat\varphi(2^{-2j}t\sL) \psi^\ast\|_1 \leq c (2^{-2j}t)^m
$$
for some constant $c$ independent of $t$ and $j$.
Now it is simple to see that the application of the preceding
estimation in the integral (\ref{integral}) for the summands over
negative $j$ implies the finiteness of the integral (\ref{integral})
for $m<s/2$. For the positive $j$ we take $m>s/2$.
To show ``$\preceq$'' holds, let $ c=\int_0^\infty
|\widehat\varphi(t)|^2 dt/t$. This is finite by the choice of
$\widehat\varphi$. By dilation invariance property of the measure
$dt/t$, for any $\lambda>0$ we can rewrite $ c=\int_0^\infty
|\widehat\varphi(\lambda t)|^2 ~\frac{dt}{t}$. For simplicity, we assume that
$c=1$. By substituting $\lambda\mapsto \sL$ and using the spectral
theory for the sub-Laplacian $\sL$, we derive the following Calder\`on
formula from the integral:
\begin{align}\label{identity-relation} I= \int_0^\infty
|\widehat\varphi |^2(t\sL) ~\frac{dt}{t} = \int_0^\infty
\widehat\varphi(t\sL)^* \widehat\varphi(t\sL) ~\frac{dt}{t}
\end{align}
where $I$ is the identity,
and the integral converges
in the strong sense. Applying (\ref{identity-relation}) to $\widehat\psi_j(\sL)f$
and employing
$\widehat\psi_j(\sL)\widehat\varphi(t\sL)=\widehat\varphi(t\sL)\widehat\psi_j(\sL)$, we have
\begin{align}\label{something}
\widehat\psi_j(\sL)f
= \int_0^\infty
\widehat\varphi(t\sL)^*\widehat\psi_j(\sL) \widehat\varphi(t\sL)f ~\frac{dt}{t}
= \int_1^4 \sum_{l\in\ZZ} \widehat\varphi(2^{2l}t\sL)^*\widehat\psi_j(\sL)
\widehat\varphi(2^{2l}t\sL)f ~\frac{dt}{t}
\end{align}
If $\psi_j$ is the distribution kernel of $\widehat\psi_j(\sL)$, then the
distribution kernel of $\widehat\varphi(2^{2l}t\sL)^*\widehat\psi_j(\sL) $ is
$\widehat\varphi(2^{2l}t\sL)^*\psi_j $ and for any
$g\in \cSZd$ we have
$$ \widehat\varphi(2^{2l}t\sL)^*\widehat\psi_j(\sL)g=g\ast \widehat\varphi(2^{2l}t\sL)^*\psi_j ~ \quad\quad (\text{in the dual sense}).
$$
In particular, for $g= \widehat\varphi(2^{2l}t\sL) f $, in
(\ref{something}) we have
\begin{align}\label{something-2}
\widehat\psi_j(\sL)f = \int_1^4
\sum_{l\in\ZZ} \widehat\varphi(2^{2l}t\sL) f \ast \widehat\varphi(2^{2l}t\sL)^*
\psi_j~\frac{dt}{t}~.
\end{align}
First, by applying Minkowski's inequality for integrals and series
and then Young's inequality for $p\geq 1$, from (\ref{something-2}) we
deduce the following.
\begin{align}\notag
\|\widehat\psi_j(\sL) f\|_p \leq
\int_1^4 \sum_{l\in\ZZ} \|\widehat\varphi(2^{2l}t\sL) f\|_p ~ \|
\widehat\varphi(2^{2l}t\sL)^* \psi_j\|_1 ~\frac{dt}{t}~.
\end{align}
Therefore for any $q\geq 1$
\begin{align} \label{est_single_Delta-1}
\|\widehat\psi_j(\sL) f\|_p^q \preceq
\int_1^4 \left(\sum_{l\in\ZZ} \|\widehat\varphi(2^{2l}t\sL) f\|_p ~ \|
\widehat\varphi(2^{2l}t\sL)^* \psi_j\|_1 \right)^q~ ~\frac{dt}{t}~.
\end{align}
If in Lemma \ref{exchang-dilation-parameters} we let $r=2^{2l}t$ and
$s=2^{-2j}$, then
${\widehat\varphi}(2^{2l}t\sL)^*\widehat\psi(2^{-2j}\sL)= {\widehat\varphi}(2^{2(l+j)}t\sL)^*\widehat\psi(\sL)$
and consequently
$\widehat\varphi(2^{2l}t\sL)^* \psi_j=\widehat\varphi(2^{2(l+j)}t\sL)^*\psi$.
Using this in (\ref{est_single_Delta-1}), taking sum over $j$ with weights $2^{jsq}$ and applying Fubini's theorem yields
\begin{align*}
\sum_{j\in\ZZ}2^{jsq}
\|\widehat\psi_j(\sL) f\|_p^q & \leq \int_1^4 \sum_{j\in\ZZ}2^{jsq}
\left(\sum_{l\in\ZZ} \|\widehat\varphi(2^{2l}t\sL)
f\|_p~\|{\widehat\varphi}(2^{2(l+j)}t\sL)^* \psi\|_1\right)^q ~\frac{dt}{t}\\\notag & = \int_1^4 \sum_{j\in\ZZ} \left(\sum_{l\in\ZZ} 2^{-ls}
\|\widehat\varphi(2^{2l}t\sL) f\|_p ~ ~ 2^{(j+l)s}
\|{\widehat\varphi}(2^{2(l+j)}t\sL)^* \psi\|_1\right)^q ~\frac{dt}{t} ~.
\end{align*}
The summation over $l$ is a convolution
of two sequences at $-j$. Therefore by Young's inequality for $q$
\begin{align}\label{irgend-etwas}
\sum_{j\in\ZZ}2^{jsq}
\|\widehat\psi_j(\sL) f \|_p^q \leq \int_1^4 \sum_{l\in\ZZ} 2^{-lsq}
\|\widehat\varphi(2^{2l}t\sL) f\|_p^q ~ ~ \left(\sum_{l\in\ZZ} 2^{-ls}
\|\widehat\varphi(2^{-2l}t\sL)^* \psi\|_1\right)^q ~\frac{dt}{t}~.
\end{align} Note that the left side of (\ref{irgend-etwas}) is $\|f\|_{\dot B_{p,q}^s}^q$. And, by a similar argument used
above, one can show that the sum $\sum_{l\in\ZZ} 2^{-ls}\|\widehat\varphi(2^{-2l}t\sL)^*
\psi\|_1$ is finite.
With these and that $1\leq t\leq 4$, in
(\ref{irgend-etwas}) we proceed as below.
\begin{align*}\|f\|_{\dot B_{p,q}^s}^q&\preceq \int_1^4 \sum_{l\in\ZZ} 2^{-lsq} \|\widehat\varphi(2^{2l}t\sL)
f\|_p^q ~\frac{dt}{t} \\\notag &\preceq \int_1^4 \sum_{l\in\ZZ}
(2^{2l}t)^{-sq/2} \|\widehat\varphi(2^{2l}t\sL) f\|_p^q ~\frac{dt}{t}\\\notag &=
\int_0^\infty t^{-sq/2} \|\widehat\varphi(t\sL) f\|_p^q ~\frac{dt}{t}~.
\end{align*}
This completes the proof of $``\preceq"$.
Using the standard convention, the proof of the theorem for the cases
$q=\infty$ and $p=\infty$ is simple and we omit it here.
\end{proof}
We shall say a compactly supported function $\widehat\phi\in \mathcal S(\RR^+)$ is a cut off function on $\RR^+$ if $\widehat\phi\equiv 1$ in an open subinterval of its support.
The following corollary generalizes the results of Theorem 4.4 in \cite{FuMa}.
\begin{corollary} Let $ \widehat\varphi $
be any cut off function on $\RR^+$. Then for any $\widehat\psi\in \cS(\RR^+)$
\begin{align*}
\| f \|_{{\dot B}_{p,q}^s} \asymp
\left( \int_0^\infty \|\widehat\psi(t\sL)\widehat\varphi (t\sL)f \|_p^q \frac{dt}{t}\right)^{1/q}~ .
\end{align*}
In particular one takes $\widehat\psi(\lambda)= \lambda^k e^{-\lambda}$ for any $k\in \NN$.
\end{corollary}
\section{Besov spaces as coorbits}\label{se-BSC}
\subsection{Some representation theory of stratified Lie groups}
In this section we will look at some representation
theoretic results for the semidirect product $\RR^+\ltimes G$
where $\RR^+$ acts on $G$ by dilations.
For any $a>0$ and any function
$f$ on ${G}$, define the
dilation of $f$ as follows
\begin{align*}
D_a f (x)= a^{-Q/2} f(a^{-1}x),
\end{align*}
where $Q$ is the homogenous degree of $G$.
Let $\pi$ denote the quasi-regular representation of the
semidirect product $\RR^+\ltimes G$
defined by
\begin{equation}
\label{eq:repn}
\pi(a,x)f=\ell_xD_a f, \ ~~ a>0, \ x\in G
\end{equation}
for all $f\in L^2(G)$.
The representation $\pi$ induces a family of infinitesimal operators
$\pi^\infty(X)$ for $X$ in the Lie algebra of $\RR^+\ltimes G$
\begin{equation*}
\pi^\infty(X)f = \lim_{t\to 0} \frac{\pi(\exp(tX))f-f}{t},
\end{equation*}
with limit in $L^2(G)$. We
use $\pi^\infty$ to separate this from the
weak derivatives we have used before, yet
note that a strongly smooth function in $\cSZ$ is also
both weak and weak$^*$ differentiable.
\begin{lemma}\label{lem:repn}
$(\pi, \cS_0(G))$ and
$(\pi^\infty,\cS_0(G))$ are representations
of $\RR^+\ltimes G$ and its Lie algebra respectively.
\end{lemma}
\begin{proof}
By Propositions 1.46 and 1.25 in \cite{FollandStein82} we already
know that $\cS_0(G)$ is invariant under the left translation.
That $\cS_0(G)$ is dilation invariant follows from the
homogeneity of the norm $|\cdot |$.
From the seminorms $|\cdot |_N$ it is not
hard to see that $\cS_0(G)$ is invariant under
left differentiation, and thus we only need to
show that the differentiation arising from dilations
leaves $\cS_0(G)$ invariant.
Let $T=(1,0)$ be in the Lie algebra of $\RR^+\ltimes G$, then
there exists polynomials $p_\alpha$ such that
\begin{equation*}
\pi^\infty(T)f(x)
= \frac{d}{dt}\Big|_{t=0}\pi(\exp(tT))f(x)
=\frac{d}{dt}\Big|_{t=0} e^{-t/2} f(e^{-t}x)
= - \frac{1}{2} f(x) + \sum_{|\alpha|=1} p_\alpha(x) R^\alpha f(x),
\end{equation*}
where $p_\alpha$ are polynomials of degree $1$ in $x$.
For this statement see p. 41 in \cite{FollandStein82} coupled with
p. 25 in \cite{FollandStein82}.
This proves invariance under the infinitesimal operator
$\pi^\infty(T)$ .
\end{proof}
The Haar measure on $\RR^+\ltimes G$ is given by
$d\mu(x,a)=a^{-(Q+1)}\,dx\,da$ where $dx$ is the Haar measure on $G$.
The following convolution
identity is crucial to our main result.
\begin{lemma}
\label{lem:reprformula}
Let $u\in \cSZ$ be the distribution kernel for
$\widehat{u}$ with support away from zero satisfying
\begin{equation}\label{eq:1}
\int_0^\infty \frac{|\widehat{u}(\lambda)|^2}{\lambda} \,d\lambda = 1.
\end{equation}
Then with convolution on $\RR^+\ltimes G$ we have
\begin{equation*}
W_u(\phi)*W_u(u)(a,x) = W_u(\phi)(a,x)
\end{equation*}
for all $\phi\in \cSZd$.
\end{lemma}
\begin{proof}
Since $u,v\in\cSZ$ have vanishing moments
of all orders, then
$u*(D_a v)(x) = (u+p_1)*D_a(v+p_2)$ for all polynomials $p_1,p_2$.
In particular if we let $p_x$ be the right Taylor polynomium
of homogeneous degree $k$ for $u$ around $x$, then
\begin{equation*}
\int u(y)v(a^{-1}y^{-1}x)\,dy
= \int u(xy)v(a^{-1}y^{-1})\,dy
= \int (u(xy)-p_x(y)) v(a^{-1}y^{-1})\,dy.
\end{equation*}
By the esimate of $u(xy)-p_x(y) $ from Lemma 3.2 in \cite{FuMa} we therefore get
\begin{align}
|u*D_av(x)|
&\leq C(u,k) \int a^{-Q/2}|y|^{k+1} |v(a^{-1}y^{-1})|\,dy \notag\\
&\leq C(u,k) a^{Q/2} \int |ay|^{k+1} |v(y^{-1})|\,dy \notag\\
&\leq C(u,v,k,m) a^{Q/2+k+1} (1+|x|)^{-m}. \label{eq:4}
\end{align}
A similar inequality is obtained in (10) from \cite{FuMa}
(note the dilation there is different from $D_a$):
\begin{equation}\label{eq:5}
|u*D_a v(x)| \leq C(u,v,k,m) a^{-k-1-Q/2} (1+|x|)^{-m}.
\end{equation}
With these facts in place we are ready
to prove the reproducing formula.
Using the definition of $\pi$, this is the same as showing that
for all $\phi\in \cSd$, $a\in\RR^+$ and $x\in G$
\begin{equation*}
\int_0^\infty
\phi*(D_bu)^**(D_bu)*(D_au)^*(x)\, \frac{db}{b^{Q+1}}
= \phi*(D_au)^*(x).
\end{equation*}
All convolutions involving $u$ commute (by definition of $u$)
and thus we need to show
\begin{equation*}
\lim_{\epsilon\to 0,N\to \infty}
\int_\epsilon^N \phi*(D_au)^**(D_bu)*(D_bu)^*(x)\, \frac{db}{b^{Q+1}}
= \phi*(D_au)^*(x).
\end{equation*}
Since $(D_a u) * (D_a u)^* = a^{-Q/2} D_a(u*u^*)$ the assertion will
follow if for all $v\in\cSZ$
\begin{equation*}
g_{\epsilon,N}(x)
:=\int_\epsilon^N v*D_b(u*u^*)\, \frac{db}{b^{Q+1}}
\end{equation*}
converges in $\cSZ$
(by the $L^2$ spectral theory and \eqref{eq:1} the limit is $v$).
For $g_{\epsilon,N}$ to be Cauchy in $\cSZ$ it suffices to show that
for all $m$ and $|\alpha|\leq m$ we have
\begin{equation*}
\int_0^\infty \sup_{x\in G}
(1+|x|^m) |R^\alpha (v*D_b(u*u^*))(x)|\,\frac{db}{b^{Q+1}}
< \infty.
\end{equation*}
This can be verified by choosing $k$ and $m$ large enough in \eqref{eq:4}
(when $b\leq 1$) and \eqref{eq:5} (when $b\geq 1$).
\end{proof}
\begin{remark}
The integral \eqref{eq:1} is finite for any $\widehat u\in
\mathcal S(\RR^+)$ with support away zero.
Therefore for the lemma one needs to normalize
$\widehat u$ such that the integral is one.
\end{remark}
\subsection{Coorbit description and atomic decompositions
for Besov spaces}
In this section we show that the Besov spaces on stratified Lie groups
can be described via certain Banach space norms of wavelet transforms (see Section 1).
The Banach spaces of interest are
equivalence classes,
$L_{s}^{p,q}:=L^{p,q}_s(\RR^+\times G)$ for $1\leq p,q\leq \infty$, ~$s\in \RR$,
of measurable functions for which
\begin{equation*} \| f\|_{L^{p,q}_s} = \left( \int_{\RR^+} \left(
\int_G |f(a,x)|^p dx \right)^{q/p}a^{-qs/2} \,\frac{da}{a}
\right)^{1/q}~<\infty~.
\end{equation*}
As a corollary to Lemma ~\ref{equivalency-with-heat-kernel} (b)
and Theorem~\ref{norm-equi-general} we note:
\begin{corollary}\label{w-inequality}
Let $1\leq p,q \leq \infty$ and $s\in\RR$, then
for any vector $u\in \cSZ$ the mapping
$$\cSZ\ni v\mapsto W_u(v)\in L^{p,q}_s$$
is continuous.
\end{corollary}
The main result of this paper follows:
\begin{theorem}\label{analyze vector}
Let
$\widehat u\in \cS(\RR^+)$ be compactly
supported.
Then $u$ is an analyzing vector and,
for any $1\leq p,q\leq \infty$ and $s\in\RR$,
up to norm equivalence we have
$$B_{p,q}^{Q-2s/q}(G)=\mathrm{Co}_{\cSZ}^u L^{p,q}_s.$$
Furthermore, the frames and atomic decompositions from
Theorem~\ref{thm:7} all apply.
\end{theorem}
\begin{proof}
We first verify the conditions of Theorem~\ref{thm:coorbitsduality}
with $S=\cSZ$, $B = L_s^{p,q}(\RR^+\times G)$,
and $\pi$ as given in \eqref{eq:repn}.
By Lemma~\ref{all-vanishing-moments} it is known
that $u$ is in $\cSZ$.
Let $\phi\in \cSZd$ such that
$\langle \pi(a,x)u,\phi\rangle=0$ for
all $(a,x)\in \RR^+\times G$. By Theorem \ref{norm-equi-general}
$\phi\equiv 0$ in $\dot B_{p,q}^s$ for all $1\leq p,q\leq \infty$ and
$s\in \RR$. The continuous embedding
$B_{p,q}^s \hookrightarrow \cSZd$ implies
that $\phi\equiv 0$ in $\cSZd$. Thus $u$ is a cyclic
vector in $\cSG$.
The following H\"older inequality and
Corollary~\ref{w-inequality} imply the continuity of map (\ref{eq-doubleCont})
in Theorem~\ref{thm:coorbitsduality}:
$$\|f\ast W_u(u)\|_{L_{s'}^{\infty,\infty}}
\leq \|f\|_{L_{s}^{p,q}} \|W_u(u)\|_{L_{-s}^{p',q'}}
$$
where $1/p+1/p'=1$ and $1/q+1/q'=1$, $s'=s-2Q/q'$.
The multiplier $\widehat u$ satisfies \eqref{eq:1} (otherwise we normalize $\widehat u$), therefore the reproducing formula
$W_u(\phi)*W_u(u) =W_u(\phi)$ holds true by Lemma~\ref{lem:reprformula}. Theorem \ref{norm-equi-general} completes the proof of the first part.
For the last statement we need to verify that the conditions of
Theorem~\ref{thm:7} are satisfied. The invariance of $\cSZ$ under $\pi^\infty$
(and hence the invariance under weak derivatives) follows from
Lemma~\ref{lem:repn}.
By Young's inequality derived below
\begin{align*}
\|f\ast g\|_{L_s^{p,q}}&=
\Big\| \int_{\RR^+}\int_G \tilde g(a,x) { R_{(a,x) } f} ~ a^{-(Q+1)}dadx\Big\|_{ L_{s}^{p,q} }\\%\notag
&\leq \int_{\RR^+}\int_G |\tilde g(a,x)|~ \| { R_{(a,x) } f} \|_{L_s^{p,q}}~ a^{-(Q+1)}dadx\\%\notag
&= \|f\|_{L_{s}^{p,q}}\|\tilde g\|_{L_{2Q+s}^{1,1}}\\%\notag
&= \|f\|_{L_{s}^{p,q}}\|g\|_{L^{1,1}_{-s} }~,
\end{align*}
and Corollary~\ref{w-inequality}, for any
$|\alpha|\leq \dim(\RR^+\ltimes G)$,
all convolutions
\begin{equation*}
B\ni f\mapsto f*W_{\pi(X^\alpha)u}(u) \in B
\quad\text{and}\quad
B\ni f\mapsto f*W_{u}(\pi^*(X^\alpha)u) \in B
\end{equation*}
are continuous and thus the decompositions apply.
\end{proof} | 122,654 |
Hip-hop groups are all about the dynamics. Probably more apparent than in any other genre, rap has always exceeded as a platform where verbose identities get voiced alongside shady anti-heroes, masterminds and deranged lyricists within the confines of some verses and a chorus. When Clash opens up conversation with Flatbush’s own trio of rap servants, it’s clear straight from the off who’s playing what roles. Brash and consciously unreserved, rappers Meechy Darko and Zombie Juice bark out answers whilst in-house producer/part-time rapper Erick Arc Elliott swoons in the background, pitching in thoughtful quotes when he seems fit. It’d be way too easy to compare the Flatbush Zombies to the diversity of Ol’ Dirty Bastard’s Rottweiler persona sewn up tight with the RZA’s masterful guise. If anything though, that totally disregards how they’re swerving in their own acid-driven, nightmare-deluged lane.
Read an excerpt from our interview with Flatbush Zombies below.
What was it like trying to find your feet in Flatbush?
ZJ: Growing up in Flatbush was hell! No, growing up in Flatbush was fun but when you don’t have a lot of money you have to hustle your way around town and follow that NY State Of Mind.
M: It’s like survival of the fittest. People don’t live here for long if they can’t eat. People move in and then they move out because it’s hard and you have to hustle.
How big a part did school play in you now making music?
ZJ: Me and Meech dropped out of high school, but Erick finished school. We didn’t notice that we dropped out of high school until about two years after. That’s when we actually realised ‘holy shit, we dropped out’.
M: I couldn’t fit with school. It was the system and shit. It didn’t work for me personally and the way that I think so I had to leave. It was like jail.
If school incarcerated you, was it music that eventually set you free?
M: Definitely. I was out of school and I wasn’t doing music back then but listening to music definitely helped me. I didn’t start making music until maybe two years after we left school. Erick was making music for years though.
E: I was doing it since I was about fifteen or sixteen, then about three years ago we became the group or whatever.
So, Erick, what or who influences your beatmaking style?
E: In general, I like art. Musical influences were a big part of my shit. George Clinton, Outkast…I listened to a bit everything.
ZJ: They got swag!
And you started out making beats on the MV8000?
E: How did you know that? Because I'm a wizard…
ZJ: Hey Grand Wizarrrddd!
E: That’s pretty good. Using a drum machine is so good. I thought it was really cool to follow in the footprints of people making music using them. Ones like the MPC, but I ended up selling it in the end and got into drum programming which I enjoyed a lot better in a weird way.
‘D.R.U.G.S’ genuinely sounds like it was a lot of fun to make…
M: It was the funniest thing I ever did in my life next to losing my virginity and taking acid.
What’s acid like? I've never tried it before.
ZJ: Altered perceptions, reality you question.
M: It’s literally a mind-altering drug. It’s literally what it is; I can’t really describe it for you because it’s different for everybody, but I can’t really say because it’s the hardest question.
This is an excerpt from the April 2013 issue of Clash magazine. Find out more about the issue.
Words: Errol Anderson
Photography: Kevin Amato | 201,117 |
BELFAST, NORTHERN IRELAND — Seamus McElwaine, one of Northern Ireland's most wanted fugitives, was killed Saturday in a shootout with British soldiers. The Irish Republican Army said he was killed while on ''active service'' and offered condolences to the family. Press Association, the British domestic news agency, said the soldiers came upon two men preparing a bomb on a road near the Irish Republic border and killed one of them. McElwaine, 25, imprisoned for two killings, took part in a breakout by 37 prisoners at the Maze top security prison in 1983. Eleven remain at large. | 186,149 |
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TITLE: How do I derive the closure of $\{1/n: n \in \mathbb{N}\}$?
QUESTION [3 upvotes]: So, I have to prove that
$$\mathrm{cl}\left(\left\{\frac{1}{n}: n \in \mathbb N\right\}\right) = \left\{\frac{1}{n}: n \in \mathbb N\right\}\cup\{0\}$$
Here $\mathrm{cl}(A)$ is the closure of set $A$.
Can anyone assist me with starting this proof? To prove that these two sets are equal, do I show that A ⊆ B and B ⊆ A?
(Sorry, I'm still trying to get the hang of proving…)
REPLY [0 votes]: Stating the question:
Let $A = \{\,\tfrac{1}{n}:n = 1\,,2\,, \ldots \,\}$, and let $\overline{A}$ denote the closure of $A$.
We want to show that $\overline{A} = A \cup \{0\}$.
Defition of the closure $\overline{A}$ of $A$:
$\overline{A} = A \cup A'$, where $A'$ is the set of limit point of $A$ in $\mathbb{R}$.
Assuming that we work in $\mathbb{R}$ with the usual $|\, \cdot \,|$-metric (i.e. the distance function $(x,y) \mapsto |\,x-y\,|$). If so, a point $p \in \mathbb{R}$ is a limit point of $A$ iff for every $\epsilon > 0$ $$B(\epsilon , p) \cap (A \setminus \{p\}) \ne \varnothing,$$
where $B(\epsilon,x) = \{\,x \in \mathbb{R} : |\,x-p\,| < \epsilon \,\}$ (called the open ball of radius $\epsilon$ and center $x$.)
Something like this (or how is it done in your book?).
Solution
From this we see that our task is to show that the only limit point to $A$ is $0$.
So a good begining might be to show that $0$ is a limit point and from there go on to show that any other point $p \ne 0$ is not a limit point. | 124,486 |
Ko'olau Bed & Biscuit
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Seat adjuster, 18-way front including 4-way power lumbar adjustment
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Front Premium floor mats with V-Series logo
AVAILABLE Sueded microfiber steering wheel and shifter
AVAILABLE Cold Weather Package: heated driver and front passenger seats and automatic heated steering wheel
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AVAILABLE Platinum Package: Parking Package, Semi-Aniline leather seats, interior color Sedona Sauvage with Jet Black accents with exclusive Carbon Fiber trim and UltraView dual pane sunroof (Requires Premium Package)
EXTERIOR
Grilles, performance design front fascia with dark aero mesh
Rocker moldings, body-color, with black extensions
Moldings, black
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Wheels, 19" premium painted alloy with Pearl Nickel finish
• Tires, 19" summer-only, run-flat
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Laminated acoustic windshield and front side glass
Rear spoiler
Rear lower air diffuser with body-color insert
Additional Black accents
AVAILABLE UltraView dual pane sunroof
AVAILABLE Wheels, 19" (48.3 cm) premium painted alloy with Satin Graphite finish
• Tires, 19" summer-only, run-flat
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• Tires, 19" summer-only, run-flat
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Engine, 3.0L Twin Turbo High-Output V6
Automatic Stop/Start with disable switch
10-Speed automatic transmission
Electronic Precision Shift
Quad trapezoid bright tip exhaust
4-wheel anti-lock brakes
Black Brembo front performance brakes with V-Series logo
Performance Traction Management
Launch Control
Electronic limited slip differential
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Performance suspension
AVAILABLE All-Wheel Drive
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Market.
Advanced process control (APC) systems are a set of excellent control and optimization products, which are fully integrated into a uniquely powerful collection that meet the requirements of every application, from a small-scale single-unit plant to a large-scale multi-unit facility. It enables the organization to increase the efficiency, productivity, and to reduce the downtime of the industry.
The competitive landscape of global advanced process control market is formed by major players of the market. The automated process control has driven many areas like oil & gas, power, pharmaceuticals, food & beverages, chemical, and others. North America leads the market for global advanced process control. ABB Group (U.S.) a leader in automation and robotics recently won an order to provide overall process control to SCA’s Ostrand Pulp mill expansion. The aim of the project is to double the production capacity of bleached softwood Kraft pulp. Valmet’s (Finland) APC application has improved combustion for the biomass boiler at West Rock.
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The global advanced process control market is segmented on the basis of type, revenue source, application, and region. On the basis of the type the segment is further classified into advanced regulatory control, multivariable predictive control, inferential control, sequential control, and compressor control. On the basis of revenue source the segment is further divided into software and services. APC caters wide area of applications like oil & gas, power, pharmaceuticals, food & beverages, chemical, and others.
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Search results for "swa"
Shadow Modeling
This digital 3-D model of Boston reveals the shadows cast by new construction
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LOHA, SOM, and Kevin Daly Architects collaborate on new student housing at UCSB
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News
Work Stride and Novartis honored for Most Valuable Collaboration
Tue, 16 Apr 2019
Photo: Work Stride co-developer Lillie Shockney celebrates with Dawn Aubel, Director of Global Patient Relations for Novartis
Work Stride: Managing Cancer at Work and Novartis were recognized as the Most Valuable Collaboration at the 2019 eyeforpharma Awards on April 16th. This award recognizes innovative collaborative initiatives (between two or more pharmaceutical companies or between pharmaceutical companies and other entities) that have made a meaningful contribution to improving people’s lives and which clearly bring a new proposition to customers and/or healthcare systems, create new value and redefine what is possible in the pharmaceutical industry with a degree of lateral thinking.
eyeforpharma Philadelphia is the most significant global meeting for pharma executives to meet, learn, network and do business to embrace value-based care, build new relationships and destroy silos. More than 800 industry decision makers from Sales, Marketing, Patient Engagement, Patient Advocacy, Medical Affairs, Market Access and RWE functions come together with patients, payers, healthcare providers, health start-ups and cutting-edge solution providers.
The video below highlights our award-winning collaboration.
Click here to learn more about Work Stride: Managing Cancer at Work. | 47,536 |
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There is plenty of room to relax in our Penthouse Suite! All of these suites include a sofa sleeper for extra guests. They also each have a fireplace, for those chilly Kentucky evenings.
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Our Penthouse Suites feature an additional sleeping area on the first floor which includes a queen sized bed. A full bathroom is located just off of the sleeping area.
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Our fully-equipped kitchens feature stainless steel appliances and granite countertops. Appliances include a full-sized refrigerator, microwave, oven, stove, coffeemaker, and dishwasher. We'll even do your grocery shopping for you!
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The most important meal of the day is on us! Enjoy our HomeTouch™ breakfast buffet, served everyday in the Gatehouse area.
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Unsworth to be guilty of the accusations", Unsworth's complaint said , according to Reuters. Unsworth is seeking more than $75,000 in damages, and demands that Musk - who recently called him a "child rapist" - stop making any further allegations.
Still, American consumers could start feeling the cost in everyday goods, as the latest move brings all Chinese imports subject to a new tariff to US$250 billion, roughly half of China's shipments to the USA previous year. President Donald Trump on Monday said tariffs had put the U.S. "in a very strong bargaining position". When Cohn tried, using data and history, to argue against Trump's conviction that trade deficits with other countries made the USA the loser, the book reports Trump ...
Arun Jaitley , Finance Minister of India briefed the media and said that the merger has been conceived in such a way that the combined lender will not end up being weaker than the existing individual entities. The government is now looking to offload its majority stake in IDBI Bank to Life Insurance Corp. of India . "No employee will face any adverse service conditions after the amalgamation".
The White House said in a statement that Trump had been clear that he and his administration would continue to take action to address China's trade practices and encouraged Beijing to address US concerns. The Office of the U.S. Trade Representative charged in a March report that China is using predatory tactics to obtain foreign technology, including hacking U.S.
However, senior administration officials told reporters the initial list announced in July was reduced by 300 product lines after the administration received 6,000 written comments from consumers and businesses. Trump has said he will no longer allow China to take advantage of the United States on trade, though opposition to escalating tariffs has swelled in recent weeks within US business circles.
His latest warning came amid growing expectations that the White House will impose 10 per cent tariffs on a $200 billion list of about 1,000 Chinese exports, from toys and tyres to seafood and furniture. Census Bureau data. The Chinese foreign ministry said Thursday that it was invited to hold new talks. USA officials say those violate Beijing's market-opening commitments and worry they might erode United States industrial leadership.
Elon Musk , Tesla Motors chairman and CEO, has promised that by the end of the year, the electric carmaker will turn an annual profit for the first time in its 15-year history. Unsworth told CNN , Elon's sub was a " PR stunt " that "had absolutely no chance of working". The suit, which blasts the "unlawful, unsupportable, and reprehensive acccusations" by Musk that Unsworth "is a pedophile and child rapist ", was filed in a California court.
Kent Landers, a spokesman from Coca-Cola , declined to comment about Aurora. Aurora's shares surged on the news, jumping as much as 23 percent Monday in NY to $8. It is the non-psychoactive chemical component of marijuana used for medicinal goal which doesn't produce the high that usually comes from tetrahydrocannabinol (THC).
The company, as we all know, has introduced flamethrowers and hats in the past and is no stranger on taking unconventional approaches. Though it remains unclear if it is Musk himself "test-driving" the machine or some other Boring Company employee, the billionaire did note that this was one of the new machines used by the company for burrowing and building a vast tunnel network under the ground.
In order to bring their product on top of search results, there are a number of tricks sellers have deployed against Amazon's automated system. According to the Journal piece, brokers in China identify Amazon staffers who have accounts on the WeChat messaging platform, and approach them with offers for payment.
Flood waters from the Trent River inundate a park in Pollocksville, North Carolina. North Carolina Governor Roy Cooper warned Saturday that the storm was more risky now that it was on land. The rain was unrelenting in Cheraw, a town of about 6,000 people in northeastern SC. "We feel fully supported". It was expected to reach 24 feet by midday Sunday.
Contaminated strawberries have been found in all of Australia's mainland states. Police began looking into the potentially unsafe berries after a Facebook user on Saturday wrote that their friend took a bite of a strawberry and ate "half a sewing needle", according to the Palm Beach Post .
In Pune petrol was retailing at pumps for Rs 89.31 per litre, and diesel for Rs 77 per litre, both record highs. Since oil prices have started rising, following USA sanctions on Iraq and the slide of the rupee against the dollar, there have been calls to reduce tax.
U.S. officials are pressing allies in Europe, Asia and the Middle East, including Saudi Arabia , to adhere to the sanctions, which are aimed at pressuring Iran to negotiate a new agreement to halt its nuclear programmes. Tanker tracking compiled by Bloomberg indicates that OPEC's fourth-largest exporter is already having to store barrels amid dwindling demand. China typically cuts South Pars condensate imports sharply during the summer months between the second and third quarter because ...
The discovery comes only days after a man found two needles in a punnet of Queensland strawberries sold at a supermarket in Jamestown this week. The Sydney Morning Herald on Monday (Sep 17) quoted Queensland Police Commissioner Ian Stewart as saying that police had received a report of a metal object in a banana.
The couple scoop up the media and marketing company that boasts 175 million United States consumers each month from Meredith Corporation , which ended up with the brand as part of its purchase of Time Inc in January this year. Print advertising is on the decline and Time cut circulation down from 3 million to 2.3 million earlier this year, according to the Alliance for Audited Media.
Amongst a recession and slump in the pound, Carney has stated that the United Kingdom housing market could see prices fall by as much as a third. Carney is due to speak in Dublin at 1000GMT. "Carney has made himself a laughing stock in the City with such an outrageous warning", said Richard Tice, a Brexit supporter who is co chair of the "Leave Means Leave" group.
Duke Energy has shut down its Brunswick Nuclear Plant as it was directly in the predicted path of Hurricane Florence . "Those power plants are, one, obviously hardened". The locations of nuclear power plants in North and SC. "This is no ordinary storm and customers could be without power for a very long time - not days, but weeks", said Fowler.
Australian authorities are warning its citizens against eating certain brands of strawberries after some have been found to contain hidden sewing needles inside. The strawberry industry in Queensland , Australia, is suffering after needles were found in strawberries. "It is a very, very broad picture and we can't speculate in any way, shape or form", said Terry Lawrence , Queensland acting chief superintendent.
After 123 years, Henri Bendel is closing its doors...for good. L Brands acquired Bendel in 1985 and led its expansion into 11 states. "We have chose to stop operating Bendel to improve company profitability and focus on our larger brands that have greater growth potential", CEO Leslie Wexner said in a company statement.
Queensland's Chief Health Officer Dr Jeannette Young said anyone else who had bought the brands of strawberries without signs of tampering should return them to the store or throw them away. Sewing needles have been found hidden inside strawberries sold at Woolworths, sparking a recall of two brands that had been sold in Queensland , NSW and Victoria .
An admission of any past drug use is grounds for a lifetime ban from the US, although some banned people can successfully apply for waivers. "We don't recognise that as a legal business", Mr Owen said..
The systems that allow the 1200 to do this have been designed by a team led by graduate engineer Stefan Hans. As BMW R1200GS is the total self driving motorcycle so it is highly focused on safety. ...
Light, sweet crude for October delivery dropped $1.78, or 2.5%, to $68.59 a barrel on the New York Mercantile Exchange, its worst day since August 15 after it closed at its highest level since July 20 a session earlier..
If a Waffle House closes , officials believe that area must have been hit hard. The restaurant chain appears to be taking the job seriously, activating its own storm center as Hurricane Florence approaches . Waffle House said on Twitter that its "Storm Center" was monitoring Hurricane Florence , which is expected to hit the coast between North Carolina and SC late Thursday or early Friday and could impact 230 of its restaurants.
The same issue prompted the firm to recall about 800,000 pickup trucks previous year. GM reported that if EPS is lost and suddenly returns, the driver of the vehicle could have difficulty steering. The company says it has 30 reports of crashes with two injuries, but no deaths. GM said in a government filing that "hydrogen gas can remain trapped in the piston body", which may make the brakes feel soft and take longer to stop the vehicle.
)
The alert test will be performed on the National Wireless Emergency Alert system next week on Thursday, Sept. 20 at 10:18 a.m., with a EAN (Emergency Action Notification) test at 10:20 a.m. "..
Paid review fraud - when companies or individuals "sell" fake reviews to business owners - is a violation of the law in many jurisdictions. The company wants to make it clear that " writing fake reviews constitutes criminal conduct under laws relating to impersonation fraud", citing the European Union's Unfair Commercial Practices Directive , which prohibits "the practice of falsely representing oneself as a consumer".
Further, independent voters back Democrats over Republicans by 50-35 percent, according to the poll. As Newsweek reports, when asked if they believed the 45th president is intelligent, just 51 percent answered that they do, while 42 percent answered that they did not..
He added that the lira was experiencing "fake volatility", repeating his view that the currency's fall was part of a foreign plot and the result of an economic war. "The central bank is independent and makes its own decisions", he said. It said: "Accordingly, the committee has chose to implement a strong monetary tightening to support price stability".
Bezos solicited ideas on Twitter a year ago for ways to donate some of his wealth. He cited his work at his space company, Blue Origin, and his investment, estimated now at $1 billion a year, "in the future of our planet and civilization through the development of foundational space infrastructure".
The announcement followed reports by American and European chambers of commerce that foreign companies in China have been hurt by earlier tariff hikes by both sides in the fight over Beijing's technology policy. But Kudlow was non-committal over the chances of a breakthrough, adding: "I guarantee nothing". "China has indeed received an invitation from the U.S. | 68,835 |
America is one of the few countries that the has most comprehensive digital systems for immigration and visa procedures. The process to get US visa is pretty complex and getting a visa can be tough for people from certain countries. When you travel to USA, they have specific entry and exit requirements to their country ranging from how their Customs and Border Protection(CBP) Officers screen incoming passengers to enter US to how they handle incoming luggage at Customs. You need to carry all documents supporting your travel to US.
What is US Port of Entry (PoE) ? What typically happens here ?
In America, locations such as international airports, land entry checkpoints or seaports that are used to screen, check visa, interview and allow incoming passengers or travellers to enter America are called as “Port of Entry (PoE)” locations. It is first point of entry for anyone planning to enter US. Usually most of the large international airports are considered as port of entry. Also, you would see many other port of entry locations designated at the land border of Mexico and Canada, where the border is shared. Any passenger or traveller entering US need to go through Port of Entry process at these locations and also go through customs check and have their luggage screened. You need to declare goods, your history as needed in the customs form as well. Below are the basic requirements for entry at PoE.
Port of Entry Requirements to enter USA
If you are traveling to US, you need to meet the below requirements at port of entry :
- Need to have valid US Visa related to your intended travel purpose, US Green Card / Permanent Residency card, be a US citizen or have passport from Visa Waiver Countries and go through ESTA process.
- Need a valid Passport for your entire duration of travel in America and preferably 6 months beyond your travel.
Documents Checklist for US Port of Entry
- Valid Passport
- Depending on your Country of origin and status you need one of the below
- Valid US Visa Stamp in Passport
- Green Card or Permanent Residency Card
- Passport from Visa Waiver Countries for ESTA process
- Completed US Customs Declaration Card. ( Not needed for anyone eligible for Automated Passport Control ( APC) )
- Copy of your return flight ticket, if you are a visitor.
- Supporting documentation related to the US visa you are entering.
- The supporting documents vary by the visa you plan to use to enter US. For example, if you are entering for temporary work on H1B, you need to have all documents related to the same like offer letters, USCIS H1B approval notice, LCA, additional supporting documents of H1B petition.
- Copy of your Hotel stay details or details of address, where you plan to stay.
- Copy of your conference / business purpose details, if you are entering for business
Port of Entry Procedures for USA Visitors, US Visa Holders, Residents
Below are the typical process steps to enter US as a foreigner. Some of these procedures are done before you arrive at US port of entry as part of the standard verification of visa, passport and other documents.
Travel to USA – High Level Process at US Port of Entry (PoE)
Step 1 – US Visa, Passport Check by Airline
If you are flying to US, first step is that your airline would ask for a valid US visa. You need to show your valid US visa that is in good state. Any visa stamps that are damaged are not valid. If your US visa is in old expired passport, but your visa in it is valid, then you can still use that visa to enter US, provided the visa is not damaged. Also, they will check your passport and verify, it all matches. If you are traveling by road, it would not matter as there is no pre-check.
Step 2 – Fill out Customs Declaration Form
Once you are on the flight, closer to your flight landing, airline will distribute US Customs Declaration form, you need to fill it out. Check out How to Fill out US Customs Declaration Form. If you are traveling using ESTA (Visa Waiver Program), B1/B2 or D Visa, or a Canadian Citizen, including Green Card Holders, then you would not fill the physical customs declarations form. You will fill out Customs form digitally at Automated Passport Control (APC). APC kiosks are available at major international airports in USA.
Step 3 – (Optional ) Fill out I-94 Arrival / Departure Card
The arrival departure I-94 card process is fully automated and digital. But, in some cases, when you enter US using land entry point or when the online system does not work, you may be asked to complete I-94 form. Read How to Fill out I-94 form in US
Step 4 – Arrive at Airport, Seaport – Stand in Immigration Line
Once you arrive at the US Port of entry after getting down from the flight or ship, you will walk along with fellow passengers and need to look for signs of immigration. You cannot really miss it as your exit from airport will be directed in such a way that you end up at the immigration section. Anyways, once you are there, depending on your visa type, you will need to pick a line. There will be separate lines for US Citizens, people from Visa Waiver Countries, Green Card holders, etc. If you are eligible for Automated Passport Control Kiosk Processing, you should get into that line, otherwise, take the general CBP Interview line. There will be signs indicating where to go and also there will be support personnel asking you regarding the visa and directing you to the right line. If you are in doubt, ask the support person and get into the right line. Make sure you keep all the completed forms, visa documents with you for inspection and checking.
Step 5 –Automated Passport Control ( APC ) process (Applicable to few visa types )
If you are entering US at a major international airport or seaport as a Green Card Holder, Canadian Citizen, using ESTA ( Visa Waiver Program), B1 /B2 or D visa, then you will go through Automated Passport Control (APC) Kiosk process. At APC Kiosk, you will be scanning your visa, system asks few questions, asks you to fill customs form in the system, takes a picture and gives you a receipt to take it to CBP Officer. Read Automated Passport Control (APC) Kiosk Experience at US Airport. If you do not qualify for APC Process, then you skip this and go to normal CBP officer interview.
Step 6 – CBP Officer Interview at PoE
If you have used the APC Kiosk and have a receipt, then you would stand in a different line. If not, you will go through regular CBP interview line.
- CBP Officer Interview, Entry Stamp in Passport after APC Kiosk: If you have utilized APC Kiosk process, you will need to give our the receipt to the CBP Officer and they will check your visa, passport and the receipt printed. They will ask you questions regarding your travel such as your purpose of travel, what you do, where you live, etc. These are short interviews to validate your intent. They will put a stamp on your passport with date you entered, class of visa and let you go to collect baggage, if they are convinced.
- CBP Officer Interview, Entry Stamp in Passport without APC Kiosk : If you do not qualify for APC process, then you will need to go through a much more detailed process. You need to give your passport and Visa ( if in different passport) to the CBP officer. They will check the system and ask you for your reason for travel and documents supporting your visa that you plan to use for entry. For instance, if you are entering on H1B or L1 visa, they would ask USCIS approval notices, employer letters, LCA, etc. You need to be prepared to answer regarding your role, what you do and how long you intend to stay in US and all other details related to your visa. Read US Port of Entry Questions by CBP Officer. If they are convinced, they will ask you for biometrics scan of both of your hands, then take a picture and then stamp on your passport. The CBP Stamp has the Port of Entry code, date you entered in US and they write class of US Visa you are entering on and how long it is valid on it and then let you go to collect baggage. Check Sample CBP Officer Stamps on Passport at PoE. If the CBP officer is not convinced, you may be taken to a separate area for secondary inspection, where they will ask you more details and do detail interview. If you cannot convince the CBP officer and they find out that you have done any fraud or violated anything, they can deny your entry to US and send you back to your home country.
Step 7 – US Customs Clearance
After your CBP interview, you will need to pick up your baggage and take it for customs clearance. If you are bringing any items that were supposed to be declared such as soil, plants, etc. then you need to go to the Customs inspection process. The Customs officer will inspect the goods that you are bringing in by scanning your bags, manually inspecting them and then allow or deny you to carry the items that you have declared. You should declare everything as required on the US Customs form, not declaring goods that you are not allowed to carry without declaration can create issues for you. Customs officer would usually check for food items, agricultural items such as soil, plants, etc.
- Health Screening ( Optional ) : Sometimes, there maybe a Public Health Officer doing visual checks of passenger and you maybe subject to health screen, if you exhibit any symptoms of certain diseases. If you are arriving in US from a certain country that has any public current health risks, you may be subject to this special screening. This is an optional step and not everyone will go through this. Read Official Public Health Screening Guide by CDC
Step 8 – Exit Airport / Land Entry Point / Seaport
Once you are done with customs process, you are free to exit the airport, land entry point or sea port. Though it may seem complex, the process can be quite seamless, if you have everything in place and all your documents are genuine.
Check US Admission Stamp – Visa, Entry, Admit Until dates
It is very important for you to verify and check the US Port of entry stamp on your passport by CBP officer. It should have the correct date of entry, correct visa class you are using to enter US and correct admit until date ( basically the validity date for you to remain in US). If CBP officer writes any wrong information, it will impact your status in US, how long you can stay, including applications for benefit or a document at places such as SSN office, Driving license, etc. Make sure you check and correct them there itself, if any issues.
Common FAQs
Below are some of the common FAQs on US Port of entry procedures
Extension for Stay in US beyond Admit Until Date in Passport Stamp ?
In general, you need to depart US by the ‘Admit until’ date written on the Passport Stamp. The admit until date is your deadline to exit or your visa status expiration date . In some situations, you may need to extend your stay for health reasons, need to stay with your family or any other reason. In such case, you need to file Form I-539 with USCIS with all the documentation to extend your status. If you are visitor planning to extending your stay, you will use the same form.
Your US Visa Stamp Validity vs Admit Until Date in Stamp at Port of Entry by CBP officer
Having a US visa valid for 5 years or 10 years does not give you the right to stay in US continuously for that munch amount of time. US visa stamp on your passport is only a entry document that allows you to arrive at Port of Entry. The CBP officer at Port of Entry determines how long you can stay in US based on your visa type and writes the same on the stamp that they put on your passport. It is your responsibility to maintain status when you enter US based on your visa type by not engaging in any unauthorized activity. Read US Visa vs. Status – Expired Visa – FAQs
Travel with-in US after Port of Entry Screening by CBP Officer
You are free to travel to any state or city within America after you are screened and admitted by a CBP officer by putting a stamp on your passport. You do not have to go through any immigration check points, if you are within US. When you fly within US, you will go through domestic airport and it does not have immigration points. You will only go through general security screening, but there will not be any CBP officers asking your visa status. Also, you do not have to go through any immigration process, when you move between states with in US.
US Exit Stamp on Passport at Port of Entry / Exit Point ?
Your passport will NOT be stamped, when you exit US. The airline kiosk or check-in counter will validate your visa for the destination you plan to go and your US visa that you used to enter US and issue you boarding passes at airport. The process is pretty similar at Land check point and sea port, they will just verify your entry stamp and check your visa to enter other country and let you go. Nothing will be stamped. If you were given an I-94 card, then you will need to hand over the card to the airline staff at the boarding gate. | 320,200 |
Disabled residents of New Orleans had Article III standing to sue Uber for refusing to make uberWAV (wheelchair accessable rides) available in New Orleans. The residents didn’t have to sign up to use Uber’s app (and thus agree to its arbitration clause) to have Article III standing since doing so would have been a futile gesture. As there was no uberWAV service in New Orleans, there was no way that disabled New Orleans residents could use Uber’s services, whether or not they signed up for the app. This distinguished New Orleans from Chicago, where uberWAV was available and where it would be necessary for the plaintiff to show that despite the availability of the service, he or she couldn’t access it on the same terms as non-disabled persons. Also, the claim satisfied the causation and redressability prongs of standing. Though for uberWAV to work, drivers with wheelchair accessable cars would need to sign up with Uber, the first step was for Uber to offer them that chance. The decision also rejects Uber’s argument that plaintiff should be equitably estopped from avoiding the arbitration agreement that was part of the terms of the Uber app since plaintiff’s claims did not depend on or arise from the app or its terms and conditions. | 173,870 |
Ryanair strike dates 2019: When is next Ryanair strike? List of ALL walkout action in 2019
RYANAIR passengers faced a swathe of walk-outs in 2018, and 209 looks set to bring more of the same. So when is the next strike?
Ryanair cabin crew in Spain are set to walk out for three dates in January, which could affect tens of thousands of passengers. The walkouts have been called by two Spanish unions, USO and SITCPLA. The unions said the walkouts were based on a refusal to offer contracts for crew under Spanish regulations.
Ryanair was accused of “refusing to abide by the Spanish constitution” by the unions for the budget airline’s failure to provide sufficient work contracts.
According to Spanish media outlets, the strikes are currently planned for January 8, 10 and 13, each due to last 24 hours.
However, the unions have given the airline ten days to "avoid" the industrial action.
A union spokesperson said: "There is a period of ten days for the management of Ryanair to reconsider and follow, once and for all, the legality in Spain."
Ryanair strike dates 2019 (Image: Getty)
Claims by the unions supporting the 1,800 staff based in Spain suggest they have “worse working conditions” than others due to Ryanair’s disregard of Spanish legislation.
A statement from the two unions said: “From SITCPLA and the USO, we hope that the company will reconsider and agree to comply with Spanish legislation.”
These latest threats of industrial action come after Ryanair suffered crippling strikes in the summer of 2018.
Irish pilots went on strike over a number of weeks, with thousands of passengers affected.
Ryanair strike dates 2019: The strikes are currently planned for January 8, 10 and 13 (Image: Getty)
On 28 September, 250 flights were cancelled across Germany, Italy, the Netherlands, Portugal, Spain and Belgium.
Approximately 30,000 passengers were affected by the strike.
In September, Ryanair chief executive Michael O'Leary said he was “hopeful and optimistic” that the issues wouldn’t continue to damage the airline after the strikes contributed to a profit warning.
The profit warning in October indicated the strikes cost the airline some £100 million.
Ryanair strike dates 2019: Ryanair chief executive Michael O'Leary (Image: Getty)
Ryanair only recognised unions for the first time in December when it managed to avoid Christmas strikes.
The airline boss pointed to breakthroughs in talks with unions in Ireland and Italy and said Ryanair "hopes to make similar progress in other countries in the coming months."
On Thursday, Ryanair revealed its December traffic saw a growth of 12 per cent, with 10.3million customers travelling on over 57,000 scheduled flights last month.
Express.co.uk has contacted Ryanair for comment. | 261,376 |
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Never Miss Another Great Deal | 300,945 |
(CNSNews.com) - If Democrats celebrating their party's national convention in Los Angeles want Americans to believe they support family values, they should return all campaign contributions they've received from pornographers, a national women's organization said Monday.
Vice President Al Gore cannot distance himself from Democratic Rep. Loretta Sanchez' efforts to host a fundraiser at the Playboy Mansion in Los Angeles unless Gore - and other Democratic candidates and organizations - return $100,000 donated to them by Playboy since 1992, Concerned Women for America said.
Playboy's contributions also come at a high price to the health of many women and men who produce and buy the magazine, said Brenda MacKillop, a former Bunny who worked at the Playboy Club in Los Angeles from 1973 to 1976.
MacKillop, who speaks to groups about her religious conversion and testifies against pornography's negative effects on producers and consumers, said she was "extremely suicidal and sought psychiatric help during the time I lived in a sexually promiscuous lifestyle as a Bunny."
Other women who worked as Bunnies or posed nude for the magazine became suicidal and chronically depressed after they were exposed to such crimes as stalkings and date-rape, MacKillop said.
"I experienced everything from date-rape to physical abuse to group sex during my time with Playboy," she said.
The former Bunny also witnessed drug-taking at the mansion, knew of colleagues who contacted and passed on venereal diseases to customers or had abortions after getting pregnant by customers.
"Playboy is more than a pornographic magazine," she said. "It is a philosophy that enticed me to throw aside the Judeo-Christian ethics I grew up with, of no premarital sex and no adultery, to practice 'recreational' sex with no commitments."
Dr. Judith Reisman, an author of books on Playboy and the porn industry, said a study she conducted for the U.S. Department of Justice on images of children and crime in Playboy and other girlie magazines exposes these magazines as child pornographers.
Playboy has produced child pornography, has sexualized children under age 18 in photos, cartoons and illustrations, and Playboy Press published photos of a nude 10-year-old Brooke Shields, Reisman said.
Family groups strongly criticize President Clinton and Al Gore for accepting campaign contributions from publishers of magazines such as Playboy. They lambaste Clinton for not living up to a 1992 campaign promise he made to Morality in Media, a media watchdog group, to crack down on child pornography.
"Be assured that aggressive enforcement of federal obscenity laws by the Justice Department - particularly by the Child Exploitation and Obscenity Section - will be a priority in the Clinton-Gore administration," Clinton wrote in an October 1992 letter.
But statistics and other evidence demonstrate little or no prosecution of illegal pornography by the Janet Reno Justice Department, said James Lambert, a columnist with the American Family Association's Internet page.
In fact, obscenity prosecutions have decreased 86 percent since Clinton and Gore were elected, Lambert said, citing a 1997 report from Syracuse University. "Pornographers and advocacy publications like Adult Video News have even admitted that 'adult obscenity enforcement by the federal government is practically non-existent,'" Lambert said.
Over the weekend, Sanchez canceled the "Hispanic Unity USA Party," scheduled to take place Tuesday night at the Playboy Mansion of publisher Hugh Hefner.
Sanchez gave in to concerns by convention organizers that the portrayal of convention delegates side-by-side with scantily-clad Bunnies could undermine their attempts to portray the party as pro-family.
Calls by CNSNews.com to Playboy for comment were not returned. | 383,503 |
You are currently browsing the monthly archive for August 2010.
It had been awhile since I posted here. Have not been very creative lately not no anything blog worthy to share in my life, so it is best not to bore everyone.
Today, I finished these for a nice lady and her support.
Hope her recipent will like it.
polymer chocolates
How about you?
What others have to say… | 128,223 |
TITLE: periodic function, find g(6)
QUESTION [3 upvotes]: If $f$ is periodic, $g$ is polynomial function, $f(g(x))$ is periodic, $g(2)=3$, and $g(4)=7$, then $g(6)$ is
A) 13
B) 15
C) 11
D) none of these
The answer is c) 11, but I did not understand how $g(x)$ was considered linear polynomial (because i got answer when $g(x)$ is $2n -1$), isn't there any $g(x)$ with degree greater than 1 make $f(g(x))$ periodic?
Why? How will you solve the problem?
REPLY [0 votes]: If $g$ is a polynomial one shouldn't assume $g(x)$ is periodic and so the condition that $f(g(x))$ is periodic is quite unusual.
If $g(x)=mx+b$ is linear, and $p$ is the period of $f$ it's easy to see that $f(g(x+\frac pm)) = f(mx +p + b)=f(g(x) + p) = f(g(x))$ and $f(g(x))$ is periodic.
If $g$ isn't linear can we still have $f(g)$ be periodic? Not really.
This is a bit handwavey but I hope it is intuitively clear.
Let's suppose the period of $f$ is $m$. So $f(x+m) = f(x)$. Now if $k$ is not a muliple of $m$ that does not mean that $f(x+k) \ne f(x)$ but in general that will not be the case. And if you view all potential values $x$ you'd find $f(x+k) \ne f(x)$ "almost everywhere".
Now if $f(g(x))$ has a period of $h$ then $f(g(x+h)) = f(g(x))$ and if $g(x+h) - g(x) = k$ then $f(g(x) + k) = f(g(x))$ which means that $g(x+h) - g(x)$ is a multiple of $m$ "almost everywhere".
Okay but if there is a $g(x)$ where $g(x+h) = g(x) + km$ for some multiple $km$ or $m$ and $g$ is not linear then we can find some some $e > 0$ where $g(x+e)- g(x) \ne g((x+h) + e)-g(x+e)$ (because $g$ is not linear) and were $g(x+e)-g(x) < m$ and $g(x+h+e) -g(x+h) < m$ (because $e$ can be arbitrarily smalland polynomials are continuous).
And that means that the difference $g((x+e) +h)$ and $g(x+e)$ is not a multiple of $m$ and that in general "almost nowhere" does $f(g(x+e+h)) = f(g(x))$.
Okay, that was really hand wavey and Rhys Hughes proof is better but I think for the problem my type of thinking (or in similar terms) isn't meant to be proven but taken to be intuitively "obvious".
Frankly I don't like this problem.... | 182,884 |
\begin{document}
\title[Asymptotic stability of the compressible Euler-Maxwell equations ]
{Asymptotic stability of stationary
solutions to the compressible Euler-Maxwell equations }
\author{Qingqing Liu}
\address{(QQL)
The Hubei Key Laboratory of Mathematical Physics, School of
Mathematics and Statistics, Central China Normal University, Wuhan,
430079, P. R. China} \email{[email protected]}
\author{Changjiang Zhu*}
\address{(CJZ)
The Hubei Key Laboratory of Mathematical Physics, School of
Mathematics and Statistics, Central China Normal University, Wuhan,
430079, P. R. China} \email{[email protected]}
\thanks{*Corresponding author. Email: [email protected] }
\date{\today}
\keywords{Compressible Euler-Maxwell equations, stationary
solutions, asymptotic stability}
\subjclass[2000]{35Q35, 35P20}
\begin{abstract}
In this paper, we are concerned with the compressible Euler-Maxwell
equations with a nonconstant background density (e.g. of ions) in
three dimensional space. There exist stationary solutions when the
background density is a small perturbation of a positive constant
state. We first show the asymptotic stability of solutions to the
Cauchy problem near the stationary state provided that the initial
perturbation is sufficiently small. Moreover the convergence rates
are obtained by combining the $L^p$-$L^q$ estimates for the
linearized equations with time-weighted estimate.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
The dynamics of two separate compressible fluids of ions and
electrons interacting with their self-consistent electromagnetic
field in plasma physics can be described by the compressible 2-fluid
Euler-Maxwell equations \cite{Besse,Rishbeth}. In this paper, we
consider the following one-fluid compressible Euler-Maxwell system
when the background density $n_{b}$ is a function of spatial
variable and the electrons flow is isentropic (see
\cite{Duan,UK,USK} when $n_{b}=const.$), taking the form of
\begin{eqnarray}\label{1.1}
&&\left\{\begin{aligned}
&\partial_t n+\nabla\cdot(nu)=0,\\
&\partial_t u+u \cdot \nabla u+\frac{1}{n}\nabla
p(n)=-(E+u\times B)-\nu u,\\
&\partial_t E-\nabla\times B=nu,\\
&\partial_t B+\nabla \times E=0,\\
&\nabla \cdot E=n_{b}(x)-n, \ \ \nabla \cdot B=0.
\end{aligned}\right.
\end{eqnarray}
Here, $n=n(t,x)\geq 0 $ is the electron density, $ u=u(t,x)\in
\mathbb{R}^{3}$ is the electron velocity, $ E=E(t,x)\in
\mathbb{R}^{3}$, $ B=B(t,x)\in \mathbb{R}^{3}$, for $ t>0, \ x \in
\mathbb{R}^{3} $, denote electron and magnetic fields respectively.
Initial data is given as
\begin{eqnarray}\label{1.2}
[n,u,E,B]|_{t=0}=[n_{0},u_{ 0},E_{0},B_0],\ \ \ x\in\mathbb{R}^{3},
\end{eqnarray}
with the compatible conditions
\begin{eqnarray}\label{1.3}
\nabla \cdot E_0=n_{b}(x)-n_{0}, \ \ \nabla \cdot B_0=0, \ \ \
x\in\mathbb{R}^{3}.
\end{eqnarray}
The pressure function $ p(\cdot)$ of the flow depending only on the
density satisfies the power law $p(n)=A n^{\gamma}$ with constants
$A>0$ and the adiabatic exponent $\gamma >1 $. Constant $\nu>0$ is
the velocity relaxation frequency. In this paper, we set $ A=1,\
\nu=1$ without loss of generality. $n_{b}(x)$ denotes the stationary
background ion density satisfying
\begin{eqnarray*}
n_{b}(x)\rightarrow n_{\infty}, \ \ \ \ \textrm{as}\ \ \ \
|x|\rightarrow \infty,
\end{eqnarray*}
for a positive constant state $n_{\infty}>0$. Throughout this paper,
we take $n_{\infty}=1$ for simplicity.
In comparison with the Euler-Maxwell system studied in \cite{Duan},
where the background density is a uniform constant, the naturally
existing steady states of system \eqref{1.1} are no longer constants
$[1,0,0,0]$. The stationary equations to the Cauchy problem
\eqref{1.1}-\eqref{1.2} are given as
\begin{eqnarray}\label{sta.eq0}
\left\{\begin{aligned} &\frac{1}{n_{st}}\nabla
p(n_{st})=-E_{st},\\
&\nabla \times E_{st}=0,\\
&\nabla \cdot E_{st}=n_{b}(x)-n_{st}.
\end{aligned}\right.
\end{eqnarray}
First, in this paper, we prove the existence of the stationary
solutions to the Cauchy problem \eqref{1.1}-\eqref{1.2} under
some conditions on the background density $n_{b}(x)$. For this
purpose, let us define the weighted norm
$\|\cdot\|_{W_{k}^{m,2}}$ by
\begin{eqnarray}\label{def.norm}
\|g\|_{W_{k}^{m,2}}=\left(\sum_{|\alpha|\leq
m}\int_{\mathbb{R}^{3}}(1+|x|)^{k}|\partial^{\alpha}_{x}g(x)|^2dx\right)^{\frac{1}{2}},
\end{eqnarray}
for suitable $g=g(x)$ and integers $m\geq0$, $k\geq0$.
Actually, one has the following theorem.
\begin{theorem}\label{sta.existence}
For integers $m\geq 2$ and $k\geq 0$, suppose that
$\|n_{b}-1\|_{W_{k}^{m,2}}$ is small enough. Then the stationary
problem \eqref{sta.eq0} admits a unique solution $(n_{st},E_{st})\in
L^\infty(0, T; W_{k}^{m,2})$ satisfying
\begin{eqnarray}\label{sta.pro}
\|n_{st}-1\|_{{W_{k}^{m,2}}}\leq C \|n_{b}-1\|_{W_{k}^{m,2}},\ \ \
\|E_{st}\|_{{W_{k}^{m-1,2}}}\leq C \|n_{b}-1\|_{W_{k}^{m,2}},
\end{eqnarray}
for some constant $C$.
\end{theorem}
There have been extensive investigations into the simplified
Euler-Maxwell system where all the physical parameters are set to
unity. For the one-fluid Euler-Maxwell system, by using the
fractional Godunov scheme as well as the compensated compactness
argument, Chen-Jerome-Wang in \cite{Chen} proved global existence of
weak solutions to the initial-boundary value problem in one space
dimension for arbitrarily large initial data in $L^{\infty}$. Jerome
in \cite{Jerome} established a local smooth solution theory for the
Cauchy problem over $\mathbb{R}^3$ by adapting the classical
semigroup-resolvent approach of Kato in \cite{Kato}. Recently, Duan
in \cite{Duan} proved the existence and uniqueness of global
solutions in the framework of smooth solutions with small amplitude,
moreover the detailed analysis of Green's function to the linearized
system was made to derive the optimal time-decay rates of perturbed
solutions. The similar results are independently given by
Ueda-Wang-Kawashima in \cite{USK} and Ueda-Kawashima in \cite{UK} by
using the pure time-weighted energy method. For the the original
two-fluid Euler-Maxwell systems
with various parameters, the limits as some parameters go to zero
have been studied recently.
Peng-Wang in \cite{Peng ,PW1,PW2} justified the convergence of the one-fluid compressible
Euler-Maxwell system to the incompressible Euler system,
compressible Euler-Poisson system and an electron
magnetohydrodynamics system for well-prepared smooth initial data.
These asymptotic limits are respectively called non-relativistic
limit, the quasi-neutral limit and the limit of their combination.
Recently, Hajjej and Peng in \cite{HP} considered the
zero-relaxation limits for periodic smooth solutions of
Euler-Maxwell systems. For the 2-fluid Euler-Maxwell system,
depending on the choice of physical parameters, especially the
coefficients of $u_{\pm}$ were assumed $\nu_{+}=\nu_{-}$,
Duan-Liu-Zhu in \cite{DLZ} obtained the existence and the time-decay
rates of the solutions. Much more studies have been made for the
Euler-Poisson system when the magnetic field is absent; see
\cite{Guo,GuoPausader,Luo,Deng,Smoller,Chae} and references therein
for discussion and analysis of the different issues such as the
existence of global smooth irrotational flow \cite{Guo} for an
electron fluid and \cite{GuoPausader} for the ion dynamics, large
time behavior of solutions \cite{Luo}, stability of star solutions
\cite{Deng,Smoller} and finite time blow-up \cite{Chae}.
However, there are few results on the global existence of solutions
to the Euler-Maxwell system when the non-moving ions provide a
nonconstant background $n_{b}(x)$, whereas in many papers related to
one-fluid Euler-Maxwell system $n_{b}=1$. In this paper, we prove
that there exists a stationary solution when the background density
is a small perturbation of a positive constant state and we show the
asymptotic stability of the stationary solution and then obtain the
convergence rate of the global solution towards the stationary
solution. The main result is stated as follows. Notations will be
explained at the end of this section.
\begin{theorem}\label{Corolary}
Let $ N\geq 3$ and $ \eqref{1.3}$ hold. Suppose
$\|n_{b}-1\|_{W_{0}^{N+1,2}}$ is small enough. Then there are $
\delta_{0}>0$, $ C_{0}>0$ such that if
\begin{eqnarray*}
\|[n_{0}-n_{st},u_{0},E_{0}-E_{st},B_{0}]\|_{N} \leq \delta_{0},
\end{eqnarray*}
then, the Cauchy problem $\eqref{1.1}$-$\eqref{1.2}$ admits a unique
global solution $[n(t,x),u(t,x),E(t,x),B(t,x)] $ satisfying
\begin{eqnarray*}
[n-n_{st},u,E-E_{st},B]\in C([0,\infty);H^{N}(\mathbb{R}^{3}))\cap
{\rm Lip}([0,\infty);H^{N-1}(\mathbb{R}^{3})),
\end{eqnarray*}
and
\begin{eqnarray*}
\sup_{t \geq 0}\|[n(t)-n_{st},u(t),E(t)-E_{st},B(t)]\|_{N}\leq C_{0}
\|[n_{0}-n_{st},u_{0},E_{0}-E_{st},B_{0}]\|_{N}.
\end{eqnarray*}
Moreover, there are $\delta_{1}>0$, $ C_{1}>0$ such that if
\begin{eqnarray*}
\|[n_{0}-n_{st},u_{0},E_{0}-E_{st},B_{0}]\|_{N+3}+\|[u_{0},E_{0}-E_{st},B_{0}]\|_{L^{1}}\leq
\delta_{1},
\end{eqnarray*}
and $\|n_{b}-1\|_{W_{0}^{N+4,2}}$ is small enough, then the solution
$[n(t,x),u(t,x),E(t,x),B(t,x)] $ satisfies that for any $ t \geq
0$,
\begin{eqnarray}\label{UN.decay}
\|[n(t)-n_{st},u(t),B(t),E(t)-E_{st}]\|_{N} \leq C_{1}
(1+t)^{-\frac{3}{4}},
\end{eqnarray}
\begin{eqnarray}\label{UhN.decay}
\|\nabla[n(t)-n_{st},u(t),B(t),E(t)-E_{st}]\|_{N-1} \leq C_{1}
(1+t)^{-\frac{5}{4}}.
\end{eqnarray}
More precisely, if
\begin{eqnarray*}
\|[n_{0}-n_{st},u_{0},E_{0}-E_{st},B_{0}]\|_{6}+\|[u_{0},E_{0}-E_{st},B_{0}]\|_{L^{1}}\leq
\delta_{1},
\end{eqnarray*}
and $\|n_{b}-1\|_{W_{0}^{7,2}}$ is small enough, we have
\begin{eqnarray}\label{sigmau.decay}
\|[n(t)-n_{st},u(t)]\| \leq C_{1} (1+t)^{-\frac{5}{4}},
\end{eqnarray}
\begin{eqnarray}\label{EB.decay}
\|[E(t)-E_{st},B(t)]\|\leq C_{1}(1+t)^{-\frac{3}{4}}.
\end{eqnarray}
If
\begin{eqnarray*}
\|[n_{0}-n_{st},u_{0},E_{0}-E_{st},B_{0}]\|_{7}+\|[u_{0},E_{0}-E_{st},B_{0}]\|_{L^{1}}\leq
\delta_{1},
\end{eqnarray*}
and $\|n_{b}-1\|_{W_{0}^{8,2}}$ is small enough, then $E(t)$
satisfies
\begin{eqnarray}\label{E.decay}
\|E(t)-E_{st}\|\leq C_{1}(1+t)^{-\frac{5}{4}}.
\end{eqnarray}
\end{theorem}
The proof of existence in Theorem \ref{Corolary} is based on the
classical energy method. As in \cite{Duan}, the key point is to
obtain the uniform-in-time {\it a priori} estimates in the form of
$$
\mathcal{E}_N(\bar{V}(t))+\lambda
\int_0^t\mathcal{D}_N(\bar{V}(s))\,ds\leq \mathcal{E}_N(\bar{V}_0),
$$
where $\bar{V}(t)$ is the perturbation of solutions, and
$\mathcal{E}_N(\cdot)$, $\mathcal{D}_N(\cdot)$ denote the energy
functional and energy dissipation rate functional. Here if we make
the energy estimates like what Duan did in \cite{Duan}, it is
difficult to control the highest-order derivative of $\bar{E}$
because of the regularity-loss type in the sense that
$[\bar{E},\bar{B}]$ is time-space integrable up to $N-1$ order only.
In this paper, we modify the energy estimates by choosing a weighted
function $1+\sigma_{st}+\Phi(\sigma_{st})$ which plays a vital role
in closing the energy estimates.
Furthermore, for the convergence rates of perturbed solutions in
Theorem 1.1, we can not analyze the corresponding linearized system
of \eqref{1.1} around the steady state $[n_{st},0,E_{st},0]$
directly. In this case, the Fourier analysis fails due to the
difficulty of variant coefficients. Here, the main idea follows from
\cite{Duan} for combining energy estimates with the linearized
results in \cite{Duan}. In the process of obtaining the fastest
decay rates of the perturbed solution, the great difficulty is to
deal with these linear nonhomogeneous sources including $\rho_{st}$,
which can not bring enough decay rates. Whereas in \cite{Duan}, the
nonhomogeneous sources are at least quadratically nonlinear. To
overcome this difficulty, we make iteration for the inequalities
\eqref{sec5.ENV0} and $\eqref{sec5.high}$ together. In theorem
\ref{Corolary}, we only capture the same time-decay properties of
$u,\ E-E_{st}$ and $B$ as \cite{Duan} except $n-n_{st}$.
$\|n-n_{st}\|$ decays as $(1+t)^{-\frac{5}{4}}$ in the fastest way,
because the nonhomogeneous sources containing $\rho_{st}$ decay at
most the same as $\sqrt{\mathcal{E}^h_N(\cdot)}$.
The similar work was done for Vlasov-Poisson-Boltzmann system, where
the background density is also a function of spatial variable. Duan
and Yang in \cite{RY} considered the stability of the stationary
states which were given by an elliptic equation with the exponential
nonlinearity. The optimal time-decay of the Vlasov-Poisson-Boltzmann
system in $\mathbb{R}^{3}$ was obtained by Duan and Strain in
\cite{DS}. We also mention the work Duan-Ukai-Yang-Zhao in
\cite{RSYZ}, Duan-Liu-Ukai-Yang in \cite{DLUY} for the study of
optimal convergence rates of the compressible Navier-Stokes
equations with potential forces. Their proofs were based on the
combination of spectral analysis and energy estimates. Recently,
Duan-Ukai-Yang in \cite{RSY} developed a method of the combination
of the spectral analysis and energy estimates to deal with the
optimal time decay for study of equations of gas motion.
We further remark the result in \cite{RY}, the existence of solution
to the elliptic equation $\Delta \phi=e^{\phi}-\bar{\rho}(x)$ has
been proved when $\|\bar{\rho}-1\|_{W_{k}^{m,\infty}}$ is
sufficiently small, where the weighted norm
$\|\cdot\|_{W_{k}^{m,\infty}}$ is defined by
\begin{eqnarray}\label{def.norm1}
\|g\|_{W_{k}^{m,\infty}}=\sup_{x\in\mathbb{R}^{3}}(1+|x|)^{k}\sum_{|\alpha|\leq
m}|\partial^{\alpha}_{x}g(x)|
\end{eqnarray}
for suitable $g=g(x)$ and integers $m\geq0$, $k\geq0$, the stability
of the perturbed solutions can be proved when
$\|\bar{\rho}-1\|_{W_{2}^{N+1,\infty}}$ is sufficiently small. We
can also prove the stability of stationary solutions in the
framework of \cite{RY} if $\|n_{b}-1\|_{W_{0}^{N+1,\infty}}$ is
sufficiently small. In order to obtain the same convergence rates,
$\|n_{b}-1\|_{W_{2}^{N+4,\infty}}$ should be sufficiently small in
the process of dealing with $\rho_{st}\bar{u}$ as in Section
\ref{sec4},
$$
\|\rho_{st} \bar{u}\|_{L^1} \leq \|\rho_{st}\|\left\|\bar{u}
\right\| \leq C \|\rho_{st}\|_{W_{2}^{N+4,\infty}}\| \bar{u}\|.
$$
Notice that $W_{2}^{N+4,\infty}\subseteq W_{0}^{N+4,2}$, it seems to
be better to consider the existence of steady states in the weighted
Sobolev space $W_{k}^{m,2}$.
Let us introduce some notations for the use throughout this paper. $C$ denotes some
positive (generally large) constant and $ \lambda$ denotes some
positive (generally small) constant, where both $C$ and $ \lambda$
may take different values in different places. For two quantities
$a$ and $b$, $a\sim b$ means $\lambda a \leq b \leq
\frac{1}{\lambda} a $ for a generic constant $0<\lambda<1$. For any
integer $m\geq 0$, we use $H^{m}$, $\dot{H}^{m}$ to denote the usual
Sobolev space $H^{m}(\mathbb{R}^{3})$ and the corresponding
$m$-order homogeneous Sobolev space, respectively. Set $L^{2}=H^{m}$
when $m = 0$. For simplicity, the norm of $ H^{m}$ is denoted by
$\|\cdot\|_{m} $ with $\|\cdot \|=\|\cdot\|_{0}$. We use $
\langle\cdot, \cdot \rangle$ to denote the inner product over the
Hilbert space $ L^{2}(\mathbb{R}^{3})$, i.e.
\begin{eqnarray*}
\langle f,g \rangle=\int_{\mathbb{R}^{3}} f(x)g(x)dx,\ \ \ \ f =
f(x),\ \ g = g(x)\in L^2(\mathbb{R}^{3}).
\end{eqnarray*}
For a multi-index $\alpha =
[\alpha_1, \alpha_2, \alpha_3]$, we denote $\partial^{\alpha} =
\partial^{\alpha_{1}}_ {x_1}\partial^{\alpha_{2}}_ {x_2} \partial^{\alpha_{3}}_ {x_3} $.
The length of $ \alpha$ is $|\alpha| = \alpha_1 + \alpha_2 +
\alpha_3$. For simplicity, we also set
$\partial_{j}=\partial_{x_{j}}$ for $j = 1, 2, 3$.
We conclude this section by stating the arrangement of the rest of
this paper. In Section 2, we prove the existence of the stationary
solution. In Section 3, we reformulate the Cauchy problem under
consideration and obtain asymptotic stability of solutions near the
stationary state provided that the initial perturbation is
sufficiently small. In Section 4, we study the time-decay rates of
solutions to the stationary solutions by combining the $L^p$-$L^q$
time-decay property of the linearized homogeneous system with
time-weighted estimate.
\vspace{5mm}
\section{Existence of stationary solution}\label{sec2}
In this section, we will prove the existence of stationary solutions
to $\eqref{sta.eq0}$ by using the contraction mapping theorem. From
$\eqref{sta.eq0}_2$, there exists $\phi_{st}$ such that
$E_{st}=\nabla \phi_{st}$, it turns equation $\eqref{sta.eq0}$ into
\begin{eqnarray}\label{sta.eq1}
\left\{\begin{aligned} &\frac{1}{n_{st}}\nabla
p(n_{st})=-\nabla\phi_{st},\\
&\Delta\phi_{st}=n_{b}(x)-n_{st}.
\end{aligned}\right.
\end{eqnarray}
We introduce the nonlinear transformation (cf. \cite{Deng})
\begin{eqnarray}\label{sta.tra}
Q_{st}=\frac{\gamma}{\gamma-1}(n_{st}^{\gamma-1}-1).
\end{eqnarray}
From $\eqref{sta.eq1}$ and $\eqref{sta.tra}$, we derive the
following elliptic equation
\begin{eqnarray}\label{sta.ellip}
\Delta
Q_{st}=\left(\frac{\gamma-1}{\gamma}Q_{st}+1\right)^{\frac{1}{\gamma-1}}-n_{b}(x).
\end{eqnarray}
For convenience, we replace $Q_{st}$ by $\phi$ in the following.
Equation $\eqref{sta.ellip}$ can be rewritten as the integral form
\begin{eqnarray*}
\phi=T(\phi)=G*\left(\left(\frac{\gamma-1}{\gamma}\phi+1\right)^{\frac{1}{\gamma-1}}-\frac{1}{\gamma}\phi
-n_{b}(x)\right),
\end{eqnarray*}
where $G=G(x)$ given by
\begin{eqnarray*}
G(x)=-\frac{1}{4\pi|x|}e^{-\tfrac{1}{\sqrt{\gamma}}|x|}
\end{eqnarray*}
is the fundamental solution to the linear elliptic equation $
\Delta_{x}G-\frac{1}{\gamma}G=0$. Thus \eqref{sta.ellip} admits a
solution if and only if the nonlinear mapping $T$ has a fixed point.
Define
\begin{eqnarray*}
\mathscr{B}_{m,k}(B)=\{\phi\in
W^{m,2}_{k}(\mathbb{R}^{3});\|\phi\|_{W^{m,2}_{k}}\leq
B\|n_{b}-1\|_{W^{m,2}_{k}},\ m\geq2\}
\end{eqnarray*}
for some constant $B$ to be determined later. Next, we prove that if
$\|n_{b}-1\|_{W^{m,2}_{k}} $ is small enough, there exists a constant $B$ such that
$T:\mathscr{B}_{m,k}(B)\rightarrow \mathscr{B}_{m,k}(B) $ is a
contraction mapping. In fact, for simplicity, let us denote
\begin{eqnarray*}
g(x)=\left(\frac{\gamma-1}{\gamma}x+1\right)^{\frac{1}{\gamma-1}}-\frac{1}{\gamma}x-1.
\end{eqnarray*}
Then it holds that
\begin{eqnarray}\label{T.phi}
T(\phi)(x)=-\int_{\mathbb{R}^{3}}\frac{1}{4\pi|x-y|}e^{-\tfrac{1}{\sqrt{\gamma}}|x-y|}
[g(\phi(y))-(n_{b}(y)-1)]dy.
\end{eqnarray}
Taking derivatives $ \partial_{x}^{\alpha}$ on both sides of $
\eqref{T.phi}$, one has
\begin{eqnarray}\label{T.estimate}
\arraycolsep=1.5pt
\begin{array}[b]{rl}
\partial_{x}^{\alpha}T(\phi)(x)=&\displaystyle-(-1)^{|\alpha|}
\int_{\mathbb{R}^{3}}\frac{1}{4\pi|x-y|}e^{-\tfrac{1}{\sqrt{\gamma}}|x-y|}
[\partial^{\alpha}_{y}
g(\phi(y))-\partial^{\alpha}_{y}(n_{b}(y)-1)]dy\\[5mm]
=&-(-1)^{|\alpha|}G*(\partial^{\alpha}g(\phi)-\partial^{\alpha}(n_{b}-1)).
\end{array}
\end{eqnarray}
Here let's list some properties of the operator $G*$.
\begin{lemma}\label{Pro.OG}
For any $k\geq 0$, it holds that
\begin{eqnarray}\label{pro.Gdecay}
\int_{\mathbb{R}^{3}}\frac{1}{|y|}e^{-\tfrac{1}{\sqrt{\gamma}}|y|}\frac{1}{(1+|x-y|)^{k}}dy\leq
\frac{C_{k}}{(1+|x|)^{k}},
\end{eqnarray}
and for any $f\in W_{k}^{m,2}$,
\begin{eqnarray}\label{pro.G}
\|(1+|x|)^{\frac{k}{2}}(G*f)\|\leq C_{k}^{\frac{1}{2}}\|G\|_{L^{1}}^{\frac{1}{2}}\|(1+|x|)^{\frac{k}{2}}f\|.
\end{eqnarray}
\end{lemma}
\textit{Proof.} \eqref{pro.Gdecay} has been proved in \cite{RY}. We
only prove \eqref{pro.G} by using $\eqref{pro.Gdecay}$.
\begin{eqnarray*}
\arraycolsep=1.5pt
\begin{array}[b]{rl}
\displaystyle\left| \int_{\mathbb{R}^{3}}G(x-y)f(y)dy \right|
\leq & \displaystyle \int_{\mathbb{R}^{3}}\frac{|G(x-y)|^{\frac{1}{2}}}{(1+|y|)^{\frac{k}{2}}}
|G(x-y)|^{\frac{1}{2}}(1+|y|)^{\frac{k}{2}}|f(y)|dy\\[5mm]
\leq & \displaystyle \left(\int_{\mathbb{R}^{3}}\frac{|G(x-y)|}{(1+|y|)^{k}}dy\right)^{\frac{1}{2}}
\left(\int_{\mathbb{R}^{3}}|G(x-y)|(1+|y|)^{k}|f(y)|^2dy\right)^{\frac{1}{2}}\\[5mm]
\leq & \displaystyle
\frac{C_{k}^{\frac{1}{2}}}{(1+|x|)^{\frac{k}{2}}}
\left(\int_{\mathbb{R}^{3}}|G(x-y)|(1+|y|)^{k}|f(y)|^2dy\right)^{\frac{1}{2}}.
\end{array}
\end{eqnarray*}
Then
\begin{eqnarray*}
\arraycolsep=1.5pt
\begin{array}[b]{rl}
& \displaystyle \int_{\mathbb{R}^{3}}(1+|x|)^{k}\left| \int_{\mathbb{R}^{3}}G(x-y)f(y)dy \right|^2dx\\[5mm]
\leq & \displaystyle C_{k} \int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}|G(x-y)|(1+|y|)^{k}|f(y)|^2dydx\\[5mm]
= & \displaystyle C_{k} \int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}|G(x-y)|(1+|y|)^{k}|f(y)|^2dxdy\\[5mm]
= & \displaystyle C_{k} \int_{\mathbb{R}^{3}}(1+|y|)^{k}|f(y)|^2 dy \int_{\mathbb{R}^{3}}|G(x-y)|dx\\[5mm]
=& C_{k}\|G\|_{L^{1}}\|(1+|x|)^{\frac{k}{2}}f\|^2.
\end{array}
\end{eqnarray*}
\textbf{Remark:} When $k=0$, $C_{k}=\|G\|_{L^{1}}$, \eqref{pro.G} is in accordance with Young inequality.
By \eqref{pro.G} and \eqref{T.estimate}, one has
\begin{eqnarray*}
\arraycolsep=1.5pt
\begin{array}[b]{rl}
& \|(1+|x|)^{\frac{k}{2}}\partial_{x}^{\alpha}T(\phi)(x)\|\\[3mm]
\leq & C\|(1+|x|)^{\frac{k}{2}}\partial^{\alpha}g(\phi)\|
+C\|(1+|x|)^{\frac{k}{2}}\partial^{\alpha}(n_{b}-1))\|.
\end{array}
\end{eqnarray*}
By the definition $\eqref{def.norm}$ of the norm $
\|\cdot\|_{W^{m,2}_{k}}$, one has
\begin{eqnarray}\label{T.nb}
\arraycolsep=1.5pt
\begin{array}[b]{rl}
\|T(\phi)(x)\|_{W^{m,2}_{k}}
= & \displaystyle \left(\sum_{|\alpha|\leq m}\|(1+|x|)^{\frac{k}{2}}\partial_{x}^{\alpha}T(\phi)(x)\|^2\right)^{\frac{1}{2}}\\[5mm]
\leq & \displaystyle C\left(\sum_{|\alpha|\leq m}\|(1+|x|)^{\frac{k}{2}}\partial^{\alpha}g(\phi)\|^2\right)^{\frac{1}{2}}
+C\left(\sum_{|\alpha|\leq m}\|(1+|x|)^{\frac{k}{2}}\partial^{\alpha}(n_{b}-1)\|^2\right)^{\frac{1}{2}}\\[5mm]
\leq & \displaystyle C\left(\sum_{|\alpha|\leq m}\|(1+|x|)^{\frac{k}{2}}\partial^{\alpha}g(\phi)\|^2\right)^{\frac{1}{2}}
+C\|n_{b}-1\|_{W^{m,2}_{k}}.
\end{array}
\end{eqnarray}
On the other hand, note
\begin{eqnarray*}
g(\phi)=\int_{0}^{1}\int_{0}^{\theta}g''(\tau\phi)d\tau d\theta
\phi^{2}\triangleq h(\phi)\phi^{2},
\end{eqnarray*}
where
$g''(x)=\frac{2-\gamma}{\gamma^{2}}\left(\frac{\gamma-1}{\gamma}x+1\right)^{\frac{3-2\gamma}{\gamma-1}}$.
It is straightforward to check that
\begin{eqnarray*}
\|(1+|x|)^{\frac{k}{2}}\partial^{\alpha}(h(\phi)\phi^{2})\|\leq
\sum_{\beta_{1}+\beta_{2}+\beta_{3}=\alpha}C_{\beta_1,\beta_2,\beta_{3}}^{\alpha}
\|(1+|x|)^{\frac{k}{2}}\partial^{\beta_{1}}h(\phi)\partial^{\beta_2}\phi\partial^{\beta_3}\phi\|.
\end{eqnarray*}
In addition, one has the following claim.
\textbf{Claim}:
\begin{eqnarray}\label{T.gphi}
\|(1+|x|)^{\frac{k}{2}}\partial^{\beta_{1}}h(\phi)\partial^{\beta_2}\phi\partial^{\beta_3}\phi\|\leq
C \|\phi\|^2_{W^{m,2}_{k}}.
\end{eqnarray}
\textit{Proof of claim:} We prove \eqref{T.gphi} by two cases.
\textbf{Case 1.} $\beta_{1}=0$. In this case,
$|\beta_{2}|+|\beta_{3}|\leq m$, thus one can suppose
$|\beta_{2}|\leq [\frac{m}{2}]$ by the symmetry of $\beta_{2}$ and
$\beta_{3}$. This deduces
\begin{eqnarray*}
\arraycolsep=1.5pt
\begin{array}[b]{rl}
\|(1+|x|)^{\frac{k}{2}}h(\phi)\partial^{\beta_2}\phi\partial^{\beta_3}\phi\|
\leq &
\|h(\phi)\|_{L^{\infty}}\|\partial^{\beta_2}\phi\|_{L^{6}}\|(1+|x|)^{\frac{k}{2}}\partial^{\beta_3}\phi\|_{L^{3}}\\[3mm]
\leq & C \|\nabla
\partial^{\beta_2}\phi\|\|(1+|x|)^{\frac{k}{2}}\partial^{\beta_3}\phi\|_{1}\\[3mm]
\leq & C \|\phi\|^2_{W^{m,2}_{k}}.
\end{array}
\end{eqnarray*}
Here, we have used that $h(\cdot)$ is a continuous function in the
argument,
\begin{eqnarray*}
\|\phi\|_{L^{\infty}}\leq C\|\nabla \phi\|_{H^{1}}\leq
C\|\phi\|_{W^{m,2}_{k}}\leq C \|n_{b}-1\|_{W^{m,2}_{k}}\ll 1,
\end{eqnarray*}
and $m\geq 2$.
\textbf{Case 2}. $|\beta_{1}|\geq 1$. Notice that
\begin{eqnarray*}
\partial^{\beta_{1}}h(\phi)=\sum_{l=1}^{|\beta_{1}|}h^{(l)}(\phi)
\sum_{\gamma_{1}+\gamma_{2}+\cdots\gamma_{l}=\beta_{1}}
C_{\gamma_{1},\gamma_{2},\cdots\gamma_{l}}\Pi_{i=1}^{l}\partial^{\gamma_{i}}\phi,
\end{eqnarray*}
\eqref{T.gphi} can be similarly obtained because $h^{(m)}(\phi)$ is
also bounded.
Putting $\eqref{T.gphi}$ into \eqref{T.nb}, and using the above
estimates, one has
\begin{eqnarray}\label{T.phi.est}
\|T(\phi)(x)\|_{W^{m,2}_{k}}\leq C B^2
\|n_{b}-1\|_{W^{m,2}_{k}}^2+C\|n_{b}-1\|_{W^{m,2}_{k}}.
\end{eqnarray}
Finally, for any $\phi_{1}=\phi_{1}(x)$ and $\phi_{2}=\phi_{2}(x)$,
it holds that
\begin{eqnarray*}
T(\phi_{1})-T(\phi_{2})=G*(g(\phi_{1})-g(\phi_{2}))
\end{eqnarray*}
with
\begin{eqnarray*}
g(\phi_{1})-g(\phi_{2})=\int_{0}^{1}g'(\theta\phi_{1}+(1-\theta)\phi_{2})d\theta(\phi_{1}-\phi_{2}).
\end{eqnarray*}
Notice that for any $\phi=\phi(x)$,
\begin{eqnarray*}
\arraycolsep=1.5pt
\begin{array}{rcl}
g'(\phi)&=&\displaystyle\frac{1}{\gamma}\left(\frac{\gamma-1}{\gamma}\phi+1\right)^{\frac{2-\gamma}{\gamma-1}}
-\frac{1}{\gamma}\\[3mm]
&=&\displaystyle\int_{0}^{1}\frac{2-\gamma}{\gamma^{2}}
\left(\frac{\gamma-1}{\gamma}\theta\phi+1\right)^{\frac{3-2\gamma}{\gamma-1}}d\theta\phi.
\end{array}
\end{eqnarray*}
Then the same computations as for $\eqref{T.phi.est}$ yield
\begin{eqnarray}\label{T.contract}
\arraycolsep=1.5pt
\begin{array}{rl}
&\|T(\phi_{1})-T(\phi_{2})\|_{W_{k}^{m,2}}\\[3mm]
\leq & C
(\|\phi_{1}\|_{W^{m,2}_{k}}+\|\phi_{2}\|_{W^{m,2}_{k}})\|\phi_{1}-\phi_{2}\|_{W^{m,2}_{k}}.
\end{array}
\end{eqnarray}
Combining $\eqref{T.phi.est}$ with $ \eqref{T.contract}$, the
standard argument implies that $T$ has a unique fixed point $\phi$
in $ \mathscr{B}_{m,k}(B)$ for a proper constant $B$ provided that
$\|n_{b}-1\|_{W^{m,2}_{k}}$ is small enough. This completes Theorem
\ref{sta.existence}.
Let us conclude this section with a remark. The existence of
solutions to the elliptic equation \eqref{sta.ellip} can also be
proved in the framework of \cite{RY} when
$\|n_{b}-1\|_{W^{m,\infty}_{k}}$ is sufficiently small. We consider
the existence when $\|n_{b}-1\|_{W^{m,2}_{k}}$ is sufficiently
small in order to derive the more general conclusion. In fact, in
the process of dealing with the stability and convergence rates,
only the smallness of $\|n_{b}-1\|_{W^{m,2}_{0}}$ is assumed, and
the space decay at infinity of $n_{b}(x)-1$ is not needed.
\vspace{6mm}
\section{Stability of stationary solution}
\vspace{4mm}
\subsection{Reformulation of the problem} Let $[n,u,E,B]$ be a smooth
solution to the Cauchy problem of the Euler-Maxwell system
(\ref{1.1}) with given initial data (\ref{1.2}) satisfying
(\ref{1.3}). Set
\begin{eqnarray}\label{2.1}
&&\left\{
\begin{aligned}
&\sigma(t,x)=\frac{2}{\gamma-1}\left\{\left[n\left(\frac{t}{\sqrt{\gamma}},x\right)\right]
^{\frac{\gamma-1}{2}}-1\right\}, \ \ \
v=\frac{1}{\sqrt{\gamma}}u\left(\frac{t}{\sqrt{\gamma}},x\right),
\\[5mm]
&\ \
\tilde{E}=\frac{1}{\sqrt{\gamma}}E\left(\frac{t}{\sqrt{\gamma}},x\right),\ \
\ \tilde{B}=\frac{1}{\sqrt{\gamma}}B\left(\frac{t}{\sqrt{\gamma}},x\right).
\end{aligned}\right.
\end{eqnarray}
Then, $V:=[\sigma,v,\tilde{E},\tilde{B}]$ satisfies
\begin{equation}\label{2.2}
\left\{
\begin{aligned}
&\partial_t \sigma+\left(\frac{\gamma-1}{2}\sigma+1\right)\nabla\cdot v+v\cdot \nabla \sigma=0,\\
&\partial_t v+v \cdot \nabla
v+\left(\frac{\gamma-1}{2}\sigma+1\right)\nabla \sigma=-\left(\frac{1}{\sqrt{\gamma}}\tilde{E}+v\times \tilde{B}\right)
-\frac{1}{\sqrt{\gamma}}v,\\
&\partial_t\tilde{E}-\frac{1}{\sqrt{\gamma}}\nabla\times\tilde{B}
=\frac{1}{\sqrt{\gamma}}v+\frac{1}{\sqrt{\gamma}}[\Phi(\sigma)+\sigma]v,\\
&\partial_t \tilde{B}+\frac{1}{\sqrt{\gamma}}\nabla \times \tilde{E}=0,\\
&\nabla \cdot
\tilde{E}=-\frac{1}{\sqrt{\gamma}}[\Phi(\sigma)+\sigma]
+\frac{1}{\sqrt{\gamma}}(n_{b}(x)-1), \ \ \nabla
\cdot \tilde{B}=0, \ \ \ t>0,\ x\in\mathbb{R}^{3},
\end{aligned}\right.
\end{equation}
with initial data
\begin{eqnarray}\label{2.3}
V|_{t=0}=V_{0}:=[\sigma_{0},v_{0},\tilde{E}_{0},\tilde{B}_{0}],\ \
x\in\mathbb{R}^{3}.
\end{eqnarray}
Here, $\Phi(\cdot)$ is defined by
\begin{eqnarray}\label{def.phi}
\Phi(\sigma)=\left(\frac{\gamma-1}{2}\sigma+1\right)^{\frac{2}{\gamma-1}}-\sigma-1,
\end{eqnarray}
and $V_{0}=[\sigma_{0},v_{0},\tilde{E}_{0},\tilde{B}_{0}]$ is given
from $[n_{0},u_{0},E_{0},B_0]$ according to the transform
(\ref{2.1}), and hence $V_{0}$ satisfies
\begin{eqnarray}\label{2.4}
\nabla\cdot\tilde{E}_0=-\frac{1}{\sqrt{\gamma}}[\Phi(\sigma_{0})+\sigma_{0}]
+\frac{1}{\sqrt{\gamma}}(n_{b}(x)-1),\ \ \ \
\nabla \cdot \tilde{B}_0=0,\ \ \ x\in\mathbb{R}^{3}.
\end{eqnarray}
On the other hand, set
\begin{eqnarray}\label{sta.tran}
\sigma_{st}(x)=\frac{2}{\gamma-1}\left\{n_{st}(x)^{\frac{\gamma-1}{2}}-1\right\},
\ \ \ \
\tilde{E}_{st}=\frac{1}{\sqrt{\gamma}}E_{st}(x).
\end{eqnarray}
Then, $[\sigma_{st},\tilde{E}_{st}]$ satisfies
\begin{eqnarray}\label{sta.eq}
\left\{\begin{aligned}
& \left(\frac{\gamma-1}{2}\sigma_{st}+1\right)\nabla\sigma_{st}=-\frac{1}{\sqrt{\gamma}}\tilde{E}_{st},\\
&\frac{1}{\sqrt{\gamma}}\nabla\times \tilde{E}_{st}=0,\\
&\nabla \cdot
\tilde{E}_{st}=\frac{1}{\sqrt{\gamma}}(n_{b}(x)-1)-\frac{1}{\sqrt{\gamma}}(\Phi(\sigma_{st})+\sigma_{st}).
\end{aligned}\right.
\end{eqnarray}
Based on the existence result proved in Section 2, we will study the
stability of the stationary state
$[\sigma_{st},0,\tilde{E}_{st},0]$. Set the perturbations
$[\bar{\sigma},\bar{v},\bar{E},\bar{B}]$ by
\begin{eqnarray*}
\bar{\sigma}=\sigma-\sigma_{st},\ \ \bar{v}=v,\ \
\bar{E}=\tilde{E}-\tilde{E}_{st},\ \ \bar{B}=\tilde{B}.
\end{eqnarray*}
Combining \eqref{2.2} with \eqref{sta.eq}, then
$\bar{V}:=[\bar{\sigma},\bar{v},\bar{E},\bar{B}]$ satisfies
\begin{equation}\label{sta.equ}
\left\{
\begin{aligned}
&\partial_t \bar{\sigma}+(\frac{\gamma-1}{2}\bar{\sigma}+1)\nabla\cdot \bar{v}+\bar{v}\cdot \nabla \bar{\sigma}
+\bar{v}\cdot \nabla \sigma_{st}+\frac{\gamma-1}{2}\sigma_{st}\nabla\cdot \bar{v}=0,\\
&\partial_t \bar{v}+\bar{v} \cdot \nabla
\bar{v}+(\frac{\gamma-1}{2}\bar{\sigma}+1)\nabla \bar{\sigma}
+\frac{\gamma-1}{2}\bar{\sigma}\nabla \sigma_{st}+\frac{\gamma-1}{2}\sigma_{st}\nabla\bar{\sigma}=
-(\frac{1}{\sqrt{\gamma}}\bar{E}+\bar{v}\times \bar{B})
-\frac{1}{\sqrt{\gamma}}\bar{v},\\
&\partial_t\bar{E}-\frac{1}{\sqrt{\gamma}}\nabla\times \bar{B}
=\frac{1}{\sqrt{\gamma}}\bar{v}+
\frac{1}{\sqrt{\gamma}}[\Phi(\bar{\sigma}+\sigma_{st})+\bar{\sigma}+\sigma_{st}]\bar{v},\\
&\partial_t \bar{B}+\frac{1}{\sqrt{\gamma}}\nabla \times \bar{E}=0,\\
&\nabla \cdot
\bar{E}=-\frac{1}{\sqrt{\gamma}}[\Phi(\bar{\sigma}+\sigma_{st})-\Phi(\sigma_{st})]
-\frac{1}{\sqrt{\gamma}}\bar{\sigma}, \ \ \nabla
\cdot \bar{B}=0, \ \ t>0,\ x\in\mathbb{R}^{3},
\end{aligned}\right.
\end{equation}
with initial data
\begin{eqnarray}\label{sta.equi}
\bar{V}|_{t=0}=\bar{V}_{0}:=[\sigma_{0}-\sigma_{st},v_{0},\tilde{E}_{0}-\tilde{E}_{st},\tilde{B}_{0}],\
\ x\in\mathbb{R}^{3}.
\end{eqnarray}
Here, $\Phi(\cdot)$ is defined by \eqref{def.phi},
and $\bar{V}_{0}$ satisfies
\begin{eqnarray}\label{sta.equC}
\nabla \cdot
\bar{E}_{0}=-\frac{1}{\sqrt{\gamma}}[\Phi(\bar{\sigma}_{0}+\sigma_{st})-\Phi(\sigma_{st})]
-\frac{1}{\sqrt{\gamma}}\bar{\sigma}_{0}, \ \ \nabla
\cdot \bar{B}_{0}=0, \ \ t>0,\ x\in\mathbb{R}^{3}.
\end{eqnarray}
In what follows, we suppose the integer $N \geq 3$. Besides, for
$\bar{V}=[\bar{\sigma},\bar{v},\bar{E},\bar{B}]$, we define the full
instant energy functional $\mathcal {E}_{N}(\bar{V}(t))$, the
high-order instant energy functional $\mathcal
{E}_{N}^{h}(\bar{V}(t))$, and the dissipation rates $\mathcal
{D}_{N}(\bar{V}(t))$, $\mathcal {D}_{N}^{h}(\bar{V}(t))$ by
\begin{equation}\label{de.E}
\arraycolsep=1.5pt
\begin{array}{rl}
\mathcal{E}_{N}(\bar{V}(t))=&\displaystyle\sum_{|\alpha|\leq
N}\int_{\mathbb{R}^3}(1+\sigma_{st}+\Phi(\sigma_{st}))
(|\partial^{\alpha}\bar{\sigma}|^2+|\partial^{\alpha}\bar{v}|^2)dx+\|[\bar{E},\bar{B}]\|_{N}^{2}\\[5mm]
&\displaystyle+\kappa_{1}\sum_{|\alpha|\leq N-1} \langle
\partial^{\alpha}\bar{v},\nabla\partial^{\alpha}\bar{\sigma}\rangle+\kappa_{2}\sum_{|\alpha|\leq N-1}\langle
\partial^{\alpha}\bar{v},\partial^{\alpha}\bar{E}\rangle\\[5mm]
&\displaystyle-\kappa_{3}\sum_{|\alpha|\leq N-2}\langle \nabla
\times
\partial^{\alpha}\bar{E},\partial^{\alpha}\bar{B}\rangle,
\end{array}
\end{equation}
and
\begin{equation}\label{de.Eh}
\begin{aligned}
\mathcal{E}_{N}^{h}(\bar{V}(t))&=\sum_{1\leq|\alpha|\leq
N}\int_{\mathbb{R}^3}(1+\sigma_{st}+\Phi(\sigma_{st}))
(|\partial^{\alpha}\bar{\sigma}|^2+|\partial^{\alpha}\bar{v}|^2)dx+\|\nabla[\bar{E},\bar{B}]\|_{N-1}^{2}\\
&+\kappa_{1}\sum_{1\leq|\alpha|\leq N-1}\langle
\partial^{\alpha}\bar{v},\nabla\partial^{\alpha}\bar{\sigma}\rangle+\kappa_{2}\sum_{1\leq|\alpha|\leq N-1}\langle
\partial^{\alpha}\bar{v},\partial^{\alpha}\bar{E}\rangle\\[3mm]
&-\kappa_{3}\sum_{1\leq |\alpha|\leq N-2}\langle \nabla
\times\partial^{\alpha}\bar{E},\partial^{\alpha}\bar{B}\rangle,
\end{aligned}
\end{equation}
respectively, where $0<\kappa_{3}\ll\kappa_{2}\ll\kappa_{1}\ll 1$
are constants to be properly chosen in the later proof. Notice that
since all constants $\kappa_i$ $(i=1,2,3)$ are small enough, one has
\begin{equation*}
\mathcal {E}_{N}(\bar{V}(t))\sim
\|[\bar{\sigma},\bar{v},\bar{E},\bar{B}] \|_{N}^{2},\quad \mathcal
{E}_{N}^{h}(\bar{V}(t))\sim \|\nabla
[\bar{\sigma},\bar{v},\bar{E},\bar{B}] \|_{N-1}^{2}.
\end{equation*}
We further define the dissipation rates $\mathcal
{D}_{N}(\bar{V}(t))$, $\mathcal {D}_{N}^{h}(\bar{V}(t))$ by
\begin{eqnarray}\label{de.D}
\arraycolsep=1.5pt
\begin{array}{rl}
\mathcal {D}_{N}(\bar{V}(t))=\displaystyle \sum_{|\alpha|\leq
N}\int_{\mathbb{R}^3}(1+\sigma_{st}&+\Phi(\sigma_{st}))|\partial^{\alpha}\bar{v}|^{2}dx\\[3mm]
&+\|\bar{\sigma}\|_{N}^{2}+\|\nabla[\bar{E},\bar{B}]\|_{N-2}^{2}+\|\bar{E}\|^{2},
\end{array}
\end{eqnarray}
and
\begin{eqnarray}\label{de.Dh}
\arraycolsep=1.5pt
\begin{array}{rl}
\mathcal {D}_{N}^{h}(\bar{V}(t))=\displaystyle
\sum_{1\leq|\alpha|\leq
N}\int_{\mathbb{R}^3}(1+\sigma_{st}&+\Phi(\sigma_{st}))|\partial^{\alpha}\bar{v}|^{2}dx\\[3mm]
&+\|\nabla\bar{\sigma}\|_{N-1}^{2}+\|\nabla^2[\bar{E},\bar{B}]\|_{N-3}^{2}+\|\nabla\bar{E}\|^{2}.
\end{array}
\end{eqnarray}
Then, concerning the reformulated Cauchy problem
$\eqref{sta.equ}$-$\eqref{sta.equi}$, one has the following global
existence result.
\begin{proposition}\label{pro.2.1}
Suppose that $\|n_{b}-1\|_{W_{0}^{N+1,2}}$ is small enough and
$\eqref{sta.equC}$ holds for given initial data
$\bar{V}_{0}=[\sigma_{0}-\sigma_{st},v_{0},\tilde{E}_0-\tilde{E}_{st},\tilde{B}_{0}]$.
Then, there are $\mathcal {E}_{N}(\cdot) $ and $\mathcal
{D}_{N}(\cdot)$ in the form $\eqref{de.E} $ and $\eqref{de.D}$ such
that the following holds true:
If $\mathcal {E}_{N}(\bar{V}_{0})>0$ is small enough, the Cauchy
problem $\eqref{sta.equ}$-$\eqref{sta.equi}$ admits a unique global
nonzero solution
$\bar{V}=[\sigma-\sigma_{st},v,\tilde{E}-\tilde{E}_{st},\tilde{B}] $
satisfying
\begin{eqnarray}\label{V.satisfy}
\bar{V} \in C([0,\infty);H^{N}(\mathbb{R}^{3}))\cap {\rm
Lip}([0,\infty);H^{N-1}(\mathbb{R}^{3})),
\end{eqnarray}
and
\begin{eqnarray}\label{pro.2.1j}
\mathcal {E}_{N}(\bar{V}(t))+\lambda\int_{0}^{t}\mathcal
{D}_{N}(\bar{V}(s))ds\leq \mathcal {E}_{N}(\bar{V}_{0})
\end{eqnarray}
for any $t\geq 0$.
\end{proposition}
Moreover, solutions obtained in Proposition $ \ref{pro.2.1}$ indeed
decay in time with some rates under some extra regularity and
integrability conditions on initial data. For that, given
$\bar{V}_{0}=[\sigma_{0}-\sigma_{st},v_{0},\tilde{E}_0-\tilde{E}_{st},\tilde{B}_{0}]$,
set $\epsilon_{m}(\bar{V}_0)$ as
\begin{eqnarray}\label{def.epsi}
\epsilon_{m}(\bar{V}_0)=\|\bar{V}_{0}\|_{m}+\|[v_{0},\tilde{E}_0-\tilde{E}_{st},\tilde{B}_{0}]\|_{L^{1}},
\end{eqnarray}
for the integer $m \geq 6$. Then one has the following proposition.
\begin{proposition}\label{pro.2.2}
Suppose that $\|n_{b}-1\|_{W_{0}^{N+4,2}}$ is small enough and
$\eqref{sta.equC}$ holds for given initial data
$\bar{V}_{0}=[\sigma_{0}-\sigma_{st},v_{0},\tilde{E}_0-\tilde{E}_{st},\tilde{B}_{0}]$.
If $\epsilon_{N+3}(\bar{V}_{0})>0$ is small enough, then the
solution
$\bar{V}=[\sigma-\sigma_{st},v,\tilde{E}-\tilde{E}_{st},\tilde{B}] $
satisfies
\begin{eqnarray}\label{V.decay}
\|\bar{V}(t)\|_{N} \leq C
\epsilon_{N+3}(\bar{V}_{0})(1+t)^{-\frac{3}{4}},
\end{eqnarray}
and
\begin{eqnarray}\label{nablaV.decay}
\|\nabla \bar{V}(t)\|_{N-1} \leq C
\epsilon_{N+3}(\bar{V}_{0})(1+t)^{-\frac{5}{4}}
\end{eqnarray}
for any $t\geq 0$.
\end{proposition}
\subsection{a priori estimates} In this subsection, we prove that the stationary
solution obtained in Section \ref{sec2} is stable under small
initial perturbation.
We begin to use the refined energy
method to obtain some uniform-in-time {\it a priori} estimates for
smooth solutions to the Cauchy problem
(\ref{sta.equ})-(\ref{sta.equi}). To the end, let us denote
\begin{equation}\label{def.delta}
\delta=\|\sigma_{st}\|_{W_{0}^{N+1,2}}=\left(\sum_{|\alpha|\leq
N+1}\int_{\mathbb{R}^3}|\partial^{\alpha}_{x}\sigma_{st}|^2dx\right)^{\frac{1}{2}}
\end{equation}
for simplicity of presentation. A careful look at the proof of
Theorem \ref{sta.existence} shows that
\begin{eqnarray*}
\sigma_{st}&=&\frac{2}{\gamma-1}\left\{n_{st}^{\frac{\gamma-1}{2}}-1\right\}\\
&=&\frac{2}{\gamma-1}\left\{\left(\frac{\gamma-1}{\gamma}Q_{st}+1\right)^{\frac{1}{2}}-1\right\}\\
&=&\frac{2}{\gamma}\dfrac{Q_{st}}{\left(\frac{\gamma-1}{\gamma}Q_{st}+1\right)^{\frac{1}{2}}+1}\sim
Q_{st}.
\end{eqnarray*}
It follows that $\delta \leq C\|Q_{st}\|_{W_{0}^{N+1,2}}\leq
C\|n_{b}-1\|_{W_{0}^{N+1,2}}$ is small enough. Notice that
(\ref{sta.equ}) is a quasi-linear symmetric hyperbolic system. The
main goal of this subsection is to prove
\begin{theorem}\label{estimate}(\textrm{a priori estimates}).
Let $0<T\leq \infty$ be given. Suppose
$\bar{V}:=[\bar{\sigma},\bar{v},\bar{E},\bar{B}]\in
C([0,T);H^{N}(\mathbb{R}^{3}))$ is smooth for $T>0$ with
\begin{eqnarray}\label{3.1}
\sup_{0\leq t<T}\|\bar{V}(t)\|_{N}\leq 1,
\end{eqnarray}
and assume that $\bar{V}$ solves the system (\ref{sta.equ}) for
$t\in(0,T)$. Then, there are $\mathcal {E}_{N}(\cdot) $ and
$\mathcal {D}_{N}(\cdot)$ in the form $\eqref{de.E} $ and
$\eqref{de.D}$ such that
\begin{eqnarray}\label{3.2}
&& \frac{d}{dt}\mathcal {E}_{N}(\bar{V}(t))+\lambda\mathcal
{D}_{N}(\bar{V}(t))\leq
C[\mathcal {E}_{N}(\bar{V}(t))^{\frac{1}{2}}+\mathcal {E}_{N}(\bar{V}(t))+\delta]\mathcal {D}_{N}(\bar{V}(t))
\end{eqnarray}
for any $0\leq t<T$.
\end{theorem}
\begin{proof}
The proof is divided into five steps.
\medskip
\textbf{ Step 1.} It holds that
\begin{equation}\label{3.3}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\left(\sum_{|\alpha|\leq
N}\int_{\mathbb{R}^3}(1+\sigma_{st}+\Phi(\sigma_{st}))
(|\partial^{\alpha}\bar{\sigma}|^2+|\partial^{\alpha}\bar{v}|^2)dx+\|[\bar{E},\bar{B}]\|_{N}^{2}\right)\\
&+\frac{1}{\sqrt{\gamma}}\sum_{|\alpha|\leq
N}\int_{\mathbb{R}^3}(1+\sigma_{st}+\Phi(\sigma_{st}))|\partial^{\alpha}\bar{v}|^{2}dx\\
\leq &
C(\|\bar{V}\|_{N}+\delta)(\|[\bar{\sigma},\bar{v}]\|^{2}+\|\nabla[\bar{\sigma},\bar{v}]\|_{N-1}^{2}
+\|\nabla \bar{E}\|_{N-2}^2).
\end{aligned}
\end{equation}
In fact, applying $\partial^{\alpha}$ to the first two equations of
(\ref{sta.equ}) for $|\alpha|\leq N$ and multiplying them by
$(1+\sigma_{st}+\Phi(\sigma_{st}))\partial^{\alpha}\bar{\sigma}$ and
$(1+\sigma_{st}+\Phi(\sigma_{st}))\partial^{\alpha}\bar{v}$
respectively, taking integrations in $x$ and then using integration
by parts give
\begin{eqnarray}\label{3.4}
&& \begin{aligned}
&\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^3}(1+\sigma_{st}+\Phi(\sigma_{st}))
(|\partial^{\alpha}\bar{\sigma}|^2+|\partial^{\alpha}\bar{v}|^2)dx+\frac{1}{\sqrt{\gamma}}
\langle \partial^{\alpha}\bar{E},(1+\sigma_{st}+\Phi(\sigma_{st}))\partial^{\alpha}\bar{v}\rangle\\
&+\frac{1}{\sqrt{\gamma}}\int_{\mathbb{R}^3}(1+\sigma_{st}+\Phi(\sigma_{st}))|\partial^{\alpha}\bar{v}|^{2}dx
=-\sum_{\beta<\alpha}C^{\alpha}_{\beta}I_{\alpha,\beta}(t)+I_{1}(t).
\end{aligned}
\end{eqnarray}
Here,
$I_{\alpha,\beta}(t)=I_{\alpha,\beta}^{(\sigma)}(t)+I_{\alpha,\beta}^{(v)}(t)$
with
\begin{eqnarray*}
\arraycolsep=1.5pt
\begin{array}{rl}
\displaystyle
I_{\alpha,\beta}^{(\sigma)}(t)=& \displaystyle\langle
\partial^{\alpha-\beta}\bar{v} \cdot \nabla \partial^{\beta}\bar{\sigma},
(1+\sigma_{st}+\Phi(\sigma_{st}))\partial^{\alpha}\bar{\sigma}\rangle\\[3mm]
&\displaystyle+\frac{\gamma-1}{2}\langle\partial^{\alpha-\beta}\bar{\sigma}
\partial^{\beta}\nabla\cdot\bar{v} ,(1+\sigma_{st}+\Phi(\sigma_{st}))\partial^{\alpha}\bar{\sigma}\rangle\\[3mm]
& \displaystyle+\frac{\gamma-1}{2}\langle
\partial^{\alpha-\beta}\sigma_{st} \partial^{\beta}\nabla\cdot\bar{v} ,
(1+\sigma_{st}+\Phi(\sigma_{st}))\partial^{\alpha}\bar{\sigma}
\rangle\\[3mm]
& \displaystyle +\langle
\partial^{\alpha-\beta}\bar{v}\cdot \partial^{\beta}\nabla \sigma_{st} ,(1+\sigma_{st}+\Phi(\sigma_{st}))\partial^{\alpha}\bar{\sigma}\rangle,
\end{array}
\end{eqnarray*}
\begin{eqnarray*}
\arraycolsep=1.5pt
\begin{array}{rl}
\displaystyle
I_{\alpha,\beta}^{(v)}(t)=& \displaystyle\langle
\partial^{\alpha-\beta}\bar{v} \cdot \nabla \partial^{\beta}\bar{v} ,
(1+\sigma_{st}+\Phi(\sigma_{st}))\partial^{\alpha}\bar{v}\rangle\\[3mm]
&\displaystyle+\frac{\gamma-1}{2}\langle\partial^{\alpha-\beta}\bar{\sigma}
\nabla\partial^{\beta}\bar{\sigma},(1+\sigma_{st}+\Phi(\sigma_{st}))\partial^{\alpha}\bar{v}\rangle\\[3mm]
&\displaystyle+\frac{\gamma-1}{2}\langle
\partial^{\alpha-\beta}\sigma_{st} \nabla \partial^{\beta}\bar{\sigma}
,(1+\sigma_{st}+\Phi(\sigma_{st}))\partial^{\alpha}\bar{v}\rangle\\[3mm]
&\displaystyle+\langle
\partial^{\alpha-\beta}\bar{v}\times \partial^{\beta}\bar{B}
,(1+\sigma_{st}+\Phi(\sigma_{st}))\partial^{\alpha}\bar{v}\rangle\\[3mm]
&\displaystyle+\frac{\gamma-1}{2}\langle
\partial^{\alpha-\beta}\bar{\sigma} \nabla \partial^{\beta}\sigma_{st} ,(1+\sigma_{st}+\Phi(\sigma_{st}))
\partial^{\alpha}\bar{v} \rangle
\end{array}
\end{eqnarray*}
and
\begin{eqnarray*}
\arraycolsep=1.5pt
\begin{array}{rl}
I_{1}(t)=&
\displaystyle\frac{1}{2}\langle \nabla \cdot \bar{v},
(1+\sigma_{st}+\Phi(\sigma_{st}))(|\partial^{\alpha}\bar{\sigma}|^{2}+|\partial^{\alpha}\bar{v}|^{2})
\rangle\\[3mm]
& \displaystyle+ \frac{\gamma-1}{2}\langle \nabla
\bar{\sigma}\cdot\partial^{\alpha}\bar{v}
,(1+\sigma_{st}+\Phi(\sigma_{st}))\partial^{\alpha}\bar{\sigma}
\rangle-\langle \bar{v}\times
\partial^{\alpha}\bar{B},(1+\sigma_{st}+\Phi(\sigma_{st}))\partial^{\alpha}\bar{v}\rangle\\[3mm]
& \displaystyle+\frac{\gamma-1}{2}\langle \nabla\sigma_{st}
\partial^{\alpha}\bar{v},(1+\sigma_{st}+\Phi(\sigma_{st}))\partial^{\alpha}\bar{\sigma}
\rangle-\frac{\gamma-1}{2}\langle \bar{\sigma}\partial^{\alpha}
\nabla\sigma_{st},(1+\sigma_{st}+\Phi(\sigma_{st}))\partial^{\alpha}\bar{v}
\rangle\\[3mm]
&- \displaystyle \langle \bar{v}\cdot\partial^{\alpha}
\nabla\sigma_{st},(1+\sigma_{st}+\Phi(\sigma_{st}))\partial^{\alpha}\bar{\sigma}
\rangle\\[3mm]
&\displaystyle+\left\langle
\left(\frac{\gamma-1}{2}\bar{\sigma}+1\right)\partial^{\alpha}\bar{v},
\nabla(1+\sigma_{st}+\Phi(\sigma_{st}))\partial^{\alpha}\bar{\sigma}
\right\rangle\\[3mm]
& \displaystyle +\frac{\gamma-1}{2}\langle \sigma_{st}
\partial^{\alpha}\bar{v},\nabla(1+\sigma_{st}+\Phi(\sigma_{st}))\partial^{\alpha}\bar{\sigma}
\rangle\\[3mm]
&\displaystyle+\frac{1}{2}\langle\bar{v},\nabla(1+\sigma_{st}+\Phi(\sigma_{st}))
(|\partial^{\alpha}\bar{\sigma}|^2+|\partial^{\alpha}\bar{v}|^{2})\rangle\triangleq\sum_{j=1}^{9}I_{1,j}(t).
\end{array}
\end{eqnarray*}
When $|\alpha|=0$, it suffices to estimate $I_{1}(t)$ by
\begin{eqnarray*}
\begin{aligned}
I_{1}(t)\leq &C \|\nabla \cdot \bar{v}\|(\|\bar{v}\|_{L^{6}}\|\bar{v}\|_{L^{3}}
+\|\bar{\sigma}\|_{L^{6}}\|\bar{\sigma}\|_{L^{3}})
+C \|\nabla \bar{\sigma}\|\|\bar{v}\|_{L^{6}}\|\bar{\sigma}\|_{L^{3}}
+C \|\bar{B}\|_{L^{\infty}}\|\bar{v}\|^{2}\\
&+C \|\nabla\sigma_{st}\|\left\|\bar{\sigma}\right\|_{L^{6}}
\|\bar{v}\|_{L^3}+C\|\sigma_{st}\|_{L^{\infty}}\|\nabla\sigma_{st}\|\left\|\bar{\sigma}\right\|_{L^{6}}
\|\bar{v}\|_{L^3}\\
&+\|\bar{v}\|_{L^{\infty}}
\|\nabla\sigma_{st}\|(\|\bar{\sigma}\|_{L^{6}}\|\bar{\sigma}\|_{L^{3}}
+\|\bar{v}\|_{L^{6}}\|\bar{v}\|_{L^{3}})
\\
\leq & C (\|[\bar{\sigma},\bar{v}]\|_{H^{1}}+\delta+\delta\|\nabla\bar{v}\|_{H^1})(\|\nabla
[\bar{\sigma},\bar{v}]\|^{2}+\|[\bar{\sigma},\bar{v}]\|^{2})+ C \|\nabla
\bar{B}\|_{H^{1}}\|\bar{v}\|^{2},
\end{aligned}
\end{eqnarray*}
which is further bounded by the r.h.s. term of (\ref{3.3}). When
$|\alpha|\geq 1$, for $I_{1}(t)$, the similarity of $I_{1,1}(t)$ and
$I_{1,2}(t)$ shows that we can estimate them together as follows
\begin{eqnarray*}
\begin{aligned}
I_{1,1}(t)+I_{1,2}(t)\leq & C
\|\nabla\cdot\bar{v}\|_{L^{\infty}}\|(1+\sigma_{st}+\Phi(\sigma_{st}))\|_{L^{\infty}}
\|\nabla[\bar{\sigma},\bar{v}]\|_{N-1}^2\\
&+C\|\nabla\bar{\sigma}\|_{L^{\infty}}\|(1+\sigma_{st}+\Phi(\sigma_{st}))\|_{L^{\infty}}
\|\nabla[\bar{\sigma},\bar{v}]\|_{N-1}^2\\
\leq & C
\|[\bar{\sigma},\bar{v}]\|_{N}\|\nabla[\bar{\sigma},\bar{v}]\|_{N-1}^2.
\end{aligned}
\end{eqnarray*}
For $I_{1,3}(t)$, $I_{1,5}(t)$ and $I_{1,6}(t)$, there are no
derivative of $\bar{\sigma}$ or $\bar{v}$, then we use $L^{\infty}$
of $\bar{v}$ or $\bar{\sigma}$,
\begin{eqnarray*}
\begin{aligned}
I_{1,3}(t)+I_{1,5}(t)+I_{1,6}(t)\leq & C
\|\bar{v}\|_{L^{\infty}}\|\partial^{\alpha}
\bar{B}\|\|(1+\sigma_{st}+\Phi(\sigma_{st}))\|_{L^{\infty}}
\|\nabla\bar{v}\|_{N-1}\\
&+C \|\bar{\sigma}\|_{L^{\infty}}\|\partial^{\alpha}
\nabla\sigma_{st}\|\|(1+\sigma_{st}+\Phi(\sigma_{st}))\|_{L^{\infty}}
\|\nabla\bar{v}\|_{N-1}\\
&+C \|\bar{v}\|_{L^{\infty}}\|\partial^{\alpha}
\nabla\sigma_{st}\|\|(1+\sigma_{st}+\Phi(\sigma_{st}))\|_{L^{\infty}}
\|\nabla\bar{\sigma}\|_{N-1}\\
\leq &
C(\delta+\|\bar{B}\|_{N})\|\nabla[\bar{\sigma},\bar{v}]\|_{N-1}^2.
\end{aligned}
\end{eqnarray*}
For other terms of $I_{1}(t)$, both $\bar{\sigma}$ and $\bar{v}$
contain the derivative, one can use the $L^{2}$ of these terms and
$L^{\infty}$ of others. Combining the above two estimates, one has
\begin{eqnarray*}
I_{1}(t)\leq C (\|[\bar{\sigma},\bar{v},\bar{B}]\|_{N}
+\delta+\delta\|\nabla \bar{v}\|_{H^1})\|\nabla
[\bar{\sigma},\bar{v}]\|_{N-1}^{2},
\end{eqnarray*}
which is bounded by the r.h.s. term of (\ref{3.3}). On the other
hand, since each term in $I_{\alpha,\beta}(t)$ is the integration of
the four-terms product in which there is at least one term
containing the derivative, one has
\begin{eqnarray*}
I_{\alpha,\beta}(t)\leq C (\|[\bar{\sigma},\bar{v},\bar{B}]\|_{N}
+\delta+\delta\|\nabla \bar{v}\|_{H^1})\|\nabla
[\bar{\sigma},\bar{v}]\|_{N-1}^{2},
\end{eqnarray*}
which is also bounded by the r.h.s. term of (\ref{3.3}).
From (\ref{sta.equ}), energy estimates on $\partial^{\alpha}\bar{E}$
and $\partial^{\alpha}\bar{B}$ with $|\alpha| \leq N$ give
\begin{eqnarray}\label{3.5}
&& \begin{aligned}
&\frac{1}{2}\frac{d}{dt}\|\partial^{\alpha}[\bar{E},\bar{B}]\|^{2}
-\frac{1}{\sqrt{\gamma}}\langle
(1+\sigma_{st}+\Phi(\sigma_{st}))\partial
^{\alpha}\bar{v},\partial^{\alpha}\bar{E}\rangle\\
=&\frac{1}{\sqrt{\gamma}}\langle \partial
^{\alpha}[(\Phi(\bar{\sigma}+\sigma_{st})-\Phi(\sigma_{st}))\bar{v}],\partial^{\alpha}\bar{E}\rangle+
\frac{1}{\sqrt{\gamma}}\langle \partial
^{\alpha}[\bar{\sigma}\bar{v}],\partial^{\alpha}\bar{E}\rangle\\
&+\frac{1}{\sqrt{\gamma}}\sum_{\beta<\alpha}C_{\beta}^{\alpha}\langle \partial^{\alpha-\beta}(1+\sigma_{st}+\Phi(\sigma_{st}))
\partial^{\beta}\bar{v},\partial^{\alpha}\bar{E}\rangle\\
=&I_{2,1}(t)+I_{2,2}(t)+\sum_{\beta<\alpha}C_{\beta}^{\alpha}I_{2,\beta}(t).
\end{aligned}
\end{eqnarray}
In a similar way as before, when $|\alpha|=0$, it suffices to
estimate $I_{2,1}(t)+ I_{2,2}(t)$ by
\begin{eqnarray*}
I_{2,1}(t)+ I_{2,2}(t)\leq C \|\nabla
\bar{\sigma}\|\cdot\|\bar{v}\|_{1}\|\bar{E}\|.
\end{eqnarray*}
When $|\alpha|>0$, $I_{2,1}(t)$ and $I_{2,2}(t)$ can be estimated in
a similar way as in \cite{Duan},
\begin{eqnarray*}
I_{2,1}(t)+ I_{2,2}(t)\leq C \|\nabla \bar{\sigma}\|_{N-1}\|\nabla \bar{v}\|_{N-1}\|\bar{E}\|_{N}.
\end{eqnarray*}
When $|\alpha|>0$, for each $\beta$ with $\beta<\alpha$,
$I_{2,\beta}$ is estimated by three cases.
\textsl{Case 1.} $|\alpha|=N$. In this case, integration by parts
shows that
\begin{eqnarray*}
&& \begin{aligned}
I_{2,\beta}(t) \leq & C \delta \|\nabla \bar{v}\|_{N-1}\|\nabla\bar{E}\|_{N-2}\\
\leq & C \delta \|\nabla \bar{v}\|_{N-1}^2+C \delta \|\nabla \bar{E}\|_{N-2}^2.
\end{aligned}
\end{eqnarray*}
\textsl{Case 2.} $|\alpha|<N $ and $|\beta|\geq 1$ which imply
$|\alpha-\beta|\leq N-2$. It holds that
\begin{eqnarray*}
&& \begin{aligned}
I_{2,\beta}(t) \leq & C\|\partial^{\alpha-\beta}(1+\sigma_{st}+\Phi(\sigma_{st}))\|_{L^{\infty}}
\|\partial^{\beta}\bar{v}\|\|\partial^{\alpha}\bar{E}\|\\
\leq & C\|\nabla\partial^{\alpha-\beta}(1+\sigma_{st}+\Phi(\sigma_{st}))\|_{H^{1}}\|\nabla \bar{v}\|_{N-1}\|\nabla\bar{E}\|_{N-2}\\
\leq & C \delta \|\nabla \bar{v}\|_{N-1}^2+C \delta \|\nabla
\bar{E}\|_{N-2}^2.
\end{aligned}
\end{eqnarray*}
\textsl{Case 3.} $|\alpha|<N $ and $|\beta|=0$. In this case, there
is no derivative of $\bar{v}$, one can use $L^{\infty}$ of $\bar{v}$
to estimate $I_{2,\beta}(t)$,
\begin{eqnarray*}
&& \begin{aligned}
I_{2,\beta}(t) \leq & C\|\partial^{\alpha-\beta}(1+\sigma_{st}+\Phi(\sigma_{st}))\|
\|\bar{v}\|_{L^{\infty}}\|\partial^{\alpha}\bar{E}\|\\
\leq & C \delta \|\nabla \bar{v}\|_{N-1}^2+C \delta \|\nabla
\bar{E}\|_{N-2}^2,
\end{aligned}
\end{eqnarray*}
which is bounded by the r.h.s. term of (\ref{3.3}). Then (\ref{3.3})
follows by taking summation of (\ref{3.4}) and (\ref{3.5}) over
$|\alpha| \leq N$. Then the time evolution of the full instant
energy $\|V(t)\|_{N}^{2}$ has been obtained but its dissipation rate
only contains the contribution from the explicit relaxation variable
$\bar{v}$. In a parallel way as \cite{Duan}, by introducing some
interactive functionals, the dissipation from contributions of the
rest components $\bar{\sigma}$, $\bar{E}$, and $\bar{B}$ can be
recovered in turn.
\medskip
\textbf{Step 2.} It holds that
\begin{eqnarray}\label{step2}
&&\begin{aligned}
&\frac{d}{dt}\mathcal {E}_{N,1}^{int}(\bar{V})+\lambda\|\bar{\sigma}\|^{2}_{N} \\
\leq & C\|\nabla\bar{v}\|_{N-1}^{2}+C(\|[\bar{\sigma}, \bar{v},\bar{B}]\|_{N}^{2}+\delta)
\|\nabla[\bar{\sigma},\bar{v}]\|_{N-1}^{2},
\end{aligned}
\end{eqnarray}
where $\mathcal {E}_{N,1}^{int}(\cdot)$ is defined by
\begin{eqnarray*}
\mathcal {E}_{N,1}^{int}(\bar{V})=\sum_{|\alpha|\leq N-1}\langle
\partial^{\alpha}\bar{v},\nabla\partial^{\alpha}\bar{\sigma}\rangle.
\end{eqnarray*}
In fact, the first two equations of $ \eqref{sta.equ}$ can be
rewritten as
\begin{eqnarray}\label{3.7}
&&\partial_t \bar{\sigma}+\nabla \cdot \bar{v}=f_{1},
\end{eqnarray}
\begin{eqnarray}\label{3.9}
\partial_t \bar{v}+\nabla
\bar{\sigma}+\frac{1}{\sqrt{\gamma}}\bar{E}=f_{2}-\frac{1}{\sqrt{\gamma}}\bar{v},
\end{eqnarray}
where
\begin{eqnarray}\label{f1f2}
&& \left\{
\begin{aligned}
& f_{1}:=-\bar{v}\cdot
\nabla\bar{\sigma}-\frac{\gamma-1}{2}\bar{\sigma}\nabla \cdot \bar{v}
-\bar{v}\cdot
\nabla \sigma_{st}-\frac{\gamma-1}{2}\sigma_{st}\nabla \cdot \bar{v},\\
& f_{2}:=-\bar{v}\cdot \nabla
\bar{v}-\frac{\gamma-1}{2}\bar{\sigma}\nabla \bar{\sigma}-\bar{v}\times \bar{B}
-\frac{\gamma-1}{2}\sigma_{st}\nabla \bar{\sigma}-\frac{\gamma-1}{2}\bar{\sigma}\nabla\sigma_{st}.
\end{aligned}\right.
\end{eqnarray}
Let $|\alpha|\leq N-1$. Applying $\partial^{\alpha}$ to (\ref{3.9}),
multiplying it by $\partial^{\alpha}\nabla \bar{\sigma}$, taking
integrations in $x$ and then using integration by parts and also the
final equation of (\ref{sta.equ}), replacing
$\partial_{t}\bar{\sigma}$ from (\ref{3.7}) give
\begin{eqnarray*}
\arraycolsep=1.5pt
\begin{array}{rl}
&\displaystyle
\frac{d}{dt}\langle
\partial^{\alpha}\bar{v},\nabla
\partial^{\alpha}\bar{\sigma}\rangle
+\|\nabla\partial^{\alpha}\bar{\sigma}\|^{2}+\frac{1}{\gamma}\|
\partial^{\alpha}\bar{\sigma}\|^2\\[3mm]
=&\displaystyle -\frac{1}{\gamma}\langle
\partial^{\alpha}\left(\Phi(\bar{\sigma}+\sigma_{st})-\Phi(\sigma_{st})\right),
\partial^{\alpha}\bar{\sigma}\rangle+\langle\partial^{\alpha}f_{2},\nabla\partial^{\alpha}\bar{\sigma}\rangle\\[3mm]
&-\displaystyle\frac{1}{\sqrt{\gamma}}\langle\partial^{\alpha}\bar{v},\nabla\partial^{\alpha}\bar{\sigma}\rangle
+\|\nabla \cdot
\partial^{\alpha}\bar{v}\|^{2}-\langle\partial^{\alpha}f_{1},\nabla \cdot
\partial^{\alpha}\bar{v}\rangle.
\end{array}
\end{eqnarray*}
Then, it follows from Cauchy-Schwarz inequality that
\begin{eqnarray}\label{3.11}
\arraycolsep=1.5pt
\begin{array}{rl}
&\displaystyle
\frac{d}{dt}\langle
\partial^{\alpha}\bar{v},\nabla
\partial^{\alpha}\bar{\sigma}\rangle
+\lambda(\|\nabla\partial^{\alpha}\bar{\sigma}\|^{2}+\|
\partial^{\alpha}\bar{\sigma}\|^2)\\[3mm]
\leq & C \displaystyle \|\nabla \cdot
\partial^{\alpha}\bar{v}\|^{2}+C(\|\partial^{\alpha}
\left(\Phi(\bar{\sigma}+\sigma_{st})-\Phi(\sigma_{st})\right) \|^{2}
+\|\partial^{\alpha}f_{1}\|^{2}+\|\partial^{\alpha}f_{2}\|^{2}).
\end{array}
\end{eqnarray}
Noticing that $\Phi(\sigma)$ is smooth in $\sigma$ with
$\Phi'(0)=0$, one has from (\ref{f1f2}) that
\begin{eqnarray*}
\arraycolsep=1.5pt
\begin{array}{rl}
&\|\partial^{\alpha}
\left(\Phi(\bar{\sigma}+\sigma_{st})-\Phi(\sigma_{st})\right) \|^{2}
+\|\partial^{\alpha}f_{1}\|^{2}+\|\partial^{\alpha}f_{2}\|^{2}\\[3mm]
\leq &C(\|[\bar{\sigma},\bar{v},\bar{B}]\|^{2}_{N}+\delta)\|
\nabla[\bar{\sigma},\bar{v}]\|_{N-1}^{2}.
\end{array}
\end{eqnarray*}
Here, if there is no derivative on $\bar{\sigma}$ or $\bar{v}$, then
use the $L^{\infty}$ of $\bar{\sigma}$ or $\bar{v}$. Plugging this
into (\ref{3.11}) taking summation over $|\alpha|\leq N-1$ yield
(\ref{step2}).
\medskip
\textbf{Step 3.} It holds that
\begin{equation}\label{step3}
\begin{aligned}
\dfrac{d}{dt}\mathcal {E}_{N,2}^{int}(\bar{V})+\lambda\|\bar{E}\|^{2}_{N-1} \leq
C&\|[\bar{\sigma},\bar{v}]\|_{N}^{2}+C\|\bar{v}\|_{N}\|\nabla
\bar{B}\|_{N-2}\\
&+C(\|[\bar{\sigma},\bar{v},\bar{B}]\|_{N}^{2}+\delta)
\|\nabla[\bar{\sigma},\bar{v}]\|_{N-1}^{2},
\end{aligned}
\end{equation}
where $\mathcal {E}_{N,2}^{int}(\cdot)$ is defined by
\begin{eqnarray*}
\mathcal {E}_{N,2}^{int}(\bar{V})=\sum_{|\alpha|\leq N-1}\langle
\partial^{\alpha}\bar{v},\partial^{\alpha}\bar{E}\rangle.
\end{eqnarray*}
Applying $\partial^{\alpha}$ to (\ref{3.9}), multiplying it by
$\partial^{\alpha}\bar{E}$, taking integrations in $x$ and using
integration by parts and replacing $ \partial_{t}\bar{E}$ from the
third equation of (\ref{sta.equ}) give
\begin{equation*}
\arraycolsep=1.5pt
\begin{array}{rl}
&\dfrac{d}{dt}\langle
\partial^{\alpha}\bar{v},\partial^{\alpha}\bar{E}\rangle+
\dfrac{1}{\sqrt{\gamma}}\|\partial^{\alpha}\bar{E}\|^{2}\\[3mm]
=& \dfrac{1}{\sqrt{\gamma}}\|\partial^{\alpha}\bar{v}\|^{2}+
\dfrac{1}{\sqrt{\gamma}}\langle
\partial^{\alpha}\bar{v},\nabla \times \partial^{\alpha}\bar{B}\rangle+\dfrac{1}{\sqrt{\gamma}}\langle
\partial^{\alpha}\bar{v},
\partial^{\alpha}[\Phi(\bar{\sigma}+\sigma_{st})\bar{v}+(\bar{\sigma}+\sigma_{st})\bar{v}]\rangle\\[3mm]
&-\langle\partial^{\alpha}\nabla
\bar{\sigma}+\dfrac{1}{\sqrt{\gamma}}\partial^{\alpha}\bar{v},\partial^{\alpha}\bar{E}\rangle
+\langle\partial^{\alpha}f_{2},
\partial^{\alpha}\bar{E}\rangle,
\end{array}
\end{equation*}
which from the Cauchy-Schwarz inequality further implies
\begin{equation*}
\arraycolsep=1.5pt
\begin{array}{rl}
&\dfrac{d}{dt}\langle
\partial^{\alpha}\bar{v},\partial^{\alpha}\bar{E}\rangle+
\lambda\|\partial^{\alpha}\bar{E}\|^{2}\\[3mm]
\leq &
C\|[\bar{\sigma},\bar{v}]\|_{N}^{2}+C\|\bar{v}\|_{N}\|\nabla
\bar{B}\|_{N-2}+C(\|[\bar{\sigma},\bar{v},\bar{B}]\|_{N}^{2}+\delta)
\|\nabla[\bar{\sigma},\bar{v}]\|_{N-1}^{2}.
\end{array}
\end{equation*}
Thus $\eqref{step3}$ follows from taking summation of the above
estimate over $|\alpha|\leq N-1$.
\medskip
\textbf{Step 4.} It holds that
\begin{equation}\label{step4}
\begin{aligned}
\frac{d}{dt}\mathcal {E}_{N,3}^{int}(\bar{V})+\lambda\|\nabla\bar{B}\|^{2}_{N-2}
\leq & C\|[\bar{v},\bar{E}]\|_{N-1}^{2}\\
&+C(\|\bar{\sigma}\|_{N}^{2}+\delta)\|\nabla \bar{v}\|_{N-1}^{2},
\end{aligned}
\end{equation}
where $\mathcal {E}_{N,3}^{int}(\cdot)$ is defined by
\begin{eqnarray*}
\mathcal {E}_{N,3}^{int}(\bar{V})=-\sum_{|\alpha|\leq N-2}\langle
\nabla \times
\partial^{\alpha}\bar{E},\partial^{\alpha}\bar{B}\rangle.
\end{eqnarray*}
In fact, for $|\alpha|\leq N-2$, applying $\partial^{\alpha}$ to the
third equation of $\eqref{sta.equ}$, multiplying it by
$-\partial^{\alpha}\nabla \times \bar{B}$, taking integrations in
$x$ and using integration by parts and replacing $
\partial_{t}\bar{B}$ from the fourth equation of $\eqref{sta.equ}$
implie
\begin{equation*}
\arraycolsep=1.5pt
\begin{array}{rl}
& -\dfrac{d}{dt}\langle
\partial^{\alpha}\bar{E},\nabla \times \partial^{\alpha}\bar{B}\rangle+
\dfrac{1}{\sqrt{\gamma}}\|\nabla\times \partial^{\alpha}\bar{B}\|^{2} \\[3mm]
=&\dfrac{1}{\sqrt{\gamma}}\|\nabla\times\partial^{\alpha}\bar{E}\|^{2}-\dfrac{1}{\sqrt{\gamma}}
\langle \partial^{\alpha}\bar{v},\nabla \times
\partial^{\alpha}\bar{B}\rangle
-\dfrac{1}{\sqrt{\gamma}}\langle
\partial^{\alpha}[\Phi(\bar{\sigma}+\sigma_{st})\bar{v}+(\bar{\sigma}+\sigma_{st})\bar{v}],\nabla \times
\partial^{\alpha}\bar{B}\rangle,
\end{array}
\end{equation*}
which gives $\eqref{step4}$ by further using Cauchy-Schwarz
inequality and taking summation over $|\alpha|\leq N-2$, where we
also used
\begin{eqnarray*}
\|\partial^{\alpha}\partial_{i}\bar{B}\|=\|\partial_{i}\Delta^{-1}\nabla
\times(\nabla\times\partial^{\alpha}\bar{B}) \|\leq\|\nabla\times
\partial^{\alpha}\bar{B}\|
\end{eqnarray*}
for each $1\leq i\leq 3$, due to the fact
$\partial_{i}\Delta^{-1}\nabla$ is bounded from $L^{p}$ to itself
for $1<p<\infty$, cf. \cite{Stein}.
\medskip
\textbf{Step 5.} Now, following the four steps above, we are ready to prove
$\eqref{3.2}$. Let us define
\begin{eqnarray*}
\mathcal {E}_{N}(\bar{V}(t))=\sum_{|\alpha|\leq
N}\int_{\mathbb{R}^3}(1+\sigma_{st}+\Phi(\sigma_{st}))
(|\partial^{\alpha}\bar{\sigma}|^2+|\partial^{\alpha}\bar{v}|^2)dx+\|[\bar{E},\bar{B}]\|_{N}^{2}
+\sum_{i=1}^{3}\kappa_{i}\mathcal {E}^{int}_{N,i}(\bar{V}(t)),
\end{eqnarray*}
that is,
\begin{equation}\label{3.12}
\arraycolsep=1.5pt
\begin{array}{rl}
\mathcal{E}_{N}(\bar{V}(t))=&\displaystyle\sum_{|\alpha|\leq
N}\int_{\mathbb{R}^3}(1+\sigma_{st}+\Phi(\sigma_{st}))
(|\partial^{\alpha}\bar{\sigma}|^2+|\partial^{\alpha}\bar{v}|^2)dx+\|[\bar{E},\bar{B}]\|_{N}^{2}\\[3mm]
&\displaystyle+\kappa_{1}\sum_{|\alpha|\leq N-1} \langle
\partial^{\alpha}\bar{v},\nabla\partial^{\alpha}\bar{\sigma}\rangle+\kappa_{2}\sum_{|\alpha|\leq N-1}\langle
\partial^{\alpha}\bar{v},\partial^{\alpha}\bar{E}\rangle\\[3mm]
&\displaystyle-\kappa_{3}\sum_{|\alpha|\leq N-2}\langle \nabla
\times
\partial^{\alpha}\bar{E},\partial^{\alpha}\bar{B}\rangle
\end{array}
\end{equation}
for constants $0<\kappa_{3}\ll\kappa_{2}\ll\kappa_{1}\ll 1$ to be
determined. Notice that as long as $ 0<\kappa_{i}\ll 1$ is small
enough for $i=1,2,3$, and $\sigma_{st}+\Phi(\sigma_{st}) $ depending
only on $ x$ is sufficiently small compared with $1$, then
$\mathcal{E}_{N}(\bar{V}(t))\sim \|\bar{V}(t)\|^{2}_{N}$ holds true.
Moreover, letting $0<\kappa_{3}\ll\kappa_{2}\ll\kappa_{1}\ll 1$ with
$\kappa_{2}^{3/2}\ll\kappa_{3}$, the sum of $\eqref{3.3}\times
\kappa_{1}$, $\eqref{step2}\times \kappa_{2}$, $\eqref{step4}\times
\kappa_{3}$ implies that there are $\lambda>0$, $C>0$ such that
$\eqref{3.2}$ holds true with $\mathcal {D}_{N}(\cdot)$ defined in
$\eqref{de.D}$. Here, we have used the following Cauchy-Schwarz
inequality:
\begin{eqnarray*}
2 \kappa_{2} \|\bar{v}\|_{N}\|\nabla \bar {B}\|_{N-2}\leq
\kappa_{2}^{1/2}\|\bar{v}\|_{N}^{2}+\kappa_{2}^{3/2}\|\nabla\bar{B}\|^{2}_{N-2}.
\end{eqnarray*}
Due to $\kappa_{2}^{3/2}\ll \kappa_{3}$, both terms on the r.h.s. of
the above inequality were absorbed. This completes the proof of
Theorem $\ref{estimate}$.
\end{proof}
Since $\eqref{sta.equ}$ is a quasi-linear symmetric hyperbolic
system, the short-time existence can be proved in much more general
case as in \cite{Kato}; see also (Theorem 1.2, Proposition 1.3, and
Proposition 1.4 in Chapter 16 of \cite{Taylor}). From Theorem
\ref{estimate} and the continuity argument, it is easy to see that $
\mathcal {E}_{N}(\bar{V}(t)) $ is bounded uniformly in time under
the assumptions that $\mathcal {E}_{N}(\bar{V}_{0})>0$ and
$\|n_{b}-1\|_{W_{0}^{N+1,2}}$ are small enough. Therefore, the
global existence of solutions satisfying \eqref{V.satisfy} and
\eqref{pro.2.1j} follows in the standard way; see also \cite{Duan}.
This completes the proof of Proposition \ref{pro.2.1}.\qed
\vspace{5mm}
\section{Decay in time for the non-linear system}\label{sec4}
In this section, we are devoted to the rate of the convergence of
solution to the equilibrium $[n_{st},0,E_{st},0]$ for the system
\eqref{1.1} over $\mathbb{R}^3$. In fact by setting
\begin{eqnarray*}
\bar{\rho}=n-n_{st},\ \ \bar{u}=u,\ \ E_{1}=E-E_{st},\ \ B_{1}=B,
\end{eqnarray*}
and
\begin{eqnarray*}
\rho_{st}=n_{st}-1,
\end{eqnarray*}
then $\bar{U}:=[\bar{\rho},\bar{u},E_{1},B_{1}]$ satisfies
\begin{equation}\label{rhost}
\left\{
\begin{aligned}
&\partial_t \bar{\rho}+\nabla\cdot \bar{u}=g_{1} ,\\
&\partial_t \bar{u}+\bar{u} + E_{1} +\gamma \nabla\bar{\rho}=g_{2},
\\
&\partial_t E_{1}-\nabla\times B_{1}-\bar{u}=g_{3},\\
&\partial_t B_{1}+\nabla \times E_{1}=0,\\
&\nabla \cdot E_{1}=-\bar{\rho}, \ \ \nabla \cdot B_{1}=0, \ \ \ t>0,\ x\in\mathbb{R}^{3},\\
\end{aligned}\right.
\end{equation}
with initial data
\begin{eqnarray}\label{rhosti}
\begin{aligned}
\bar{U}|_{t=0}=\bar{U}_{0}:=&[\bar{\rho}_{0},\bar{u}_{0},E_{1,0},B_{1,0}]\\
=&[n_0-n_{st},u_0,E_{0}-E_{st},B_0], \ \
\ x\in\mathbb{R}^{3},
\end{aligned}
\end{eqnarray}
satisfying the compatible conditions
\begin{eqnarray}\label{rhostC}
\nabla \cdot E_{1,0}=-\bar{\rho}_{0}, \ \ \nabla \cdot B_{1,0}=0.
\end{eqnarray}
Here the nonlinear source terms take the form of
\begin{equation}\label{sec5.ggg}
\arraycolsep=1.5pt \left\{
\begin{aligned}
& g_{1}=-\nabla\cdot[(\bar{\rho}+\rho_{st}) \bar{u}],\\
&\begin{array}[b]{rcl}
g_{2}&=&-\bar{u} \cdot \nabla \bar{u}-\bar{u}\times B_{1}
-\gamma [(\bar{\rho}+1+\rho_{st})^{\gamma-2}-1]\nabla\bar{\rho}\\
&&-\gamma
[(1+\bar{\rho}+\rho_{st})^{\gamma-2}-(1+\rho_{st})^{\gamma-2}]\nabla\rho_{st},
\end{array}\\
& g_{3}=(\bar{\rho}+\rho_{st}) \bar{u}.
\end{aligned}\right.
\end{equation}
In what follows, we will denote $[\rho,u,E,B]$ as the solution to
the the following linearized equation of \eqref{rhost}:
\begin{equation}\label{DJ}
\left\{
\begin{aligned}
&\partial_t \rho+\nabla\cdot u=0,\\
&\partial_t u+u+ E +\gamma \nabla\rho=0,\\
&\partial_t E-\nabla\times B-u=0,\\
&\partial_t B+\nabla \times E=0,\\
&\nabla \cdot E=-\rho, \ \ \nabla \cdot B=0, \ \ \ t>0, \ \ x\in\mathbb{R}^{3},\\
\end{aligned}\right.
\end{equation}
with given initial data
\begin{eqnarray}\label{2.61}
U|_{t=0}=\bar{U}_{0}:=[\bar{\rho}_{0},\bar{u}_{0},E_{1,0},B_{1,0}],
\ \ \ x\in\mathbb{R}^{3},
\end{eqnarray}
satisfying the compatible conditions \eqref{rhostC}.
For the above linearized equations, the $L^{p}$-$L^{q}$ time-decay
property was proved by Duan in \cite{Duan}. We list only some
special $L^{p}$-$L^{q}$ time decay properties in the following
proposition.
\begin{proposition}\label{thm.decay}
Suppose $U(t)=e^{tL}\bar{U}_{0}$ is the solution
to the Cauchy problem \eqref{DJ}-\eqref{2.61} with the initial data
$\bar{U}_{0}=[\bar{\rho}_{0},\bar{u}_{0},E_{1,0},B_{1,0}] $
satisfying \eqref{rhostC}. Then, $U=[\rho,u,E,B]$ satisfies the
following time-decay property:
\begin{eqnarray}\label{col.decay1}
&& \left\{
\begin{aligned}
& \|\rho(t)\|\leq C e^{-\frac{t}{2}}\|[\bar{\rho}_{0},\bar{u}_{0}]\|,\\
& \|u(t)\| \leq C e^{-\frac{t}{2}}\|\bar{\rho}_{0}\|+C(1+t)^{-\frac{5}{4}}
\|[\bar{u}_{0}, E_{1,0},B_{1,0}]\|_{L^1\cap \dot{H}^{2}},\\
&\|E(t)\|\leq C (1+t)^{-\frac{5}{4}}
\|[\bar{u}_{0}, E_{1,0},B_{1,0}]\|_{L^1\cap \dot{H}^{3}},\\
&\|B(t)\|\leq C (1+t)^{-\frac{3}{4}}
\|[\bar{u}_{0}, E_{1,0},B_{1,0}]\|_{L^1\cap \dot{H}^{2}},
\end{aligned}\right.
\end{eqnarray}
and
\begin{eqnarray}\label{col.decayinfty1}
&& \left\{
\begin{aligned}
& \|\rho(t)\|_{\infty}\leq C e^{-\frac{t}{2}}\|[\bar{\rho}_{0},\bar{u}_{0}]\|_{L^{2}\cap\dot{H}^{2}},\\
& \|u(t)\|_{\infty} \leq C e^{-\frac{t}{2}}\|\bar{\rho}_{0}\|_{L^{2}\cap\dot{H}^{2}}+C(1+t)^{-2}
\|[\bar{u}_{0}, E_{1,0},B_{1,0}]\|_{L^1\cap \dot{H}^{5}},\\
&\|E(t)\|_{\infty}\leq C (1+t)^{-2}
\|[\bar{u}_{0}, E_{1,0},B_{1,0}]\|_{L^1\cap \dot{H}^{6}},\\
&\|B(t)\|_{\infty}\leq C (1+t)^{-\frac{3}{2}}
\|[\bar{u}_{0}, E_{1,0},B_{1,0}]\|_{L^1\cap \dot{H}^{5}},
\end{aligned}\right.
\end{eqnarray}
and, moreover,
\begin{eqnarray}\label{col.EB}
&& \left\{
\begin{aligned}
&\|\nabla B(t)\|\leq C (1+t)^{-\frac{5}{4}}
\|[\bar{u}_{0}, E_{1,0},B_{1,0}]\|_{ L^1 \cap \dot{H}^{4}},\\
& \|\nabla^{N}[E(t),B(t)]\|\leq C(1+t)^{-\frac{5}{4}}
\|[\bar{u}_{0}, E_{1,0},B_{1,0}]\|_{ L^1 \cap \dot{H}^{N+3}}.
\end{aligned}\right.
\end{eqnarray}
\end{proposition}
In what follows, since we shall apply the linear $L^{p}$-$L^{q}$
time-decay property of the homogeneous system \eqref{DJ}, we need
the mild form of the non-linear Cauchy problem
\eqref{rhost}-\eqref{rhosti}. From now on, we always denote
$\bar{U}=[\bar{\rho},\bar{u},E_{1},B_{1}]$ to the non-linear Cauchy
problem $\eqref{rhost}$-$\eqref{rhosti}$. Then, by Duhamel's
principle, the solution $\bar{U}$ can be formally written as
\begin{eqnarray}\label{sec5.U}
\bar{U}(t)=e^{tL}\bar{U}_{0}+\int_{0}^{t}e^{(t-s)L}[g_{1}(s),g_{2}(s),g_{3}(s),0]d
s,
\end{eqnarray}
where $e^{tL}\bar{U}_{0}$ denotes the solution to the Cauchy problem
$\eqref{DJ}$-$\eqref{2.61}$ without nonlinear sources.
The following two lemmas give the full and high-order energy
estimates.
\begin{lemma}\label{lem.V}
Let $\bar{V}=[\bar{\sigma},\bar{v},\bar{E},\bar{B}]$ be the solution
to the Cauchy problem $\eqref{sta.equ}$--$ \eqref{sta.equi}$ with
initial data
$\bar{V}_{0}=[\bar{\sigma}_{0},\bar{v}_{0},\bar{E}_{0},\bar{B}_{0}]$
satisfying $\eqref{sta.equC}$. Then, if $\mathcal
{E}_{N}(\bar{V}_{0})$ and $\|n_{b}-1\|_{W_{0}^{N+1,2}}$ are
sufficiently small,
\begin{eqnarray}\label{sec5.ENV0}
\dfrac{d}{dt}\mathcal {E}_{N}(\bar{V}(t))+\lambda \mathcal {D}_{N}(\bar{V}(t))
\leq 0
\end{eqnarray}
holds for any $t>0$, where $\mathcal {E}_{N}(\bar{V}(t))$, $\mathcal
{D}_{N}(\bar{V}(t))$ are defined in the form of $\eqref{de.E}$ and
$\eqref{de.D}$, respectively.
\end{lemma}
\begin{proof}
It can be seen directly from the proof of Theorem \ref{estimate}.
\end{proof}
\begin{lemma}\label{estimate2}
Let $\bar{V}=[\bar{\sigma},\bar{v},\bar{E},\bar{B}]$ be the solution
to the Cauchy problem $\eqref{sta.equ}$-$\eqref{sta.equi}$ with
initial data
$\bar{V}_{0}=[\bar{\sigma}_{0},\bar{v}_{0},\bar{E}_{0},\bar{B}_{0}]$
satisfying $\eqref{sta.equC}$ in the sense of Proposition
$\ref{pro.2.1}$. Then if $ \mathcal {E}_{N}(\bar{V}_{0})$ and
$\|n_{b}-1\|_{W_{0}^{N+1,2}}$ are sufficiently small, there are the
high-order instant energy functional $\mathcal {E}_{N}^{h}(\cdot)$
and the corresponding dissipation rate $\mathcal {D}_{N}^{h}(\cdot)$
such that
\begin{eqnarray}\label{sec5.high}
&& \frac{d}{dt}\mathcal {E}_{N}^{h}(\bar{V}(t))+\lambda\mathcal
{D}^{h}_{N}(\bar{V}(t))\leq 0,
\end{eqnarray}
holds for any $ t \geq 0$.
\end{lemma}
\begin{proof}
The proof can be done by modifying the proof of Theorem
$\ref{estimate}$ a little. In fact, by letting the energy estimates
made only on the high-order derivatives, then corresponding to
$\eqref{3.3}$, $\eqref{step2}$, $\eqref{step3}$ and $\eqref{step4}$,
it can be re-verified that
\begin{equation*}
\arraycolsep=1.5pt
\begin{array}{rl}
&\displaystyle \frac{1}{2}\frac{d}{dt}\left(\sum_{1\leq|\alpha|\leq
N}\int_{\mathbb{R}^3}(1+\sigma_{st}+\Phi(\sigma_{st}))
(|\partial^{\alpha}\bar{\sigma}|^2+|\partial^{\alpha}\bar{v}|^2)dx+\|\nabla[\bar{E},\bar{B}]\|_{N-1}^{2}\right)\\[5mm]
&\displaystyle+\frac{1}{\sqrt{\gamma}}\sum_{1\leq|\alpha|\leq
N}\int_{\mathbb{R}^3}(1+\sigma_{st}+\Phi(\sigma_{st}))|\partial^{\alpha}\bar{v}|^{2}dx\\[5mm]
\leq & \displaystyle
C(\|\bar{V}\|_{N}+\delta)(\|\nabla[\bar{\sigma},\bar{v}]\|_{N-1}^{2}
+\|\nabla \bar{E}\|_{N-2}^2),
\end{array}
\end{equation*}
\begin{eqnarray*}
\frac{d}{dt}\sum_{1\leq|\alpha|\leq N-1}\langle
\partial^{\alpha}\bar{v},\nabla\partial^{\alpha}\bar{\sigma}\rangle+\lambda\|\nabla\bar{\sigma}\|^{2}_{N-1}
\leq C\|\nabla^2\bar{v}\|_{N-2}^{2}+C(\|[\bar{\sigma}, \bar{v},\bar{B}]\|_{N}^{2}+\delta)
\|\nabla[\bar{\sigma},\bar{v}]\|_{N-1}^{2},
\end{eqnarray*}
\begin{eqnarray*}
\begin{aligned}
\dfrac{d}{dt}\sum_{1\leq|\alpha|\leq N-1}\langle
\partial^{\alpha}\bar{v},\partial^{\alpha}\bar{E}\rangle+\lambda\|\nabla\bar{E}\|^{2}_{N-2} \leq
C&\|\nabla[\bar{\sigma},\bar{v}]\|_{N-1}^{2}+C\|\nabla\bar{v}\|_{N-1}\|\nabla^2
\bar{B}\|_{N-3}\\
&+C(\|[\bar{\sigma},\bar{v},\bar{B}]\|_{N}^{2}+\delta)
\|\nabla[\bar{\sigma},\bar{v}]\|_{N-1}^{2},
\end{aligned}
\end{eqnarray*}
and
\begin{eqnarray*}
&&\begin{aligned}
& -\frac{d}{dt}\sum_{1\leq |\alpha|\leq N-2}\langle \nabla
\times\partial^{\alpha}\bar{E},\partial^{\alpha}\bar{B}\rangle+\lambda\|\nabla^{2}\bar{B}\|^{2}_{N-3}\\
\leq & C\|\nabla^2\bar{E}\|_{N-3}^{2}
+C\|\nabla\bar{v}\|_{N-3}^2+C(\|\bar{\sigma}\|_{N}^{2}+\delta)\|\nabla \bar{v}\|_{N-1}^{2}.
\end{aligned}
\end{eqnarray*}
Here, the details of proof are omitted for simplicity. Now, similar
to $\eqref{3.12}$, let us define
\begin{equation}\label{def.high}
\begin{aligned}
\mathcal{E}_{N}^{h}(\bar{V}(t))&=\sum_{1\leq|\alpha|\leq
N}\int_{\mathbb{R}^3}(1+\sigma_{st}+\Phi(\sigma_{st}))
(|\partial^{\alpha}\bar{\sigma}|^2+|\partial^{\alpha}\bar{v}|^2)dx+\|\nabla[\bar{E},\bar{B}]\|_{N-1}^{2}\\
&+\kappa_{1}\sum_{1\leq|\alpha|\leq N-1}\langle
\partial^{\alpha}\bar{v},\nabla\partial^{\alpha}\bar{\sigma}\rangle+\kappa_{2}\sum_{1\leq|\alpha|\leq N-1}\langle
\partial^{\alpha}\bar{v},\partial^{\alpha}\bar{E}\rangle\\[3mm]
&-\kappa_{3}\sum_{1\leq |\alpha|\leq N-2}\langle \nabla
\times\partial^{\alpha}\bar{E},\partial^{\alpha}\bar{B}\rangle.
\end{aligned}
\end{equation}
Similarly, one can choose $0<\kappa_{3}\ll\kappa_{2}\ll\kappa_{1}\ll
1$ with $\kappa_{2}^{3/2}\ll\kappa_{3}$ such that $\mathcal
{E}_{N}^{h}(\bar{V}(t))\sim \|\nabla \bar{V}(t)\|_{N-1}^{2}$ because
$\sigma_{st}+\Phi(\sigma_{st}) $ depends only on $ x$ sufficiently
small compared with $1$. Furthermore, the linear combination of
previously obtained four estimates with coefficients corresponding
to $\eqref{def.high}$ yields $\eqref{sec5.high}$ with $\mathcal
{D}_{N}^{h}(\cdot)$ defined in $\eqref{de.Dh}$. This completes the
proof of Lemma \ref{estimate2}.
\end{proof}
Now, we begin with the time-weighted estimate and iteration for the
Lyapunov inequality $\eqref{sec5.ENV0}$. Let $\ell \geq 0$.
Multiplying $\eqref{sec5.ENV0}$ by $(1+t)^{\ell}$ and taking
integration over $[0,t]$ give
\begin{eqnarray*}
\begin{aligned}
& (1+t)^{\ell}\mathcal {E}_{N}(\bar{V}(t))+\lambda
\int_{0}^{t}(1+s)^{\ell}\mathcal {D}_{N}(\bar{V}(s))d s \\
\leq & \mathcal {E}_{N}(\bar{V}_{0})+ \ell
\int_{0}^{t}(1+s)^{\ell-1}\mathcal {E}_{N}(\bar{V}(s))d s.
\end{aligned}
\end{eqnarray*}
Noticing
\begin{eqnarray*}
\mathcal {E}_{N}(\bar{V}(t))
\leq C (D_{N+1}(\bar{V}(t))+\|
\bar{B}\|^{2}),
\end{eqnarray*}
it follows that
\begin{eqnarray*}
\begin{aligned}
& (1+t)^{\ell}\mathcal {E}_{N}(\bar{V}(t))+\lambda
\int_{0}^{t}(1+s)^{\ell}\mathcal {D}_{N}(\bar{V}(s))d s \\
\leq & \mathcal {E}_{N}(\bar{V}_{0})+ C \ell
\int_{0}^{t}(1+s)^{\ell-1}\|
\bar{B}(s)\|^{2}d s+ C\ell\int_{0}^{t}(1+s)^{\ell-1}\mathcal {D}_{N+1}(\bar{V}(s))d
s.
\end{aligned}
\end{eqnarray*}
Similarly, it holds that
\begin{eqnarray*}
\begin{aligned}
& (1+t)^{\ell-1}\mathcal {E}_{N+1}(\bar{V}(t))+\lambda
\int_{0}^{t}(1+s)^{\ell-1}\mathcal {D}_{N+1}(\bar{V}(s))d s \\
\leq & \mathcal {E}_{N+1}(\bar{V}_{0})+ C (\ell-1)
\int_{0}^{t}(1+s)^{\ell-2}\|
\bar{B}(s)\|^{2}ds
+ C(\ell-1)\int_{0}^{t}(1+s)^{\ell-2}\mathcal {D}_{N+2}(\bar{V}(s))d s,
\end{aligned}
\end{eqnarray*}
and
\begin{eqnarray*}
\mathcal {E}_{N+2}(\bar{V}(t))+\lambda \int_{0}^{t}\mathcal
{D}_{N+2}(\bar{V}(s))d s \leq \mathcal {E}_{N+2}(\bar{V}_{0}).
\end{eqnarray*}
Then, for $1<\ell<2$, it follows by iterating the above estimates
that
\begin{eqnarray}\label{sec5.ED}
\begin{aligned}
& (1+t)^{\ell}\mathcal {E}_{N}(\bar{V}(t))+\lambda
\int_{0}^{t}(1+s)^{\ell}\mathcal {D}_{N}(\bar{V}(s))d s \\
\leq & C \mathcal {E}_{N+2}(\bar{V}_{0})+ C
\int_{0}^{t}(1+s)^{\ell-1}\|
\bar{B}(s)\|^{2}d s.
\end{aligned}
\end{eqnarray}
Similarly, for $2<\kappa<3$, the time-weighted estimate and
iteration for the Lyapunov inequality $\eqref{sec5.high}$ give
\begin{eqnarray*}
\begin{aligned}
& (1+t)^{\kappa}\mathcal {E}_{N}^h(\bar{V}(t))+\lambda
\int_{0}^{t}(1+s)^{\kappa}\mathcal {D}_{N}^h(\bar{V}(s))d s \\
\leq & C \mathcal {E}_{N+3}^h(\bar{V}_{0})+ C
\int_{0}^{t}(1+s)^{\kappa-1}\|
\nabla\bar{B}(s)\|^{2}d s.
\end{aligned}
\end{eqnarray*}
Here the smallness of $\|n_{b}-1\|_{W_{0}^{N+4,2}}$ has been used in
the process of iteration for the Lyapunov inequalities
$\eqref{sec5.ENV0}$ and $\eqref{sec5.high}$. Taking $\kappa=l+1$, it
holds that
\begin{multline}\label{sec5.EhD}
(1+t)^{l+1}\mathcal {E}_{N}^h(\bar{V}(t))+\lambda
\int_{0}^{t}(1+s)^{l+1}\mathcal {D}_{N}^h(\bar{V}(s))d s \\
\leq C \mathcal {E}_{N+3}^h(\bar{V}_{0})+ C
\int_{0}^{t}(1+s)^{l}\|
\nabla\bar{B}(s)\|^{2}d s\\
\leq C \mathcal {E}_{N+3}^h(\bar{V}_{0})+ C \int_{0}^{t}(1+s)^{\ell}\mathcal {D}_{N}(\bar{V}(s))d s.
\end{multline}
Combining $\eqref{sec5.ED}$ with $\eqref{sec5.EhD}$, we have
\begin{multline}\label{sec5.EDEhD}
(1+t)^{\ell}\mathcal {E}_{N}(\bar{V}(t))+
\int_{0}^{t}(1+s)^{\ell}\mathcal {D}_{N}(\bar{V}(s))d s\\
+(1+t)^{l+1}\mathcal
{E}_{N}^h(\bar{V}(t))+
\int_{0}^{t}(1+s)^{l+1}\mathcal {D}_{N}^h(\bar{V}(s))d s \\
\leq C \mathcal {E}_{N+3}(\bar{V}_{0})+ C \int_{0}^{t}(1+s)^{\ell-1}\|
\bar{B}(s)\|^{2}d s.
\end{multline}
For this time, to estimate the integral term on the r.h.s. of
$\eqref{sec5.EDEhD}$, let's define
\begin{eqnarray}\label{sec5.def}
\mathcal {E}_{N,\infty}(\bar{V}(t))=\sup\limits_{0\leq s \leq t} \
\left\{(1+s)^{\frac{3}{2}}\mathcal
{E}_{N}(\bar{V}(s))+(1+s)^{\frac{5}{2}}\mathcal
{E}_{N}^h(\bar{V}(s))\right\},
\end{eqnarray}
\begin{eqnarray}\label{sec5.defL}
L_{0}(t)=\sup\limits_{0\leq s \leq t}
(1+s)^{\frac{5}{2}}\|[\bar{\rho},\bar{u}]\|^{2}.
\end{eqnarray}
Then, we have the following
\begin{lemma}\label{lem.Bsigma}
For any $t\geq0$, it holds that:
\begin{eqnarray}\label{lem.tildeB}
&&\begin{aligned} \|\bar{B}(t)\|^2\leq C
(1+t)^{-\frac{3}{2}}\left(\|[\bar{\sigma}_{0},\bar{v}_{0}]\|^{2}+
\|[\bar{v}_{0},\right.& \bar{E}_{0},\bar{B}_{0}]\|^2_{L^1\cap \dot{H}^{2}}\\
&\left.+[\mathcal {E}_{N,\infty}(\bar{V}(t))]^2+\delta^2 \mathcal
{E}_{N,\infty}(\bar{V}(t))\right).
\end{aligned}
\end{eqnarray}
\end{lemma}
\begin{proof}
Applying the fourth linear estimate on $B$ in $\eqref{col.decay1}$
to the mild form \eqref{sec5.U} gives
\begin{eqnarray}\label{sec5.decayB}
&&\begin{aligned} \|B_{1}(t)\|\leq C (1+t)^{-\frac{3}{4}}
\|[\bar{u}_{0},& E_{1,0},B_{1,0}]\|_{L^1\cap \dot{H}^{2}}\\
&+C
\int_{0}^{t}(1+t-s)^{-\frac{3}{4}}\|[g_{2}(s),g_{3}(s)]\|_{L^{1}\cap\dot{H}^{2}}ds.
\end{aligned}
\end{eqnarray}
Applying the $L^{2}$ linear estimate on $u$ in $\eqref{col.decay1}$
to the mild form $\eqref{sec5.U}$,
\begin{eqnarray}\label{baruL2}
&&\begin{aligned}
\|\bar{u}(t)\| \leq C(1+t)^{-\frac{5}{4}}(
&\|\bar{\rho}_{0}\|+\|[\bar{u}_{0}, E_{1,0},B_{1,0}]\|_{L^{1}\cap\dot{H}^{2}})\\
&+C \int_{0}^{t}(1+t-s)^{-\frac{5}{4}}\left(\|g_{1}(s)\|+\|[g_{2}(s),g_{3}(s)]\|_{L^{1}\cap
\dot{H}^{2}}\right)ds.
\end{aligned}
\end{eqnarray}
Applying the $L^{2}$ linear estimate on $\rho$ in
$\eqref{col.decay1}$ to $\eqref{sec5.U}$, one has
\begin{eqnarray}\label{rhoL2}
\|\bar{\rho}(t)\|\leq C
e^{-\frac{t}{2}}\|[\bar{\rho}_{0},\bar{u}_{0}]\|+ C
\int_{0}^{t}e^{-\frac{t-s}{2}}\|[g_{1}(s),g_{2}(s)]\|d s.
\end{eqnarray}
Recall the definition $\eqref{sec5.ggg}$ of $g_{1}$, $g_{2}$ and
$g_{3}$,
\begin{eqnarray*}
\begin{aligned}
& g_{1}(s)=-\rho_{st} \nabla \cdot \bar{u}-\bar{\rho} \nabla \cdot
\bar{u}-\bar{u} \cdot \nabla \rho_{st}-\bar{u} \cdot \nabla
\bar{\rho},\\
& g_{2}(s)\sim \bar{u} \cdot \nabla \bar{u} + \bar{u}\times B_{1} +\bar{\rho}\nabla \bar{\rho}+ \rho_{st}\nabla \bar{\rho} +\bar{\rho}\nabla \rho_{st},\\
& g_{3}(s)=\bar{\rho} \bar{u}+\rho_{st} \bar{u}.
\end{aligned}
\end{eqnarray*}
Firstly, we estimate those terms including $\rho_{st}$. It follows that
\begin{eqnarray*}
\begin{aligned}
&\|\rho_{st}\nabla \cdot \bar{u}\|\leq
\|\rho_{st}\|_{L^{\infty}}\|\nabla \bar{u}\|,\ \ \ \ \ \|\bar{u}
\cdot \nabla \rho_{st}\|\leq
\|\nabla\rho_{st}\|\|\bar{u}\|_{L^{\infty}}\leq
\|\nabla\rho_{st}\|\|\nabla\bar{u}\|_{H^{1}},\\
&\|\rho_{st}\nabla \bar{\rho}\|_{L^1}\leq \|\rho_{st}\|\|\nabla
\bar{\rho}\|,\ \ \ \ \ \|\rho_{st}\nabla \bar{\rho}\|\leq
\|\rho_{st}\|_{L^{\infty}}\|\nabla\bar{\rho}\|\leq
\|\nabla\rho_{st}\|_{H^{1}}\|\nabla\bar{\rho}\|,\\
& \|\bar{\rho}
\nabla \rho_{st}\|_{L^1}\leq \|\nabla\rho_{st}\|\|\bar{\rho}\|,\ \ \
\ \|\nabla\rho_{st}\bar{\rho}\|\leq
\|\bar{\rho}\|_{L^{\infty}}\|\rho_{st}\|\leq
\|\rho_{st}\|\|\nabla\bar{\rho}\|_{H^{1}},\\
&\|\rho_{st} \bar{u}\|_{L^1} \leq \|\rho_{st}\|\|\bar{u}\|,\\
\end{aligned}
\end{eqnarray*}
and for $|\alpha|=2$, one has
\begin{eqnarray*}
&&\begin{aligned} \|\partial^{\alpha}(\rho_{st}\nabla
\bar{\rho})\|\leq & \|\rho_{st}\partial^{\alpha}\nabla\bar{\rho}\|
+\|\partial^{\alpha}(\rho_{st}\nabla
\bar{\rho})-\rho_{st}\partial^{\alpha}\nabla\bar{\rho}\|\\[3mm]
\leq &
\|\rho_{st}\|_{L^{\infty}}\|\partial^{\alpha}\nabla\bar{\rho}\|
+C\|\nabla\rho_{st}\|_{H^{|\alpha|-1}}\|\nabla\bar{\rho}\|_{L^{\infty}}+C\|\nabla
\rho_{st}\|_{L^{\infty}}\|\nabla \bar{\rho}\|_{H^{|\alpha|-1}}\\[3mm]
\leq & C\delta\|\nabla \bar{\rho}\|_{H^{2}},
\end{aligned}
\end{eqnarray*}
where we have used the estimate $\|\partial^{\alpha}(f g)-f
\partial^{\alpha}g\|\leq C\|\nabla f\|_{H^{k-1}}\|g\|_{L^{\infty}}+C\|\nabla f\|_{L^{\infty}}\|g\|_{H^{k-1}}$, for any $|\alpha|=k$.
Similarly, it holds that
\begin{eqnarray*}
\begin{aligned}
\|\partial^{\alpha}(\rho_{st} \bar{u})\|\leq & \|\bar{u}
\partial^{\alpha}\rho_{st}\| +\|\partial^{\alpha}(\rho_{st}\bar{u})-\bar{u} \partial^{\alpha}\rho_{st}\|
\leq C \delta \|\nabla \bar{u}\|_{H^{2}},
\end{aligned}
\end{eqnarray*}
\begin{eqnarray*}
\begin{aligned}
& \|\partial^{\alpha}(\bar{\rho} \nabla \rho_{st})\|\leq C \delta
\|\nabla \bar{\rho}\|_{H^{2}}.
\end{aligned}
\end{eqnarray*}
It is straightforward to verify that for any $0\leq s\leq t$,
\begin{eqnarray}\label{dec.g2g3}
\begin{aligned}
\|[g_{2}(s),g_{3}(s)]\|_{L^{1}}\leq & C \|\bar{u}\|\|\nabla \bar{u}\|+\|\bar{u}\|\|B_{1}\|+\|\bar{\rho}\|\|\bar{u}\|+\|\bar{\rho}\|\|\nabla \bar{\rho}\|\\
& +C(\|\rho_{st}\nabla \bar{\rho}\|_{L^1}+\|\rho_{st} \bar{u}\|_{L^1}+\|\bar{\rho} \nabla \rho_{st}\|_{L^1} )\\
&\leq C \mathcal {E}_{N}(\bar{U}(s))+ C \delta \sqrt{\mathcal
{E}_{N}^h(\bar{U}(s))}+C \delta \|[\bar{\rho},\bar{u}]\|,
\end{aligned}
\end{eqnarray}
\begin{eqnarray}\label{dec.g2g3H}
\begin{aligned}
\|[g_{2}(s),g_{3}(s)]\|_{\dot{H}^{2}}\leq & C\mathcal
{E}_{N}(\bar{U}(s))+ C \delta \sqrt{\mathcal {E}_{N}^h(\bar{U}(s))},
\end{aligned}
\end{eqnarray}
and
\begin{eqnarray}\label{dec.g1g2}
\begin{aligned}
\|[g_{1}(s),g_{2}(s)]\|\leq & C\mathcal {E}_{N}(\bar{U}(s))+ C
\delta \sqrt{\mathcal {E}_{N}^h(\bar{U}(s))}.
\end{aligned}
\end{eqnarray}
Notice that $ \mathcal {E}_{N}(\bar{U}(s))\leq C \mathcal
{E}_{N}(\bar{V}(\sqrt{\gamma}s))$. From $\eqref{sec5.def}$ and
$\eqref{sec5.defL}$, for any $0\leq s\leq t$,
\begin{eqnarray*}
\mathcal {E}_{N}(\bar{V}(\sqrt{\gamma}s))\leq
(1+\sqrt{\gamma}s)^{-\frac{3}{2}}\mathcal
{E}_{N,\infty}(\bar{V}(\sqrt{\gamma}t)),
\end{eqnarray*}
\begin{eqnarray*}
\mathcal {E}_{N}^h(\bar{V}(\sqrt{\gamma}s))\leq
(1+\sqrt{\gamma}s)^{-\frac{5}{2}}\mathcal
{E}_{N,\infty}(\bar{V}(\sqrt{\gamma}t)),
\end{eqnarray*}
\begin{eqnarray*}
\|[\bar{\rho},\bar{u}](s)\|\leq \sqrt{L_{0}(t)}(1+s)^{-\frac{5}{4}}.
\end{eqnarray*}
Then, it follows that for $0\leq s \leq t$,
\begin{multline*}
\|[g_{2}(s),g_{3}(s)]\|_{L^{1}}\leq
C(1+\sqrt{\gamma}s)^{-\frac{3}{2}}\mathcal
{E}_{N,\infty}(\bar{V}(\sqrt{\gamma}t))\\
+C \delta (1+\sqrt{\gamma}s)^{-\frac{5}{4}}\sqrt{\mathcal
{E}_{N,\infty}(\bar{V}(\sqrt{\gamma}t))}+C\delta
\sqrt{L_{0}(t)}(1+s)^{-\frac{5}{4}},
\end{multline*}
\begin{eqnarray*}
\begin{aligned}
\|[g_{2}(s),g_{3}(s)]\|_{\dot{H}^{2}}\leq &
C(1+\sqrt{\gamma}s)^{-\frac{3}{2}}\mathcal
{E}_{N,\infty}(V(\sqrt{\gamma}t))\\
&+C \delta (1+\sqrt{\gamma}s)^{-\frac{5}{4}}\sqrt{\mathcal
{E}_{N,\infty}(\bar{V}(\sqrt{\gamma}t))},
\end{aligned}
\end{eqnarray*}
\begin{eqnarray*}
\begin{aligned}
\|[g_{1}(s),g_{2}(s)]\|\leq &
C(1+\sqrt{\gamma}s)^{-\frac{3}{2}}\mathcal
{E}_{N,\infty}(V(\sqrt{\gamma}t))\\
&+C \delta (1+\sqrt{\gamma}s)^{-\frac{5}{4}}\sqrt{\mathcal
{E}_{N,\infty}(\bar{V}(\sqrt{\gamma}t))}.
\end{aligned}
\end{eqnarray*}
Putting the above inequalities into \eqref{sec5.decayB},
$\eqref{baruL2}$ and \eqref{rhoL2} respectively gives
\begin{multline}\label{sec5.decayB1}
\|B_{1}(t)\|\leq C (1+t)^{-\frac{3}{4}}\Big\{
\|[\bar{u}_{0}, E_{1,0},B_{1,0}]\|_{L^1\cap \dot{H}^{2}}+\mathcal
{E}_{N,\infty}(\bar{V}(\sqrt{\gamma}t))\\
+\delta\sqrt{L_{0}(t)}+\delta \sqrt{\mathcal
{E}_{N,\infty}(\bar{V}(\sqrt{\gamma}t))}\Big\},
\end{multline}
\begin{multline}\label{sec5.decayu}
\|\bar{u}(t)\|\leq C (1+t)^{-\frac{5}{4}}\Big\{
\|\bar{\rho}_{0}\|+\|[\bar{u}_{0}, E_{1,0},B_{1,0}]\|_{L^{1}\cap\dot{H}^{2}}+\mathcal
{E}_{N,\infty}(\bar{V}(\sqrt{\gamma}t))\\
+\delta\sqrt{L_{0}(t)}+\delta \sqrt{\mathcal
{E}_{N,\infty}(\bar{V}(\sqrt{\gamma}t))}\Big\},
\end{multline}
\begin{eqnarray}\label{sec5.decayrho}
\begin{aligned}
\|\bar{\rho}(t)\|\leq C (1+t)^{-\frac{5}{4}}\Big\{
\|[\bar{\rho}_{0},u_{0}]\|+\mathcal
{E}_{N,\infty}(\bar{V}&(\sqrt{\gamma}t))\\
&+\delta \sqrt{\mathcal
{E}_{N,\infty}(\bar{V}(\sqrt{\gamma}t))}\Big\}.
\end{aligned}
\end{eqnarray}
The definition of $L_{0}(t)$, \eqref{sec5.decayu} and
\eqref{sec5.decayrho} further imply that
\begin{multline}\label{bou.L}
L_{0}(t)\leq C\|[\bar{\rho}_{0},u_{0}]\|^{2}+C\|[\bar{u}_{0},
E_{1,0},B_{1,0}]\|_{L^{1}\cap\dot{H}^{2}}^{2}\\
+C\left[\mathcal
{E}_{N,\infty}(\bar{V}(\sqrt{\gamma}t))\right]^{2}+C\delta^{2}\mathcal
{E}_{N,\infty}(\bar{V}(\sqrt{\gamma}t)),
\end{multline}
where we have used that $\delta$ is small enough. Plugging the above
estimate into \eqref{sec5.decayB1} implies $\eqref{lem.tildeB}$,
since $\|\bar{ B}(t)\|\leq C \| B_{1}(t/\sqrt{\gamma})\|$ and
$[\bar{\rho},\bar{u},E_{1},B_{1}]$ is equivalent with
$[\bar{\sigma},\bar{v},\bar{E},\bar{B}]$ up to a positive constant.
This completes the proof of Lemma $\ref{lem.Bsigma}$.
\end{proof}
Now, the rest is to prove the uniform-in-time bound of $\mathcal
{E}_{N,\infty}(\bar{V}(t))$ which yields the time-decay rates of the
Lyapunov functionals $\mathcal
{E}_{N}(\bar{V}(t))$ and $\mathcal
{E}_{N}^h(\bar{V}(t))$ thus $\|\bar{V}(t)\|_{N}^{2}$, $\|\nabla\bar{V}(t)\|_{N-1}^{2}$.
In fact, by taking $\ell =\frac{3}{2}+\epsilon$ in
$\eqref{sec5.EDEhD}$ with $\epsilon>0$ small enough, one has
\begin{multline*}
(1+t)^{\frac{3}{2}+\epsilon}\mathcal {E}_{N}(\bar{V}(t))+
\int_{0}^{t}(1+s)^{\frac{3}{2}+\epsilon}\mathcal {D}_{N}(\bar{V}(s))d s\\
+(1+t)^{\frac{5}{2}+\epsilon}\mathcal
{E}_{N}^h(\bar{V}(t))+
\int_{0}^{t}(1+s)^{\frac{5}{2}+\epsilon}\mathcal {D}_{N}^h(\bar{V}(s))d s \\
\leq C \mathcal {E}_{N+3}(\bar{V}_{0})+ C \int_{0}^{t}(1+s)^{\frac{1}{2}+\epsilon}\|
\bar{B}(s)\|^{2}d s.
\end{multline*}
Here, using $\eqref{lem.tildeB}$ and the fact $\mathcal
{E}_{N,\infty}(\bar{V}(t))$ is non-decreasing in $t$, it further holds
that
\begin{eqnarray*}
\begin{aligned}
\int_{0}^{t}(1+s)^{\frac{1}{2}+\epsilon}\|
\bar{B}(s)\|^{2}d s\leq
C(1+t)^{\epsilon}\Big\{\|[\bar{\sigma}_{0},&\bar{v}_{0}]\|^{2}+
\|[\bar{v}_{0},\bar{E}_{0},\bar{B}_{0}]\|^2_{L^1\cap \dot{H}^{2}}\\
&+[\mathcal
{E}_{N,\infty}(\bar{V}(t))]^2 +\delta ^2 \mathcal
{E}_{N,\infty}(\bar{V}(t))\Big\}.
\end{aligned}
\end{eqnarray*}
Therefore, it follows that
\begin{multline*}
(1+t)^{\frac{3}{2}+\epsilon}\mathcal
{E}_{N}(\bar{V}(t))+(1+t)^{\frac{5}{2}+\epsilon}\mathcal
{E}_{N}^h(\bar{V}(t))\\
+ \int_{0}^{t}(1+s)^{\frac{3}{2}+\epsilon}\mathcal
{D}_{N}(\bar{V}(s))d s
+ \int_{0}^{t}(1+s)^{\frac{5}{2}+\epsilon}\mathcal {D}_{N}^h(\bar{V}(s))d s \\
\leq C \mathcal {E}_{N+3}(\bar{V}_{0})+ C (1+t)^{\epsilon}\left(\|[\bar{\sigma}_{0},\bar{v}_{0}]\|^{2}+
\|[\bar{v}_{0},\bar{E}_{0},\bar{B}_{0}]\|^2_{L^1\cap \dot{H}^{2}}\right.\\
\left.+[\mathcal
{E}_{N,\infty}(\bar{V}(t))]^2 +\delta^2 \mathcal
{E}_{N,\infty}(\bar{V}(t))\right),
\end{multline*}
which implies
\begin{multline*}
(1+t)^{\frac{3}{2}}\mathcal
{E}_{N}(\bar{V}(t))+(1+t)^{\frac{5}{2}}\mathcal
{E}_{N}^h(\bar{V}(t))
\leq C \Big\{ \mathcal {E}_{N+3}(\bar{V}_{0})+
\|[\bar{v}_{0},\bar{E}_{0},\bar{B}_{0}]\|^2_{L^1}\\
+[\mathcal
{E}_{N,\infty}(\bar{V}(t))]^2 +\delta ^2 \mathcal
{E}_{N,\infty}(\bar{V}(t))\Big\},
\end{multline*}
and thus
\begin{eqnarray}\label{ENb}
\mathcal {E}_{N,\infty}(\bar{V}(t))
\leq C \left( \epsilon_{N+3}(\bar{V}_{0})^{2}+
\mathcal {E}_{N,\infty}(\bar{V}(t))^{2}\right).
\end{eqnarray}
Here, we have used that $\delta$ is small enough. Recall the
definition of $\epsilon_{N+3}(\bar{V}_{0})$, since
$\epsilon_{N+3}(\bar{V}_{0})>0$ is sufficiently small, $\mathcal
{E}_{N,\infty}(\bar{V}(t)) \leq C \epsilon_{N+3}(\bar{V}_{0})^{2}$
holds true for any $t\geq 0$, which implies
\begin{eqnarray}\label{UN}
\|\bar{V}(t)\|_{N} \leq C \mathcal {E}_{N}(\bar{V}(t))^{1/2}
\leq C \epsilon_{N+3}(\bar{V}_{0})(1+t)^{-\frac{3}{4}},
\end{eqnarray}
\begin{eqnarray}\label{nablaUN}
\|\nabla\bar{V}(t)\|_{N-1} \leq C \mathcal
{E}_{N}^{h}(\bar{V}(t))^{1/2}
\leq C \epsilon_{N+3}(\bar{V}_{0})(1+t)^{-\frac{5}{4}}.
\end{eqnarray}
The definition of $L_{0}(t)$, the uniform-in-time bound of $\mathcal
{E}_{N,\infty}(\bar{V}(t))$ and $\eqref{bou.L}$ show that
\begin{eqnarray*}
\|[\bar{\rho},\bar{u}](t)\|
\leq C \epsilon_{N+3}(\bar{V}_{0})(1+t)^{-\frac{5}{4}}.
\end{eqnarray*}
In addition, applying the $L^{2}$ linear estimate on $E$ in
$\eqref{col.decay1}$ to the mild form $\eqref{sec5.U}$,
\begin{eqnarray*}
&&\begin{aligned} \|E_{1}(t)\|\leq C (1+t)^{-\frac{5}{4}}
\|[\bar{u}_{0},& E_{1,0},B_{1,0}]\|_{L^1\cap \dot{H}^{3}}\\
&+C \int_{0}^{t}(1+t-s)^{-\frac{5}{4}}\|[g_{2}(s),g_{3}(s)]\|_{L^{1}\cap
\dot{H}^{3}}ds.
\end{aligned}
\end{eqnarray*}
Since by $\eqref{UN}$ and $\eqref{nablaUN}$, similar to obtaining
$\eqref{dec.g2g3}$ and $\eqref{dec.g2g3H}$, we have
\begin{eqnarray*}
\begin{aligned}
&\|[g_{2}(s),g_{3}(s)]\|_{L^{1}\cap \dot{H}^{3}}\leq
C\|\bar{U}(t)\|^{2}_{4}+ C\delta\|\nabla
\bar{U}(t)\|_{3}+C\delta\|[\bar{\rho},\bar{u}]\|\leq
C\epsilon_{7}(\bar{V}_{0})(1+t)^{-\frac{5}{4}},
\end{aligned}
\end{eqnarray*}
it follows that
\begin{eqnarray}\label{uL2}
\|E_{1}(t)\| \leq C\epsilon_{7}(\bar{V}_{0})(1+t)^{-\frac{5}{4}}.
\end{eqnarray}
This completes Theorem \ref{Corolary}.
\medskip
\medskip
\noindent{\bf Acknowledgements:}\ \ The first author Qingqing Liu
would like to thank Dr. Renjun Duan for his guidance and continuous
help. The research was supported by the National Natural Science
Foundation of China $\#$11071093, the PhD specialized grant of the
Ministry of Education of China $\#$20100144110001, and the Special
Fund for Basic Scientific Research of Central Colleges
$\#$CCNU10C01001, $\#$CCNU12C01001.
\bigbreak | 93,989 |
TITLE: Probability of an average of naturals being also natural
QUESTION [3 upvotes]: Probably not much of a hard question, but still annoys me that I can't find an easy answer.
Let's imagine we have an audience of n people and each can give a
score from 0 to m (equally probable) and that score is an integer number. What is
the probability that the average score for these people will be also
integer?
I did try some basic calculations but did not progress too much here so I got nothing to "show". And if anything, this question is more of an .. academic interest to me. Perhaps there is a solution known already?
REPLY [1 votes]: I don't know if an easy closed formula exists generally.
If you want to calculate the probabilites numerically, note that it doesn't matter what score each person gives, it only matters in which reminder class$\pmod n$ the score lies. So each player chooses a reminder class$\pmod n$ according to a probability distribution $(p_0, p_1,\ldots,p_{n-1})$ (where $p_0$ is the probability that a score is from reminder class $0$ is chosen, a.s.o).
Since the scores where equally probable, with $m+1=qn+r, 0\le r < n$ we get for $i=0,1,\ldots,n-1$: $p_i=\frac{q+1}{m+1}$ if $i<r$ and $p_i=\frac{q}{m+1}$ otherwise.
Note that for $r=0$ this comes out as a nice symmetrical $p_i=\frac1n$ for all $i$.
So given 2 distributions $s_1,s_2$ of reminder classes, what is the distribution of the sum of 2 random variables $x_1, x_2$ distributed according to $s_1$ and $s_2$, resp?
That's easy to calculate:
$$P(x_1+x_2 \equiv i\pmod n) = \sum_{k=0}^{n-1}s_1[k]s_2[i-k],$$
where the index $i-k$ is taken$\pmod n$.
Given this formula, one can calculate the distribution$\pmod n$ of the sum of the scores from 2 persons, then use the formula again to add a third person, a.s.o. After all $n$ persons have been added, the probability that the sum is in reminder class $0$ is the answer to the question.
In the cases where the distribution is uniformly over all reminder classes, the formula above shows that the sum is also uniformly distributed over the reminder classes. That means adding more and more persons will not change that and the final answer is $p=\frac1n$.
It also shows that for fixed $n$ that $\lim_{m \to \infty} p = \frac1n$, because we have a fixed set of continuous calculations to do to find the answer, and the initial probability distribution goes to $(\frac1n,\ldots,\frac1n)$. In other words, if you have $n=10$ but allow scores up to $m=1000$, it doesn't matter much that the probability to choose a number of reminder class $0$ is $\frac{101}{1001}$ and not $\frac1{10}$. | 207,992 |
Four weeks of wall-to-wall football in Glorious Technicolor make the World Cup an unbeatable time to bury bad news, but that is an accusation which cannot be levelled at the new Conservative-Liberal Democrat government, which has hit the ground cutting. Every day it announces the scrapping of another planned investment because of the financial emergency it insists we are in, and sport, for all the health and social benefits the coalition acknowledges it brings, is a long way from spared.
While £9.375bn of public spending is still largely ring-fenced to build the venues and run the four weeks of London's Olympic and Paralympic Games in 2012, Sebastian Coe's pledge, always shaky, that our Olympics will inspire a new generation to take up sport, is jeopardised further by the massive ongoing cuts.
Hugh Robertson, the sports minister, earned a reputation in his six years as shadow sports secretary for diligence and genuineness in his desire to improve our nation's sporting ill-health. Yet, after just two months in office, he has already scrapped free swimming for under-16s and over-60s, cancelled an associated £25m swimming pool refurbishment programme, and committed to implementing the 25% cuts required across departments by George Osborne's Treasury.
A change to distribution of Lottery money will see £50m more come to sport annually from 2012, which Robertson hopes will compensate for the savings his department will make, but he accepts that the scale of cuts, particularly by local authorities, will undermine the provision of sports.
"This is a very difficult time and sport will take a major hit across the country," the minister acknowledges. "Everything is overshadowed by the state of the economy and the budget deficit we inherited. I am doing everything in my power to mitigate it, and [to] protect sport funding."
All agree that the public finances must be reordered, in a recession caused not by Labour's supposed extravagance – spending money on schools, swimming pools and other public services – but by the banks' thunderous negligence. However the new government has made a policy choice that the deficit must be overturned in five years and 78% must be clawed back in cuts, just 22% in increased tax.
Sport, as ever, is a perfect barometer for the effect on the nation. The top income-tax rate of 50% kicks in at earnings above £150,000; there is no higher rate for super-earners such as footballers paid as much as £10m a year, their agents, Premier League chief executives on more than £1m, or the chairman Sir Dave Richards on his £350,000.
Yet sports facilities and schemes, often in the poorest areas, will suffer a thousand small cuts – in a nation where only around a third of people do the Department of Health's recommended daily exercise – half an hour for adults, an hour for under-16s.
The Building Schools for the Future programme, cancelled by the education secretary Michael Gove, earmarked an estimated 11% of its £55bn total budget, £6.05bn, for modern sports facilities to replace clapped-out old gyms. Most schools would have been required to make their new sports halls and Astroturf pitches available for community use in the evenings and at weekends. That has gone now, although Gove promises to review the needs of the 700 or so schools whose rebuilds have been scrapped.
Most public sports facilities are still maintained, subsidised and staffed by local authorities, which spent around £1bn doing so last year even though they are still not required by law to provide the option of sport.
As in the 80s, sport is certain to be a victim again, as councils must reduce their overall spending by £1.2bn this year, and by 25% by 2015. Our facilities have still not recovered from the underinvestment the last time the Conservatives were in power – a 2003 report by Davis Langdon Consulting found £550m required to bring the nation's 1,718 sports centres and swimming pools up to a reasonable standard, and little of that work has been done – but cuts will fall on them again.
"Sport is a discretionary service so is very vulnerable," says Simon Henig, Durham county council's leader and spokesman for sport and culture in the Local Government Association's Labour Group. "These cuts are extremely worrying, that we will take a great leap backwards, and all the efforts to encourage people to be more active, which were starting to work with joined-up thinking, will go. It is being done in such haste."
When, in June, Robertson announced the scrapping of the free swimming programme – "not a decision that gives me any pleasure" – he said "new research" had shown the scheme "has not delivered value for money." He concluded: "With a crippling deficit to tackle, this has become a luxury we can no longer afford."
Yet that research, an evaluation of the scheme's first year carried out by the consultant PriceWaterhouseCoopers, did not say the initiative had not delivered value for money. In fact, its report said: "In economic terms the free swimming programme has been relatively successful."
The researchers found that across the country senior citizens and young people went swimming fully seven million more times because of the government's subsidy – £40m from five departments – which made it free. The vast majority of those additional swims, 5.5 million, were made by under-16s, who by definition did not swim previously because of the cost.
That helped 32.9% of them do the recommended hour's exercise every day, up from 20.7%. Among the over-60s, 78.4% were doing their recommended half an hour a day, 12% up.
The government homed in on the finding that 11 million swims were "deadweight", taken free by people who previously paid for them, and the money spent on the scheme was not yet being repaid in long‑term health.
Yet the researchers concluded that, after just one year: ".
That gave free swimming the potential to repay, in improved health, every penny invested – and so justify the previous government's ambition that it would create one fitting legacy of the 2012 Olympics. Instead, the scheme has been scrapped, and the £25m earmarked to upgrade pools withdrawn.
Robertson maintains that despite cuts being his government's priority, he will argue to keep his budget, for the benefits sport brings.
The Lottery change will bring in extra money, and he is determined that refocusing the 2012 pledge, which he will announce shortly, will lead Sport England to try to use the Olympics more effectively to increase participation.
Jeremy Hunt, minister for culture, media and sport, has said the £9.375bn Olympic budget is "not sacrosanct", and a £27m cut was imposed on the Olympic Delivery Authority but, in truth, the vast bulk of that budget is guaranteed.
The 2012 Olympics will be London's, and Britain's, advert to the global village, demonstrating to the mass television audience and those in the new 80,000-seat stadium, for which there is still no viable after-use, a carnival of British can-do.
The government will not want the world to see the hangdog pools which will not now be refurbished, the schools we cannot rebuild, or that we have begun charging children from poorer families £2 apiece to go swimming once again because allowing them in for free was "a luxury we can no longer afford".
• This article was amended on 16 July 2010. The original said Simon Henig was Durham city council's leader. This has been corrected. | 391,765 |
More people than ever are opting to work in a business for themselves and kick the traditional career path. The pandemic and fallout also kicked up the desire to work from home, inspiring even more self-employment.
Welcome to the wonderful world of business, where everything is roses and sunshine! Or is it?
I think a lot of people have a misconception about working for themselves. It feels like you just get to do whatever you like, and money falls in your lap. But not so fast. There are some things in having a business where you don’t have a choice yet are essential to your success.
After spending more than half my life as a company owner, I promise you there’s actual work to do as a business owner. Probably a lot more than you thought.
Look, I want you to succeed in your endeavor, and I know you can. So, in this post, I’m sharing a few things to help you do better business and grow faster.
I’ve learned this stuff over many years, and sometimes the hard way. But, I promise, this list will serve you well!
And maybe, keep you grounded and realistic about being a business owner.
For me, owning a company is one of the best decisions of my life. But not everyone is cut out for it.Look, I want you to succeed in your business, and I know you can. So, in this post, I’m sharing a few things to help you do better business and grow faster. #smallbusiness Click To Tweet
Your Business and Brand
For starters, you are your business and your brand. The actions you take and the way you conduct business reflect on who you are as a brand. Ultimately, customer relationships and trust in you and your services stem from how you operate as a business.
Realizing your actions and abilities do reflect your brand is a reminder of aspiring to a positive leadership role. You want a shiny brand image, don’t you? It’s good to earn trust and respect for your company, right?
One way to accomplish this is to provide impeccable service to your customers as your first priority. It doesn’t matter whether you provide a product or service; what matters is the satisfaction and delight clients feel.
If your product or service makes people happy, then they will appreciate you as a business. In return, let customers know you appreciate them as well. If you remember to say thank you in business, it goes a long way.
No matter how fantastic you are at marketing your company, or writing copy, or whatever, there first has to be value in what you offer. You can’t sell a sub-par product, nor should you want to—the intrinsic value of what you sell matters.
Keep improving.
For me, I always strive to be “the best in the business,” at whatever I’m doing. Setting your standards high for yourself, your services, and your products serves your company well. It also sets an excellent example for your team as you grow.If you remember to say thank you in business, it goes a long way. #smallbusinesstips Click To Tweet
Say What You Mean
People learn to trust you when you keep your word. The biggest example of this in business is to meet or exceed expectations consistently. No less than above and beyond is good enough.
This means you need to make deadlines, be on time or early for meetings, and show up for whatever you agree to do. You need to show you are reliable. And, when you are reliable, you gain repeat business as clients start to count on you.
The truth is, there’s no late in business, and if you adopt that mindset from the start, you’re already ahead. Also, this is another value for your company to model for crew members.
Set the standards and walk the talk!
It’s good business to say “yes” to customer requests, but it’s not worth it to promise beyond your capabilities. Transparency and honesty are the easiest way to handle such situations.
One possible solution is to build networks with related businesses and service providers to have a subcontracting, rental, or freelance system to collaborate with and share business.
Learn how other company owners handle things. Network with other owners where you are comfortable to ask questions or get opinions. Joining your local Chamber of Commerce may prove helpful for meeting business owners locally.
Finding out how others do things helps you find your way. Eventually, your own best practices develop from a mix of what you discover, test, or learn.
Take it all in, but don’t be afraid to be one-of-a-kind. As you grow, your voice emerges, and you gain confidence in a model that suits you uniquely.Learn how other business owners handle things. Network with other owners where you are comfortable to ask questions or get opinions. via @SueAnnBubacz Click To Tweet
Stay the Course
Business building is a roller coaster. There are always ups and downs. Screams and bends. But don’t fall off.
Great businesses rarely emerge overnight. You need to have some business grit to make it to the finish—or don’t even get on the ride!
One thing I learned early on is not to evaluate results day-to-day. Take a more comprehensive view for a more accurate picture like weekly, month-to-month, or quarterly, even.
Watching sales too closely makes you crazy and distorts your ability to do your best work in some cases. This is especially true when you’re just starting.
Every step forward is progress, and moving forward with the many parts of your business takes time.
Therefore, concentrate on reaching the big picture and take action steps to get there. Don’t get hung up on day-to-day sales! Seriously.
In fact, you’re much better off paying attention to customer service, operations, and product improvement. But, don’t worry, cash flow will follow.Therefore, concentrate on reaching the big picture and take action steps to get there. Don’t get hung up on day-to-day sales! #smallbusinessowner Click To Tweet
The Success of Your Business
I notice how often success comes your way when you are merely working away at your business. Much more so than when you are preoccupied with earnings. Customer relationships, customer care, and customer service are the best places to emphasize for your company rather than focusing on the bottom line.
Sure, if you aren’t making money, you aren’t a business. But, emphasizing only making money in your everyday work, more so than doing good business, is often a huge mistake. Honing your business skills, perfecting products and services, and giving your clients the attention they need and deserve gets you sales.
Plan and control taking your business higher!
2 thoughts on “There are Some Things in Business Where You Don’t Have a Choice”
Hello Sue-Ann,
Exceptional customer service is the best way to make yourself standout. In a good way 🙂 And, it usually isn’t that hard. As for staying the course, that sometimes takes a whole lot of effort.
SharlaAnn
Hi SharlaAnn, I’d agree with you about exceptional service to make you and or your business stand out today. Thanks for coming by on this one and have a great weekend SharlaAnn. | 102,446 |
PHOTO: electoral politics.
In Charlotte, several campaign signs in front of polling locations were covered in hammer and sickles as well as revolutionary slogans supporting the election boycott such as “Don’t Vote! Fight for Revolution!” Incendiary also received a photograph of graffiti in Charlotte reading, “Elections No! Revolution Yes!”
In Austin, there has been a wide campaign against roadside election signs, with many being targeted with graffiti and others slashed in half or otherwise destroyed. Elsewhere in the city, Heidi Sloan, member of the Democratic Socialists of America (DSA) running for the Democratic nomination for US House of Representatives, was targeted with dozens of her own campaign signs, covered in hammer and sickles and large X’s, placed in her yard.
Super Tuesday is the largest single day for primary elections, with 14 states holding primaries or caucuses to nominate each party’s respective candidate for the 2020 presidential election. Today is extremely important for the two frontrunners in the tightening race between former Vice-President Joe Biden and Vermont Senator Bernie Sanders for the Democratic nomination, as approximately one-third of the total delegates are up for grabs.
The Democratic Party itself is going through an identity crisis, with two major factions battling each other over the best way to prop up US imperialism and bring it into a new generation as it faces continued and deepening crisis.
The party elite have closed ranks around Biden over the past few days, with several former candidates including Robert Francis ‘Beto’ O’Rourke, Amy Klobuchar, and Pete Buttigieg endorsing him last night at a campaign rally in Dallas. Candidates that had previously railed against Biden in numerous debates are now cozying up to him in a bid to secure their own futures within the party. Biden represents the longstanding Democratic establishment, who seek to fortify support for US imperialism and the reactionary state with watered-down ‘inclusivity’ rhetoric, relatively meager government spending in comparison to the total expenditures of the US state, and partnerships with private capital.
On the other hand, the so-called ‘progressive’ wing of the party represented by Sanders have sought to co-opt vaguely revolutionary slogans and the name of Socialism to restore the masses’ dwindling faith in capitalism and bourgeois elections. While his supporters often paint themselves as enemies of the status-quo and the richest ‘one percent,’ their movement is is little more than a facelift for the dying Democrat Party, importing policies that would be considered moderate to even European Social Democrats (themselves bourgeois agents) and rebranding it as ‘Democratic Socialism.’ | 376,927 |
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\begin{document}
\thispagestyle{empty}
\begin{center}
{ \LARGE\bf The convergence of the sums of independent random variables under the sub-linear expectations $^{\ast}$}
\end{center}
\begin{center} {\sc
Li-Xin Zhang\footnote{This work was supported by grants from the NSF of China (Grant No. 11731012), the 973 Program (Grant No. 2015CB352302), Zhejiang Provincial Natural Science Foundation (Grant No. LY17A010016) and the Fundamental Research Funds for the Central Universities.
}
}\\
{\sl \small School of Mathematical Sciences, Zhejiang University, Hangzhou 310027} \\
(Email:[email protected])\\
\end{center}
\renewcommand{\abstractname}{~}
\begin{abstract}
\centerline{\bf Abstract}
Let $\{X_n;n\ge 1\}$ be a sequence of independent random variables on a probability space $(\Omega, \mathcal{F}, \pr)$ and $S_n=\sum_{k=1}^n X_k$. It is well-known that the almost sure convergence, the convergence in probability and the convergence in distribution of $S_n$ are equivalent. In this paper, we prove similar results for the independent random variables under the sub-linear expectations, and give a group of sufficient and necessary conditions for these convergence. For proving the results, the Levy and Kolmogorov maximal inequalities for independent random variables under the sub-linear expectation are established. As an application of the maximal inequalities, the sufficient and necessary conditions for the central limit theorem of independent and identically distributed random variables are also obtained.
{\bf Keywords:} sub-linear expectation; capacity; independence; Levy maximal inequality; central limit theorem.
{\bf AMS 2010 subject classifications:} 60F15; 60F05
\end{abstract}
\baselineskip 22pt
\renewcommand{\baselinestretch}{1.7}
\section{ Introduction and main results}\label{sect1}
\setcounter{equation}{0}
The convergence of the sums of independent random variables are well-studied. For example, it is well-known that, if $\{X_n;n\ge 1\}$ is a sequence of independent random variables on a probability space $(\Omega, \mathcal{F}, \pr)$, then that the infinite series $\sum_{n=1}^{\infty}X_n$ is convergent almost surely, that it is convergent in probability and that it is convergent in distribution are equivalent. In this paper, we consider this elementary equivalence under the sub-linear expectations.
The general framework of the sub-linear expectation is introduced by Peng \cite{PengG-Expectation06,peng2008a,peng2009survey} in a general function space by relaxing the linear property of the classical linear expectation
to the sub-additivity
and positive homogeneity (cf. Definition~\ref{def1.1} below).
The sub-linear expectation does not depend on the probability measure, provides
a very flexible framework to model distribution uncertainty problems and produces many interesting properties different from those of the linear expectations. Under Peng's framework, many limit theorems have been being gradually established recently, including the central limit theorem and weak law of large numbers (cf. Peng \cite{peng2008a,peng2010}), the small derivation and Chung's law of the iterated logarithm (cf. Zhang \cite{Zhang Donsker}), the strong law of large numbers (cf. Chen \cite{chen2016strong}, Chen et al \cite{Chen Z 2013}, Hu \cite{Hu C 2018}, Zhang \cite{Zhang Rosenthal}, Zhang and Lin \cite{ZhangLin}), and the law of the iterated logarithm (cf. Chen \cite{Chen2014LIL}, Zhang \cite{Zhang Exponential}).
For the convergence of the infinite series $\sum_{n=1}^{\infty}X_n$, Xu and Zhang \cite{XuZhang2018} gave sufficient conditions of the almost sure convergence for independent random variables under the sub-linear expectation via a three-series theorem, recently. In this paper, we will consider the necessity of these conditions and the equivalence of the almost sure convergence, the convergence in capacity and the convergence in distribution. In the classical probability space, the Levy maximal inequalities are basic to the study of the almost sure behavior of sums of independent random variables and a key to show that the convergence in probability of $\sum_{n=1}^{\infty} X_n$ implies its almost sure convergence. We will establish Levy type inequalities under the sub-linear expectation. For showing that the convergence in distribution of $\sum_{n=1}^{\infty} X_n$ implies its convergence in probability, the characteristic function is a basic tool. But, under the sub-linear expectation, there is no such tools. We will find a new way to show a similar implication under the sub-linear expectations basing on a Komlogorov type maximal inequality.
As for the central limit theorem, it is well-known that the finite variances and mean zeros are sufficient and necessary for $\frac{\sum_{k=1}^n X_k}{\sqrt{n}}$ to converge in distribution to a normal random variable if $\{X_n;n\ge 1\}$ is a sequence of independent and identically distributed random variables on a classical probability space $(\Omega, \mathcal{F}, \pr)$. Under the sub-linear expectation, Peng \cite{peng2008a, peng2010} proved the cental limit theorem under the finite $(2+\alpha)$-th moment. By applying a moment inequality and the truncation method, Zhang \cite{Zhang Exponential} and Lin and Zhang \cite{LinZhang2017} showed that the moment condition can be weakened to the finite second moment. A nature question is whether the finite second moment is necessary. In this paper, by applying the maximal inequalities, we will obtain the sufficient and necessary conditions for the central limit theorem.
In the remainder of the section, we state some natation. In the next section, we will establish the maximal inequalities for random variables under the sub-linear expectation. The results on the convergence of the infinite series of random variables will given in Section \ref{Sect Convergece}. The sufficient and necessary conditions for the central limit theorem are given in Section \ref{Sect CLT}.
We use the framework and notations of Peng \cite{peng2008a}. Let $(\Omega,\mathcal F)$
be a given measurable space and let $\mathscr{H}$ be a linear space of real functions
defined on $(\Omega,\mathcal F)$ such that if $X_1,\ldots, X_n \in \mathscr{H}$ then $\varphi(X_1,\ldots,X_n)\in \mathscr{H}$ for each
$\varphi\in C_{l,Lip}(\mathbb R_n)$, where $C_{l,Lip}(\mathbb R_n)$ denotes the linear space of local Lipschitz
functions $\varphi$ satisfying
\begin{eqnarray*} & |\varphi(\bm x) - \varphi(\bm y)| \le C(1 + |\bm x|^m + |\bm y|^m)|\bm x- \bm y|, \;\; \forall \bm x, \bm y \in \mathbb R_n,&\\
& \text {for some } C > 0, m \in \mathbb N \text{ depending on } \varphi. &
\end{eqnarray*}
$\mathscr{H}$ is considered as a space of ``random variables''. In this case we denote $X\in \mathscr{H}$. In the paper, we also denote $C_{b,Lip}(\mathbb R_n)$ the space of bounded Lipschitz
functions, $C_b(\mathbb R_n)$ the space of bounded continuous functions, and $C_b^{1}(\mathbb R_n)$ the space of bounded continuous functions with bounded continuous derivations on $\mathbb R_n$.
\begin{definition}\label{def1.1} A sub-linear expectation $\Sbep$ on $\mathscr{H}$ is a function $\Sbep: \mathscr{H}\to \overline{\mathbb R}$ satisfying the following properties: for all $X, Y \in \mathscr H$, we have
\begin{description}
\item[\rm (a)] Monotonicity: If $X \ge Y$ then $\Sbep [X]\ge \Sbep [Y]$;
\item[\rm (b)] Constant preserving: $\Sbep [c] = c$;
\item[\rm (c)] Sub-additivity: $\Sbep[X+Y]\le \Sbep [X] +\Sbep [Y ]$ whenever $\Sbep [X] +\Sbep [Y ]$ is not of the form $+\infty-\infty$ or $-\infty+\infty$;
\item[\rm (d)] Positive homogeneity: $\Sbep [\lambda X] = \lambda \Sbep [X]$, $\lambda\ge 0$.
\end{description}
Here $\overline{\mathbb R}=[-\infty, \infty]$. The triple $(\Omega, \mathscr{H}, \Sbep)$ is called a sub-linear expectation space. Give a sub-linear expectation $\Sbep $, let us denote the conjugate expectation $\cSbep$of $\Sbep$ by
$$ \cSbep[X]:=-\Sbep[-X], \;\; \forall X\in \mathscr{H}. $$
\end{definition}
From the definition, it is easily shown that $\cSbep[X]\le \Sbep[X]$, $\Sbep[X+c]= \Sbep[X]+c$ and $\Sbep[X-Y]\ge \Sbep[X]-\Sbep[Y]$ for all
$X, Y\in \mathscr{H}$ with $\Sbep[Y]$ being finite. Further, if $\Sbep[|X|]$ is finite, then $\cSbep[X]$ and $\Sbep[X]$ are both finite.
\begin{definition}\label{def1.2}
\begin{description}
\item[ \rm (i)] ({\em Identical distribution}) Let $\bm X_1$ and $\bm X_2$ be two $n$-dimensional random vectors defined
respectively in sub-linear expectation spaces $(\Omega_1, \mathscr{H}_1, \Sbep_1)$
and $(\Omega_2, \mathscr{H}_2, \Sbep_2)$. They are called identically distributed, denoted by $\bm X_1\overset{d}= \bm X_2$ if
$$ \Sbep_1[\varphi(\bm X_1)]=\Sbep_2[\varphi(\bm X_2)], \;\; \forall \varphi\in C_{b,Lip}(\mathbb R_n), $$
where $C_{b,Lip}(\mathbb R_n)$ is the space of bounded Lipschitz functions.
\item[\rm (ii)] ({\em Independence}) In a sub-linear expectation space $(\Omega, \mathscr{H}, \Sbep)$, a random vector $\bm Y =
(Y_1, \ldots, Y_n)$, $Y_i \in \mathscr{H}$ is said to be independent to another random vector $\bm X =
(X_1, \ldots, X_m)$ , $X_i \in \mathscr{H}$ under $\Sbep$ if
$$ \Sbep [\varphi(\bm X, \bm Y )] = \Sbep \big[\Sbep[\varphi(\bm x, \bm Y )]\big|_{\bm x=\bm X}\big], \;\; \forall \varphi\in C_{b,Lip}(\mathbb R_m \times \mathbb R_n). $$
Random variables $\{X_n; n\ge 1\}$
are said to be independent, if $X_{i+1}$ is independent to $(X_1,\ldots, X_i)$ for each $i\ge 1$.
\end{description}
\end{definition}
In Peng \cite{peng2008a,peng2010,peng2010b}, the space of the test function $\varphi$ is $C_{l,Lip}(\mathbb R_n)$.
Here, the test function $\varphi$ in the definition is limit in the space of bounded Lipschitz functions. When the considered random variables have finite moments of each order, i.e., $\Sbep[|X|^p]<\infty$ for each $p>0$, then the space of test functions $ C_{b,Lip}(\mathbb R_n)$ can be equivalently extended to $C_{l,Lip}(\mathbb R_n)$.
A function $V:\mathcal{F}\to [0,1]$ is called a capacity if $V(\emptyset)=0$, $V(\Omega)=1$ and $V(A\cup B)\le V(A)+V(B)$ for all $A, B\in \mathcal{F}$. Let $(\Omega, \mathscr{H}, \Sbep)$ be a sub-linear space. We denote a pair $(\Capc,\cCapc)$ of capacities by
$$ \Capc(A):=\inf\{\Sbep[\xi]: I_A\le \xi, \xi\in\mathscr{H}\}, \;\; \cCapc(A):= 1-\Capc(A^c),\;\; \forall A\in \mathcal F, $$
where $A^c$ is the complement set of $A$.
Then
$$
\Sbep[f]\le \Capc(A)\le \Sbep[g], \;\;\cSbep[f]\le \cCapc(A) \le \cSbep[g],\;\;
\text{ if } f\le I_A\le g, f,g \in \mathscr{H}.
$$
It is obvious that $\Capc$ is sub-additive, i.e., $\Capc(A\bigcup B)\le \Capc(A)+\Capc(B)$. But $\cCapc$ and $\cSbep$ are not. However, we have
$$
\cCapc(A\bigcup B)\le \cCapc(A)+\Capc(B) \;\;\text{ and }\;\; \cSbep[X+Y]\le \cSbep[X]+\Sbep[Y]
$$
due to the fact that $\Capc(A^c\bigcap B^c)=\Capc(A^c\backslash B)\ge \Capc(A^c)-\Capc(B)$ and $\Sbep[-X-Y]\ge \Sbep[-X]-\Sbep[Y]$.
Further,
if $X$ is not in $\mathscr{H}$, we define $\Sbep[X]$ by
$$ \Sbep[X]=\inf\{\Sbep[Y]: X\le Y, \; Y\in \mathscr{H}\}. $$
Then $\Capc(A)=\Sbep[I_A]$.
\begin{definition}\label{def1.3}
(I) A function $V: \mathcal{F}\to [0, 1]$ is called to be countably sub-additive if
\[
V\Big(\bigcup_{n=1}^{\infty}A_n\Big)\leq \sum_{n=1}^{\infty}V(A_n),\ \ \forall A_n\in \mathcal{F}.
\]
(II) A function $V: \mathcal{F}\to [0, 1]$ is called to be continuous if it satisfies:\\
(i) Continuity from below: $V(A_n)\uparrow V(A)$ if $A_n\uparrow A$, where $A_n, A\in \mathcal{F}$.\\
(ii) Continuity from above: $V(A_n)\downarrow V(A)$ if $A_n \downarrow A$, where $A_n, A\in \mathcal{F}$.
\end{definition}
It is easily seen that a continuous capacity is countably sub-additive.
\section{Maximal inequalities}\label{Sect Inequality}
\setcounter{equation}{0}
In this section, we establish several inequalities on the maximal sums. The first one is the Levy maximal inequality.
\begin{lemma}\label{LevyIneq} Let $X_1,\cdots, X_n$ be independent random variables in a sub-linear expectation space $(\Omega, \mathscr{H}, \Sbep)$, $S_k=\sum_{i=1}^k X_i$, and $0<\alpha<1$ be a real number. If there exist real constants $\beta_{n, k}$ such that
$$ \Capc\left(S_k-S_n\ge \beta_{n,k}+\epsilon\right)\le \alpha, \text{ for all } \epsilon>0 \text{ and } k=1,\cdots ,n, $$
then
\begin{equation}\label{eqLIQ1} (1-\alpha) \mathbb{V}\left(\max_{k\le n}(S_k -\beta_{n,k})> x+\epsilon\right)\le \mathbb{V}\left(S_n>x\right), \text{ for all }x>0, \epsilon>0.
\end{equation}
If there exist real constants $\beta_{n, k}$ such that
$$ \Capc\left(|S_k-S_n|\ge \beta_{n,k}+\epsilon\right)\le \alpha, \text{ for all } \epsilon>0 \text{ and } k=1,\cdots ,n, $$
then
\begin{equation}\label{eqLIQ2} (1-\alpha) \mathbb{V}\left(\max_{k\le n}(|S_k| -\beta_{n,k})> x+\epsilon\right)\le \mathbb{V}\left(|S_n|>x\right), \text{ for all }x>0, \epsilon>0.
\end{equation}
\end{lemma}
{\bf Proof.} We only give the proof of (\ref{eqLIQ1}) since the proof of (\ref{eqLIQ2}) is similar.
Let $g_{\epsilon}(x)$ be a function with
\begin{equation}\label{eqproofLIQ.1} g_{\epsilon} \in C_b^1(\mathbb R) \; \text{ and } I_{\{x\ge \epsilon \}}\leq g_{\epsilon}(x)\leq I_{\{x\ge \epsilon/2\}} \text{ for all } x,
\end{equation}
where $0<\epsilon<1/2$, $ C_b^1(\mathbb R)$ is the space of bounded continuous function having bounded continuous derivations. Denote $Z_k=g_{\epsilon}\left(S_k-\beta_{n,k}-x\right)$, $Z_0=0$ and $\eta_k=\prod_{i=1}^k(1-Z_i)$. Then $S_n-S_m$ is independent to $(Z_1,\ldots,Z_m)$, and
\begin{align*}
&(1-\alpha) I\{\max_{k\le n}(S_k -\beta_{n,k})> x+\epsilon\}\\
= & (1-\alpha)\left[1-\prod_{k=1}^n I\{ S_k-\beta_{n,k}- x\le \epsilon\}\right] \\
\le & (1-\alpha)\left[1-\eta_n\right]=(1-\alpha)\left[\sum_{m=1}^n \eta_{m-1} Z_m\right]\\
=& \sum_{m=1}^n \eta_{m-1} Z_m I\{S_m-S_n<\beta_{n,m}+\epsilon/2\}\\
& +\sum_{m=1}^n \eta_{m-1}Z_m\left[1-\alpha-I\{S_m-S_n<\beta_{n,m}+\epsilon/2\}\right]\\
\le & \sum_{m=1}^n \eta_{m-1}Z_m I\{S_n>x\}
+\sum_{m=1}^n \eta_{m-1}Z_m\left[-\alpha+I\{S_m-S_n\ge \beta_{n,m}+\epsilon/2\}\right]\\
=& I\{ S_n >x\}
+\sum_{m=1}^n \eta_{m-1}Z_m\left[-\alpha+g_{\epsilon/2}\left( S_m-S_n-\beta_{n,m} \right) \right],
\end{align*}
where the second inequality above is due to the fact that on the event $\{Z_m\ne 0\}$ and $\{S_m-S_n< \beta_{n,m}+\epsilon/2\}$ we have $S_n\ge S_m-(S_m-S_n)>x$. Note
$$ \Sbep\left[g_{\epsilon/2}\left(S_m-S_n-\beta_{n,m}\right) \right]
\le \mathbb{V}\left(S_m-S_n\ge \beta_{n,m}+\epsilon/4\right)\le \alpha. $$
By the independence,
\begin{align*}
& \Sbep\left[\eta_{m-1}Z_m\left[-\alpha+g_{\epsilon/2}\left(S_m-S_n-\beta_{n,m}\right) \right]\right]\\
=&\Sbep\left[\eta_{m-1}Z_m\left\{-\alpha+\Sbep\left[g_{\epsilon/2}\left(S_m-S_n-\beta_{n,m}\right) \right]\right\}\right]\le 0.
\end{align*}
By the sub-additivity of $\Sbep$, it follows that
\begin{align*}
& (1-\alpha) \mathbb{V}\left(\max_{k\le n}(S_k -\beta_{n,k})> x+\epsilon\right)\\
\le & \mathbb{V}\left(S_n>x\right)
+\sum_{m=1}^n \Sbep\left[ \eta_{m-1}Z_m\left[-\alpha+g_{\epsilon/2}\left( S_m-S_n-\beta_{n,m} \right) \right]\right]
\\
\le & \mathbb{V}\left(S_n>x\right).
\end{align*}
The proof is completed. $\Box$
The second lemma is on the Kolmogorov type inequality.
\begin{lemma}\label{KolIneq} Let $X_1,\cdots, X_n$ be independent random variables in a sub-linear expectation space $(\Omega, \mathscr{H}, \Sbep)$. Let $S_k=\sum_{i=1}^k X_i$.
\begin{description}
\item[\rm (i) ] Suppose $|X_k|\le c$, $k=1,\cdots, n$. Then
\begin{equation}\label{eqKIQ1} \mathbb{V}\left(\max_{k\le n}|S_k|> x \right)\ge 1 -\frac{(x+c)^2+2x\sum_{k=1}^n \big\{\big(\Sbep[X_k]\big)^++\big(\Sbep[-X_k]\big)^+\big\}}{\sum_{k=1}^n \Sbep[X_k^2]},
\end{equation}
for all $x>0$.
\item[\rm (ii)] Suppose $ X_k \le c$, $\Sbep[X_k]\ge 0$, $k=1,\cdots, n$. Then
\begin{equation}\label{eqKIQ2.1} \mathbb{V}\left(\max_{k\le n} S_k>x \right)\ge 1 -\frac{ x+c }{\sum_{k=1}^n \Sbep[X_k]}\; \text{ for all }x>0.
\end{equation}
\end{description}
\end{lemma}
{\bf Proof.} (i) Let $g_{\epsilon}$ be defined as in (\ref{eqproofLIQ.1}). Denote $Z_k=g_{\epsilon}\left(|S_k|-x\right)$, $Z_0=0$, $\eta_k=\prod_{i=1}^k(1-Z_i)$.
Then $I\{|S_k|\ge x+\epsilon\}\le Z_k\le I\{|S_k|> x\}$. Also, $|S_{k-1}|< x+\epsilon$ and $|S_k|< |S_{k-1}|+|X_k|\le x+\epsilon+c$ on the event $\{\eta_{k-1}\ne 0\}$. So
\begin{align*}
S_{k-1}^2 \eta_{k-1} + 2S_{k-1}X_k \eta_{k-1} +X_k^2 \eta_{k-1}
=& S_k^2 \eta_k +S_k^2 \eta_{k-1} Z_k \\
\le &S_k^2 \eta_k +(x+\epsilon+c)^2\left[\eta_{k-1} -\eta_k \right].
\end{align*}
Taking the summation over $k$ yields
\begin{align*}
&\left(\sum_{k=1}^n \Sbep[X_k^2]\right)\eta_n +\sum_{k=1}^n\left(X_k^2-\Sbep[X_k^2]\right) \eta_{k-1}
\le \sum_{k=1}^nX_k^2 \eta_{k-1} \\
\le & S_n^2 \eta_n +(x+\epsilon+c)^2\left[1-\eta_n \right]-2\sum_{k=1}^n S_{k-1}X_k \eta_{k-1} \\
\le &(x+\epsilon)^2 \eta_n +(x+\epsilon+c)^2\left[1-\eta_n \right]-2\sum_{k=1}^n S_{k-1}X_k \eta_{k-1} \\
\le & (x+\epsilon+c)^2 -2\sum_{k=1}^n S_{k-1}X_k \eta_{k-1} .
\end{align*}
Write $B_n^2=\sum_{k=1}^n \Sbep[X_k^2]$. It follows that
\begin{align*} & 1-\frac{ (x+\epsilon+c)^2 }{B_n^2}+\frac{ \sum_{k=1}^n\left(X_k^2-\Sbep[X_k^2]\right) \eta_{k-1} }{B_n^2}\\
\le & 1-\eta_n +\frac{2}{B_n^2}\sum_{k=1}^n \left[X_k S_{k-1}^-\eta_{k-1} -X_kS_{k-1}^+\eta_{k-1} \right].
\end{align*}
Note $$\Sbep[X_k S_{k-1}^-\eta_{k-1} ]=\Sbep[X_k]\Sbep[ S_{k-1}^-\eta_{k-1} ]\le (x+\epsilon) \big(\Sbep[X_k]\big)^+,$$
$$\Sbep[-X_k S_{k-1}^+\eta_{k-1} ]=\Sbep[-X_k]\Sbep[ S_{k-1}^+\eta_{k-1} ]\le (x+\epsilon) \big(\Sbep[-X_k]\big)^+$$
and
\begin{align}\label{eqpppoofKIQ1}
& \Sbep\left[\sum_{k=1}^n\left(X_k^2-\Sbep[X_k^2]\right) \eta_{k-1} \right] \nonumber\\
= & \Sbep\left[\Sbep\left[\sum_{k=1}^n\left(X_k^2-\Sbep[X_k^2]\right) \eta_{k-1} \Big|X_1,\cdots,X_{n-1} \right]\right] \nonumber\\
= & \Sbep\left[\sum_{k=1}^{n-1}\left(X_k^2-\Sbep[X_k^2]\right) \eta_{k-1} +\eta_{n-1}\Sbep[X_n^2-\Sbep[X_n^2] ] \right]\nonumber\\
=& \Sbep\left[\sum_{k=1}^{n-1}\left(X_k^2-\Sbep[X_k^2]\right) \eta_{k-1} \right]=\cdots =0.
\end{align}
It follows that
\begin{align*}
& 1-\frac{ (x+\epsilon+c)^2 }{B_n^2}-\frac{ 2(x+\epsilon)\sum_{k=1}^n \big\{\big(\Sbep[X_k]\big)^++\big(\Sbep[-X_k]\big)^+\big\} }{B_n^2} \\
& \;\; \le \Sbep\left[ 1-\eta_n\right]\le \Capc\left(\max_{k\le n} |S_k|> x\right).
\end{align*}
By letting $\epsilon\to 0$, we obtain (\ref{eqKIQ1}). The proof of (i) is completed.
(ii) Redefine $Z_k$ and $\eta_k$ by $Z_k=g_{\epsilon}\left(S_k-x\right)$, $Z_0=0$, $\eta_k=\prod_{i=1}^k(1-Z_i)$.
Then $I\{S_k\ge x+\epsilon\}\le Z_k\le I\{S_k> x\}$. Also, $ S_{k-1}< x+\epsilon$ and $ S_k = S_{k-1} + X_k < x+\epsilon+c$ on the event $\{\eta_{k-1}\ne 0\}$. So
$$
S_{k-1} \eta_{k-1} + X_k \eta_{k-1}
= S_k \eta_k +S_k \eta_{k-1} Z_k
\le S_k \eta_k +(x+\epsilon+c) \eta_{k-1} Z_k.
$$
Taking the summation over $k$ yields
\begin{align*}
&\left(\sum_{k=1}^n \Sbep[X_k ]\right)\eta_n +\sum_{k=1}^n\left(X_k -\Sbep[X_k ]\right) \eta_{k-1} \\
\le & \sum_{k=1}^nX_k \eta_{k-1}
\le S_n \eta_n +(x+\epsilon+c) \left[1-\eta_n \right] \\
\le &(x+\epsilon) \eta_n +(x+\epsilon+c) \left[1-\eta_n \right]
\le (x+\epsilon+c) .
\end{align*}
Write $e_n =\sum_{k=1}^n \Sbep[X_k]$. It follows that
\begin{align*} 1-\frac{ (x+\epsilon+c) }{e_n}+\frac{ \sum_{k=1}^n\left(X_k-\Sbep[X_k]\right) \eta_{k-1} }{e_n}
\le 1-\eta_n.
\end{align*}
Note
\begin{align*}
\Sbep\left[\sum_{k=1}^n\left(X_k -\Sbep[X_k ]\right) \eta_{k-1} \right]
= \Sbep\left[\sum_{k=1}^{n-1}\left(X_k -\Sbep[X_k ]\right) \eta_{k-1} \right]=\cdots =0,
\end{align*}
similar to (\ref{eqpppoofKIQ1}). It follows that
$$ 1-\frac{ x+\epsilon+c }{e_n } \le \Sbep\left[ 1-\eta_n \right]\le \Capc\left(\max_{k\le n} S_k > x\right). $$
By letting $\epsilon\to 0$, we obtain (\ref{eqKIQ1}). The proof is completed. $\Box$
The following lemma on the bounds of the capacities via moments will be used in the paper.
\begin{lemma}[ \cite{Zhang Exponential}]\label{moment_v}
Let $X_1, X_2, \ldots, X_n$ be independent
random variables in $(\Omega, \mathscr{H}, \Capc)$. If $\Sbep[X_{k}] \leq 0$, $k=1,\ldots, n$, then there exists a constant $C>0$ such that
\begin{equation*}
\Capc(S_n\geq x)\leq C\frac{\sum_{k=1}^{n}\Sbep[X_k^2]}{x^2} \;\text{ for all } \; \forall x>0.
\end{equation*}
\end{lemma}
\section{The convergence of infinite series}\label{Sect Convergece}
\setcounter{equation}{0}
Our results on the convergence of the series $\sum_{n=1}^{\infty}$ are stated as three theorems. The first one gives the equivalency between the almost sure convergence and the convergence in capacity.
\begin{theorem}\label{th1} Let $\{X_n;n\geq1\}$ be a sequence of independent random variables in a sub-linear expectation space $(\Omega, \mathscr{H}, \Sbep)$, $S_n=\sum_{k=1}^n X_k$, and $S$ be a random variable in the measurable space $(\Omega, \mathcal{F})$.
\begin{description}
\item[(i)] If $\Capc$ is countably sub-additive, and
\begin{equation}\label{eqth1.1} \Capc\left(|S_n-S|\ge \epsilon\right)\to 0 \text{ as } n\to \infty \text{ for all } \epsilon>0,
\end{equation}
then
\begin{equation}\label{eqth1.2} \Capc\left(\left\{\omega: \lim_{n\to \infty} S_n(\omega)\ne S(\omega)\right\}\right)=0.
\end{equation}
When (\ref{eqth1.2}) holds, we call that $\sum_{n=1}^{\infty}X_n$ is almost surely convergent in capacity, and when (\ref{eqth1.1}) holds, we call that $\sum_{n=1}^{\infty}X_n$ is convergent in capacity.
\item[(ii)] If $\Capc$ is continuous, then (\ref{eqth1.2}) implies (\ref{eqth1.1}).
\end{description}
\end{theorem}
The second theorem gives the equivalency between the convergence in capacity and the convergence in distribution.
\begin{theorem}\label{th2} Let $\{X_n;n\geq1\}$ be a sequence of independent random variables in a sub-linear expectation space $(\Omega, \mathscr{H}, \Sbep)$, $S_n=\sum_{k=1}^n X_k$.
\begin{description}
\item[(i)] If there is a random variable $S$ in the measurable space $(\Omega, \mathcal{F})$ such that
\begin{equation}\label{eqth2.1} \Capc\left(|S_n-S|\ge \epsilon\right)\to 0 \text{ as } n\to \infty \text{ for all } \epsilon>0,
\end{equation}
and $S$ is tight under $\Sbep$, i.e., $\Sbep\left[I_{\{|S|\le x\}^c}\right]=\Capc(|S|> x)\to 0$ as $x\to \infty$,
then
\begin{equation}\label{eqth2.2} \Sbep\left[\phi(S_n)\right]\to \Sbep\left[\phi(S)\right],\;\; \phi\in C_b(\mathbb{R}),
\end{equation}
where $C_b(\mathbb R)$ is the space of bounded continuous functions on $\mathbb R$. When (\ref{eqth2.2}) holds, we call that $\sum_{n=1}^{\infty}X_n$ is convergent in distribution.
\item[(ii)] Suppose that there is a sub-linear space $(\widetilde{\Omega}, \widetilde{\mathscr{H}}, \widetilde{\mathbb E})$ and a random variable $\widetilde{S}$ on it such that $\widetilde{S}$ is tight under $\widetilde{\mathbb E}$, i.e., $\widetilde{\Capc}(|\widetilde{S}|> x)\to 0$ as $x\to \infty$, and
\begin{equation}\label{eqth2.3} \Sbep\left[\phi(S_n)\right]\to \widetilde{\mathbb E} \left[\phi(\widetilde{S})\right],\;\; \phi\in C_{b}(\mathbb{R}),
\end{equation}
then $S_n$ is a Cauchy sequence in capacity $\Capc$, namely
\begin{equation}\label{eqth2.4} \Capc\left(|S_n-S_m|\ge \epsilon \right)\to 0 \text{ as } n,m\to \infty \text{ for all } \epsilon>0.
\end{equation}
Furthermore, if $\Capc$ is countably sub-additive, then on the measurable space $(\Omega, \mathcal F)$ there is a random variable $S$ which is tight under $\Sbep$, such that (\ref{eqth1.1}) and (\ref{eqth1.2}) hold.
\end{description}
\end{theorem}
Recently, Xu and Zhang \cite{XuZhang2018} gave sufficient conditions for $\sum_{n=1}^{\infty} X_n$ to be convergent almost surely in capacity via three series theorem. The third theorem of us gives the sufficient and necessary conditions for $S_n$ to be a Cauchy sequence in capacity.
For any random variable $X$ and constant $c$, we denote $X^c=(-c)\vee(X\wedge c)$.
\begin{theorem}\label{th4} Let $\{X_n;n\geq1\}$ be a sequence of independent random variables in $(\Omega, \mathscr{H}, \Sbep)$, $S_n=\sum_{k=1}^n X_k$. Then $S_n$ will be a Cauchy sequence in capacity $\Capc$ if the following three conditions hold for some $c>0$.
\begin{description}
\item[\rm (S1) ] $\sum\limits_{n=1}^{\infty}\Capc(|X_n|>c)<\infty$,
\item[\rm (S2) ] $\sum\limits_{n=1}^{\infty}\Sbep[X_n^c]$ and $ \sum\limits_{n=1}^{\infty}\Sbep[-X_n^c]$ are both convergent,
\item[\rm (S3) ] $\sum\limits_{n=1}^{\infty}\Sbep\left[ \big(X_n^c-\Sbep[X_n^c]\big)^2\right] <\infty$ or/and $\sum\limits_{n=1}^{\infty}\Sbep\left[ \big(X_n^c+\Sbep[-X_n^c]\big)^2\right] <\infty$.
\end{description}
Conversely, if $S_n$ is a Cauchy sequence in capacity $\Capc$, then (S1),(S2) and (S3) will hold for all $c>0$.
\end{theorem}
From Theorem \ref{th4}, we have the following three series theorem on the sufficient and necessary conditions for the almost sure convergence of $\sum_{n=1}^{\infty} X_n$.
\begin{corollary}\label{three series} Let $\{X_n;n\geq1\}$ be a sequence of independent random variables in $(\Omega, \mathscr{H}, \Sbep)$. Suppose that $\Capc$ is countably sub-additive. Then $\sum_{n=1}^{\infty}X_n$ will converge almost surely in capacity if the three conditions (S1),(S2) and (S3) in Theorem \ref{th4} hold for some $c>0$.
Conversely, if $\Capc$ is continuous and $\sum_{n=1}^{\infty}X_n$ is convergent almost surely in capacity, then (i),(ii) and (iii) will hold for all $c>0$.
\end{corollary}
The sufficiency of (S1), (S2) and (S3) is proved by Xu and Zhang \cite{XuZhang2018}, and also follows from Theorem \ref{th4} and the second part of conclusion of Theorem \ref{th2} (ii). The necessity follows from Theorem \ref{th4} and Theorem \ref{th1} (ii).
\bigskip
The prove Theorems \ref{th1} and \ref{th2}. We need some more lemmas.
The first lemma is a version of Theorem 9 of Peng \cite{peng2010b}.
\begin{lemma}\label{lem1} Let $\{\bm Y_n; n\ge 1\}$ be a sequence of $d$-dimensional random variables in a sub-linear expectation space $(\Omega, \mathscr{H}, \Sbep)$. Suppose that $\bm Y_n$ is asymptotically tight, i.e.,
$$ \limsup_{n\to\infty}\Sbep\left[I_{\{\bm Y_n\|\le x\}^c}\right]=\limsup_{n\to\infty} \Capc\left(\|\bm Y_n\|> x\right)\to 0 \; \text{ as } x\to \infty. $$
Then for any subsequence $\{\bm Y_{n_k}\}$ of $\{\bm Y_n\}$, there exist further a subsequence $\{\bm Y_{n_{k^{\prime}}}\}$ of $\{\bm Y_{n_k}\}$ and a sub-linear expectation space $(\overline{\Omega}, \overline{\mathscr{H}}, \overline{\mathbb E})$ with a $d$-dimensional random variable $\bm Y$ on it such that
$$ \Sbep\left[\phi\left(\bm Y_{n_{k^{\prime}}}\right)\right]\to \overline{\mathbb E}\left[\phi(\bm Y)\right] \text{ for any } \phi\in C_b(\mathbb R^d) $$
and $\bm Y$ is tight under $\overline{\mathbb E}$.
\end{lemma}
{\bf Proof.} Let
$$ \mathbb E\left[\phi\right]=\limsup_{n\to \infty}\Sbep\left[\phi(\bm Y_n)\right], \;\; \phi\in C_b(\mathbb R^d). $$
Then $\mathbb E$ is a sub-linear expectation on the function space $C_b(\mathbb R^d)$ and is tight in sense that for any $\epsilon>0$, there is a compact set $K=\{\bm x:\|\bm x\|\le M\}$ for which
$\mathbb E\left[I_{K^c}\right]<\epsilon$. With the same argument as in the proof of Theorem 9 of Peng \cite{peng2010b}, there is a countable subset $\{\varphi_j\}$ of $C_b(\mathbb R^d)$ such that for each $\phi\in C_b(\mathbb R^d)$ and any $\epsilon>0$ one can find a $\varphi_j$ satisfying
\begin{equation}\label{eqprooflem1.1}\mathbb E\left[|\phi-\varphi_j|\right]<\epsilon. \end{equation}
On the other hand, for each $\varphi_j$, the sequence $\Sbep\left[\varphi_j(\bm Y_n)\right]$ is bounded and so there is a Cauchy subsequence. Note that the set $\{\varphi_j\}$ is countable. By the diagonal choice method, one can find a sequence $\{n_k\}\subset \{n\}$ such that $\Sbep\left[\varphi_j(\bm Y_{n_k})\right]$ is a Cauchy sequence for each $\varphi_j$. Now, we show that $\Sbep\left[\phi(\bm Y_{n_k})\right]$ is a Cauchy sequence for any $\phi\in C_b(\mathbb R^d)$. For any $\epsilon>0$, choose a $\varphi_j$ such that (\ref{eqprooflem1.1}) holds. Then
\begin{align*}
& \left|\Sbep\left[\phi(\bm Y_{n_k})\right]-\Sbep\left[\phi(\bm Y_{n_l})\right]\right| \\
\le & \left|\Sbep\left[\varphi_j(\bm Y_{n_k})\right]-\Sbep\left[\varphi_j(\bm Y_{n_l})\right]\right| \\
&+ \Sbep\left[\left|\phi(\bm Y_{n_k}) - \varphi_j(\bm Y_{n_k})\right|\right] + \Sbep\left[\left|\phi(\bm Y_{n_l}) -\varphi_j(\bm Y_{n_l})\right|\right].
\end{align*}
Taking the limits yields
$$
\limsup_{k,l\to\infty} \left|\Sbep\left[\phi(\bm Y_{n_k})\right]-\Sbep\left[\phi(\bm Y_{n_l})\right]\right|
\le 0+ 2\mathbb E\left[|\phi-\varphi_j|\right]<2\epsilon.
$$
Hence $\Sbep\left[\phi(\bm Y_{n_k})\right]$ is a Cauchy sequence for any $\phi\in C_b(\mathbb R^d)$, and then
\begin{equation}\label{eqprooflem1.2} \lim_{k\to \infty} \Sbep\left[\phi(\bm Y_{n_k})\right] \; \text{ exists and is finite for any } \phi\in C_b(\mathbb R^d).
\end{equation}
Now, let $\overline{\Omega}=\mathbb R^d$, $\overline{\mathscr{H}}=C_{l,lip}(\mathbb R^d)$. Define
$$ \overline{\mathbb E}\left[\varphi\right]=\limsup_{k\to \infty} \Sbep\left[\varphi(\bm Y_{n_k})\right], \;\; \varphi\in C_{l,lip}(\mathbb R^d). $$
Then $(\overline{\Omega}, \overline{\mathscr{H}},\overline{\mathbb E})$ is a sub-linear expectation space. Define the random variable $\bm Y$ by $\bm Y(\bm x)=\bm x$, $\bm x\in \overline{\Omega}$.
From (\ref{eqprooflem1.2}) it follows that
$$ \lim_{k\to \infty} \Sbep\left[\phi(\bm Y_{n_k})\right]=\overline{\mathbb E}\left[\phi(\bm Y)\right] \text{ for any } \phi\in C_b(\mathbb R^d).
$$
The proof is completed. $\Box$
\begin{lemma}\label{lem2} Let $X$ and $Y$ be random variables in a sub-linear expectation space $(\Omega, \mathscr{H}, \Sbep)$. Suppose that $Y$ and $X$ are independent ($Y$ is independent to $X$, or $X$ is independent to $Y$), and $X$ is tight, i.e. $\Capc(|X|\ge x)\to 0$ as $x> \infty$. If $X+Y\overset{d}=X$, then $\Capc(|Y|\ge \epsilon)=0$ for all $\epsilon>0$.
\end{lemma}
{\bf Proof.} Without loss of generality, we assume that $Y$ is independent to $X$. We can find a sub-linear expectation space $(\Omega^{\prime}, \mathscr{H}^{\prime}, \Sbep^{\prime})$ on which there are independent random variables $X_1, Y_1,Y_2, \cdots, Y_n,\cdots$ such that $X_1\overset{d}=X$, $Y_i\overset{d}=Y$, $i=1,2,\cdots,$. Without loss of generality, assume $(\Omega^{\prime}, \mathscr{H}^{\prime}, \Sbep^{\prime})=(\Omega, \mathscr{H}, \Sbep)$. Let $S_k=\sum_{j=1}^k Y_k$. Then $X_1+S_k\overset{d}=X$. So,
\begin{align}\label{eqprooflem2.1} \max_{k\le n} \Capc(|S_k|>x_0)\le & \max_{k\le n} \Capc(|X_1+S_k|>x_0/2)+ \Capc(|X_1|>x_0/2) \nonumber
\\
\le & \Sbep\left[g_{1/2}\left(\frac{|X_1+S_k|}{x_0}\right)\right]+ \Sbep\left[g_{1/2}\left(\frac{|X_1|}{x_0}\right)\right]\nonumber\\
=& 2 \Sbep\left[g_{1/2}\left(\frac{|X|}{x_0}\right)\right]\le 2\Capc(|X|\ge x_0/4)<1/4
\end{align}
for $x_0$ large enough, where $g_{\epsilon}$ is defined as in (\ref{eqproofLIQ.1}).
By Lemma \ref{LevyIneq},
\begin{equation}\label{eqprooflem2.2} \Capc(\max_{k\le n}|S_k|>2x_0+\epsilon)\le \frac{4}{3} \max_n \Capc(|S_n|>x_0) \le \frac{4}{3} \cdot 2 \Capc(|X|\ge x_0/4)<\frac{1}{3} .
\end{equation}
It follows that for any $\epsilon>0$,
$$ \Capc(\max_{k\le n}|Y_k|>4x_0+2\epsilon) <\frac{1}{3}. $$
Let $Z_k=g_{\epsilon} (|Y_k|-4x_0-2\epsilon)$, where $g_{\epsilon}$ is defined as in (\ref{eqproofLIQ.1}). Denote $q=\Capc(|Y_1|>4x_0+3\epsilon)$. Then $Z_1,Z_2\cdots, Z_n$ are independent and identically distributed with $\{|Y_k|>4x_0+3\epsilon\}\le Z_k\le \{|Y_k|>4x_0+2\epsilon\}$ and $\Sbep[Z_1]\ge \Capc(|Y_1|>4x_0+3\epsilon)=q$. Then by Lemma \ref{KolIneq} (ii),
\begin{equation} \label{eqprooflem2.3} \frac{1}{3}> \Capc\left(\sum_{k=1}^n Z_k\ge 1\right)
\ge 1-\frac{1+1}{\sum_{k=1}^n \Sbep[Z_k]}\ge 1-\frac{2}{nq}.
\end{equation}
The above inequality holds for all $n$, which is impossible unless $q=0$. So we conclude that
$$\Capc(|Y_1|>4x_0+\epsilon) = 0 \;\; \text{ for any } \epsilon>0. $$
Now, let $\widetilde{Y}_k=(-5x_0)\vee Y_k \wedge (5x_0)$, $\widetilde{S}_k=\sum_{i=1}^k \widetilde{Y}_i$. Then $\widetilde{Y}_1, \cdots, \widetilde{Y}_n$ are independent and identically distributed bounded random variables, $\Capc(\widetilde{Y}_k\ne Y_k)=0$ and $\Capc(\widetilde{S}_k\ne S_k)=0 $. If $\Sbep[\widetilde{Y}_1]>0$, then by Lemma \ref{KolIneq} (ii) again,
$$
\Capc(\max_{k\le n}S_k\ge 3x_0)=\Capc(\max_{k\le n}\widetilde{S}_k\ge 3x_0)\ge 1 -\frac{3x_0+5x_0}{n \Sbep[\widetilde{Y}_1]}, $$
which contradicts to (\ref{eqprooflem2.2}) when $n> 12 x_0/\Sbep[\widetilde{Y}_1]$. Hence, $\Sbep[\widetilde{Y}_1]\le 0$. Similarly, $\Sbep[-\widetilde{Y}_1]\le 0$. We conclude that $\Sbep[\widetilde{Y}_1]=\Sbep[-\widetilde{Y}_1]= 0$. Now, if $ \Sbep[\widetilde{Y}_1^2]\ne 0$, then by Lemma \ref{KolIneq} (i) we have
$$
\Capc(\max_{k\le n}|S_k|\ge 3x_0)\ge 1 -\frac{(3x_0+5x_0)^2}{n \Sbep[\widetilde{Y}_1^2]}, $$
which contradicts to (\ref{eqprooflem2.2}) when $n> 96x_0^2/\Sbep[\widetilde{Y}_1^2]$. We conclude that $\Sbep[\widetilde{Y}_1^2]=0$.
Finally, for any $\epsilon>0$ ($\epsilon<5x_0$),
$$ \Capc\left(|Y|\ge \epsilon\right)\le \frac{\Sbep[Y^2\wedge(5x_0)^2]}{\epsilon^2} = \frac{\Sbep[\widetilde{Y}_1^2]}{\epsilon^2}=0. $$
The proof is completed. $\Box$
\bigskip
{\bf Proof of Theorem \ref{th1}.} (i) Let $\epsilon_k=1/2^k$, $\delta_k=1/4^k$. By (\ref{eqth1.1}), there exits a sequence $n_1<n_2<\cdots<n_k\to \infty$, such that
\begin{equation}\label{eqproofth1.1} \max_{n\ge n_k} \Capc\left(|S_n-S|\ge \epsilon_k\right)<\delta_k.
\end{equation}
By the countably sub-additivity of $\Capc$, we have
\begin{align*}
&\Capc\left(\limsup_{k\to \infty}|S_{n_k}-S|>0\right)
\le
\Capc\left(\bigcap_{m=1}^{\infty} \bigcup_{k=m}^{\infty} \{|S_{n_k}-S|\ge \epsilon_k\}\right)\\
\le & \sum_{k=m}^{\infty}\Capc\left( |S_{n_k}-S|\ge \epsilon_k \right)\le \sum_{k=m}^{\infty}\delta_k \to 0 \text{ as } m\to \infty.
\end{align*}
By (\ref{eqproofth1.1}), $\max_{n\ge n_k} \Capc\left(|S_n-S_{n_{k+1}}|\ge 2\epsilon_k\right)<2\delta_k<1/2$. Apply the Levy inequality (\ref{eqLIQ2}) yields
\begin{equation}\label{eqproofth1.2} \Capc\left(\max_{n_k\le n\le n_{k+1}}|S_n-S_{n_k}|> 5\epsilon_k\right)\le 2 \Capc\left(|S_{n_{k+1}}-S_{n_k}|> 2\epsilon_k\right)<4 \delta_k.
\end{equation}
By the countably sub-additivity of $\Capc$ again,
\begin{align*}
&\Capc\left(\limsup_{k\to \infty}\max_{n_k\le n\le n_{k+1}}|S_n-S_{n_k}|>0\right)\\
\le &
\Capc\left(\bigcap_{m=1}^{\infty} \bigcup_{k=m}^{\infty} \{\max_{n_k\le n\le n_{k+1}}|S_n-S_{n_k}|\ge 5\epsilon_k\}\right)\\
\le & \sum_{k=m}^{\infty}\Capc\left( \max_{n_k\le n\le n_{k+1}}|S_n-S_{n_k}|\ge 5\epsilon_k \right)\le 4\sum_{k=m}^{\infty}\delta_k \to 0 \text{ as } m\to \infty.
\end{align*}
It follows that
\begin{align*}
& \Capc\left(\limsup_{n\to \infty}|S_n-S|>0\right)\\
\le & \Capc\left(\limsup_{k\to \infty}|S_{n_k}-S|>0\right)+\Capc\left(\limsup_{k\to \infty}\max_{n_k\le n\le n_{k+1}}|S_n-S_{n_k}|>0\right)=0.
\end{align*}
(\ref{eqth1.2}) follows.
(ii) From (\ref{eqth1.2}) and the continuity of $\Capc$, it follows that for any $\epsilon>0$,
\begin{align*}
0\ge \Capc\left(\bigcap_{n=1}^{\infty}\bigcup_{m=n}^{\infty}\{|S_m-S|\ge \epsilon\}\right)= & \lim_{n\to \infty}\Capc\left(\bigcup_{m=n}^{\infty}\{|S_m-S|\ge \epsilon\}\right) \\
\ge & \limsup_{n\to \infty}\Capc\left( |S_n-S|\ge \epsilon \right).
\end{align*}
(\ref{eqth1.1}) follows. The proof is completed. $\Box$
{\bf Proof of Theorem \ref{th2}.} (i) We first show that (\ref{eqth2.2}) holds for any bounded uniformly continuous function $\phi$. For any $\epsilon>0$, there is a $\delta>0$ such that $|\phi(x)-\phi(y)|<\epsilon$ when $|x-y|<\delta$. It follows that
$$ \left|\Sbep\left[\phi(S_n)\right]-\Sbep\left[\phi(S)\right]\right|\le \epsilon+ 2\sup_x|\phi(x)|\Capc\left(|S_n-S|\ge \delta\right). $$
By letting $n\to \infty$ and the arbitrariness of $\epsilon>0$, we obtain (\ref{eqth2.2}). Now, suppose that $\phi$ is a bounded continuous function. Then for any $N>1$, $\phi((-N)\vee x\wedge N)$ is a bounded uniformly continuous function. Hence
$$ \lim_{n\to \infty}\Sbep\left[\phi((-N)\vee S_n\wedge N)\right]=\Sbep\left[\phi((-N)\vee S\wedge N)\right]. $$
On the other hand,
$$\left|\Sbep\left[\phi((-N)\vee S\wedge N)\right]-\Sbep\left[\phi( S )\right]\right|\le 2\sup_x\big|\phi(x)\big|\Capc\left(|S|>N\right)\to 0 \text{ as } N\to \infty, $$
and
\begin{align*}
&\limsup_{n\to \infty} \left|\Sbep\left[\phi((-N)\vee S_n\wedge N)\right]-\Sbep\left[\phi( S_n )\right]\right|\\
\le & 2\sup_x|\phi(x)|\limsup_{n\to \infty} \Capc\left(|S_n|\ge N\right)\le 2\sup_x\big|\phi(x)\big|\limsup_{n\to \infty} \Sbep\left[g_1\left(\frac{|S_n|}{N}\right)\right]\\
=& 2\sup_x|\phi(x)| \Sbep\left[g_1\left(\frac{|S|}{N}\right)\right]\le 2\sup_x\big|\phi(x)\big|\limsup_{n\to \infty} \Capc\left(|S|\ge N/2\right)\to 0 \text{ as } N\to \infty,
\end{align*}
where $g_{\epsilon}$ is defined as in (\ref{eqproofLIQ.1}). Hence, (\ref{eqth2.2}) holds for a bounded continuous function $\phi$.
(ii) Note
$$ \Capc\left(|S_n-S_m|\ge 2x\right)\le \Capc\left(|S_n|\ge x\right)+\Capc\left(|S_m|\ge x\right). $$
It follows that
\begin{align*}
&\limsup_{m\ge n\to \infty}\Capc\left(|S_n-S_m|\ge 2x\right)\le 2\limsup_{n\to \infty} \Capc\left(|S_n|\ge x\right) \\
\le & 2\limsup_{n\to \infty} \Sbep\left[g_1\left(\frac{|S_n|}{x}\right)\right]=\widetilde{\mathbb E} \left[g_1\left(\frac{|\widetilde{S}|}{x}\right)\right]
\le 2\Capc\left(|\widetilde{S}|\ge x/2\right) \to 0 \text{ as } x\to \infty.
\end{align*}
Write $\bm Y_{n,m}=(S_n, S_m-S_n)$, then the sequence $\{\bm Y_{n,m}; m\ge n\}$ is asymptotically tight, i.e.,
$$\limsup_{m\ge n\to \infty} \Capc\left(\|\bm Y_{n,m}\|\ge x\right)\to 0\; \text{ as } x \to \infty. $$
By Lemma \ref{lem1}, for any subsequence $(n_k,m_k)$ of $(n,m)$, there is further a subsequence $(n_{k^{\prime}},m_{k^{\prime}})$ of $(n_k,m_k)$ and a sub-linear expectation space $(\overline{\Omega}, \overline{\mathscr{H}}, \overline{\mathbb E})$ with a random vector $\bm Y=(Y_1,Y_2)$ such that
\begin{equation}\label{eqproofth2.3} \Sbep\left[\phi\left(\bm Y_{n_{k^{\prime}},m_{k^{\prime}}}\right)\right]\to \overline{\mathbb E}\left[\phi(\bm Y)\right],\;\;\phi\in C_b(\mathbb R^2).
\end{equation}
Note that $S_{m_{k^{\prime}}}- S_{n_{k^{\prime}}}$ is independent to $S_{n_{k^{\prime}}}$. By Lemma 4.4 of Zhang \cite{Zhang Donsker}, $Y_2$ is independent to $Y_1$ under $ \overline{\mathbb E}$.
Let $\phi\in C_{b,Lip}(\mathbb R)$. By (\ref{eqproofth2.3}),
\begin{equation}\label{eqproofth2.4}\Sbep\left[\phi\left(S_{m_{k^{\prime}}}\right)\right]\to \overline{\mathbb E}\left[\phi(Y_1+Y_2)\right], \;\; \Sbep\left[\phi\left(S_{n_{k^{\prime}}}\right)\right]\to \overline{\mathbb E}\left[\phi(Y_1)\right]
\end{equation}
and
\begin{equation}\label{eqproofth2.5} \Sbep\left[\phi\left(S_{m_{k^{\prime}}}-S_{n_{k^{\prime}}}\right)\right]\to \overline{\mathbb E}\left[\phi(Y_2)\right].
\end{equation}
On the other hand, by (\ref{eqth2.3}),
\begin{equation}\label{eqproofth2.6} \Sbep\left[\phi\left(S_{m_{k^{\prime}}}\right)\right] \to \widetilde{\mathbb E} \left[\phi(\widetilde{S})\right]
\; \text{ and } \Sbep\left[\phi\left(S_{n_{k^{\prime}}}\right)\right] \to \widetilde{\mathbb E} \left[\phi(\widetilde{S})\right].
\end{equation}
Combing (\ref{eqproofth2.4}) and (\ref{eqproofth2.6}) yields
$$ \overline{\mathbb E}\left[\phi(Y_1+Y_2)\right] =\overline{\mathbb E}\left[\phi(Y_1)\right]=\widetilde{\mathbb E} \left[\phi(\widetilde{S})\right],\;\;\phi\in C_{b,Lip}(\mathbb R). $$
Hence, by Lemma \ref{lem2}, we obtain $\overline{\mathbb V}(|Y_2|\ge \epsilon)=0$ for all $\epsilon>0$. By choosing $\phi\in C_{b,Lip}(\mathbb R)$ such that $I_{|x|\ge \epsilon}\le \phi(x)\le I_{|x|\ge \epsilon/2}$ in
(\ref{eqproofth2.5}), we have
$$
\limsup_{k^{\prime}\to \infty} \Capc\left(\left|S_{m_{k^{\prime}}}-S_{n_{k^{\prime}}}\right|\ge \epsilon\right)\le \overline{\mathbb V}(|Y_2|\ge \epsilon/2)=0.
$$
So, we conclude that for any subsequence $(n_k,m_k)$ of $(n,m)$, there is a further a subsequence $(n_{k^{\prime}},m_{k^{\prime}})$ of $(n_k,m_k)$ such that
$$ \Capc\left(\left|S_{m_{k^{\prime}}}-S_{n_{k^{\prime}}}\right|\ge \epsilon\right)\to 0 \text{ for all }\epsilon>0. $$
Hence (\ref{eqth2.4}) is proved.
Next, suppose that $\Capc$ is countably sub-additive. Let $\epsilon_k=1/2^k$, $\delta_k=1/3^k$. By (\ref{eqth2.4}), there is a sequence $n_1<n_2<\cdots<n_k<\cdots$ such that
$$ \Capc\left(|S_{n_{k+1}}-S_{n_k}|\ge \epsilon_k\right)\le \delta_k. $$
Let $A=\{\omega: \sum_{k=1}^{\infty}|S_{n_{k+1}}-S_{n_k}|<\infty\}$. Then
\begin{align*} \Capc\left(A^c\right)\le & \Capc\left(\sum_{k=K}^{\infty}|S_{n_{k+1}}-S_{n_k}|\ge \sum_{k=K}^{\infty}\epsilon_k\right)\\
\le & \sum_{k=K}^{\infty} \Capc\left(|S_{n_{k+1}}-S_{n_k}|\ge \epsilon_k\right)\le \sum_{k=K}^{\infty}\delta_k \to 0 \text{ as } K\to \infty.
\end{align*}
Define $S=\lim_{k\to \infty} S_{n_k}$ on $A$, and $S=0$ on $A^c$. Then
\begin{align*} \Capc\left(|S-S_{n_k}|\ge 1/2^{k-1}\right)\le &\Capc(A^c)+ \Capc\left(A, \sum_{i=k}^{\infty}|S_{n_{i+1}}-S_{n_i}|\ge \sum_{i=k}^{\infty}\epsilon_i\right)\\
\le & \sum_{i=k}^{\infty} \Capc\left(|S_{n_{i+1}}-S_{n_i}|\ge \epsilon_i\right)\le \sum_{i=k}^{\infty}\delta_i \to 0 \text{ as } k\to \infty.
\end{align*}
On the other hand, by (\ref{eqth2.4}),
$$\Capc\left(|S_n-S_{n_k}|\ge \epsilon\right)\to 0 \text{ as } n, n_k\to \infty. $$
Hence
$$\Capc\left(|S_n-S|\ge \epsilon\right)\le \Capc\left(|S_n-S_{n_k}|\ge \epsilon/2\right)+\Capc\left(|S-S_{n_k}|\ge \epsilon/2\right)\to 0. $$
(\ref{eqth1.1}) is proved. Further,
\begin{align*}
\Capc\left(|S|\ge 2M\right)\le & \limsup_n \Capc\left(|S_n|\ge M\right)+\limsup_n \Capc\left(|S_n-S|\ge M\right)\\
\le & \widetilde{\mathbb V}\left(|\widetilde{S}|\ge M/2\right)\to 0
\text{ as } M\to \infty.
\end{align*}
So, $S$ is tight.
Finally, (\ref{eqth1.2}) follows from Theorem \ref{th1}. $\Box$
\bigskip
For showing Theorem \ref{th4}, we need a more lemma.
\begin{lemma}\label{lem3} Let $\{X_n; n\ge 1\}$ be a sequence of independent random variables in a sub-linear expectation space $(\Omega, \mathscr{H}, \Sbep)$ with $ |X_k| \le c$, $\Sbep[X_k]\ge 0$ and $\Sbep[-X_k]\ge 0$, $k=1,2,\cdots$. Let $S_k=\sum_{i=1}^k X_i$. Suppose
\begin{equation}\label{eqlem3.1} \lim_{x\to \infty}\lim_{n\to \infty} \mathbb{V}\left(\max_{k\le n} |S_k|>x \right)<1.
\end{equation}
Then $\sum_{n=1}^{\infty}\Sbep[X_n]$, $\sum_{n=1}^{\infty}\Sbep[-X_n]$ and $\sum_{n=1}^{\infty}\Sbep[X_n^2]$ are convergent.
\end{lemma}
{\bf Proof.} By (\ref{eqlem3.1}), there exist $0<\beta<1$, $x_0>0$ and $n_0$, such that
$$\mathbb{V}\left(\max_{k\le n} |S_k|>x \right) <\beta, \;\;\text{for all } x\ge x_0, \; n\ge n_0. $$
By (\ref{eqKIQ2.1}),
$$ \sum_{k=1}^n \Sbep[X_k]\le \frac{x+c}{1-\beta},\;\; \text{for all } x\ge x_0, \; n\ge n_0. $$
So $\sum_{k=1}^{\infty} \Sbep[X_k]$ is convergent. Similarly, $\sum_{k=1}^{\infty} \Sbep[-X_k]$ is convergent.
Now, by (\ref{eqKIQ1}),
\begin{align*} \sum_{k=1}^n \Sbep[X_k^2] \le & \frac{(x+c)^2+2x\sum_{k=1}^n \big\{\big(\Sbep[X_k]\big)^++\big(\Sbep[-X_k]\big)^+\big\}}{1-\beta} \\
\le & \frac{(x+c)^2+2x\sum_{k=1}^{\infty} \big\{ \Sbep[X_k] + \Sbep[-X_k] \big\}}{1-\beta},\;\; \text{for all } x\ge x_0, \; n\ge n_0.
\end{align*}
So $\sum_{n=1}^{\infty} \Sbep[X_n^2]$ is convergent. The proof is completed.
$\Box$
\bigskip
{\bf Proof of Theorem \ref{th4}.} (i) By Lemma \ref{moment_v} and the condition (S3),
\begin{align*}
& \Capc\left(S_n-S_m-\sum_{k=m+1}^n \Sbep[X_k]\ge \epsilon\right)\\
\le & C\frac{\sum_{k=m+1}^n\Sbep\left[(X_k-\Sbep[X_k])^2\right]}{\epsilon^2}\to 0 \text{ as } n\ge m\to \infty.
\end{align*}
The convergence of $\sum_{n=1}^{\infty} \Sbep[X_n] $ implies $\sum_{k=m+1}^n \Sbep[X_k]\to 0$. It follows that
$$ \lim_{n\ge m\to \infty}\Capc\left(S_n-S_m \ge \epsilon\right)=0\; \text{ for all }\epsilon>0. $$
On the other hand, note $\Sbep[X_k]+\Sbep[-X_k]\ge 0$. The condition (S2) implies $\sum_{n=1}^{\infty}\left(\Sbep[X_k]+\Sbep[-X_k]\right)<\infty$, and then $\sum_{n=1}^{\infty}\left(\Sbep[X_k]+\Sbep[-X_k]\right)^2<\infty$. Hence, by the condition (S3) and the fact that $\Sbep\left[(-X_k-\Sbep[-X_k])^2\right]\le \Sbep\left[(X_k-\Sbep[X_k])^2\right]+(\Sbep[X_k]+\Sbep[-X_k])^2$,
$$\sum_{n=1}^{\infty}\Sbep\left[(-X_n-\Sbep[-X_n])^2\right]<\infty. $$
By considering $-X_n$ instead of $X_n$, we have
$$ \lim_{n\ge m\to \infty}\Capc\left(-S_n+S_m \ge \epsilon\right)=0\; \text{ for all }\epsilon>0. $$
It follows that (\ref{eqth2.4}) holds, i.e., $S_n$ is a Cauchy sequence in capacity $\Capc$.
(ii) Suppose that $S_n$ is a Cauchy sequence in capacity $\Capc$. Similar to (\ref{eqproofth1.2}), by applying the Levy inequality (\ref{eqLIQ2}) we have
\begin{equation}\label{eqproofth4.1} \lim_{n\ge m\to \infty} \Capc\left(\max_{m\le k\le n} |S_k-S_m|> \epsilon \right)=0 \text{ for all } \epsilon>0.
\end{equation}
Then
\begin{equation}\label{eqproofth4.2}\lim_{n\ge m\to \infty} \Capc\left(\max_{m\le k\le n} |X_k|\ge c \right)=0 \text{ for all } c>0.
\end{equation}
Write $v_k=\Capc\left( |X_k|\ge 2c \right)$. Similar to (\ref{eqprooflem2.3}), we have for $m_0$ large enough and all $n\ge m\ge m_0$,
\begin{align*}
\frac{1}{3}>\Capc\left(\max_{m\le k\le n} |X_k|\ge c \right)\ge 1-\frac{2}{ \sum_{k=m+1}^n v_k}.
\end{align*}
It follows that $\sum_{k=1}^{\infty} v_k<\infty$. The condition (S1) is satisfied for all $c>0$.
Next, we consider (S3). Write $X_n^c=(-c)\vee X_n\wedge c$ and $S_n^c=\sum_{k=1}^n X_k^c$. Note on the event $\{\max_{m\le k\le n} |X_k|< c\}$, $X_k^c=X_k$, $k=m+1,\cdots, n$. By (\ref{eqproofth4.1}) and (\ref{eqproofth4.2}),
\begin{equation}\label{eqproofth4.3} \lim_{n\ge m\to \infty} \Capc\left(\max_{m\le k\le n} |S_k^c-S_m^c|> \epsilon \right)=0 \text{ for all } \epsilon>0.
\end{equation}
Let $Y_1, Y_1^{\prime}, Y_2, Y_2^{\prime},\cdots, Y_n, Y_n^{\prime}, \cdots $ be independent random variables under the sub-linear expectation $\Sbep$ with $Y_k\overset{d}=Y_k^{\prime}\overset{d}= X_k^c$, $k=1,2,\cdots$. Then
$$\{Y_{m+1},\cdots, Y_n \} \overset{d}=\{Y_{m+1}^{\prime},\cdots, Y_n^{\prime}\} \overset{d}=\{X_{m+1}^c,\cdots, X_n^c\}. $$
Let $T_k=\sum_{i=1}^k Y_i$ and $T_k^{\prime}=\sum_{i=1}^k Y_i^{\prime}$. By (\ref{eqproofth4.3}),
\begin{align}\label{eqproofth4.4} & \lim_{n\ge m\to \infty} \Capc\left(\max_{m\le k\le n} |T_k-T_m|> \epsilon \right)\nonumber \\
= &\lim_{n\ge m\to \infty} \Capc\left(\max_{m\le k\le n} |T_k^{\prime}-T_m^{\prime}|> \epsilon \right)=0 \text{ for all } \epsilon>0.
\end{align}
Write $\widetilde{Y}_n=Y_n-Y_n^{\prime}$ and $\widetilde{T}_n=\sum_{k=1}^n \widetilde{Y}_k$. Then $\{\widetilde{Y}_n;n\ge 1\}$ is a sequence of independent random variables with $\Capc(|\widetilde{Y}_n|> 3c)=0$. Without loss of generality, we can assume $|\widetilde{Y}_n|\le 3c$ for otherwise we can replace $\widetilde{Y}_n$ by $(-3c)\vee \widetilde{Y}_n \wedge(3c)$. By (\ref{eqproofth4.4}),
$$ \lim_{n\ge m\to \infty} \mathbb{V}\left(\max_{m\le k\le n} |\widetilde{T}_k-\widetilde{T}_m|>2\epsilon \right)=0 \text{ for all } \epsilon>0.$$
Note $\Sbep[-\widetilde{Y}_k]=\Sbep[\widetilde{Y}_k]=(\Sbep[X_k^c]+\Sbep[-X_k^c])/2\ge 0$. By Lemma \ref{lem3},
$$ \sum_{n=1}^{\infty} \left(\Sbep[X_n^c]+\Sbep[-X_n^c]\right)\;\text{ and } \sum_{n=1}^{\infty} \Sbep[\widetilde{Y}_n^2] \text{ are convergent}. $$
Note
\begin{align*}
\Sbep\left[\widetilde{Y}_n^2|Y_n\right]\ge & \big(Y_n-\Sbep[Y_n]\big)^2+\Sbep\left[\big(Y_n^{\prime}-\Sbep[Y_n^{\prime}]\big)^2\right]\\
&+2\big(Y_n-\Sbep[Y_n]\big)^-\cSbep\left[Y_n^{\prime}-\Sbep[Y_n^{\prime}] \right].
\end{align*}
So
\begin{align*}
\Sbep\left[\widetilde{Y}_n^2\right]\ge & 2\Sbep\big[ \big(X_n^c-\Sbep[X_n^c]\big)^2\big]-2 \{\Sbep[X_n^c]+\Sbep[-X_n^c]\} \Sbep\left[\big(X_n^c-\Sbep[X_n^c]\big)^-\right]
\\
\ge & 2\Sbep\big[ \big(X_n^c-\Sbep[X_n^c]\big)^2\big]-2c \{\Sbep[X_n^c]+\Sbep[-X_n^c]\}.
\end{align*}
It follows that
\begin{equation} \label{eqproofth4.5}
\sum_{n=1}^{\infty}\Sbep\big[ \big(X_n^c-\Sbep[X_n^c]\big)^2\big]<\infty. \end{equation}
Since $\Sbep\big[ \big(-X_n^c-\Sbep[-X_n^c]\big)^2\big]\le \Sbep\big[ \big(X_n^c-\Sbep[X_n^c]\big)^2\big]+ \big(\Sbep[X_n^c+\Sbep[-X_n^c]\big)^2$, we also have
$$ \sum_{n=1}^{\infty}\Sbep\big[ \big(-X_n-\Sbep[-X_n]\big)^2\big]<\infty. $$
The condition (S3) is proved.
Finally, we consider (S2). For any $\epsilon>0$, when $m,n$ are large enough, $\sum_{k=m+1}^n \big(\Sbep[X_n^c]+\Sbep[-X_n^c]\big)<\epsilon$. By (\ref{eqproofth4.5}) and Lemma \ref{moment_v},
\begin{align*}
&\Capc\left(S_n^c-S_m^c-\sum_{k=m+1}^n \frac{\Sbep[X_k^c]-\Sbep[-X_k^c]}{2}\ge \epsilon\right)\\
= & \Capc\left(S_n^c-S_m^c-\sum_{k=m+1}^n \Sbep[X_k^c] \ge \epsilon-\sum_{k=m+1}^n \frac{\Sbep[-X_k^c]+\Sbep[X_k^c]}{2}\right)\\
\le & C \frac{\sum_{k=m+1}^n \Sbep\big[ \big(X_k^c-\Sbep[X_k^c]\big)^2\big]}{(\epsilon/2)^2}\to 0 \text{ as } n\ge m\to \infty.
\end{align*}
Similarly, by considering $-X_k^c$ instead of $X_k^c$ we have
\begin{align*}
\Capc\left(-S_n^c+S_m^c-\sum_{k=m+1}^n \frac{\Sbep[-X_k^c]-\Sbep[X_k^c]}{2}\ge \epsilon\right) \to 0 \text{ as } n\ge m\to \infty.
\end{align*}
It follows that, for any $\epsilon>0$,
$$
\Capc\left(\left|S_n^c-S_m^c-\sum_{k=m+1}^n \frac{\Sbep[X_k^c]-\Sbep[-X_k^c]}{2}\right|\ge \epsilon\right) \to 0 \text{ as } n\ge m\to \infty,
$$
which, together with (\ref{eqproofth4.3}), implies
$$ \sum_{k=m+1}^n \frac{\Sbep[X_k^c]-\Sbep[-X_k^c]}{2} \to 0 \text{ as } n\ge m\to \infty. $$
Hence, $\sum_{n=1}^{\infty} \big(\Sbep[X_k^c]-\Sbep[-X_k^c]\big)$ is convergent. Note that $\sum_{n=1}^{\infty} \big(\Sbep[X_k^c]+\Sbep[-X_k^c]\big)$ is convergent. We conclude that both
$\sum_{n=1}^{\infty} \Sbep[X_k^c]$ and $\sum_{n=1}^{\infty} \Sbep[-X_k^c]$ are convergent. The proof of (ii) is completed. $\Box$.
\section{Central limit theorem}\label{Sect CLT}
\setcounter{equation}{0}
In this section, we consider the sufficient and necessary conditions for the central limit theorem. We first recall the definition of G-normal random variables which is introduced by Peng \cite{peng2008a, peng2010}.
\begin{definition}\label{def4.1} ({\em G-normal random variable})
For $0\le \underline{\sigma}^2\le \overline{\sigma}^2<\infty$, a random variable $\xi$ in a sub-linear expectation space $(\widetilde{\Omega}, \widetilde{\mathscr H}, \widetilde{\mathbb E})$ is called a normal $N\big(0, [\underline{\sigma}^2, \overline{\sigma}^2]\big)$ distributed random variable (written as $\xi \sim N\big(0, [\underline{\sigma}^2, \overline{\sigma}^2]\big)$ under $\widetilde{\mathbb E}$), if for any $\varphi\in C_{l,Lip}(\mathbb R)$, the function $u(x,t)=\widetilde{\mathbb E}\left[\varphi\left(x+\sqrt{t} \xi\right)\right]$ ($x\in \mathbb R, t\ge 0$) is the unique viscosity solution of the following heat equation:
\begin{equation}\label{eqheatequation}\partial_t u -G\left( \partial_{xx}^2 u\right) =0, \;\; u(0,x)=\varphi(x),
\end{equation}
where $G(\alpha)=\frac{1}{2}(\overline{\sigma}^2 \alpha^+ - \underline{\sigma}^2 \alpha^-)$.
\end{definition}
That $\xi$ is a normal distributed random variable is equivalent to that, if $\xi^{\prime}$ is an independent copy of $\xi$ (i.e., $\xi^{\prime}$ is independent to $\xi$ and $\xi\overset{d}=\xi^{\prime})$, then
\begin{equation}\label{eqnormal} \widetilde{\mathbb E}\left[\varphi(\alpha \xi+\beta \xi^{\prime})\right]
=\widetilde{\mathbb E}\left[\varphi\big(\sqrt{\alpha^2+\beta^2}\xi\big)\right], \;\;
\forall \varphi\in C_{l,Lip}(\mathbb R) \text{ and } \forall \alpha,\beta\ge 0,
\end{equation}
(cf. Definition II.1.4 and Example II.1.13 of Peng \cite{peng2010}). We also write $\eta\overset{d}= N\big(0, [\underline{\sigma}^2, \overline{\sigma}^2]\big)$ if $\eta\overset{d}=\xi$ (as defined in Definition \ref{def1.2} (i)) and $\xi \sim N\big(0, [\underline{\sigma}^2, \overline{\sigma}^2]\big)$ (as defined in Definition \ref{def4.1}). By definition, $\eta\overset{d}=\xi$ if and only if for any $\varphi\in C_{b,Lip}(\mathbb R)$, the function $u(x,t)=\Sbep\left[\varphi\left(x+\sqrt{t} \eta\right)\right]$ ($x\in \mathbb R, t\ge 0$) is the unique viscosity solution of the equation (\ref{eqheatequation}). In the sequel, without loss of generality, we assume that the sub-linear expectation spaces $(\widetilde{\Omega}, \widetilde{\mathscr H}, \widetilde{\mathbb E})$ and $(\Omega, \mathscr{H},\Sbep)$ are the same.
Let $\{X_n; n\ge 1\}$ be a sequence of independent and identically distributed random variables in a sub-linear expectation space $(\Omega, \mathscr{H},\Sbep)$, $S_n=\sum_{k=1}^nX_k$. Peng \cite{peng2008a, peng2010} proved that, if $\Sbep[X_1]=\Sbep[-X_1]=0$ and $\Sbep[|X_1|^{2+\alpha}]<\infty$ for some $\alpha>0$, then
\begin{equation}\label{cltpeng}
\lim_{n\to \infty} \Sbep\left[\varphi\left(\frac{S_n}{\sqrt{n}}\right)\right]=\Sbep\left[\varphi(\xi )\right], \forall \varphi\in C_b(\mathbb R),
\end{equation}
where $\xi\sim N\left(0,[\underline{\sigma}^2,\overline{\sigma}^2]\right)$, $\overline{\sigma}^2=\Sbep[X_1^2]$ and $\underline{\sigma}^2=\cSbep[X_1^2]$. Zhang \cite{Zhang Exponential} showed that
$\Sbep[|X_1|^{2+\alpha}]<\infty$ can be weakened to $\Sbep[(X_1^2-c)^+]\to 0$ as $c\to\infty$ by applying the moment inequalities of sums of independent random variables and the truncation method. A nature question is whether $\Sbep[X_1^2]<\infty$ and $ \Sbep[X_1]=\Sbep[-X_1]=0$ are sufficient and necessary for (\ref{cltpeng}). The following theorem is our main result.
\begin{theorem}\label{thclt} Let $\{X_n; n\ge 1\}$ be a sequence of independent and identically distributed random variables in a sub-linear expectation space $(\Omega, \mathscr{H},\Sbep)$, $S_n=\sum_{k=1}^nX_k$. Suppose that
\begin{description}
\item[\rm (i) ] $\lim_{c\to\infty} \Sbep[X_1^2\wedge c]$ is finite;
\item[\rm (ii)] $x^2\Capc\left(|X_1|\ge x\right)\to 0$ as $x\to \infty$;
\item[\rm (iii)] $\lim_{c\to \infty}\Sbep\left[(-c)\vee X_1\wedge c)\right]=\lim_{c\to \infty}\Sbep\left[(-c)\vee (- X_1)\wedge c)\right]=0$.
\end{description}
Write $\overline{\sigma}^2=\lim_{c\to\infty} \Sbep[X_1^2\wedge c]$ and $\underline{\sigma}^2=\lim_{c\to\infty} \cSbep[X_1^2\wedge c]$. Then for any $\varphi\in C_b(\mathbb R)$,
\begin{equation}\label{clt1}
\lim_{n\to \infty} \Sbep\left[\varphi\left(\frac{S_n}{\sqrt{n}}\right)\right]=\Sbep\left[\varphi(\xi )\right],
\end{equation}
where $\xi\sim N\left(0,[\underline{\sigma}^2,\overline{\sigma}^2]\right)$.
Conversely, if (\ref{clt1}) holds for any $\varphi\in C_b^1(\mathbb R)$ and a random variable $\xi$ with $x^2\Capc\left(|\xi|\ge x\right)\to 0$ as $x\to \infty$, then
(i),(ii) and (iii) hold and $\xi\overset{d}= N\left(0,[\underline{\sigma}^2,\overline{\sigma}^2]\right)$.
\end{theorem}
Before prove the theorem, we give some remarks on the conditions. Note that $\Sbep[X_1^2\wedge c]$ and $\cSbep[X_1^2\wedge c]$ are non-decreasing in $c$. So, $\overline{\sigma}^2$ and $\underline{\sigma}^2$ are well-defined and nonnegative, and are finite if the condition (i) is satisfied. It is easily seen that, for $c_1>c_2>0$,
\begin{equation}\label{eqproofclt6}\left|\Sbep[X_1^{c_1}]-\Sbep[X_1^{c_2}]\right|\le \Sbep[(|X_1|\wedge c_1-c_2)^+]\le \frac{\overline{\sigma}^2}{c_2}.
\end{equation}
So, the condition (i) implies that $ \lim_{c\to \infty}\Sbep[X_1^{c}]$ and $ \lim_{c\to \infty}\Sbep[-X_1^{c}]$ exist and are finite.
If $\Sbep$ is a continuous sub-linear expectation, i.e., $\Sbep[X_n]\nearrow \Sbep[X]$ whenever $0\le X_n\nearrow X$, and $\Sbep[X_n]\searrow 0$ whenever $X_n\searrow 0$, $\Sbep[X_n]<\infty$,
then (i) is equivalent to $\Sbep[X_1^2]<\infty$, (iii) is equivalent to $\Sbep[X_1]=\Sbep[-X_1]=0$, and (ii) is automatically implied by $\Sbep[X_1^2]<\infty$. In general, the condition $\Sbep[X_1^2]<\infty$ and (i) with (ii) do not imply each other. However, it is easily verified that, if $\Sbep[(X_1^2-c)^+]\to 0$ as $c\to \infty$, then (i) and (ii) are satisfied and (iii) is equivalent to $\Sbep[X_1]=\Sbep[-X_1]=0$.
\bigskip
To prove Theorem \ref{thclt}, we need a more lemma.
\begin{lemma}\label{lem4.1} Let $X_{n1},\cdots X_{nn}$ be independent random variables in a sub-linear expectation space $(\Omega, \mathscr{H},\Sbep)$ with
$$ \frac{1}{\sqrt{n}}\sum_{k=1}^n \left\{\left|\Sbep[X_{nk}]\right|+\left|\Sbep[-X_{nk}]\right|\right\}\to 0, $$
$$ \frac{1}{n}\sum_{k=1}^n \left\{\big|\Sbep[X_{nk}^2]-\overline{\sigma}^2\big|+\big|\cSbep[X_{nk}^2]-\underline{\sigma}^2\big|\right\}\to 0 $$
and
$$ \frac{1}{n^{3/2}}\sum_{k=1}^n \Sbep[|X_{nk}|^3]\to 0. $$
Then
$$\lim_{n\to \infty} \Sbep\left[\varphi\left(\frac{\sum_{k=1}^n X_{nk}}{\sqrt{n}}\right)\right]=\Sbep\left[\varphi(\xi )\right], \forall \varphi\in C_b(\mathbb R), $$
where $\xi\sim N\left(0,[\underline{\sigma}^2,\overline{\sigma}^2]\right)$.
\end{lemma}
This lemma can be proved by refining the arguments of Li and Shi \cite{LiShi10} and can also follow from the Lindeberg central limit theorem \cite{Zhang Lindeberg}. We omit the proof here.
\bigskip
{\bf Proof of Theorem \ref{thclt}. } We first prove the sufficient part, i.e., (i),(ii) and (iii) $\implies$ (\ref{clt1}). Let $X_{nk}= (-\sqrt{n})\vee X_k \wedge \sqrt{n}$. Then
for any $\epsilon>0$,
$$\frac{1}{n^{3/2}} \sum_{k=1}^n \Sbep[|X_{nk}|^3]=\frac{1}{n^{1/2}} \Sbep[|X_{n1}|^3]\le \epsilon \overline{\sigma}^2+n\Capc\left(|X_1|\ge \epsilon\sqrt{n}\right)\to 0
$$
as $n\to \infty$ and then $\epsilon\to 0$,
by the condition (ii). Also,
\begin{align*}
& \frac{1}{n}\sum_{k=1}^n \left\{\big|\Sbep[X_{nk}^2]-\overline{\sigma}^2\big|+\big|\cSbep[X_{nk}^2]-\underline{\sigma}^2\big|\right\}\\
=& \big|\Sbep\left[X_1^2\wedge n\right]
-\overline{\sigma}^2\big|+\big|\cSbep\left[X_1^2\wedge n\right]
-\underline{\sigma}^2\big|\to 0,
\end{align*}
by (i). Note by (ii) and (i),
\begin{align*}
&\frac{1}{\sqrt{n}} \sum_{k=1}^n \left|\Sbep[X_{nk}]\right|=\sqrt{n} \left|\Sbep[X_{n1}]\right|\\
= & \sqrt{n} \lim_{c\to \infty} \left|\Sbep[X_{n1}]-\Sbep\left[(-c\sqrt{n})\vee X_1 \wedge (c\sqrt{n})\right]\right]\\
\le & \sqrt{n}\lim_{c\to \infty} \Sbep\left[ \left(|X_1| \wedge (c\sqrt{n})-x\sqrt{n}\right)^+\right]+ \sqrt{n}\Sbep\left[ \left(|X_1| \wedge (x\sqrt{n})- \sqrt{n}\right)^+\right]\\
\le & \frac{\overline{\sigma}^2}{x}+ x n\Capc\left(|X_1|\ge \sqrt{n}\right)\to 0 \; \text{ as } n\to \infty \text{ and then } x\to \infty,
\end{align*}
and similarly,
$$ \frac{1}{\sqrt{n}}\sum_{k=1}^n \left|\Sbep[-X_{nk}]\right|\to 0. $$
The conditions in Lemma \ref{lem4.1} are satisfied. We obtain
$$\lim_{n\to \infty} \Sbep\left[\varphi\left(\frac{\sum_{k=1}^n X_{nk}}{\sqrt{n}}\right)\right]=\Sbep\left[\varphi(\xi )\right]. $$
It is obvious that
$$\Sbep\left[\left|\varphi\left(\frac{\sum_{k=1}^n X_{nk}}{\sqrt{n}}\right)-\varphi\left(\frac{S_n}{\sqrt{n}}\right)\right|\right]\le \sup_x|\varphi(x)|n\Capc\left(|X_1|\ge \sqrt{n}\right) \to 0. $$
(\ref{clt1}) is proved.
Now, we consider the necessary part. Letting $\varphi=g_{\epsilon}\big(|x|-t\big)$ yields
$$ \limsup_{n\to \infty}\Capc\left(\frac{|S_n|}{\sqrt{n}}\ge t+\epsilon\right)\le \Capc\left(|\xi|\ge t \right) \text{ for all } t>0, \epsilon>0. $$
So
$$ \limsup_{n\ge m\to \infty}\max_{m\le k,l\le n} \Capc\left(\frac{|S_k-S_l|}{\sqrt{n}}\ge 2t+\epsilon\right)\le 2\Capc\left(|\xi|\ge t \right) \text{ for all } t>0, \epsilon>0. $$
Choose $t_0$ such that $\Capc\left(|\xi |\ge t_0\right)<1/(32)$. Applying the Levy maximal inequality (\ref{eqLIQ2}) yields
\begin{equation}\label{eqproofclt1} \limsup_{n\ge m\to \infty} \Capc\left(\frac{\max_{m\le k\le n}|S_k-S_m|}{\sqrt{n}}\ge 4t\right)< \frac{64}{31}\Capc\left(|\xi|\ge t \right) \text{ for all } t>t_0. \end{equation}
Hence
\begin{equation}\label{eqproofclt2} \limsup_{n\ge m\to \infty} \Capc\left(\frac{\max_{m\le k\le n}|X_k|}{\sqrt{n}}\ge 8t\right)< \frac{64}{31}\Capc\left(|\xi|\ge t \right) \text{ for all } t>t_0.
\end{equation}
Let $t_1>t_0$ and $m_0$ such that
\begin{equation}\label{eqproofclt3} \Capc\left(\frac{\max_{m\le k\le n}|S_k-S_m|}{\sqrt{n}}> 4t_1\right)< \frac{2}{31} \text{ for all } m\ge m_0
\end{equation}
and
\begin{equation}\label{eqproofclt4} \Capc\left(\frac{\max_{m\le k\le n}|X_k|}{\sqrt{n}}> 8t_1\right)< \frac{4}{31} \text{ for all } m\ge m_0.
\end{equation}
Write $Y_{nk}=(-8t_1)\vee\left(\frac{X_k}{\sqrt{n}}\right)\wedge(8t_1)$. Then by (\ref{eqproofclt3}) and (\ref{eqproofclt4}),
\begin{equation}\label{eqproofclt5} \Capc\left( \max_{m\le k\le n}\big|\sum_{j=m+1}^k Y_{nj}\big| > 4t_1\right)< \frac{2}{31} + \frac{4}{31}<\frac{1}{5} \text{ for all } m\ge m_0
\end{equation}
If $\Sbep[Y_{n1}]>0$, then by Lemma \ref{KolIneq} (ii),
$$\frac{1}{5}>1-\frac{4t_1+8t_1}{(n-m) \Sbep[Y_{n1}]}. $$
Hence $(n-m) \big(\Sbep[Y_{n1}])^+\le 15t_1$. Similarly, $(n-m) \big(\Sbep[-Y_{n1}])^+\le 15t_1$. Hence, by Lemma \ref{KolIneq} (i), it follows that
$$ \frac{1}{5}> 1-\frac{ (4t_1+8t_1)^2+8t_1 \big\{(n-m) \big(\Sbep[Y_{n1}])^++(n-m) \big(\Sbep[-Y_{n1}])^+\big\}}{ (n-m) \Sbep[Y_{n1}^2]}. $$
We conclude that $(n-m)\Sbep[Y_{n1}^2]\le \frac{5}{4}(12^2+240) t_1^2$. Choose $m=n/2$ and let $n\to \infty$. We have
$$ \lim_{c\to \infty}\Sbep[X_1^2\wedge c]=\lim_{n\to \infty} n \Sbep[Y_{n1}^2]\le \frac{5}{2}(12^2+240) t_1^2. $$
(i) is proved. Note that (i) implies that $ \lim_{c\to \infty}\Sbep[X_1^{c}]$ exists and is finite. Then
$$ \lim_{c\to \infty}\Sbep[X_1^{c}]=\limsup_{n\to \infty}\sqrt{n}\Sbep[Y_{n1}] \le \limsup_{n\to \infty}\frac{30t_1}{\sqrt{n}}=0. $$
Similarly, $ \lim_{c\to \infty}\Sbep[-X_1^{c}]$ exists, is finite and not positive. Note $\Sbep[-X_1^{c}]+\Sbep[X_1^{c}]\ge 0$.
Hence (iii) follows.
Finally, we show (ii). For any given $0<\epsilon<1/2$, by the condition $x^2\Capc(|\xi|\ge x)\to 0$, one can choose $t_1>t_0$ such that
$\frac{64}{31}\Capc(|\xi|\ge t_1)\le \frac{\epsilon}{9^3t_1^2}<1/2$. Then by (\ref{eqproofclt2}), there is $m_0$ such that
$$ \Capc\left(\frac{\max_{m\le k\le n}|X_k|}{\sqrt{n}}\ge 8t_1\right)< \frac{\epsilon}{9^3t_1^2}, \; n\ge m\ge m_0. $$
Choose $Z_k=g_{\epsilon}\big(\frac{|X_k|}{8t_1\sqrt{n}}-1\big)$ such that $I\{|X_k|\ge 9t_1\sqrt{n}\}\le Z_k\le I\{|X_k|\ge 8t_1\sqrt{n}\}$. Let $q_n=\Capc\left(|X_1|\ge 9t_1\sqrt{n}\right)$. Then
\begin{align*}
\Capc\left(\frac{\max_{m\le k\le n}|X_k|}{\sqrt{n}}\ge 8t_1\right)\ge & \Sbep\left[1-\prod_{k=m+1}^n(1-Z_k)\right]\\
=& 1-\prod_{k=m+1}^n(1-\Sbep[Z_k])\ge 1-e^{-(n-m)q_n}.
\end{align*}
It follows that
$$ n\Capc\left(|X_1|\ge 9t_1\sqrt{n}\right)\le 2(n-m)q_n< 2\times 2\times \frac{\epsilon}{9^3t_1^2}\text{ for } m=[n/2]\ge m_0. $$
Hence
$$ (9t_1\sqrt{n})^2 \Capc\left(|X_1|\ge 9t_1\sqrt{n}\right)< \frac{4 \epsilon}{9}, \;\; n\ge 2m_0. $$
When $x\ge 9t_1 \sqrt{2m_0}$, there is $n$ such that $9t_1\sqrt{n}\le x\le 9t_1\sqrt{n+1}$. Then
$$ x^2\Capc\left(|X_1|\ge x\right)\le (9t_1\sqrt{n+1})^2 \Capc\left(|X_1|\ge 9t_1\sqrt{n}\right)\le \frac{8 \epsilon}{9}. $$
It follows that $\limsup_{x\to \infty}x^2\Capc\left(|X_1|\ge x\right)<\epsilon$. (ii) is proved. The proof is now completed. $\Box$
\begin{remark} From the proof, we can find that
$$ \lim_{x\to \infty}\limsup_{n\to \infty}\Capc\left(\frac{|S_n|}{\sqrt{n}}\ge x\right)=0 $$
implies (i) and (ii). One may conjecture that,
\begin{description}
\item[\rm C1] if (\ref{clt1}) holds for any $\varphi\in C_b^1(\mathbb R)$ and a tight random variable $\xi$ (i.e., $\Capc\left(|\xi|\ge x\right)\to 0$ as $x\to \infty$), then (i), (ii) and (iii) holds and $\xi\overset{d}= N\left(0,[\underline{\sigma}^2,\overline{\sigma}^2]\right)$.
\end{description}
An equivalent conjecture is that,
\begin{description}
\item[\rm C2] if $\xi$ and $\xi^{\prime}$ are independent and identically distributed tight random variables, and
\begin{equation}\label{eqnormal2} \Sbep\left[\varphi(\alpha \xi+\beta \xi^{\prime})\right]
=\Sbep\left[\varphi\big(\sqrt{\alpha^2+\beta^2}\xi\big)\right], \;\;
\forall \varphi\in C_b(\mathbb R) \text{ and } \forall \alpha,\beta\ge 0,
\end{equation}
then $\xi\overset{d}= N\big(0,[\underline{\sigma}^2,\overline{\sigma}^2]\big)$, where $\overline{\sigma}^2=\lim_{c\to \infty}\Sbep[\xi^2\wedge c]$ and $\underline{\sigma}^2=\lim_{c\to \infty}\cSbep[\xi^2\wedge c]$.
\end{description}
It should be noted that the conditions (\ref{eqnormal}) and (\ref{eqnormal2}) are different. The condition (\ref{eqnormal}) implies that $\xi$ have finite moments of each order, but non information about the moments of $\xi$ is hidden in (\ref{eqnormal2}). As Theorem \ref{thclt}, the conjecture C2 is true when $x^2\Capc\left(|\xi|\ge x\right)\to 0$ as $x\to \infty$. In fact, let $X_1,X_2, \cdots, $ be independent random variables with $X_k\overset{d}=\xi$. Then by (\ref{eqnormal2}), $\frac{S_n}{\sqrt{n}}\overset{d}=\xi$. By the necessary part of Theorem \ref{thclt}, the conditions (i), (ii) and (iii) are satisfied. Then by the sufficient part of the theorem, $\xi\overset{d}=N\big(0,[\underline{\sigma}^2,\overline{\sigma}^2]\big)$. We don't known whether
conjectures C1 and C2 are true without assuming any moment conditions. It is very possible that they are not true in general. But finding a counterexample is not an easy task.
\end{remark}
\bigskip | 151,549 |
This tag are a little darker than your typical Valentine. I started by layering the tag with Pumice Stone Distress ink. Stampers Anonymous "Evidence" stamp is inked with Black Soot Distress. The script stamp was done in Pumice Stone too.
The heart was given a coat of Festive Berries Distress Stain. I then misted it and crumpled it up. I dried it and then ran Black Soot and Antique Photo all over to pick up the creases and wrinkles. The Tattered Wings were given the same treatment minus the distress stain.
The edges were roughed up using the Tonic Paper Distresser.
A memo pin to make it look like a specimen and it's done.
Dark and lovely.
I'm entering this one into the Stampotique Designers Challenge 133 Heart in your Art Challenge and the Fussy and Fancy Valentine Challenge. Last, but not least, I will enter into the Simon Says Stamp Wednesday Have a Heart Challenge
What have you been crafting?
Jess
What a lovely deep red colour that heart is!
Thank you for adding a heart to your art and joining us at stampotique
Thank you Jackie!
Gorgeous tag! I love everything on it.
Thanks for joining us at Stampotique DC this week :)
xx Arwen
Thank you Arwen. I love this tag too.
Wonderful tag Jess!! So very unique!! Well done...thank you for joining us at Fussy and Fancy "Be My Valentine"!! Good luck and hope to see you again!!
Thanks Beth! I didn't want to do a typical Valentine look.
Gorgeous tag, Jess! I love the heart with wings. Thanks so much for playing along with the Simon Says Stamp Wednesday Challenge: Have a Heart!
Thank you Emily. I really enjoy the SSS challenges.
Lovely tag. Thanks for joining us at Stampotique.
Thank you. I had fun making this one.
Love the vintage touch you have given this beautiful tag, Jess. I see the underlying emotion in this piece...definitely heart in this art! Thanks for joining us at Stampotique!
Thank you for the compliments. I really love this tag.
I love this tag, Jess. It looks precious.
Thanks for joining us at Stampotique DC!
Tanja
Thank you Tanja!
This is just gorgeous - so different from a usual Valentine, yet stunning! Thanks for joining in the Fussy and Fancy Challenge.
Hugs, Vannessa
F& F DT
Thank you for the compliments, Vannessa
Jess
Gorgeous Tag!!! I love the wings behind the heart. Thank you for joining us at Stampotique originals;.
This is a great Tag, every time I see it on our F&F Blog I think to myself, Oh I like that! xx | 84,208 |
History of the Golden Gate Bridge
A symposium was held to commemorate the 75th anniversary of the opening of the Golden Gate Bridge. Topics included the background of… read more
A symposium was held to commemorate the 75th anniversary of the opening of the Golden Gate Bridge. Topics included the background of the area before the bridge was built, details of its construction, and its impact on the region. The panelists showed slides as they discusses themes such as preservation, commerce, labor, geography, and demographics that are typically overlooked in standard accounts of the Golden Gate Bridge, which have traditionally focused on the bridge’s importance as an artistic and engineering feat. The panelists responded to questions from members of the audience. John King moderated.
“Spanning Space and Time: The Golden Gate Bridge and the Transformation of the Bay Area” was a program at the California Historical Society in partnership with San Francisco Architectural Heritage and the National Trust for Historic Preservation. close
*The transcript for this program was compiled from uncorrected Closed Captioning.
People in this video
- Gray Brechin Founder University of California, Berkeley->California’s Living New Deal Project
- Michael "Mike" Buhler Executive Director San Francisco Architectural Heritage
- Anthea M. Hartig Executive Director California Historical Society
- John King Critic San Francisco Chronicle->Urban Design
- Catherine Powell Director San Francisco State University->Labor Archives and Research Center
- Richard Walker Professor Emeritus University of California, Berkeley->Geography Department
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TITLE: Name of integration technique where product term is near constant over the interval?
QUESTION [2 upvotes]: Consider
$$\int_{\theta-\epsilon}^{\theta+\epsilon} g(x)f(x) dx$$
where f(x) is near constant on the interval $(\theta-\epsilon, \theta+\epsilon)$, and g(x) is not. It follows:
$$\int_{\theta-\epsilon}^{\theta+\epsilon} g(x)f(x) dx \approx f\left(\theta\right)\int_{\theta-\epsilon}^{\theta+\epsilon} g(x) dx$$
I can't find reference to this technique, is this valid? Is there maybe a better way to express this? Maybe some quadrature rule I've missed?
REPLY [1 votes]: It's not generally true, you need to refine what you mean by $\approx$ relative to $\epsilon$ and the derivative of $g$ otherwise pathological choices can be made. E.g. let $\theta = 0$, $f(x) = mx$ for some very small $m$ (i.e. "$m \approx 0$"), let $g(x) = \frac{1}{\epsilon^{100} m}x$, then one integral is $0$ but the other is probably pretty big (depending on $\epsilon$). | 202,526 |
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Keywords: Seasons, Vineland , Hospice Nurse - RN (Vineland,NJ), Healthcare , Vineland, New Jersey | 232,610 |
In mathematics, the Prékopa–Leindler inequality is an integral inequality closely related to the reverse Young's inequality, the Brunn–Minkowski inequality and a number of other important and classical inequalities in analysis. The result is named after the Hungarian mathematicians András Prékopa and László Leindler.
Let 0 < λ < 1 and let f, g, h : R<sup>n</sup> → [0, +∞) be non-negative real-valued measurable functions defined on n-dimensional Euclidean space R<sup>n</sup>. Suppose that these functions satisfy
{{NumBlk|:|$h \left( (1-\lambda)x + \lambda y \right) \geq f(x)^{1 - \lambda} g(y)^\lambda $|}}
for all x and y in R<sup>n</sup>. Then
$$
\| h\|_{1} := \int_{\mathbb{R}^n} h(x) \mathrm{d} x \geq \left( \int_{\mathbb{R}^n} f(x) \mathrm{d} x \right)^{1 -\lambda} \left( \int_{\mathbb{R}^n} g(x) \mathrm{d} x \right)^\lambda =: \| f\|_1^{1 -\lambda} \| g\|_1^\lambda.
$$
Recall that the essential supremum of a measurable function f : R<sup>n</sup> → R is defined by
$$
\mathop{\mathrm{esssup}}_{x \in \mathbb{R}^{n}} f(x) = \inf \left\{ t \in [- \infty, + \infty] \mid f(x) \leq t \text{ for almost all } x \in \mathbb{R}^{n} \right\}.
$$
This notation allows the following essential form of the Prékopa–Leindler inequality: let 0 < λ < 1 and let f, g ∈ L<sup>1</sup>(R<sup>n</sup>; [0, +∞)) be non-negative absolutely integrable functions. Let
$$
s(x) = \mathop{\mathrm{esssup}}_{y \in \mathbb{R}^n} f \left( \frac{x - y}{1 - \lambda} \right)^{1 - \lambda} g \left( \frac{y}{\lambda} \right)^\lambda.
$$
Then s is measurable and
$$
\| s \|_1 \geq \| f \|_1^{1 - \lambda} \| g \|_1^\lambda.
$$
The essential supremum form was given in. Its use can change the left side of the inequality. For example, a function g that takes the value 1 at exactly one point will not usually yield a zero left side in the "non-essential sup" form but it will always yield a zero left side in the "essential sup" form.
It can be shown that the usual Prékopa–Leindler inequality implies the Brunn–Minkowski inequality in the following form: if 0 < λ < 1 and A and B are bounded, measurable subsets of R<sup>n</sup> such that the Minkowski sum (1 - λ)A + λB is also measurable, then
$$
\mu \left( (1 - \lambda) A + \lambda B \right) \geq \mu (A)^{1 - \lambda} \mu (B)^{\lambda},
$$
where μ denotes n-dimensional Lebesgue measure. Hence, the Prékopa–Leindler inequality can also be used to prove the Brunn–Minkowski inequality in its more familiar form: if 0 < λ < 1 and A and B are non-empty, bounded, measurable subsets of R<sup>n</sup> such that (1 - λ)A + λB is also measurable, then
$$
\mu \left( (1 - \lambda) A + \lambda B \right)^{1 / n} \geq (1 - \lambda) \mu (A)^{1 / n} + \lambda \mu (B)^{1 / n}.
$$
The Prékopa–Leindler inequality is useful in the theory of log-concave distributions, as it can be used to show that log-concavity is preserved by marginalization and independent summation of log-concave distributed random variables. Suppose that H(x,y) is a log-concave distribution for (x,y) ∈ R<sup>m</sup> × R<sup>n</sup>, so that by definition we have
{{NumBlk|:|$H \left( (1 - \lambda)(x_1,y_1) + \lambda (x_2,y_2) \right) \geq H(x_1,y_1)^{1 - \lambda} H(x_2,y_2)^{\lambda},$|}}
and let M(y) denote the marginal distribution obtained by integrating over x:
$$
M(y) = \int_{\mathbb{R}^m} H(x,y) dx.
$$
Let y<sub>1</sub>, y<sub>2</sub> ∈ R<sup>n</sup> and 0 < λ < 1 be given. Then equation () satisfies condition () with h(x) = H(x,(1 - λ)y<sub>1</sub> + λy<sub>2</sub>), f(x) = H(x,y<sub>1</sub>) and g(x) = H(x,y<sub>2</sub>), so the Prékopa–Leindler inequality applies. It can be written in terms of M as
$$
M((1-\lambda) y_1 + \lambda y_2) \geq M(y_1)^{1-\lambda} M(y_2)^\lambda,
$$
which is the definition of log-concavity for M.
To see how this implies the preservation of log-convexity by independent sums, suppose that X and Y are independent random variables with log-concave distribution. Since the product of two log-concave functions is log-concave, the joint distribution of (X,Y) is also log-concave. Log-concavity is preserved by affine changes of coordinates, so the distribution of (X + Y, X − Y) is log-concave as well. Since the distribution of X+Y is a marginal over the joint distribution of (X + Y, X − Y), we conclude that X + Y has a log-concave distribution.
The Prékopa–Leindler inequality can be used to prove results about concentration of measure.
Theorem Let $ A \subseteq \mathbb{R}^n $, and set $ A_{\epsilon} = \{ x : d(x,A) < \epsilon \} $. Let $ \gamma(x) $ denote the standard Gaussian pdf, and $ \mu $ its associated measure. Then $ \mu(A_{\epsilon}) \geq 1 - \frac{ e^{ - \epsilon^2/4}}{\mu(A)} $.
The proof of this theorem goes by way of the following lemma:
Lemma In the notation of the theorem, $ \int_{\mathbb{R}^n} \exp ( d(x,A)^2/4) d\mu \leq 1/\mu(A) $.
This lemma can be proven from Prékopa–Leindler by taking $ h(x) = \gamma(x), f(x) = e^{ \frac{ d(x,A)^2}{4}} \gamma(x), g(x) = 1_A(x) \gamma(x) $ and $ \lambda = 1/2 $. To verify the hypothesis of the inequality, $ h( \frac{ x + y}{2} ) \geq \sqrt{ f(x) g(u)} $, note that we only need to consider $ y \in A $, in which case $ d(x,A) \leq ||x - y|| $. This allows us to calculate:
$$
(2 \pi)^n f(x) g(x) = \exp( \frac{ d(x,A) }{4} - ||x||^2/2 - ||y||^2/2 ) \leq \exp( \frac{ ||x - y||^2 }{4} - ||x||^2/2 - ||y||^2/2 ) = \exp ( - ||\frac{x + y}{2}||^2 ) = (2 \pi)^n h( \frac{ x + y}{2})^2.
$$
Since $ \int h(x) dx = 1 $, the PL-inequality immediately gives the lemma.
To conclude the concentration inequality from the lemma, note that on $ \mathbb{R}^n \setminus A_{\epsilon} $, $ d(x,A) > \epsilon $, so we have $ \int_{\mathbb{R}^n} \exp ( d(x,A)^2/4) d\mu \geq ( 1 - \mu(A_{\epsilon})) \exp ( \epsilon^2/4) $. Applying the lemma and rearranging proves the result.
| 216,947 |
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\begin{document}
\begin{center}
\begin{LARGE}
{\bf On singularities of dynamic response functions in the massless regime of the XXZ spin-1/2 chain}
\end{LARGE}
\vspace{1cm}
\vspace{4mm}
{\large Karol K. Kozlowski \footnote{e-mail: [email protected]}}
\\[1ex]
Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, F-69342 Lyon, France \\[2.5ex]
\par
\end{center}
\vspace{40pt}
\centerline{\bf Abstract} \vspace{1cm}
\parbox{12cm}{\small}
This work extracts, by means of an exact analysis, the singular behaviour of the dynamical response functions -the Fourier transforms of dynamical two-point functions-
in the vicinity of the various excitation thresholds in the massless regime of the XXZ spin-1/2 chain. The analysis yields the edge exponents and associated amplitudes
which describe the local behaviour of the response function near a threshold. The singular behaviour is derived starting from first principle considerations:
the method of analysis \textit{does not rely, at any stage}, on
some hypothetical correspondence with a field theory or other phenomenological approaches.
The analysis builds on the massless form factor expansion for the response functions of the XXZ chain obtained recently by the author.
It confirms the non-linear Luttinger based predictions relative to the power-law behaviour and of the associated edge exponents
which arise in the vicinity of the dispersion relation of one massive excitation (hole, particle or bound state).
In addition, the present analysis shows that, due to the lack of strict convexity of the particles dispersion relation
and due to the presence of slow velocity branches of the bound states, there exist excitation thresholds with a different structure of edge exponents. These origin from multi-particle/hole/bound state excitations
maximising the energy at fixed momentum.
\vspace{40pt}
\tableofcontents
\section{An outline of the problem and main results}
\subsection{The XXZ chain}
Due to the substantial progress which took place in experimental condensed matter physics, one-dimensional models of quantum many body physics evolved from a status of purely theoretical toy-models
of many body physics to
concrete compounds exhibiting a genuine one-dimensional behaviour. Even more remarkably, there exist a plethora of compounds
whose properties are grasped, within a very good precision, by one-dimensional quantum integrable Hamiltonians.
The most prominent example is probably given by the XXZ spin-$\tf{1}{2}$ chain in an external longitudinal magnetic $h$. The Hamiltonian of the model takes the form
\beq
\op{H} \, = \, J \sum_{a=1}^{L} \Big\{ \sigma^x_a \,\sigma^x_{a+1} +
\sigma^y_a\,\sigma^y_{a+1} + \De \,\sigma^z_a\,\sigma^z_{a+1}\Big\} \, - \, \f{h}{2} \sul{a=1}{L} \sg_{a}^{z} \; .
\label{ecriture hamiltonien XXZ}
\enq
Here $J>0$ represents the so-called exchange interaction, $\De$ is the anisotropy parameter,
$h>0$ the external magnetic field and $L \in 2\mathbb{N}$ corresponds to the number of sites. $\op{H}$ acts on the Hilbert space $\mf{h}_{XXZ}=\otimes_{a=1}^{L}\mf{h}_a$ with $\mf{h}_a \simeq \Cx^2$,
$\sg^{w}$, $w=x,y,z$, are the Pauli matrices and the operator $\sg_a^{w}$ acts as the Pauli matrix $\sg^{w}$
on $\mf{h}_a$ and as the identity on all the other spaces, \textit{viz}.
\beq
\sg_a^{w} \; = \; \underbrace{ \e{id}\otimes \cdots \otimes \e{id} }_{ a-1 \; \e{times} } \otimes \; \sg^{w} \otimes \underbrace{ \e{id}\otimes \cdots \otimes \e{id} }_{ L-a \; \e{times} } \;.
\enq
Finally, the model is subject to periodic boundary conditions, \textit{viz.} $\sg_{a+L}^{\ga}=\sg_{a}^{\ga}$.
\vspace{2mm}
Crystals such as $\e{K}\e{Cu}\e{F}_3$ \cite{NaglerTennantCowleyPerringSatijaTestOfXXXDispersionRelationForKCuF3} or
$\e{Cu}(\e{C}_4\e{H}_4\e{N}_2)(\e{N}\e{O}_3)_2$ \cite{HammarStoneReichBroholmGibsonLandeeOshikawaCu(C4H4N2)(NO3)2AsXXXMagnetIdentified} have been identified
to be well-grasped by the isotropic XXX Hamiltonian, \textit{viz}. the Hamiltonian $\op{H}$ given in \eqref{ecriture hamiltonien XXZ} when $\De$ is set to $1$.
In its turn, the behaviour of $\e{CsCoCl}_3$ has been found to be well-captured \cite{GoffTennantNaglerCsCoCl3IndetifiedasMassiveXXZChain} by the XXZ antiferromagnetic Heisenberg chain with $\De \simeq 10$
while certain aspects of the behaviour of the spin-ladder compound $(\e{C}_5\e{H}_{12}\e{N})_2\e{Cu}\e{Br}_4$ are well-described \cite{BinerBoehmCauxGudelHabichtKieferKramerLauchliMcMorrowMesotNormandRueggRonnowStahnThielemann(C5H12N)2CuBr4AsDeltaHalfXXZChain}
by an effective XXZ Hamiltonian with $\De=1/2$.
Most experiments on the above and many other effectively one-dimensional materials measure the Fourier transforms of two-point correlation functions -the so-called dynamic response functions (DRF)- and typically rely on techniques
such as Bragg \cite{StamperetalMeasureStructureFactorbyBraggSpectroscopyonBEC} or photoemission spectroscopy or inelastic neutron scattering
\cite{LakeTennantCauxBarthelSchollwockNaglerFrostKCUF3DSFComparisionFromExpMEasureAndABA,StoneReichBroholmLefmannRischelLandeeTurnbullVeryClearDSFMeasureForXXXMagnetCu(C4H4N2)(NO3)2}.
In fact, most experiments take place at rather low temperatures, what effectively means that they measure, with good accuracy, the zero-temperature DRF.
In the case of the XXZ chain, the zero temperature DRFs\symbolfootnote[4]{The connectedness of the correlator allows one to regularise the convergence of the transforms at infinity.} take the form
\beq
\msc{S}^{(\ga)}(k,\om) \, = \, \sul{ m \in \mathbb{Z} }{} \Int{ \R }{} \big< (\sg_1^{\ga})^{\dagger}\!(t) \, \sg_{m+1}^{\ga} (0) \big>_{\e{c}} \cdot \ex{ \i(\om t - k m) } \dd t \;.
\label{definition dyn resp fct general}
\enq
Above, $\dagger$ stands for the Hermitian conjugation, and
the integrand refers to the presumably existing\symbolfootnote[3]{The results of \cite{KitanineMailletTerrasElementaryBlocksPeriodicXXZ,KozProofOfDensityOfBetheRoots,YangYangXXZproofofBetheHypothesis} put together entail
the existence of the limit at $t=0$ in that they provide a rigorous derivation of a well-defined multiple integral representation for the reduced density matrix of the chain. An appropriate trace thereof
allows one to compute $\big< (\sg_1^{\ga})^{\dagger}\!(0) \, \sg_{m+1}^{\ga} (0) \big>_{\e{c}}$. Note that the existence of the limit at $t=0$ also follows from the general theory developed in \cite{RuelleRigorousResultsForStatisticalMechanics}.
It is also fairly easy to see that the limit \eqref{definition comme limite volume infini correlateurs connexes} exists for extracted subsequences in $L$.} infinite volume limit of the connected dynamical two-point function at zero temperature
\beq
\big< (\sg_1^{\ga})^{\dagger}\!(t) \, \sg_{m+1}^{\ga}(0) \big>_{\e{c}} \; = \;
\lim_{L \tend + \infty} \bigg\{ \Big( \Om, (\sg_1^{\ga})^{\dagger}\!(t) \, \sg_{m+1}^{\ga} (0) ,\Om \Big) \, - \, \Big| \Big( \Om, \sg_{1}^{\ga} ,\Om \Big) \Big|^2 \bigg\} \;.
\label{definition comme limite volume infini correlateurs connexes}
\enq
Here $\Om$ stands for the model's ground state while the time and space evolution of a spin operator takes the form
\beq
\sg^{\ga}_{m+1}(t)\, = \, \ex{\i m \op{P} + \i \op{H} t} \cdot \sg^{\ga}_1 \cdot \ex{ -\i t \op{H} -\i m \op{P}} \;,
\enq
where $\op{P}$ is the momentum operator and, hence, $\ex{\i \op{P} } $ the translation operator by one-site.
\subsubsection{Singularities of response functions}
Taken that dynamic response functions are natural experimental observables, there is a clear demand to build effective and reliable theoretical
tools allowing for their study, at least in some limiting regimes, and providing a satisfactory explanation of the experimental observations.
Typically, dynamic response functions in one-dimensional models are observed to exhibit a singular structure in the momentum $k$ -- frequency $\om$ plane.
Namely, at fixed momentum $k$, they exhibit a power-law behaviour $(\de \om)^{\mu}$ in $\de \om=\om-\mc{E}(k)$, this in the vicinity of certain curves $(k,\mc{E}(k))$.
The curves $k\mapsto \mc{E}(k)$ correspond to dispersion relations of the excitations that are at the root of generating the given non-analytic behaviour.
The edge exponent $\mu$ governing a given singularity may be positive or negative. The range of possible values of the edge exponent $\mu$ strongly depends on whether the model is in a massive or massless phase
and, in the latter case, on the universality class governing the massless regime.
In fact, the singular structure of the DRF, and in particular the form taken by the edge exponents is deeply connected with the critical exponents driving the long-distance and large-time power-law decay
of the real space correlators. In the massive case, one expects, unless some non-generic accident happens, that this decay is driven by Gaussian saddle-points, be it in one or several dimensions.
Thus, in the massive case, the edge exponents are expected to be of the form $-\tf{1}{2}+n$, $n\geq 0$ an integer, the typical behaviour being
either a square root divergence or a square root cusp in the vicinity of the dispersion curves $k\mapsto \mc{E}(k)$. The situation appears to be much richer in a massless model precisely due to the existence
of infinitely many zero energy excitations. The latter generate a non-trivial tower of critical exponents which give rise
to edge exponents $\mu$ that, generically, exhibit a dependence on the momentum $k$, can be positive or negative and which are, generically, non-rational.
Ideally, one would like to have at one's disposal tools allowing one to unravel the mentioned singularity structure of the DRFs for a generic, not necessarily integrable,
one-dimensional model at zero temperature. The approach should also provide accurate and explicit enough predictions.
\subsection{The main achievement of the work}
A reasonable path for achieving the goal described above appears to start by devising exact tools allowing one to fully describe the singularity structure of the DRF
in at least some instances of quantum integrable models; indeed, then, one can hope to rely on the exact solvability of the model which provides one with numerous
additional algebraic properties allowing one to simplify the calculations. As will be discussed below, such calculations could have been carried out, at least in part, for some examples of quantum integrable models.
However, what would be really useful for the purpose of unravelling a larger picture would be to construct tools and a framework of analysis allowing one to stay as close as possible to objects
and pictures usually used in condensed matter physics. The success of such an approach could then allow, by extrapolating the features responsible for the emergence of singularities in an
integrable model, to devise an exact phenomenological approach allowing one to grasp the universal part of the structure of DRFs, at least, in certain classes of non-integrable models.
By exact phenomenological approach, I mean one being able to produce an exact and analysable to the end expression for the DRF in which the building blocks
will be given by specific to the model -but not explicit- functions and such that the part responsible for the singular behaviour of the DRF is
captured by a universal structure common to all models belonging to the universality class of interest.
In conjunction with the representations that were obtained in my previous work \cite{KozMasslessFFSeriesXXZ}, this is precisely the program that is achieved in this work.
By starting from the massless form factor\symbolfootnote[4]{I remind that a "form factor" refers to a matrix element of some local operator taken between two Eigenstates of the model's
Hamiltonian. Such objects are well-defined in finite volume $L$ as it is the case for the XXZ Hamiltonian \eqref{ecriture hamiltonien XXZ}. See \cite{KitanineMailletTerrasFormfactorsperiodicXXZ}
where finite-size determinant representation for these objects have been obtained} based representation, which I
obtained in \cite{KozMasslessFFSeriesXXZ} for the zero temperature DRF of the massless XXZ spin-1/2 chain, I develop a method of rigorous analysis of the behaviour of
each multiple integral present in the series. While the construction of the series that was carried out in \cite{KozMasslessFFSeriesXXZ} relies on a certain amount of hypotheses that are yet to be proven to hold,
the analysis of each multiple integral carried out in this work is rigorous.
This allows me to extract the singular behaviour of the DFRs for the XXZ chain and hence determine, through an exact approach, the value of edge exponents $\mu$, singularities curves
$k \mapsto \mc{E}(k)$ and amplitudes characterising the singularities in the $(k,\om)$ plane. Doing so, allows me to:
\begin{itemize}
\item[{\bf i)}] test and confirm the predictions, issuing form the existing heuristic methods, in respect to the structure of the subset of the singularities associated with one particle/hole/bound state excitations;
\item[{\bf ii)}] fully analyse the effect of multi-particle/hole/bound state processes in the generation of the excitation thresholds. These thresholds take origin in that the velocity of the excitations is \textit{not} monotonously increasing
and, more importantly, in that particles, holes or bound states may share same values of their velocities.
Such multi-particle thresholds were, so far, mostly unaccounted for within the existing heuristic methods and not all of the effects present at such thresholds were
fully grasped.
\end{itemize}
I stress that this is the first \textit{ab inicio} calculation of the singularities of the response functions in the XXZ spin-$1/2$ chain, an interacting integrable model containing bound states.
An important point is that the approach developed in the present work is \textit{universal} in the sense given earlier. I argued in \cite{KozMasslessFFSeriesXXZ} that the massless form factor series
expansions of the zero temperature DRF
that I obtained for the massless regime of the XXZ spin-$1/2$ chain has, in fact, a universal form that should be shared by all models belonging to the Luttinger liquid universality class.
I refer to that paper for a more precise discussion of that fact.
As a consequence, the techniques of analysis -up to trivial modifications- developed in this work will allow one to grasp the singular structure of DRFs in models belonging to the Luttinger liquid universality class.
Of course, these will then only be phenomenological results since, for a general model, one does not have an explicit access to form factor densities of local operators
or to dispersion relations of the elementary excitations. However, such an approach is not so uncommon in physics and, more importantly,
the approximations made to get the result are genuinely constructive and do not rely on this or that heuristics which, in concrete situations, might turn out to be
complicated to verify or even simply to have an intuition of. Furthermore, the data (form factor densities, dispersion relations) on which the phenomenological approach builds can, in principle,
be computed perturbatively in the vicinity of a free theory, at least on formal grounds \cite{CauxGlazmanImambekovShasiNonUniversalPrefactorsFromFormFactors}.
\subsection{The principal theorems}
On technical grounds, the main achievement of this work are the two theorems given below. These results allow one to grasp the small parameter asymptotic expansion
of a class of integrals which, upon specialisation, correspond to the one arising in the series expansion for the dynamic response functions obtained in \cite{KozMasslessFFSeriesXXZ}.
In order to state these theorems, I first need to introduce a specific class of smooth functions. This definition involves smooth functions on closed set,
see Definition \ref{Definition fct lisse sur ferme} for a precise characterisation of the concept.
\begin{defin}
\label{definition smooth class on K}
Given $K$ a compact subset $\R^n$ for some $n\in \mathbb{N}^*$, a function $\msc{G}$ on $K\times \R^+\times \R^+$ is said to be in the smooth class of $K$ associated with functions $d_{\pm}$
and constant $\tau \in \intoo{0}{1}$, if there exists a decomposition
\bem
\msc{G}\big( \bs{x}, u , v \big) \, = \, d_{+}(\bs{x}) \, d_{-}(\bs{x}) \, \msc{G}^{(1)}\big( \bs{x} \big) \; + \; d_{-}(\bs{x}) \, \msc{G}^{(2)}\big( \bs{x}, u \big) \cdot [u]^{1-\tau} \\
\; + \; d_{+}(\bs{x}) \, \msc{G}^{(3)}\big( \bs{x}, v \big) \cdot [v]^{1-\tau} \; + \; \msc{G}^{(4)}\big( \bs{x}, u , v \big) \cdot [u\, v]^{1-\tau} \;,
\label{ecriture decomposition smooth class K}
\end{multline}
where $\msc{G}^{(1)}$ is smooth on $ K $, $\msc{G}^{(2)}, \msc{G}^{(3)}$ are smooth and bounded on $ K \times \R^+$,
$\msc{G}^{(4)}$ is smooth and bounded on $ K \times \R^+\times \R^+$.
\vspace{2mm}
These functions are such that, for any $(\bs{s},\ell_u,\ell_v) \in \mathbb{N}^{n} \times\mathbb{N}\times \mathbb{N}$,
$s\in \intn{1}{4}$ and $\eps>0$
\begin{itemize}
\item[Hi)] $ \pl{a=1}{ n } \Dp{ x_a }^{s_a } \, \cdot \, \Dp{u}^{\ell_u} \, \Big\{ \msc{G}^{(s)}\big( \bs{x}, u \big) [u]^{1-\tau} \Big\} \, = \, \e{O}\Big( [u]^{1-\tau-\ell_u} \Big) $
uniformly in $\bs{x}\in K$, $u\in \intof{ 0 }{ \eps^{-1} }$ and for $s=2,3$;
\item[Hii)] $ \pl{ a=1 }{ n }\Dp{ x_a }^{s_a } \, \cdot \, \Dp{u}^{\ell_u} \, \cdot \, \Dp{v}^{\ell_v} \, \Big\{ \msc{G}^{(4)}\big( \bs{x}, u , v \big) [u v]^{1-\tau} \Big\}
\, = \, \e{O}\Big( [u]^{1-\tau-\ell_u} [v]^{1-\tau-\ell_v} \Big)$ uniformly in $\bs{x}\in K$, $(u,v)\in \intof{ 0 }{ \eps^{-1} }^2$.
\end{itemize}
\vspace{2mm}
Finally, if $n \geq 2$, the functions $\msc{G}^{(s)}$, $s\in \intn{1}{4}$, along with any of their partial derivatives, all vanish on $\Dp{}K$, $\Dp{}K \times \R^+$, $\Dp{}K \times \R^+\times \R^+$.
\end{defin}
\vspace{2mm}
The first theorem deals with the case of one-dimensional integrals.
\vspace{2mm}
\begin{theorem}
\label{Theorem Principal cas 1D}
Let $a<b$ be two reals. Let $\mf{z}_{\pm}(\la)$ be two real-holomorphic functions in a neighbourhood of the interval $\msc{J}=\intff{a}{b}$, such that
\begin{itemize}
\item all the zeroes of $\mf{z}_{\pm}$ on $\msc{J}$ are simple;
\item[$\bullet$] $\mf{z}_{+}$ and $\mf{z}_{-}$ admit a unique common zero $\la_0 \in \e{Int}(\msc{J})$ that, furthermore, is such that $\mf{z}_{+}^{\prime}(\la_0) \not= \mf{z}_{-}^{\prime}(\la_0)$.
\end{itemize}
Let $\De_{\ups}$ be real analytic on $\e{Int}(\msc{J})$ and such that $\De_{\ups} \geq 0$. Let $\msc{G}$ be in the smooth class of $\msc{J}$
associated with the functions $\De_{\pm}$ and with a constant $\tau$.
Then, for $\mf{x}\not=0$ and small enough,
\beq
\la \mapsto \msc{G}\Big(\la, \wh{\mf{z}}_{+}(\la), \wh{\mf{z}}_{-}(\la)\Big) \cdot \pl{\ups= \pm }{}
\Big\{ \Xi\big( \; \wh{\mf{z}}_{\ups}(\la) \big) \cdot \big[ \, \wh{\mf{z}}_{\ups}(\la) \big]^{ \De_{\ups}(\la)-1 } \Big\} \in L^{1}\big( \msc{J} \big)
\enq
where $\wh{\mf{z}}_{\pm}(\la)=\mf{z}_{\pm}(\la)+ \mf{x}$. Let $\mc{I}(\mf{x})$ denote the integral
\beq
\mc{I}(\mf{x})\, = \, \Int{ \msc{J} }{} \msc{G}\Big(\la, \wh{\mf{z}}_{+}(\la), \wh{\mf{z}}_{-}(\la)\Big) \cdot \pl{\ups= \pm }{}
\Big\{ \Xi\big( \; \wh{\mf{z}}_{\ups}(\la) \big) \cdot \big[ \, \wh{\mf{z}}_{\ups}(\la) \big]^{ \De_{\ups}(\la)-1 } \Big\} \cdot \dd \la \;.
\enq
\vspace{2mm}
Assume that $ \de_{\pm} \, = \, \De_{\pm}(\la_0) >0$.
\vspace{2mm}
\noindent {\bf a) } If $\mf{z}_{+}^{\prime}(\la_0)\cdot \mf{z}_{-}^{\prime}(\la_0) <0$, then $\mc{I}(\mf{x})$ admits the $\mf{x} \tend 0$ asymptotic expansion
\beq
\mc{I}(\mf{x})\, = \, \Xi\Big( \mf{z}_{+}^{\prime}(\la_0) \cdot \mf{X} \Big) \cdot \Bigg\{
\f{ \msc{G}^{(1)}(\la_0) \cdot \de_+ \de_- \cdot | \mf{X} |^{\de_+ + \de_- - 1} }
{ |\, \mf{z}_{+}^{\prime}(\la_0) |^{\de_-} \cdot | \, \mf{z}_{-}^{\prime}(\la_0) |^{\de_+} }
\cdot \f{ \Ga\big( \de_+ \big) \cdot \Ga\big(\de_- \big) }{ \Ga\big( \de_++\de_-\big) }
\, + \, \e{O}\Big( |\mf{x}|^{ \de_+ + \de_- - \tau} \Big) \Bigg\}
\, + \, f_{<}(\mf{x})
\enq
where
\beq
\mf{X} \, = \, \mf{x} \cdot \big[ \mf{z}_{+}^{\prime}(\la_0)-\mf{z}_{-}^{\prime}(\la_0) \big] \;,
\enq
$\msc{G}^{(1)}$ is as appearing in \eqref{ecriture decomposition smooth class K}
and $f_{<}$ is a smooth function of $ \mf{x}$. Furthermore, if $\mf{z}_{\pm}$ have no zeroes on $\msc{J}$ other than $\la_0$, then $f_{<}=0$.
\vspace{2mm}
\noindent {\bf b) } If $\mf{z}_{+}^{\prime}(\la_0)\cdot \mf{z}_{-}^{\prime}(\la_0) >0$, then $\mc{I}(\mf{x})$ admits the $\mf{x} \tend 0$ asymptotic expansion
\bem
\mc{I}(\mf{x})\, = \,
\f{ \msc{G}^{(1)}(\la_0) \cdot \de_+ \, \de_- \cdot | \mf{X} |^{\de_+ + \de_- - 1} }
{ |\, \mf{z}_{+}^{\prime}(\la_0) |^{\de_-} \cdot | \, \mf{z}_{-}^{\prime}(\la_0) |^{\de_+} }
\cdot \Ga\big( \de_+ \big) \cdot \Ga\big(\de_- \big) \cdot \Ga\big( 1- \de_+ - \de_- \big) \\
\times \bigg\{ \Xi(\mf{x}) \tfrac{1}{\pi} \sin\big[ \pi \de_{\mf{p}} \big] \, + \, \Xi(-\mf{x}) \tfrac{1}{\pi} \sin\big[ \pi \de_{-\mf{p}} \big] \bigg\}
\, + \, \e{O}\Big( |\mf{x}|^{ \de_+ + \de_- - \tau } \Big)
\; + \; f_{>}(\mf{x})
\end{multline}
where $ \mf{X} $ and $\de_{\pm}$ are as above,
\beq
\mf{p}\, = \, - \e{sgn} \big[ \mf{z}_{+}^{\prime}(\la_0) \big] \cdot \e{sgn} \big[ \mf{z}_{+}^{\prime}(\la_0)-\mf{z}_{-}^{\prime}(\la_0) \big]
\enq
and $f_{>}$ is a smooth function of $ \mf{x}$.
\end{theorem}
The second theorem, deals with a multi-dimensional analogue of the integral given in \eqref{definition integrale 1D type Beta genralise}. Its statement demands to introduce a few notations and objects. One assumes to be given:
\begin{itemize}
\item a strictly positive real $\op{v} >0$;
\item a choice of signs $\zeta_r\in \{ \pm \}$;
\item a collection of compact intervals $\msc{I}_{r}$, $r=1,\dots, \ell$ ;
\item smooth functions $\mf{u}_r$ on $\msc{I}_r$ such that $\mf{u}^{\prime}_r$ is strictly monotonous on $\msc{I}_r$, and such that
\beq
\mf{u}^{\prime}_r(k) \not= \pm \op{v} \qquad \e{for} \qquad k \in \e{Int}\big(\msc{I}_{r} \big) \;.
\enq
\end{itemize}
The intervals $\msc{I}_r$ are such that they partition as
\beq
\msc{I}_{r}\; = \; \msc{I}_{r}^{(\e{in})}\sqcup \msc{I}_{r}^{(\e{out})} \qquad \e{with} \qquad \msc{I}_{1}\; = \; \msc{I}_{1}^{(\e{in})}
\enq
so that
\beq
\mf{u}_{r}^{\prime}\Big( \e{Int}\big(\msc{I}_{r}^{(\e{out})} \big) \Big) \, \cap \, \mf{u}_{1}^{\prime}\Big( \e{Int}\big(\msc{I}_{1}^{(\e{in})} \big) \Big) \, = \, \emptyset
\qquad \e{and} \qquad
\mf{u}_{r}^{\prime}\Big( \e{Int}\big(\msc{I}_{r}^{(\e{in})} \big) \Big) \, = \, \mf{u}_{1}^{\prime}\Big( \e{Int}\big(\msc{I}_{1}^{(\e{in})} \big) \Big) \;.
\enq
The above ensures that there exist homeomorphisms
\beq
t_r \, : \, \msc{I}_{1}^{(\e{in})} \tend \msc{I}_{r}^{(\e{in})} \qquad \e{such} \; \e{that} \qquad
\mf{u}_1^{\prime}(k) \; = \; \mf{u}_r^{\prime}\big( t_r(k) \big) \;.
\enq
\vspace{2mm}
One defines the macroscopic "momentum" and "energy" as
\beq
\mc{P}(k) \; = \; \, \sul{r=1}{\ell }n_r \, \zeta_r \, t_r(k) \qquad \e{and} \qquad \mc{E}(k) \; = \; \sul{r=1}{\ell } n_r \, \zeta_r \, \mf{u}_r\big( t_r(k) \big) \;\; , \qquad k\in \msc{I}_{1} \; .
\enq
It is assumed that $k\mapsto \mc{P}(k)$ is strictly monotonous on $\e{Int}(\msc{I}_1)$.
\begin{theorem}
\label{Theorem Principal}
Let $\msc{I}_{\e{tot}}= \msc{J}_1^{n_1}\times \cdots \times \msc{J}_{\ell}^{n_{\ell}}$ and
$ \De_{\pm} $ be smooth positive functions on $ \msc{I}_{\e{tot}} $ admitting smooth square roots on $ \msc{I}_{\e{tot}} $.
Let $\msc{G}$ be in the smooth class of $ \msc{I}_{\e{tot}} $ associated with the functions $\De_{\pm}$ and a constant $\tau \in \intoo{0}{1}$,
\textit{c.f.} Definition \ref{definition smooth class on K}.
Finally, let
\beq
\mf{z}_{\ups}(\bs{p})\;=\; \mc{E}_0 \, -\, \sul{ r=1}{\ell}\sul{a=1}{n_r} \zeta_r \mf{u}_r\big( p_a^{(r)} \big) \, + \; \ups \op{v} \, \bigg\{ \mc{P}_0 - \sul{ r=1}{\ell}\sul{a=1}{n_r} \zeta_r p_a^{(r)} \bigg\} \; ,
\quad \ups \in \{ \pm \} ,
\enq
with $\zeta_r \in \{\pm 1\}$ and where $(\mc{P}_0,\mc{E}_0) \in \R^2$.
\noindent Let $\mc{I}\big( \mf{x} \big)$ correspond to the multiple integral
\beq
\mc{I}\big( \mf{x} \big) \, = \,\pl{r=1}{\ell} \bigg\{ \Int{ \msc{I}_r^{n_r} }{} \dd \bs{p}^{(r)} \bigg\} \; \msc{G}_{\e{tot}}(\bs{p})
\qquad with \qquad
\bs{p} \; = \; \big( \bs{p}^{(1)} , \dots , \bs{p}^{(\ell)} \, \big) \; \; , \;\; \bs{p}^{(r)} \in \msc{I}_r^{n_r} \;,
\enq
where
\beq
\msc{G}_{\e{tot}}(\bs{p}) \, = \, \msc{G}\Big( \bs{p}, \mf{z}_{+}(\bs{p})+ \mf{x} , \mf{z}_{-}(\bs{p})+ \mf{x} \Big) \cdot \pl{\ups=\pm }{} \bigg\{ \Xi\Big( \, \mf{z}_{\ups}(\bs{p})+ \mf{x} \Big)
\cdot \Big[ \, \mf{z}_{\ups }(\bs{p})+ \mf{x} \Big]^{ \De_{\ups}(\bs{p}) -1 } \bigg\} \cdot \pl{r=1}{\ell} \pl{a<b}{n_r} \big( p_a^{(r)}-p_b^{(r)} \big)^2 \;.
\enq
The type of $\mf{x}\tend 0$ asymptotic expansion of $\mc{I}(\mf{x})$ depends on the value of $(\mc{P}_0,\mc{E}_0)$.
\vspace{2mm}
{\bf a)} \texttt{The regular case.}
\vspace{2mm}
\noindent If the two conditions given below hold
\beq
\big( \mc{P}_0,\mc{E}_0 \big) \, \not\in \, \Big\{ \big( \mc{P}(k),\mc{E}(k) \big) \; : \; k \in \msc{I}_1 \Big\}
\enq
and
\beq
\underset{ \substack{ \a \in \Dp{}\msc{I}_{1} \\ \ups=\pm} }{\e{min}} \big| \mc{E}_0\,-\, \mc{E}(\a) + \ups \op{v}\, (\mc{P}_0\,-\, \mc{P}(\a))\big| \; > \; 0
\label{ecriture condition positivite energie impulsion macroscopique thm principal}
\enq
then $\msc{G}_{\e{tot}} \in L^1\big( \msc{I}_{\e{tot}} \big)$ and $\mc{I}\big( \mf{x} \big) $ is smooth in $\mf{x}$, for $|\mf{x}|$ small enough.
\vspace{2mm}
{\bf b)} \texttt{The singular case.}
\vspace{2mm} \noindent Let $k_0 \in \e{Int}(\msc{I}_1)$
\beq
\De_{\ups}^{(0)} \, = \, \De_{\ups}\big( \bs{t}(k_0) \big) \qquad and \qquad
\vth \; = \; \f{1}{2} \sul{r=1}{\ell}n_r^2 \; - \; \f{3}{2} \, + \, \De_{+}^{(0)} \, + \, \De_{-}^{(0)} \;,
\enq
with
\beq
\bs{t}(k_0)\, = \, \big( \bs{t}_1(k_0), \dots, \bs{t}_{\ell}(k_0) \big) \in \R^{\ov{\bs{n}}_{\ell}} \qquad with \qquad \bs{t}_r(k_0) \, = \, \big( t_r(k_0), \dots, t_r(k_0) \big) \in \R^{n_{r}} \;.
\enq
If
\beq
\big( \mc{P}_0,\mc{E}_0 \big) \, = \, \big( \mc{P}(k_0),\mc{E}(k_0) \big) \;,
\quad \vth \not\in \mathbb{N} \;, \quad and \quad \De_{ \pm }^{(0)} >0
\enq
then $\msc{G}_{\e{tot}} \in L^1\big( \msc{I}_{\e{tot}} \big)$ and $\mc{I}(\mf{x})$ admits the $\mf{x}\tend 0^+$ asymptotic expansion:
\bem
\mc{I}\big( \mf{x} \big) \; = \;
\f{ \De_{ +}^{(0)} \, \De_{ - }^{(0)}\, \mc{G}^{(1)}\big( \bs{t}(k_0) \big) \cdot \big(2 \op{v} \big)^{\De_{ +}^{(0)} + \De_{-}^{(0)} - 1 } }
{ \sqrt{ | \mc{P}^{\prime}(k_0) | } \cdot \pl{\ups=\pm }{} \big| \op{v} - \ups \mf{u}_1^{\prime}(k_0) \big|^{ \De_{ \ups }^{(0)} } }
\cdot \Ga\big( \De_{ +}^{(0)} \big) \Ga\big( \De_{ -}^{(0)} \big) \Ga\big( - \vth \big) \cdot
\pl{r=1}{\ell} \Bigg\{ \f{ G(2+n_r) \cdot \big( 2\pi\big)^{\frac{ n_r - \de_{r,1} }{2} } }{ \big|\mf{u}_{r}^{\prime\prime}(t_r(k_0))\big|^{ \frac{1}{2} ( n_r^2 - \de_{r,1} ) } } \Bigg\} \\
\times
|\mf{x}|^{ \vth } \cdot \Bigg\{ \Xi(\mf{x}) \f{ \sin \big[ \pi \nu_{+}\big] }{\pi} \, + \, \Xi(-\mf{x})\f{ \sin \big[ \pi \nu_{-}\big] }{\pi} \Bigg\}
\, + \, \mf{r}(\mf{x}) \, + \, \e{O} \Big( |\mf{x}|^{ \vth + 1 -\tau } \Big)\;.
\end{multline}
Above $\mf{r}(\mf{x})$ is smooth in $\mf{x}$, for $|\mf{x}|$ small enough.
Finally,
\beq
\nu_{ \pm } \; = \; \f{1}{2}\sul{ \substack{ r=1 \, : \, \\ \veps_r= \mp 1} }{ \ell } n_r^2 \; - \; \f{ 1 \mp \mf{s} }{ 4 }
\; + \hspace{-4mm} \sul{ \substack{ \ups=\pm \, : \, \\ \pm [ \op{v} - \ups \mf{u}_1^{\prime}(k_0) ]>0 } }{} \hspace{-4mm} \De_{ \ups }^{(0)}
\enq
where $\mf{s}=-\e{sgn}\Big( \frac{ \mc{P}^{\prime}(k_0) }{ \mf{u}_{1}^{\prime\prime}(k_0) } \Big) $ and $\veps_r=-\zeta_r \e{sgn}\Big(\mf{u}^{\prime\prime}_{r}(t_r(k_0)) \Big) $\;.
\end{theorem}
\subsection{Outline of the paper}
This paper is organised as follows. This is the Introduction. Sub-section \ref{SousSSection Histoire de analyse fct rep dyn} to come reviews the
various developments that took place in the analysis of the dynamic response functions of one-dimensional models.
Section \ref{Section main results} contains a short review of the structure of excitations in the model followed by
a discussion of the obtained results in the simple case of the singular structure of the at most two-particle/hole contribution to the longitudinal DRF $\msc{S}^{(z)}(k,\om)$.
Finally, this section closes on the description of the series of multiple integrals representation for $\msc{S}^{(\ga)}(k,\om)$
derived in \cite{KozMasslessFFSeriesXXZ}. In particular, I discuss the various properties enjoyed by the integrands of the multiple integrals building up the series.
The singular behaviour of each multiple integral, \textit{viz}. summands arising in the series, is then extracted, for the most typical excitations, in Section \ref{Section Edge singular behaviour des fcts spectrales}.
All the technical details necessary for obtaining these results are relegated to several appendices.
Appendix \ref{Appendix Fcts speciales} lists the main notations contained in this work. Appendix \ref{Appendix auxiliary theorems} recalls the statements of four theorems,
the Morse lemma, the Weierstrass and the Malgrange preparation theorems as well as the Whitney extension theorem. All these will
be used in the core of the analysis developed in Appendices \ref{Appendix DA integrales unidimensionnelles} and \ref{Appendix DA integrales multidimensionnelles}.
Appendix \ref{Appendix Observables XXZ} recalls the properties of interest of certain observables in the XXZ chain.
Sub-appendix \ref{Appendix Lin Int Eqns Defs et al} recalls the linear integral equation based description of the observables
in the XXZ chain. Sub-appendix \ref{Appendix Section phase oscillante dpdte de la vitesse} discusses the properties of the
velocity of the particles and hole excitations that play an important role in the analysis.
Appendices \ref{Appendix DA integrales unidimensionnelles} and \ref{Appendix DA integrales multidimensionnelles} are devoted to a detailed analysis of the
asymptotic behaviour of auxiliary integrals whose understanding is necessary for obtaining the \textit{per se} singular behaviour of the DRF studied in Section
\ref{Section Edge singular behaviour des fcts spectrales}.
Appendix \ref{Appendix DA integrales unidimensionnelles} is devoted to the analysis of the asymptotics of a generalisation of one-dimensional Euler $\be$-integrals while
Appendix \ref{Appendix DA integrales multidimensionnelles} carries such an analysis relatively to a multi-dimensional generalisation of one-dimensional $\be$ integrals.
The rigorous analysis developed in Appendices \ref{Appendix DA integrales unidimensionnelles} and \ref{Appendix DA integrales multidimensionnelles} constitutes
the main technical achievement of this work. Finally, Appendices \ref{Appendix Section Auxiliary results} and \ref{Appendix Section Asymptotics of model integral}
develop several technical results that are needed so as to carry out the analysis developed in Appendix \ref{Appendix DA integrales multidimensionnelles}.
\subsection{Some history of the analysis of dynamic response functions}
\label{SousSSection Histoire de analyse fct rep dyn}
\subsubsection{Heuristic approaches}
There is clearly little hope, for a generic one-dimensional model, to extract the singular structure of DRFs by means of direct, \textit{ab inicio}, calculations.
Still, over the years, there emerged various approximation techniques allowing one to analyse certain features of such a singular behaviour.
In the massive case, the singularities of the DRFs appear to be controlled by Van Hove singularities and this completely catches the aforementioned behaviour.
A whole lot more attention was dedicated to the massless case where one expects a much richer behaviour and where no such simple explanation exists.
To start with, one can argue that the equal-time long-distance asymptotics of the correlators in a massless model should be grasped by putting the model in
correspondence with a Luttinger liquid \cite{LutherPeschelCriticalExponentsXXZZeroFieldLuttLiquid} or, more generally, with a conformal field theory (CFT) \cite{BeliavinPolyakovZalmolodchikovCFTin2DQFT}.
Mappings of this kind are built by looking at the momentum and energy of the low-lying excited states above the
ground state of the model \cite{BloteCardyNightingalePredictionL-1correctionsEnergyAscentralcharge,CardyConformalDimensionsFromLowLSpectrum}
from where one can read-off the scaling dimensions of the operators on the CFT side which give access to the critical exponents arising in the equal-time long-distance
asymptotic behaviour of the zero temperature correlation functions in the original model. In its turn this allows one to argue the
behaviour of the Fourier transforms in the vicinity of the point $(k,\om)=(0,0)$.
The situation becomes much more involved if one would like to grasp, at least qualitatively, the behaviour of DRFs in the whole $(k,\om)$ plane.
Indeed, then, it becomes necessary to take into account certain of the non-linearities in the spectrum of the model's excitations.
A first phenomenological description of the DRF's singularities in the $(k,\om)$ plane was argued by Beck, Bonner and Müller \cite{BeckBonnerMullerFenomenologicalFormDSFinXXX} in 1979. The approach
was substantially developed one year later by these authors and Thomas \cite{BeckBonnerMullerThomasSpectralFctsXXXGeneralFeatures}
for the XXX Heisenberg spin-$1/2$ chain at zero magnetic field. These authors also proposed heuristic reasonings based on selection rules so as to predict
some of the features of the DRF in the presence of a non-zero magnetic field.
A substantial progress towards the setting of an operative phenomenological approach occurred, however, only in the mid '00. In 2006, Glazmann, Kamenev, Khodas and Pustilnik \cite{GlazmanKamenevKhodasPustilnikNLLLTheoryAndSpectralFunctionsFremionsFirstAnalysis}
managed to take into account the non-linearities in the dispersion relation of one-dimensional spinless fermions and argued, in the case of the density structure factor\symbolfootnote[3]{The latter corresponds to
$\msc{S}^{(z)}(k,\om)$ in the case of the XXZ chain, \textit{c.f.} \eqref{definition dyn resp fct general}.},
the presence of a singular behaviour along single particle $k\mapsto \mf{e}_p(k)$ or hole $k\mapsto \mf{e}_h(k)$ excitations thresholds characterised by a non-trivial, \textit{viz}.
differing from a half-integer, edge exponents $\mu$.
Next year, the authors generalised their approach in \cite{GlazmanKamenevKhodasPustilnikNLLLTheoryAndSpectralFunctionsFremionsBetterStudy} so as to encompass other DRFs
and computed perturbatively the edge exponents arising in the $\de$-function Bose gas in \cite{GlazmanKamenevKhodasPustilnikDSFfor1DBosons}.
More explicit results appeared later on. Pustilnik \cite{PustilnikDSFForCalogero} built on the expressions for the exact form factors in the Calogero-Sutherland model
so as to unravel the singular behaviour of the density structure factor in that model. Further, building on the explicit expressions for the spectrum of excitations provided by the Bethe Ansatz,
Glazmann and Imambekov \cite{GlazmanImambekovComputationEdgeExpExact1DBose} proposed closed expressions for the edge exponents arising in the $\de$-function Bose gas,
while Cheianov and Pustilnik \cite{CheianovPustilnikXXZLoweredgeNLLLEdgeExp} argued the expression for the edge exponents associated with the lower threshold -corresponding to one hole excitations-
in the massless regime of the XXZ spin-1/2 chain.
Such kinds of predictions for the edge exponents were generalised, in 2008-09 by Affleck, Pereira and White \cite{AffleckPereiraWhiteEdgeSingInSpin1-2,AffleckPereiraWhiteSpectralFunctionsfor1DLatticeFermionsBoundStatesContributions},
to various other thresholds present in the XXZ chain.
In \cite{GlazmanImambekovDvPMTCompletTheoryNNLL}, Glazmann and Imambekov advocated the manifestation of various universal behaviours in the amplitudes appearing in front of the power-law
behaviour $(\de \om)^{\mu}$ of the DRF, hence providing a firm ground to the so-called non-linear Luttinger liquid theory supposed to govern the edge singular behaviour of
dynamic response functions in massless models.
I refer to the review \cite{GlazmanImambekovSchmidtReviewOnNLLuttingerTheory} and references therein for a broader discussion of that approach.
Similarly to the case of the edge exponents, Caux, Imambekov, Panfil and Shashi \cite{CauxImambekovPanfilShashiHeuristicsPrefactorsForEdgeExpInIntModels},
by building on recent techniques pioneered in \cite{KozKitMailSlaTerEffectiveFormFactorsForXXZ,KozKitMailSlaTerThermoLimPartHoleFormFactorsForXXZ,SlavnovFormFactorsNLSE} and allowing one to study the large-volume behaviour of form factors
of local operators in quantum integrable models, argued the expressions of the amplitudes in front of the singular power-law behaviour of the DRF in the case of the XXZ spin-1/2 chain, the $\de$-function Bose gas and
the Calogero-Sutherland model.
One should also mention that, more recently, the lower thresholds present in DRFs of the spin-$\tf{1}{2}$ XXX Heisenberg chain where analysed,
by Campbell, Carmelo, Machado and Sacramento \cite{CampbellCarmeloMachadoSacramentoLowerTresholdsXXXLongAndTransDSFPseudoFermionDynTheor}, within the pseudofermion dynamical theory
and by taking the Bethe Ansatz issued input for the energies.
\subsubsection{Exact approaches}
The heuristic approaches described above appear quite powerful.
It is necessary to check and test the limits of applicability of the mentioned methods \textit{versus} results stemming from exact, \textit{ab inicio},
calculations of DRF and the extraction of their singularities, carried out on quantum integrable models.
Obtaining such exact results constituted a hard and long-standing problem, despite that numerous techniques of exact computations of correlation functions have been developed
after the invention of the algebraic Bethe Ansatz \cite{FaddeevSklyaninTakhtajanSineGordonFieldModel} on the one hand
and of the vertex operator approach \cite{DavisFodaJimboMiwaNakayashikiDiagonalizationXXZinfiniteDelta>1} on the other hand.
First results relative to DRFs appeared for free fermion equivalent models. The density structure factor, \textit{viz}. the longitudinal Fourier transform $\msc{S}^{(z)}(k,\om)$,
of the XX chain was computed in a closed form by Beck, Bonner, M\"{u}ller and Thomas \cite{BeckBonnerMullerThomasSpectralFctsXXXGeneralFeatures} in 1981.
The case of transverse response functions was much harder, even for the XX chain, due to the much more involved
structure of the transverse correlators. An analysis of the power-law divergencies in $\om$ for the transverse frequency Fourier transform $\int_{0}^{2\pi}\msc{S}^{(x)}(k,\om)\cdot \dd k $
of the XX chain was achieved in 1984 by M\"{u}ller and Shrock \cite{MullerShrockDynamicCorrFnctsTIandXXAsymptTimeAndFourier} by exploiting the
connection between the associated two-point function and Painlevé transcendents.
The development of the vertex operator approach \cite{DavisFodaJimboMiwaNakayashikiDiagonalizationXXZinfiniteDelta>1} in the
mid '90s allowed for a substantial progress in the computation of the correlation functions in an interacting, \textit{viz}.
away from the free fermion point, quantum integrable model, namely for the XXZ chain in its massive regime, \textit{i.e.} the Hamiltonian \eqref{ecriture hamiltonien XXZ} for $\De>1$, this in presence of a zero external magnetic field $h=0$.
In 1995, Jimbo and Miwa \cite{JimboMiwaFormFactorsInMassiveXXZ} obtained 2n-fold multiple integral representations for the form factors of local operators
of the chain taken between the ground state and an excited state containing $2n$-spinon excitations.
Although initially obtained for the XXZ chain at $\De>1$, these integral representation admitted a regular $\De \tend 1^+$ limit, hence yielding the corresponding expressions for the XXX Heisenberg chain.
The construction of integral representations for the form factors opened the possibility to estimate the DRF of the XXZ chain at $\De \geq 1$
by taking explicitly the space and time Fourier transforms of the form factor series. Doing so allowed Bougourzi, Karbach and M\"{u}ller \cite{BougourziKarbachMuller2SpinonDSFMassiveXXZFromVopExplicit}
to obtain, in 1998, the two-spinon sector contribution $\msc{S}^{(x)}_2(k,\om)$ to the transverse dynamic response function $\msc{S}^{(x)}(k,\om)$ in the massive regime of the XXZ chain.
This analysis was revisited and corrected by Caux, Mossel and Perez-Castillo \cite{CauxMosselPerezCastillo2SpinonDSFMassiveXXZReloaded} in 2008
what allowed them to explain the presence of an asymmetry in these DRF.
Relatively to singularities, the bottom line of these investigations is that $\msc{S}^{(x)}_2(k,\om)$ exhibits square root cusps or singularities along two-spinon excitations thresholds -as expected from a DRF of a massive model-.
The two-spinon contribution to $\msc{S}^{(x)}(k,\om)$ in the case of the XXX chain was computed by Bougourzi, Couture, Kacir
\cite{BougourziCoutureKacirAll2SpinonDSFforXXXFromVOpsColseFormulae} in 1996. Building on these results, Bougourzi, Fledderjohan, Karbach, M\"{u}ller and M\"{u}tter \cite{BougourziFledderjohanKarbachMullerMutterDynamicTwoSpinonStructureFactor}
have shown in 1997 that the two-spinon sector saturates \textit{ca}. 73\% of the total intensity of $\msc{S}^{(x)}(k,\om)$. They carried as well
a thorough analysis of the singularity structure of this DRF, showing the presence of a square root cusp behaviour on the upper two-spinon treshold and
a square root divergence on the lower-treshold (plus a logarithmic behaviour).
Although the complexity of the integral representations for the higher than 2 spinon sector form factors makes the computations more involved,
Abada, Bougourzi, SiLakhal \cite{AbadaBougourziSiLakhalFourSpinonDSFXXXMultInt} and, later, Caux and Hagemans \cite{CauxHagemansDSFXXXFourSpinonContributionDeeperAnalysis}
still managed to deal with the four spinon contributions to the XXX DRFs.
Finally, in 2012, Caux, Konno, Sorrel and Weston \cite{CauxKonnoSorrelWestonFFofMasslessXXZfromXYZResults} managed to compute explicitly the two-spinon contribution to the XXZ chain directly in the massless regime
and at $h=0$ by using earlier results of Lashkievich and Pugai \cite{LashkevichPugaiFormFactorsEightVertex} and later rewritings thereof.
Here, much in the spirit of the results for the XXX case, the DRF were obtained by first starting from the integral representations for the two-spinon form factors in a massive model (the XYZ chain
in this case) and then by taking an appropriate massless scaling limit thereof. Again, the analysis unraveled the presence of square root cusps or divergences, depending on the spinon tresholds.
However, due to the much more involved structure on the XYZ chain side, no result exists so far for the higher than two spinon contribution to the form factor series in the massless
regime of the XXZ chain at $h=0$.
\subsubsection{Numerical and Bethe Ansatz based approaches}
All the exact results mentioned so far were obtained in a zero external magnetic field. The obtention of exact results in the presence of a non-zero magnetic field
turned out to be much more involved. Nonetheless, it was possible to estimate the response functions numerically.
First numerical plots of the longitudinal and transverse DRF in the XXX chain at $h\not=0$ were obtained by Karbach and M\"{u}ller
\cite{KarbachMullerDSFXXXSelectionRuleAndPlotsFromCBA} in 2000 and then by Biegel, Karbach and M\"{u}ller \cite{BiegelKarbachMullerDSFXXXNumericsFromFormCBAEigenvectors} in 2002.
The plots were obtained by means of a brute force numerical evaluation of the matrix elements of
local operators which, in their turn, were computed by using the coordinate Bethe Ansatz representation for the Eigenfunctions of the chain.
A qualitative and quantitative step forward of the numerical approach was enabled by the construction of determinant representations for the form factors
of local operators in the XXZ chain by Kitanine, Maillet and Terras \cite{KitanineMailletTerrasFormfactorsperiodicXXZ} in 1998. Using such representations which
remarkably simplified the numerics, Biegel, Karbach and M\"{u}ller \cite{BiegelKarbachMullerDSFXXXLongItudandTransverseAtqPiPisur2VariousExcitationClasses}
obtained in 2002 plots of the longitudinal and transverse response functions at fixed momentum $k \in \big\{ \pi, \tfrac{1}{2} \pi \big\}$
for the XXX chain at finite magnetic field and by distinguishing the contributions of various classes of excitations. Again, for the same values of the momentum,
the two-spinon contribution to the longitudinal response function, at various values of the anisotropy $1>\De>0$ of the XXZ chain,
was evaluated numerically by Biegel, Karbach and M\"{u}ller \cite{BiegelKarbachMullerDSFXXZTransverseAtqPiPisur2VariousvaluesDelta} in 2003.
Then, Sato, Shiroishi and Takahashi \cite{SatoShiroishiTakahashiDSFLongitudXXZABA} obtained in 2004 plots at fixed momentum $k=\tf{\pi}{2}$ and energy-momentum plots of
the two-spinon contribution to the longitudinal response function at half-saturation field in the massless regime of the XXZ chain, this for various values of the anisotropy.
In 2005, Caux and Maillet \cite{CauxMailletDynamicalCorrFunctXXZinFieldPlots} and then Caux, Maillet and Hagemans \cite{CauxHagemansMailletDynamicalCorrFunctXXZinFieldPlots}
obtained $(k,\om)$ plots of the multi-particle and bound state contribution to the longitudinal $\msc{S}^{(z)}(k,\om)$ and transverse $\msc{S}^{(+)}(k,\om)$ DRF.
Similar numerics related to the $\msc{S}^{(-)}(k,\om)$ response function for the XXX chain were carried out by Kohno \cite{KohnoDynamDominantStringExcitationXXZChainVariousDSFXXXFiniteandZeroh} in 2009,
for various values of $h$. In particular, this work has shown that, for the $\msc{S}^{(-)}(k,\om)$ DRF, the two and three string bound states carry a certain non-negligible part of the spectral weight,
as opposed to the $\msc{S}^{(z)}(k,\om)$ and $\msc{S}^{(+)}(k,\om)$ response functions where most of the spectral weight is carried by particle-hole excitations.
A similar type of numerical analysis was performed in 2006 for the DRFs of the $\de$-function Bose gas by Caux and Calabrese \cite{CauxCalabreseDynamicalStructureFactoBoseGas} and in 2007 by
Caux, Calabrese and Slavnov \cite{CauxCalabreseSlavnovSpectralFunctionBoseGas}.
\subsubsection{The restricted sum approach}
A breakthrough in the exact analysis of certain regimes of form factor expansions of two-point functions in the massless regime of
the XXZ chain was achieved by Kitanine, Maillet, Slavnov, Terras and myself \cite{KozKitMailSlaTerRestrictedSums} in 2011.
In that work, we proposed a way to sum up the expansion of XXZ's static two-point correlation functions over the
so-called critical\symbolfootnote[2]{Expectation values of local operators taken between the ground state
and the low-lying excited states exhibiting a conformal structure of their energies.} form factor.
By heuristically arguing that only such form factors should contribute to the leading order of the large-distance
asymptotic behaviour of the two-point functions in this chain, we have been able to compute the amplitude and critical exponent of the
leading term associated to every harmonic arising in the long-distance behaviour.
Owing to the sole presence of particle-hole excitations in the $\delta$-function Bose gas, we have extended \cite{KozKitMailSlaTerRestrictedSumsEdgeAndLongTime} in 2012
the above analysis so as to encompass the case of dynamic two-point functions of that model.
We managed to extract, on the basis of first principle arguments, the leading long-time and large-distance asymptotic behaviour
of two-point functions while also providing the leading amplitude and critical exponent of every oscillating harmonic
(oscillating term at a given frequency and momentum) arising in the asymptotics.
The method of analysis we employed also allowed us to investigate the singularity structure of the edge exponents for the
dynamic response functions hence confirming, through an \textit{ab inicio} analysis, the predictions stemming from
the non-linear Luttinger liquid approach. Although successful for that particular case, the analysis
left several open questions. In itself, the method used in \cite{KozKitMailSlaTerRestrictedSums,KozKitMailSlaTerRestrictedSumsEdgeAndLongTime}
only allows one to argue the various asymptotic regimes of the correlators
(be it the long-distance/time or the edge singular behaviour of DRFs). In particular, it
does not provide one with a way to write down a closed form for a massless form factor expansion in the thermodynamic limit and invokes certain heuristics in the
handlings of the asymptotic analysis. Furthermore, the applicability of the method to the case of integrable models containing bound states was open.
These points were recently solved by myself in \cite{KozMasslessFFSeriesXXZ}. There I managed to circumvent the various problems
associated with defining form factor series expansions for massless models and constructed an explicit form factor expansion
representation for the dynamical two-point functions in the massless regime of the XXZ spin-$1/2$ chain at non-zero magnetic field.
This representation was enough to take the Fourier transforms explicitly and led to a series of multiple integral representation
for the DRFs of the model. The series will be starting point for the analysis carried out in the present work.
The main goal of this paper is to provide a thorough analysis of the edge singularities in the dynamic response functions of the XXZ chain
at finite magnetic field and throughout the massless regime, this on the basis of first-principle based calculation:
the work starts from the series of multiple integrals representation for the DRFs obtained, on the level of the microscopic model, in \cite{KozMasslessFFSeriesXXZ}.
It then carries out rigorously -the well-definiteness and some of the properties of the representation obtained in \cite{KozMasslessFFSeriesXXZ} being taken for granted- only those approximations that are
consistent with the limiting regimes considered.
As a consequence, the analysis carried out in this work does not relies, at any point of our calculations, upon some conjectural or heuristically argued correspondence with a
simplified effective model such as a CFT, a Luttinger liquid or its non-linear generalisation.
Furthermore, although obtained for the massless regime of the XXZ chain, taken the "universal" nature of the massless form factor expansion based representation for the
DRFs and that the analysis developed in this work solely uses this universal structure, the results will hold -provided one accepts the validity of the
phenomenological form of massless form factor expansions advocated in \cite{KozMasslessFFSeriesXXZ}- for any massless one-dimensional quantum Hamiltonian belonging to the Luttinger
liquid universality class.
\section{Main results}
\label{Section main results}
\subsection{The setting and some generalities on the model}
I shall focus on the so-called massless anti-ferromagnetic regime at positive magnetic field which corresponds to
$-1<\De <1$ and $h_{\e{c}}>h>0$, where the critical field $h_{\e{c}}$ takes the form $h_{\e{c}}=4J(1+\De)$. $h_{\e{c}}$ is the saturation field above which the model becomes
ferromagnetic. Then, it appears convenient to parametrise the anisotropy $\De$ introduced in \eqref{ecriture hamiltonien XXZ}
as
\beq
\De= \cos(\zeta) \quad \e{with} \quad \zeta \in \intoo{0}{\pi} \, .
\label{ecriture reparametrisation anisotropie}
\enq
In the thermodynamic limit, the Bethe Ansatz analysis ensures that, for this range of parameters, the excited states above the ground state are built from
a pile up of elementary dressed excitations of different types: holes and $r$-strings.
For given value of $\zeta$, only certain values of $r$ are possible for the $r$-strings and it is convenient to collect these in the set $\mf{N} = \{ r_1, \dots, r_{ |\mf{N}|} \}$.
The set $\mf{N}$ is finite when $\tf{\zeta}{\pi}$ is rational and infinite otherwise \cite{TakahashiThermodynamics1DSolvModels}.
Furthermore, independently of the value of $\zeta$, there always exists $1$-strings excitations (\textit{viz}. $r_1=1$).
The $1$-string excitations correspond to so-called particle excitations. Among all possible $r$-string excitations, only the particles -\textit{i.e.} $1$-strings- may generate
massless excitation, \textit{i.e.} carrying a zero energy. A given excited state will be made up of $n_h\in \mathbb{N}$ holes, $n_{r_k}\in \mathbb{N}$ $r_k$-strings and left/right Fermi boundary Umklapp excitations with deficiencies $\ell_{\pm}\in \mathbb{Z}$.
These integers satisfy to the constraint
\beq
n_h= \sul{r \in \mf{N} }{} r n_r \, + \, \sul{\ups=\pm}{} \ell_{\ups} \;.
\label{ecriture contraintes entiers trous et strings}
\enq
It is convenient to collect the integers labelling the number of excitations of each kind into a single vector
\beq
\bs{n} \, = \, \big(\ell_+, \ell_- ; n_h, n_{r_1},\dots, n_{r_{|\mf{N}|}} \big)\;.
\label{definition premiere apparition vecteur n}
\enq
Owing to the constraint \eqref{ecriture contraintes entiers trous et strings}, there are only finitely many non-zero entries in $\bs{n}$.
$\bs{n}$ being fixed, the $n_h$ holes will carry momenta $t_1,\dots, t_{n_h}$ which take values in $\msc{I}_h=\intff{-p_F}{p_F}$, $p_{F} \in \intff{0}{\tf{\pi}{2}}$ being the Fermi momentum,
and the $n_{r}$ $r$-strings will carry momenta $k^{(r)}_1,\dots, k_{n_r}^{(r)}$ taking values in $\msc{I}_r=\intff{ p_-^{(r)} }{ p_+^{(r)} }$.
I refer to Appendix \ref{Appendix Lin Int Eqns Defs et al} for more precise definitions of these intervals. It appears convenient to gather the momenta carried by the various elementary excitations into the single vector
\beq
\bs{\mf{K}} \, = \, \big(\ell_{+},\ell_{-} ; \bs{t}, \bs{k}^{(r_1)},\cdots , \bs{k}^{(r_{ |\mf{N}|})} \big) \quad \e{with} \quad \bs{t} \in \msc{I}_h^{n_h} \quad \e{and} \quad \bs{k}^{(r )} \in \msc{I}_r^{n_r} \;.
\label{ecriture vecteur des impuslions des excitations diverses}
\enq
This notation should be understood as follows. If $n_h=0$, resp. $n_r=0$ with $r \in \mf{N}$, then the associated vectors $\bs{t}$, resp. $\bs{k}^{(r)}$, are to be read as
$\emptyset$, meaning that there is simply no component of the hole or of this $r$-string momenta in $\bs{\mf{K}}$, since there are no excitations of this type
in the given excited state. The use such a notation allows one to keep the precise track, on the level of the vector $\bs{\mf{K}}$, of the types of excitations which are present and those which are absent.
I stress that formally $\bs{\mf{K}}$ may contain infinitely many components with such a conventions, but only finitely many of them correspond to non-empty sets since, for fixed $\ell_{\pm}$
and $n_h$, there is only a finite number of integers $n_r$ that are non-zero. Hence $\bs{\mf{K}}$
makes sense as an inductive limit. Furthermore, effectively speaking, $\bs{\mf{K}}$ is built up from vector momenta $\bs{t}$, resp. $\bs{k}^{(r)}$ with $r \in \mf{N}$,
such that $n_h\not=0$, resp. $n_r\not=0$.
A given excited state in a sector of relative spin $\op{s}_{\ga}$ above the ground state and associated with a vector momentum $\bs{\mf{K}}$ has a total excitation momentum
\beq
\mc{P}(\bs{\mf{K}}) \, = \, \sul{ r \in \mf{N} }{} \sul{a=1}{n_r} k_a^{(r)} \, + \, p_{F}\sul{\ups=\pm}{} \ups \ell_{\ups}
+ \pi \op{s}_{\ga} - \sul{a=1}{n_h} t_a
\label{definition impulsion excitation}
\enq
and carries a total excitation energy
\beq
\mc{E}(\bs{\mf{K}}) \, = \, \sul{ r \in \mf{N} }{} \sul{a=1}{n_r} \mf{e}_r\big( k_a^{(r)} \big) \,- \, \sul{a=1}{n_h} \mf{e}_1(t_a) \;.
\label{definition energie excitation}
\enq
The functions $\mf{e}_a$ correspond to the dispersion relation of the various excitations, $\mf{e}_r$ for the $r$-strings, $-\mf{e}_1$ for the holes. They are defined as solutions to linear integral equations,
see Appendix \ref{Appendix Lin Int Eqns Defs et al}, equations \eqref{definition fct mathfrak e 1}-\eqref{definition fct mathfrak e r} for more details.
Again, by convention, sums that are subordinate to $n_r=0$ or $n_h=0$ are simply understood to be absent.
The velocity of a given $r$-string excitation with momentum $k$ is defined as $\mf{v}_r(k)=\mf{e}_r^{\prime}(k)$. Moreover, $\mf{v}_1(k)$ gives the velocity, depending on the domain where $k$ evolves, of $1$-strings (particles)
if $k \in \msc{I}_1$ or holes if $k \in \msc{I}_{h}$.
Particles, holes, and more generally strings, may share the same value of their velocities. In particular, one can prove, \textit{c.f.} Proposition \ref{Proposition proprietes fondamentales de la vitesse des particules trous}
in Appendix \ref{Appendix Section phase oscillante dpdte de la vitesse},
that for certain regimes of the model's parameters, that there exists an interval $\intff{K_m}{K_M}\subset \intoo{ p_-^{(1)} }{ p_+^{(1)} }$ and a diffeomorphism
\beq
\mf{t}:\intff{K_m}{K_M}\tend \msc{I}_h \quad \e{such}\; \e{that} \quad \mf{v}_1(k)=\mf{v}_1(\mf{t}(k)) \, .
\label{definition isomorphisme mf t}
\enq
It is conjectured that this property holds for any regime of the parameters and this is backed by an extensive numerical analysis.
\subsection{The behaviour of the longitudinal dynamic response function in the two hole excitations in $\msc{S}^{(z)}(k,\om)$}
\label{SousSection description edge exponents}
\begin{figure}
\begin{center}
\includegraphics[width=.5\textwidth]{two_particle_hole_extremas.eps}
\caption{\label{Figure relation dispersion 2 particules trous} Singularity curves issued from the sectors involving up to two particles, two-holes and no $r$-strings with $r\geq 2$ for $\Delta=0.57$ and
in presence of a magnetic field $h$ which fixes the \textit{per} site magnetisation $\mf{m}=1-2D$ such that $D=0.21$.
Continuous curves correspond to one massive -hole or particle- excitation. Dotted curves correspond to a collective, coordinated, multi-particle-hole excitation. This excitation is such that all particles
and holes building it up have the same velocity.}
\end{center}
\end{figure}
The excitations thresholds giving rise to singularities of the longitudinal dynamic response function $\msc{S}^{(z)}(k,\om)$ and built up from excitations containing
at most two holes and/or two $1$-strings are depicted in Figure \ref{Figure relation dispersion 2 particules trous}.
The curves $\mc{C}_{h}^{(a)}$, $a=1,2$, resp. $\mc{C}_{p}^{(b)}$, $b=1,\dots,4$, correspond to one hole, resp. particle, excitation above the ground state.
The curves $\mc{C}_{p-h}^{(a)}$, $a=1,2$ correspond to a joint particle-hole excitation where the particle and hole both have the same velocity. Finally, the curve $\mc{C}_{2p-h}$ is built up from a
two particle - one hole excitation, all having the same velocity.
All the particles or holes building up the excitations in the curves depicted in Figure \ref{Figure relation dispersion 2 particules trous} are massive -\textit{viz}. carry a finite excitation energy-
with the exception of the $\om=0$ line and of the junctures between the curves that are drawn in continuous and
dotted lines. The present approach is unable to analyse the singularity structure at these points. Below, $0<\tau<1$ is arbitrary and can be taken as small as necessary.
More precisely, the results established in Section \ref{Section Edge singular behaviour des fcts spectrales} entail that
\begin{itemize}
\item $\mc{C}_h^{(1)}$ is realised as a one hole excitation with $\ell_+=1, \ell_-=0$. It takes the parametric form
\beq
(\mc{P}_0,\mc{E}_0)= \big( p_F-t_0,-\mf{e}_1(t_0) \big) \quad \e{with} \quad t_0\in \intoo{-p_F}{p_F}.
\enq
Along this curve, the response function behaves as
\beq
\msc{S}^{(z)}(\mc{P}_0,\mc{E}_0+\de \om) \; = \; \msc{S}^{(z)}_{h;\e{reg}}(\de \om) \; + \; \mc{A}^{(h)} \cdot (\de \om)^{\De^{(h)}} \cdot \Xi(\de\om)+\e{O}\Big( (\de\om)^{\De^{(h)}+1-\tau} \Big) \;.
\enq
$\msc{S}^{(z)}_{h;\e{reg}}(\de \om)$ is smooth in $\de \om$ while the critical exponent takes the form $\De^{(h)}=\de_+^{(h)}+\de_-^{(h)}-1$.
$\de_{\pm}^{(h)}$ are expressed in terms of the dressed phase, \textit{c.f.} \eqref{ecriture phase habillee dans rep impuslion}, as
\beq
\de_+^{(h)} \, = \, \Big( \vp_1(p_F,t_0)-\vp_1(p_F,p_F)-1\Big)^2 \quad , \quad
\de_-^{(h)}\, = \, \Big( \vp_1(-p_F,t_0)-\vp_1(-p_F,p_F)\Big)^2 \;.
\enq
Finally, the amplitude $\mc{A}^{(h)}$ is closely related to the properly renormalised in the volume form factor squared $\msc{F}^{(z)}\big( \bs{\mf{K}}_{0}^{(h)} \big) $ of the operator $\sg^{z}$ taken between the ground state
and the excited state associated with $\mc{C}^{(1)}_{p}$:
\beq
\mc{A}^{(h)} \; = \;
\f{ (2\pi)^2 \cdot \msc{F}^{(z)}\big( \bs{\mf{K}}_{0}^{(h)} \big) }
{ \Ga\Big( \de_{+}^{(h)} + \de_{-}^{(h)} \Big) \cdot \big[ \op{v}_F+\mf{v}_1(t_0) \big]^{ \de_{-}^{(h)} } \cdot \big[ \op{v}_F-\mf{v}_1(t_0) \big]^{ \de_{+}^{(h)} } } \;.
\label{ecriture amplitude Ah}
\enq
The precise definition of $ \msc{F}^{(z)}\big( \bs{\mf{K}}_{0}^{(h)} \big) $ is given in \eqref{definition facteur de forme convenablement renormalise}
and $\bs{\mf{K}}_{0}^{(h)}\, = \, \Big( \ell_{+}=1, \ell_{-}=0 ; \bs{t}= t_0 , \emptyset, \dots \Big)$.
All building blocks of $\mc{A}^{(h)} $ other than the renormalised form factor $ \msc{F}^{(z)}\big( \bs{\mf{K}}_{0}^{(h)} \big) $ correspond to the universal part of the
amplitude associated with this hole excitation branch. Finally, $\op{v}_F=\mf{v}_1(q)$ is the velocity of the excitations on the right Fermi boundary.
\item $\mc{C}_p^{(1)}$ is realised as a one particle excitation with $\ell_+=-1, \ell_-=0$. It takes the parametric form
\beq
(\mc{P}_0,\mc{E}_0)= \big( k_0-p_F,\mf{e}_1(k_0) \big) \quad \e{with} \quad k_0\in \intoo{p_F}{ K_m } \;.
\enq
Along this curve, the response function behaves as
\bem
\msc{S}^{(z)}(\mc{P}_0,\mc{E}_0+\de \om) \; = \; \msc{S}^{(z)}_{p;\e{reg}}(\de \om) \; + \; \mc{A}^{(p)} \cdot |\de \om|^{\De^{(p)}}
\bigg\{ \Xi(\de \om) \f{\sin[ \pi \de_-^{(p)}] }{ \pi } + \Xi(-\de \om) \f{\sin [\pi \de_+^{(p)}] }{ \pi } \bigg\} \\
+\e{O}\Big( (\de\om)^{\De^{(p)}+1-\tau} \Big) \;.
\end{multline}
$\msc{S}^{(z)}_{p;\e{reg}}(\de \om)$ is smooth in $\de \om$.
The critical exponent takes the form $\De^{(p)}=\de_+^{(p)}+\de_-^{(p)}-1$ and $\de_{\pm}^{(p)}$ are expressed in terms of the dressed phase, \textit{c.f.} \eqref{ecriture phase habillee dans rep impuslion}, as
\beq
\de_+^{(p)} \, = \, \Big( 1-\vp_1(p_F,k_0)+\vp_1(p_F,p_F)\Big)^2 \quad , \quad
\de_-^{(p)}\, = \, \Big( \vp_1(-p_F,p_F)-\vp_1(-p_F,k_0)\Big)^2 \;.
\enq
The amplitude $\mc{A}^{(p)}$ takes the form
\beq
\mc{A}^{(p)} \, = \, \f{ (2 \pi)^2 \cdot \Ga\Big( 1 - \de_{+}^{(p)} - \de_{-}^{(p)} \Big) }
{ \big| \op{v}_F+\mf{v}_1\big( k_0 \big) \big|^{ \de_{-}^{(p)} } \cdot \big| \op{v}_F-\mf{v}_1\big( k_0\big) \big|^{ \de_{+}^{(p)} } } \cdot \msc{F}^{(z)}\big( \bs{\mf{K}}^{(p)}_0\big) \;.
\label{ecriture amplitue particule Ap}
\enq
$ \msc{F}^{(z)}\big( \bs{\mf{K}}_{0}^{(p)} \big) $ has the same interpretation as given above, is defined in \eqref{definition facteur de forme convenablement renormalise}
and is parameterised by the vector momentum $\bs{\mf{K}}_{0}^{(p)}\, = \, \Big( \ell_{+}=-1, \ell_{-}=0 ; \bs{t}=\emptyset, \bs{k}^{(1)}= k_0, \emptyset ,\dots \Big)$.
All the other building blocks of $\mc{A}^{(p)} $ correspond to the universal part of this particle branch amplitude.
\item $\mc{C}_{ph}^{(1)}$ is realised as an excitation with $\ell_+=0, \ell_-=0$, and containing a particle and a hole, both having the same velocity. It takes the parametric form
\beq
(\mc{P}_0,\mc{E}_0)\, = \, \Big( k_0 - \mf{t}(k_0), \mf{e}_1(k_0) - \mf{e}_1(\mf{t}(k_0)) \Big) \quad \e{with} \quad k_0\in \intoo{ K_m }{K_M} \,
\enq
and where $\mf{t}$ has been introduced in \eqref{definition isomorphisme mf t}.
Along this curve, the response function behaves as
\bem
\msc{S}^{(z)}(\mc{P}_0,\mc{E}_0+\de \om) \; = \; \msc{S}^{(z)}_{ph;\e{reg}}(\de \om) \; + \; \mc{A}^{(ph)} \cdot |\de \om|^{\De^{(ph)}}
\bigg\{ \Xi(\de \om) \f{\cos[ \pi \De^{(ph)}] }{ \pi } + \Xi(-\de \om) \f{1 }{ \pi } \bigg\} \\
+\e{O}\Big( (\de\om)^{ \De^{(ph)}+ 1 - \tau } \Big) \;.
\end{multline}
$\msc{S}^{(z)}_{ph;\e{reg}}(\de \om)$ is smooth in $\de \om$. The critical exponent takes the form $\De^{(ph)}=\de_+^{(ph)}+\de_-^{(ph)}-\tf{1}{2}$ and $\de_{\pm}^{(ph)}$
are expressed in terms of the dressed phase, \textit{c.f.} \eqref{ecriture phase habillee dans rep impuslion}, as
\beq
\de_+^{(ph)} \, = \, \Big( \vp_1(p_F,\mf{t}(k_0) ) - \vp_1(p_F,k_0) \Big)^2 \quad , \quad
\de_-^{(ph)}\, = \, \Big( \vp_1(-p_F,\mf{t}(k_0)) - \vp_1(-p_F,k_0)\Big)^2 \;.
\label{eqn edge + et - ph excitation}
\enq
Finally, the amplitude $\mc{A}^{(ph)}$ takes the form
\beq
\mc{A}^{(ph)} \, = \,
\f{ (2 \pi)^2 }{ \sqrt{ 1-\mf{t}^{\prime}(k_0) } }
\cdot \;\; \bigg( \f{ 2 \pi }{ \mf{v}^{\prime}_1\big( \mf{t}(k_0 ) \big) } \bigg)^{ \f{1}{2}}
\cdot \f{ \Ga\Big( -\De^{(ph)} \Big) }
{ \big| \op{v}_F+\mf{v}_1\big( k_0 \big) \big|^{ \de_{-}^{(ph)} } \big| \op{v}_F-\mf{v}_1\big( k_0 \big) \big|^{ \de_{+}^{(ph)} } }
\cdot \msc{F}^{(z)}\big( \bs{\mf{K}}^{(ph)}_0 \big) \;.
\enq
$ \msc{F}^{(z)}\big( \bs{\mf{K}}_{0}^{(ph)} \big) $ has the same interpretation
and $\bs{\mf{K}}_{0}^{(ph)}\, = \, \Big(\ell_{+}=0, \ell_{-}=0 ; \bs{t}=\mf{t}(k_0), \bs{k}^{(1)}=k_0, \emptyset,\dots \Big)$.
All the other building blocks of $\mc{A}^{(ph)} $ correspond to the universal part of this equal velocity particle-hole branch amplitude.
\item $\mc{C}_{p}^{(4)}$ is realised as a one particle excitation with $\ell_+=0, \ell_-=-1$. It takes the parametric form
\beq
(\mc{P}_0,\mc{E}_0)\, = \, \big( k_0 +p_F, \mf{e}_1(k_0) \big) \quad \e{with} \quad k_0\in \intoo{K_M}{2\pi - 3 p_{F} } \; .
\enq
Along this curve, the response function behaves as
\bem
\msc{S}^{(z)}(\mc{P}_0,\mc{E}_0+\de \om) \; = \; \msc{S}^{(z)}_{p;\e{reg}}(\de \om) \; + \; \mc{A}^{(p)} \cdot |\de \om|^{\De^{(p)}}
\bigg\{ \Xi(\de \om) \f{\sin[ \pi \de_+^{(p)}] }{ \pi } + \Xi(-\de \om) \f{\sin [\pi \de_-^{(p)}] }{ \pi } \bigg\} \\
+\e{O}\Big( (\de\om)^{\De^{(p)}+1-\tau} \Big) \;.
\end{multline}
$\msc{S}^{(z)}_{p;\e{reg}}(\de \om)$ is smooth in $\de \om$. The critical exponent takes the form $\De^{(p)}=\de_+^{(p)}+\de_-^{(p)}-1$ and $\de_{\pm}^{(p)}$ are expressed in terms of the dressed phase, \textit{c.f.} \eqref{ecriture phase habillee dans rep impuslion}, as
\beqa
\de_+^{(p)} & = & \Big( \vp_1(p_F,p_F ) - \vp_1(p_F,k_0) + \bs{1}_{I_{-}}(k_0) \e{sgn}(\pi-2\zeta)\mc{Z}(p_F) \Big)^2 \;, \\
\de_-^{(p)} & = & \Big( -1+\vp_1(-p_F,-p_F) - \vp_1(-p_F,k_0) + \bs{1}_{I_{-}}(k_0) \e{sgn}(\pi-2\zeta)\mc{Z}(p_F) \Big)^2 \;.
\eeqa
Here, $I_-=\intoo{2\pi-2p_F \e{sgn}(\pi-2\zeta) - (\pi-\zeta)-(\pi-2\zeta)\tfrac{p_F}{\pi} }{ 2\pi-p_F -2p_F \e{sgn}(\pi-2\zeta) }$. Finally, the amplitude $\mc{A}^{(p)}$ takes the same
form as in \eqref{ecriture amplitue particule Ap}, with the constants appropriately substituted.
\item $\mc{C}_{p2h}$ is realised as an excitation with $\ell_+=1, \ell_-=0$ that consists of one particle and two holes, all having the same velocity. It takes the parametric form
\beq
(\mc{P}_0,\mc{E}_0)\, = \, \Big( k_0 - 2\mf{t}(k_0)+p_F, \mf{e}_1(k_0) - 2\mf{e}_1(\mf{t}(k_0)) \, \Big) \quad \e{with} \quad k_0\in \intoo{ K_m }{K_M} \,,
\enq
and $\mf{t}$ as in \eqref{definition isomorphisme mf t}.
For this parameterisation, $\mc{P}_0$ increases on the interval $\intff{K_m-p_F}{K_M+3p_F}$.
Along this curve, the response function has the singular structure
\beq
\msc{S}^{(z)}(\mc{P}_0,\mc{E}_0+\de \om) \; = \; \msc{S}^{(z)}_{p2h;\e{reg}}(\de \om) \; + \; \mc{A}^{(p2h)} \cdot (\de \om)^{\De^{(p2h)}}
\cdot \Xi(\de \om) \cdot \f{\sin[ \pi \De^{(p2h)}] }{ \pi } +\e{O}\Big( (\de\om)^{ \De^{(p2h)} + 1 - \tau } \Big) \;.
\enq
The critical exponent takes the form $\De^{(p2h)}=\de_+^{(p2h)}+\de_-^{(p2h)}+1$ and $\de_{\pm}^{(p2h)}$ are expressed in terms of the dressed phase, \textit{c.f.} \eqref{ecriture phase habillee dans rep impuslion}, as
\beqa
\de_+^{(p2h)} & = & \Big(-1 + 2 \vp_1(p_F,\mf{t}(k_0) ) - \vp_1(p_F,k_0) - \vp_1(p_F,p_F)\Big)^2 \;, \label{eqn edge + p2h excitation} \\
\de_-^{(p2h)} & = & \Big( 2\vp_1(-p_F,\mf{t}(k_0)) - \vp_1(-p_F,k_0) - \vp_1(-p_F,p_F)\Big)^2 \;.
\label{eqn edge - p2h excitation}
\eeqa
Finally,
\beq
\mc{A}^{(p2h)} \, = \,
\f{ - (2 \pi)^{3} }{ \sqrt{ 1-2 \mf{t}^{\prime}(k_0) } } \cdot \;\; \bigg( \f{ 1 }{ \mf{v}^{\prime}_1\big( \mf{t}(k_0 ) \big) } \bigg)^2 \\
\cdot \f{ \Ga\Big( -\De^{(p2h)} \Big) }
{ \big| \op{v}_F+\mf{v}_1\big( k_0 \big) \big|^{ \de_{-}^{(p2h)} } \big| \op{v}_F-\mf{v}_1\big( k_0 \big) \big|^{ \de_{+}^{(p2h)} } }
\cdot \msc{F}^{(z)}\big( \bs{\mf{K}}^{(p2h)}_0 \big) \;.
\enq
$ \msc{F}^{(z)}\big( \bs{\mf{K}}_{0}^{(p2h)} \big) $ has the same interpretation
and $\bs{\mf{K}}_{0}^{(p2h)}\, = \, \Big( \ell_{+}=1, \ell_{-}=0 ; \bs{t}=(\mf{t}(k_0),\mf{t}(k_0)), \bs{k}^{(1)}=k_0,\emptyset,\dots \Big)$.
All the other building blocks of $\mc{A}^{(p2h)} $ correspond to the universal part of this equal velocity one particle two hole branch amplitude.
\end{itemize}
The curves appearing in Fig.~\ref{Figure relation dispersion 2 particules trous} are symmetric in respect to the $k=\pi$ axis.
This symmetry also applies relatively to the behaviour along these curves. Thus, the cases that were not listed above can be inferred by this symmetry operation.
Also, one should observe that certain curves are realised as $2p_F$ or $2(\pi-p_F)$ translations of other curves. This is reminiscent of the possibility, in the model, to realise zero energy excitations carrying a
non-zero discrete momentum which is an integer multiple of $2p_F$. $C_p^{(2)}$ is deduced from $\mc{C}^{(1)}_p$ by adding a particle on the right end of the Fermi zone and a hole on the left end what corresponds to
$(\ell_+,\ell_-)=(-1,0)\hookrightarrow (\ell_+^{\prime},\ell_-^{\prime})=(0,1)$. This, however, changes the values of the critical exponents.
The excitation thresholds $\mc{C}_{h}^{(a)}$, $a=1,2$ and $\mc{C}_{p}^{(b)}$, $b=1,\dots,4$, along with the associated universal structure of the singular behaviour have been argued in the literature
by means of heuristic approaches: the non-linear Luttinger liquid \cite{CheianovPustilnikXXZLoweredgeNLLLEdgeExp} in what concerns $\mc{C}_h^{(a)}$,
\cite{AffleckPereiraWhiteEdgeSingInSpin1-2,AffleckPereiraWhiteSpectralFunctionsfor1DLatticeFermionsBoundStatesContributions} relatively to $\mc{C}_{h}^{(a)}$, $\mc{C}_{p}^{(b)}$
and the pseudofermion dynamic theory \cite{CampbellCarmeloMachadoSacramentoLowerTresholdsXXXLongAndTransDSFPseudoFermionDynTheor} relatively to $\mc{C}_h^{(a)}$.
The present analysis does confirm these predictions on the basis of rigorous considerations.
The thresholds corresponding to the curves $\mc{C}_{ph}$ and $\mc{C}_{p2h}$ have never been discussed within the aforementioned approaches.
These excitation thresholds are characterised by a different structure of edge exponents as clearly appears in \eqref{eqn edge + et - ph excitation}
and \eqref{eqn edge + p2h excitation}-\eqref{eqn edge - p2h excitation}. On physical grounds, these thresholds issue from
the presence of excited states built up from various excitations (particles, holes and/or r-strings), all having equal velocities.
The singular structure of the dynamic response functions in the vicinity of multi-hole/r-string excitations, $r \in \mf{N}_{\e{st}}$
is discussed in Theorem \ref{Theorem DRF behaviour of multi string hole threshold}. Finally, although it is
not detailed in the body of the paper, the structure of the behaviour of dynamic response functions in the vicinity of equal velocity multi-particle/hole/r-string thresholds
can be readily worked out by appropriately adjusting the results of the main theorem established in this paper,
Theorem \ref{Theorem Principal}.
One should mention, that solely the work \cite{AffleckPereiraWhiteSpectralFunctionsfor1DLatticeFermionsBoundStatesContributions} considered
the thresholds generated by a joint multi-particle massive excitation. In \cite{AffleckPereiraWhiteSpectralFunctionsfor1DLatticeFermionsBoundStatesContributions},
the authors argued heuristically the expression for the edge exponents in the case of an equal velocity excitation built up from two holes and one two-string.
They also asserted that the singularity is only one-sided. They did not discuss the form of the amplitude though. The present analysis recovers all these features
and provides much more thorough information on the amplitude. The presence of one sided singularities does not hold, however, for generic $r$-string excitations.
\subsection{The series representation for the dynamic response functions}
Under certain assumptions, I have derived in \cite{KozMasslessFFSeriesXXZ} a series of multiple integral representation for the dynamic response functions of the
XXZ spin-$1/2$ chain in the massless regime $-1<\De<1$ and at finite magnetic field $h_{\e{c}}>h>0$ . The derivation of the representation relied on the assumption that it is licit to exchange certain limits with summations,
that the remainders were uniformly summable and that the resulting series was convergent. The rest of the handling were rigorous. I shall not discuss here further the rigour of the obtained series.
\textit{In the present work, I shall take for granted the existence and well-definiteness of the series of multiple integrals representing the DRF}.
The justification of the exchange of limits procedures used in its derivations along with the convergence of the series is left for future investigations and will quite probably demand to
invent new mathematical tools adapted for dealing with such questions.
The present work carries out \textit{a rigorous analysis of the singularity structure of each summands in the series} representing $\msc{S}^{(\ga)}(k,\om)$.
Developing a technique allowing one for a rigorous analysis of a class of multiple integrals containing, upon specialisations, the integrals of interest constitutes the main achievement of this work.
The series of multiple integrals obtained in \cite{KozMasslessFFSeriesXXZ} takes the form
\beq
\msc{S}^{(\ga)}(k,\om) \; = \; \sul{ \bs{n} \in \mf{S} }{} \msc{S}^{(\ga)}_{\bs{n}}(k,\om)
\label{ecriture series complete representant la DSF}
\enq
where the summation runs through all the allowed choices of hole, $r$-string and Umklapp integers, all gathered in a single vector $\bs{n}$, as in \eqref{definition premiere apparition vecteur n}, while
\beq
\mf{S} \, = \, \Big\{ (\ell_{+},\ell_{-}; n_h, n_{r_1},\dots, n_{ r_{ |\mf{N}|} })\; : \; \ell_{\pm}\in \mathbb{Z}\; , \; n_h, n_r \in \mathbb{N} \; \quad \e{and} \quad
\; n_h= \sul{r \in \mf{N} }{} r n_r \, + \, \sul{\ups=\pm}{} \ell_{\ups} \Big\} \;.
\label{ecriture range summation dans series DRF}
\enq
A given summand $\msc{S}^{(\ga)}_{\bs{n}}(k,\om) $ represents the contribution to the dynamic response function of all the excited states whose number of excitations of each type
is equal to the corresponding entry of the vector $\bs{n}$. It is given by the multidimensional integral
\bem
\msc{S}^{(\ga)}_{\bs{n}}(k,\om) \, = \,
\Int{ \big( \msc{J}_h^{(\eps)} \big)^{n_h} }{} \dd^{n_h}t \cdot \pl{ r \in \mf{N} }{} \; \bigg\{ \Int{ \big( \msc{J}_r^{(\eps)} \big)^{n_r} }{} \dd^{n_r}k^{(r)} \bigg\} \cdot \; \mc{F}^{(\ga)}\big( \bs{\mf{K}} \big) \\
\times \sul{s\in \mathbb{Z} }{} \pl{\ups= \pm }{} \bigg\{ \Xi\Big(\, \wh{\mf{y}}_{\ups}\big( \bs{\mf{K}}; s\big) \Big) \cdot \Big[ \, \wh{\mf{y}}_{\ups}\big( \bs{\mf{K}}; s\big) \Big]^{ \De_{\ups}(\bs{\mf{K}}) -1 } \bigg\}
\cdot \bigg( 1 + \mf{r}\big( \bs{\mf{K}};s \big) \bigg) \;.
\label{ecriture contrib excitation donnee facteur structure}
\end{multline}
Just as earlier on, by convention, if a hole $n_h$ or an $r$-string $n_r$ integer is zero, then the associated integration, and \textit{a fortiori} integration variables, are simply absent.
The integration variables are collected in the vector $\bs{\mf{K}}$ that was introduced in \eqref{ecriture vecteur des impuslions des excitations diverses}.
In the definition of this vector, it should be understood that, if $n_{r_k}=0$, than the corresponding vector $\bs{k}^{(r )}$ is simply absent. Thus, due to the summation constraint in \eqref{ecriture range summation dans series DRF},
for each $\bs{n}$, there are only finitely many $\bs{k}^{(r)}$ vectors present in $\bs{\mf{K}}$, \textit{c.f.} the discussion which followed after \eqref{ecriture vecteur des impuslions des excitations diverses}.
I now describe, in detail, the different building blocks of the multiple integral.
\subsubsection*{$\bullet$ The integration domain and the regulator $\eps>0$}
The integration variables run through the slightly deformed domains
\beq
\msc{J}_h^{(\eps)} \; = \; \intff{-p_F+\eps}{ p_F-\eps } \quad , \quad
\msc{J}_1^{(\eps)} \; = \; \intff{p_{-}^{(1)}+\eps}{ p_{+}^{(1)}-\eps }
\label{definition domaines integrations eps deformees}
\enq
with $p_{\pm}^{(1)}$ being parameterised in terms of $\zeta$, introduced in \eqref{ecriture reparametrisation anisotropie}, as $p_{+}^{(1)} = 2\pi -p_F -2p_F\e{sgn}(\pi-2\zeta)$, $p_{-}^{(1)} = p_F $
More generally, $\msc{J}_r^{(\eps)}=\msc{I}_r=\intff{ p^{(r)}_- }{ p^{(r)}_+ }$ for any $r\geq 2$ and the explicit form for $p_{\pm}^{(r)}$ can be inferred from
the content of Appendix \ref{Appendix Lin Int Eqns Defs et al}.
Recall that massless excitations are realised by particles and/or holes whose momenta collapse, in the thermodynamic limit, on the left and right endpoint of the Fermi zone, \textit{i.e.}
the points $\pm p_F$ for the holes and the points $p_{\pm}^{(1)}$ for the particles. In their turn, the massive excitations
carry a finite excitation energy in the thermodynamic limit. Thus, massive particles and/or holes have their momenta located uniformly away from the endpoints of the Fermi zone.
The integral representation \eqref{ecriture contrib excitation donnee facteur structure} involves a small but otherwise arbitrary parameter $\eps>0$.
The latter was introduced in \cite{KozMasslessFFSeriesXXZ} as a regulator defining a separating scale between the massive and massless particle and hole type excitations in the model.
The matter is that the contributions of the massless modes cannot be summed by means of a Lebesgue-measure based integral and demand a very different treatment.
Their leading effect is already taken into account and manifests itself in the dependence on the functions $\wh{\mf{y}}_{\ups}\big( \bs{\mf{K}}; s\big)$, \textit{c.f.} \eqref{definition des fonctions hat y ups}.
The $\eps$-dependence appears explicitly on the level of the domain of integration \eqref{definition domaines integrations eps deformees},
while the rest of $\eps$-dependence is contained in the remainder $\mf{r}\big( \bs{\mf{K}};s \big)$, \textit{c.f.} the later discussion.
The whole series \eqref{ecriture series complete representant la DSF} does not depend on the regulator $\eps$. One cannot take the
$\eps\tend 0^+$ individually in each multiple integral due to the presence of non-integrable singularities in the integrals and
a non-uniformness in of the control on the remainder in the $\eps\tend 0^+$ limit. However, one can always consider $\eps$ to be as small as necessary for the
purpose of the analysis, as long as it remains fixed.
\subsubsection*{$\bullet$ The integrand}
The integrand in \eqref{ecriture contrib excitation donnee facteur structure} is built up from two contributions: the static part $\mc{F}^{(\ga)}\big( \bs{\mf{K}} \big)$,
and the dynamic part built up from the functions $\wh{\mf{y}}_{\pm}\big( \bs{\mf{K}} ; s\big) $ and $\De_{\pm}( \bs{\mf{K}} )$.
Note that it is precisely the dynamic part that introduces singularities in the integrand and, as such, is the one responsible for the existence of an edge singular behaviour of the
DRFs.
\vspace{2mm}
$\circledast$ {\bf The dynamic part}
\vspace{2mm}
The function
\beq
\wh{\mf{y}}_{\ups}\big( \bs{\mf{K}}; s\big) \; = \; \om - \mc{E}(\bs{\mf{K}}) + \ups \op{v}_{F} \big[ k - \mc{P}(\bs{\mf{K}}) + 2\pi s \big]
\label{definition des fonctions hat y ups}
\enq
is the only building block of the integrand that depends on the momentum $k$ and the energy $\om$.
Its expression involves the relative excitation momentum $\mc{P}(\bs{\mf{K}})$ and relative excitation energy $\mc{E}(\bs{\mf{K}})$ which were defined, resp.,
in \eqref{definition impulsion excitation} and \eqref{definition energie excitation}. It also involves $\op{v}_F=\mf{v}_1(q)$, the velocity of the excitations on the Fermi boundary.
The exponents $\De_{\pm}( \bs{\mf{K}} ) \geq 0$ are smooth functions of $\bs{\mf{K}}$. Their explicit expression
can be found in equation \eqref{definition shifted sfift function} of Appendix \ref{Appendix Lin Int Eqns Defs et al} and just above it.
The dynamic part is summed up over $s$ in \eqref{ecriture contrib excitation donnee facteur structure}. This summation is, in fact, finite. Indeed, for fixed
$\bs{n} \in \mf{S}$, the functions $\mc{P}(\bs{\mf{K}})$ and $\mc{E}(\bs{\mf{K}})$ are bounded on the integration domain from below and above. Thus, for $(k,\om)$ belonging to
any compact subset of $\R^2$, there will exist finitely many $s\in \mathbb{Z}$ such that both $ \wh{\mf{y}}_{\pm}\big( \bs{\mf{K}}; s\big) >0$.
In fact, the summation over $s$ in \eqref{ecriture contrib excitation donnee facteur structure} simply translates the fact that the spectral function is a $2\pi$ periodic function of $k$, owing to the
discrete nature of the XXZ chain.
\vspace{2mm}
$\circledast$ {\bf The static part}
\vspace{2mm}
The static part is a smooth function of $\bs{\mf{K}}$, at least when the latter ranges through the integration domain given in \eqref{ecriture contrib excitation donnee facteur structure}.
It is expressed as
\beq
\mc{F}^{(\ga)}\big( \bs{\mf{K}} \big) \;=\; \f{ (2\pi)^2 \cdot \msc{F}^{(\ga)}\big( \bs{\mf{K}} \big) \cdot \big[ 2\op{v}_F \big]^{-\De_{+} ( \bs{\mf{K}} ) -\De_{-} (\bs{\mf{K}} ) +1 } }
{ n_h! \cdot \pl{ r \in \mf{N} }{} n_r! \cdot \Ga\Big(\De_{+} \big( \bs{\mf{K}} \big) \Big) \cdot \Ga\Big(\De_{-} \big( \bs{\mf{K}} \big) \Big)} \;.
\enq
$\msc{F}^{(\ga)}\big( \bs{\mf{K}} \big)$ corresponds to the properly renormalised in the volume, thermodynamic limit of the form factor squared
of the spin operator $\sg_1^{\ga}$ taken between the ground state $\Om$ and the state $ \Ups_{\bs{\mf{K}}} $ which is the
Eigenstate of $\op{H}$ satisfying to the constraints:
\begin{itemize}
\item[ {\bf i) } ] in the thermodynamic limit, $ \Ups_{\bs{\mf{K}}} $ is parameterised in terms of elementary excitations whose momenta are gathered in the vector $\bs{\mf{K}}$;
\item[ {\bf ii) } ] $ \Ups_{\bs{\mf{K}}} $ has the lowest possible, compatible with {\bf i)}, relative excitation energy above the ground state in finite volume $L$.
\end{itemize}
This properly normalised form factor reads
\beq
\msc{F}^{(\ga)}\big( \bs{\mf{K}} \big) \; = \; \lim_{L\tend +\infty} \bigg\{ \, \Big(\f{ L }{ 2\pi} \Big)^{ \tau(\bs{\mf{K}}) } \cdot \Big| \Big(\Ups_{\bs{\mf{K}}} , \sg_1^{\ga} \, \Om \Big) \Big|^2 \bigg\}
\quad \e{with} \quad
\tau(\bs{\mf{K}}) \; = \; \De_{+} ( \bs{\mf{K}} ) + \De_{-} (\bs{\mf{K}} ) + n_h + \sul{r \in \mf{N} }{} r n_r \;.
\label{definition facteur de forme convenablement renormalise}
\enq
The explicit expression for $\msc{F}^{(\ga)}\big( \bs{\mf{K}} \big)$ can be found in \cite{KozProofOfAsymptoticsofFormFactorsXXZBoundStates}.
\vspace{2mm}
$\circledast$ {\bf The remainder}
\vspace{2mm}
Finally, $\mf{r}(\bs{\mf{K}};s)$ is a remainder term. It is controlled as
\beq
\mf{r}(\bs{\mf{K}};s) \; = \; \e{O}\Big( \sum_{ \ups = \pm } \big|\, \wh{\mf{y}}_{\ups}\big( \bs{\mf{K}}; s\big) \big|^{1-\tau} \Big)
\enq
and this estimation is uniform throughout the integration domain. The parameter $\tf{1}{2}>\tau>0$ is arbitrary provided that it is taken small enough.
The control on the remainder is also differentiable in respect to the parameters $(\om,k)$, in the sense of Definition \ref{defintion reste differentiable}.
However, the control on the remainder is \textit{not} uniform in respect to $\eps \tend 0^+$.
In fact, one expects that the optimal control on $\mf{r}(\bs{\mf{K}};s)$ is provided by the sharper bound
\beq
\mf{r}(\bs{\mf{K}};s) = \e{O}\Big( \sum_{ \ups = \pm } \big|\, \wh{\mf{y}}_{\ups}\big( \bs{\mf{K}}; s\big) \big|\ln \big|\, \wh{\mf{y}}_{\ups}\big( \bs{\mf{K}}; s\big) \big| \Big) \, .
\enq
\vspace{2mm}
$\circledast$ {\bf An additional property of the integrand }
\vspace{2mm}
As argued in \cite{KozLongDistanceLargeTimeXXZ}, the series \eqref{ecriture series complete representant la DSF} taken as a whole
has a built-in mechanism which enforces the complete cancellation between the contributions of an immediate vicinity of the boundaries of integration
\beq
\Dp{} \Big\{ \big( \msc{J}_h^{(\eps)} \big)^{n_h} \times_{r \in \mf{N} }^{} \big( \msc{J}_r^{(\eps)} \big)^{n_r} \Big\}
\label{definition bord domaine integration}
\enq
arising in each of the multiple integrals \eqref{ecriture contrib excitation donnee facteur structure}. This cancellation property effectively results in
that the form factor density $\msc{F}^{(\ga)}\big( \bs{\mf{K}} \big)$ can be considered as a function vanishing smoothly on the boundary \eqref{definition bord domaine integration}.
\section{The edge singular behaviour of dynamic response functions}
\label{Section Edge singular behaviour des fcts spectrales}
This section gathers various theorems capturing the singular behaviour of the dynamic response function issuing from various excitation sectors in the model's spectrum.
The statements follows from an application of the general theorems proven throughout Appendix \ref{Appendix DA integrales unidimensionnelles} and \ref{Appendix DA integrales multidimensionnelles}.
All the theorems stated below, take as a hypothesis the smooth vanishing of the integrands on the boundary of integration which was discussed above.
The precise and rigorous establishing of this property, beyond the arguments given in \cite{KozLongDistanceLargeTimeXXZ}, is left for further study.
Also, some of these rely on the properties stated in Conjecture \ref{Conjecture diffeo liee a la vitesse}, which can be proven in certain cases of the
coupling constants $\De$ and $h$, \textit{c.f.} Appendix \ref{Appendix Section phase oscillante dpdte de la vitesse}.
\subsection{The one free rapidity sector}
In this subsection, I extract the singular behaviour of the dynamical response functions associated with one massive excitation, namely an excitation consisting either of one hole or one particle far from the Fermi boundaries, or
one $r$-string with $r\in \mf{N}\setminus \{1\}$. Such an excitation can be accompanied by any value of the left or right Umklapp integers $\ell_{\pm}$ that are compatible with
the constraint \eqref{ecriture contraintes entiers trous et strings}.
\subsubsection{The one-hole contributions}
For the present purpose, it is convenient to parameterise the momentum-energy $(k,\om)$ combination as
\beq
k=\mc{P}_0 \; \; \e{where} \;\; \mc{P}_0 \; = \; \pi \op{s}_{\ga} \, + \, p_{F}\sul{\ups=\pm}{} \ups \ell_{\ups} - t_0 -2\pi s_{0}
\quad \e{and} \quad \; \om=\de \om + \mc{E}_0
\label{ecriture parametrisation impulsion energie hole spectral fct}
\enq
where $t_0 \in \intfo{-\pi}{\pi}$, $s_0 \in \mathbb{Z}$, and $\ell_{\pm}$ are subject to the constraint $\sul{\ups=\pm}{}\ell_{\ups}=1$.
The one-hole DRF takes the form
\bem
\msc{S}^{(\ga)}_{\bs{n}_h}\big(\mc{P}_0,\mc{E}_0 + \de \om\big) \; = \; \Int{ \msc{J}_h^{(\eps)} }{} \dd t \; \mc{F}^{(\ga)}\big( \bs{\mf{K}}^{(h)} \big)
\cdot \sul{s\in \mathbb{Z} }{}\pl{\ups= \pm }{} \bigg\{ \Xi\Big( \de \om + \mf{y}_{\ups}^{(h)}\big( t ;s \big) \Big) \cdot \Big[ \de \om + \mf{y}_{\ups}^{(h)}\big( t ;s \big) \Big]^{ \De_{\ups}(\bs{\mf{K}}^{(h)} ) -1 } \bigg\}
\\
\times
\Big( 1+\mf{r}\big( \bs{\mf{K}}^{(h)}; s \big) \Big) \;.
\label{ecriture TF S un trou}
\end{multline}
Here, I have set
\beq
\bs{n}_{h}\, = \, (\ell_+,\ell_-; n_h=1, 0,\dots )
\quad \e{and} \quad
\bs{\mf{K}}^{(h)} \, = \, \Big( \ell_{+}, \ell_{-} ; \bs{t}= t , \emptyset, \dots \Big) \;.
\enq
As a consequence, there
are no $\bs{k}^{(r)}$ vectors present in $\bs{\mf{K}}^{(h)}$.
The vector $\bs{\mf{K}}^{(h)}$ only involves the momentum $t$ of one hole excitation and the Umklapp integers. The $\mf{y}^{(h)}_{\ups}$ functions appearing in \eqref{ecriture TF S un trou} take the form
\beq
\mf{y}_{\ups}^{(h)}\big( t ; s \big) \; = \; \mf{e}_1(t) + \mc{E}_0 \, + \, \ups \op{v}_{F}\big[ t - t_0 +2\pi (s-s_0) \big] \;.
\enq
Finally, the remainder satisfies
\beq
\mf{r}\big( \bs{\mf{K}}^{(h)}; s \big) \, = \, \e{O}\Big( \sum_{ \ups = \pm } \big|\, \de \om + \mf{y}_{\ups}^{(h)}\big( t ;s \big) \big|^{1-\tau} \Big) \;,
\enq
and the control is differentiable in the sense of Definition \ref{defintion reste differentiable}.
\begin{theorem}
\label{Theorem exictation a un trou}
Assume that
\begin{itemize}
\item[{\bf i)} ] $t_0 \not\in \intff{-p_F}{p_F}$, \textit{i.e.} does not belong to the range of available momenta for a hole, in which case $\mc{E}_0$ can take any value;
\item[ {\bf ii)} ] $t_0 \in \intff{-p_F}{p_F}$ is within the range of momenta of a hole excitation and that the subsidiary condition holds $\mc{E}_0\not= -\mf{e}_{1}(t_0)$ .
\end{itemize}
Then $\msc{S}^{(\ga)}_{\bs{n}_h}\big(\mc{P}_0,\mc{E}_0+ \de \om\big)$ is smooth in $\de \om$ belonging to a neighbourhood of $0$.
\vspace{2mm}
Assume that
\beq
t_0\in \intoo{-p_F}{p_F} \; , \quad \mc{E}_0=-\mf{e}_1(t_0) \quad and \quad \de_{\ups}^{(h)} \, = \, \De_{\ups}( \bs{\mf{K}}_0^{(h)} ) >0
\enq
where $\bs{\mf{K}}_{0}^{(h)}\, = \, \Big(\ell_{+}, \ell_{-}; \bs{t}= t_0 , \emptyset ,\dots \Big)$.
Then, one has the $\de \om \tend 0$ asymptotic expansion
\bem
\msc{S}^{(\ga)}_{\bs{n}_h}\big(\mc{P}_0,\mc{E}_0+ \de \om\big) \; = \; \f{ \Xi(\de \om) \cdot \big( \de \om \big)^{ \de_{+}^{(h)} + \de_{-}^{(h)} - 1 } }{ \Ga\Big( \de_{+}^{(h)} + \de_{-}^{(h)} \Big) }
\f{ (2\pi)^2 \cdot \msc{F}^{(\ga)}\big( \bs{\mf{K}}_{0}^{(h)} \big) }{ \big[ \op{v}_F+\mf{v}_1(t_0) \big]^{ \de_{-}^{(h)} } \cdot \big[ \op{v}_F-\mf{v}_1(t_0) \big]^{ \de_{+}^{(h)} } } \\
+\; \e{O}\Big( |\de \om|^{ \de_{+}^{(h)} + \de_{-}^{(h)} -\tau} \Big)
\; + \; \msc{S}^{(\ga)}_{h;\e{reg}}(\de\om) \;.
\label{ecriture comportement singulier DRF a un trou}
\end{multline}
The function $\msc{S}^{(\ga)}_{h;\e{reg}}(\de\om)$ appearing above is smooth in the neighbourhood of the origin.
\end{theorem}
I recall that $ \mf{v}_1$ appearing in \eqref{ecriture comportement singulier DRF a un trou} is defined as
\beq
\mf{v}_1(t) \, = \, \mf{e}^{\prime}_1(t) \;.
\label{rappel definition vitesse trous}
\enq
Further, one should observe that since $\De_{\ups}$ is an analytic function of the rapidity $t_0$ and that $\De_{\ups} \geq 0$ by construction, the constraint of the theorem is always satisfied
for a generic choice of parameters.
\Proof
Consider the contribution to $\msc{S}^{(\ga)}_{\bs{n}_h}\big(\mc{P}_0,\mc{E}_0+ \de \om\big)$ stemming from the integrals in \eqref{ecriture TF S un trou} associated with
picking $s\not=s_0$. Then, the functions
$ \mf{y}_{\pm}^{(h)}\big( t ; s \big)$ cannot share a common zero on $\msc{J}^{(0)}_h$. Assume the contrary. Then, denoting this zero as $t^{\prime} \in \intoo{-p_F}{p_F}$ one would have that
\beq
0 \, = \, \mf{y}_{+}^{(h)}(t^{\prime};s) \, - \, \mf{y}_{-}^{(h)}(t^{\prime};s) \, = \, 2\op{v}_F \big(t^{\prime} - t_0 +2\pi (s-s_0) \big) \;.
\label{ecriture equation zero joint de z hole}
\enq
However, $|t^{\prime} - t_0|\leq p_F+\pi < 2\pi$, hence producing a contradiction. Observe that one has
\beq
\Dp{t} \mf{y}_{\ups}^{(h)} \big( t ; s \big) \; = \; \mf{v}_1(t) + \ups \op{v}_F \not= 0
\label{ecriture derivee y ups h}
\enq
with $\mf{v}_1$ as defined in \eqref{rappel definition vitesse trous}. As discussed in Appendix \ref{Appendix Section phase oscillante dpdte de la vitesse},
one has $ | \mf{v}_1(t) | < \op{v}_F$ for $t \in \intoo{-p_F}{p_F}$ what ensures that $\mf{y}_{\ups}^{(h)}(t ;s)$ has at most one zero on
$\msc{J}_{h}^{(\eps)}$ and that the latter is simple. A straightforward application of Lemma \ref{Lemme integrale type beta reguliere} then ensures that integrals subordinate to $s\not=s_0$
only produce smooth functions of $\de \om$ in the neighbourhood of $0$.
It remains to focus on the $s=s_0$ case. First, consider the situation subordinate to the cases {\bf i)} and {\bf ii)}.
If one is in case {\bf i)}, then due to \eqref{ecriture equation zero joint de z hole} the functions $\mf{y}_{\pm}^{(h)}(t ;s_0)$ cannot share a zero on
$\msc{J}_h^{(\eps)}$. In case {\bf ii)}, \eqref{ecriture equation zero joint de z hole} would impose that a common zero $t^{\prime}$ necessarily coincides with $t_0$. The latter would then impose that
$0= \mf{y}_{+}^{(h)}(t_0;s) \, = \, \mc{E}_0 + \mf{e}_1(t_0) $ what leads to a contradiction.
Thus, since in both cases the functions $\mf{y}_{\ups}^{(h)}$ do not share a common zero on $\msc{J}_{h}^{(\eps)}$, one has, by Lemma \ref{Lemme integrale type beta reguliere}, that
$\msc{S}^{(\ga^{\prime} \ga)}_{\bs{n}_h}\big(\mc{P}_0,\mc{E}_0 + \de \om\big) $ is smooth in $\de \om$ around $0$.
Finally, I focus on the last case $t_0 \in \intoo{-p_F}{p_F}$ with $\mc{E}_0=-\mf{e}_1(t_0)$. Since $t_0 \in \intoo{-p_F}{p_F}$, one can invoke the freedom of choosing the regulator $\eps>0$
so that $t_0 \in \e{Int}(\msc{J}_h^{(\eps)})$.
$t_0$ is clearly a common zero to $t \mapsto \mf{y}_{\ups}^{(h)} \big( t ; s_0 \big)$. It is the only one on $\msc{J}_h^{(\eps)}$ owing to \eqref{ecriture equation zero joint de z hole}.
Furthermore, due to \eqref{ecriture derivee y ups h}, one has
$ \Dp{t} \mf{y}_{+}^{(h)} \big( t_0 ; s\big) \cdot \Dp{t} \mf{y}_{-}^{(h)} \big( t_0 ;s \big) <0$ and $\Dp{t} \mf{y}_{+}^{(h)} \big( t_0 ; s\big) - \Dp{t} \mf{y}_{-}^{(h)} \big( t_0 ;s \big)=2\op{v}_F\not= 0$.
All is set so as to apply Theorem \ref{Proposition integrale principale spectral function} and thus, in the $\de \om \tend 0$ limit, one indeed gets \eqref{ecriture comportement singulier DRF a un trou}.
\qed
\subsubsection{The one $r$-string contributions}
For the purpose of discussing the contribution of one $r$-string excitations to the DRF, it appears convenient to parametrise the momentum-energy $(k,\om)$ variables of the response function as
\beq
k=\mc{P}_0 \; \; \e{where} \;\; \mc{P}_0 \; = \; \pi \op{s}_{\ga} \, + \, p_{F}\sul{\ups=\pm}{} \ups \ell_{\ups} + k^{(r)}_0 -2\pi s_0
\quad \e{and} \quad \; \om=\de \om + \mc{E}_0 \;.
\label{ecriture parametrisation impulsion energie r string spectral fct}
\enq
Above, $k^{(r)}_0 \in \msc{I}_r$ while the Umklapp integers are subject to the constraints $\sul{\ups=\pm}{} \ell_{\ups} \, = \, -r$ and $s_0 \in \mathbb{Z}$.
The associated one $r$-string, $r \in \mf{N}$, DRF takes the form
\bem
\msc{S}^{(\ga)}_{\bs{n}_r}\big(\mc{P}_0,\mc{E}_0+ \de \om\big)\; = \; \Int{ \msc{J}_r }{} \dd k^{(r)} \; \mc{F}^{(\ga)}\big( \bs{\mf{K}}^{(r)} \big)
\cdot \pl{\ups= \pm }{} \bigg\{ \Xi\Big( \de \om + \mf{y}_{\ups}^{(r)}\big( k^{(r)} ; s \big) \Big) \cdot \Big[ \de \om + \mf{y}_{\ups}^{(r)}\big( k^{(r)} ; s \big) \Big]^{ \De_{\ups}(\bs{\mf{K}}^{(r)}) -1 } \bigg\}
\\
\times
\Big( 1+ \mf{r}\big( \bs{\mf{K}}^{(r)};s \big) \Big) \;.
\label{ecriture TF S un r string}
\end{multline}
Here,
\beq
\left\{ \ba{ccc}
\bs{n}_{r} & = & (\ell_+,\ell_-; n_h=0, n_{1}=0,\dots,0, n_r=1,0,\dots ) \vspace{2mm}\\
\bs{\mf{K}}^{(r)} & = & \Big(\ell_{+},\ell_{-}; \bs{t}=\emptyset,\bs{k}^{(1)}=\emptyset,\dots, \emptyset, \bs{k}^{(r)}=k^{(r)} , \emptyset,\dots \Big) \ea \right.
\enq
where the notation means that the only rapidity that is present in $\bs{\mf{K}}^{(r)}$ is the rapidity $k^{(r)}$ of one $r$-string
while, Umklapp integers being set apart, the only non-zero integer in $\bs{n}_r$ is the one counting the $r$-string excitations, and it is set to one.
Also \eqref{ecriture TF S un r string} involves the functions
\beq
\mf{y}_{\ups}^{(r)}\big( k^{(r)} ; s \big) \; = \; \mc{E}_0 \, - \, \mf{e}_r\big( k^{(r)} \big) \, + \, \ups \op{v}_{F}\big[ k_0^{(r)} - k^{(r)} +2\pi (s-s_0) \big] \;.
\enq
Finally, the remainder satisfies
\beq
\mf{r}\big( \bs{\mf{K}}^{(r)}; s \big) \, = \, \e{O}\Big( \sum_{ \ups = \pm } \big|\, \de \om + \mf{y}_{\ups}^{(r)}\big( k^{(r)} ;s \big) \big|^{1-\tau} \Big) \;,
\enq
and the control on $\mf{r}\big( \bs{\mf{K}}^{(r)}; s \big)$ it is differentiable in the sense of Definition \ref{defintion reste differentiable}.
\begin{theorem}
\label{Theorem exictation a une r corde}
Let $k_0^{(r)}(s)=k_0^{(r)}+2\pi(s-s_0)$. Assume that
\begin{itemize}
\item[{\bf i)} ] $k_0^{(r)}(s) \not\in \msc{I}_r$, this for any $s$, in which case $\mc{E}_0$ can take any value;
\item[ {\bf ii)} ] $k_0^{(r)}(s) \in \msc{I}_r$, at least for one $s$ and that, for any such $s$, one has $\mc{E}_0\not= \mf{e}_r\big( k_0^{(r)}(s) \big)$ .
\end{itemize}
Then $\msc{S}^{(\ga^{\prime} \ga)}_{\bs{n}_r}\big(\mc{P}_0,\mc{E}_0+ \de \om\big)$ is smooth in $\de \om$ belonging to a neighbourhood of $0$.
\vspace{5mm}
\noindent Assume that
\begin{itemize}
\item $k_0^{(r)}(s) \in \e{Int}(\msc{I}_r)$ for at least for one $s$,
\item $\mc{E}_0=\mf{e}_r\big( k^{(r)}_0 + 2\pi (s-s_0) \big)$ for the same value of $s$.
\item $\de_{\ups}^{(r)}(s) \, = \, \De_{\ups}\Big( \bs{\mf{K}}^{(r)}_0(s) \Big) >0$, where
\beq
\bs{\mf{K}}_{0}^{(r)}(s)\, = \, \Big(\ell_{+},\ell_{-}; \bs{t}=\emptyset,\bs{k}^{(1)}=\emptyset,\dots, \emptyset, \bs{k}^{(r)}=k^{(r)}_0(s) , \emptyset,\dots \Big) \;.
\enq
\end{itemize}
\vspace{3mm}
\noindent {$\blacklozenge$ Case 1 : \quad If $| \mf{v}_r\big(k_0^{(r)}(s) \big) | > \op{v}_F$}
\vspace{3mm}
\noindent then, agreeing upon
\beq
\eta(s) \, = \,- \e{sgn}\Big\{ \mf{v}_r\Big( k_0^{(r)}(s) \Big) \Big\}
\enq
one has the asymptotic expansion
\bem
\msc{S}^{(\ga)}_{\bs{n}_r}\big(\mc{P}_0,\mc{E}_0+ \de \om\big) \; = \; \sul{ \substack{ s\; : \; \mc{E}_{0}=\mf{e}_{r}(k_0^{(r)}(s)) \\ k_0^{(r)}(s) \in \msc{I}_r } }{}
\f{ (2 \pi)^2 \cdot \msc{F}^{(\ga)}\big( \bs{\mf{K}}^{(r)}_0(s) \big) \cdot \Ga\Big( 1 - \de_{+}^{(r)}(s) - \de_{-}^{(r)}(s) \Big) }
{ \Big| \op{v}_F+\mf{v}_r\Big( k^{(r)}_0(s) \Big) \Big|^{ \de_{-}^{(r)}(s) } \cdot \Big| \op{v}_F-\mf{v}_r\Big( k^{(r)}_0(s) \Big) \Big|^{ \de_{+}^{(r)}(s) } } \cdot
\big| \de \om \big|^{ \de_{+}^{(r)}(s) + \de_{-}^{(r)}(s) - 1 } \\
\times \Bigg\{ \Xi(\de \om) \f{ \sin\big[ \pi \de_{\eta(s)}^{(r)}(s) \big] }{ \pi } \; + \; \Xi(-\de \om) \f{ \sin\big[ \pi \de_{-\eta(s)}^{(r)}(s) \big] }{ \pi } \Bigg\}
\quad + \; \msc{S}^{(\ga)}_{r;\e{reg}}(\de\om) \; + \; \e{O}\Big( |\de \om|^{\de_{+}^{(r)}(s) + \de_{-}^{(r)}(s) -\tau} \Big) \;.
\end{multline}
$\msc{S}^{(\ga)}_{r;\e{reg}}(\de\om)$ is smooth in the neighbourhood of the origin, and
\beq
\mf{v}_r(t) \, = \, \mf{e}^{\prime}_r(t) \;.
\label{rappel definition vitesse r cordes}
\enq
\vspace{3mm}
\noindent {$\blacklozenge$ Case 2 : \quad If $| \mf{v}_r\big(k_0^{(r)}(s) \big) | < \op{v}_F$}
\vspace{3mm}
\noindent then, under the same conventions as above
\bem
\msc{S}^{(\ga)}_{\bs{n}_r}\big(\mc{P}_0,\mc{E}_0+ \de \om\big) \; = \; \sul{ \substack{ s\; : \; \mc{E}_{0}=\mf{e}_{r}(k_0^{(r)}(s)) \\ k_0^{(r)}(s) \in \msc{I}_r } }{}
\f{ (2 \pi)^2 \cdot \msc{F}^{(\ga)}\big( \bs{\mf{K}}^{(r)}_0(s) \big) }{ \Ga\Big( \de_{+}^{(r)}(s) + \de_{-}^{(r)}(s) \Big) }
\cdot \f{ \big| \de \om \big|^{ \de_{+}^{(r)}(s) + \de_{-}^{(r)}(s) - 1 } \Xi(\de \om) }
{ \big| \op{v}_F+\mf{v}_r\Big( k^{(r)}_0(s) \Big) \big|^{ \de_{-}^{(r)}(s) } \cdot \big| \op{v}_F-\mf{v}_r\Big( k^{(r)}_0(s) \Big) \big|^{ \de_{+}^{(r)}(s) } } \\
\; + \; \msc{S}^{(\ga)}_{r;\e{reg}}(\de\om) \; + \; \e{O}\Big( |\de \om|^{\de_{+}^{(r)}(s) + \de_{-}^{(r)}(s) -\tau} \Big) \;.
\end{multline}
\end{theorem}
\Proof
The analysis is quite similar to the one-hole excitation case.
Cases ${\bf i)}$ and ${\bf ii)}$ are dealt with by means of Lemma \ref{Lemme integrale type beta reguliere}.
It remains to focus on the case where $k_0^{(r)}(s) \in \e{Int}(\msc{I}_r)$. Again, if $r=1$, then one adjusts $\eps>0$ so that $k_0^{(r)}(s) \in \e{Int}(\msc{J}_r^{ (\eps) }) $.
By hypothesis, one has that $\mc{E}_0=\mf{e}_r\big( k_0^{(r)}(s) \big)$, for at least one $s$.
Then, one treats each integral subordinate to a value of $s$ separately. Only integrals subordinate to values of $s$ such that
$k_0^{(r)}(s) \in \e{Int}(\msc{I}_r^{(\eps)})$ and $\mc{E}_0=\mf{e}_r\big( k_0^{(r)}(s) \big)$ will give rise to a non-smooth
behaviour when $\de \om \tend 0$. For any such value of $s$, just as for the one-hole case, one concludes that $k_0^{(r)}(s)$ is the only simultaneous zero of $\mf{y}_{\pm}^{(r)}(\cdot; s)$ on $\msc{J}_r^{(\eps)}$
and that
\beq
\Dp{k^{(r)}} \mf{y}_{\ups}^{(r)} \big( k^{(r)} ; s \big) \; = \; - \big[ \mf{v}_r( k^{(r)} ) \, + \, \ups \op{v}_F \big] \;.
\enq
\textit{A priori}, and this is supported by numerical investigations, \textit{c.f.} Appendix \ref{Appendix Section phase oscillante dpdte de la vitesse},
$|\mf{v}_r|$ may or may not be smaller than $\op{v}_F$, namely depending on the choice of the anisotropy $\zeta$, the values of the magnetic field $h$ -and hence the endpoint of the
Fermi zone-, the value of $r \in \mf{N}$ and, finally, the value of $k_0^{(r)}(s)$ both situations may occur, namely
\beq
| \mf{v}_r\big( k_0^{(r)}(s) \big) | < \op{v}_F \qquad \e{or} \qquad | \mf{v}_r\big( k_0^{(r)}(s) \big) | > \op{v}_F \, .
\enq
\vspace{3mm}
\noindent In case 1 listed in the statement, \textit{viz}. $| \mf{v}_r\big(k_0^{(r)}(s) \big) | > \op{v}_F$, one observes that
\beq
\Dp{k^{(r)}} \mf{y}_{+}^{(r)} \big( k_0^{(r)}(s) ; s \big) \cdot \Dp{k^{(r)}} \mf{y}_{-}^{(r)} \big( k_0^{(r)}(s) ; s \big) \; > \; 0
\enq
and that
\beq
-\e{sgn}\Big\{ \Dp{k^{(r)}} \mf{y}_{+}^{(r)} \big( k_0^{(r)} (s) ; s \big)
\cdot \big[ \Dp{k^{(r)}} \mf{y}_{+}^{(r)} \big( k_0^{(r)}(s) ; s \big) \, - \, \Dp{k^{(r)}} \mf{y}_{-}^{(r)} \big( k_0^{(r)}(s) ; s \big) \big] \Big\}
\; = \; - \e{sgn}\Big\{ \mf{v}_r\Big( k_0^{(r)}(s) \Big) \Big\} \;.
\enq
This is all that is needed so as to apply Theorem \ref{Proposition integrale principale spectral function} to the situation of interest.
\vspace{3mm}
Finally, when $| \mf{v}_r\big(k_0^{(r)}(s) \big) | < \op{v}_F$ the analysis parallels the one exposed for the contribution of the one hole excitation sector. The details are left to the interested reader. \qed
\subsection{The multi-hole/$r$-string excitation sector}
Below, I will consider the contribution to the DRF issuing form the sector containing multiple hole and multiple $r$-strings all having the same value of $r \in \mf{N}$.
Although it will be not discussed here, the case of multiple hole and various numbers $r$-strings can be treated analogously and
leads to a similar structure of singularities. Likewise, one may derive the behaviour in the sector built up only from multi-particle excitations.
Such results may be easily extracted from the main structural theorem governing the asymptotics of the
class of multiple integrals of interest to the analysis of dynamic response functions, Theorem \ref{Theorem Principal}
which is established in the Appendix.
\subsubsection{Excitations built up from holes and, possibly, particles}
For the purpose of the present section, it is convenient to parametrise the momentum-energy $(k,\om)$ combination as
\beq
k=\mc{P}_0 \; \; \e{where} \;\; \mc{P}_0 \; = \; \pi \op{s}_{\ga} \, + \, p_{F}\sul{\ups=\pm}{} \ups \ell_{\ups} + \mf{q}_0 -2\pi s_{0}
\quad \e{and} \quad \; \om=\de \om + \mc{E}_0
\enq
where $\ell_{\pm}$ are subject to the constraint $\sul{\ups=\pm}{}\ell_{\ups}=n_h-n_p$. The integers $n_p, n_h$ are assumed to satisfy
\beq
n_h \geq 1 \quad \e{and} \quad n_p+n_h \geq 2 \; .
\enq
In this case of interest, the contribution of these types of excitations to the dynamical response function takes the form
\bem
\msc{S}^{(\ga)}_{\bs{n}_{hp}}\big(\mc{P}_0,\mc{E}_0 + \de \om\big) \; = \; \Int{ \big( \msc{J}_h^{(\eps)}\big)^{n_h} }{} \hspace{-2mm} \dd^{n_h} \bs{t} \; \Int{ \big( \msc{J}_1^{(\eps)}\big)^{n_p} }{} \hspace{-2mm} \dd^{n_p} \bs{k}
\; \; \mc{F}^{(\ga)}\big( \bs{\mf{K}}^{(hp)} \big) \\
\times \sul{s\in \mathbb{Z} }{}\pl{\ups= \pm }{}
\bigg\{ \Xi\Big( \de \om + \mf{y}_{\ups}^{(hp)}\big( \bs{\mf{K}}^{(hp)} ;s \big) \Big) \cdot \Big[ \de \om + \mf{y}_{\ups}^{(hp)}\big( \bs{\mf{K}}^{(hp)} ;s \big) \Big]^{ \De_{\ups}(\bs{\mf{K}}^{(hp)} ) -1 } \bigg\}
\cdot \Big( 1+\mf{r}\big( \bs{\mf{K}}^{(hp)}; s \big) \Big) \;.
\end{multline}
Here, I have set
\beq
\bs{n}_{hp}\, = \, (\ell_+,\ell_-;n_h, n_{1}=n_p, 0,\dots )
\quad \e{and} \quad
\bs{\mf{K}}^{(hp)} \, = \, \Big( \ell_{+}, \ell_{-} ; \bs{t} , \bs{k}^{(1)}=\bs{k},\emptyset,\dots \Big)
\enq
and agree upon
\beq
\mf{y}_{\ups}^{(hp)}\big( \bs{\mf{K}}^{(hp)} ;s \big) \, = \, \mc{E}_{0}\, - \, \sul{a=1}{n_p} \mf{e}_1(k_a) \, + \, \sul{a=1}{n_h} \mf{e}_1(t_a)
\, + \, \ups \op{v}_F \Big( \mf{q}_0 - \, \sul{a=1}{n_p} k_a \, + \, \sul{a=1}{n_h} t_a + 2\pi (s-s_0) \Big) \;.
\enq
Also, recall, \textit{c.f.} Appendix \ref{Appendix Section phase oscillante dpdte de la vitesse}, that for a reduced range of the model's parameters and as a conjecture more generally,
there exists a strictly decreasing diffeomorphism $\mf{t}:\intff{K_m}{K_M}\tend \intff{-p_F}{p_F}$ such that
$\mf{v}_1(k)=\mf{v}_1( \mf{t}(k))$. Set $\mf{p}=\mf{t}^{-1}$ for its inverse.
Further, given $t \in \intff{ - p_F }{ p_F }$, let
\beq
\mc{P}(t) \, = \, n_p \mf{p}(t) - n_h t \quad \e{and} \quad
\mc{E}(t) \, = \, n_p \mf{e}_1( \mf{p}(t) ) - n_h \mf{e}_1(t) \;.
\enq
Finally, the remainder satisfies
\beq
\mf{r}\big( \bs{\mf{K}}^{(hp)}; s \big) \, = \, \e{O}\Big( \sum_{ \ups = \pm } \big|\, \de \om + \mf{y}_{\ups}^{(hp)}\big( \bs{\mf{K}}^{(hp)} ;s \big) \big|^{1-\tau } \Big) \;,
\enq
and the control on $\mf{r}\big( \bs{\mf{K}}^{(hp)}; s \big) $ is differentiable in the sense of Definition \ref{defintion reste differentiable}.
\begin{theorem}
\label{Theorem DRF behaviour of multi particle hole threshold}
Let $\mf{q}_0(s)=\mf{q}_0+2\pi(s-s_0)$ and assume that $n_p+n_h \geq 2$ with $n_h \geq 1$. Also, assume that Conjecture \ref{Conjecture diffeo liee a la vitesse} holds if the
set of parameters of the model does not enter into the specifications of Theorem \ref{Proposition proprietes fondamentales de la vitesse des particules trous}.
\vspace{2mm}
If, for any $s \in \mathbb{Z}$,
\beq
\big( \mf{q}_0(s) , \mc{E}_0 \big) \not\in \Big\{ \big(\mc{P}(t), \mc{E}(t) \big) \; : \; t \in \intff{-p_F}{p_F} \Big\} \;,
\enq
then $\msc{S}^{(\ga)}_{\bs{n}_{hp}}\big(\mc{P}_0,\mc{E}_0 + \de \om\big)$ is a smooth function in $\de \om$ belonging to a neighbourhood of the origin.
\vspace{3mm}
\noindent Assume that, for at least one $s \in \mathbb{Z}$
\beq
\big( \mf{q}_0(s) , \mc{E}_0 \big) \, = \, \Big( \mc{P}(t_0(s)), \mc{E}(t_0(s)) \Big) \quad for \; a \quad t_0(s) \in \intoo{-p_F}{p_F} \;,
\label{definition rapidite k0 de s dans DRF multi points}
\enq
and that for any such $s$ it holds $\de_{\ups}^{(hp)}(s)>0$ where
\beq
\de_{\ups}^{(hp)}(s) \, = \, \De_{\ups}( \bs{\mf{K}}^{(hp)}_0(s)) \quad with \quad
\bs{\mf{K}}_{0}^{(hp)}\, = \,\Big( \ell_{+},\ell_{-} ; \bs{t_0(s)},\bs{\mf{p}( t_0(s) ) } , \emptyset ,\dots \Big) \;.
\enq
There $ \bs{ t_0(s) } \, = \, \Big( t_0(s) ,\cdots, t_0(s) \Big)\in \R^{n_h}$ and $ \bs{\mf{p}( t_0(s) ) } = \Big( \mf{p}( t_0(s) ),\cdots, \mf{p}( t_0(s) ) \Big)\in \R^{n_p}$.
Then, the multi particle-hole spectral function has the $\de \om \tend 0$ asymptotic expansion
\bem
\msc{S}^{(\ga)}_{\bs{n}_{hp}}\big(\mc{P}_0,\mc{E}_0 + \de \om\big)\; = \; \msc{S}^{(\ga)}_{\bs{n}_{hp};\e{reg}}(\de\om) \; + \;
\hspace{-3mm} \sul{ s\; : \; \exists t_0(s) }{}
\f{ \msc{F}^{(\ga)}\big( \bs{\mf{K}}^{(hp)}_0(s) \big) }{ \sqrt{ |\mc{P}^{\prime}(t_0(s)) | } } \cdot
\Bigg( \f{-1 }{ \mf{v}^{\prime}_1\Big( \mf{p}(t_0(s)) \Big) } \Bigg)^{ \f{n_p^2}{2}} \hspace{-2mm} \cdot \;\; \Bigg( \f{ 1 }{ \mf{v}^{\prime}_1\Big( t_0(s) \Big) } \Bigg)^{ \f{n_h^2-1}{2}} \\
\times G(n_p+1) \, G(n_h+1) \cdot \f{ ( \sqrt{2 \pi} )^{3+n_p+n_h } \Ga\Big( -\de_{+}^{(hp)}(s) - \de_{-}^{(hp)}(s) -\tfrac{n_p^2+n_h^2 -3}{2} \Big) }
{ \big| \op{v}_F+\mf{v}_1\Big( t_0(s) \Big) \big|^{ \de_{-}^{(hp)}(s) } \big| \op{v}_F-\mf{v}_1\Big( t_0(s) \Big) \big|^{ \de_{+}^{(hp)}(s) } }
\cdot |\de \om |^{ \de_{+}^{(hp)}(s) + \de_{-}^{(hp)}(s) +\tfrac{n_p^2+n_h^2-3}{2}} \\
\bigg\{ \Xi(\de \om) \f{1}{\pi} \sin \Big( \pi \Big[ \de_{+}^{(hp)}(s) + \de_{-}^{(hp)}(s) \Big] \Big) + \Xi(- \de \om) \f{1}{\pi} \sin \Big(\pi \f{n_p^2+n_h^2-1}{2} \Big) \bigg\}
\; + \; \e{O}\Big( |\de \om |^{ \de_{+}^{(hp)}(s) + \de_{-}^{(hp)}(s) + \tfrac{n_p^2+n_h^2-1}{2} + \tau } \Big) \;.
\label{ecriture DA singulier de S pour multi part trou}
\end{multline}
$ \msc{S}^{(\ga)}_{\bs{n}_{hp};\e{reg}}(\de\om)$ is smooth in the neighbourhood of the origin.
Finally, the summation over $s$ in \eqref{ecriture DA singulier de S pour multi part trou} runs through all solutions $t_0(s)$ to \eqref{definition rapidite k0 de s dans DRF multi points}.
This summation contains at most two terms and, for generic parameters, it only contains one term.
\end{theorem}
\Proof
This is a direct consequence of Theorem \ref{Theorem Principal}. In order to identify quantities with the notations of that theorem,
one should identify the quantities given in Section \ref{Appendix DA integrales multidimensionnelles} of the appendix:
\beq
\ell=2\; , \quad \msc{I}_1 \, = \, \msc{J}_h^{(\eps)} \;, \quad \msc{I}_2 \, = \, \msc{J}_1^{(\eps)} \;, \quad (n_1,n_2)=(n_h,n_p)
\enq
in what concerns the intervals. Further,
\beq
\big(\bs{p}^{(1)}, \bs{p}^{(2)} \big) \, = \, \big(\bs{t} , \bs{k} \big) \; , \quad \big(\mf{u}_1, \mf{u}_{2} \big) \, = \, \big(\mf{e}_1 , \mf{e}_1 \big) \;, \quad
\big( \zeta_1,\zeta_2 \big) \, = \, (-1,1) \;.
\enq
From there one infers that one has
\beq
\mc{P}^{\prime}(t) \, = \, -\big( n_h - n_p\mf{p}^{\prime}(t) \big)<0\quad \e{on} \quad \intoo{-p_F}{p_F}
\enq
since $\mf{p}$ is strictly decreasing. Also, it follows directly from the definition of $\mf{p}$ that
\beq
\mc{E}^{\prime}(t) \, = \, \mf{v}_1(t) \cdot \mc{P}^{\prime}(t)
\enq
so that $t\mapsto \mc{E}(t)$ is strictly increasing on $\intoo{-p_F}{0}$ and strictly decreasing on $\intoo{0}{p_F}$.
Furthermore, one has that
\beq
\Dp{t}\Big\{ \mc{E}_0-\mc{E}(t) \pm \op{v}_F\big( \mf{q}_0(s)-\mc{P}(t) \big) \Big\} \; = \; -\mc{P}^{\prime}(t) \cdot \Big( \mf{v}_1(t)\pm \op{v}_F \Big) \not=0
\label{calcul derive grosse fct Z ups cas multiparticlues}
\enq
on $\intoo{-p_F}{p_F}$.
I first focus on the regular case, namely when, for any $s \in \mathbb{Z}$,
\beq
\big( \mf{q}_0(s) , \mc{E}_0 \big) \not\in \Big\{ \big(\mc{P}(t), \mc{E}(t) \big) \; : \; t \in \intff{-p_F}{p_F} \Big\} \;.
\enq
Then observe that \eqref{calcul derive grosse fct Z ups cas multiparticlues} implies that $t\mapsto \mc{E}_0-\mc{E}(t) \pm \op{v}_F\big( \mf{q}_0(s)-\mc{P}(t) \big)$ are both strictly monotonous
on $\msc{J}_h^{(\eps)}$. Thus, should one of these functions vanish on $\Dp{}\msc{J}_h^{(\eps)}$, it is enough to slightly change $\eps>0$, which is a free parameter in the problem (as long as it is small
and strictly positive) so as to have a non-vanishing function. Thus, automatically, condition \eqref{ecriture condition positivite energie impulsion macroscopique thm principal} stated
in Theorem \ref{Theorem Principal} is satisfied. Then, the results of that theorem ensures the smooth behaviour
of $\de\om \mapsto \msc{S}^{(\ga)}_{\bs{n}_{hp}}\big(\mc{P}_0,\mc{E}_0 + \de \om\big)$ around $0$.
\vspace{3mm}
In the case when there exist at least one $s \in \mathbb{Z}$ such that $\big( \mf{q}_0(s) , \mc{E}_0 \big) \, = \, \Big( \mc{P}(t_0(s)), \mc{E}(t_0(s)) \Big) $ for a $t_0(s) \in \intoo{-p_F}{p_F}$,
one needs to identify additional constants. First, however, one fixes $\eps$ such that $t_0(s) \in \e{Int}(\msc{I}_h^{(\eps)})$ for any $s$ compatible with the mentioned constraint.
One should also observe that the variations of $t\mapsto \mc{E}(t)$ and $t\mapsto \mc{P}(t)$ on $\intoo{-p_F}{p_F}$ entail that
there may at most exist two different $s$ such that the previous equality holds. It is also evident that, in the generic case, only one such $s$ will exist.
Since $\mf{u}_1^{\prime\prime}(t)=\mf{v}_1^{\prime}(t)>0$ on $\intoo{-p_F}{p_F}$, while $\mf{u}_{2}^{\prime\prime}( \mf{p}(t) ) = \tf{ \mf{v}_1^{\prime}(t) }{ \mf{p}^{\prime}(t) } <0$ on $\intoo{-p_F}{p_F}$,
it follows that
\beq
\left\{ \ba{cccccc} \veps_1 & = - \zeta_1 \e{sgn}\Big( \mf{u}_1^{\prime\prime}(t_0(s)) \Big) & = & \e{sgn}\Big( \mf{v}_1^{\prime}(t_0(s)) \Big) & =1 \vspace{2mm} \\
\veps_2 & = - \zeta_2 \e{sgn}\Big( \mf{u}_2^{\prime\prime}\big[ \mf{p}(t_0(s)) \big] \Big) & = & -\e{sgn}\Big( \mf{v}_1^{\prime}\big[ \mf{p}(t_0(s)) \big] \Big) & =-1 \ea \right.
\enq
and $\mf{s}= -\e{sgn} \bigg( \f{\mc{P}^{\prime}(t_0) }{ \mf{u}_1^{\prime\prime}(t_0(s)) } \bigg) > 0 $\;.
All parameters being identified, it remains to apply the results of Theorem \ref{Theorem Principal}
to each $s \in \mathbb{Z}$ such that a $t_0(s)$ exists.
Finally, the $n_h \geq 2 $ and $n_p=0$ case is treated along much the same lines. The results boils down to \eqref{ecriture DA singulier de S pour multi part trou}
with $n_p$ being set to $0$. \qed
\subsubsection{Excitations built up from holes and a fixed $r$-string species}
Below, $r\in \mf{N}_{\e{st}}$ is assumed to be fixed.
For the purpose of the present section, it is convenient to parametrise the momentum-energy $(k,\om)$ combination as
\beq
k=\mc{P}_0 \; \; \e{where} \;\; \mc{P}_0 \; = \; \pi \op{s}_{\ga} \, + \, p_{F}\sul{\ups=\pm}{} \ups \ell_{\ups} + \mf{q}_0 -2\pi s_{0}
\quad \e{and} \quad \; \om=\de \om + \mc{E}_0
\enq
where $\ell_{\pm}$ are subject to the constraint $\sul{\ups=\pm}{}\ell_{\ups}=n_h-n_{\e{st}}$. The integers $n_{\e{st}}, n_h$ are assumed to satisfy
\beq
n_h \geq 1 \quad \e{and} \quad n_{\e{st}} + n_h \geq 2 \; .
\enq
In this case of interest, the contribution of these types of excitations to the dynamical response function takes the form
\bem
\msc{S}^{(\ga)}_{\bs{n}_{hr}}\big(\mc{P}_0,\mc{E}_0 + \de \om\big) \; = \;
\Int{ \big( \msc{J}_h^{(\eps)}\big)^{n_h} }{} \hspace{-2mm} \dd^{n_h} \bs{t} \; \Int{ \msc{J}_r^{ n_{\e{st}} } }{} \hspace{-2mm} \dd^{ n_{\e{st}} } \bs{k}
\; \; \mc{F}^{(\ga)}\big( \bs{\mf{K}}^{(hr)} \big) \\
\times \sul{s\in \mathbb{Z} }{}\pl{\ups= \pm }{}
\bigg\{ \Xi\Big( \de \om + \mf{y}_{\ups}^{(hr)}\big( \bs{\mf{K}}^{(hr)} ;s \big) \Big) \cdot \Big[ \de \om + \mf{y}_{\ups}^{(hr)}\big( \bs{\mf{K}}^{(hr)} ;s \big) \Big]^{ \De_{\ups}(\bs{\mf{K}}^{(hr)} ) -1 } \bigg\}
\cdot \Big( 1+\mf{r}\big( \bs{\mf{K}}^{(hr)}; s \big) \Big) \;.
\end{multline}
Here, I have set
\beq
\bs{n}_{hr}\, = \, (\ell_+,\ell_-;n_h, 0,\dots, 0, n_{r}=n_{\e{st}}, 0,\dots )
\quad \e{and} \quad
\bs{\mf{K}}^{(hr)} \, = \, \Big( \ell_{+}, \ell_{-} ; \bs{t} ,\emptyset , \dots, \emptyset, \bs{k}^{(r)}=\bs{k},\emptyset,\dots, \Big)
\enq
and agree upon
\beq
\mf{y}_{\ups}^{(hr)}\big( \bs{\mf{K}}^{(hr)} ;s \big) \, = \, \mc{E}_{0}\, - \, \sul{a=1}{ n_{\e{st}} } \mf{e}_r(k_a) \, + \, \sul{a=1}{n_h} \mf{e}_1(t_a)
\, + \, \ups \op{v}_F \Big( \mf{q}_0 - \, \sul{a=1}{ n_{\e{st}} } k_a \, + \, \sul{a=1}{n_h} t_a + 2\pi (s-s_0) \Big)
\enq
Also, recall, \textit{c.f.} Appendix \ref{Appendix Section phase oscillante dpdte de la vitesse}, that for a reduced range of the model's parameters and as a conjecture more generally,
there exists a diffeomorphism $\mf{h}^{(r)}:\intff{-p_F}{p_F} \tend \intff{ K_m^{(r)} }{ K_M^{(r)} }$ such that
$\mf{v}_1(t)=\mf{v}_r( \mf{h}^{(r)}(t))$.
Further, given $t \in \intff{ - p_F }{ p_F }$, let
\beq
\mc{P}(t) \, = \, n_{\e{st}} \mf{h}^{(r)}(t) - n_h t \quad \e{and} \quad
\mc{E}(t) \, = \, n_{\e{st}} \mf{e}_r( \mf{h}^{(r)}(t) ) - n_h \mf{e}_1(t) \;.
\enq
Finally, the remainder satisfies
\beq
\mf{r}\big( \bs{\mf{K}}^{(hr)}; s \big) \, = \, \e{O}\Big( \sum_{ \ups = \pm } \big|\, \de \om + \mf{y}_{\ups}^{(hr)}\big( \bs{\mf{K}}^{(hr)} ;s \big) \big|^{1-\tau } \Big) \;,
\enq
and the control on $\mf{r}\big( \bs{\mf{K}}^{(hr)}; s \big) $ is differentiable in the sense of Definition \ref{defintion reste differentiable}.
\begin{theorem}
\label{Theorem DRF behaviour of multi string hole threshold}
Let $\mf{q}_0(s)=\mf{q}_0+2\pi(s-s_0)$ and assume that $n_{\e{st}}+n_h \geq 2$ with $n_h \geq 1$. Also, assume that Conjecture \ref{Conjecture diffeo liee a la vitesse} holds if the
set of parameters of the model does not enter into the specifications of Theorem \ref{Proposition proprietes fondamentales de la vitesse des particules trous}.
Finally, assume that $t \mapsto \mc{P}(t)$ is a diffeomorphism on $\intff{-p_F}{p_F}$.
\vspace{2mm}
If, for any $s \in \mathbb{Z}$,
\beq
\big( \mf{q}_0(s) , \mc{E}_0 \big) \not\in \Big\{ \big(\mc{P}(t), \mc{E}(t) \big) \; : \; t \in \intff{-p_F}{p_F} \Big\} \;,
\enq
then, for $\eps>0$ small enough, $\msc{S}^{(\ga)}_{\bs{n}_{hr}}\big(\mc{P}_0,\mc{E}_0 + \de \om\big)$ is a smooth function in $\de \om$ belonging to a neighbourhood of the origin.
\vspace{3mm}
\noindent Assume that, for at least one $s \in \mathbb{Z}$,
\beq
\big( \mf{q}_0(s) , \mc{E}_0 \big) \, = \, \Big( \mc{P}(t_0(s)), \mc{E}(t_0(s)) \Big) \quad for \; a \quad t_0(s) \in \intoo{-p_F}{p_F} \;,
\label{definition rapidite k0 de s dans DRF multi hole r string}
\enq
and that, for any such $s$ it holds $\de_{\ups}^{(hr)}(s)$ with
\beq
\de_{\ups}^{(hr)}(s) \, = \, \De_{\ups}( \bs{\mf{K}}^{(hr)}_0(s)) \quad and \quad
\bs{\mf{K}}_{0}^{(hr)}\, = \,\Big( \ell_{+},\ell_{-} ; \bs{t_0(s)},\emptyset, \dots, \emptyset, \underset{ r-\e{string} \, \e{position} }{ \bs{\mf{h}^{(r)}( t_0(s) ) } } , \emptyset ,\dots \Big) \;.
\enq
There $ \bs{ t_0(s) } \, = \, \Big( t_0(s) ,\cdots, t_0(s) \Big)\in \R^{n_h}$ and $ \bs{\mf{h}^{(r)}( t_0(s) ) } = \Big( \mf{h}^{(r)}( t_0(s) ),\cdots, \mf{h}^{(r)}( t_0(s) ) \Big)\in \R^{ n_{\e{st}} }$.
Then, the multi r-string-hole spectral function has the $\de \om \tend 0$ asymptotic expansion
\bem
\msc{S}^{(\ga)}_{\bs{n}_{hr}}\big(\mc{P}_0,\mc{E}_0 + \de \om\big)\; = \; \msc{S}^{(\ga)}_{\bs{n}_{hr};\e{reg}}(\de\om) \; + \;
\hspace{-3mm} \sul{ s\; : \; \exists t_0(s) }{}
\f{ \msc{F}^{(\ga)}\big( \bs{\mf{K}}^{(hr)}_0(s) \big) }{ \sqrt{ |\mc{P}^{\prime}(t_0(s)) | } } \cdot
\Bigg( \tfrac{ 1 }{ \big| \mf{v}^{\prime}_1\big( \mf{h}^{(r)}(t_0(s)) \big) \big| } \Bigg)^{ \f{n_{\e{st}}^2}{2}} \hspace{-2mm} \cdot \;\; \Bigg( \tfrac{ 1 }{ \mf{v}^{\prime}_1\big( t_0(s) \big) } \Bigg)^{ \f{n_h^2-1}{2}} \\
\times G( n_{\e{st}} + 1) \, G(n_h+1) \cdot \f{ ( \sqrt{2 \pi} )^{3+n_{\e{st}}+n_h } \Ga\Big( -\de_{+}^{(hr)}(s) - \de_{-}^{(hr)}(s) -\tfrac{n_{\e{st}}^2+n_h^2 -3}{2} \Big) }
{ \big| \op{v}_F+\mf{v}_1\Big( t_0(s) \Big) \big|^{ \de_{-}^{(hr)}(s) } \big| \op{v}_F-\mf{v}_1\Big( t_0(s) \Big) \big|^{ \de_{+}^{(hr)}(s) } }
\cdot |\de \om |^{ \de_{+}^{(hr)}(s) + \de_{-}^{(hr)}(s) +\tfrac{n_{\e{st}}^2+n_h^2-3}{2}} \\
\bigg\{ \Xi(\de \om) \f{1}{\pi} \sin \Big( \pi \nu_+^{(hr)}(s) \Big) + \Xi(- \de \om) \f{1}{\pi} \sin \Big(\pi \nu_-^{(hr)}(s) \Big) \bigg\}
\; + \; \e{O}\Big( |\de \om |^{ \de_{+}^{(hr)}(s) + \de_{-}^{(hr)}(s) + \tfrac{n_{\e{st}}^2+n_h^2-1}{2} + \tau } \Big) \;.
\label{ecriture DA singulier de S pour multi hole r string}
\end{multline}
$ \msc{S}^{(\ga)}_{\bs{n}_{hr};\e{reg}}(\de\om)$ is smooth in the neighbourhood of the origin.
Further, one has
\beq
\nu_+^{(hr)}(s) \; = \; \de_{+}^{(hr)}(s) + \de_{-}^{(hr)}(s)+\frac{1}{2}\de_{-,\sg_r} n_{\e{st}}^2-\f{1-\mf{s}_r}{2}
\quad , \qquad
\nu_-^{(hr)}(s) \; = \; \frac{1}{2}\de_{+,\sg_r} n_{\e{st}}^2 \, + \, \f{ 1}{2} n_h^2 + \f{1-\mf{s}_r}{2}
\enq
where
\beq
\sg_r \, = \, \e{sgn}\Big[ \mf{v}^{\prime}_1\Big( \mf{h}^{(r)}(t_0(s)) \Big) \Big] \qquad and \qquad \mf{s}_r \; = \; - \e{sgn}\Big[ \mc{P}^{\prime}(t_0(s)) \Big] \;.
\enq
Finally, the summation over $s$ in \eqref{ecriture DA singulier de S pour multi hole r string} runs through all solutions $t_0(s)$ to \eqref{definition rapidite k0 de s dans DRF multi hole r string}.
This summation contains at most two terms and, for generic parameters, it only contains one term.
\end{theorem}
The proof follows closely the case of the multi-hole multi-particle sector, so that I omit the details.
\section{Conclusion}
This work developed a technique allowing one to extract, on rigorous grounds, the asymptotic behaviour in certain parameters of a family of multiple integrals. These results are detailed
in Sections \ref{Appendix DA integrales unidimensionnelles}, \ref{Appendix DA integrales multidimensionnelles} of the appendix. The multiple integrals studied in these sections, upon specialisation, contain the multiple
integrals which define the coefficients of the series giving the
massless form factor expansion issued representation for the DRF in the XXZ chain that was derived in \cite{KozMasslessFFSeriesXXZ}.
Hence, the analysis I developed allowed, upon relying on additional properties that were argued in \cite{KozLongDistanceLargeTimeXXZ,KozMasslessFFSeriesXXZ},
to give a precise characterisation of the singular behaviour in the $(k,\om)$ plane of the series' coefficients.
In doing so, this work provides a test and confirmation of the predictions,
issuing from the non-linear Luttinger liquid approach, for some of the singularities of the DRF, namely those
issuing from a one massive excitation process. Furthermore, the work showed the existence of other singularity lines: the ones issuing from multi-particle/hole/r-string excitations and which correspond to
configurations of the various momenta that maximise the multi-excitation energy at fixed momentum.
Such multi-species singularity curves generate structurally different edge exponents and universality constants. The edge exponents associated with one such
"mixed" excitation were discussed in \cite{AffleckPereiraWhiteSpectralFunctionsfor1DLatticeFermionsBoundStatesContributions}, but all the other cases were not
considered in the literature. Furthermore, the work \cite{AffleckPereiraWhiteSpectralFunctionsfor1DLatticeFermionsBoundStatesContributions} only focused on the exponents
and so did not provide any expression for the universal part of the amplitude. Thus, the expression for the universal part of the
amplitude is new.
\section*{Acknowledgment}
K.K.K. acknowledges support from CNRS and ENS de Lyon. The author is indebted to J.-S. Caux, F. Göhmann, J.M. Maillet, G. Niccoli for stimulating discussions
on various aspects of the project.
\appendix
\section*{Appendix}
\section{Main notations}
\label{Appendix Fcts speciales}
\subsection*{Sets}
\begin{itemize}
\item Given a set $A$, $\e{Int}(A)$ stands for its interior, $\ov{A}$ for its closure and $\Dp{}A$ for its boundary.
\item Given a finite set $A$, $|A|$ stands for its cardinal.
\item $\mathbb{N}=\{0,1, 2, \dots\}$, $\R^{+}=\intoo{0}{+\infty}$, $\R^{*}=\R\setminus \{0\}$.
\item $\intn{1}{n}=\{1,\dots, n\}$ and $\mf{S}_{n}$ stands for the permutation group of $\intn{1}{n}$.
\item $\sqcup$ refers to the disjoint union of sets.
\item $\de_{a,b}$ stands for the Kronecker symbol: $\de_{a,b}=1$ if $a=b$ and $\de_{a,b}=0$ otherwise.
\item Given $\ell$ integers $n_1,\dots, n_{\ell}$, it is understood that
\beq
\ov{\bs{n}}_{\ell} \; = \; \sul{ r = 1 }{ \ell } n_{r} \;.
\label{definition bs ne ell}
\enq
\end{itemize}
\subsection*{Vectors and related objects}
\begin{itemize}
\item $^{\op{t}}$ stands for the transposition of a matrix or vector, depending on the context.
\item $\op{I}_{n}$ stands for the identity matrix on $\R^n$, and it will sometimes also be denoted as $\e{id}$.
\item Vectors are denoted in bold, \textit{viz}. $ \bs{x} \in \R^n$ corresponds to the vector $\bs{x}=(x_1,\dots,x_n)$. The dimensionality of the vector is
always undercurrent by the context.
\item If the vector space has a natural Cartesian product structure $\pl{r=1}{\ell} \R^{n_{r}}$, then any vector $\bs{x}$ is represented as
\beq
\bs{x} \; = \; \big( \bs{x}^{(1)} , \dots , \bs{x}^{(\ell)} \, \big)\qquad \e{with} \qquad \bs{x}^{(r)}\, = \, \big( x_1^{(r)}, \cdots, x_{n_r}^{(r)} \big) \in \R^{n_r} \;.
\label{notation vecteur dans produit cartesien}
\enq
\item Vectors with omitted coordinates are denoted as :
\beq
\bs{x}^{(r)}_{[a]} \, = \, \big( x_1^{(r)} , \dots, x_{a-1}^{(r)},x_{a+1}^{(r)},\dots , x_{n_r}^{(r)} \, \big) \quad \e{and} \quad
\bs{x}_{[r,a]} \, = \, \Big( \bs{x}^{(1)} , \dots , \bs{x}^{(r)}_{[a]}, \dots , \bs{x}^{(\ell)} \big) \;.
\label{notation vecteurs reduits}
\enq
\item Given $\bs{\a}, \bs{\be} \in \pl{r=1}{\ell} \mathbb{N}^{n_{r}}$, it is understood that
\beq
| \bs{\a} | \, = \, \sul{r=1}{ \ell } \sul{a=1}{n_{r}} \a_{a}^{(r)} \qquad \e{and} \qquad
\bs{\a} \geq \bs{\be} \quad \Longleftrightarrow \quad \forall (r,a) \quad \a_{a}^{(r)} \geq \be_{a}^{(r)} \;.
\label{notation norme et ordre partiel sur entiers vecteurs}
\enq
\item Given $\bs{\a} \in \pl{r=1}{\ell} \mathbb{N}^{n_{r}}$ and $\bs{x} \in \pl{r=1}{\ell} \R^{n_{r}}$, one has
\beq
\bs{x}^{ \bs{\a} } \, = \, \pl{r=1}{\ell}\pl{a=1}{n_r} \big[ x_{a}^{(r)} \big]^{ \a_{a}^{(r)} } \;.
\label{notation exposant polynomial vectoriel}
\enq
\end{itemize}
\subsection*{Functions}
\begin{itemize}
\item Given a set $A$, $\bs{1}_{A}$ stands for the indicator function of $A$.
\item $\Xi$ refers to the Heaviside step function, \textit{viz}. $\Xi=\bs{1}_{\R^+}$.
\item $\Ga$ refers to the Gamma function which allows one to express the Euler $\be$-integral as
\beq
\Int{0}{1} t^{x-1} (1-t)^{y-1} \cdot \dd t \, = \, \f{ \Ga(x) \Ga(y) }{ \Ga(x + y) } \;.
\label{ecriture formule integrale pour fct beta}
\enq
\item $G$ stands for the Barnes \cite{BarnesDoubleGaFctn1,BarnesDoubleGaFctn2} function.
\item The Gaudin-Mehta integral is expressed in terms of the Barnes function as:
\beq
\Int{ \R^n }{} \dd \bs{y} \; \ex{- (\bs{y},\bs{y})} \pl{a<b}{n} (y_a-y_b)^2 \; =\; \Big( \f{1}{2} \Big)^{\f{1}{2} n^2 } \big( 2\pi \big)^{ \f{n}{2} } G(2+n) \;.
\label{formule integrale Gaudin-Mehta}
\enq
\item Given $S \subset \R^{n} $ measurable and a function $f:S \tend \R$,
\beq
\norm{f}_{L^{\infty}(S)} \, = \, \e{supess}\Big\{|f(\bs{x})| \, : \, \bs{x}\in S \Big\} \;.
\enq
\item Given $g:U\times V \tend W$, with $U\subset \R^n$, $V\subset \R^m$ and $W\subset \R^o$ the totally even part of a function
in respect to a set of variables is defined as
\beq
\Big[ g(\bs{z}, \bs{v}) \Big]_{ \bs{z}-\e{even} } \; = \; \f{1}{ 2^{n} } \sul{ \substack{ \eps_a=\pm \\ a=1,\dots, n } }{ } g(\bs{z}^{(\eps)}, \bs{v})
\qquad \e{with} \qquad
\bs{z}^{(\eps)}\; = \; \big(\eps_1 z_1, \dots, \eps_d z_d \big) \;.
\label{definition even part of a function}
\enq
\item Given a smooth function $f:U\tend V$ between two open subsets $U\subset \R^n$ and $V\subset \R^m$, $D_{\bs{x}}^{(k)}f$ denotes the $k^{\e{th}}$-order differential of $f$
at the point $\bs{x}\in U$. When $k=1$, it is simply denoted as $D_{\bs{x}}f$.
\end{itemize}
\begin{defin}
\label{defintion reste differentiable}
Given smooth functions $f,g$ on an open neighbourhood of a point $\bs{y} \in \R^n$, one says that a $\e{O}$-remainder relation $f=\e{O}(g)$ when $\bs{x} \tend \bs{y}$ is
differentiable if, for each $\bs{\ell}=(\ell_1,\dots, \ell_n) \in \mathbb{N}^n$ there exists a smooth function $\psi_{\bs{\ell}}$ in the vicinity of $\bs{y}$ and a constant $C_{\bs{\ell}}>0$
such that
\beq
\pl{a=1}{n} \Dp{x_a}^{\ell_a} \cdot f(\bs{x}) \; \leq \; C_{\bs{\ell}} \cdot \pl{a=1}{n} \Dp{x_a}^{\ell_a} \cdot [\psi_{\bs{\ell}} g \big](\bs{x})
\label{Defintion reste differentiable}
\enq
on some open neighbourhood of $\bs{y}$.
\end{defin}
Note that the use of $\psi_{\bs{\ell}}$ in this definition allows one to encompass a situation when $g$ does not depend explicitly on some of the variables.
\section{Auxiliary theorems}
\label{Appendix auxiliary theorems}
The proof of theorems \ref{Theorem Morse Lemma}, \ref{Theorem Weierstrass preparation theorem} and \ref{Theorem Malgrange preparation theorem}
can be found in \cite{GolubitskyGuilleminStableMappingAndSings}. The proof of Theorem \ref{Theoreme extension de Whitney} can be found in \cite{BierstoneWhitneyExtensionTheoremAndOtherDiffFctsProperties}.
\begin{theorem}{\bf Morse Lemma}
\label{Theorem Morse Lemma}
Let $f : U\tend \R $ be a smooth function on an open set $U\subset \R^n$. Let $\bs{p} \in U$ be a non-degenerate critical point of $f$.
Let $\op{M}$ be the matrix associated with the bilinear form $ D_{\bf{p}}^2f$:
\beq
\big( \bs{v}, \op{M} \bs{w} \big) \, = \, D_{\bf{p}}^2f\big( \bs{v}, \bs{w} \big) \; .
\enq
Then, there exists an open neighbourhood $V_0$ of $\bs{0}\in \R^n$ and a smooth diffeomorphism onto $g: V_0 \mapsto U_0 \subset U$ such that
\begin{itemize}
\item $\bs{0}\in V_0$ and $g(\bs{0})=\bs{p} \in U_0$;
\item $f\circ g (\bs{x}) \, = \, (\bs{x}, \op{M} \bs{x})$ on $V_0$.
\end{itemize}
Here $(\cdot,\cdot)$ is the canonical scalar product on $\R^n$.
\end{theorem}
\begin{theorem}{\bf Weierstrass preparation theorem}
\label{Theorem Weierstrass preparation theorem}
Let $f$ be a holomorphic function on an open set $V \subset \Cx^n$. Let $\bs{y}\in V$ and $d \in \mathbb{N}$ be such that
\beq
\big( \Dp{z_n}^{k}f \big) (\bs{y}) \, = \, 0 \quad for \quad k=0,\dots, d-1, \quad and \quad
\big( \Dp{z_n}^{d}f\big) (\bs{y}) \, \not= \, 0 \;.
\enq
Then there exists
\begin{itemize}
\item open neighbourhoods $U_0 \subset \Cx^{n-1}$ of $\bs{y}_{[n]}=(y_1,\dots,y_{n-1})$ and $W_0 \subset \Cx $ of $y_n$
such that $V_0=U_0\times W_0 \subset V$;
\item a holomorphic, non-vanishing, function h on $V_0$;
\item a Weierstrass polynomial
\beq
\mc{W}(\bs{z}) \, = \, (z_n-y_n)^d \, + \, \sul{k=0}{d-1} (z_n-y_n)^k \, a_k(\bs{z}_{[n]}) \qquad with \quad
\bs{z}_{[n]}\, = \, (z_1,\dots,z_{n-1})\; ,
\enq
and $a_k$, $k \in \intn{0}{d-1}$ , being holomorphic functions on $U_0$ satisfying $a_k(\bs{y}_{[n]})=0$;
\end{itemize}
such that one has the factorisation
\beq
f = \mc{W} \cdot h \quad on \quad V_0=U_0\times W_0 \, .
\enq
\end{theorem}
\begin{theorem}{\bf Malgrange preparation theorem}
\label{Theorem Malgrange preparation theorem}
Let $f$ be a smooth function on an open set $V \subset \R^n$. Let $\bs{y}\in V$ and $d \in \mathbb{N}$ be such that
\beq
\big( \Dp{z_n}^{k}f\big)(\bs{y}) \, = \, 0 \quad for \quad k=0,\dots, d-1, \quad and \quad
\big( \Dp{z_n}^{d}f \big)(\bs{y}) \, \not= \, 0 \;.
\enq
Then there exists
\begin{itemize}
\item open neighbourhoods $U_0 \subset \R^{n-1}$ of $\bs{y}_{[n]}=(y_1,\dots,y_{n-1})$ and $W_0 \subset \R $ of $y_n$
such that $V_0=U_0\times W_0 \subset V$;
\item a smooth, non-vanishing, function h on $V_0$;
\item a Weierstrass polynomial
\beq
\mc{W}(\bs{x}) \, = \, (x_n-y_n)^d \, + \, \sul{k=0}{d-1} (x_n-y_n)^k \, a_k(\bs{x}_{[n]}) \qquad with \quad
\bs{x}_{[n]}\, = \, (x_1,\dots,x_{n-1})\; ,
\enq
and the $a_k$'s all being smooth on $U_0$ and such that $a_k(\bs{y}_{[n]})=0 $;
\end{itemize}
such that one has the factorisation
\beq
f = \mc{W} \cdot h \quad on \quad V_0=U_0\times W_0 \, .
\enq
\end{theorem}
\begin{defin}
\label{Definition fct lisse sur ferme}
Let $F$ be a closed set in $\R^n$ such that $F \; = \; \ov{\e{Int}(F)}$. A function $f$ is said to be a smooth function on $F$ if
\begin{itemize}
\item $f$ is smooth on $\e{Int}(F)$;
\item for any $\bs{k} \in \mathbb{N}^n$, $f^{(\bs{k})}\equiv \pl{a=1}{N}\Dp{x_a}^{k_a} f$ extends continuously to $F$;
\item for any $\bs{a} \in \Dp{} F$, $f$ admits an all order Taylor series expansion, \textit{viz}. for any $m\geq 0$ it holds
\beq
f(\bs{x}) \; = \; \sul{ \substack{ \bs{k} \in \mathbb{N}^n \; : \; \\ |\bs{k}| \leq m } }{} f^{(\bs{k})}(a) \big(\bs{x}-\bs{a} \big)^{\bs{k}} \; + \, R_a^{m}[f](x)
\qquad \e{with} \qquad
R_a^{m}[f](x) \, = \, \e{o}\Big( ||\bs{x}-\bs{a} ||^m \Big) \;.
\enq
\end{itemize}
\end{defin}
This definition of smoothness can be stated, in greated generality, in the language of jets where it translates itself in the jet associated to a given function being a Whitney field.
\begin{theorem}{\bf Whitney extension theorem}
\label{Theoreme extension de Whitney}
Let $U\subset \R^n$ be open and $X \subset U$ be closed in $\R^n$. Any $f$ smooth on $X$ admits a smooth extension into a function $f_{\mf{e}}$ to $U$, with $f_{\mf{e}}^{(\bs{k})}=f^{(\bs{k})}$ on $X$.
\end{theorem}
\section{Observables in the infinite XXZ chain}
\label{Appendix Observables XXZ}
\subsection{Solutions to linear integral equations}
\label{Appendix Lin Int Eqns Defs et al}
The observables describing the thermodynamic limit of the spin-$1/2$ XXZ chain are characterised by means of a collection of functions
solving linear integral equations. These equations are driven by an operator $\op{K}_{\eta,Q}$ on $L^2(\intff{-Q}{Q})$ characterised by the integral kernel $K(\la,\mu)=K(\la-\mu\mid \eta)$ with
\beq
K(\la\mid \eta ) \, = \, \f{ \sin(2\eta) }{ 2\pi \sinh(\la + \i \eta) \sinh(\la - \i \eta) } \;.
\label{ecriture fonction K de lambda et eta}
\enq
To introduce all of the functions of interest to this work, one starts by defining the $Q$-dependent dressed energy which allows one to construct the Fermi zone of the model.
It is defined as the solution to the linear integral equation
\beq
\veps(\la\mid Q) \, + \, \Int{-Q}{Q} K\big(\la-\mu\mid \zeta \big) \, \veps(\la\mid Q) \cdot \dd \mu \; = \; h - 4 \pi J \sin(\zeta) K\big( \la \mid \tfrac{1}{2}\zeta \big) \;.
\label{definition energie habille et energie nue}
\enq
Note that the unique solvability of \eqref{definition energie habille et energie nue} follows from $\op{K}_{\zeta,Q}$ having its spectral radius $<1$.
The endpoint of the Fermi zone is defined as the unique \cite{KozDugaveGohmannThermoFunctionsZeroTXXZMassless} positive solution $q$ to $\veps(q\mid q)=0$.
Then, the function $\veps_1(\la)\equiv \veps(\la\mid q)$ corresponds to the dressed energy of the particle-hole excitations of the model.
The functions
\beq
\veps_r(\la)\; = \; r h - 4 \pi J \sin(\zeta) K\big( \la \mid \tfrac{r}{2}\zeta \big) \, -\, \Int{-q}{q} K_{r}\big(\la-\mu \big) \veps_1(\mu) \cdot \dd \mu
\label{definition r energi habille}
\enq
with
\beq
K_{r}(\la) \, = \, K\Big(\la \mid \tfrac{1}{2} \zeta(r+1) \Big) \, + \, K\Big(\la \mid \tfrac{1}{2} \zeta(r-1) \Big)
\enq
correspond to the dressed energies of the $r$-bound state excitations.
For any $0<\zeta<\tf{\pi}{2}$ and under some additional constraints for $\tf{\pi}{2}< \zeta < \pi$, one can show \cite{KozProofOfStringSolutionsBetheeqnsXXZ}
that $\veps_{r}(\la+\i\de \tf{\pi}{2}) > c_r>0$ for any $\la\in \R$, and $\de \in \{0, 1\}$.
However, this lower bound should hold throughout the whole massless regime $0<\zeta<\pi$, irrespectively of some additional constraints.
This property has been checked to hold by numerical study of the solutions to \eqref{definition r energi habille}, \textit{c.f.} \cite{KozProofOfStringSolutionsBetheeqnsXXZ}.
In order to introduce the dressed momenta of the $r$-bound states and of the particle-hole excitations, I first need to
define the $r$-bound state bare phases $\theta_r$ :
\beq
\theta_r(\la) \, = \, 2\pi \Int{ \Ga_{\la} }{} K_r(\mu-0^+ ) \cdot \dd \mu \quad \e{for} \;\; r \geq 2
\quad \e{and} \qquad \theta_1(\la) \, = \, \theta(\la\mid \zeta)
\enq
with
\beq
\theta(\la\mid \eta) \, = \, 2\pi \Int{ \Ga_{\la} }{} K(\mu-0^+\mid \eta ) \cdot \dd \mu \;.
\enq
The contour of integration corresponds to the union of two segments $ \Ga_{\la} \; = \; \intff{ 0 }{ \i \Im(\la) }\cup \intff{\i \Im(\la) }{ \la } $ and the $-0^+$ prescription indicates that the poles
of the integrand at $\pm \i \eta +\i \pi \mathbb{Z}$ should be avoided from the left.
Then, the function
\bem
p_r(\la)\; = \; \theta\big(\la\mid \tfrac{r}{2}\zeta \big) \, -\, \Int{-q}{q} \theta_{r}\big(\la-\mu \big) p^{\prime}_1(\mu) \cdot \f{ \dd \mu }{2\pi} \\
\, + \, \pi \ell_r(\zeta)-p_{F}m_r(\zeta)
-2p_{F} \sul{ \sg=\pm }{} \big(1\, - \, \de_{\sg,-}\de_{r,1} \big) \e{sgn}\Big( 1- \tfrac{2}{\pi} \cdot \wh{\tfrac{r+\sg}{2}\zeta} \Big) \cdot \bs{1}_{ \mc{A}_{r,\sg} } (\la)\;,
\label{definition r moment habille}
\end{multline}
extended by $\i \pi$-periodicity to $\Cx$, corresponds to the dressed momentum of the $r$-bound states. Above, I have introduced
\beq
\ell_r(\zeta)=1-r+2 \lfloor \tfrac{r \zeta}{2\pi} \rfloor \qquad \e{and} \qquad
m_r(\zeta)=2-r - \de_{r,1} + 2 \sul{ \ups = \pm }{} \lfloor \zeta \tfrac{r + \ups }{2\pi} \rfloor \;.
\enq
Furthermore, I agree upon
\beq
\wh{\eta} \, = \, \eta - \pi \lfloor \tfrac{ \eta }{ \pi } \rfloor \quad \e{and} \qquad \mc{A}_{r,\sg} \, = \,
\Big\{ \la \in \Cx \; : \; \tfrac{\pi}{2}\geq |\Im(\la) | \geq \e{min}\big( \wh{\tfrac{r+\sg}{2}\zeta} , \pi- \wh{\tfrac{r+\sg}{2}\zeta} \big) \Big\} \;.
\enq
In order to obtain $p_r$, one should first solve the linear integro-differential equation for $p_1$ and then
use $p_1$ to define $p_r$ by \eqref{definition r moment habille}. $p_1$ corresponds to the dressed momentum of the particle-hole excitations and $p_F=p_1(q)$ corresponds to the Fermi momentum.
$1$-strings have their rapidities $\la \in \big\{ \R \setminus \intff{-q}{q} \big\} \cup \big\{ \R+\i\tf{\pi}{2}\big\}$ while $r\geq 2$ strings, $r \in \mf{N}\setminus \{1\}$,
have their rapidities $\la \in \R+\i\de_{r}\tf{\pi}{2}$ for a $\de_r=0$ or $1$, depending on the value of $r$ and $\zeta$. See, \textit{e.g.} \cite{TakahashiThermodynamics1DSolvModels}
for more details on the string parities.
One can show \cite{KozProofOfStringSolutionsBetheeqnsXXZ} under similar conditions on $\zeta$ as for the dressed energy that, for any $\la \in \R$,
\beq
\big| p^{\prime}_{r}\big(\la +\i \de_{r} \tfrac{\pi}{2} \big) \big| \; > \; 0 \quad \e{when} \quad r \in \mf{N}\setminus \{1\}
\qquad \e{and} \qquad
\e{min}\Big( p^{\prime}_{1}\big(\la\big) \, , \, - p^{\prime}_{1}\big(\la + \i \tfrac{\pi}{2} \big) \Big) \; > \; 0 \;.
\label{ecriture equation positivite pr prime}
\enq
Again, a numerical investigation indicates that \eqref{ecriture equation positivite pr prime} does, in fact, hold irrespectively of the value of $\zeta$.
It is convenient to introduce a piecewise shifted deformation of $p_1$:
\beq
\wh{p}_1(\la) \; = \; p_1(\la) \, -\, 2p_F \e{sgn}\big(\pi-2\zeta \big) \bs{1}_{\intoo{-\infty}{-q}}(\la) +2\pi \bs{1}_{\intoo{-\infty}{-q}\cup \{\R+\i\tfrac{\pi}{2}\}}(\la)
\enq
which is a diffeomorphism from the oriented concatenation of sets
\beq
\intfo{q}{+\infty}\cup \big\{-\R+\i\tfrac{\pi}{2}\big\}\cup\intof{-\infty}{-q} \quad
\e{onto} \quad \intff{p_F}{2\pi -p_F-\, 2p_F \e{sgn}\big(\pi-2\zeta \big)} \;.
\enq
The image of $\big\{-\R+\i\tfrac{\pi}{2}\big\}\cup\big\{ \R \setminus \intff{-q}{q} \big\}$ under $\wh{p}_1$ defines the range $ \msc{I}_1 = \intff{ p_-^{(1)} }{ p_+^{(1)} }$
with $p_-^{(1)}=p_F$ and $p_+^{(1)}=2\pi -p_F-\, 2p_F \e{sgn}\big(\pi-2\zeta \big)$, where the particles' momenta evolve. Likewise, the image of $\R+\i \de_r \tf{\pi}{2}$
under $p_r$ defines the range $ \msc{I}_r = \intff{ p_-^{(r)} }{ p_+^{(r)} }$ where the $r$-string momenta evolve. $ p_{\pm}^{(r)}$ can be
readily computed by taking the $\la -\i \de_r \tf{\pi}{2}\tend \pm \infty$ limits in \eqref{definition r moment habille}. However,
since their explicit values do not play a role, we do not provide them here.
\vspace{3mm}
The dressed energies of the excitations in the momentum representation are defined as:
\beqa
\mf{e}_{1}(k) & = & \veps_1\circ \wh{p}_1^{\, -1}(k) \quad \e{for} \quad k \in \msc{I}_1
\label{definition fct mathfrak e 1} \\
\mf{e}_{r}(k) & = & \veps_r\circ p_r^{-1}(k) \quad \e{for} \quad k \in \msc{I}_r \quad \e{and} \quad r \in \mf{N}\setminus \{1\} \;.
\label{definition fct mathfrak e r}
\eeqa
The $r$-bound dressed phase is defined as the solution to
\beq
\phi_{r}(\la,\mu) \, = \, \f{ 1 }{ 2 \pi } \theta_{r}\big( \la -\mu \big) \, - \, \Int{-q}{q} K(\la-\nu)\, \phi_{r}(\nu, \mu ) \cdot \dd \nu \; + \; \f{m_{r}(\zeta)}{2}
\label{definition dressed phase}
\enq
and the dressed charge solves
\beq
Z(\la)\, + \, \Int{-q}{q} K(\la-\mu)\, Z(\mu ) \cdot \dd \mu \, = \, 1 \;.
\label{definition dressed charge}
\enq
The dressed charge is related to the dressed phase by the below identities \cite{KorepinSlavnovNonlinearIdentityScattPhase}:
\beq
\phi_1(\la,q) \, - \, \phi_1(\la,-q) \, + \, 1 \; = \; Z(\la) \quad \e{and} \quad 1+\phi_1(q,q) - \phi_1(-q,q) \, = \, \f{1}{ Z(q) } \;.
\label{ecriture identites entre phase et charge habilles}
\enq
Similarly to the dressed energy in the momentum representation, it is convenient to introduce the momentum representation of the $r$-bound dressed phase:
\beq
\vp_{r}(s,k) \, = \, \phi_{r}\Big(\wh{p}_1^{\, -1}(s), \wh{p}_r^{-1}(k) \Big) \quad \e{for} \quad s \in \intff{ -p_F }{ 2\pi - p_F - 2 p_F\e{sgn}(\pi-2\zeta) }
\quad \e{and} \quad
k \in \msc{I}_r \,.
\label{ecriture phase habillee dans rep impuslion}
\enq
Here, one should understand that $\wh{p}_r=p_r$ if $r\geq 2$. Also, one sets $\mc{Z}=Z\circ \wh{p}_1^{\,-1}$.
Then, the exponents $\De_{\pm}( \bs{\mf{K}} )$ governing the dynamic part of the DRF are expressed as $ \De_{\pm}( \bs{\mf{K}} )=\vth_{\ups}^2( \bs{\mf{K}} ) $
where
\bem
\vth_{\ups}( \bs{\mf{K}} ) \, = \, \, - \, \ups \ell_{\ups} \, + \, \tfrac{ 1 }{ 2 } \op{s}_{\ga} \mc{Z}( p_F ) \, + \, \sul{ a=1 }{ n_h } \vp_1( \ups p_F , t_{a} )
\; - \; \sul{ r \in \mf{N} }{ } \sul{ a=1 }{ n_r } \vp_{r}(\ups p_F ,k_a^{(r)} ) \\
\; - \hspace{-1mm} \sul{ \ups^{\prime} \in \{ \pm \} }{}\ell_{ \ups^{\prime} } \vp_1(\ups p_F , \ups^{\prime} p_F \, )
+ \, \e{sgn}(\pi-2\zeta) \cdot n_1(\bs{\mf{K}}) \mc{Z}(p_F) \;.
\label{definition shifted sfift function}
\end{multline}
Here $ n_1(\bs{\mf{K}}) \, = \, \# \Big\{ k_a^{(1)}\; : \; \wh{p}_1^{\, -1}( k_a^{(1)} ) \in \intoo{-\infty}{-q} \Big\}$.
\subsection{The velocity of individual excitations}
\label{Appendix Section phase oscillante dpdte de la vitesse}
The velocity $\mf{v}_{r}$ of an $r$-string excitation if $r \in \mf{N}_{\e{st}}$ and of a particle/hole excitation if $r=1$ is defined by
\beq
\mf{v}_r(k) \, = \; \mf{e}_{r}^{\prime}(k) \;. \qquad \e{In} \; \e{particular} \qquad \op{v}_{F} = \mf{e}^{\prime}_1(p_F)
\enq
is the Fermi velocity, namely the velocity of a particle or of a hole excitation located directly on the right edge of the Fermi zone $\intff{-p_F}{p_F}$ in the momentum representation.
$\mf{v}_1$ is defined, originally, on
\beq
\intff{-p_F}{2\pi-p_F-2p_F \e{sgn}(\pi-2\zeta) }
\enq
and it is easy to see that it extends to a $2\pi- 2p_F \e{sgn}(\pi-2\zeta)$ periodic function
on $\R$.
Furthemore, $\mf{v}_1$ enjoys the symmetry
\beq
\mf{v}_1\big( k \big) \; = \; -\mf{v}_1\big(2\pi-2p_F \e{sgn}(\pi-2\zeta) -k \big) \;.
\enq
These properties follow easily from its definition.
Also $\mf{v}_1$ is a continuous function on $\R$ that is piecewise smooth. The points where smoothness may fail correspond to the two momenta $\wh{p}_{1}^{\,-1}\big( \pm \infty \big) \, = \, \wh{p}_{1}^{\,-1}\big( \pm \infty + \i\tf{\pi}{2} \big)$.
One can easily prove for $p_F$ small enough, or for $\zeta$ belonging to a sufficiently small open neighbourhood of $\tf{\pi}{2}$, the below proposition
characterising some of the properties of $\mf{v}_1$.
\begin{prop}
\label{Proposition proprietes fondamentales de la vitesse des particules trous}
There exists $p_F^{(0)}$ and $\de \zeta^{(0)}$ such that, if either of the two bounds holds
\beq
0\leq p_F<p_F^{(0)} \qquad or \qquad \big| \zeta - \tf{\pi}{2} \big| < \de \zeta^{(0)} \; ,
\enq
then
\begin{itemize}
\item $|\mf{v}_1|<\op{v}_F$ on $\intoo{-p_F}{p_F}$;
\item there exists $ P_m \in \intoo{ p_F }{ \pi - 2 p_F \e{sgn}(\pi-2\zeta) }$ such that $\mf{v}_1$ is strictly increasing on
\beq
\intoo{ -p_F }{ P_m } \cup \intoo{ P_M }{ 2\pi-p_F-2p_F \e{sgn}(\pi-2\zeta) }
\enq
with $P_M= 2\pi - 2 p_F \e{sgn}(\pi-2\zeta) - P_m $, and strictly decreasing on $\intoo{P_m } { P_M }$;
\item there exists an interval
\beq
\intoo{K_m}{K_M} \subset \intoo{p_F}{2\pi-p_F-2p_F \e{sgn}(\pi-2\zeta) } \qquad with \qquad K_M \, = \, 2\pi \, - \, 2p_F \e{sgn}(\pi-2\zeta) \,- \, K_m
\label{definition intervalle de egalite vitesses particule trou}
\enq
such that
\beq
\big| \mf{v}_1(k) \big| < \op{v}_F \;\; \e{for} \;\; k \in \intoo{K_m}{K_M}
\enq
and
\beq
\big| \mf{v}_1(k) \big| > \op{v}_F \;\; \e{for} \;\; k \in \intoo{p_F}{2\pi-p_F-2p_F \e{sgn}(\pi-2\zeta)} \setminus \intff{K_m}{K_M} \, ;
\enq
\item there exists a strictly decreasing homeomorphism
\beq
\mf{t}\; : \; \intff{K_m}{K_M} \mapsto \intff{-p_F}{p_F} \quad such \; that \quad \mf{t}(k) \;is \, the\, unique\, solution \, to \quad
\mf{v}_{1}(k)=\mf{v}_1\big( \mf{t}(k) \big)
\label{ecriture propriete fonction t}
\enq
with $k \in \intff{K_m}{K_M}$ and $\mf{t}(k) \in \intff{-p_F}{p_F}$. The map $\mf{t}$ is smooth on $\intoo{K_m}{K_M}$;
\item there exists a strictly decreasing homeomorphisms $\mf{p}_L$, $\mf{p}_R$
\beq
\mf{p}_{L}\; : \; \intff{p_F}{P_m} \mapsto \intff{P_m}{K_m} \;\; , \qquad \mf{p}_{R}\; : \; \intff{P_M}{2\pi-p_F-2p_F\e{sgn}(\pi-2\zeta) } \mapsto \intff{K_M}{P_M},
\enq
such that $\mf{p}_{L}(k)$, resp. $\mf{p}_{R}(k)$, is the unique solution to $\mf{v}_{1}(k)=\mf{v}_1\big( \mf{p}_{L}(k) \big)$,
resp. $\mf{v}_{1}(k)=\mf{v}_1\big( \mf{p}_{R}(k) \big)$, on their respective range. The maps $\mf{p}_{R,L}$ are smooth on the interior of their domains.
\item $k\mapsto \mf{v}_{r}$ is a diffeomorphism from $\msc{J}_r=\intoo{-p_-^{(r)} }{ p_+^{(r)} }$ onto $\intoo{-\op{v}^{(r)} }{ \op{v}^{(r)} }$
with $ \op{v}^{(r)} > \op{v}_F$. Furthermore, there exists $K_m^{(r)}, K_M^{(r)} \in \msc{J}_r$ and a diffeomorphism
\beq
\mf{h}^{(r)} \; : \; \intff{-p_F}{p_F} \tend \intff{ K_m^{(r)} }{ K_M^{(r)} } \qquad such \, that \qquad
\mf{v}_1(t) \, = \, \mf{v}_r \Big(\mf{h}^{(r)}( t ) \Big) \;.
\enq
\end{itemize}
\end{prop}
In fact, one can check by means of numerical analysis (\textit{c.f.} Fig.\ref{Figure Vitesse particule trous}) that the properties listed
in Proposition \ref{Proposition proprietes fondamentales de la vitesse des particules trous} above hold true for any $p_F\in \intff{0}{\tf{\pi}{2}}$ and
$\zeta \in \intoo{0}{\pi}$. Thus the conjecture:
\begin{conj}
\label{Conjecture diffeo liee a la vitesse}
The conclusions of Proposition \ref{Proposition proprietes fondamentales de la vitesse des particules trous} hold true irrespectively of the values of $\zeta \in \intoo{0}{\pi}$ or
$p_F\in \intff{0}{\tf{\pi}{2}}$.
\end{conj}
\begin{figure}[t]
$\ba{cc}
\includegraphics[width=.4\textwidth]{Velocity_Delta=0,57_D=0,21.eps}
&
\includegraphics[width=.4\textwidth]{Velocity_Delta=0,604_D=0,30.eps}
\ea$
\caption{\label{Figure Vitesse particule trous} Velocity $\mf{v}_1$ plotted for $\Delta=0.57$ and magnetic field $h$ such that the per-site magnetisation $\mf{m}=1-2D$
is parameterised by $D=0.21$ (\textit{lhs}) and for $\Delta=-0.60$ and $D=0.30$ for the \textit{rhs}.}
\end{figure}
\section{Asymptotics of a one-dimensional $\be$-like integral}
\label{Appendix DA integrales unidimensionnelles}
In this appendix, I establish the main theorem in the one-dimensional case, \textit{viz}. Theorem \ref{Theorem Principal cas 1D}.
For convenience, I recall the statement of the theorem below and then expose the proof. The latter is based on two auxiliary lemmata
which are discussed in a separate section.
\subsection{The structural theorem in the one-dimensional case}
\begin{theorem}
\label{Proposition integrale principale spectral function}
Let $a<b$ be two reals. Let $\mf{z}_{\pm}(\la)$ be two real-holomorphic functions in a neighbourhood of the interval $\msc{J}=\intff{a}{b}$, such that
\begin{itemize}
\item all the zeroes of $\mf{z}_{\pm}$ on $\msc{J}$ are simple;
\item[$\bullet$] $\mf{z}_{+}$ and $\mf{z}_{-}$ admit a unique common zero $\la_0 \in \e{Int}(\msc{J})$ that, furthermore, is such that $\mf{z}_{+}^{\prime}(\la_0) \not= \mf{z}_{-}^{\prime}(\la_0)$.
\end{itemize}
Let $\De_{\ups}$ be real analytic on $\e{Int}(\msc{J})$ and such that $\De_{\ups} \geq 0$. Let $\msc{G}$ be in the smooth class of $\msc{J}$
associated with the functions $\De_{\pm}$ and with a constant $\tau$, c.f. Definition \ref{definition smooth class on K}.
Then, for $\mf{x}\not=0$ and small enough,
\beq
\la \mapsto \msc{G}\Big(\la, \wh{\mf{z}}_{+}(\la), \wh{\mf{z}}_{-}(\la)\Big) \cdot \pl{\ups= \pm }{}
\Big\{ \Xi\big( \; \wh{\mf{z}}_{\ups}(\la) \big) \cdot \big[ \, \wh{\mf{z}}_{\ups}(\la) \big]^{ \De_{\ups}(\la)-1 } \Big\} \in L^{1}\big( \msc{J} \big)
\enq
where $\wh{\mf{z}}_{\pm}(\la)=\mf{z}_{\pm}(\la)+ \mf{x}$. Let $\mc{I}(\mf{x})$ denote the integral
\beq
\mc{I}(\mf{x})\, = \, \Int{ \msc{J} }{} \msc{G}\Big(\la, \wh{\mf{z}}_{+}(\la), \wh{\mf{z}}_{-}(\la)\Big) \cdot \pl{\ups= \pm }{}
\Big\{ \Xi\big( \; \wh{\mf{z}}_{\ups}(\la) \big) \cdot \big[ \, \wh{\mf{z}}_{\ups}(\la) \big]^{ \De_{\ups}(\la)-1 } \Big\} \cdot \dd \la \;.
\label{definition integrale 1D type Beta genralise}
\enq
\vspace{2mm}
Assume that $ \de_{\pm} \, = \, \De_{\pm}(\la_0) >0$.
\vspace{2mm}
\noindent {\bf a) } If $\mf{z}_{+}^{\prime}(\la_0)\cdot \mf{z}_{-}^{\prime}(\la_0) <0$, then $\mc{I}(\mf{x})$ admits the $\mf{x} \tend 0$ asymptotic expansion
\beq
\mc{I}(\mf{x})\, = \, \Xi\Big( \mf{z}_{+}^{\prime}(\la_0) \cdot \mf{X} \Big) \cdot \Bigg\{
\f{ \msc{G}^{(1)}(\la_0) \cdot \de_+ \de_- \cdot | \mf{X} |^{\de_+ + \de_- - 1} }
{ |\, \mf{z}_{+}^{\prime}(\la_0) |^{\de_-} \cdot | \, \mf{z}_{-}^{\prime}(\la_0) |^{\de_+} }
\cdot \f{ \Ga\big( \de_+ \big) \cdot \Ga\big(\de_- \big) }{ \Ga\big( \de_++\de_-\big) }
\, + \, \e{O}\Big( |\mf{x}|^{ \de_+ + \de_- - \tau} \Big) \Bigg\}
\, + \, f_{<}(\mf{x})
\label{ecriture Da integrale I de omege cas zeros entrelaces}
\enq
where
\beq
\mf{X} \, = \, \mf{x} \cdot \big[ \mf{z}_{+}^{\prime}(\la_0)-\mf{z}_{-}^{\prime}(\la_0) \big] \;,
\label{defintion tau et de pm petits}
\enq
$\msc{G}^{(1)}$ is as appearing in \eqref{ecriture decomposition smooth class K}
and $f_{<}$ is a smooth function of $ \mf{x}$. Furthermore, if $\mf{z}_{\pm}$ have no zeroes on $\msc{J}$ other than $\la_0$, then $f_{<}=0$.
\vspace{2mm}
\noindent {\bf b) } If $\mf{z}_{+}^{\prime}(\la_0)\cdot \mf{z}_{-}^{\prime}(\la_0) >0$, then $\mc{I}(\mf{x})$ admits the $\mf{x} \tend 0$ asymptotic expansion
\bem
\mc{I}(\mf{x})\, = \,
\f{ \msc{G}^{(1)}(\la_0) \cdot \de_+ \, \de_- \cdot | \mf{X} |^{\de_+ + \de_- - 1} }
{ |\, \mf{z}_{+}^{\prime}(\la_0) |^{\de_-} \cdot | \, \mf{z}_{-}^{\prime}(\la_0) |^{\de_+} }
\cdot \Ga\big( \de_+ \big) \cdot \Ga\big(\de_- \big) \cdot \Ga\big( 1- \de_+ - \de_- \big) \\
\times \bigg\{ \Xi(\mf{x}) \tfrac{1}{\pi} \sin\big[ \pi \de_{\mf{p}} \big] \, + \, \Xi(-\mf{x}) \tfrac{1}{\pi} \sin\big[ \pi \de_{-\mf{p}} \big] \bigg\}
\, + \, \e{O}\Big( |\mf{x}|^{ \de_+ + \de_- - \tau } \Big)
\; + \; f_{>}(\mf{x})
\end{multline}
where $ \mf{X} $ and $\de_{\pm}$ are as above,
\beq
\mf{p}\, = \, - \e{sgn} \big[ \mf{z}_{+}^{\prime}(\la_0) \big] \cdot \e{sgn} \big[ \mf{z}_{+}^{\prime}(\la_0)-\mf{z}_{-}^{\prime}(\la_0) \big]
\label{definition parametre eta integrale type fct beta}
\enq
and $f_{>}$ is a smooth function of $ \mf{x}$.
\end{theorem}
\Proof
The hypothesis on $\mf{z}_{\pm}(\la)$ ensure that these functions have a holomorphic inverse in a neighbourhood of any of their zeroes.
As a consequence, any zero of $\wh{\mf{z}}_{\pm}$ is holomorphic in $\mf{x}$ small enough. Thus, the integral can be decomposed as $\mc{I}(\mf{x})=\sul{k=1}{n}\mc{I}_{k}(\mf{x})$,
where
\beq
\mc{I}_k(\mf{x})\, = \, \Int{ a_k(\mf{x}) }{ b_k(\mf{x}) } \msc{G}\Big(\la, \wh{\mf{z}}_{+}(\la), \wh{\mf{z}}_{-}(\la)\Big) \cdot \pl{\ups= \pm }{} \Big\{ \big[ \, \wh{\mf{z}}_{\ups}(\la) \big]^{ \De_{\ups}(\la)-1 } \Big\} \cdot \dd \la \;.
\label{ecriture integrale Ik}
\enq
The endpoints $ a_k(\mf{x})$, $k \geq 2$, and $b_k(\mf{x})$, $k \leq n-1$, all correspond to a zero of $\wh{\mf{z}}_{+}$ or $\wh{\mf{z}}_{-}$.
Furthermore, if $a_1(\mf{x})>a$ and/or $b_n(\mf{x})<b$, then these also correspond to a zero of $\wh{\mf{z}}_{+}$ or $\wh{\mf{z}}_{-}$.
However, it may be that $a_1(\mf{x})=a$ and/or $b_n(\mf{x})=b$, where I remind that $\msc{J}=\intff{a}{b}$. Then $a_1(\mf{x})=a$ and/or $b_n(\mf{x})=b$ may or may not correspond to
zeroes of $\wh{\mf{z}}_{\ups}$.
The fact that $\msc{G}$ belongs to the smooth class of $\msc{J}$ with functions $\De_{\pm}$ and a constant $\tau$ ensures that the integrals $\mc{I}_k(\mf{x})$ are well-defined.
Indeed, problems with the $L^{1}$-nature of its integrand could, in principle, arise if some zero of $\wh{\mf{z}}_{\ups}$ coincides with a zero of $\De_{\ups}$.
However, observe that due to the smooth class property and the hypotheses stated above (also \textit{c.f.} equation \eqref{decomposition zero mu pm}), the zeroes of $\wh{\mf{z}}_{+}$
and $\wh{\mf{z}}_{-}$ are all distinct and simple, at least provided that $\mf{x}$ is small enough. Furthermore, one has the decomposition
\bem
\msc{G}\Big(\la, \wh{\mf{z}}_{+}(\la), \wh{\mf{z}}_{-}(\la)\Big) \cdot \pl{\ups= \pm }{} \Big\{ \big[ \, \wh{\mf{z}}_{\ups}(\la) \big]^{ \De_{\ups}(\la)-1 } \Big\} \; = \;
\msc{G}^{(1)}\big(\la\big) \cdot \pl{\ups= \pm }{} \Big\{ \De_{\ups}(\la) \big[ \, \wh{\mf{z}}_{\ups}(\la) \big]^{ \De_{\ups}(\la)-1 } \Big\} \\
\; + \; \msc{G}^{(2)}\Big(\la, \wh{\mf{z}}_{+}(\la) \Big) \cdot \De_{-}(\la) \, \big[ \, \wh{\mf{z}}_{-}(\la) \big]^{ \De_{-}(\la)-1 } \cdot \big[ \, \wh{\mf{z}}_{+}(\la) \big]^{ \De_{+}(\la)-\tau }
\; + \; \msc{G}^{(3)}\Big(\la, \wh{\mf{z}}_{-}(\la) \Big) \cdot \De_{+}(\la) \, \big[ \, \wh{\mf{z}}_{+}(\la) \big]^{ \De_{+}(\la)-1 } \cdot \big[ \, \wh{\mf{z}}_{-}(\la) \big]^{ \De_{-}(\la)-\tau } \\
\; + \; \msc{G}^{(4)}\Big(\la, \wh{\mf{z}}_{+}(\la), \wh{\mf{z}}_{-}(\la) \Big) \cdot \pl{\ups= \pm }{} \Big\{ \big[ \, \wh{\mf{z}}_{\ups}(\la) \big]^{ \De_{\ups}(\la)-\tau } \Big\} \;.
\label{ecriture decomposition integrande Ik}
\end{multline}
By the above, $\wh{\mf{z}}_{\ups}$ vanishes linearly at its zeroes. $\De_{\ups}$ being holomorphic, it vanishes at least linearly at its zeroes.
These two properties ensure the $L^{1}\big( \intff{ a_k(\mf{x}) }{ b_k(\mf{x}) } \big)$ nature of the integrand in \eqref{ecriture integrale Ik}.
\vspace{2mm}
In the following, I denote by $\mu_{\pm}(\mf{x})$ the zeroes of $\wh{\mf{z}}_{\pm}(\la)$ such that $\mu_{\pm}(0)=\la_0$. If neither $a_{k}(\mf{x})$ nor $b_{k}(\mf{x})$ coincides with $\mu_{\pm}(\mf{x})$,
then the endpoint $a_{k}(0)$, resp. $b_{k}(0)$, is at most a simple
zero of one of the functions $\mf{z}_{\pm}$, but not of both. The latter is a direct consequence of the assumed properties of the functions $\mf{z}_{\pm}$.
Hence, $\mc{I}_k(\mf{x})$ corresponds to the class of integrals
studied in Lemma \ref{Lemme integrale type beta reguliere} and, as such, is smooth in $\mf{x}$ small enough. Its contribution is thus included
in one of the functions $f_{<}(\mf{x})$ or $f_{>}(\mf{x})$, depending on the case of interest.
\vspace{2mm}
It thus remains to focus on the integral containing, as one of its endpoints, the zero $\mu_{\pm}(\mf{x})$. For convenience, denote
this integral by $\mc{J}_{\la_0}(\mf{x})$.
\vspace{2mm}
As already stated, the zeroes $\mu_{\pm}(\mf{x})$ are analytic functions of $\mf{x}$, at least for $\mf{x}$ small enough.
Furthermore, it is readily checked that
\beq
\mu_{\pm}(\mf{x}) \; = \; \la_0 \, - \, \f{ \mf{x} }{ \mf{z}_{\pm}^{\prime}(\la_0) } \; + \; \e{O}\big( \mf{x}^2\big) \;.
\label{decomposition zero mu pm}
\enq
This ensures that $\mu_{+}(\mf{x})\not= \mu_{-}(\mf{x})$ for $\mf{x}$ small enough. Being holomorphic, $\wh{\mf{z}}_{\ups}(\la)$ admits the factorisation:
\beq
\wh{\mf{z}}_{\ups}(\la) \, = \, \big( \la-\mu_{\ups}(\mf{x}) \big) \cdot h_{\ups}(\la , \mf{x} ) \qquad \e{with} \qquad h_{\ups} \Big( \mu_{\ups}(\mf{x}) , \mf{x} \Big) \; = \; \mf{z}_{\ups}^{\prime} \Big( \mu_{\ups}(\mf{x}) \Big) \;.
\label{ecriture factorisation fct z pm hat}
\enq
By the Weierstrass preparation theorem, \textit{c.f.} Theorem \ref{Theorem Weierstrass preparation theorem}, $h_{\ups}$ is holomorphic in $\la$ and $\mf{x}$, at least for $|\mf{x}|$ small enough.
In order to proceed further, one has to distinguish between the cases ${\bf a)}$ and ${\bf b)}$ outlined in the statement of the theorem.
\vspace{2mm}
\noindent {\bf $\bullet$ Case a): $\mf{z}_{+}^{\prime}(\la_0)\cdot \mf{z}_{-}^{\prime}(\la_0) <0$ }
\vspace{2mm}
Let $ \vsg= \e{sgn}\big[ \mf{z}_{+}^{\prime}(\la_0) \big] $. Then,
\beq
\wh{\mf{z}}_{+}(\la_0) >0 \; \; \e{on} \; \; \vsg \intoo{ \mu_{+}(\mf{x}) }{ \nu_{+}^{(\vsg)}(\mf{x}) } \quad \e{while}\quad
\wh{\mf{z}}_{-}(\la_0) >0 \; \; \e{on} \;\; \vsg \intoo{ \nu_{-}^{(\vsg)}(\mf{x}) }{ \mu_{-}(\mf{x}) }
\label{ecriture tableau positivite fcts z pm}
\enq
where $\nu_{\pm}^{(\vsg)}(\mf{x})$ is the closest zero of $\wh{\mf{z}}_{\pm}$ to $\mu_{\pm}(\mf{x})$ such that
the function satisfies to the above properties\symbolfootnote[4]{In case there are no more zeroes, one should simply take $\nu_{+}^{(+)}(\mf{x})=b$ and $\nu_{+}^{(-)}(\mf{x})=a$ or
$\nu_{-}^{(+)}(\mf{x})=a$ or $\nu_{-}^{(-)}(\mf{x})=b$ depending on the situation, where I remind that $\msc{J} \, = \, \intff{a}{b}$.}.
The $\vsg$ pre-factor in front of the intervals means that the interval is always oriented from the smallest to the largest element.
One can convince oneself that
\beq
\nu_{\pm}^{(\vsg)}(\mf{x}) \, = \, \mu_{\pm}(\mf{x}) \pm \vsg \de \nu_{\pm}^{(\vsg)}(\mf{x)} \quad \e{where} \quad
\de \nu_{\pm}^{(\vsg)}(\mf{x)} \, > \, C
\enq
for some $\mf{x}$-independent constant $C>0$.
The above means that the neighbourhood of $\la_0$ will produce non-vanishing contributions to $\mc{J}_{\la_0}(\mf{x})$ only if $\mu_{\vsg}(\mf{x}) < \mu_{-\vsg}(\mf{x})$. Provided this inequality holds,
the integration in $\mc{J}_{\la_0}(\mf{x})$ runs through the interval $\intff{ \mu_{\vsg}(\mf{x}) }{ \mu_{-\vsg}(\mf{x}) }$.
Since one has
\beq
\mu_{-\vsg}(\mf{x}) - \mu_{\vsg}(\mf{x}) \, = \, \mf{x} \cdot \f{ \mf{z}_{\vsg}^{\prime}(\la_0) - \mf{z}_{-\vsg}^{\prime}(\la_0) }{ - \mf{z}_{+}^{\prime}(\la_0)\cdot \mf{z}_{-}^{\prime}(\la_0) }
\Big( 1 \, + \, \e{O}\big( \mf{x} \big) \Big) \; ,
\enq
the condition $\mu_{\vsg}(\mf{x}) < \mu_{-\vsg}(\mf{x})$ can be recast, for $|\mf{x}|$ small enough, as $\mf{z}_{+}^{\prime}(\la_0) \cdot \mf{X} >0 $ where $\mf{X}$ is as defined in \eqref{defintion tau et de pm petits}.
Thence, upon inserting the factorisation \eqref{ecriture factorisation fct z pm hat}
into $\mc{J}_{\la_0}(\mf{x})$, the integral can be recast, for $|\mf{x}|$ small enough, as
\beq
\mc{J}_{\la_0}(\mf{x}) \, = \, \Xi\Big( \mf{z}_{+}^{\prime}(\la_0) \cdot \mf{X} \Big) \cdot \Int{ \mu_{\vsg}(\mf{x}) }{ \mu_{-\vsg}(\mf{x}) }
\msc{H}(\la) \cdot \pl{\ups = \pm }{} \Big\{ \ups \vsg \big[\la -\mu_{\ups}(\mf{x}) \big] \Big\}^{\De_{\ups}(\la)-1} \cdot \dd \la
\enq
with
\beq
\msc{H}(\la) \; = \; \msc{G}\Big(\la, \wh{\mf{z}}_{+}(\la), \wh{\mf{z}}_{-}(\la)\Big)
\cdot \pl{\ups=\pm}{} \Big\{ \ups \vsg h_{\ups}( \la , \mf{x} ) \Big\}^{\De_{\ups}(\la)-1} \;.
\enq
The representation \eqref{ecriture decomposition integrande Ik}, the properties of the functions $\msc{G}^{(k)}$ and the fact that the $\msc{G}$ independent-part of the integrand has constant sign, all lead together to the decomposition
\beq
\mc{J}_{\la_0}(\mf{x}) \, = \, \check{\mc{J}}\big[ H,\De_{+},\De_{-}\big](\mf{x}) \; + \; \sul{\ups=\pm}{} \e{O}\Big( \check{\mc{J}}\big[1,\De_{+} + (1-\tau)\de_{\ups,+} \, ,\De_{-} + (1-\tau) \de_{\ups,-}\big](\mf{x}) \Big)
\enq
where $\de_{a,b}$ stands for the Kronecker symbol. Here
\beq
\check{\mc{J}}\big[H,\De_{+},\De_{-}\big](\mf{x}) \, = \, \Xi\Big( \mf{z}_{+}^{\prime}(\la_0) \cdot \mf{X} \Big) \cdot \Int{ \mu_{\vsg}(\mf{x}) }{ \mu_{-\vsg}(\mf{x}) }
H(\la) \cdot \pl{\ups = \pm }{} \Big\{ \ups \vsg \big[\la -\mu_{\ups}(\mf{x}) \big] \Big\}^{\De_{\ups}(\la)-1} \cdot \dd \la
\enq
with
\beq
H(\la) \; = \; \De_{+}(\la) \, \De_{-}(\la) \cdot \msc{G}^{(1)}(\la)
\cdot \pl{\ups=\pm}{} \Big\{ \ups \vsg h_{\ups}( \la , \mf{x} ) \Big\}^{\De_{\ups}(\la)-1} \;.
\enq
Then, the change of variables
\beq
t \; = \; \f{ \la - \mu_{\vsg}(\mf{x}) }{ \mu_{-\vsg}(\mf{x}) - \mu_{\vsg}(\mf{x}) }
\enq
recasts the integral as
\beq
\check{\mc{J}}\big[H,\De_{+},\De_{-}\big](\mf{x}) \, = \, \Xi\Big( \mf{z}_{+}^{\prime}(\la_0) \cdot \mf{X} \Big) \cdot
\Int{ 0 }{ 1 } t^{ \wt{\De}_{\vsg}(t)-1} \cdot \big( 1 - t \big)^{ \wt{\De}_{-\vsg}(t)-1} \cdot \wt{H}(t) \cdot \dd t
\enq
where
\beq
\wt{H}(t) \; = \; H\Big( \mu_{\vsg}(\mf{x}) +t \cdot \big[\mu_{-\vsg}(\mf{x}) - \mu_{\vsg}(\mf{x})\big] \Big) \cdot
\Big( \mu_{-\vsg}(\mf{x}) - \mu_{\vsg}(\mf{x}) \Big)^{ \wt{\De}_+(t)+\wt{\De}_-(t) - 1}
\enq
and
\beq
\wt{\De}_{\ups}(t) \; = \; \De_{\ups}\Big( \mu_{\vsg}(\mf{x}) +t \cdot \big[\mu_{-\vsg}(\mf{x}) - \mu_{\vsg}(\mf{x})\big] \Big) \,.
\enq
Being smooth, all functions have an expansions in $\mf{x}$ that is uniform in $t \in \intff{0}{1}$. This fact ensures that the leading asymptotic expansion of the
integral is obtained by setting the argument $t$ of all functions to $0$, leading to
\beq
\check{\mc{J}}\big[H,\De_{+},\De_{-}\big](\mf{x}) \, = \, \Xi\Big( \mf{z}_{+}^{\prime}(\la_0) \cdot \mf{X} \Big) \cdot \wt{H}(0) \cdot
\f{ \Ga\Big(\, \wt{\De}_{\vsg}(0) \Big) \cdot \Ga\Big( \, \wt{\De}_{-\vsg}(0) \Big) }{ \Ga\Big( \, \wt{\De}_{\vsg}(0) + \wt{\De}_{-\vsg}(0) \Big) }
\cdot \Big( 1 \, + \, \e{O}\big( \mf{x} \ln \mf{x} \big) \Big) \;.
\enq
Note that the $\e{O}\big( \mf{x} \ln \mf{x} \big) $ remainder issues from the expansion of the exponents in $\wt{H}(t)$.
One can simplify the formula further. One has $ \wt{\De}_{\ups}(0)= \de_{\ups} + \e{O}\big( \mf{x} \big) $ with $\de_{\pm}$
as in \eqref{defintion tau et de pm petits} as well as
\beq
\wt{H}(0) \; = \;\f{ \de_{+} \de_{-} \cdot \msc{G}^{(1)}(\la_0) \cdot | \mf{X} |^{\de_+ + \de_- - 1} }
{ |\, \mf{z}_{+}^{\prime}(\la_0) |^{\de_-} \cdot | \, \mf{z}_{-}^{\prime}(\la_0) |^{\de_+} } \, + \, \e{O}\Big( |\mf{x}|^{\de_+ + \de_- } \ln |\mf{x}| \Big)
\enq
what allows one to conclude regarding to \eqref{ecriture Da integrale I de omege cas zeros entrelaces}.
\vspace{2mm}
\noindent {\bf $\bullet$ Case b): $\mf{z}_{+}^{\prime}(\la_0)\cdot \mf{z}_{-}^{\prime}(\la_0) > 0$ }
\vspace{2mm}
Still agreeing upon $ \vsg= \e{sgn}\big[ \mf{z}_{+}^{\prime}(\la_0) \big] $, and keeping the same definition of $\nu_{\pm}^{(\vsg)}(\mf{x})$, one gets that
\beq
\wh{\mf{z}}_{\pm }(\la_0) >0 \; \; \e{on} \; \; \vsg \intoo{ \mu_{\pm}(\mf{x}) }{ \nu_{\pm}^{(\vsg)}(\mf{x}) } \;.
\enq
Here, as earlier, the $\vsg$ prefactor indicates that the interval is oriented from its smallest to its largest element.
It is easy to convince oneself that, in the present case of interest,
\beq
\nu_{\pm}^{(\vsg)}(\mf{x}) \, = \, \mu_{\pm}(\mf{x}) + \vsg \de \nu_{\pm}^{(\vsg)}(\mf{x)} \quad \e{where} \quad
\de \nu_{\pm}^{(\vsg)}(\mf{x)} >C
\enq
for some $\mf{x}$-independent constant $C>0$.
Thus, after imposing the positivity constraints and using that $|\mf{x}|$ is small, the integral $\mc{J}_{\la_0}(\mf{x})$ runs through $\vsg \intff{ b_{\vsg} }{ c_{\vsg} }$ where
\beq
b_{\vsg} \, = \, \vsg \, \e{max}\Big( \vsg \mu_{+}(\mf{x}) , \vsg \mu_{-}(\mf{x}) \Big) \qquad \e{and} \qquad
c_{\vsg} \, = \, \vsg \, \e{min}\Big( \vsg \nu_{+}^{(\vsg)}(\mf{x}) ,\vsg \nu_{-}^{(\vsg)}(\mf{x}) \Big)\;.
\enq
The integral of interest can then be decomposed as $\mc{J}_{\la_0}(\mf{x} ) \, = \,\mc{J}_{\la_0}^{(1)}(\mf{x} ) \, + \, \mc{J}_{\la_0}^{(2)}(\mf{x} )$
\beq
\mc{J}_{\la_0}^{(1)}(\mf{x} )\; = \; \vsg \hspace{-2mm} \Int{ b_{\vsg} } { b_{\vsg} + \vsg \de } \hspace{-2mm}
\msc{G}\Big(\la, \wh{\mf{z}}_{+}(\la), \wh{\mf{z}}_{-}(\la)\Big) \pl{\ups= \pm }{} \big[ \, \wh{\mf{z}}_{\ups}(\la) \big]^{ \De_{\ups}(\la)-1 }\hspace{-1mm} \cdot \dd \la
\enq
and
\beq
\mc{J}_{\la_0}^{(2)}(\mf{x} )\; = \; \vsg \hspace{-2mm} \Int{ b_{\vsg} + \vsg \de }{ c_{\vsg} } \hspace{-2mm} \msc{G}\Big(\la, \wh{\mf{z}}_{+}(\la), \wh{\mf{z}}_{-}(\la)\Big)
\pl{\ups= \pm }{} \big[ \, \wh{\mf{z}}_{\ups}(\la) \big]^{ \De_{\ups}(\la)-1 } \hspace{-1mm} \cdot \dd \la \; ,
\enq
where $\de>0$ is taken small enough.
$\mc{J}_{\la_0}^{(2)}(\mf{x} )$ is a smooth function of $\mf{x}$. This can be seen as follows. If $ c_{\vsg} \in \Dp{}\msc{J}$ and if the endpoints of $\msc{J}$ are not
zeroes of $\mf{z}_{\pm}$, then the integrand in $\mc{J}_{\la_0}^{(2)}(\mf{x})$ can be expanded into
powers of $\mf{x}$ owing to
\beq
\e{inf}\Big\{ \mf{z}_{\pm}(s) \; : \; s \in \intff{ b_{\vsg} + \vsg \de }{ c_{\vsg} } \Big\}>C^{\prime} \; ,
\enq
for some $C^{\prime}>0$.
This entails the claim. Otherwise, $ c_{\vsg} $ coincides with a zero $\wh{\mf{z}}_{\pm}$. One treats the part of the integral corresponding to an integration over a domain
uniformly away from $c_{\vsg}$, exactly as in the first case. Then, the neighbourhood of $c_{\vsg}$ can be treated as in Lemma \ref{Lemme integrale type beta reguliere}, and the claim follows.
\vspace{2mm}
It thus remains to focus on $\mc{J}_{\la_0}^{(1)}(\mf{x} )$. Recalling the definition \eqref{definition parametre eta integrale type fct beta} of $\mf{p}$, one can readily check that, for $\mf{x}$
small enough,
\beq
b_{\vsg} \, = \, \mu_{-\e{sgn}(\mf{x}) \mf{p}}(\mf{x}) \qquad \e{and} \qquad
\vsg \, \e{min}\Big( \vsg \mu_{+}(\mf{x}) , \vsg \mu_{-}(\mf{x}) \Big) \, = \, \mu_{ \e{sgn}(\mf{x}) \mf{p}}(\mf{x}) \;.
\enq
After the change of variables $\la=b_{\vsg} + \vsg t$, by using the factorisation \eqref{ecriture factorisation fct z pm hat} and setting
\beq
a_{\vsg} \; = \; \vsg \Big( \mu_{-\e{sgn}(\mf{x}) \mf{p} }(\mf{x}) \, - \, \mu_{ \e{sgn}(\mf{x}) \mf{p} }(\mf{x}) \Big) \, \geq 0 \, ,
\enq
one gets that
\beq
\mc{J}_{\la_0}^{(1)}(\mf{x} ) \; = \; \Int{ 0 }{ \de } \msc{H}\Big( t, t u(t),(t+a_{\vsg}) v(t) \Big) \cdot t^{A(t)} \cdot (t+a_{\vsg})^{ B(t) } \cdot \dd t \;.
\enq
Above, we have set
\beq
A(t) \, = \, \De_{-\e{sgn}(\mf{x}) \mf{p} }(b_{\vsg} +\vsg t)-1 \; ,\quad B(t) \, = \, \De_{\e{sgn}(\mf{x}) \mf{p}}(b_{\vsg} +\vsg t)-1 \;
\enq
and
\beq
u(t) \, = \, \vsg h_{-\e{sgn}(\mf{x}) \mf{p}} (b_{\vsg} +\vsg t , \mf{x}) \; , \quad
v(t) \, = \, \vsg h_{ \e{sgn}(\mf{x}) \mf{p}} (b_{\vsg} +\vsg t , \mf{x}) \; .
\enq
Finally,
\beq
\msc{H}(t, x, y ) \, = \, \msc{G}\Big( b_{\vsg} +\vsg t, x , y\Big)
\cdot \pl{\ups=\pm}{} \Big\{ \vsg h_{\ups}(b_{\vsg} +\vsg t , \mf{x}) \Big\}^{ \De_{\ups }(b_{\vsg} +\vsg t)-1 } \;.
\enq
The properties of $\msc{G}$ entail that
\beq
\msc{H}(t, x, y ) \, = \, H(t) \; + \; \e{O}\Big( x^{1- \tau} \, + \, y^{1- \tau}\Big)
\enq
with a differentiable remainder in the sense of Definition \ref{Defintion reste differentiable} and where
\beq
H(t) \, = \, \Big(\De_{+} \De_{-} \msc{G}^{(1)} \Big)\big( b_{\vsg} +\vsg t \big) \cdot \pl{\ups=\pm}{} \Big\{ \vsg h_{\ups}(b_{\vsg} +\vsg t , \mf{x}) \Big\}^{ \De_{\ups }(b_{\vsg} +\vsg t)-1 } \;.
\enq
One is now in position to apply the result of Lemma \ref{Lemme integrale beta auxiliaire} given below. This yields that
\bem
\mc{J}^{(1)}_{\la_0}(\mf{x}) \; = \; - \tfrac{ H(0) }{ \pi } \big( a_{\vsg} \big)^{ 1+A(0)+B(0) } \sin[\pi B(0)]
\cdot \Ga\Big(1+A(0)\Big) \Ga\Big(1+B(0)\Big)\Ga\Big(-1-A(0)-B(0)\Big) \\
\, + \, \e{O}\Big( a_{\vsg}^{2+A(0)+B(0)} \cdot \ln a_{\vsg} \Big) \; + \; r(a_{\vsg}) \;,
\end{multline}
where $r$ is some smooth function. At this stage, it remains to use that
\beq
\De_{\ups}( b_{\vsg} ) = \de_{\ups}+\e{O}(\mf{x}) \qquad \e{and} \qquad
a_{\vsg} = |\mf{X}| \cdot \big[ \mf{z}^{\prime}_{+}(\la_0) \cdot \mf{z}^{\prime}_{-}(\la_0) \big]^{-1} \cdot \big( 1 + \e{O}(\mf{x}) \big)
\enq
with $\mf{X}$ as given in \eqref{defintion tau et de pm petits}, so as to conclude. \qed
\subsection{Auxiliary lemmata}
\begin{lemme}
\label{Lemme integrale beta auxiliaire}
Let $1> \de>0$ be fixed and $f(t),A(t), B(t)$ be smooth real valued functions on $\intff{0}{\de}$ admitting the expansion around zero
\beq
f(t) \, = \, f_0 \, + \, \e{O}(t) \; ,
\qquad A(t) \, = \, a_0 \, + \, \e{O}(t) \; ,
\qquad B(t) \, = \, b_0 \, + \, \e{O}(t) \;,
\enq
where $a_0>-1$ and $b_0>-1$ are such that $a_0+b_0 \not\in \mathbb{N}$. Further, let $\msc{F}$ be smooth on $\intff{0}{\de}\times \R^+\times \R^+$ and such that, for $x,y$ bounded
\beq
\msc{F}\big(t; x, y \big) \; = \; f(t) \, + \, \e{O}\big( x^{\a}+y^{\a} \big) \quad for\; some \quad 0<\a<1
\enq
with a differentiable remainder, \textit{c.f.} Definition \ref{defintion reste differentiable}. Let $u,v$ be smooth on $\intff{0}{\de}$ and such that $u(t),v(t)>0$.
Then, the integral
\beq
\mc{J}[\msc{F},A,B]( \mf{x} ) \;= \; \Int{0}{\de} \msc{F}\big(t; t \, u(t) , \, (t+\mf{x})\, v(t) \big) \cdot t^{ A(t) } \cdot (t+ \mf{x} )^{ B(t) } \cdot \dd t
\enq
has the $ \mf{x} \tend 0^+$ asymptotic expansion
\bem
\mc{J}[\msc{F}, A, B]( \mf{x} ) \;= \; -f_0 \tfrac{ \sin [ \pi b_0 ] }{ \pi } \Ga\Big( 1+a_0 \Big)\Ga\Big( 1+b_0 \Big)\Ga\Big( -1-a_0-b_0 \Big) \cdot \mf{x}^{ 1+a_0+b_0 } \\
\, + \, r( \mf{x}) \; + \; \e{O}\Big( \mf{x}^{1 + a_0 + b_0 + \a } \Big) \;.
\end{multline}
where the function $r$ is smooth in $\mf{x}$, and does depend on $\de, A, B$.
\end{lemme}
\Proof
Observe that $\mf{x} \mapsto \mc{J}[\msc{F}, A, B]( \mf{x} )$ is smooth in $\mf{x} \in \R^+$ and that the $\mf{x}$ derivatives are obtained by differentiating under the integral.
Let $n\in \mathbb{N}$ be such that
\beq
-2 < 1+a_0+b_0-n <-1 \;.
\label{ecriture choix entier n}
\enq
Observe that the hypotheses on the differentiability of the remainder ensures that
\beq
\Dp{\mf{x}}^{n} \Big\{ \msc{F}\big(t; t \, u(t) , \, (t+\mf{x})\, v(t) \big) \cdot (t+ \mf{x} )^{ B(t) } \Big\} \; = \; \wt{f}(t) (t+ \mf{x} )^{ \wt{B}(t) } \, + \, \e{O}\Big( (t+ \mf{x} )^{ \wt{B}(t) +\a } \Big)
\, + \, \e{O}\Big( t^{\a} (t+ \mf{x} )^{ \wt{B}(t) } \Big)
\enq
with
\beq
\wt{f}(t)\, = \, f(t) \cdot \f{ \Ga\big( B(t)+1 \big) }{ \Ga\big( 1 + B(t) - n \big) } \qquad \e{and} \qquad \wt{B}(t) \, = \, B(t) -n \;.
\enq
Furthermore, being smooth functions on $\intff{0}{\de}$, one has that
\beq
\wt{f}(t) \, = \, \sul{k=0}{p} \wt{f}_k t^k \, + \, \e{O}(t^{p+1})
\qquad A(t) \, = \, \sul{k=0}{p} a_k t^k \, + \, \e{O}(t^{p+1})
\qquad \wt{B}(t) \, = \, \sul{k=0}{p} \wt{b}_k t^k \, + \, \e{O}(t^{p+1}) \;.
\enq
From there and the fact that $u(t), v(t)>c$ for some $c>0$, one readily deduces that
\beq
\wt{f}(t) \cdot t^{A(t) } \cdot (t+ \mf{x})^{ \wt{B}(t) } \; = \; t^{ a_0 } \cdot (t+ \mf{x})^{ \wt{b}_0 } \cdot \bigg( \wt{f}_0 \, + \, \e{O}\Big( t \cdot \e{max}\big\{ | \ln t | , |\ln (t + \mf{x})|, 1 \big\} \Big) \bigg) \;.
\enq
Then, since $\ln t$, $\ln(t+ \mf{x})$ have constant sign on $\intff{0}{\de}$ provided that $|\mf{x}|$ is small enough, straightforward bounds lead to
\bem
\Dp{ \mf{x} }^{n} \Big\{\, \mc{J}[\msc{F}, A, B]( \mf{x} ) \, \Big\} \;= \; \wt{f}_0 \, \mc{T}(a_0,\wt{b}_0)
\; + \; \e{O}\bigg( \, \big| \Dp{a} \mc{T}(a, b) \big| \, + \, \big| \Dp{b}\mc{T}(a,b) \big| \, + \, \big| \mc{T}(a,b) \big| \, \bigg)_{ \big| \substack{ a=a_0+1 \\ b=\wt{b}_0 } } \\
\; + \; \e{O}\bigg( \big| \mc{T}(\a + a_0,\wt{b}_0) \big| \, + \, \big| \mc{T}( a_0,\wt{b}_0+\a) \big| \bigg)
\label{ecriture dvpmt derivee nieme integrale J 1D}
\end{multline}
where
\beq
\mc{T}(a,b) \; = \; \Int{0}{ \de } t^{ a} \cdot (t+ \mf{x})^{b} \cdot \dd t \; = \; \mf{x}^{ a + b + 1} \Int{0}{ \tf{\de}{ \mf{x}} } t^{ a } \cdot (t+1)^{ b } \cdot \dd t \;.
\enq
Note that, if need be, one may always slightly decrease the value of $\a$ so that $\a+a_0+b_0\not \in \mathbb{Z}$ while preserving the differentiability of the remainder.
The change of variables $v=\tf{t}{(t+1)}$ recasts the integral as
\beq
\mc{T}(a,b) \; = \; \mf{x}^{a + b + 1} \Int{0}{ \tfrac{\de}{\de+ \mf{x}} } v^{a} \cdot (1-v)^{-a-b-2} \cdot \dd v \;.
\enq
It remains to expand the model integral
\beq
\wt{\mc{T}}\big(x,y; z \big)\; = \; \Int{0}{z} t^{x-1} \cdot (1-t)^{y-1} \cdot \dd t \quad, \; \Re(x)>0\, ,
\enq
around $z=1$. Let $p\in \mathbb{N}$ be such that $\Re(y)+p>0$. Then, using the expansion for $|t-1|<1$
\beq
t^{x-1}\,=\,[1+(t-1)]^{x-1} = \sul{ n \geq 0 }{ } C_{n}(x)(1-t)^n \quad \e{with} \quad C_0(x)=1 \, ,
\label{definition des coefs Cn de x}
\enq
one has that
\bem
\wt{\mc{T}}\big(x,y; z \big)\; = \; \Int{0}{z} \Big( t^{x-1} - \sul{n=0}{p-1} C_n(x) (1-t)^n \Big) \cdot (1-t)^{y-1} \cdot \dd t
\, + \, \sul{n=0}{p-1} \f{1-(1-z)^{y+n}}{y+n} C_n(x) \\
= \; -\sul{n \geq 0 }{ } C_n(x) \f{(1-z)^{y+n}}{y+n} \, + \, \wt{\mc{T}}_0
\end{multline}
where
\beq
\wt{\mc{T}}_0 \; = \; \sul{n=0}{p-1} \f{ C_n(x) }{ y + n } \, + \, \Int{0}{1} \Big(t^{x-1} - \sul{n=0}{p-1} C_n(x) (1-t)^n \Big) (1-t)^{y-1} \cdot \dd t
\; = \; \f{ \Ga(x) \Ga(y) }{ \Ga(x+y) }
\enq
has been computed by meromorphic continuation in $y$. The above expansion ensures that there exists a function $h$ that is smooth in $\mf{x}$ belonging to a neighbourhood of $0$, and in $a>-1$ and $b \not \in \mathbb{Z}$, such that
\beq
\mc{T}(a,b) \; = \; - \mf{x}^{a + b + 1} \f{ \sin [\pi b] }{ \pi } \Ga(a+1)\Ga(b+1)\Ga(-a - b - 1) \, + \, \f{ (\de+ \mf{x})^{ a+ b + 1} }{ a + b +1 } \; + \; \mf{x}\cdot h( \mf{x} ) \;.
\label{ecriture DA T alpha beta}
\enq
Owing to the choice of the integer $n$ in \eqref{ecriture choix entier n}, all integrals $\mc{T}(a,b)$ appearing in \eqref{ecriture dvpmt derivee nieme integrale J 1D} diverge in the $\mf{x}\tend 0$ limit.
Thence, upon using the relation between $\wt{f}_0, \wt{b}_{0}$ and their un-tilded counterparts, one gets
\bem
\Dp{ \mf{x} }^{n} \Big\{\, \mc{J}[\msc{F}, A, B]( \mf{x} ) \, \Big\} \;= \; (-1)^{n+1} \mf{x}^{ a_0 + b_0 + 1-n} f_0 \f{ \sin [\pi b_0 ] }{ \pi } \Ga( a_0+1)\Ga( b_0 +1)\Ga(n-a_0 - b_0 - 1) \\
\; + \; \e{O}\bigg( \, \mf{x}^{ a_0 + b_0 + 1 + \a -n} \bigg) \;.
\end{multline}
Then, $n$-fold integration in respect to $\mf{x}$ entails the claim.
\begin{lemme}
\label{Lemme integrale type beta reguliere}
Let $\mf{z}_{\pm}(\la)$ be two real-holomorphic functions in a neighbourhood of $\intff{a}{b}$, $ a \, < \, b $, such that
\begin{itemize}
\item[$\bullet$] $ \mf{z}_{\pm} >0$ on $\intoo{a}{b}$;
\item[$\bullet$] $a$, resp. $b$, is, either such that both $\mf{z}_{\pm}(a)>0$, resp. $\mf{z}_{\pm}(b)>0$, or such that it is
a simple zero of $\mf{z}_{\eps_{\ell}}$, resp. $\mf{z}_{\eps_{r}}$, ($\eps_{\ell/r}\in \{\pm 1\}$) but not a zero of the other function.
\end{itemize}
Let $\, \wh{\mf{z}}_{\pm}(\la)=\mf{z}_{\pm}(\la)+\mf{x}$ and let
\begin{itemize}
\item $a(\mf{x})=a$, resp. $b(\mf{x})=b$, in the case when both $\mf{z}_{\pm}(a)>0$, resp. $\mf{z}_{\pm}(b)>0$;
\item $a(\mf{x})$, resp. $b(\mf{x})$, be the zero of $\, \wh{\mf{z}}_{\eps_{\ell}}$, resp. $\wh{\mf{z}}_{\eps_{r}}$, such that $a(\mf{x})=a+\e{O}(\mf{x})$, resp. $b(\mf{x}) \, = \, b + \e{O}(\mf{x})$, if
$\mf{z}_{\eps_{\ell}}(a)=0$, resp. $\mf{z}_{\eps_{r}}(b)=0$.
\end{itemize}
\noindent Let $\De_{\ups}\geq 0$ be smooth on $\intff{ a(\mf{x}) }{ b(\mf{x}) }$ uniformly in $\mf{x}$ small enough. Let
$\msc{G}$ be in the smooth class of $\intff{ a(\mf{x}) }{ b(\mf{x}) }$ with functions $\De_{\pm}$ and constant $\tau$.
Then, the integral
\beq
\mc{J}(\mf{x})\, = \, \Int{ a(\mf{x}) }{ b(\mf{x}) } \msc{G}\Big(\la, \wh{\mf{z}}_{+}(\la) , \wh{\mf{z}}_{-}(\la) \Big) \cdot \pl{\ups= \pm }{} \Big\{ \big[ \, \wh{\mf{z}}_{\ups}(\la) \big]^{ \De_{\ups}(\la)-1 } \Big\} \cdot \dd \la
\enq
is a smooth function of $\mf{x}$ small enough. In particular, it admits a Taylor series expansion around $\mf{x}=0$.
\end{lemme}
\Proof
To start with, consider the simpler situation when $\De_{\pm}>0$ on $\intff{ a(\mf{x}) }{ b(\mf{x}) }$.
First consider the case when $a$ and $b$ are both a zero of one of the functions $\mf{z}_{\pm}$.
Then, let $\eps_{\ell/r} \in \{ \pm 1\}$ be such that $\mf{z}_{\eps_{\ell}}(a)=0$, $\mf{z}_{\eps_{r}}(b)=0$. In such a case, for any $\eta>0$ and small enough, the hypotheses of the lemma ensure that
there exists a constant $c>0$ such that
\beq
\left\{ \ba{ccc} \mf{z}_{-\eps_{\ell}}(\la) \, > \, c & \e{on} & \intff{a}{a+\eta} \\
\mf{z}_{-\eps_{r}}(\la) \, > \, c \ & \e{on} & \intff{b-\eta}{b} \ea \right.
\qquad \e{and} \qquad \mf{z}_{\pm}(\la) \, > \, c \;\; \e{on} \; \intff{a+\eta}{b-\eta} \;.
\label{ecriture conditions positivite}
\enq
Since $a$, resp. $b$, is a simple zero of $\mf{z}_{\eps_{\ell}}(\la)$, resp. $\mf{z}_{\eps_{r}}(\la)$, the function is a local biholomorphism in the neighbourhood of that point.
Hence, the zeroes $a(\mf{x})$ and $b(\mf{x})$ are analytic in $\mf{x}$ small enough and one has the factorisation
\beq
\wh{\mf{z}}_{\eps_{\ell}}(\la) \, =\, (\la-a(\mf{x})) \cdot h_{ \ell }(\la , \mf{x}) \qquad \e{and} \qquad
\wh{\mf{z}}_{\eps_{r}}(\la) \, =\, (b(\mf{x})-\la) \cdot h_{ r }(\la , \mf{x})
\enq
with $ h_{ r/\ell }(\la , \mf{x}) >0$ and analytic in $\la$ and $\mf{x}$ by the Weierstrass preparation theorem \ref{Theorem Weierstrass preparation theorem}.
Finally, the inverse $ \big( \, \wh{\mf{z}}_{\eps_{ \ell/r}}\big)^{-1}$ takes the explicit form
$ \big( \, \wh{\mf{z}}_{\eps_{ \ell/r}}\big)^{-1}(t) \, = \, \mf{z}_{\eps_{ \ell/r}}^{-1}(t-\mf{x}) $
Thence, picking some $\eta>0$ small enough, one can decompose the original integral as
\beq
\mc{J}(\mf{x})\, = \, \mc{J}_{\ell}(\mf{x}) \, + \, \mc{J}_{c}(\mf{x}) \, + \, \mc{J}_{r}(\mf{x})
\quad \e{with} \quad
\mc{J}_{c}(\mf{x}) \, = \hspace{-3mm} \Int{ \mf{z}_{\eps_{ \ell}}^{-1}(\eta-\mf{x}) }{ \mf{z}_{\eps_{ r }}^{-1}(\eta-\mf{x}) } \hspace{-3mm} G_{c}(\la,\mf{x}) \cdot \dd \la
\enq
and
\beq
\mc{J}_{\ell}(\mf{x}) \, = \hspace{-3mm} \Int{a(\mf{x}) }{ \mf{z}_{\eps_{ \ell}}^{-1}(\eta-\mf{x}) } \hspace{-3mm} G_{\ell}(\la,\mf{x}) \cdot
\big[ \, \wh{\mf{z}}_{\eps_{\ell}}(\la) \big]^{ \De_{ \eps_{\ell} }(\la)-1 }
\cdot \mf{z}_{\eps_{\ell}}^{\prime}(\la)\cdot \dd \la
\; \quad , \quad \;
\mc{J}_{r}(\mf{x}) \, = \hspace{-3mm} \Int{ \mf{z}_{\eps_{ r }}^{-1}(\eta-\mf{x}) }{b(\mf{x}) } \hspace{-3mm} G_{r}(\la,\mf{x})
\cdot \big[ \, \wh{\mf{z}}_{\eps_{r}}(\la) \big]^{ \De_{ \eps_{r} }(\la)-1 }
\cdot \mf{z}_{\eps_{r}}^{\prime}(\la)\cdot \dd \la \;.
\enq
Above, I have set
\beq
G_{c}(\la,\mf{x}) \; = \; \msc{G}\Big(\la, \wh{\mf{z}}_{+}(\la) , \wh{\mf{z}}_{-}(\la) \Big) \cdot \pl{\ups= \pm }{} \Big\{ \big[ \, \wh{\mf{z}}_{\ups}(\la) \big]^{ \De_{\ups}(\la)-1 } \Big\}
\enq
and
\beq
G_{\ell/r}(\la,\mf{x}) \; = \; \msc{G}\Big(\la, \wh{\mf{z}}_{+}(\la) , \wh{\mf{z}}_{-}(\la) \Big) \cdot
\big[ \, \wh{\mf{z}}_{-\eps_{\ell/r}}(\la) \big]^{ \De_{-\eps_{\ell/r}}(\la)-1 } \cdot \f{ 1 }{ \mf{z}_{\eps_{\ell/r}}^{\prime}(\la) }
\;.
\enq
The lower bounds \eqref{ecriture conditions positivite}, the smoothness of $G_{c}(\la;\mf{x})$ and the fact that the integration runs through a compact,
all together ensure that $\mc{J}_{c}(\mf{x})$ is smooth in $\mf{x}$. Furthermore, a change of variables recasts $ \mc{J}_{a}(\mf{x})$, with $a\in \{\ell, r\}$ as
\beq
\mc{J}_{a}(\mf{x}) \, = \vsg_{a} \Int{ 0 }{ \eta } G_{a}\Big( \mf{z}_{\eps_{ a }}^{-1}(s-\mf{x}) , \mf{x} \Big) \cdot
s ^{ \De_{ \eps_{a} }\circ \mf{z}_{\eps_{ a }}^{-1}(s-\mf{x}) -1 }
\dd s
\enq
with $\vsg_{\ell}=1$ and $\vsg_{r}=-1$. The same arguments as for $\mc{J}_{c}(\mf{x})$ then allow one to conclude.
\vspace{2mm}
The remaining cases of possible values of $\mf{z}_{\pm}(a)$
and $\mf{z}_{\pm}(b)$ can be treated quite similarly.
\vspace{2mm}
It remains to discuss the situation when one allows for $\De_{\ups}$ to vanish. The latter case remains unchanged relatively to $\mc{J}_{c}(\mf{x})$.
As for $\mc{J}_{a}(\mf{x})$, $a \in \big\{ \ell, r \big\}$, by the properties of a smooth class function, one may recast
\beq
G_{a}\Big( \mf{z}_{\eps_{ a }}^{-1}(s-\mf{x}) , \mf{x} \Big) \; = \; \De_{\eps_a}\circ \mf{z}_{\eps_{ a }}^{-1}(s-\mf{x}) \cdot G_{a}^{(1)}\big( s-\mf{x} \big)
\; + \; G_{a}^{(2)}\big( s-\mf{x} \big) \cdot s^{1-\tau}
\enq
with $G_a^{(1)}$, $G_a^{(2)}$ smooth. Thus,
\bem
\mc{J}_{a}(\mf{x}) \, = \vsg_{a} \Int{ 0 }{ \eta } G_{a}^{(2)}\big( s-\mf{x} \big) \cdot s ^{ \De_{ \eps_{a} }\circ \mf{z}_{\eps_{ a }}^{-1}(s-\mf{x}) - \tau } \cdot \dd s
\, - \, \vsg_{a} \Int{ 0 }{ \eta } \Dp{s} \Big\{ G_{a}^{(2)}\big( s-\mf{x} \big) \cdot v^{ \De_{ \eps_{a} }\circ \mf{z}_{\eps_{ a }}^{-1}(s-\mf{x}) } \Big\}_{\mid v = s } \cdot \dd s \\
\, + \, \vsg_a G_{a}^{(2)}\big( \eta-\mf{x} \big) \cdot \eta ^{ \De_{ \eps_{a} }\circ \mf{z}_{\eps_{ a }}^{-1}(\eta-\mf{x}) }
\, - \, \vsg_a G_{a}^{(2)}\big( -\mf{x} \big) \cdot { s^{ \De_{ \eps_{a} }\circ \mf{z}_{\eps_{ a }}^{-1}( s-\mf{x}) } }_{\mid s=0} \;.
\end{multline}
Note that the last term issuing from the integration by parts is present only if $\De_{ \eps_{a} }\circ \mf{z}_{\eps_{ a }}^{-1}( -\mf{x})=0$ and in that case,
the contribution is also smooth in $\mf{x}$. Smoothness of all the other terms is clear. \qed
\section{Asymptotics of multi-dimensional $\be$-like integral}
\label{Appendix DA integrales multidimensionnelles}
\subsection{General assumptions}
\label{SousSection hypothese gles sur consituants integrale modele}
It is convenient to introduce a few notations and objects that will be used throughout this section.
One assumes to be given:
\begin{itemize}
\item a strictly positive real $\op{v} >0$;
\item a collection of compact intervals $\msc{I}_{r}$, $r=1,\dots, \ell$ ;
\item smooth functions $\mf{u}_r$ on $\msc{I}_r$ such that $\mf{u}^{\prime}_r$ is strictly monotonous on $\msc{I}_r$, and such that
\beq
\mf{u}^{\prime}_r(k) \not= \pm \op{v} \qquad \e{for} \qquad k \in \e{Int}\big(\msc{I}_{r} \big) \;.
\label{ecriture propriete de vitesse defferente de vF}
\enq
\end{itemize}
Taken the physical interpretation that is discussed in the core of the paper,
\begin{itemize}
\item $k \mapsto \mf{u}_r(k)$ corresponds to the momentum-energy dispersion curve associated with a single particle excitation of "type" $r$;
\item $\mf{u}^{\prime}_{r}$ corresponds to the velocity of this excitation;
\item $\msc{I}_r$ is the domain, in momentum space, where the dispersion curve $k \mapsto \mf{u}_r(k)$ is strictly convex or concave, \textit{viz}. where $\mf{u}^{\prime\prime}_{r}$ has constant sign in its interior.
\end{itemize}
Given $n_r \in \mathbb{N}^{*}$, $r=1,\dots, \ell$, define the compact subset $\msc{I}_{\e{tot}}$ of $\R^{ \ov{\bs{n}}_{\ell} }$ with $ \ov{\bs{n}}_{\ell} \, = \, \sum_{r=1}^{\ell}n_r$, as
\beq
\msc{I}_{\e{tot}}\; = \; \pl{r=1}{\ell} \msc{I}_r^{n_r} \;.
\label{definition de intervalle multidimensionnels I}
\enq
It is assumed that the intervals $\msc{I}_r$ partition as
\beq
\msc{I}_{r}\; = \; \msc{I}_{r}^{(\e{in})}\sqcup \msc{I}_{r}^{(\e{out})}
\label{ecriture partition in out intervalle r}
\enq
with $\msc{I}_{1}\; = \; \msc{I}_{1}^{(\e{in})}$, \textit{i.e.} $\msc{I}_{1}^{(\e{out})} = \emptyset$. The partition is such that
\beq
\mf{u}_{r}^{\prime}\Big( \e{Int}\big(\msc{I}_{r}^{(\e{out})} \big) \Big) \, \cap \, \mf{u}_{1}^{\prime}\Big( \e{Int}\big(\msc{I}_{1}^{(\e{in})} \big) \Big) \, = \, \emptyset
\qquad \e{and} \qquad
\mf{u}_{r}^{\prime}\Big( \e{Int}\big(\msc{I}_{r}^{(\e{in})} \big) \Big) \, = \, \mf{u}_{1}^{\prime}\Big( \e{Int}\big(\msc{I}_{1}^{(\e{in})} \big) \Big) \;.
\enq
The hypothesis of strict monotonicity of $\mf{u}^{\prime}_{r}$ ensures that all the sets $\mf{u}_{r}^{\prime}\Big( \e{Int}\big(\msc{I}_{r}^{(\e{in})} \big) \Big)$ are in one-to-one correspondence. More precisely, there exist homeomorphisms
\beq
t_r \, : \, \msc{I}_{1}^{(\e{in})} \tend \msc{I}_{r}^{(\e{in})} \qquad \e{such} \; \e{that} \qquad
\mf{u}_1^{\prime}(k) \; = \; \mf{u}_r^{\prime}\big( t_r(k) \big) \;.
\enq
The hypotheses on $\mf{u}_{r}$ ensure that $t_r$ is a smooth diffeomorphism from $\e{Int}\big(\msc{I}_{1}^{(\e{in})} \big)$ onto $\e{Int}\big(\msc{I}_{r}^{(\e{in})} \big)$.
It will appear useful, sometimes, to denote $t_1(k)=k$. The partitioning \eqref{ecriture partition in out intervalle r} splits the momentum range of type $r$
excitations into an interval $ \msc{I}_{r}^{(\e{in})} $ associated with momenta of type $r$ excitations having a velocity that is also shared by type "1" excitations,
and an interval $ \msc{I}_{r}^{(\e{out})} $ whose associated velocities never coincide with those of type "1" excitations.
\vspace{2mm}
Given a choice of signs $\zeta_r \in \{\pm 1\}$, one defines the associated macroscopic "momentum" and "energy"
\beq
\mc{P}(k) \; = \; \, \sul{r=1}{\ell }n_r \, \zeta_r \, t_r(k) \qquad \e{and} \qquad \mc{E}(k) \; = \; \sul{r=1}{\ell } n_r \, \zeta_r \, \mf{u}_r\big( t_r(k) \big) \;\; , \qquad k\in \msc{I}_{1} \; ,
\label{definition P et E de max multi particule excitation}
\enq
of an agglomeration of equal velocity particles of different types. It is assumed in the following that $k\mapsto \mc{P}(k)$ is strictly monotonous on $\e{Int}(\msc{I}_1)$, \textit{i.e.} that
\beq
k \; \mapsto \; \mc{P}^{\prime}(k) \, =\, \sul{r=1}{\ell} \zeta_r \, n_r \, t_r^{\prime}(k)
\label{ecriture hypothese non vanishing impusion macro}
\enq
does not vanish on $\e{Int}\big(\msc{I}_{1}^{(\e{in})} \big)$.
Finally, it is convenient to represent vectors in block form relatively to the Cartesian product structure of $\msc{I}_{\e{tot}}$, \textit{c.f.} \eqref{definition de intervalle multidimensionnels I},
\beq
\bs{p} \; = \; \big( \bs{p}^{(1)} , \dots , \bs{p}^{(\ell)} \, \big)\qquad \e{with} \qquad \bs{p}^{(r)}\, = \, \big( p_1^{(r)}, \cdots, p_{n_r}^{(r)} \big) \in \R^{n_r} \;.
\label{definition vecteur p}
\enq
Also, given a vector $\bs{p}$ as above, it will be useful to introduce a special notation for a related vector where some of the components of $\bs{p}$ have been dropped:
\beq
\bs{p}^{(r)}_{[a]} \, = \, \big( p_1^{(r)} , \dots, p_{a-1}^{(r)},p_{a+1}^{(r)},\dots , p_{n_r}^{(r)} \, \big) \quad \e{and} \quad
\bs{p}_{[r,a]} \, = \, \Big( \bs{p}^{(1)} , \dots , \bs{p}^{(r)}_{[a]}, \dots , \bs{p}^{(\ell)} \big) \;.
\label{definition vecteur p avec composantes omises}
\enq
In the following, $k_0\in \e{Int}(\msc{J}_1)$ will single out a point in $\e{Int}(\msc{J}_1)$. Analogously to the above way of writing vectors, one denotes
\beq
\bs{t}(k_0)\, = \, \big( \bs{t}_1(k_0), \dots, \bs{t}_{\ell}(k_0) \big) \in \R^{\ov{\bs{n}}_{\ell}} \qquad \e{with} \qquad \bs{t}_r(k_0) \, = \, \big( t_r(k_0), \dots, t_r(k_0) \big) \in \R^{n_{r}}
\label{ecriture definition vecteur tk0}
\enq
as well as
\beq
\bs{t}_{[\ell,n_{\ell}]}(k_0)\, = \, \big( \bs{t}_1(k_0), \dots, \bs{t}_{\ell,[n_{\ell}]}(k_0) \big) \in \R^{\ov{\bs{n}}_{\ell}-1} \qquad
\e{with} \qquad \bs{t}_{\ell,[n_{\ell}]}(k_0) \, = \, \big( t_{\ell}(k_0), \dots, t_{\ell}(k_0) \big) \in \R^{n_{\ell}-1}
\enq
and where all the other $\bs{t}_r(k_0)$'s are as given in \eqref{ecriture definition vecteur tk0}.
Finally, the set of all possible labels $(r,a)$ arising in the coordinates of $\bs{p}$ given in \eqref{definition vecteur p} is denoted as:
\beq
\mc{M} \; = \; \Big\{ (r,a) \; : \; r\in \intn{1}{\ell} \; \e{and} \; a\in \intn{1}{n_r} \Big\} \;.
\enq
Sometimes, the notation
\beq
\mc{M}_{[\ell, n_{\ell}]} \; = \; \mc{M} \setminus \{(\ell, n_{\ell})\}
\label{definition ensemble M elle n ell enleve}
\enq
will be used.
\vspace{3mm}
It is easily seen that properties $Hi)-Hii)$ of a function $\msc{G}$ on $K \times \R^+ \times \R^+$
that is in the smooth class of $K$ and associated with functions $d_{\pm}$ and a constant $\tau$, \textit{c.f.} Definition \ref{definition smooth class on K},
entail that, for any $(\bs{s},\ell_u,\ell_v)$ as above and for fixed $\eps>0$, it holds
\begin{itemize}
\item[H1)] $\big( \bs{x}, u , v \big) \mapsto \pl{ a=1 }{ n } \Dp{ x_a }^{s_a } \, \cdot \, \Dp{u}^{\ell_u} \, \msc{G}^{(s)}\big( \bs{x}, u , v \big)$ is bounded on $ K \times \intff{ \eps }{ \eps^{-1} } \times \intff{0}{\eps^{-1}}$;
\item[H2)] $\big( \bs{x}, u , v \big) \mapsto \pl{ a=1 }{ n }\Dp{ x_a }^{s_a } \, \cdot \, \Dp{v}^{\ell_v} \, \msc{G}^{(s)}\big( \bs{x}, u , v \big)$ is bounded on $ K \times \intff{0}{\eps^{-1}}\times \intff{ \eps }{ \eps^{-1} }$;
\item[H3)] $\big( \bs{x}, u , v \big) \mapsto\pl{ a=1}{ n }\Dp{ x_a }^{s_a } \, \cdot \, \msc{G}^{(s)}\big( \bs{x}, u , v \big)$ is bounded on $ K \times \intff{0}{\eps^{-1}}^2$.
\end{itemize}
Note that depending on the values of $s\in\intn{1}{4}$, the $u$ or $v$ variables may or may not be effectively present in the above equations, \textit{viz}. one should understand
in the formulae above $\msc{G}^{(1)}(\bs{x}, u, v) = \msc{G}^{(1)}(\bs{x})$ \textit{etc}.
Furthermore, when $n\geq 2$, all the functions appearing in $H1)-H3)$ vanish upon replacing $K \hookrightarrow \Dp{}K$.
\subsection{The structural theorem in the multidimensional setting}
\begin{theorem}
\label{Theorem principal caractere lisse et non lisse des integrals multi particules}
Let $\msc{I}_{\e{tot}}$ be as defined in \eqref{definition de intervalle multidimensionnels I} and
$ \De_{\pm} $ be smooth positive functions on $ \msc{I}_{\e{tot}} $ admitting smooth square roots on $ \msc{I}_{\e{tot}} $.
Let $\msc{G}$ be in the smooth class of $ \msc{I}_{\e{tot}} $ assoiated with the functions $\De_{\pm}$ and a constant $\tau \in \intoo{0}{1}$,
according to Definition \ref{definition smooth class on K}.
Finally, let
\beq
\mf{z}_{\ups}(\bs{p})\;=\; \mc{E}_0 \, -\, \sul{ (r,a) \in \mc{M} }{ } \zeta_r \mf{u}_r\big( p_a^{(r)} \big) \, + \; \ups \op{v} \, \bigg\{ \mc{P}_0 - \sul{ (r,a) \in \mc{M} }{ } \zeta_r p_a^{(r)} \bigg\} \; ,
\quad \ups \in \{ \pm \} ,
\label{definition fonctions zpm modele}
\enq
with $\zeta_r \in \{\pm 1\}$ as given in \eqref{definition P et E de max multi particule excitation} and where $(\mc{P}_0,\mc{E}_0) \in \R^2$.
\noindent Let $\mc{I}\big( \mf{x} \big) $ be given by the multiple integral
\beq
\mc{I}\big( \mf{x} \big) \, = \, \Int{ \msc{I}_{\e{tot}} }{} \dd \bs{p} \; \msc{G}_{\e{tot}}(\bs{p})
\enq
where
\beq
\msc{G}_{\e{tot}}(\bs{p}) \, = \, \msc{G}\Big( \bs{p}, \mf{z}_{+}(\bs{p})+ \mf{x} , \mf{z}_{-}(\bs{p})+ \mf{x} \Big) \cdot \pl{\ups=\pm }{} \bigg\{ \Xi\Big( \, \mf{z}_{\ups}(\bs{p})+ \mf{x} \Big)
\cdot \Big[ \, \mf{z}_{\ups }(\bs{p})+ \mf{x} \Big]^{ \De_{\ups}(\bs{p}) -1 } \bigg\} \cdot V(\bs{p}) \;,
\label{definition Gtot dans appendix}
\enq
and
\beq
V(\bs{p}) \; = \; \pl{r=1}{\ell} \pl{a<b}{n_r} \big( p_a^{(r)}-p_b^{(r)} \big)^2 \; .
\label{defintion V Vdm product squared}
\enq
The type of $\mf{x}\tend 0$ asymptotic expansion of $\mc{I}(\mf{x})$ depends on the value of $(\mc{P}_0,\mc{E}_0)$.
\vspace{2mm}
{\bf a)} \texttt{The regular case.}
\vspace{2mm}
\noindent If the two conditions given below hold
\beq
\big( \mc{P}_0,\mc{E}_0 \big) \, \not\in \, \Big\{ \big( \mc{P}(k),\mc{E}(k) \big) \; : \; k \in \msc{I}_1 \Big\}
\label{ecriture hypothese forme energie impulsion reguliere}
\enq
and
\beq
\underset{ \substack{ \a \in \Dp{}\msc{I}_{1} \\ \ups=\pm} }{\e{min}} \big| \mc{E}_0\,-\, \mc{E}(\a) + \ups \op{v}\, (\mc{P}_0\,-\, \mc{P}(\a))\big| \; > \; 0
\label{Hypothese non vanishing of Z ups on boundary of I1}
\enq
then $\msc{G}_{\e{tot}} \in L^1\big( \msc{I}_{\e{tot}} \big)$ and $\mc{I}\big( \mf{x} \big) $ is smooth in $\mf{x}$, for $|\mf{x}|$ small enough.
\vspace{2mm}
{\bf b)} \texttt{The singular case.}
\vspace{2mm} \noindent Let $k_0 \in \e{Int}(\msc{I}_1)$ and recalling $\bs{t}(k_0)$ as defined in \eqref{ecriture definition vecteur tk0}, let
\beq
\De_{\ups}^{(0)} \, = \, \De_{\ups}\big( \bs{t}(k_0) \big) \qquad and \qquad
\vth \; = \; \f{1}{2} \sul{r=1}{\ell}n_r^2 \; - \; \f{3}{2} \, + \, \De_{+}^{(0)} \, + \, \De_{-}^{(0)} \;.
\label{definition de cal theta 0}
\enq
If
\beq
\big( \mc{P}_0,\mc{E}_0 \big) \, = \, \big( \mc{P}(k_0),\mc{E}(k_0) \big) \;,
\quad \vth \not\in \mathbb{N} \;, \quad and \quad \De_{ \pm }^{(0)} >0
\label{ecriture forme de impulsion energie singuliere}
\enq
then $\msc{G}_{\e{tot}} \in L^1\big( \msc{I}_{\e{tot}} \big)$ and $\mc{I}(\mf{x})$ admits the $\mf{x}\tend 0^+$ asymptotic expansion:
\bem
\mc{I}\big( \mf{x} \big) \; = \;
\f{ \De_{ +}^{(0)} \, \De_{ - }^{(0)}\, \mc{G}^{(1)}\big( \bs{t}(k_0) \big) \cdot \big(2 \op{v} \big)^{\De_{ +}^{(0)} + \De_{-}^{(0)} - 1 } }
{ \sqrt{ | \mc{P}^{\prime}(k_0) | } \cdot \pl{\ups=\pm }{} \big| \op{v} - \ups \mf{u}_1^{\prime}(k_0) \big|^{ \De_{ \ups }^{(0)} } }
\cdot \Ga\big( \De_{ +}^{(0)} \big) \Ga\big( \De_{ -}^{(0)} \big) \Ga\big( - \vth \big) \cdot
\pl{r=1}{\ell} \Bigg\{ \f{ G(2+n_r) \cdot \big( 2\pi\big)^{\frac{ n_r - \de_{r,1} }{2} } }{ \big|\mf{u}_{r}^{\prime\prime}(t_r(k_0))\big|^{ \frac{1}{2} ( n_r^2 - \de_{r,1} ) } } \Bigg\} \\
\times
|\mf{x}|^{ \vth } \cdot \Bigg\{ \Xi(\mf{x}) \f{ \sin \big[ \pi \nu_{+}\big] }{\pi} \, + \, \Xi(-\mf{x})\f{ \sin \big[ \pi \nu_{-}\big] }{\pi} \Bigg\}
\, + \, \mf{r}(\mf{x}) \, + \, \e{O} \Big( |\mf{x}|^{ \vth + 1 -\tau } \Big)\;.
\end{multline}
Above $\mf{r}(\mf{x})$ is smooth in $\mf{x}$, for $|\mf{x}|$ small enough.
Finally,
\beq
\nu_{ \pm } \; = \; \f{1}{2}\sul{ \substack{ r=1 \, : \, \\ \veps_r= \mp 1} }{ \ell } n_r^2 \; - \; \f{ 1 \mp \mf{s} }{ 4 }
\; + \hspace{-4mm} \sul{ \substack{ \ups=\pm \, : \, \\ \pm [ \op{v} - \ups \mf{u}_1^{\prime}(k_0) ]>0 } }{} \hspace{-4mm} \De_{ \ups }^{(0)}
\enq
where $\mf{s}=-\e{sgn}\Big( \frac{ \mc{P}^{\prime}(k_0) }{ \mf{u}_{1}^{\prime\prime}(k_0) } \Big) $ and $\veps_r=-\zeta_r \e{sgn}\Big(\mf{u}^{\prime\prime}_{r}(t_r(k_0)) \Big) $\;.
\end{theorem}
It is to be expected that the conditions $\De_{ \pm }^{(0)} >0$ are only technical and can be relaxed down to $\De_{ \pm }^{(0)} \geq 0$, upon some improvement
of the method of analysis.
\Proof
\vspace{1mm}
\subsubsection*{$\bullet$ $L^{1}(\msc{J}_{\e{tot}})$ character}
The integrand is smooth with the exception of the points where $ \mf{z}_{\ups }(\bs{p})+ \mf{x}=0$. Thus, to conclude on its $L^{1}(\msc{J}_{\e{tot}})$ integrability
it is only necessary to focus on its local behaviour in the vicinity of these points. The local behaviour of the integrand around these points, after an appropriate
change of variables that rectifies this behaviour, is thoroughly investigated in the core of the proof. It is the integrations over such vicinity
that generate the non-smooth behaviour in $\mf{x}$. These integrals reduce to the "local" integrals described in \eqref{ecriture integrale I sg parallele cas u lower v 1ere reduction}
and \eqref{ecriture integrale I sg parallele cas u bigger v 1ere reduction} whose study can be reduced to reasoning on one-dimensional integrals
by means of appropriate changes of variables. On the level of these representations, it is easy to see that the local $L^{1}$ character, in virtue of
$\msc{G}$ being in the smooth class of $\De_{\pm}$, reduces to $\De_{\pm}\geq 0$.
\subsubsection*{$\bullet$ A preliminary decomposition into totally collinear and non-collinear parts}
The first step of the analysis consists in decomposing the integral into those parts which may, under certain conditions on $(\mc{P}_0,\mc{E}_0)$, generate a non-smooth behaviour in $\mf{x}$
and those parts which will always, independently of the value of $(\mc{P}_0,\mc{E}_0)$, produce a smooth behaviour.
This is achieved by decomposing the integration domain into portions where one can directly apply Lemma \ref{Lemme integrale multidimensionnelle auxiliaire reguliere}, hence guaranteeing
smoothness in $\mf{x}$ of their contribution,
and those portions which require further study.
\vspace{2mm}
Given $\eta>0$ small enough, one has the below decomposition of $\msc{I}_{\e{tot}}$
\beq
\msc{I}_{\e{tot}}\, = \, \mc{D}^{(\perp)}_{\eta} \sqcup \mc{D}^{(\sslash)}_{\eta}
\enq
where
\beq
\mc{D}^{(\perp)}_{\eta} \, = \, \Big\{ \, \bs{p}\in \msc{I}_{\e{tot}} \; : \; \exists \, (r,a) \not= (1,1) \;\;\e{such}\; \e{that} \;\; \big| \mf{u}_1^{\prime}\big( p_1^{(1)} \big) \, - \, \mf{u}_r^{\prime}\big( p_a^{(r)} \big) \big| \geq \eta \, \Big\}
\label{definition domaine particules ext}
\enq
contains vectors $\bs{p}$ where at least one variable is associated with a different velocity than the one carried by the first component $p_1^{(1)}$ of $\bs{p}$ and
\beq
\mc{D}^{(\sslash)} _{\eta} \, = \, \Big\{ \, \bs{p}\in \msc{I}_{\e{tot}} \; : \; \forall (r,a) \in \mc{M} \;\; \big| \mf{u}_1^{\prime}\big( p_1^{(1)} \big) \, - \, \mf{u}_r^{\prime}\big( p_a^{(r)} \big) \big| < \eta \, \Big\}
\label{definition domaine particules out}
\enq
contains vectors all of whose components have almost equal velocities. Let $\vp^{(\sslash)}$ be smooth and such that
\beq
0\leq \vp^{(\sslash)} \leq 1 \quad, \qquad \vp^{(\sslash)}\, = \, 1 \quad \e{on} \quad \ov{\mc{D}^{(\sslash)}_{\eta/2}} \quad \e{and} \quad \vp^{(\sslash)}\, = \, 0 \;\; \e{on} \; \; \mc{D}^{(\perp)}_{\eta} \;.
\enq
Then set $ \vp^{(\perp)} \, = \, 1 - \vp^{(\sslash)}$. This entails that $ \vp^{(\perp)} \, \not= \, 0 $ only on $\mc{D}^{(\perp)}_{\tf{\eta}{2}}$
so that one has a partition of unity on $\msc{I}_{\e{tot}}$ : $ \vp^{(\sslash)} + \vp^{(\perp)} \, = \, 1 $ which induces the decomposition
of $\mc{I}\big( \mf{x} \big) \, = \, \mc{I}^{(\perp)}\big( \mf{x} \big) \, + \, \mc{I}^{(\sslash)}\big( \mf{x} \big) $ with
\beq
\mc{I}^{(\perp)}\big( \mf{x} \big) \; = \Int{ \mc{D}^{(\perp)}_{ \tf{\eta}{2} } }{} \hspace{-1mm} \dd \bs{p} \; \msc{G}_{\e{tot}}^{(\perp)} (\bs{p})
\qquad \e{and} \qquad
\mc{I}^{(\sslash)}\big( \mf{x} \big) \; = \Int{ \mc{D}^{(\sslash)}_{ \eta } }{} \hspace{-1mm} \dd \bs{p} \; \msc{G}_{\e{tot}}^{(\sslash)} (\bs{p}) \;.
\enq
Above and in the following, we agree upon
\beq
\msc{G}_{\e{tot}}^{(\perp)}(\bs{p}) \, = \, \vp^{(\perp)}(\bs{p}) \cdot \msc{G}_{\e{tot}} (\bs{p}) \qquad \e{and} \qquad
\msc{G}_{\e{tot}}^{(\sslash)}(\bs{p}) \, = \, \vp^{(\sslash)}(\bs{p}) \cdot \msc{G}_{\e{tot}} (\bs{p}) \;.
\enq
\subsubsection*{$\bullet$ The integral $\mc{I}^{(\perp)}$}
I establish below that $\mc{I}^{(\perp)}\big( \mf{x} \big)$ solely generates a smooth behaviour in $\mf{x}$ small enough.
Given $ \bs{h} \in \mc{D}^{(\perp)}_{ \tf{\eta}{2} }$, by definition, there exists $(r,a) \not= (1,1)$ such that $\big| \mf{u}_1^{\prime}\big( h_1^{(1)} \big) \, - \, \mf{u}_r^{\prime}\big( h_a^{(r)} \big) \big| \geq \tf{\eta}{2}$.
Then, the map
\beq
f_{[r,a]}\big( \bs{p}\big)\, = \, \Big(\bs{p}_{1}^{(1)},\dots ,\bs{p}_{[a]}^{(r)}, \dots, \bs{p}^{(\ell)}, \mf{z}_+\big( \bs{p} \big), \mf{z}_-\big( \bs{p} \big) \Big)
\enq
satisfies
\beq
\det\Big[ D_{\bs{p}} f_{[r,a]} \Big] \; = \; 2 \op{v} \, \zeta_1 \, \zeta_r \cdot (-1)^{m_{r,a} } \cdot \big[ \, \mf{u}_r^{\prime}\big( p_a^{(r)} \big) \, - \, \mf{u}_1^{\prime}\big( p_1^{(1)} \big) \, \big]
\quad \e{with} \quad
m_{r,a}=a+\sul{b=1}{r-1}n_b\;.
\enq
Hence, for all $\bs{h}\in \mc{D}^{(\perp)}_{ \tf{\eta}{2} }$,
\beq
\Big| \det\Big[ D_{\bs{h}} f_{[r,a]} \Big] \Big| \, \geq \, \op{v} \eta \; .
\enq
One is thus in position to apply Lemma \ref{Lemme integrale multidimensionnelle auxiliaire reguliere}
so as to conclude that $\mf{x} \mapsto \mc{I}^{(\perp)}\big( \mf{x} \big)$ is smooth in $|\mf{x}|$ small enough.
\noindent As a consequence, it only remains to focus on the $\mf{x}\tend 0$ behaviour of $\mc{I}^{(\sslash)}\big( \mf{x} \big)$.
\subsubsection*{$\bullet$ Behaviour of $\mc{I}^{(\sslash)}$ in the regular case}
This corresponds to case {\bf a)} appearing in the statement of the theorem. Since $\msc{I}_1$ is compact and $k\mapsto (\mc{P}(k),\mc{E}(k))$ is continuous,
where $\mc{P}(k),\mc{E}(k)$ are as defined in \eqref{definition P et E de max multi particule excitation},
hypotheses \eqref{ecriture hypothese forme energie impulsion reguliere} and \eqref{Hypothese non vanishing of Z ups on boundary of I1} entail that there exists $\varrho>0$ such that
\beq
\underset{k \in \msc{I}_1 }{\e{inf}} \Big\{ \;
\e{d}\Big( \big( \mc{P}_0,\mc{E}_0 \big) \, , \, \big( \mc{P}(k),\mc{E}(k) \big) \Big) \; \Big\} \, > \, \varrho \qquad \e{and} \qquad
\underset{ \substack{ \a \in \Dp{}\msc{I}_{1} \\ \ups=\pm} }{\e{min}} \big| \mc{E}_0\,-\, \mc{E}(\a) + \ups \op{v} (\mc{P}_0\,-\, \mc{P}(\a))\big| \, > \, \varrho \;.
\label{ecriture borne inf sur distance a la courbe impulsion energie}
\enq
It is useful to recast $\mf{z}_{\ups}(\bs{p})$ as:
\beq
\mf{z}_{\ups}(\bs{p}) \; = \; \mc{Z}_{\ups}\big(p_1^{(1)} \big) \, + \, \de \mf{z}_{\ups}\big( \bs{p} \big)
\enq
where
\beq
\mc{Z}_{\ups}(k)
= \mc{E}_0\, -\, \mc{E}(k) + \, \ups \op{v} \big[ \mc{P}_0 \, - \, \mc{P}(k)\big]
\enq
and
\beq
\de \mf{z}_{\ups}\big( \bs{p} \big) \, = \,
-\sul{ (r,a) \in \mc{M} }{ } \zeta_r\mf{w}_{\ups}^{(r)}\Big( p_a^{(r)} ; t_r(p_1^{(1)}) \Big) \;.
\enq
Here, I have introduced
\beq
\mf{w}_{\ups}^{(r)}(k ; p) \, = \, \mf{u}_r(k)\, - \, \mf{u}_r(p)\, + \, \ups \op{v} \big( k - p\big) \;.
\label{definition fonction w frak}
\enq
One has that $\mc{Z}_{\ups}$ is smooth on $\e{Int}\big( \msc{I}_1 \big)$ and
\beq
\mc{Z}^{\prime}_{\ups}(k)\,=\, - \bigg\{ \sul{r=1}{\ell} n_r \, \zeta_r \, t^{\prime}_r(k) \bigg\} \cdot \big(\mf{u}_1^{\prime}(k)+\ups \op{v} \big) \;.
\enq
Thus owing to hypothesis \eqref{ecriture propriete de vitesse defferente de vF} and \eqref{ecriture hypothese non vanishing impusion macro},
$\mc{Z}_{\ups}^{\prime}$ does not vanish on $\e{Int}\big( \msc{I}_1 \big)$, so that $\mc{Z}_{\ups}$ is strictly monotonous on $\msc{I}_1$.
This entails that $\mc{Z}_{\ups}$ has at most one zero on $\msc{I}_1$.
\vspace{2mm}
\noindent There are several cases to discuss depending on whether $\mc{Z}_{\ups}$ has a zero or not on $\msc{I}_1$.
\vspace{2mm}
{\bf i)} $\big(\mc{P}_0,\mc{E}_{0} \big)$ is such that both $\mc{Z}_{\pm}$ do not vanish on $\msc{I}_1$.
\vspace{2mm}
\noindent In such a case, there exists $C_0$, such that $|\mc{Z}_{\pm}(k)| \geq 2 C_0 $, for any $k \in \msc{I}_1$.
I now establish that this property entails the non-vanishing of $\mf{z}_{\ups}$ on $\mc{D}^{(\sslash)}_{\eta}$. For that purpose, observe that since $\mf{u}^{\prime}_r$ is strictly monotonous on $\msc{I}_r$ and continuous,
it is continuously invertible on its image.
This allows one to recast
\beq
\de \mf{z}_{\ups}\big( \bs{p} \big) \, = \,
-\sul{ (r,a) \in \mc{M} }{ } \zeta_r \, \wt{\mf{w}}_{\ups}^{(r)}\Big( \mf{u}_r^{\prime}\big( p_a^{(r)} \big) ; \underbrace{ \mf{u}_r^{\prime}\big( t_r(p_1^{(1)}) \big) }_{= \mf{u}_1^{\prime}\big( p_1^{(1)} \big) } \Big)
\enq
where
\beq
\wt{\mf{w}}_{\ups}^{(r)}\big( k ; p \big) \; = \; \mf{u}_r\circ \big( \mf{u}_r^{\prime} \big)^{-1}(k)\, - \,\mf{u}_r\circ \big( \mf{u}_r^{\prime} \big)^{-1} (p)
\, + \, \ups \, \op{v}_F \, \Big( \, \big( \mf{u}_r^{\prime} \big)^{-1}(k) - \big( \mf{u}_r^{\prime} \big)^{-1}(p) \Big) \;.
\enq
Since $\mf{u}_r^{\prime} \big( \msc{I}_{r} \big) \times \mf{u}_1^{\prime} \big( \msc{I}_{1}\big) $ is compact, $\wt{\mf{w}}_{\ups}^{(r)}$
is uniformly continuous on this set. This entails that there exists $s_{\eta}$, with $s_{\eta} \tend 0^+$ when $\eta \tend 0^+$ such that,
\beq
\e{uniformly} \; \e{in} \quad \bs{p} \in \mc{D}^{(\sslash)}_{\eta} \qquad \e{it} \,\e{holds} \qquad \de \mf{z}_{\ups}\big( \bs{p} \big) \, = \, \e{O}\big( s_{\eta} \big) \; .
\label{ecriture estimation sur delta z ups}
\enq
Then, by taking $\eta$ small enough in \eqref{definition domaine particules ext}-\eqref{definition domaine particules out},
one gets that for any $\bs{p}\in \mc{D}^{(\sslash)}_{\eta}$
\beq
\big| \mf{z}_{\ups}(\bs{p}) \big| \, > \, \Big| \, | \mc{Z}^{\prime}_{\ups}( p_1^{(1)} ) | - | \de \mf{z}_{\ups}\big( \bs{p} \big) | \, \Big| \, > \, C_0 \;.
\enq
This lower bound is enough so as to conclude, by derivation under the integral theorems, that $\mc{I}^{(\sslash)}(\mf{x})$ is smooth, provided that $|\mf{x}|$ is small enough.
\vspace{2mm}
{\bf ii)} $\big(\mc{P}_0,\mc{E}_{0} \big)$ is such that least one of the two functions $\mc{Z}_{\pm}$ vanishes on $\msc{I}_1$.
\vspace{2mm}
First of all, by \eqref{ecriture borne inf sur distance a la courbe impulsion energie}, $\mc{Z}_{\ups}$ cannot vanish on $\Dp{}\msc{I}_1$, and
hence, by continuity, on an open neighbourhood thereof. Thus, if a zero exists, it is at a finite distance from the boundary of $\msc{I}_1$.
Furthermore, $\mc{Z}_{\pm}$ cannot share a common zero on $\e{Int}\big(\msc{I}_1 \big)$.
Indeed, if that were the case, then one would have $\mc{Z}_{\pm}(k)=0$ for some $k \in \e{Int}\big( \msc{I}_1 \big)$.
This would then entail that
\beq
\left\{ \ba{ccc} 0\, = \, \mc{Z}_{+}(k)\,-\, \mc{Z}_{-}(k)& = & 2 \op{v} \, \big[\mc{P}_0-\mc{P}(k) \big] \vspace{2mm} \\
0\, = \, \mc{Z}_{+}(k)\,+\, \mc{Z}_{-}(k)& = & 2\big[\mc{E}_0-\mc{E}(k) \big] \ea \right.
\;.
\label{ecriture difference impulsions et energie nulle pour zero commun Z pm}
\enq
However, such a vanishing contradicts \eqref{ecriture borne inf sur distance a la courbe impulsion energie}.
Denote by $k_{\ups}\in \e{Int}\big(\msc{I}_1 \big)$ the zeroes of $\mc{Z}_{\ups}$, if these exists.
Let $\mc{N}_{\ups}$ be an open neighbourhood of $k_{\ups}$ in $\e{Int}(\msc{I}_1)$ such that
\beq
\ov{\mc{N}}_{\ups} \subset \e{Int}(\msc{I}_1) \qquad \e{and} \qquad
\ov{\mc{N}}_{+}\cap \ov{\mc{N}}_{-} \, = \, \emptyset \;,
\enq
where the last condition only applies if both zeros exist and can be made possible since $k_+\not= k_-$ as argued earlier. Then set
\beq
\mc{K}_{\ups} \; = \; \Big\{ \, \bs{p}\in \msc{I}_{\e{tot}} \; : \; p_{1}^{(1)}\in \mc{N}_{\ups} \quad \e{and} \quad \forall (r,a) \in \mc{M} \quad
\big| \mf{u}_1^{\prime}\big( p_1^{(1)} \big) \, - \, \mf{u}_r^{\prime}\big( p_a^{(r)} \big) \big| < \eta \, \Big\} \;.
\enq
By construction, $k_{\ups} \not \in \e{pr}_{[1,1]}\Big( \mc{D}^{(\sslash)}_{\eta} \setminus \mc{K}_{\ups} \Big)$, where $\e{pr}_{[1,1]}$ is the projection on the first coordinate.
Thus, $\mc{Z}_{\ups}$ does not vanish on $\e{pr}_{[1,1]}\Big( \mc{D}^{(\sslash)}_{\eta} \setminus \mc{K}_{\ups} \Big)$.
Recall that, uniformly on $\mc{D}^{(\sslash)}_{\eta}$, one has $\de \mf{z}_{\ups}\big( \bs{p} \big) = \e{O}\big( s_{\eta} \big)$, with $s_{\eta}\tend 0$ as $\eta\tend 0^{+}$.
Reducing $\eta$ if necessary, one concludes, as before, that there exists a constant $C>0$ such that
\beq
\big| \mf{z}_{\ups}(\bs{p}) \big| > C \qquad \e{for}\, \e{any} \quad \bs{p} \in \mc{D}^{(\sslash)}_{\eta} \setminus \mc{K}_{\ups} \;.
\label{ecriture condition positivite z pm frak}
\enq
It remains to deal with the behaviour of $\mf{z}_{\ups}$ inside of $\mc{K}_{\ups}$. The map
\beq
f^{(\ups)}_{ [1,1] } \; : \; \bs{p} \, \mapsto \, \big( \bs{p}_1^{(1)},\bs{p}^{(2)}, \dots, \bs{p}^{(\ell)}, \mf{z}_{\ups}(\bs{p}) \big)
\label{ecriture chngement vars f ups 11}
\enq
satisfies
\beq
\det\Big[ D_{\bs{p}} f^{(\ups)}_{ [1,1] } \Big] \, = \, \pl{r=1}{\ell} (-1)^{ n_r } \, \cdot \, \zeta_{1} \Big( \ups \op{v} \, + \, \mf{u}_1^{\prime}\big( p_1^{(1)} \big) \Big) \;.
\enq
Since $\mf{u}_1^{\prime}(k) \not= \pm \op{v} $ on $\e{Int}\big( \msc{I}_1 \big)$, it follows that $\det\Big[ D_{\bs{p}} f^{(\ups)}_{ [1,1] } \Big] \not=0$
on $\ov{\mc{K}}_{\ups}$. Upon reducing $\eta$ if necessary, by compactness of $\ov{\mc{K}}_{\ups}$ and smoothness of $f^{(\ups)}_{ [1,1] }$ on an open neighbourhood of $\ov{\mc{K}}_{\ups}$,
there exists:
\begin{itemize}
\item points $\bs{p}_{k} \in \ov{\mc{K}}_{\ups}$ , $k=1, \dots, m_{\ups}$;
\item open neighbourhoods $U_{\ups;k}$ of $\bs{p}_{k}$ forming a finite open cover of $\ov{\mc{K}}_{\ups}$ such that
\beq
\ov{\mc{K}}_{\ups} \subset \cup_{k=1}^{m_{\ups}} U_{\ups; k} \subset \msc{I}_{\e{tot}} \; ;
\enq
\item open sets $V_{\ups;k}$ and constants $\de_{\ups;k}>0$ satisfying $\mf{z}_{\ups}(\bs{p}_k)\pm \de_{\ups;k} \not=0$;
\end{itemize}
such that
\beq
f^{(\ups)}_{ [1,1] } \; : \; U_{\ups; k} \; \tend \; V_{\ups; k}\times \intoo{ \mf{z}_{\ups}(\bs{p}_k) -\de_{\ups;k} }{ \de_{\ups;k} + \mf{z}_{\ups}(\bs{p}_k) }
\enq
is a diffeomorphism onto and that its inverse $\big( f^{(\ups)}_{ [1,1] } \big)^{-1}$ extends smoothly to a neighbourhood of
\beq
V_{\ups;k}\times \intoo{ \mf{z}_{\ups}(\bs{p}_k) - \de_{\ups;k}}{ \de_{\ups;k} + \mf{z}_{\ups}(\bs{p}_k) } \; .
\enq
Note that, if both zeroes exists, the neighbourhoods $U_{\ups}=\bigcup_{k=1}^{m_{\ups}} U_{\ups;k}$ can and are chosen such that $U_{+}\cap U_{-}=\emptyset$.
Denote by $\big\{ \vp_{\ups;k} \}_{ k=1 }^{ m_{\ups} }$ the partition of unity associated with the open cover $U_{\ups, k}$.
\vspace{2mm}
Below, I only discuss the case when both $\mc{Z}_{+}$ and $\mc{Z}_{-}$ have a zero. All other cases are treated analogously.
\vspace{2mm}
By using that the integrand vanishes outside of $\mc{D}_{\eta}^{(\sslash)}$, one decomposes the integral as
\beq
\mc{I}^{(\sslash)}(\mf{x}) \, = \, \mc{I}^{(\sslash)}_{\infty}(\mf{x})\, +\, \mc{I}^{(\sslash)}_{+}(\mf{x})\, + \, \mc{I}^{(\sslash)}_{-}(\mf{x}) \;.
\enq
There
\beq
\mc{I}^{(\sslash)}_{\infty}(\mf{x}) \, = \hspace{-3mm} \Int{ \mc{D}^{(\sslash)}_{\eta} \setminus\big\{ U_{+}\cup U_{-} \big\} }{}
\hspace{-4mm} \msc{G}_{ \e{tot }}^{(\sslash)}( \bs{p} ) \cdot \dd \bs{p}
\enq
and
\beq
\mc{I}_{ \ups }^{(\sslash)}(\mf{x}) \, = \, \sul{k=1}{m_{\ups}} \Int{ V_{\ups;k} }{} \; \dd\bs{ v } \hspace{-3mm} \Int{ \mf{z}_{\ups}(\bs{p}_k) -\de_{\ups;k} +\mf{x} }{ \mf{z}_{\ups}(\bs{p}_k) +\de_{\ups;k} +\mf{x} } \hspace{-6mm} \dd u
\;\; \wt{ \mc{G} }_{ \ups ; k}( \bs{v},u-\mf{x} ) \cdot \Xi\big(u \big) \cdot
\big[ u \big]^{ \wt{\De}_{\ups}^{(\ups)}( \bs{v},u -\mf{x} )-1 } \; .
\label{ecriture integrale I ups sslash}
\enq
Above, I have introduced
\bem
\wt{ \mc{G} }_{ \ups ;k }( \bs{v},u ) \, = \, \vp_{\ups;k}\circ\big( f^{(\ups)}_{ [1,1]} \big)^{-1}( \bs{v},u )
\cdot \msc{G}_{\ups}\Big(\big( f^{(\ups)}_{ [1,1]} \big)^{-1}( \bs{v},u ) , u +\mf{x}, \wt{\mf{z}}_{-\ups}( \bs{v},u )+\mf{x} \Big) \cdot \big| \det \Big[ D_{ ( \bs{v},u ) } \big( f^{(\ups)}_{ [1,1]} \big)^{-1} \Big]\big| \\
\times \Xi\Big( \mf{x} +\wt{\mf{z}}_{-\ups}( \bs{v},u )\Big)
\cdot \Big[ \mf{x} + \wt{\mf{z}}_{-\ups}( \bs{v},u ) \Big]^{ \wt{\De}_{-\ups}^{(\ups)}( \bs{v},u )-1 }
\label{expression explicite fct tile ups k G}
\end{multline}
and used the shorthand notation
\beq
\msc{G}_{+}\Big( \bs{p} , u, v \Big) \; = \; \msc{G}_{\e{tot}}^{(\sslash)}\Big( \bs{p} , u, v \Big) \qquad , \qquad
\msc{G}_{-}\Big( \bs{p} , u, v \Big) \; = \; \msc{G}_{\e{tot}}^{(\sslash)}\Big( \bs{p} , v, u \Big)
\enq
as well as
\beq
\wt{\De}_{\pm}^{(\ups)}( \bs{v},u ) \, = \, \De_{\pm}\circ \big( f^{(\ups)}_{ [1,1] } \big)^{-1}( \bs{v},u ) \qquad \e{and} \qquad
\wt{\mf{z}}_{-\ups}( \bs{v},u ) \, = \, \mf{z}_{-\ups} \circ \big( f^{(\ups)}_{ [1,1] } \big)^{-1}( \bs{v},u )\; .
\enq
One can now conclude, individually for each integral.
\begin {itemize}
\item The bound \eqref{ecriture condition positivite z pm frak} along with the smoothness of the integrand allows one to conclude
that $\mc{I}^{ (\sslash) }_{ \infty } (\mf{x})$ are smooth in $\mf{x}$ belonging to some open neighbourhood of $0$.
\item Regarding to $\mc{I}^{ (\sslash) }_{\ups}(\mf{x})$, one should focus on the contribution of each summand $k$. There are three cases to consider.
If $ \mf{z}_{\ups}(\bs{p}_k) + \de_{\ups;k} +\mf{x}<0$, the associated integral simply vanishes for $\mf{x}$ small enough
and there is nothing more to do. If $ \mf{z}_{\ups}(\bs{p}_k) -\de_{\ups;k} >0$, then properties $H1)-H3)$ of a smooth class function as given in Definition \ref{definition smooth class on K},
the fact that the Jacobian determinant in \eqref{expression explicite fct tile ups k G} never vanishes and has thus a constant sign, the smoothness of the other
building blocks and the lower bound \eqref{ecriture condition positivite z pm frak} relatively to $\mf{z}_{-\ups}$ allow one to apply derivation under the integral theorems
so as to conclude that the corresponding integral generates a smooth behaviour in $\mf{x}$ for $|\mf{x}|$ small enough.
Finally, if $ \mf{z}_{\ups}(\bs{p}_k) + \de_{\ups;k} >0$ and $ \mf{z}_{\ups}(\bs{p}_k) -\de_{\ups;k} <0$, then for $\mf{x}$ small enough, the corresponding contribution reduces to
\beq
\mc{I}_{\ups;k} \, = \, \Int{ V_{\ups;k} }{} \; \dd\bs{ v } \hspace{-3mm} \Int{ 0 }{ \mf{z}_{\ups}(\bs{p}_k) +\de_{\ups;k} +\mf{x} } \hspace{-6mm} \dd u
\;\; \wt{ \mc{G} }_{ \ups ; k}( \bs{v}, u - \mf{x} ) \cdot
\big[ u \big]^{ \wt{\De}_{\ups}^{(\ups)}( \bs{v},u - \mf{x} ) -1 } \;.
\enq
By virtue of the decomposition for smooth class functions on $\msc{J}_{\e{tot}}$ associated with $\De_{\pm}$ and the parameter $\tau$, one has the decomposition
\beq
\wt{ \mc{G} }_{ \ups ;k }( \bs{v},u -\mf{x} ) \, = \,\wt{ \mc{G} }_{ \ups ;k }^{(1)}( \bs{v},u-\mf{x} ) \cdot [u]^{1-\tau} \, + \, \wt{\De}_{\pm}^{(\ups)}( \bs{v},u -\mf{x} ) \cdot \wt{ \mc{G} }_{ \ups ;k }^{(2)}( \bs{v},u -\mf{x} )
\enq
with $\wt{ \mc{G} }_{ \ups ;k }^{(a)}$ being smooth and bounded in all of their arguments. This allows one for the rewriting
\bem
\mc{I}_{\ups;k} \, = \, \Int{ V_{\ups;k} }{} \; \dd\bs{ v } \hspace{-3mm} \Int{ 0 }{ \mf{z}_{\ups}(\bs{p}_k) +\de_{\ups;k} +\mf{x} } \hspace{-6mm} \dd u
\Bigg\{ \wt{ \mc{G} }_{ \ups ; k}^{(1)}( \bs{v}, u - \mf{x} ) \cdot \big[ u \big]^{ \wt{\De}_{\ups}^{(\ups)}( \bs{v},u - \mf{x} ) -\tau }
\, - \, \Dp{u} \Big[ \wt{ \mc{G} }_{ \ups ; k}^{(2)}( \bs{v}, u - \mf{x} ) \cdot \big[ s \big]^{ \wt{\De}_{\ups}^{(\ups)}( \bs{v},u - \mf{x} ) } \Big]_{\mid s=u} \Bigg\} \\
+ \Int{ V_{\ups;k} }{} \hspace{-1mm} \dd\bs{ v } \Bigg\{ \wt{ \mc{G} }_{ \ups ; k}^{(2)}( \bs{v}, \mf{z}_{\ups}(\bs{p}_k) +\de_{\ups;k} )
\cdot \big[ \mf{z}_{\ups}(\bs{p}_k) +\de_{\ups;k} + \mf{x} \big]^{ \wt{\De}_{\ups}^{(\ups)}( \bs{v}, \mf{z}_{\ups}(\bs{p}_k) +\de_{\ups;k} ) }
\, - \, \wt{ \mc{G} }_{ \ups ; k}^{(2)}( \bs{v}, - \mf{x} ) \cdot \Big[ \big( s \big)^{ \wt{\De}_{\ups}^{(\ups)}( \bs{v}, - \mf{x} ) } \Big]_{\mid s=0} \Bigg\} \;.
\label{ecriture integrale I ups k}
\end{multline}
Here, one should note that the terms corresponding to taking the $ s \tend 0$ limit only appear if the exponent $\wt{\De}_{\ups}^{(\ups)}$ is vanishing on a set of positive measure.
Due to the mentioned properties of the integrand, one may apply derivation under the integral theorems in the above representation so as to infer that the above integral is smooth in $\mf{x}$ small enough.
\end{itemize}
\subsubsection*{$\bullet$ Behaviour of $\mc{I}^{(\sslash)}$ in the singular case}
This corresponds to case $\bf{b)}$ appearing in the statement of the theorem and is more tricky to deal with. Extracting the $\mf{x}\tend 0$ asymptotics demands several transformations
on the integral $\mc{I}^{ (\sslash) }(\mf{x})$. I first start with a preliminary decomposition.
Assume that $(\mc{P}_0,\mc{E}_0)$ takes the form \eqref{ecriture forme de impulsion energie singuliere} for some $k_0 \in \e{Int}(\msc{I}_1)$.
Then, one can recast $\mf{z}_{\ups}(\bs{p})$ as:
\beq
\mf{z}_{\ups}(\bs{p}) \; = \; - \sul{ (r,a) \in \mc{M} }{ } \zeta_{r} \mf{w}_{\ups}^{(r)} \Big( p_a^{(r)} ; t_r(k_0) \Big)
\enq
where $\mf{w}_{\ups}^{(r)}$ is as in \eqref{definition fonction w frak}. Then, owing to the proximity of the velocities of the integration variables
\beq
\mf{u}_1^{\prime}( p_2^{(1)} ),\dots, \mf{u}_1^{\prime}( p_{n_1}^{(1)} ) \quad \e{to} \quad \mf{u}_1^{\prime}( p_1^{(1)} )
\quad \e{and} \quad \mf{u}_r^{\prime}( p_1^{(r)} ),\dots, \mf{u}_r^{\prime}( p_{n_r}^{(r)} ) \quad \e{to} \quad \mf{u}_r^{\prime}\big( t_r(p_1^{(1)}) \big) \; , \quad \e{for} \; r \in \intn{2}{\ell} \;,
\enq
it is convenient to decompose further $\mf{z}_{\ups}(\bs{p}) $ as
\beq
\mf{z}_{\ups}(\bs{p}) \; = \; Z_{\ups}\big( p_1^{(1)} , k_0 \big) \, + \, \de \mf{z}_{\ups}\big( \bs{p} \big)
\qquad \e{with} \qquad
Z_{\ups}(k_1,k_0) \, = \, \mc{E}(k_0)-\mc{E}(k_1) + \ups \op{v}\, \big[ \mc{P}(k_0) - \mc{P}(k_1) \big]
\enq
and
\beq
\de \mf{z}_{\ups}\big( \bs{p} \big) \; = \; - \sul{ (r,a) \in \mc{M} }{} \hspace{-2mm} \zeta_r \, \mf{w}_{\ups}^{(r)}\Big( p_a^{(r)} ;t_r( p_1^{ (1) } ) \Big) \;.
\enq
The previous estimates ensure that $\de \mf{z}_{\ups}\big( \bs{p} \big) =\e{O}\big( s_{\eta} \big)$ uniformly on $\mc{D}^{(\sslash)}_{\eta}$, \textit{c.f.} \eqref{ecriture estimation sur delta z ups}.
In other words, $Z_{\ups}$ grasps the dominant part of $\mf{z}_{\ups}(\bs{p}) $.
By the same arguments as earlier on, one gets that
\beq
\ups \Dp{k} Z_{\ups}\big( k , k_0 \big) \; = \; -\ups \cdot \mc{P}^{\prime}(k) \cdot \Big( \mf{u}_1^{\prime}( k )\, + \, \ups\op{v} \Big) \; \not= \; 0
\enq
so that $k \, \mapsto \, Z_{\ups}\big( k , k_0 \big)$ is strictly monotonous on $\msc{I}_1$. One can then rely on this property so as to split, by means of an appropriate
partition of unity, the integral into one over a domain corresponding to a neighbourhood of the point $\bs{t}(k_0)=\big(\bs{t}_1(k_0), \dots, \bs{t}_{\ell}(k_0) \big)$
with $ \bs{t}_r(k_0) \, = \, (t_r(k_0),\dots, t_r(k_0) ) \in \R^{n_r}$ which will generate a non-smooth behaviour in $\mf{x}$
and an integral over its complement in $\mc{D}^{(\sslash)}_{\eta}$ which will only generate a smooth behaviour. However, the steps for achieving such a decomposition depend on the magnitude of $|\mf{u}^{\prime}_1|$
respectively to $ \op{v} $: one should distinguish between the two possible situations which can arise due to hypothesis \eqref{ecriture propriete de vitesse defferente de vF}:
\beq
|\mf{u}^{\prime}_1(k)|<\op{v} \quad \e{on} \quad \e{Int}(\msc{I}_1) \qquad \e{or} \qquad |\mf{u}^{\prime}_1(k)|>\op{v} \quad \e{on} \quad \e{Int}(\msc{I}_1) \; .
\enq
\subsubsection*{$\bullet$ $|\mf{u}^{\prime}_1(k)|<\op{v}$ on $\e{Int}(\msc{I}_1)$}
Since, $\de \mf{z}_{\ups}(\bs{p})=\e{O}\big( s_{\eta} \big)$ and since $k \mapsto Z_{\ups}(k,k_0)$ is strictly monotonous, the magnitude and sign on $ \mf{z}_{\ups}(\bs{p})$ will depend on whether one is close to a zero of $Z_{\ups}$
or not.
Let
\beq
\sg \, = \, \e{sgn}\Big( \mc{P}^{\prime}(k) \, \big( \mf{u}^{\prime}_{1}(k)+ \op{v} \big) \Big) \, .
\label{definition signe Pprime u plus vF}
\enq
Hypothesis \eqref{ecriture hypothese non vanishing impusion macro} ensures that $\sg$ is constant on $\e{Int}(\msc{I}_1)$. Taking $\eta$ small enough,
the fact that $Z_{\ups}$ is strictly monotonous and that $\de \mf{z}_{\ups}(\bs{p})=\e{O}\big( s_{\eta} \big)$ both ensure that
there exists $\rho_{\eta}>0$
such that $\rho_{\eta}\tend 0^+$ when $\eta\tend 0^+$, and $\ga_{\eta}$ strictly increasing in $\eta$, $\ga_{\eta}\underset{\eta \tend 0^+}{\tend} 0$, so that
\beq
\ba{ccccc} \ups \sg \mf{z}_{\ups}\big(\bs{p}\big) &<& -\rho_{\eta} & \e{if} & p_1^{(1)} > k_0 +\ga_{\eta} \vspace{2mm} \\
\ups \sg \mf{z}_{\ups}\big(\bs{p}\big) &>& \rho_{\eta} & \e{if} & p_1^{(1)} < k_0 - \ga_{\eta} \ea
\qquad \e{provided}\; \e{that} \qquad \bs{p} \, \in \, \mc{D}^{(\sslash)}_{\eta} \;.
\enq
The above ensures that, for $\eta$ small enough and $|\mf{x}|<\rho_{\eta}$, $\mf{x} \, + \, \mf{z}_{\pm}\big(\bs{p}\big)$ will have opposite signs if $|p_1^{(1)} - k_0 | \geq \ga_{\eta}$.
The presence of the Heaviside step function in the integrand then allows one to reduce the integration domain in $\mc{I}^{(\sslash)}(\mf{x})$ leading to
\beq
\mc{I}^{(\sslash)}\big( \mf{x} \big) \, = \, \Int{ \mc{D}^{(\e{sg})}_{\eta, \ga_{\eta} } }{} \dd \bs{p}\; \msc{G}_{\e{tot}}^{(\sslash)}(\bs{p}) \;.
\enq
Above, I have introduced
\beq
\mc{D}^{(\e{sg})}_{\eta, \ga} \; = \; \Big\{ \bs{p} \in \msc{I}_{\e{tot}} \; : \; |p_1^{(1)}-k_0|<\ga \;\; \e{and} \;\;
\forall (r,a) \in \mc{M} \;\; , \;\; \big| \mf{u}_1^{\prime}\big( p_1^{(1)} \big) \, - \, \mf{u}_r^{\prime}\big( p_a^{(r)} \big) \big| < \eta \, \Big\} \;.
\label{definition de D sg eta ga}
\enq
Finally, let $\vp^{(\e{sg})}$ be smooth on $\pl{r=1}{\ell} \R^{n_r}$ and such that
\beq
0 \leq \vp^{(\e{sg})} \leq 1 \quad, \quad \vp^{(\e{sg})}=1 \quad \e{on} \quad \mc{D}^{(\e{sg})}_{ \eta, \ga_{ \eta } } \quad \e{and} \quad
\vp^{(\e{sg})}=0 \quad \e{on} \quad \mc{D}^{(\e{out})}_{ 2\eta, \ga_{2\eta} } \;.
\label{definition de vp sg indicator function}
\enq
where
\beq
\mc{D}^{(\e{out})}_{\eta, \ga } \; = \; \Big\{ \bs{p} \in \msc{I}_{\e{tot}} \; : \; |p_1^{(1)}-k_0|>\ga \;\;
\e{and} \;\; \forall \; (r,a) \in \mc{M} \;\; , \;\; \big| \mf{u}_1^{\prime}\big( p_1^{(1)} \big) \, - \, \mf{u}_r^{\prime}\big( p_a^{(r)} \big) \big| < \eta \, \Big\} \;.
\label{definition de D out eta ga}
\enq
Since, by construction, the integrand vanishes on $\mc{D}^{(\sslash)}_{2\eta}\setminus \mc{D}^{(\sslash)}_{\eta}$, one may recast $\mc{I}^{(\sslash)}\big( \mf{x} \big) $ in the form
\beq
\mc{I}^{(\sslash)}\big( \mf{x} \big) \; \equiv \; \mc{I}^{(\sslash)}_{\e{sg}}\big( \mf{x} \big) \; = \hspace{-1mm} \Int{ \mc{D}^{(\e{sg})}_{ 2\eta, \ga_{2\eta} } }{} \hspace{-2mm} \dd \bs{p}\; \msc{G}_{\e{sg}}(\bs{p})
\qquad \e{with} \qquad
\msc{G}_{\e{sg}}(\bs{p}) \, = \, \vp^{(\e{sg})}(\bs{p})\cdot \msc{G}_{\e{tot}}^{(\sslash)}(\bs{p}) \;.
\label{definition G sg caligraphique}
\enq
\subsubsection*{$\bullet$ $|\mf{u}^{\prime}_1(k)|>\op{v}$ on $\e{Int}(\msc{I}_1)$}
Define $\sg$ as in \eqref{definition signe Pprime u plus vF}. Then $\sg$ is constant on $\e{Int}(\msc{I}_1)$ by hypotheses \eqref{ecriture propriete de vitesse defferente de vF} and \eqref{ecriture hypothese non vanishing impusion macro}.
Taking $\eta$ small enough, there exists $\rho_{\eta}>0$ and $\ga_{\eta}>0$, a strictly decreasing function of $\eta$,
such that $\ga_{\eta}\underset{\eta \tend 0^+}{\tend} 0$ so that
\beq
\ba{ccccc} \sg \mf{z}_{\ups}\big(\bs{p}\big) &<& -\rho_{\eta} & \e{if} & p_1^{(1)} > k_0 +\ga_{\eta} \vspace{2mm} \\
\sg \mf{z}_{\ups}\big(\bs{p}\big) &>& \rho_{\eta} & \e{if} & p_1^{(1)} < k_0 - \ga_{\eta} \ea
\qquad \e{provided}\; \e{that} \qquad \bs{p} \, \in \, \mc{D}^{(\sslash)}_{\eta} \;.
\label{ecriture condition bornage z dans le cas vitesse superieure a Fermi}
\enq
Taken this into account, it appears convenient to introduce $\vp^{(\e{sg})}$ as in \eqref{definition de vp sg indicator function}.
Then, one gets the decomposition of the integral as $\mc{I}^{(\sslash)}\big( \mf{x} \big) \, = \, \mc{I}_{\e{sg}}^{(\sslash)}\big( \mf{x} \big) + \mc{I}_{\e{out}}^{(\sslash)}\big( \mf{x} \big) $
where
\beq
\mc{I}_{\e{sg}}^{(\sslash)}\big( \mf{x} \big) \, = \, \Int{ \mc{D}^{(\e{sg})}_{2\eta, \ga_{2\eta}} }{} \hspace{-3mm} \dd \bs{p}\; \msc{G}_{\e{sg}}(\bs{p})
\qquad \e{and} \qquad
\mc{I}_{\e{out}}^{(\sslash)}\big( \mf{x} \big) \, = \, \Int{ \mc{D}^{(\e{out})}_{ \eta, \ga_{ \eta} } }{} \hspace{-3mm} \dd \bs{p}\; \big(1-\vp^{(\e{sg})}(\bs{p}) \big) \msc{G}_{\e{tot}}(\bs{p}) \;.
\enq
There, $ \msc{G}_{\e{sg}}(\bs{p})$ is as appearing in \eqref{definition G sg caligraphique} and $\mc{D}^{(\e{out/sg})}_{\eta, \ga }$
have been defined in \eqref{definition de D sg eta ga} and \eqref{definition de D out eta ga}.
Due to the bound \eqref{ecriture condition bornage z dans le cas vitesse superieure a Fermi}, one has that
\beq
|\mf{z}_{\ups}\big(\bs{p}\big)|> \rho_{ \eta } \qquad \e{on} \qquad \mc{D}^{(\e{out})}_{ \eta, \ga_{ \eta } } \;.
\enq
This lower bound allows one to apply derivation under the integral theorems so as to infer that $\mc{I}_{\e{out}}^{(\sslash)}\big( \mf{x} \big)$ is smooth in $\mf{x}$ belonging to a
sufficiently small neighbourhood of $0$.
\subsubsection*{$\bullet$ Simplified form of $\mc{I}_{\e{sg}}^{(\sslash)}\big( \mf{x} \big)$}
The fact that the integration domain in $\mc{I}_{\e{sg}}^{(\sslash)}\big( \mf{x} \big)$ has been reduced $\mc{D}^{(\e{sg})}_{2\eta, \ga_{2\eta} }$ allows one to implement a change of variables which recasts
the integral in a simplified form. Doing so is an important step towards the analysis of its $\mf{x}\tend 0 $ behaviour.
Observe that given any $\bs{p}\in \mc{D}^{(\e{sg})}_{2\eta, \ga_{2\eta} }$, for any $(r,a)\in \mc{M}$, it holds
\beq
\big| \mf{u}^{\prime}_{r}(p_a^{(r)}) \, - \, \mf{u}^{\prime}_{r}( t_r(k_0) ) \big|
\, \leq \, \big| \mf{u}^{\prime}_{1}( p_1^{(1)} ) \, - \, \mf{u}^{\prime}_{1}(k_0 ) \big| \, + \, \big| \mf{u}^{\prime}_{r}(p_a^{(r)}) \, - \, \mf{u}^{\prime}_{1}( p_1^{(1)} ) \big|
\, < \, 2\eta \, + \, \ga_{2\eta} \cdot \norm{ \mf{u}_1^{\prime} }_{ L^{\infty}(\msc{J}_1) } \;.
\label{ecriture bornes sur distance entre pts dans image via u prime r}
\enq
Since $k_0 \in \e{Int}(\msc{J}_1)$, $t_r(k_0) \in \e{Int}(\msc{J}_r)$ and by \eqref{ecriture bornes sur distance entre pts dans image via u prime r}, upon reducing $\eta>0$ if need be,
it follows that there exists $\veps>0$ such that, for any $\bs{p} \in \mc{D}^{(\e{sg})}_{2\eta, \ga_{2\eta} }$,
\beq
\e{d}\Big( \mf{u}^{\prime}_r\big( p_a^{(r)} \big) , \Dp{} \mf{u}^{\prime}_r\big( \msc{I}_r \big) \Big) \, > \, \veps \; ,
\enq
for the canonical distance $\e{d}$ between points and subsets of $\R$. This ensures that there exists a compact $K_r \subset \e{Int}(\msc{J}_r)$ containing an open neighbourhood of $t_r(k_0)$,
such that both $p_a^{(r)}, t_r(p_1^{(1)}) \in K_r$
uniformly in $\bs{p}\in \mc{D}^{(\e{sg})}_{2\eta, \ga_{2\eta} }$. Since $\mf{u}_{r}^{\prime}$ is a smooth diffeomorphism on an open neighbourhood of $K_r$, there exist constants $c_r,C_r>0$ such that
\beq
c_r \cdot \big| \mf{u}_r^{\prime}\big( x\big) \, - \, \mf{u}_r^{\prime}\big( y \big) \big| \; \leq \, | x - y | \; \leq \; C_{r} \cdot \big| \mf{u}_r^{\prime}\big(x \big) \, - \, \mf{u}_r^{\prime}\big( y \big) \big|
\qquad \e{for} \; \e{any} \qquad x,y \in K_r\;.
\enq
Recall that $k_0 \in \e{Int}(\msc{J}_1)$ so that $t_r(k_0) \in \e{Int}(\msc{J}_r)$. Thus, upon taking $\eta$ small enough, the strict monotonicity of $\mf{u}^{\prime}_r$ on $\msc{J}_r$
and the above bounds ensure that, for any $\bs{p}\in \mc{D}^{(\e{sg})}_{2\eta, \ga_{2\eta} }$,
\beq
\big| t_r(k_0) \, - \, p_a^{(r)} \big| \, \leq \, C_r \Big\{ \big| \mf{u}^{\prime}_{r}(p_a^{(r)}) \, - \, \mf{u}^{\prime}_{1}( p_1^{(1)} ) \big| \, + \, \big| \mf{u}^{\prime}_{1}(k_0) \, - \, \mf{u}^{\prime}_{1}( p_1^{(1)} ) \big| \Big\}
\, \leq \,2 C_r \eta \, + \, \f{ C_r }{ c_1 } \ga_{2\eta} \, \equiv \, \eta_{r} \;.
\label{definition etar}
\enq
Therefore, upon denoting $B_{\eps}(x_0)=\{ x \in \R \, : \, |x-x_0| < \eps \}$ the open ball in $\R$ of radius $\eps$ centred at $x_0$, one gets
\beq
\mc{D}^{(\e{sg})}_{2\eta, \ga_{2\eta}} \subset B_{\ga_{2\eta}}(k_0) \times \pl{r=1}{\ell} \Big( B_{\eta_r}\big( t_r(k_0) \big) \Big)^{n_r-\de_{r,1} } \;,
\label{ecriture inclusion D sg dans boule multi-rayon O eta}
\enq
with $\eta_r$ as given in \eqref{definition etar}.
Define auxiliary functions on $ \pl{r=1}{\ell} \R^{n_r}$
\beq
\tilde{z}_{\ups}(\bs{x}) \; = \;- \sul{ (r,a)\in \mc{M} }{} \zeta_{r} \bigg\{ \mf{h}_r\big( x_a^{(r)} \big) +\ups \op{v} x_a^{(r)} \bigg\} \qquad \e{and} \qquad
\left\{ \ba{ccc} \mf{h}_r\big( x \big) & = & \mf{u}_{r}^{\prime\prime}\big( t_r(k_0) \big) \f{ x^2 }{ 2 } \, + \, \mf{u}_{1}^{\prime}(k_0) \, x \vspace{1mm} \\
t_r^{(0)}(x) & =& \f{ \mf{u}_{1}^{\prime\prime}\big( t_1(k_0) \big) }{ \mf{u}_{r}^{\prime\prime}\big( t_r(k_0) \big) } \, x \ea \right. \;,
\enq
along with the domain
\beq
\mc{D}^{(\e{eff})}_{ \eta } \, = \, \Bigg\{ \bs{x} \in \pl{r=1}{\ell} \R^{n_{r}} \; : \; |x_1^{(1)}| \, \leq \, C \eta \; , \;
\forall (r,a )\in \mc{M} \; : \; \big| t_r^{(0)}(x_1^{(1)})-x_a^{(r)} \big| \, \leq \, \f{ \eta }{ \big| \mf{u}_{r}^{\prime\prime}\big( t_r(k_0) \big) \big| } \Bigg\} \;.
\enq
By the above discussion, $\mc{D}^{(\e{eff})}_{ \eta }$ is an open neighbourhood of the origin.
Then, by virtue of Proposition \ref{Proposition trivialisation locale fct zups},
there exists $\mf{x}_0>0$ and $\eta^{\prime}>0$ such that there exists:
\begin{itemize}
\item smooth functions $\mf{f}_{\ups}$ on $\intoo{ -\mf{x}_0 }{ \mf{x}_0 } \, \times \, \mc{D}^{(\e{eff})}_{\eta^{\prime}}$ satisfying $\mf{f}_{\ups}(\mf{x}; \bs{x})\, = \, 1 + \e{O}\Big( \norm{\bs{x}}+|\mf{x}|\Big)$,
\item a smooth diffeomorphism $\Psi_{\mf{x}}: \mc{D}^{(\e{eff})}_{\eta^{\prime}} \; \tend \; \Psi_{\mf{x}}\Big( \mc{D}^{(\e{eff})}_{\eta^{\prime}} \Big)$ satisfying $D_{\bs{0}} \Psi_{\mf{x}} \, = \, \e{id} + \mf{x} \op{N}_{\Psi}$ with
$\norm{ \op{N}_{\Psi} } \leq C$, for some $\mf{x}$-independent $C>0$,
\end{itemize}
such that
\beq
\mf{x}+\mf{z}_{\ups} \circ \Psi_{\mf{x}}(\bs{x}) \; = \; \mf{f}_{\ups}(\mf{x};\bs{x}) \cdot \Big(\mf{x} + \tilde{z}_{\ups}(\bs{x}) \Big) \;,
\enq
and, for $|\mf{x}|<\mf{x}_0$, $\Psi_{\mf{x}}\Big( \mc{D}^{(\e{eff})}_{\eta^{\prime}} \Big) \subset \msc{J}_{\e{tot}}$ contains a $\mf{x}$-independent open neighbourhood of $\bs{t}(k_0) \in \msc{J}_{\e{tot}}$ with $\bs{t}(k_0)$
as given in \eqref{ecriture definition vecteur tk0}. Finally,
\beq
(\mf{x},\bs{x}) \; \mapsto \; \Psi_{\mf{x}}(\bs{x})
\enq
is smooth on $\intoo{ -\mf{x}_0 }{ \mf{x}_0 } \times \mc{D}^{(\e{eff})}_{\eta^{\prime}}$.
Furthermore, by virtue of Proposition \ref{Proposition factorisation jolie en zeroes de tilde z ups}, there exists an invertible linear map $\op{M}$ on $\pl{r=1}{\ell} \R^{n_r}$
and integers $m_{\pm} \in \mathbb{N}$ satisfying $m_+ + m_- + 1 \, = \, \sum_{r=1}^{\ell}n_r$ such that
\beq
\tilde{z}_{\ups}\big( \op{M} \bs{x} \big) + \mf{x} \, = \, P_{\ups}(\bs{x}) \qquad \e{with} \qquad \bs{x}=\big(y, \bs{z}^{(+)}, \bs{z}^{(-)} \big) \in \R\times \R^{m_+}\! \times \R^{m_-}
\label{ecriture effet transformation matrice M}
\enq
and
\beq
P_{\ups}(\bs{x}) \; = \; - \f{ \mf{u}^{\prime\prime}_1(k_0) }{2 \mc{P}^{\prime}(k_0) } y^2 \, - \, \big( \mf{u}^{\prime}_1(k_0) + \ups \op{v} \big) y \, + \, \vp_{\mf{x} }(\bs{z})
\enq
in which
\beq
\vp_{\mf{x} }(\bs{z}) \; = \; \mf{x} \, + \, \sul{s=1}{m_+} \big( z_s^{(+)} \big)^2 \, - \, \sul{s=1}{m_-} \big( z_s^{(-)} \big)^2 \quad \e{with} \quad \bs{z}=\big( \bs{z}^{(+)}, \bs{z}^{(-)} \big) \;.
\label{definition varphi de z et kappa}
\enq
\vspace{3mm}
To proceed further, it is convenient to introduce the auxiliary function
$F$ which is defined around the origin through its series expansion
\beq
F(x) \, = \, 1 \, + \, \f{1}{2 \op{v} } \sul{k \geq 1 }{} d_k x^k \f{ \big[ \op{v}+\mf{u}_{1}^{\prime}(k_0) \big]^{2k+1} \, - \, \big[ \mf{u}_{1}^{\prime}(k_0) - \op{v}\big]^{2k+1} }
{ \big[ \op{v}^2 - \big(\mf{u}_{1}^{\prime}(k_0) \big)^2 \big]^{2k} } \;.
\label{definition fonction G pour difference des racines de P ups}
\enq
The sequence $d_k$ appearing above can be read-off from its generating series
\beq
2 \f{1-\sqrt{1-x}}{x} \, = \, 1+\sul{k\geq 1}{} d_k x^k \;.
\enq
Then, for $\de_0>0$ and small enough, one sets
\beq
\mu(\bs{z}) \; = \; \de_0 \, \wt{F}( \vp_{\mf{x} }(\bs{z}) ) \qquad \e{with} \qquad
\wt{F}(x) \, = \, 2 \op{v} \cdot \f{ F\Big( \tfrac{ - 2 \mf{u}^{\prime\prime}_1(k_0) }{ \mc{P}^{\prime}(k_0) } x \Big) }{ \big| \big(\mf{u}_{1}^{\prime}(k_0) \big)^2 \, - \, \op{v}^2 \big| } \;.
\label{definition fonction mu de z}
\enq
Observe that for any
\beq
\bs{z}\, = \, \big( \bs{z}^{(+)}, \bs{z}^{(-)} \big) \in \big[-\de_{+}^{\tf{1}{2}} ; \de_{+}^{\tf{1}{2}} \big]^{m_+} \times \big[-\de_{-}^{\tf{1}{2}} ; \de_{-}^{\tf{1}{2}} \big]^{m_-}
\enq
it holds $ \vp_{\mf{x} }(\bs{z}) \, = \, \e{O}\big( \de_+ + \de_- + |\mf{x}| \big)$. Thus, provided that $\de_{+}, \de_{-}>0$ and $|\mf{x}|$ are all taken small enough,
$\mu(\bs{z})$ is well-defined for such $\bs{z}$'s and, owing to $F(0)=1$, it holds
\beq
\f{ \op{v} \de_0 }{ \big| \big(\mf{u}_{1}^{\prime}(k_0) \big)^2 \, - \, \op{v}^2 \big| } \, \leq \, \mu(\bs{z}) \, \leq \, \f{ 4 \op{v} \de_0 }{ \big| \big(\mf{u}_{1}^{\prime}(k_0) \big)^2 \, - \, \op{v}^2 \big| } \;.
\label{ecriture bornes sur mu de z}
\enq
Enough has now been introduced so as to allow me to define the domain
\beq
\mc{D} \; = \; \bigg\{ (y, \bs{z} ) \; : \; \bs{z} \, = \, \big( \bs{z}^{(+)},\bs{z}^{(-)} \big) \in \big[-\de_{+}^{\tf{1}{2}} ; \de_{+}^{\tf{1}{2}} \big]^{m_+} \times \big[-\de_{-}^{\tf{1}{2}} ; \de_{-}^{\tf{1}{2}} \big]^{m_-}
\;\; \e{and} \; \; y \in \intff{ - \mu(\bs{z}) }{ \mu(\bs{z}) } \bigg\} \;.
\label{definition domaine D pour reduction}
\enq
Then, the inclusion \eqref{ecriture inclusion D sg dans boule multi-rayon O eta} ensures that, given $\de_0>0, \de_{\pm}>0$ small enough, and upon diminishing, if necessary, the parameter $\eta>0$, it holds
\beq
\mc{D}^{(\e{sg})}_{2 \eta, \ga_{2 \eta}} \; \subset \; \Psi_{\mf{x}}\big( \op{M} \cdot \mc{D} \big) \; \subset \; \Psi_{\mf{x}}\Big( \mc{D}_{\eta^{\prime}}^{(\e{eff})} \Big) \;,
\enq
where $\op{M}$ is as in \eqref{ecriture effet transformation matrice M}, this uniformly in $|\mf{x}|<\mf{x}_0$.
Thus, for such a choice of parameters at play, upon using that the integrand vanishes away from $\mc{D}^{(\e{sg})}_{2 \eta, \ga_{2 \eta}}$ one
one gets that
\beq
\mc{I}_{\e{sg}}^{(\sslash)}(\mf{x}) \; = \;
\hspace{-2mm} \Int{ \Psi_{\op{M}}\big( \mc{D} \big) }{} \hspace{-3mm} \dd \bs{p}\; \msc{G}_{\e{sg}}(\bs{p}) \qquad \e{with} \qquad \Psi_{\op{M}} = \Psi_{\mf{x}} \circ \op{M}\;.
\enq
The change of variables
\beq
\bs{p}=\Psi_{\op{M}}(\bs{x}) \qquad \e{with} \qquad \bs{x} = \big( y, \bs{z}^{(+)}, \bs{z}^{(-)} \big) \in \R \times \R^{m_+} \times \R^{m_-}
\enq
recasts the integral in the form
\beq
\mc{I}_{\e{sg}}^{(\sslash)}(\mf{x}) \; = \; \Int{ \mc{D} }{} \dd \bs{x}\; \msc{F}(\bs{x}) \pl{\ups=\pm}{} \bigg\{ \Xi\Big( P_{\ups}(\bs{x}) \Big) \cdot \big[ P_{\ups}(\bs{x}) \big]^{ \mf{d}_{\ups}(\bs{x}) -1 } \bigg\}
\label{reduction finale de la partie singuliere vers integrale modele}
\enq
where $\mf{d}_{\ups}(\bs{x}) \, = \, \De_{\ups}\circ\Psi_{\op{M}}(\bs{x})$ and
\bem
\msc{F}(\bs{x}) \, = \, V\circ \Psi_{\op{M}}(\bs{x}) \cdot \msc{G}\Big( \Psi_{\op{M}}(\bs{x}) \, , \, \mf{f}_{+}(\mf{x};\op{M}\cdot \bs{x}) P_{+}(\bs{x}) \, , \, \mf{f}_{-}(\mf{x};\op{M}\cdot \bs{x}) P_{-}(\bs{x}) \Big) \\
\times \Big(\vp^{(\sslash)} \vp^{(\e{sg})} \Big)\Big( \Psi_{\op{M}}(\bs{x}) \Big) \cdot \big| \det D_{\bs{x}} \Psi_{\op{M}} \big|
\cdot \pl{ \ups = \pm }{} \Big\{ \mf{f}_{\ups}(\mf{x};\op{M}\cdot \bs{x}) \Big\}^{ \mf{d}_{\ups}(\bs{x}) -1 } \;.
\end{multline}
Also, I remind that $V$ has been defined in \eqref{defintion V Vdm product squared}.
\subsubsection*{$\bullet$ Properties of the polynomials $P_{\ups}(\bs{x})$}
One may put the integral $\mc{I}_{\e{sg}}^{(\sslash)}\big( \mf{x} \big)$ into a canonical form by focalising more on the structure of the polynomial $P_{\ups}(\bs{x})$. It is easy to see that, uniformly in
$\bs{x} \in \mc{D}$ with $\de_{\pm}$ small enough and $\mc{D}$ as defined through \eqref{definition domaine D pour reduction}, it admits the factorisation
\beq
P_{\ups}( \bs{x}) \; = \; - \f{ \mf{u}^{\prime\prime}_1(k_0) }{2 \mc{P}^{\prime}(k_0) } \cdot \Big( y-y_+^{(\ups)} \Big) \cdot \Big( y-y_-^{(\ups)} \Big) \;.
\enq
Upon setting $\sg_{\ups}=\e{sgn}\big( \mf{u}^{\prime}_1(k_0) + \ups \op{v} \big)$, one has that
\beq
y_{- \sg_{\ups}}^{(\ups)} \; = \; \f{ \vp_{ \mf{x} }(\bs{z}) }{ \mf{u}^{\prime}_1(k_0) + \ups \op{v} }
\, U\Bigg( - \f{ 2 \mf{u}^{\prime\prime}_1(k_0) \, \vp_{\mf{x}}(\bs{z}) }{ \mc{P}^{\prime}(k_0) \, \big[ \mf{u}^{\prime}_1(k_0) + \ups \op{v} \big]^2 } \Bigg) \;,
\quad U(x) \, = \, 2 \f{ 1 \, - \, \sqrt{ 1 - x } }{ x } \;,
\label{ecriture racine y ups moins sg ups}
\enq
where I remind that $\bs{z}=\big( \bs{z}^{(+)}, \bs{z}^{(-)} \big)$ and $ \vp_{\mf{x} }(\bs{z}) $ is as introduced in \eqref{definition varphi de z et kappa}. Also, it holds
\beqa
y_{ \sg_{\ups}}^{(\ups)} & = & \f{- \mc{P}^{\prime}(k_0) }{ \mf{u}^{\prime\prime}_1(k_0) } \Big( \mf{u}^{\prime}_1(k_0) + \ups \op{v} \Big)
\cdot \Bigg( 1 \, + \, \sqrt{ 1+ \tfrac{ 2 \mf{u}^{\prime\prime}_1(k_0) \vp_{\mf{x}}(\bs{z}) }{ \mc{P}^{\prime}(k_0) [ \mf{u}^{\prime}_1(k_0) + \ups \op{v} ]^2 } } \Bigg) \label{ecriture racine y ups sg ups}\\
& = & \f{- 2 \mc{P}^{\prime}(k_0) }{ \mf{u}^{\prime\prime}_1(k_0) } \Big( \mf{u}^{\prime}_1(k_0) + \ups \op{v} \Big) \cdot \Big( 1 \, + \, \e{O}\big( \de_+ + \de_- +|\mf{x}|\big) \Big) \;.
\nonumber
\eeqa
Note that the expressions \eqref{ecriture racine y ups moins sg ups} and \eqref{ecriture racine y ups sg ups} entail that $y_{\pm}^{(\ups)}$ are functions of the
variable $\bs{z}$ only through the combination $ \vp_{\mf{x} }(\bs{z})$. In the following, unless it will be necessary, this $\bs{z}$-dependence of $y_{\pm}^{(\ups)}$
will be kept implicit.
Let
\beq
\mf{s}= \e{sgn}\Big( - \tfrac{ \mf{u}^{\prime\prime}_1(k_0) }{ \mc{P}^{\prime}(k_0) } \Big) \;.
\label{definition sign mf s}
\enq
One has that $ \sg_{\ups} \, \mf{s} \, y_{\sg_{\ups}}^{(\ups)} \, > \, \sg_{\ups} \, \mf{s} \, y_{-\sg_{\ups}}^{(\ups)} $ so that
\beq
\ba{c|c|c|c|c}
& \sg_{\ups} \mf{s} >0 & \sg_{\ups} \mf{s}<0 \\ \hline
\mf{s} P_{\ups}(\bs{x}) < 0 & \Big] y_{-\sg_{\ups}}^{(\ups)} ; y_{\sg_{\ups}}^{(\ups)} \Big[^{} & \Big] y_{\sg_{\ups}}^{(\ups)} ; y_{-\sg_{\ups}}^{(\ups)} \Big[ \\ \hline
\mf{s} P_{\ups}(\bs{x}) > 0 & \R \setminus \Big[ y_{-\sg_{\ups}}^{(\ups)} ; y_{\sg_{\ups}}^{(\ups)} \Big] & \R \setminus \Big[ y_{\sg_{\ups}}^{(\ups)} ; y_{-\sg_{\ups}}^{(\ups)} \Big] \\ \hline
\ea
\label{ecriture tableau signe de P ups}
\enq
gives the domains, in the $y$ variable and for fixed $\bs{z}$, of positivity and negativity of the polynomials $P_{\ups}(\bs{x})$.
To proceed further, one should distinguish between the two cases where $|\mf{u}_1^{\prime}(k_0)|<\op{v}$ and $|\mf{u}_1^{\prime}(k_0)|>\op{v}$, since their treatment slightly differs.
\subsubsection*{$\bullet$ Joint positivity interval in the $|\mf{u}_1^{\prime}(k_0)|<\op{v}$ regime}
\hspace{4mm} $\circledast$ If $\mf{s}>0$, then by \eqref{ecriture tableau signe de P ups}, one will have
\beqa
P_+(\bs{x}) > 0 \qquad \e{on} \qquad \big]-\infty; y_{-}^{(+)} \, \big[ \cup \big] \, y_{+}^{(+)}; +\infty \big[ & \quad \e{with} \quad &
\left\{ \ba{c} y_{+}^{(+)} >0 \\ y_{-}^{(+)} \, = \, \e{O}\Big(\de_+ + \de_- +|\mf{x}| \Big) \ea \right. \\
P_-(\bs{x}) > 0 \qquad \e{on} \qquad \big]-\infty; y_{-}^{(-)} \, \big[ \cup \big] \, y_{+}^{(-)}; +\infty \big[ & \e{with} & \left\{ \ba{c} y_{-}^{(-)} <0 \\ y_{+}^{(-)} \, = \, \e{O}\Big(\de_+ + \de_- +|\mf{x}| \Big) \ea \right. \;.
\eeqa
Thus, both polynomials $P_{\pm}(\bs{x})$ will be positive on the union of intervals
\beq
\big]-\infty; y_{-}^{(-)} \, \big[ \cup \big] \, y_{+}^{(-)} \, ; \, y_{-}^{(+)} \, \big[ \cup \big] \, y_{+}^{(+)}; +\infty \big[
\enq
where the central interval is present only if the subsidiary consition $ y_{-}^{(+)} \, - \, y_{+}^{(-)} \, > \, 0 $ holds. In fact, this is the sole interval that will be included in the
$y$ integration domain present in $\mc{D}$ \eqref{definition domaine D pour reduction}, \textit{viz}. $\intff{ -\mu(\bs{z}) }{\mu(\bs{z})}$, where $\mu(\bs{z})$
given in \eqref{definition fonction mu de z} is fixed upon choosing the $\bs{z}$ variables and is small enough as in \eqref{ecriture bornes sur mu de z}.
Then, using the local positivity on this interval of the various building blocks present in the factorisation of the polynomials $P_{\ups}$, one gets that, for $\bs{x} \in \mc{D}$,
\bem
\pl{\ups=\pm}{} \bigg\{ \Xi\Big( P_{\ups}(\bs{x}) \Big) \cdot \big[ P_{\ups}(\bs{x}) \big]^{ \mf{d}_{\ups}(\bs{x}) -1 } \bigg\} \; = \; \Xi\Big( \, y_{-}^{(+)} \, - \, y_{+}^{(-)} \Big)
\cdot \bs{1}_{ \big[ \, y_{+}^{(-)} \, ; \, y_{-}^{(+)} \, \big] }(y) \\
\times \pl{\ups = \pm }{} \bigg\{ \tfrac{ - \ups \mf{u}^{\prime\prime}_1(k_0) }{2 \mc{P}^{\prime}(k_0) } \cdot \big(\, y_{\ups}^{(\ups)}- y \big) \bigg\}^{ \mf{d}_{\ups}(\bs{x}) -1 }
\cdot \pl{\ups = \pm }{} \bigg\{ \ups \big( \, y_{-\ups}^{(\ups)}- y \big) \bigg\}^{ \mf{d}_{\ups}(\bs{x}) -1 } \;.
\label{factorisation partie petite et grande factorisation polynome cas mod u lower v}
\end{multline}
\par $\circledast$ If $\mf{s}<0$, then by \eqref{ecriture tableau signe de P ups}, one will have
\beqa
P_+(\bs{x}) > 0 \qquad \e{on} \qquad \big] y_{+}^{(+)} ; y_{-}^{(+)} \, \big[ & \quad \e{with} \quad & \left\{ \ba{c} y_{+}^{(+)} <0 \\ y_{-}^{(+)} \, = \, \e{O}\Big(\de_+ + \de_- +|\mf{x}| \Big) \ea \right. \;, \\
P_-(\bs{x}) > 0 \qquad \e{on} \qquad \big] y_{+}^{(-)} ; y_{-}^{(-)} \, \big[ & \e{with} & \left\{ \ba{c} y_{-}^{(-)} >0 \\ y_{+}^{(-)} \, = \, \e{O}\Big(\de_+ + \de_- +|\mf{x}| \Big) \ea \right. \; .
\eeqa
Thus, both polynomials will be simultaneously positive if any only if $ y_{-}^{(+)} \, - \, y_{+}^{(-)} \, > \, 0 $ and then the interval of joint positivity is
\beq
\big] \, y_{+}^{(-)} \, ; \, y_{-}^{(+)} \, \big[ \;.
\enq
Upon using the local positivity, on this interval, of the various building blocks present in the factorisation of the polynomials $P_{\ups}$, one gets
that the factorisation \eqref{factorisation partie petite et grande factorisation polynome cas mod u lower v} also holds in the present case.
\subsubsection*{$\bullet$ Joint positivity interval in the $|\mf{u}_1^{\prime}(k_0)|>\op{v}$ regime}
In this regime, one has that
\beq
\sg_{\ups} \, = \, \e{sgn}\big( \mf{u}^{\prime}_{1}(k_0)+\ups \op{v} \big) = \vsg \;,
\label{definition signe varsigma}
\enq
\textit{i.e.} $\sg_{\ups}$ does not depend on $\ups\in \{\pm 1\}$.
\par $\circledast$ If $\mf{s}>0$, then by \eqref{ecriture tableau signe de P ups}, one will have
\beq
P_{\pm}(\bs{x}) > 0 \qquad \e{on} \qquad \big]-\infty; y_{-}^{(\pm)} \, \big[ \cup \big] \, y_{+}^{(\pm)}; +\infty \big[
\enq
where, for some $c>0$
\beq
\ba{cccc} y_{+}^{(\pm)} >c> 0 &, \quad y_{-}^{(\pm)} \, = \, \e{O}\Big(\de_+ + \de_- +|\mf{x}| \Big) & \e{if} & \vsg = + \vspace{2mm} \; \\
y_{-}^{(\pm)} <-c<0 &, \quad y_{+}^{(\pm)} \, = \, \e{O}\Big(\de_+ + \de_- +|\mf{x}| \Big) & \e{if} & \vsg = - \ea \;.
\enq
Thus, the polynomials $P_{\pm}(\bs{x})$ will be simultaneously positive on the union of intervals
\beq
\Big]-\infty; \e{min}\Big(y_{-}^{(+)}, y_{-}^{(-)} \Big) \, \Big[ \cup \Big] \, \e{max}\Big(y_{+}^{(+)}, y_{+}^{(-)} \Big) ; +\infty \Big[ \;.
\enq
Indeed, this is a consequence of the fact that the roots $y^{(\pm)}_{\vsg}$ have both "large" absolute value -in respect to $\mu(\bs{z})$ and this uniformly in $\bs{z}$ by virtue of \eqref{ecriture bornes sur mu de z}-,
whereas the roots $y^{(\pm)}_{-\vsg}$ are both close to the origin.
Thus, taken that the function $\mu(\bs{z})$ \eqref{definition fonction mu de z} delimiting the $y$ integration domain in $\mc{D}$ \eqref{definition domaine D pour reduction} is small enough \eqref{ecriture bornes sur mu de z},
for any fixed $\bs{z}$, the interval $\intff{ - \mu(\bs{z}) }{ \mu(\bs{z}) }$ defining the $y$-integration in \eqref{reduction finale de la partie singuliere vers integrale modele} will reduce to
\beq
J_{\vsg}(\bs{z}) \; = \; \vsg \Big] - \vsg \mu(\bs{z}) \, ; \, \vsg \e{min}\big(\vsg \, y_{-\vsg}^{(+)} , \vsg\, y_{-\vsg}^{(-)} \big) \Big[
\label{definition intervalle J vsg}
\enq
in which the prefactor $\vsg$ indicates the orientation of the interval.
Then, using the local positivity, on this interval, of the various building blocks present in the factorisation of the polynomials $P_{\ups}$, one gets that
\beq
\pl{\ups=\pm}{} \bigg\{ \Xi\Big( P_{\ups}(\bs{x}) \Big) \cdot \big[ P_{\ups}(\bs{x}) \big]^{ \mf{d}_{\ups}(\bs{x}) -1 } \bigg\} \; = \;
\bs{1}_{ J_{\vsg}(\bs{z}) }(y)
\cdot \pl{\ups = \pm }{} \bigg\{ \tfrac{ - \vsg \mf{u}^{\prime\prime}_1(k_0) }{2 \mc{P}^{\prime}(k_0) } \, \big(\, y_{\vsg}^{(\ups)}- y \big) \bigg\}^{ \mf{d}_{\ups}(\bs{x}) -1 }
\cdot \pl{\ups = \pm }{} \bigg\{ \vsg \big( \, y_{-\vsg}^{(\ups)}- y \big) \bigg\}^{ \mf{d}_{\ups}(\bs{x}) -1 } \;.
\label{factorisation partie petite et grande factorisation polynome cas mod u greater v}
\enq
\par $\circledast$ If $\mf{s}<0$, then by \eqref{ecriture tableau signe de P ups}, one will have
\beqa
P_{\ups}(\bs{x}) > 0 \qquad \e{on} \qquad \big] y_{+}^{(\ups)} ; y_{-}^{(\ups)} \, \big[
\eeqa
where, for some $c>0$,
\beq
\ba{cccc} y_{+}^{(\ups)} < -c < 0 &, \quad y_{-}^{(\ups)} \, = \, \e{O}\Big(\de_+ + \de_- +|\mf{x}| \Big) & \e{if} & \vsg = + \vspace{2mm} \\
y_{-}^{(\ups)} > c >0 &, \quad y_{+}^{(\ups)} \, = \, \e{O}\Big(\de_+ + \de_- +|\mf{x}| \Big) & \e{if} & \vsg = - \ea \;.
\enq
Thus, both polynomials will be simultaneously positive only on the interval
\beq
\Big] \, \e{max}\big(y_{+}^{(+)}, y_{+}^{(-)} \big) \, ; \, \e{min}\big(y_{-}^{(+)}, y_{-}^{(-)} \big)\, \Big[ \;.
\enq
Since $\mu(\bs{z})$ is taken small enough, \textit{c.f.} \eqref{ecriture bornes sur mu de z} and in particular such that $ 0 < \mu(\bs{z}) < |y_{\vsg}^{(\ups)}| $, $\ups=\pm$, one will have that
the presence of Heaviside functions of $P_{\ups}(\bs{x})$ will, effectively, result in a reduction of the $y$-integration domain $\intff{ - \mu(\bs{z}) }{ \mu(\bs{z}) }$ in
$\mc{D}$ \eqref{definition domaine D pour reduction} to the interval $J_{\vsg}(\bs{z})$ already introduced in \eqref{definition intervalle J vsg}.
Furthermore, using the local positivity on this interval of the various building blocks present in the factorisation of the polynomials $P_{\ups}$,
the factorisation \eqref{factorisation partie petite et grande factorisation polynome cas mod u greater v} also holds in the present case.
\subsubsection*{$\bullet$ Canonical form of $\mc{I}_{\e{sg}}^{(\sslash)}(\mf{x}) $ in the $|\mf{u}_1^{\prime}(k_0)|<\op{v}$ regime}
The factorisation \eqref{factorisation partie petite et grande factorisation polynome cas mod u lower v} entails that, irrespectively of the value of $\mf{s}$ introduced in \eqref{definition sign mf s},
$\mc{I}_{\e{sg}}^{(\sslash)}(\mf{x}) $ as given in \eqref{reduction finale de la partie singuliere vers integrale modele} now takes the form
\beq
\mc{I}_{\e{sg}}^{(\sslash)}(\mf{x}) \; = \; \pl{ \ups = \pm }{} \bigg\{ \Int{ -\sqrt{\de_{\ups}} }{ \sqrt{\de_{\ups}} } \hspace{-2mm} \dd^{m_{\ups}}z^{(\ups)} \bigg\}
\Int{ y_{+}^{(-)} }{ y_{-}^{(+)} } \dd y \cdot \Xi\Big( y_{-}^{(+)} \, - \, y_{+}^{(-)} \Big) \cdot
\, \msc{F}^{(1)}(\bs{x}) \cdot \pl{\ups = \pm }{} \bigg\{ \ups \big(\, y_{-\ups}^{(\ups)}- y \big) \bigg\}^{ \mf{d}_{\ups}(\bs{x}) -1 } \;,
\label{ecriture integrale I sg parallele cas u lower v 1ere reduction}
\enq
where $\bs{x}$ is parameterised in terms of the integration variables as in \eqref{ecriture effet transformation matrice M} and, for short,
\beq
\msc{F}^{(1)}(\bs{x}) \, = \, \msc{F} (\bs{x}) \cdot \pl{\ups = \pm }{} \bigg\{ \tfrac{ - \ups \mf{u}^{\prime\prime}_1(k_0) }{2 \mc{P}^{\prime}(k_0) } \big(\, y_{\ups}^{(\ups)}- y \big) \bigg\}^{ \mf{d}_{\ups}(\bs{x}) -1 } \;.
\enq
Note that the integration domain for the $\bs{z}^{(\pm)}$ variables is symmetric. Hence, only the totally even part of the integrand in respect to these variables does
contribute to the value of $\mc{I}_{\e{sg}}^{(\sslash)}(\mf{x})$. Hence, one has
\beq
\mc{I}_{\e{sg}}^{(\sslash)}(\mf{x}) \; = \; \pl{ \ups = \pm }{} \bigg\{ \Int{ -\sqrt{\de_{\ups}} }{ \sqrt{\de_{\ups}} } \hspace{-2mm} \dd^{m_{\ups}}z^{(\ups)} \bigg\}
\Int{ y_{+}^{(-)} }{ y_{-}^{(+)} } \dd y \cdot \Xi\Big( y_{-}^{(+)} \, - \, y_{+}^{(-)} \Big) \cdot
\, \bigg[ \msc{F}^{(1)}(\bs{x}) \cdot \pl{\ups = \pm }{} \bigg\{ \ups \big(\, y_{-\ups}^{(\ups)}- y \big) \bigg\}^{ \mf{d}_{\ups}(\bs{x}) -1 } \bigg]_{(\bs{z}^{(+)},\bs{z}^{(-)})-\e{even} }\;,
\enq
where $ \big[ \cdot \big]_{(\bs{z}^{(+)},\bs{z}^{(-)})-\e{even} }$ stands for the totally even part of a function in respect to the mentioned variables
\beq
\Big[ g(\bs{z}, \bs{v}) \Big]_{ \bs{z}-\e{even} } \; = \; \f{1}{ 2^{d} } \sul{ \substack{ \eps_a=\pm \\ a=1,\dots, d } }{ } g(\bs{z}^{(\eps)}, \bs{v})
\qquad \e{with} \qquad
\bs{z}^{(\eps)}\; = \; \big(\eps_1 z_1, \dots, \eps_d z_d \big) \;.
\enq
At this stage, one observes that
\beq
y_{-}^{(+)} \, - \, y_{+}^{(-)} \, = \, \f{ 2 \op{v} \vp_{\mf{x}}(\bs{z}) }{ \op{v}^2 - \big(\mf{u}_{1}^{\prime}(k_0) \big)^2 } F\Big( \tfrac{ - 2 \mf{u}^{\prime\prime}_1(k_0) }{ \mc{P}^{\prime}(k_0) } \vp_{\mf{x}}(\bs{z}) \Big)
\label{ecriture forme explicite difference y+ moins y-}
\enq
where $F$ is as defined in \eqref{definition fonction G pour difference des racines de P ups}. Note that the series defining $F$ is convergent since $\vp_{\mf{x}}(\bs{z}) = \e{O}\Big( \de_+ + \de_- + |\mf{x}| \Big)$
and $\de_{\pm}, |\mf{x}|$ are all taken small enough.
Furthermore, since $F(0)=1$, the estimate on $\vp_{\mf{x}}(\bs{z})$ ensures that the $F$-dependent term in \eqref{ecriture forme explicite difference y+ moins y-} will be bounded from below by a strictly positive constant,
this throughout the whole integration domain $\mc{D}$.
Further, setting,
\beq
\bs{t}=\Big( t_0, \bs{z} \Big) \quad \e{with} \quad \left\{ \ba{c}
\bs{z} \; = \; \Big( \bs{z}^{(+)}, \bs{z}^{(-)} \Big) \in \Big[-\sqrt{\de_+} ; \sqrt{\de_+} \Big]^{m_+}\!\times \Big[-\sqrt{\de_-} ; \sqrt{\de_-} \Big]^{m_-} \vspace{2mm} \\
\quad t_0 \, = \, y_{+}^{(-)} \, + \; t \cdot \big[\, y_{-}^{(+)} \, - \, y_{+}^{(-)} \big] \ea \right.
\label{definition variable integration t}
\enq
entails that
\beq
P_{\ups}(\bs{t}) \;= \; \vp_{\mf{x}}\big( \bs{z} \big) \cdot F_{\ups} \Big( \vp_{\mf{x}}\big( \bs{z} \big), t_0 \Big) \cdot
\left\{ \ba{cc} (1-t) & \ups= + \\
t & \ups= - \ea \right.
\enq
with
\beq
F_{\ups} \Big( \vp_{\mf{x}}\big( \bs{z} \big) , t_0 \Big) \; = \; \f{ - \ups \mf{u}^{\prime\prime}_1(k_0) \, \op{v} }{ \mc{P}^{\prime}(k_0) \big[ \op{v}^2 - \big(\mf{u}_{1}^{\prime}(k_0) \big)^2 \big] }
F\Big( \tfrac{ - 2 \mf{u}^{\prime\prime}_1(k_0) }{ \mc{P}^{\prime}(k_0) } \vp_{\mf{x}}(\bs{z}) \Big)
\cdot \Big( y_{\ups}^{(\ups)}- y_{-}^{(+)} \, - \; t \cdot \big[\, y_{-}^{(+)} \, - \, y_{+}^{(-)} \big] \Big) \;,
\label{definition G ups pour prefacteur f ups}
\enq
and $t_0$ as in \eqref{definition variable integration t}. Note that $F_{\ups}$ is indeed a sole function of $\vp_{\mf{x}}\big( \bs{z} \big)$ and $t_0$ since the roots $y_{\pm}^{(\ups)}$ only depend on $\vp_{\mf{x}}\big( \bs{z} \big)$, \textit{c.f.}
\eqref{ecriture racine y ups moins sg ups} and \eqref{ecriture racine y ups sg ups}.
Thus, the change of variables
\beq
y \, = \, y_{_-}^{(+)} \, + \; t \cdot \big[\, y_{-}^{(+)} \, - \, y_{+}^{(-)} \big]
\enq
recasts $\mc{I}_{\e{sg}}^{(\sslash)}(\mf{x})$ in the form
\beq
\mc{I}_{\e{sg}}^{(\sslash)}(\mf{x}) \; = \; \pl{ \ups = \pm }{} \bigg\{ \Int{ -\sqrt{\de_{\ups}} }{ \sqrt{\de_{\ups}} } \hspace{-2mm} \dd^{m_{\ups}}z^{(\ups)} \bigg\}
\Int{ 0 }{ 1 } \dd t \, \bigg[ (1-t)^{ \mf{d}_{+}(\bs{t}) -1 }\, t^{ \mf{d}_{-}(\bs{t}) -1 }
\, \msc{F}^{(2)}(\bs{t}) \cdot \Xi\Big( \vp_{\mf{x}}(\bs{z}) \Big) \cdot \Big[ \vp_{\mf{x}}(\bs{z}) \Big]^{ \mf{d}_{+}(\bs{t})+\mf{d}_{-}(\bs{t}) -1 } \bigg]_{\bs{z}-\e{even}}
\label{ecriture integrale I sg parallele cas u lower v 2eme reduction}
\enq
where $\bs{t}$ is as in \eqref{definition variable integration t} and
\beq
\msc{F}^{(2)}(\bs{t}) \, = \, \msc{F} (\bs{t}) \cdot
\f{ 2 \op{v} }{ \op{v}^2 - \big(\mf{u}_{1}^{\prime}(k_0) \big)^2 } F\Big( \tfrac{ - 2 \mf{u}^{\prime\prime}_1(k_0) }{ \mc{P}^{\prime}(k_0) } \vp_{\mf{x}}(\bs{z}) \Big)
\cdot \pl{\ups = \pm }{} \Big[ F_{\ups} \Big( \vp_{\mf{x}}\big( \bs{z} \big), t_0 \Big) \Big]^{ \mf{d}_{\ups}(\bs{t}) -1 } \;.
\enq
At this stage, one decomposes the integral into domains where a square root change of variables is well defined:
\beq
\Big[- \sqrt{\de_{\ups}} \, ;\, \sqrt{\de_{\ups}} \, \Big]^{ m_{\ups} } \, = \,
\bigsqcup_{ \bs{\eps}^{(\ups)} \in \{ \pm 1 \}^{ m_{\ups} } } \Bigg\{ \pl{a=1}{m_{\ups}} \Big\{ \eps_{a}^{(\ups)} \Big[ 0 ; \eps_a^{(\ups)} \sqrt{\de_{\ups}} \Big] \Big\} \Bigg\}
\qquad \e{with} \qquad \bs{\eps}^{(\ups)} \; = \; \big( \eps^{ (\ups) }_{ 1 }, \dots, \eps^{ (\ups) }_{ m_{\ups} } \big)
\enq
in which the sign prefactor in front of each interval indicates its orientation. Then, in each of the sets building up the partition, one sets
\beq
z_a^{(\ups)} \, = \, \eps_{a}^{(\ups)} \cdot \big[ w_a^{(\ups)} \big]^{\f{1}{2}} \qquad a=1,\dots, m_{\ups} \;.
\enq
This yields
\beq
\mc{I}_{\e{sg}}^{(\sslash)}(\mf{x}) \; = \; \sul{ \substack{ \bs{\eps}^{(\ups)} \in \{ \pm 1 \}^{ m_{\ups} } \\ \ups = \pm } }{} \mc{J}_{\bs{\eps}}\Big[\msc{F}^{(3)}, \mf{d}_{+}-1, \mf{d}_{-}-1 \Big]
\label{ecriture I sg slash decompose sur les divers eps pm}
\enq
where the building block integral is defined as
\bem
\mc{J}_{\bs{\eps}}\Big[\mc{F} , A , B \Big] \; = \; \pl{\ups= \pm }{}\Bigg\{ \Int{ 0 }{ \de_{\ups} } \f{ \dd^{ m_{\ups} } w^{(\ups)} }{ \sqrt{ w_a^{(\ups)} } } \Bigg\} \Int{0}{1}\dd t \;
\bigg[ \mc{F}\Big( \bs{u}^{(\bs{\eps})} , (1-t)\vp_{\mf{x}}\big(\bs{u}^{(\bs{\eps})}_{\bs{w}} \big) , t \vp_{\mf{x}}\big(\bs{u}^{(\bs{\eps})}_{\bs{w}} \big) \Big) \\
\times (1-t)^{ A(\bs{u}^{(\bs{\eps})}) } \cdot t^{ B(\bs{u}^{(\bs{\eps})}) } \cdot
\Xi\big[ \vp_{\mf{x}}\big(\bs{u}^{(\bs{\eps})}_{\bs{w}} \big) \big] \cdot
\big[ \vp_{\mf{x}}\big(\bs{u}^{(\bs{\eps})}_{\bs{w}} \big) \big]^{ A(\bs{u}^{(\bs{\eps})})+B(\bs{u}^{(\bs{\eps})})+1 } \bigg]_{\bs{u}^{(\bs{\eps})}_{\bs{w}}-\e{even} }\;.
\label{ecriture integrale modele pour eps decomposition de l'integrale I sg u lower v}
\end{multline}
Above, it is undercurrent
\beq
\bs{u}^{(\bs{\eps})}\, = \, \bigg(u_0^{(\bs{\eps})} ,
\bs{u}^{(\bs{\eps})}_{\bs{w}}\Big)
\quad \e{with} \quad
\bs{u}^{(\bs{\eps})}_{\bs{w}}\, = \, \Big( \bs{u}^{(\bs{\eps};+)}_{\bs{w}}, \bs{u}^{(\bs{\eps};-)}_{\bs{w}} \Big) \;, \quad
u_0^{(\bs{\eps})} \, = \, y_{-}^{(+)}\Big(\vp_{\mf{x}}(\bs{u}_{\bs{w}}^{(\bs{\eps})}) \Big) \, + \; t \cdot \Big[\, y_{-}^{(+)}\Big(\vp_{\mf{x}}(\bs{u}_{\bs{w}}^{(\bs{\eps})}) \Big)
\, - \, y_{+}^{(-)}\Big(\vp_{\mf{x}}(\bs{u}_{\bs{w}}^{(\bs{\eps})}) \Big) \Big]
\enq
and, finally,
\beq
\bs{u}^{(\bs{\eps};\ups)}_{\bs{w}} \, = \, \Big( \eps^{ (\ups) }_{ 1 } \big[ w_1^{(\ups)} \big]^{\f{1}{2}} , \dots, \eps^{ (\ups) }_{ m_{\ups} } \big[ w_{ m_{\ups} }^{(\ups)} \big]^{\f{1}{2}} \Big) \;.
\enq
Here, I have made explicit the fact that the functions $y_{\mp}^{(\pm)}$ only depend on the $\bs{u}^{(\bs{\eps})}_{\bs{w}}$ integration variables through the function $\vp_{\mf{x}}(\bs{u}_{\bs{w}}^{(\bs{\eps})})$.
In fact, after this change of variables, it holds
\beq
\vp_{\mf{x}}(\bs{u}_{\bs{w}}^{(\bs{\eps})}) \; = \; \mf{x} \; + \; \sul{a=1}{m_+} w_a^{(+)} \; - \; \sul{a=1}{m_-} w_a^{(-)} \;.
\enq
The integrand appearing in \eqref{ecriture I sg slash decompose sur les divers eps pm} takes the form
\bem
\msc{F}^{(3)}\Big( \bs{u}^{(\bs{\eps})} , (1-t)\vp_{\mf{x}}\big(\bs{u}^{(\bs{\eps})}_{\bs{w}} \big) , t \vp_{\mf{x}}\big(\bs{u}^{(\bs{\eps})}_{\bs{w}} \big) \Big) \, = \, \,
\msc{G}\Big( \Psi_{\op{M}}(\bs{u}^{(\bs{\eps})}) \, , \, (1-t) \cdot \wt{\mf{f}_{+}}( \bs{u}^{(\bs{\eps})}) \, \vp_{\mf{x}}\big( \bs{u}^{(\bs{\eps})}_{\bs{w}} \big)
\, , \, t \cdot \wt{\mf{f}_{-}}( \bs{u}^{(\bs{\eps})})\, \vp_{\mf{x}}\big( \bs{u}^{(\bs{\eps})}_{\bs{w}} \big) \Big) \hspace{2cm} \\
\times \f{ 2^{2-\ov{\bs{n}}_{\ell}} \op{v} }{ \op{v}^2 - \big(\mf{u}_{1}^{\prime}(k_0) \big)^2 } F\Big( \tfrac{ - 2 \mf{u}^{\prime\prime}_1(k_0) }{ \mc{P}^{\prime}(k_0) } \vp_{\mf{x}}(\bs{u}^{(\bs{\eps})}_{\bs{w}} ) \Big) \cdot
\Big(V \vp^{(\sslash)} \vp^{(\e{sg})} \Big)\Big( \Psi_{\op{M}}(\bs{u}^{(\bs{\eps})}) \Big) \\
\times \big| \det D_{\bs{u}^{(\bs{\eps})}} \Psi_{\op{M}} \big|
\cdot \pl{ \ups = \pm }{} \Big\{ \wt{\mf{f}}_{\ups}\big( \bs{u}^{(\bs{\eps})} \big) \Big\}^{ \mf{d}_{\ups}(\bs{u}^{(\bs{\eps})}) -1 } \;,
\end{multline}
where
\beq
\wt{\mf{f}}_{\ups}\big( \bs{u}^{(\bs{\eps})} \big) \, = \,
\mf{f}_{\ups}\big( \mf{x}; \op{M}\cdot \bs{u}^{(\bs{\eps})} \big) \cdot F_{\ups} \Big( \vp_{\mf{x}}\big( \bs{u}^{(\bs{\eps})}_{\bs{w}} \big) , u_0^{(\bs{\eps})} \Big) \;,
\enq
and I remind that $\ov{\bs{n}}_{\ell}\, = \, \sul{r=1}{\ell} n_r$.
Here, one observes that
\beq
\mf{d}_{\ups}(\bs{t}) = \De_{\ups}(\bs{t}(k_0)) + \e{O}\Big( \sqrt{\de_+} + \sqrt{\de_-} + |\mf{x}| \Big) \;.
\enq
By hypothesis one has $\De_{\ups}(\bs{t}(k_0)) >0$ so that reducing $\de_{\pm}$ and $|\mf{x}|$ if need be, one gets that $\mf{d}_{\pm}>0$
throughout the integration domain.
The expansion \eqref{ecriture I sg slash decompose sur les divers eps pm} of $\mc{I}_{\e{sg}}^{(\sslash)}(\mf{x}) $
decomposes this integral into a sum of elementary integrals \eqref{ecriture integrale modele pour eps decomposition de l'integrale I sg u lower v} whose
$\mf{x}\tend 0$ asymptotic behaviour is analysed in Lemma \ref{Lemme integrale beta multi-dim auxiliaire locale cas u less than v}. Also,
the $L^1$-nature of the integrand is part of the conclusions of that lemma. Moreover, upon invoking Lemma \ref{Lemme VdM local expansion} so as to access to the small $\norm{ \bs{u}^{(\bs{\eps})} }$ expansion of $ \msc{F}^{(3)}$,
Lemma \ref{Lemme integrale beta multi-dim auxiliaire locale cas u less than v}, specialised to the function $ \msc{F}^{(3)}$, ensures that there exists a smooth function $ \mf{x} \tend \mf{ R }_{ \bs{\eps} }( \mf{x} ) $
around $\mf{x}=0$ such that
\beq
\mc{J}_{\bs{\eps}}\Big[\msc{F}^{(3)}, \mf{d}_{+}-1, \mf{d}_{-}-1 \Big] \; = \; \De^{(0)}_{+}\,\De^{(0)}_{-}\,\msc{G}^{(1)}\big( \bs{t}(k_0) \big) \mc{J}_{\bs{\eps}}\Big[\msc{F}_{\e{eff}}, \De^{(0)}_{+}-1, \De^{(0)}_{-}-1 \Big]
\; + \; \e{O}\Big( |\mf{x}|^{\vth + 1-\tau } \Big) \; + \; \mf{R}_{ \bs{\eps} }(\mf{x})
\enq
where $\De^{(0)}_{\ups}= \De_{\ups}\big( \bs{t}(k_0) \big)$, $\vth$ is as defined in \eqref{definition de cal theta 0}, $\msc{G}^{(1)}$ corresponds to the first term of the expansion of
$\msc{G}$ as given in \eqref{ecriture decomposition smooth class K}, and
\bem
\msc{F}_{\e{eff}} \big( \bs{u}^{(\bs{\eps})} , x, y \big) \; = \; \exp\bigg\{ -\Big( \op{M}\, \bs{u}^{(\bs{\eps})} , \op{D} \cdot \op{M} \, \bs{u}^{(\bs{\eps})} \Big) \bigg\} \cdot
\Big(V \vp_{\e{eff}}^{(\sslash)} \vp_{\e{eff}}^{(\e{sg})} \Big)\Big( \op{M} \, \bs{u}^{(\bs{\eps})} \Big) \cdot |\det[\op{M}]| \\
\times \f{ 2^{2-\ov{\bs{n}}_{\ell}} \op{v} }{ \op{v}^2 - \big(\mf{u}_{1}^{\prime}(k_0) \big)^2 } F\Big( \tfrac{ - 2 \mf{u}^{\prime\prime}_1(k_0) }{ \mc{P}^{\prime}(k_0) } \vp_{\mf{x}}(\bs{z}) \Big)
\cdot \pl{ \ups = \pm }{} \Big\{ F_{\ups}\big( \bs{u}^{(\bs{\eps})} , u_0^{(\bs{\eps})} \big) \Big\}^{ \De^{(0)}_{\ups} - 1 } \;.
\end{multline}
Above, $\op{M}$ is as introduced in \eqref{ecriture effet transformation matrice M} while the diagonal matrix $\op{D}$ defining the Gaussian weight reads
\beq
\op{D} \, = \, \left( \ba{ccc} |\mf{u}^{\prime\prime}_{1}(t_1(k_0) ) | \op{I}_{n_1} & 0 & 0 \\
0 & \ddots & 0 \\
0 & 0 & |\mf{u}^{\prime\prime}_{\ell}(t_{\ell}(k_0) ) | \op{I}_{n_{\ell}} \ea \right)
\label{definition matrice D pour poid gaussien integrale modele}
\enq
In particular, $\msc{F}_{\e{eff}}$ is $x,y$ independent. Furthermore, $\vp_{\e{eff}}^{(\sslash)}$ and $\vp_{\e{eff}}^{(\e{sg})}$ are as appearing in \eqref{definition fcts approx unite sg et sslash integrale modele}
of Proposition \ref{Proposition reduction vers voisinage singularite integrale modele}. Namely, they are smooth on $\pl{r=1}{\ell}\R^{n_r}$ and such that
\beqa
0 \leq \vp_{\e{eff}}^{(\e{sg})} \leq 1 & \quad \vp_{\e{eff}}^{(\e{sg})}=1 \quad \e{on} \quad \ov{ \mc{D}^{(\e{eff})}_{\eta^{\prime}} } & \quad \vp_{\e{eff}}^{(\e{sg})}=0 \quad \e{on} \quad
\pl{r=1}{\ell}\R^{n_r} \setminus \mc{D}^{(\e{eff})}_{2\eta^{\prime}} \nonumber \\
0 \leq \vp_{\e{eff}}^{(\sslash)} \leq 1 & \quad \vp_{\e{eff}}^{(\sslash)}=1 \quad \e{on} \quad \ov{ \mc{D}^{(\sslash;\e{eff})}_{\frac{1}{2} \eta^{\prime} } } & \quad \vp_{\e{eff}}^{(\sslash)}=0 \quad \e{on} \quad
\pl{r=1}{\ell}\R^{n_r} \setminus \mc{D}^{(\sslash;\e{eff})}_{ \eta^{\prime} } \;.
\label{definition des fcts vp eff slash et sg}
\eeqa
The domain $\mc{D}^{(\e{eff})}_{\eta^{\prime}}$, resp. $ \mc{D}^{(\sslash;\e{eff})}_{ \eta^{\prime} } $, is as defined in \eqref{definition du domaine D eff}, resp. \eqref{definition domaine De eff sslash}.
Thus, by performing backwards, on the level of $\mc{J}_{\bs{\eps}}\Big[\msc{F}_{\e{eff}}, \De^{(0)}_{+}-1, \De^{(0)}_{-}-1 \Big]$,
all the transformations that were carried out starting from \eqref{reduction finale de la partie singuliere vers integrale modele}, one gets that
there exists a smooth function $\mf{x} \tend \mf{R}(\mf{x})$
around $\mf{x}=0$ such that
\beq
\mc{I}_{\e{sg}}^{(\sslash)}(\mf{x}) \; = \; \De^{(0)}_{+}\,\De^{(0)}_{-}\,\msc{G}^{(1)}\big( \bs{t}(k_0) \big) \cdot \mc{J}_{\e{sg};\e{eff}}^{(\sslash)}(\mf{x})
\; + \; \e{O}\Big( |\mf{x}|^{\vth + 1-\tau } \Big) \; + \; \mf{R} (\mf{x})
\label{ecriture reduction de I sg slash vers J sg slash}
\enq
in which
\beq
\mc{J}_{\e{sg};\e{eff}}^{(\sslash)}(\mf{x}) \; = \; \Int{ \mc{D}^{(\e{eff})}_{2\eta^{\prime}} }{} \hspace{-2mm} \dd \bs{x} \; \ex { - \, ( \bs{x} , \op{D} \bs{x} ) }
\cdot \Big(V \vp_{\e{eff}}^{(\sslash)} \vp_{\e{eff}}^{(\e{sg})} \Big)\big(\bs{x} \big) \cdot
\pl{\ups= \pm }{} \bigg\{ \Xi\big[\mf{x} + \wt{z}_{\ups}(\bs{x}) \big] \cdot \big[\mf{x} + \wt{z}_{\ups}(\bs{x}) \big]^{ \De^{(0)}_{\ups}-1} \bigg\} \;.
\enq
Now, by virtue of Proposition \ref{Proposition reduction vers voisinage singularite integrale modele} there exists a smooth function $\wt{\mf{R}} (\mf{x})$ such that
\beq
\mc{J}_{\e{sg};\e{eff}}^{(\sslash)}(\mf{x}) \; = \;\wt{\mf{R}} (\mf{x}) \, + \,
\Int{ \pl{r=1}{\ell} \R^{n_r} }{} \hspace{-2mm} \dd \bs{x} \;
\f{ \ex { - \, ( \bs{x} , \bs{x} ) } \cdot V \big(\bs{x} \big) }{ \pl{r=1}{\ell} \big| \mf{u}^{\prime\prime}_r(t_r(k_0)) \big|^{\frac{1}{2} n_r^2 } }\cdot
\pl{\ups= \pm }{} \bigg\{ \Xi\big[\mf{x} + z_{\ups}(\bs{x}) \big] \cdot \big[\mf{x} + z_{\ups}(\bs{x}) \big]^{ \De^{(0)}_{\ups}-1} \bigg\}
\label{reecriture integrale sg locale modele et son extension a R ov n ell}
\enq
in which $z_{\ups}$ is as defined in \eqref{definition z ups etape 1} where the below identification of parameters has been made
\beq
\veps_{r} \, = \, - \zeta_r \e{sgn}\Big( \mf{u}^{\prime\prime}_r(t_r(k_0)) \Big) \quad , \quad
\xi_{r} \, = \, \big| \mf{u}^{\prime\prime}_r(t_r(k_0)) \big|^{-\frac{1}{2}} \quad , \quad
\op{u} \, = \, \mf{u}^{\prime}_1(k_0) \;.
\enq
By tracking the previous transformations backwards, one gets that there exists a smooth function $ \check{\mf{R}}$ in the vicinity of the origin
\bem
\mc{I}(\mf{x}) \, = \, \Int{ \pl{r=1}{\ell} \R^{n_r} }{} \hspace{-2mm} \dd \bs{x} \;
\f{ \ex { - \, ( \bs{x} , \bs{x} ) } \cdot V \big(\bs{x} \big) }{ \pl{r=1}{\ell} \big| \mf{u}^{\prime\prime}_r(t_r(k_0)) \big|^{\frac{1}{2} n_r^2 } }\cdot
\pl{\ups= \pm }{} \bigg\{ \Xi\big[\mf{x} + z_{\ups}(\bs{x}) \big] \cdot \big[\mf{x} + z_{\ups}(\bs{x}) \big]^{ \De^{(0)}_{\ups}-1} \bigg\} \\
\; + \, \check{\mf{R}} (\mf{x}) \; + \; \e{O}\Big( |\mf{x}|^{\vth + 1-\tau } \Big) \; .
\end{multline}
Then, it remains to apply Proposition \ref{Proposition asymptotique integrale modele} so as to get the form of the $\mf{x}\tend 0$ asymptotic expansion of
the multiple integral appearing above, what yields the claimed form of the $\mf{x}\tend 0$ asymptotics of $\mc{I}(\mf{x})$ in the regime $ | \mf{u}^{\prime}_1(k_0) | \, < \, \op{v}$.
\subsubsection*{$\bullet$ Canonical form of $\mc{I}_{\e{sg}}^{(\sslash)}(\mf{x}) $ in the $|\mf{u}_1^{\prime}(k_0)| > \op{v}$ regime}
The factorised form of the products of polynomials $P_{\ups}(\bs{x})$ given in \eqref{factorisation partie petite et grande factorisation polynome cas mod u greater v}
entails that $\mc{I}_{\e{sg}}^{(\sslash)}(\mf{x}) $ takes the form
\beq
\mc{I}_{\e{sg}}^{(\sslash)}(\mf{x}) \; = \; \pl{ \ups = \pm }{} \bigg\{ \Int{ -\sqrt{\de_{\ups}} }{ \sqrt{\de_{\ups}} } \hspace{-2mm} \dd^{m_{\ups}}z^{(\ups)} \bigg\}
\Int{ J_{\vsg}(\bs{z}) }{ } \dd y
\, \msc{F}^{(1)}(\bs{x}) \cdot \pl{\ups = \pm }{} \bigg\{ \vsg \big(\, y_{-\vsg}^{(\ups)}- y \big) \bigg\}^{ \mf{d}_{\ups}(\bs{x}) -1 }
\label{ecriture integrale I sg parallele cas u bigger v 1ere reduction}
\enq
where $\vsg$ has been introduced, $J_{\vsg}(\bs{z})$ in \eqref{definition intervalle J vsg}, the argument $\bs{x}$ is expressed in terms of the integration variables $y, \bs{z}^{(\pm)}$
as in \eqref{ecriture effet transformation matrice M}, and, for short, I agree upon
\beq
\msc{F}^{(1)}(\bs{x}) \, = \, \msc{F} (\bs{x}) \cdot \pl{\ups = \pm }{} \bigg\{ \tfrac{ - \vsg \mf{u}^{\prime\prime}_1(k_0) }{2 \mc{P}^{\prime}(k_0) } \big(\, y_{\vsg}^{(\ups)}- y \big) \bigg\}^{ \mf{d}_{\ups}(\bs{x}) -1 } \;.
\enq
As earlier on, the symmetry of the $(\bs{z}^{(+)}, \bs{z}^{(-)})$ integration domain entails that
\beq
\mc{I}_{\e{sg}}^{(\sslash)}(\mf{x}) \; = \; \pl{ \ups = \pm }{} \bigg\{ \Int{ -\sqrt{\de_{\ups}} }{ \sqrt{\de_{\ups}} } \hspace{-2mm} \dd^{m_{\ups}}z^{(\ups)} \bigg\}
\Int{ J_{\vsg}(\bs{z}) }{ } \dd y
\, \bigg[ \msc{F}^{(1)}(\bs{x}) \cdot \pl{\ups = \pm }{} \bigg\{ \vsg \big(\, y_{-\vsg}^{(\ups)}- y \big) \bigg\}^{ \mf{d}_{\ups}(\bs{x}) -1 } \bigg]_{ \bs{z}-\e{even} }
\enq
where the $\bs{z}$-even part of a function is as defined in \eqref{definition even part of a function}.
At this stage, one observes that
\beq
\vsg y_{-\vsg}^{(+)} \, - \, \vsg y_{-\vsg}^{(-)} \, = \, - \vsg \vp_{\mf{x}}(\bs{z}) \cdot \wt{F}\big( \vp_{\mf{x}}(\bs{z}) \big) \quad \e{with} \quad
\wt{F}(x) \, = \, 2 \, \op{v} \f{ F\Big( \tfrac{ - 2 \mf{u}^{\prime\prime}_1(k_0) }{ \mc{P}^{\prime}(k_0) } x \Big) }{ \big(\mf{u}_{1}^{\prime}(k_0) \big)^2 \, - \, \op{v}^2 }
\label{definition tilde G}
\enq
and where $F$ is as defined in \eqref{definition fonction G pour difference des racines de P ups}. Just as earlier on, one has that, uniformly on $\mc{D}$,
it holds $\wt{F}\big( \vp_{\mf{x}}(\bs{z}) \big) >0$ so that
\beq
\mf{p} \, = \, \e{sgn}\Big( \vsg y_{-\vsg}^{(+)} \, - \, \vsg y_{-\vsg}^{(-)} \Big) \; = \; - \vsg \e{sgn}\big( \vp_{\mf{x}}(\bs{z})\big) \;,
\enq
meaning that
\beq
a_{\mf{p}} \, = \, \vsg y_{-\vsg}^{(\mf{p})} \, - \, \vsg y_{-\vsg}^{(-\mf{p})} \; = \; \big| \vp_{\mf{x}}(\bs{z}) \big| \cdot \wt{F}\big( \vp_{\mf{x}}(\bs{z}) \big) \; \geq \; 0 \;.
\enq
The change of variables
\beq
y \, = \, b_{\mf{p}}\big( \vp_{\mf{x}}(\bs{z}) \big) \, - \, \vsg t \, \wt{F}\big( \vp_{\mf{x}}(\bs{z}) \big) \qquad \e{with} \qquad
b_{\mf{p}}\big( \vp_{\mf{x}}(\bs{z}) \big) \, = \, \vsg \e{min}\Big( \vsg y_{-\vsg}^{(+)} \, , \, \vsg y_{-\vsg}^{(-)} \Big) \, = \, y_{-\vsg}^{(-\mf{p})} \;,
\enq
in the $y$-integration recasts $\mc{I}_{\e{sg}}^{(\sslash)}(\mf{x})$ in the form
\beq
\mc{I}_{\e{sg}}^{(\sslash)}(\mf{x}) \; = \; \sul{ \mf{y} = \pm }{ }\mc{I}_{\e{sg};\mf{y}}^{(\sslash)}(\mf{x})
\enq
where the two building blocks read
\beq
\mc{I}_{\e{sg};\mf{y}}^{(\sslash)}(\mf{x}) \; = \; \pl{ \ups = \pm }{} \bigg\{ \Int{ -\sqrt{\de_{\ups}} }{ \sqrt{\de_{\ups}} } \hspace{-2mm} \dd^{m_{\ups}}z^{(\ups)} \bigg\}
\Int{ 0 }{ \de_0 } \dd t \, \bigg[ t^{ \mf{d}_{-\mf{p}}(\bs{t}) -1 } \Big[ t + |\vp_{\mf{x}}\big(\bs{z}\big)| \Big]^{ \mf{d}_{\mf{p}}(\bs{t}) -1 } \cdot \Xi\big[ \mf{y} \vp_{\mf{x}}\big(\bs{z}\big) \big]
\, \msc{F}^{(2)}(\bs{t}) \bigg]_{\bs{z}-\e{even}}\;.
\enq
There,
\beq
\bs{t}=\Big(t_0 , \bs{z} \Big) \; , \qquad
\bs{z} \; = \; \Big( \bs{z}^{(+)}, \bs{z}^{(-)} \Big) \in \R^{m_+}\!\times \R^{m_-} \;, \quad
t_0 \, = \, b_{ \mf{p}}\big( \vp_{\mf{x}}(\bs{z}) \big) \, - \, \vsg t\, \wt{F}\big( \vp_{\mf{x}}(\bs{z}) \big) \;.
\enq
Also,
\beq
\msc{F}^{(2)}(\bs{t}) \; = \; \msc{F}(\bs{t}) \cdot \wt{F}\big( \vp_{\mf{x}}(\bs{z}) \big) \pl{\ups = \pm }{} \Big[ \wt{F}_{\ups}\big( \vp_{\mf{x}}(\bs{z}) ,t_0 \big) \Big]^{ \mf{d}_{\ups}(\bs{t}) -1 }
\enq
and
\beq
\wt{F}_{\ups}\big( \vp_{\mf{x}}(\bs{z}) ,t_0 \big) \; = \; \tfrac{ - \vsg \mf{u}^{\prime\prime}_1(k_0) }{2 \mc{P}^{\prime}(k_0) } \wt{F}\big( \vp_{\mf{x}}(\bs{z}) \big)
\cdot \Big( \, y_{\vsg}^{(\ups)} \, + \, t \vsg \, \wt{F}\big( \vp_{\mf{x}}(\bs{z}) \big)\, - \, b_{\mf{p}} \Big) \;.
\enq
\vspace{4mm}
Observe that it holds
\beq
\mf{y} \vp_{\mf{x}}\big(\bs{z}\big) \, = \, \mf{y}\mf{x} + \sul{\ups=\pm}{} \ups \sul{s=1}{m_{\mf{y} \ups} } \big( z_{s}^{(\mf{y}\ups)} \big)^{2} \;.
\enq
Thus, denoting
\beq
\qquad \wh{m}_{\ups} \, = \, m_{\mf{y} \ups}\;, \quad \e{and} \quad
\wh{\de}_{\ups} \, = \; \de_{\mf{y} \ups} \;,
\enq
the change of variables $\bs{z}^{(\ups)} \hookrightarrow \bs{z}^{(\ups \mf{y})}$
leads to
\beq
\mc{I}_{\e{sg};\mf{y}}^{(\sslash)}(\mf{x}) \; = \; \pl{ \ups = \pm }{} \bigg\{ \Int{ -\sqrt{ \, \wh{\de}_{\ups} } }{ \sqrt{ \, \wh{\de}_{\ups}} } \hspace{-2mm} \dd^{ \wh{m}_{\ups}}z^{(\ups)} \bigg\}
\Int{ 0 }{ \de_0 } \dd t \, \bigg[ t^{ \mf{d}_{-\mf{p}}( \, \wh{\bs{t}} \, ) -1 } \, \Big[ t + \wh{\vp_{\mf{x}}}\big(\bs{z}\big) \Big]^{ \mf{d}_{\mf{p}}( \, \wh{\bs{t}} \, ) -1 } \cdot
\Xi\big[ \, \wh{\vp_{\mf{x}}}\big(\bs{z}\big) \, \big]
\, \msc{F}^{(2)}\big( \, \wh{\bs{t}} \, \big) \bigg]_{\bs{z}-\e{even}} \;.
\enq
There
\beq
\wh{\bs{t}}=\Big(t_0 , \bs{z} \Big) \; , \qquad
\bs{z} \; = \; \Big( \bs{z}^{(\mf{y})}, \bs{z}^{(-\mf{y})} \Big) \in \R^{ \wh{m}_+ }\!\times \R^{ \wh{m}_- } \;, \quad
t_0 \, = \, b_{ \mf{p} }\big( \wh{\vp_{\mf{x}}}(\bs{z}) \big) \, - \, \vsg t\, \wt{F}\big( \wh{\vp_{\mf{x}}}(\bs{z}) \big) \;,
\enq
and
\beq
\wh{\vp_{\mf{x}}}(\bs{z}) \, = \, \mf{y} \mf{x} \, + \, \sul{s=1}{ \wh{m}_{ + } } \big( z_{s}^{(+)} \big)^{2} \, - \, \sul{s=1}{ \wh{m}_{ - } } \big( z_{s}^{(-)} \big)^{2} \;.
\enq
At this stage, one decomposes the integral into domains where a square root change of variables is well defined:
\beq
\Big[ - \big[ \; \wh{\de}_{\ups} \big]^{\f{1}{2}} \, ; \, \big[ \; \wh{\de}_{\ups} \big]^{\f{1}{2}} \; \Big]^{ \wh{m}_{\ups} } \, = \,
\bigsqcup_{ \bs{\eps}^{(\ups)} \in \{ \pm 1 \}^{ \wh{m}_{\ups} } } \Bigg\{ \pl{a=1}{ \wh{m}_{\ups}} \bigg\{ \eps_{a}^{(\ups)} \Big[ 0 ; \eps_a^{(\ups)} \cdot \big[ \; \wh{\de}_{\ups} \big]^{\f{1}{2}} \; \Big] \bigg\} \Bigg\}
\qquad \e{with} \qquad \bs{\eps}^{(\ups)} \; = \; \Big( \eps^{ (\ups) }_{ 1 }, \dots, \eps^{ (\ups) }_{ \, \wh{m}_{\ups} } \Big)
\enq
in which the sign pre-factor indicates the orientation of the interval. Then, in each of the sets building up the partition, one sets
\beq
z_a^{(\ups)} \, = \, \eps_{a}^{(\ups)} \cdot \big[ w_a^{(\ups)} \big]^{\f{1}{2}} \qquad a=1,\dots, \wh{m}_{\ups} \;.
\enq
This yields
\beq
\mc{I}_{\e{sg};\mf{y}}^{(\sslash)}(\mf{x}) \; = \; \sul{ \substack{ \bs{\eps}^{(\ups)} \in \{ \pm 1 \}^{ \wh{m}_{\ups} } \\ \ups= \pm } }{}
\chi_{\bs{\eps};\mf{y}}\Big[\msc{F}^{(3)}, \mf{d}_{-\mf{p}}-1, \mf{d}_{ \mf{p} }-1 \Big]
\label{decomposition I slash mf y sur integrales elementaires}
\enq
where the building block integral is defined as
\bem
\chi_{\bs{\eps};\mf{y}}\Big[\mc{F} , A , B \Big] \; = \; \pl{\ups= \pm }{}\Bigg\{ \Int{ 0 }{ \wh{\de}_{\ups} } \f{ \dd^{ \wh{m}_{\ups} } w^{(\ups)} }{ \sqrt{ w_a^{(\ups)} } } \Bigg\} \Int{ 0 }{ \de_0 } \dd t \;
\bigg[ \mc{F}\Big( \bs{u}^{(\bs{\eps})} , t , t + \wh{\vp_{\mf{x}}}\big(\bs{u}^{(\bs{\eps})}_{\bs{w}} \big) \Big) \\
\times \big[t + \wh{\vp_{\mf{x}}}\big(\bs{u}^{(\bs{\eps})}_{\bs{w}}\big) \big]^{ A(\bs{u}^{(\bs{\eps})}) } \cdot t^{ B(\bs{u}^{(\bs{\eps})}) } \cdot
\Xi\Big[ \wh{\vp_{\mf{x}}}\big(\bs{u}^{(\bs{\eps})}_{\bs{w}} \big) \Big] \bigg]_{\bs{u}^{(\bs{\eps})}_{\bs{w}}-\e{even}}\;.
\label{ecriture integrale modele pour eps decomposition de l'integrale I sg u greater v}
\end{multline}
Above, it is undercurrent
\beq
\bs{u}^{(\bs{\eps})}\, = \, \Big( u^{(\bs{\eps})}_0 , \bs{u}^{(\bs{\eps})}_{\bs{w}}\Big)
\quad \e{with} \quad
\bs{u}^{(\bs{\eps})}_{\bs{w}}\, = \, \Big( \bs{u}^{(\bs{\eps};\mf{y})}_{\bs{w}}, \bs{u}^{(\bs{\eps};-\mf{y})}_{\bs{w}} \Big) \;, \quad
u^{(\bs{\eps})}_0 \, = \, b_{\mf{p}}\Big( \wh{\vp_{\mf{x}}}( \bs{u}^{(\bs{\eps})}_{\bs{w}} ) \Big) \, - \, \vsg t\, \wt{F}\Big( \wh{\vp_{\mf{x}}}( \bs{u}^{(\bs{\eps})}_{\bs{w}} ) \Big) \;,
\enq
and, finally,
\beq
\bs{u}^{(\bs{\eps};\ups)}_{\bs{w}} \, = \, \Big( \eps^{ (\ups) }_{ 1 } \big[ w_1^{(\ups)} \big]^{\f{1}{2}} , \dots, \eps^{ (\ups) }_{ \wh{m}_{\ups} } \big[ w_{ \wh{m}_{\ups} }^{(\ups)} \big]^{\f{1}{2}} \Big) \;.
\enq
Also, after the change of variables, one has that
\beq
\wh{\vp_{\mf{x}}}(\bs{u}^{(\bs{\eps})}_{\bs{w}}) \; = \; \mf{y} \mf{x} \; + \; \sul{a=1}{ \wh{m}_+} w_a^{(+)} \; - \; \sul{a=1}{ \wh{m}_-} w_a^{(-)} \;.
\enq
The integrand appearing in \eqref{decomposition I slash mf y sur integrales elementaires} takes the form
\bem
\msc{F}^{(3)}\Big( \bs{u}^{(\bs{\eps})} , t , t + \wh{\vp_{\mf{x}}}\big(\bs{u}^{(\bs{\eps})}_{\bs{w}} \big) \Big) \, = \, 2^{1-\ov{\bs{n}}_{\ell}} \cdot
\wt{F}\Big( \wh{\vp_{\mf{x}}}\big(\bs{u}^{(\bs{\eps})}_{\bs{w}} \big) \Big) \cdot
\Big(V \vp^{(\sslash)} \vp^{(\e{sg})} \Big)\Big( \Psi_{\op{M}}(\bs{u}^{(\bs{\eps})}) \Big) \cdot \big| \det D_{\bs{u}^{(\bs{\eps})}} \Psi_{\op{M}} \big| \vspace{8mm} \\
\times \pl{ \ups = \pm }{} \Big\{ \, \wt{\mf{f}}_{\ups}\big( \bs{u}^{(\bs{\eps})}, u^{(\bs{\eps})}_0 \big) \Big\}^{ \mf{d}_{\ups}(\bs{u}^{(\bs{\eps})}) -1 }
\; \cdot \; \left\{ \ba{ccc} \msc{G}\Big( \Psi_{\op{M}}(\bs{u}^{(\bs{\eps})}) \, , \, \wt{\mf{f}_{+}}( \bs{u}^{(\bs{\eps})}, u^{(\bs{\eps})}_0) \cdot \big[ t \, + \, \wh{\vp_{\mf{x}}}\big(\bs{u}^{(\bs{\eps})}_{\bs{w}} \big) \big]
\, , \, t \cdot \wt{\mf{f}_{-}}( \bs{u}^{(\bs{\eps})} ,u^{(\bs{\eps})}_0 ) \Big) & \e{if} & \mf{p} = + \vspace{3mm} \\
\msc{G}\Big( \Psi_{\op{M}}(\bs{u}^{(\bs{\eps})}) \, , \, t \cdot \wt{\mf{f}_{+}}( \bs{u}^{(\bs{\eps})}, u^{(\bs{\eps})}_0 )
\, , \, \wt{\mf{f}_{-}}( \bs{u}^{(\bs{\eps})}, u^{(\bs{\eps})}_0) \cdot \big[ t \, + \, \wh{\vp_{\mf{x}}}\big(\bs{u}^{(\bs{\eps})}_{\bs{w}} \big) \big] \Big) & \e{if} & \mf{p} = - \ea \right. \;,
\end{multline}
where
\beq
\wt{\mf{f}}_{\ups}\big( \bs{u}^{(\bs{\eps})} , u^{(\bs{\eps})}_0 \big) \, = \,
\mf{f}_{\ups}\big( \op{M}\cdot \bs{u}^{(\bs{\eps})} \big) \cdot \wt{F}_{\ups} \Big( \vp_{\mf{x}}\big( \bs{u}^{(\bs{\eps})}_{\bs{w}} \big) , u^{(\bs{\eps})}_0 \Big) \;,
\enq
and I remind that $\ov{\bs{n}}_{\ell}\, = \, \sul{r=1}{\ell} n_r$.
As in the previous case, one gets that $\mf{d}_{\pm}>0$
throughout the integration domain. The expansion \eqref{decomposition I slash mf y sur integrales elementaires} of $\mc{I}_{\e{sg}}^{(\sslash)}(\mf{x}) $
decomposes this integral into a sum of elementary integrals \eqref{ecriture integrale modele pour eps decomposition de l'integrale I sg u greater v} whose
$\mf{x}\tend 0$ asymptotic behaviour is analysed in Lemma \ref{Lemme integrale beta multi-dim auxiliaire locale cas u greater than v}.
The conclusions of this lemma, specialised to the function $ \msc{F}^{(3)}$, ensure that the integrand is in $L^1$ and that there exists a smooth function $ \mf{x} \tend \mf{ R }_{ \bs{\eps} }( \mf{x} ) $
around $\mf{x}=0$ such that
\beq
\chi_{\bs{\eps};\mf{y}}\Big[\msc{F}^{(3)}, \mf{d}_{ -\mf{p} }-1, \mf{d}_{\mf{p}}-1 \Big] \; = \; \De^{(0)}_{+}\, \De^{(0)}_{-} \, \msc{G}^{(1)}\big( \bs{t}(k_0) \big)\cdot
\chi_{\bs{\eps};\mf{y}}\Big[\msc{F}_{\e{eff}}, \De^{(0)}_{-\mf{p}}-1, \De^{(0)}_{\mf{p}}-1 \Big]
\; + \; \e{O}\Big( |\mf{x}|^{\vth + 1-\tau } \Big) \; + \; \mf{R}_{ \bs{\eps} }(\mf{x})
\enq
where $\De^{(0)}_{\ups}= \De_{\ups}\big( \bs{t}(k_0) \big)$, $\vth$ is as defined in \eqref{definition de cal theta 0} and
\bem
\msc{F}_{\e{eff}} \big( \bs{u}^{(\bs{\eps})} , x, y \big) \; = \; \exp\bigg\{ -\Big( \op{M}\, \bs{u}^{(\bs{\eps})} , \op{D} \op{M} \, \bs{u}^{(\bs{\eps})} \Big) \bigg\} \cdot
\Big(V \vp_{\e{eff}}^{(\sslash)} \vp_{\e{eff}}^{(\e{sg})} \Big)\Big( \op{M} \, \bs{u}^{(\bs{\eps})} \Big) \\
\times 2^{1-\ov{\bs{n}}_{\ell}} \wt{F}\Big( \vp_{\mf{x}}(\bs{z}) \Big)
\cdot \pl{ \ups = \pm }{} \Big\{ \wt{F}_{\ups}\big( \bs{u}^{(\bs{\eps})},u_0^{(\bs{\eps})} \big) \Big\}^{ \de_{\ups} - 1 } \;.
\end{multline}
Above, $\op{M}$ is as introduced in \eqref{ecriture effet transformation matrice M} and $\op{D}$ as in \eqref{definition matrice D pour poid gaussien integrale modele}.
Finally, $\vp_{\e{eff}}^{(\sslash)}$ and $\vp_{\e{eff}}^{(\e{sg})}$ are as appearing in \eqref{definition fcts approx unite sg et sslash integrale modele}, \textit{c.f.}
also \eqref{definition des fcts vp eff slash et sg}.
By performing backwards, on the level of $ \chi_{\bs{\eps};\mf{y}}\Big[\msc{F}_{\e{eff}}, \De^{(0)}_{ -\mf{p} }-1, \De^{(0)}_{\mf{p}}-1 \Big]$,
all the transformations that were carried out starting from \eqref{reduction finale de la partie singuliere vers integrale modele}, one
arrives to the conclusions stated in \eqref{ecriture reduction de I sg slash vers J sg slash}. From there, one concludes as in the
regime $ | \mf{u}^{\prime}_1(k_0) | \, < \, \op{v} $. \qed
\section{Auxiliary results}
\label{Appendix Section Auxiliary results}
\subsection{A regularity lemma}
Given $\bs{x}\in \R^n$ with $n\geq 2$ and integers $1\leq a<b \leq n$ denote
\beq
\bs{x}_{a,b} \, = \, \big( x_1,\dots, x_{a-1},x_{a+1},\dots, x_{b-1}, x_{b+1},\dots, x_n \big) \;.
\label{definition vecteur simple ss 2 composantes}
\enq
\begin{lemme}
\label{Lemme integrale multidimensionnelle auxiliaire reguliere}
Let $K$ be a compact subset of $\R^n$, $n\geq 2$, and let $\mf{z}_{\pm}, \psi_{\pm}$ be smooth functions on $K$.
Let $0<\tau <1$ and let $\msc{G}$ be in the smooth class of $K$ with functions $\De_{\pm} \, = \, \big[ \psi_{\pm}\big]^2$
and constant $\tau$, \textit{c.f.} Definition \ref{Definition fct lisse sur ferme}
Assume that for any $\bs{x}\in K$ there exists $a<b$, $a,b \in \intn{1}{n}$, such that the differential $D_{\bs{x}}f_{a,b}$ of the map
\beq
f_{a,b} \, : \, \bs{x} \mapsto \Big( \bs{x}_{a,b}, \mf{z}_+\big(\bs{x}\big), \mf{z}_-\big(\bs{x}\big) \Big) \;,
\enq
with $\bs{x}_{a,b}$ as in \eqref{definition vecteur simple ss 2 composantes}, is invertible. Then, the integral
\beq
\mc{J}(\mf{x}) \, = \, \Int{ K }{} \dd^n x \; \msc{G}\Big( \bs{x}, \wh{\mf{z}}_{+}(\bs{x}) , \wh{\mf{z}}_{-}(\bs{x}) \Big) \cdot
\pl{\ups=\pm }{} \bigg\{ \Xi\big[ \, \wh{\mf{z}}_{\ups}(\bs{x}) \big] \cdot \big[\, \wh{\mf{z}}_{\ups}(\bs{x}) \big]^{ \De_{\ups}(\bs{x}) -1} \bigg\}
\quad with \quad
\wh{\mf{z}}_{\ups}(\bs{x})=\mf{z}_{\ups}(\bs{x})+\mf{x} \, ,
\enq
is a smooth function of $\mf{x}$, provided that $|\mf{x}|$ is small enough.
\end{lemme}
\proof
By virtue of the Whintey extension theorem, $\mf{z}_{\pm}$ and $\psi_{\pm}$ admit smooth extensions to an open neighbourhood $U_K$ of $K$. Thus, so does $\De_{\ups}=\psi_{\ups}^2$
and one obviously has that $\De_{\pm}(U_K) \subset \intff{0}{+\infty}$. One may also extend $\msc{G}$ to $U_K\times \R^+\times \R^+$ smoothly by setting
\beq
\msc{G}_{\mid \{ U_K \setminus K \} \times \R^+\times \R^+ } \, = \, 0
\enq
where the smoothness of this extension is ensured by the smooth vanising, to all orders in the derivatives, of $\msc{G}$ on $\Dp{} K \times \R^+\times \R^+$.
It follows from the hypothesis of the lemma that, for any $\bs{x}\in K$, there exists integers $a_{\bs{x}}<b_{\bs{x}}$, an open, relatively compact, neighbourhood $U_{\bs{x}}$ of $\bs{x}$,
an open, relatively compact, neighbourhood $V_{\bs{x}}$ of $\bs{x}_{a_{\bs{x}},b_{\bs{x}}}$ in $\R^{n-2}$ and $\eta_{\bs{x}} >0$
such that
\beq
f_{a_{\bs{x}},b_{\bs{x}}} \; : \; U_{\bs{x}} \; \tend \; f_{a_{\bs{x}},b_{\bs{x}}} \big( U_{\bs{x}} \big) \, = \,
V_{ \bs{x} } \, \times \, \intoo{ \mf{z}_{+}(\bs{x}) - \eta_{\bs{x}} }{ \mf{z}_{+}(\bs{x}) + \eta_{\bs{x}} } \, \times \, \intoo{ \mf{z}_{-}(\bs{x}) - \eta_{\bs{x}} }{ \mf{z}_{-}(\bs{x}) +\eta_{\bs{x}} }
\enq
is a smooth diffeomorphism onto its image. Furthermore, reducing $\eta_{\bs{x}}$ if necessary, one may always assume that $\mf{z}_{\ups}(\bs{x}) \pm \eta_{\bs{x}} \not=0$ for both values of $\ups \in \{\pm\}$.
Finally, the sets can always be chosen such that $f_{a_{\bs{x}},b_{\bs{x}}}^{-1}$ has a smooth extension to an open neighbourhood of $\ov{ f_{a_{\bs{x}},b_{\bs{x}}} \big( U_{\bs{x}} \big) }$ and such that
$U_{\bs{x}} \subset U_K$.
Then, $\cup_{\bs{x} \in K} U_{\bs{x}} \subset U_K$ is an open cover of $K$ and hence there exists a finite sub-cover $\bigcup_{k=1}^{\ell} U_{\bs{x}_k} \subset U_K$
with associated diffeomorphisms $f_{a_{\bs{x}_k},b_{\bs{x}_k}}$ mapping $U_{\bs{x}_k}$ onto $f_{a_{{\bs{x}_k}},b_{{\bs{x}_k}}} \big( U_{\bs{x}_k} \big)$.
Let $\{\vp_k\}_{k=1}^{\ell}$ be the partition of unity subordinate to the cover $\bigcup_{k=1}^{\ell} U_{\bs{x}_k} $. Then, using that the integrand extends by zero outside of
$K$, one has
\beq
\mc{J}(\mf{x}) \, = \, \Int{ U_K }{} \dd^n x \; \msc{G}\Big( \bs{x}, \wh{\mf{z}}_{+}(\bs{x}) , \wh{\mf{z}}_{-}(\bs{x}) \Big) \cdot
\pl{\ups=\pm }{} \bigg\{ \Xi\big[ \, \wh{\mf{z}}_{\ups}(\bs{x}) \big] \cdot \big[\, \wh{\mf{z}}_{\ups}(\bs{x}) \big]^{ \De_{\ups}(\bs{x}) -1} \bigg\}
\enq
what allows one to decompose the integral as
$\mc{J}(\mf{x}) \, = \, \sum_{k=1}^{\ell} \mc{J}_k(\mf{x})$ where
\beq
\mc{J}_k(\mf{x}) \; = \; \Int{ V_{\bs{x}_k} }{} \dd^{n-2} w \cdot \pl{\ups=\pm}{} \bigg\{ \Int{ \mf{x} + \mf{z}_{\ups}(\bs{x}_k) - \eta_{\bs{x}_k} }{ \mf{x}+\mf{z}_{\ups}(\bs{x}_k) + \eta_{\bs{x}_k} } \dd \mf{z}_{\ups} \bigg\}
\cdot \wt{\mc{G}}_k(\bs{u}_{\mf{x}}) \cdot \pl{\ups=\pm }{} \bigg\{ \Xi\big( \mf{z}_{\ups} \big) \cdot \big[ \mf{z}_{\ups} \big]^{ \wt{\De}_{\ups}(\bs{u}_{\mf{x}}) -1} \bigg\}
\enq
with $\bs{u}_{\mf{x}} \, = \, ( \bs{w}, \mf{z}_+ - \mf{x} , \mf{z}_- - \mf{x})$ and
\beq
\wt{\mc{G}}_k( \bs{u}_{\mf{x}} ) \, = \, \vp_{k}\Big( f_{ a_{\bs{x}_k},b_{\bs{x}_k}}^{-1}( \bs{u}_{\mf{x}} ) \Big) \cdot \big| \e{det}\big[ D_{\bs{u}_{\mf{x}}} f^{-1}_{ _{a_{\bs{x}_k},b_{\bs{x}_k}} }\big] \big|
\cdot \msc{G}\big(f_{ a_{\bs{x}_k},b_{\bs{x}_k}}^{-1}( \bs{u}_{\mf{x}} ) , \mf{z}_+, \mf{z}_- \big) \;.
\label{definition fct tilde G cal k}
\enq
Finally, $\wt{\De}_{\ups}(\bs{u} ) \, = \, \De_{\ups}\circ f_{ a_{\bs{x}_k},b_{\bs{x}_k}}^{-1}(\bs{u})$.
This representation is obtained, by restricting the integration domain to $U_{\bs{x}_k}$ due to the presence of $\vp_{k}$ followed by making the change of variables
$f_{a_{\bs{x}_k},b_{\bs{x}_k}}^{-1}\big( \, \wt{\bs{u}} \, \big) \, = \, \bs{x}$. Finally, one shifts the last two integration variables by $ -\mf{x}$.
Note that $\wt{\mc{G}}_k$ is smooth since the determinant never vanishes and has thus constant sign.
There are four cases to distinguish depending on whether $
0\in \intoo{ \mf{x} + \mf{z}_{\ups}(\bs{x}_k) - \eta_{\bs{x}_k} }{ \mf{x} + \mf{z}_{\ups}(\bs{x}_k) + \eta_{\bs{x}_k} }$ or not.
\begin{itemize}
\item[$i)$] If $\mf{z}_{\ups}(\bs{x}_k) + \eta_{\bs{x}_k}<0$ for at least one $\ups \in \{ \pm\}$, then $\mc{J}_k(\mf{x})$ vanishes for $|\mf{x}|$ small enough.
\item[$ii)$] If $\mf{z}_{\ups}(\bs{x}_k) - \eta_{\bs{x}_k}>0$ for $\ups = \pm$, then, for $|\mf{x}|$ small enough, the integral reduces to
\beq
\mc{J}_k(\mf{x}) \; = \; \Int{ V_{\bs{x}_k} }{} \dd^{n-2} w \cdot \pl{\ups=\pm}{} \bigg\{ \Int{ \mf{x} + \mf{z}_{\ups}(\bs{x}_k) - \eta_{\bs{x}_k} }{ \mf{x}+\mf{z}_{\ups}(\bs{x}_k) + \eta_{\bs{x}_k} }\hspace{-5mm} \dd \mf{z}_{\ups} \bigg\}
\cdot \wt{\mc{G}}_k(\bs{u}_{\mf{x}}) \cdot \pl{\ups=\pm }{} \Big\{ \big[ \mf{z}_{\ups} \big]^{ \wt{\De}_{\ups}(\bs{u}_{\mf{x}}) -1} \Big\} \;.
\enq
By construction, the endpoints of integration in the $\mf{z}_{\pm}$ variables are uniformly away from $0$, for $|\mf{x}|$ small enough.
The hypotheses of the lemma ensure that $\wt{\mc{G}}_k(\bs{u}_{\mf{x}})$ is smooth in $\mf{x}$ small enough
and in $(\bs{w},\mf{z}_+,\mf{z}_-)$ provided that $\bs{u}_{\mf{x}}\in f_{a_{\bs{x}_k},b_{\bs{x}_k}} \big( U_{\bs{x}_k} \big)$.
Taken that the only singularities of the integrand, which are at $\mf{z}_{\pm}=0$, are uniformly away from the integration domain and taken that
the integral runs through a compact set, derivation under the integral -and in respect to endpoints of integration- theorems entail
that $\mc{J}_k(\mf{x})$ is a smooth function of $\mf{x}$, for $|\mf{x}|$ small enough.
\item[$iii)$] If $\mf{z}_{\ups}(\bs{x}_k) - \eta_{\bs{x}_k} < 0$ and $\mf{z}_{\ups}(\bs{x}_k) + \eta_{\bs{x}_k}>0$ for both values of $\ups$, then the integral splits as
$\mc{J}_k(\mf{x}) \; = \; \sul{b=1}{4} \mc{J}_k^{(b)}(\mf{x}) $ where
\beq
\mc{J}_k^{(b)}(\mf{x}) \; = \; \Int{ V_{\bs{x}_k} }{} \dd^{n-2} w \cdot \pl{\ups=\pm}{} \bigg\{ \Int{ 0 }{ \mf{x}+\mf{z}_{\ups}(\bs{x}_k) + \eta_{\bs{x}_k} } \hspace{-5mm} \dd \mf{z}_{\ups} \bigg\}
\cdot \wt{\mc{G}}_k^{(b)}(\bs{u}_{\mf{x}}) \cdot \pl{\ups=\pm }{} \Big\{ \big[ \mf{z}_{\ups} \big]^{ \wt{\De}_{\ups}(\bs{u}_{\mf{x}}) -1 } \Big\}
\cdot \left\{\ba{cc} \wt{\De}_{+}(\bs{u}_{\mf{x}}) \, \wt{\De}_{-}(\bs{u}_{\mf{x}}) & b=1 \vspace{2mm} \\
\wt{\De}_{-}(\bs{u}_{\mf{x}}) \cdot \big[ \mf{z}_{+} \big]^{1-\tau} & b=2 \vspace{2mm} \\
\wt{\De}_{+}(\bs{u}_{\mf{x}}) \cdot \big[ \mf{z}_{-} \big]^{1-\tau} & b=3 \vspace{2mm} \\
\pl{\ups=\pm }{} \Big\{ \big[ \mf{z}_{\ups} \big]^{ 1-\tau } \Big\} & b=4 \ea \right. \;.
\nonumber
\enq
The function $\wt{\mc{G}}_k^{(b)}$ are obtained from $\wt{\mc{G}}_k$ defined in \eqref{definition fct tilde G cal k} upon substituting $\msc{G} \hookrightarrow \msc{G}^{(b)}$
with $ \msc{G}^{(b)}$ arising in the expansion \eqref{ecriture decomposition smooth class K}.
Taken that $\msc{G}^{(4)}$ fulfils property $H3)$ stated below of \eqref{definition ensemble M elle n ell enleve} and that $ \mf{z} \mapsto \mf{z}^{ \de-\tau } $ is integrable on $\intff{0}{\eps}$ for any $\de\geq 0$,
one readily concludes that $\mc{J}_k^{(4)}(\mf{x}) $ produces a smooth function of $\mf{x}$. The analysis of the remaining integrals demands one more step which I detail for
$\mc{J}_k^{(2)}(\mf{x})$, the other cases being tractable in a similar way. In the case of $\mc{J}_k^{(2)}(\mf{x})$, an integration by parts yields:
\bem
\mc{J}_k^{(2)}(\mf{x}) \; = \; - \Int{ V_{\bs{x}_k} }{} \dd^{n-2} w \cdot \pl{\ups=\pm}{} \bigg\{ \Int{ 0 }{ \mf{x}+\mf{z}_{\ups}(\bs{x}_k) + \eta_{\bs{x}_k} } \hspace{-5mm} \dd \mf{z}_{\ups} \bigg\}
\cdot \Dp{\mf{z}_{-}}\bigg\{ \wt{\mc{G}}_k^{(2)}(\bs{u}_{\mf{x}}) \cdot \big[ \mf{z}_{+} \big]^{ \wt{\De}_{+}(\bs{u}_{\mf{x}}) - \tau } \cdot \big[ s \big]^{ \wt{\De}_{-}(\bs{u}_{\mf{x}}) } \bigg\}_{\mid s = \mf{z}_-} \\
\hspace{-1cm} + \Int{ V_{\bs{x}_k} }{} \dd^{n-2} w \hspace{-6mm} \Int{ 0 }{ \mf{x}+\mf{z}_{+}(\bs{x}_k) + \eta_{\bs{x}_k} } \hspace{-6mm} \dd \mf{z}_{+}
\Bigg\{ \bigg( \wt{\mc{G}}_k^{(2)}(\bs{u}_{\mf{x}}) \cdot \big[ \mf{z}_{+} \big]^{ \wt{\De}_{+}(\bs{u}_{\mf{x}}) - \tau } \cdot \big[ \mf{z}_- \big]^{ \wt{\De}_{-}(\bs{u}_{\mf{x}}) } \bigg)_{\mid \mf{z}_-= \mf{x}+\mf{z}_{-}(\bs{x}_k) + \eta_{\bs{x}_k} }
\hspace{-8mm} - \qquad \bigg( \wt{\mc{G}}_k^{(2)}(\bs{u}_{\mf{x}}) \cdot \big[ \mf{z}_{+} \big]^{ \wt{\De}_{+}(\bs{u}_{\mf{x}}) - \tau } \cdot \big[ \mf{z}_- \big]^{ \wt{\De}_{-}(\bs{u}_{\mf{x}}) } \bigg)_{\mid \mf{z}_-=0} \Bigg\} \;.
\nonumber
\end{multline}
Note that the last term occurring in this expression is only present if $\wt{\De}_{-}(\bs{u}_{\mf{x}})_{\mid \mf{z}_-=0} =0 $ on a set of non-zero measure.
Property $H3)$ fulfilled by $\msc{G}^{(2)}$, the fact that $\msc{G}^{(2)}$ does not depend on the $v$ variables as given in \eqref{ecriture decomposition smooth class K}
and the integrability of the $\mf{z}_{\ups}$-related part of the integrand all together ensure that
the resulting integrals produce smooth contributions in $\mf{x}$ for $|\mf{x}|$ small enough.
\item[$iv)$] The situation is quite similar when only one of the $\mf{z}_{\ups}$ changes sign, \textit{viz}.
$\mf{z}_{+}(\bs{x}_k) - \eta_{\bs{x}_k} < 0$ and $\mf{z}_{+}(\bs{x}_k) + \eta_{\bs{x}_k}>0$ but $\mf{z}_{-}(\bs{x}_k) - \eta_{\bs{x}_k}>0$ or the analogous situation when $+\leftrightarrow -$.
In such a case, one should decompose the integral similarly to $iii)$ and then invoke
one of the two properties $H1)$ or $H2)$, \textit{c.f.} below of \eqref{definition ensemble M elle n ell enleve}, depending on which of among the two integration domains passes though zero,
relative to the boundedness of the function $\msc{G}$ and its partial derivatives so as to conclude on the smoothness of the associated integral.
\end{itemize}
Thus the claim. \qed
\subsection{Local rectification of $\mf{z}_{\ups}$}
\begin{prop}
\label{Proposition trivialisation locale fct zups}
Let the assumptions and notation given in Subsection \ref{SousSection hypothese gles sur consituants integrale modele} hold.
Given $k_0 \in \e{Int}{\msc{J}}_1$ and given any $ \zeta_r \in \{\pm 1\}$, let
\beq
\mf{z}_{\ups}(\bs{p})\;=\; \, - \, \sul{ (r,a) \in \mc{M} }{ } \zeta_r \mf{w}_{\ups}^{(r)}\big( p_a^{(r)} ; t_r(k_0) \big) \quad , \qquad \ups \in \{ \pm \} \;,
\enq
where
\beq
\mf{w}_{\ups}^{(r)}(k ; p) \, = \, \mf{u}_r(k)\, - \, \mf{u}_r(p)\, + \, \ups \op{v} \big( k - p\big) \qquad and \qquad \op{v} \in \R^{+*}\;.
\enq
Further, let
\beq
\tilde{z}_{\ups}(\bs{x}) \; = \;- \sul{ (r,a)\in \mc{M} }{} \zeta_{r} \bigg\{ \mf{h}_r\big( x_a^{(r)} \big) +\ups \op{v} x_a^{(r)} \bigg\} \qquad where \qquad
\mf{h}_r\big( x \big) \; = \; - \zeta_r \veps_r \f{ x^2 }{ 2 \xi_r^2 } \, + \, \op{u}\, x \;,
\label{definition fonction z tilde ups}
\enq
$\veps_{r}\in \{ \pm \}$, $\xi_r \in \R^*$ and $\op{u}$ are such that
\beq
- \f{ \zeta_r \veps_r }{ \xi_r^2 } \; = \; \mf{u}_{r}^{\prime\prime}\big( t_r(k_0) \big) \quad and \quad \op{u}=\mf{u}_{1}^{\prime}(k_0) \;.
\label{definition parametres zetar vepsr}
\enq
Finally, let
\beq
t_r^{(0)}(x) \, = \, \f{ \zeta_1 \veps_1 }{ \zeta_r \veps_r } \Big( \f{ \xi_r }{ \xi_1 } \Big)^2 x
\enq
and for $C>0$, $\eta>0$, consider the domain
\beq
\mc{D}^{(\e{eff})}_{ \eta } \, = \, \bigg\{ \bs{x} \in \pl{r=1}{\ell} \R^{n_{r}} \; : \; |x_1^{(1)}| \, \leq \, C \eta \; , \;
\forall (r,a )\in \mc{M} \; : \; \big| t_r^{(0)}(x_1^{(1)})-x_a^{(r)} \big| \, \leq \, \xi_{r}^2 \eta \bigg\} \;.
\label{ecriture domaine D eff eta prime the rectification des z ups}
\enq
Then, there exists $\mf{x}_0>0$, $\eta^{\prime}>0$ and
\begin{itemize}
\item smooth functions $\mf{f}_{\ups}$ on $ \intoo{ - \mf{x}_0 }{ \mf{x}_0 }\times \mc{D}^{(\e{eff})}_{\eta^{\prime}}$ satisfying $\mf{f}_{\ups}(\mf{x}; \bs{x})\, = \, 1 + \e{O}\Big( \norm{\bs{x}}+|\mf{x}|\Big)$,
\item a smooth diffeomorphism $\Psi_{\mf{x}}:\mc{D}^{(\e{eff})}_{\eta^{\prime}} \tend \Psi\Big( \mc{D}^{(\e{eff})}_{\eta^{\prime}} \Big)$ satisfying $D_{\bs{0}} \Psi \, = \, \e{id} + \mf{x} \op{N}_{\Psi}$ with
$\norm{ \op{N}_{\Psi} } \leq C$, for some $\mf{x}$-independent $C>0$,
\end{itemize}
such that
\beq
\mf{x}+\mf{z}_{\ups} \circ \Psi_{\mf{x}}(\bs{x}) \; = \; \mf{f}_{\ups}(\mf{x};\bs{x}) \cdot \Big(\mf{x} + \tilde{z}_{\ups}(\bs{x}) \Big) \;,
\label{ecriture fondamentale de rectification}
\enq
and $\Psi_{\mf{x}}\Big( \mc{D}^{(\e{eff})}_{\eta^{\prime}} \Big) \subset \msc{J}_{\e{tot}}$ contains a $\mf{x}$-independent open neighbourhood of $\bs{t}(k_0)$.
Furthermore, the map
\beq
(\mf{x},\bs{x}) \mapsto \Psi_{\mf{x}}(\bs{x})
\enq
is smooth on $\intoo{ - \mf{x}_0 }{ \mf{x}_0 }\times \mc{D}^{(\e{eff})}_{\eta^{\prime}}$.
\end{prop}
\Proof
\subsubsection*{$\bullet$ Canonical form of $\mf{z}_{\ups}$}
Let $I_{\ell}\subset \e{Int}(\msc{J}_{\ell})$ be a segment such that $t_{\ell}(k_0) \in \e{Int}(I_{\ell})$. Since $\Dp{k}\mf{w}_{\ups}^{(\ell)}\big( k; t_{\ell}(k_0) \big) \, = \, \mf{u}_{\ell}(k)+\ups \op{v}\not=0$
on $I_{\ell}$, $k\mapsto \mf{w}_{\ups}^{(\ell)}\big( k; t_{\ell}(k_0) \big)$ is strictly monotone on $I_{\ell}$ and thus
admits $t_{\ell}(k_0)$ as its unique zero on $I_{\ell}$. Furthermore, this also implies that there exists
\beq
c>0 \quad \e{such}\, \e{that} \quad \intff{-c}{c} \subset \mf{w}_{\ups}^{(\ell)}\big( I_{\ell} ; t(k_0) \big) \;.
\label{introduction parametre c}
\enq
Given $\eps>0$, set
\beq
\mc{B}_{\eps}\Big( \bs{t}_{[\ell, n_{\ell}]}(k_0) \Big) \; = \; \bigg\{ \bs{p}_{[\ell, n_{\ell}]} \in \prod_{r=1}^{\ell} \msc{J}_r^{n_r-\de_{r,\ell}} \; : \;
\; \; \big|p_a^{(r)} - t_r(k_0) \big| < \eps \; , \; \; \forall (a,r) \in \mc{M}_{[\ell,n_{\ell}]} \bigg\} \;,
\enq
where $\mc{M}_{[\ell,n_{\ell}]}=\mc{M}\setminus \{ (\ell,n_{\ell}) \}$ has been introduced in \eqref{definition ensemble M elle n ell enleve} while $\bs{p}_{[\ell, n_{\ell}]}$
is as defined in \eqref{definition vecteur p avec composantes omises}.
Given the function $\mf{z}_{\ups}$ of $\sul{r=1}{\ell}n_{r}$ variables, it is of use to agree to denote its analogue on $\mc{B}_{\eps}\Big( \bs{t}_{[\ell, n_{\ell}]}(k_0) \Big) $, \textit{viz}. when the last variable is deleted, as :
\beq
\mf{z}_{\ups}^{([\ell, n_{\ell}])}( \bs{p}_{[\ell, n_{\ell}]} ) \; \equiv \; \, - \, \sul{ (r,a) \in \mc{M}_{[\ell,n_{\ell}]} }{ } \zeta_r \mf{w}_{\ups}^{(r)}\big( p_a^{(r)} ; t_r(k_0) \big) \quad , \qquad \ups \in \{ \pm \} \;.
\enq
Let $\bs{t}_{[\ell, n_{\ell}]}(k_0)$ be as defined in \eqref {ecriture definition vecteur tk0} and let $V_{[\ell,n_{\ell}]}$ be any open neighbourhood of $\bs{t}_{[\ell, n_{\ell}]}(k_0)$ such that $V_{[\ell,n_{\ell}]} \subset \mc{B}_{\eps}\Big( \bs{t}_{[\ell, n_{\ell}]}(k_0) \Big)$.
Then,
\beq
\mf{x}+\mf{z}_{\ups}^{([\ell, n_{\ell}])}( \bs{p}_{[\ell, n_{\ell}]} ) \; = \; \e{O}\Big( |\mf{x}|+\eps \Big) \qquad \e{uniformly} \; \e{on} \; V_{[\ell,n_{\ell}]} \;.
\label{ecriture eqn pour zero dans fct Heaviside}
\enq
Thus, provided that $|\mf{x}|$ and $\eps$ are taken small enough, one has that, for any
\beq
\bs{p}_{[\ell, n_{\ell}]} \in V_{[\ell,n_{\ell}]}, \quad \e{it} \; \e{holds} \quad
\mf{x} \, + \, \mf{z}_{\ups}\big(\bs{p}_{[\ell, n_{\ell}]} \big) \in \intff{- \tf{c}{2} }{ \tf{c}{2} } \;,
\enq
with $c>0$ as appearing in \eqref{introduction parametre c}. Then, the monotonicity of
$k \mapsto \mf{w}_{\ups}^{(\ell)}\big(k; t_{\ell}(k_0) \big) $ on $I_{\ell}$ ensures that there exists a unique $\mc{V}_{\ups}\big( \bs{p}_{[\ell, n_{\ell}]} \big) \in I_{\ell}$
such that
\beq
\mf{x}+\mf{z}_{\ups}\big( \bs{p}_{\ups} \big) \, = \, 0 \quad \e{with} \quad \bs{p}_{\ups}\; = \; \Big( \bs{p}_{[\ell, n_{\ell}]} , \mc{V}_{\ups}\big( \bs{p}_{[\ell, n_{\ell}]} \big) \Big)
\qquad \e{for}\; \e{any} \quad \bs{p}_{[\ell, n_{\ell}]} \in V_{[\ell,n_{\ell}]} \;.
\label{definition vecteur p ups}
\enq
Here, for simplicity, the $\mf{x}$ dependence of $\mc{V}_{\ups}$ has been kept implicit.
The function $\mc{V}_{\ups}$ takes the explicit form
\beq
\mc{V}_{\ups}\big( \bs{p}_{[\ell, n_{\ell}]} \big) \; = \; \Big( \mf{w}_{\ups}^{(\ell)}\Big)^{-1}\bigg( \f{ \mf{x} \, +\, \mf{z}_{\ups}^{([\ell, n_{\ell}])}\big( \bs{p}_{[\ell, n_{\ell}]} \big) }{ \zeta_{\ell} }; t_{\ell}(k_0) \bigg) \;.
\label{ecriture forme explicite pour zero Vups}
\enq
By construction, the function $\mc{V}_{\ups}$ is smooth on $V_{[\ell,n_{\ell}]}$ as a composition of smooth functions.
By the Malgrange preparation Theorem \ref{Theorem Malgrange preparation theorem} applied to the function
\beq
\big( \mf{x}, \bs{p} \big) \; \mapsto \; \mf{x}+\mf{z}_{\ups}\big( \bs{p} \big)
\enq
of $\ov{\bs{n}}_{\ell}+1$ variables, $\ov{\bs{n}}_{\ell}=\sul{r=1}{\ell} n_r$, at the point $(0,\bs{t}(k_0))$, one concludes that there exist
\begin{itemize}
\item $\mf{x}_0>0$;
\item an open neighbourhood $V_{[\ell,n_{\ell}]}^{\prime} \subset V_{[\ell,n_{\ell}]}$ of $\bs{t}_{[\ell, n_{\ell}]}(k_0)$,
\item an open neighbourhood $I^{\prime}_{\ell}$ of $t_{\ell}(k_0)$,
\item a smooth, non-vanishing, function $h_{\ups}$ on $ \intoo{-\mf{x}_0}{ \mf{x}_0 } \times V_{[\ell,n_{\ell}]}^{\prime} \times I^{\prime}_{\ell}$ ,
\end{itemize}
such that, for $\bs{p}\in V_{[\ell,n_{\ell}]}^{\prime} \times I^{\prime}_{\ell}$ and $|\mf{x}| < \mf{x}_0$, it holds
\beq
\mf{x} + \mf{z}_{\ups}\big(\bs{p} \big) \, = \, \vsg_{\ups;\ell} \cdot \Big[ p_{n_{\ell}}^{(\ell)} - \mc{V}_{\ups}\big( \bs{p}_{[\ell, n_{\ell}]} \big) \Big] \cdot h_{\ups}(\mf{x};\bs{p})
\qquad \e{with} \quad \vsg_{\ups;\ell} \, =\, -\zeta_{\ell} \, \e{sgn}\Big( \mf{u}_{1}^{\prime}(k_0) + \ups \op{v} \Big) \;.
\label{ecriture factorisation x plus z pm}
\enq
Furthermore, given $\bs{p}_{\ups}$ as in \eqref{definition vecteur p ups}, partial differentiation of the relation \eqref{ecriture factorisation x plus z pm} in respect to $p_{n_{\ell}}^{(\ell)}$ allows one to conclude that
\beq
h_{\ups}\big( \mf{x}; \bs{p}_{\ups} \big) \; = \; \Big| \ups \op{v} + \mf{u}_{\ell}^{\prime}\Big( \mc{V}_{\ups}\big( \bs{p}_{[\ell, n_{\ell}]} \big) \Big) \Big| \, > \, 0\;.
\label{ecriture simplification locale de hups}
\enq
Let $ \bs{v} \, = \, \bs{p}_{[\ell, n_{\ell}]} \, - \,\bs{t}_{[\ell, n_{\ell}]}(k_0) $. The explicit expression for $\mc{V}_{\ups}$ given in \eqref{ecriture forme explicite pour zero Vups}
warrants that one has the small $\norm{ \bs{v} }$ expansion
\beq
\mc{V}_{\ups}\big( \bs{p}_{[\ell, n_{\ell}]} \big) \; = \; \mc{V}_{\ups;0} \, + \, L \cdot \mc{V}_{\ups;1}\,+ \, \mc{V}_{\ups;2}\big(L,Q\big) \, + \, \e{O}\big( \norm{\bs{v}}^3 \big) \;.
\label{ecriture developpement nu ups}
\enq
There, I have set
\beq
L \, = \, \f{1}{\zeta_{\ell} } \, \big( \bs{y}, \bs{v} \big) \qquad \e{with} \quad
\bs{y}^{\op{t}} \, = \, \Big( \big( \bs{y}^{(1)} \big)^{\op{t}} , \dots, \big( \bs{y}^{(\ell)} \big)^{\op{t}}\Big) \qquad \e{and} \qquad \big( \bs{y}^{(r)} \big)^{\op{t}}\, = \, \zeta_{r} \, \big( 1,\dots, 1 \big) \in \R^{ n_r - \de_{r,\ell} } \;,
\label{definition L et Q et vecteur y}
\enq
while
\beq
Q \, = \, \bigg(\bs{v}, \left(\ba{ccc} \ddots & 0 & 0 \\
0 & \tfrac{ \zeta_{r} }{ \zeta_{\ell} } \mf{u}^{\prime\prime}_{r}(t_r(k_0)) \op{I}_{n_r-\de_{\ell, r} } & 0 \\ 0 & 0 & \ddots \ea \right) \bs{v} \bigg) \;.
\enq
Above, $\op{I}_n$ denotes the identity matrix on $\R^n$, $(\cdot, \cdot)$ denotes the canonical scalar product on $\pl{r=1}{\ell}\R^{ n_{r}-\de_{r,\ell}}$ and $^{\op{t}}$ denotes the transposition.
The first three coefficients of the expansion \eqref{ecriture developpement nu ups} are given by
\beq
\mc{V}_{\ups;0} \, = \, \Big(\mf{w}_{\ups}^{(\ell)} \Big)^{-1}\big( \mf{x} ; t_{\ell}(k_0) \big) \, = \, t_{\ell}(k_0) \, + \, \f{ \tf{ \mf{x} }{ \zeta_{\ell}} }{ \ups \op{v} + \mf{u}_1^{\prime}\big( k_0 \big) } \, + \, \e{O}\big( \mf{x}^2 \big) \;,
\enq
\beq
\mc{V}_{\ups;1} \, = \, - \f{ \ups \op{v} + \mf{u}_1^{\prime}\big( k_0 \big) }{ \ups \op{v} + \mf{u}_{\ell}^{\prime}\big(\mc{V}_{\ups}^{(0)}\big) } \, = \,
-1 + \f{ \mf{x} \cdot \mf{u}_{\ell}^{\prime\prime}\big( t_{\ell}(k_0) \big) }{ \zeta_{\ell} \big[ \ups \op{v} + \mf{u}_1^{\prime}\big( k_0 \big) \big]^2 } \, + \, \e{O}\big( \mf{x}^2 \big)
\enq
and
\bem
\mc{V}_{\ups;2}\big(L,Q\big) \, = \, - \f{ Q }{ 2 \Big( \ups \op{v} + \mf{u}_{\ell}^{\prime}\big( \mc{V}_{\ups}^{(0)} \big) \, \Big) }
- \mf{u}_{\ell}^{\prime\prime}\big(\mc{V}_{\ups}^{(0)} \big)\cdot \f{ L^2 \, \big[ \ups \op{v} + \mf{u}_1^{\prime}\big( k_0 \big) \big]^2 }{ 2 \big[ \ups \op{v} + \mf{u}_{\ell}^{\prime}\big( \mc{V}_{\ups}^{(0)} \big) \big]^3 } \\
\, = \, - \f{ Q + L^2 \cdot \mf{u}_{\ell}^{\prime\prime}\big( t_{\ell}(k_0) \big) }{ 2 \Big( \ups \op{v} + \mf{u}_1^{\prime}\big( k_0 \big) \Big) } \; + \; \e{O}\Big(\mf{x} \cdot \big[ |Q| +L^2 \big] \Big)\; .
\end{multline}
These expansions ensure that
\beq
\mc{V}_{-;0}-\mc{V}_{+;0} \; = \; - \f{ 2 \mf{x} \, \op{v} \cdot \zeta_{\ell}^{-1} }{ \op{v}^2-\big( \mf{u}_1^{\prime}(k_0) \big)^2 } \; + \; \e{O}\big(\mf{x}^2\big) \; , \qquad
\mc{V}_{-;1}-\mc{V}_{+;1} \; = \; 4 \op{v} \zeta_{\ell}^{-1} \mf{x} \cdot \f{ \mf{u}_{\ell}^{\prime\prime}\big( t_{\ell}(k_0) \big) \cdot \mf{u}_1^{\prime} (k_0) }
{ \big[ \op{v}^2 - \big( \mf{u}_1^{\prime} (k_0) \big)^2 \big]^2 } \; + \; \e{O}\big(\mf{x}^2\big)
\enq
and
\beq
\mc{V}_{-;2}\big(L,Q\big) \, - \, \mc{V}_{+;2}\big(L,Q\big) \; = \; \f{ \op{v} }{ \op{v} ^2 - \big( \mf{u}_1^{\prime} (k_0) \big)^2 }\cdot \Big( Q + L^2 \cdot \mf{u}_{\ell}^{\prime\prime}\big( t_{\ell}(k_0) \big) \Big)
\; + \; \e{O}\Big(\mf{x} \cdot \big[ |Q| + L^2 \big] \Big)\; .
\enq
Thus, provided that $|\mf{x}|$ is small enough, there exist smooth functions $\mc{U}_1, \mc{U}_2$ on $V_{[\ell,n_{\ell}]}^{\prime}$ such that
\beq
\mc{V}_{-}\big( \bs{p}_{[\ell, n_{\ell}]} \big)\, - \, \mc{V}_{+}\big( \bs{p}_{[\ell, n_{\ell}]} \big) \; = \; \mf{x} \, \mc{U}_1\big( \bs{p}_{[\ell, n_{\ell}]} \big) \, + \, \mc{U}_2\big( \bs{p}_{[\ell, n_{\ell}]} \big) \;.
\enq
The functions $\mc{U}_{a}$, which may depend on $\mf{x}$, are such that
\beq
\mc{U}_1\big( \bs{p}_{[\ell, n_{\ell}]} \big) \; = \; \f{ -2\op{v} \zeta_{\ell} }{ \op{v} ^2 \, - \, \big( \mf{u}_1^{\prime}(k_0) \big)^2 } \; + \;\e{O}\big( |\mf{x}| + \norm{ \bs{v} } \big) \; , \quad
\mc{U}_2\big(\bs{p}_{[\ell, n_{\ell}]} \big) \; = \; \f{ - \zeta_{\ell} \, \op{v} \, \big(\bs{v},\op{M} \bs{v} \big) }{ \op{v}^2 - \big( \mf{u}_1^{\prime}(k_0) \big)^2 } \; + \;\e{O}\big( \norm{\bs{v}}^3 \big) \;,
\label{ecriture des dvpmnts de U1 et U2 en t de k0}
\enq
where I remind that $\bs{v} = \bs{p}_{[\ell, n_{\ell}]} - \bs{t}_{[\ell, n_{\ell}]}(k_0) $, $(\cdot , \cdot )$ is the canonical
scalar product on $\pl{r=1}{\ell}\R^{ n_{r}-\de_{r,\ell}}$ and the matrix $\op{M}$ takes the form
\beq
\op{M} \; = \; - \zeta_{\ell} \mf{u}^{\prime\prime}_{\ell}(t_{\ell}(k_0)) \; \bs{y}\cdot \bs{y}^{\op{t}} \, + \, \left(\ba{ccc} \ddots & 0 & 0 \\
0 & - \zeta_{r} \mf{u}^{\prime\prime}_{r}(t_r(k_0)) \op{I}_{n_r-\de_{\ell, r} } & 0 \\ 0 & 0 & \ddots \ea \right)
\label{definition matrice M}
\enq
with $\bs{y}$ as given by \eqref{definition L et Q et vecteur y}. Since $\op{M}$ is symmetric, it is diagonalisable and has real eigenvalues.
For further utility, one still needs to establish that these are non-vanishing. For that purpose, it is enough to show that $\op{M}$ has a non-zero determinant.
\noindent Upon factorising the diagonal part, one gets that
\beq
\det\big[ \op{M} \big] \; = \; \pl{r=1}{\ell}\Big\{ - \zeta_{r} \mf{u}^{\prime\prime}_{r}(t_r(k_0)) \Big\}^{ n_{r}-\de_{r,\ell}} \cdot \det\big[ \e{id} + \bs{w}\cdot \bs{y}^{\op{t}} \big]
\enq
with $\bs{y}$ as in \eqref{definition L et Q et vecteur y},
\beq
\bs{w}^{\op{t}} \, = \, \Big( \big( \bs{w}^{(1)} \big)^{\op{t}} , \dots, \big( \bs{w}^{(\ell)} \big)^{\op{t}}\Big)
\qquad \e{and} \qquad \big( \bs{w}^{(r)} \big)^{\op{t}}\, = \, \zeta_{\ell} \cdot \f{ \mf{u}^{\prime\prime}_{\ell}(t_{\ell}(k_0)) }{ \mf{u}^{\prime\prime}_{r}(t_r(k_0)) } \cdot \big( 1,\dots, 1 \big) \in \R^{ n_r - \de_{r,\ell} } \;.
\enq
The determinant can be computed explicitly and, upon using the relation $\tf{ \mf{u}^{\prime\prime}_{\ell}(t_{\ell}(k_0)) }{ \mf{u}^{\prime\prime}_{r}(t_r(k_0)) } = \tf{ t_{r}^{\prime}(k_0) }{ t_{\ell}^{\prime}(k_0) }$,
which follows from a differentiation of $ \mf{u}^{\prime}_{\ell}(t_{\ell}(k)) = \mf{u}^{\prime}_{r}(t_r(k)) $ at $k=k_0$, one eventually obtains that
\beq
\det\big[ \op{M} \big] \; = \; - \pl{r=1}{\ell}\Big\{ - \zeta_{r} \mf{u}^{\prime\prime}_{r}(t_r(k_0)) \Big\}^{ n_{r} } \cdot \f{ \mc{P}^{\prime}(k_0) }{ \mf{u}_1^{\prime \prime}(k_0) } \, \not = 0
\enq
since, by hypothesis \eqref{ecriture hypothese non vanishing impusion macro}, $\mc{P}^{\prime}(k_0)\not= 0$.
The above ensures that
\beq
\f{ \mc{U}_2\big( \bs{t}_{[\ell, n_{\ell}]}(k_0) \big) }{ \mc{U}_1\big( \bs{t}_{[\ell, n_{\ell}]}(k_0) \big) } \, = \, 0 \; \; , \quad
D_{\bs{t}_{[\ell, n_{\ell}]}(k_0)}\bigg( \f{ \mc{U}_2 }{ \mc{U}_1 } \bigg) \, = \, 0 \quad \e{and} \quad
D_{\bs{t}_{[\ell, n_{\ell}]}(k_0)}^{2} \bigg( \f{ \mc{U}_2 }{ \mc{U}_1 } \bigg) (\bs{s},\bs{s}^{\prime})\, = \, \big[ \mf{c}(\mf{x}) \big]^2 \cdot \big(\bs{s}, \op{M} \bs{s}^{\prime} \big)
\enq
for $\bs{s},\bs{s}^{\prime} \in \pl{r=1}{\ell}\R^{ n_{r}-\de_{r,\ell}}$ and where $\mf{c}(\mf{x})=1+\e{O}(\mf{x})$ is smooth in the neighbourhood of $\mf{x}=0$.
Thus, by virtue of the Morse lemma, Theorem \ref{Theorem Morse Lemma}, followed by a dilatation of variables by $\mf{c}(\mf{x})$, one infers that there exists
\begin{itemize}
\item an open neighbourhood $V^{\prime \prime }_{[\ell, n_{\ell}]}\subset V^{\prime}_{[\ell, n_{\ell}]}$ of $\bs{t}_{[\ell, n_{\ell}]}(k_0)$,
\item an open neighbourhood $W_{\bs{\phi}}$ of $\bs{0}$ in $\pl{r=1}{\ell}\R^{ n_{r}-\de_{r,\ell}}$,
\item a smooth diffeormorphism $\bs{\phi}_{\mf{x}}: \, W_{\bs{\phi}} \; \tend \; V^{\prime\prime}_{[\ell, n_{\ell}]} $, with $\bs{\phi}_{\mf{x}}(\bs{0})=\bs{t}_{ [\ell, n_{\ell}] }(k_0)$
\end{itemize}
such that
\beq
\f{ \mc{U}_2\big( \bs{\phi}_{\mf{x}}(\bs{v}) \big) }{ \mc{U}_1\big( \bs{\phi}_{\mf{x}}(\bs{v}) \big) } \; = \; \big(\bs{v}, \op{M} \bs{v} \big)\;.
\label{ecriture rectification de Morse U2 sur U1}
\enq
In particular, one readily infers from \eqref{ecriture rectification de Morse U2 sur U1} that $ \Big(D_{\bs{0}}\bs{\phi}_{\mf{x}} \Big)^{\op{t}}\cdot \op{M} \cdot D_{\bs{0}}\bs{\phi}_{\mf{x}} \, = \, 2 \, \op{M}$.
It is clear that the size of all the domains appearing above may be taken to be $\mf{x}$-independent, at least provided that $|\mf{x}|$ is small enough, say $|\mf{x}|<\mf{x}_0$, and that then
\beq
(\mf{x}, \bs{v} ) \; \mapsto \; \bs{\phi}_{\mf{x}}(\bs{v}) \;,
\enq
is smooth on $\intoo{ - \mf{x}_0 }{ \mf{x}_0 } \times W_{\bs{\phi}}$. Clearly, upon adjusting the parameters, one may take $\mf{x}_0$ as introduced earlier on.
\subsubsection*{$\bullet$ Canonical form of $\tilde{z}_{\ups}$}
The very same reasoning applied to the function $\tilde{z}_{\ups}(\bs{x})$, as defined in \eqref {definition fonction z tilde ups}, ensures that there exist
\begin{itemize}
\item an open neighbourhood $V^{(0)}_{[\ell,n_{\ell}]}$ of $ \bs{0} \in \pl{r=1}{\ell}\R^{ n_{r}-\de_{r,\ell}}$,
\item a segment $I_{\ell}^{(0)}$ containing an open neighbourhood of $0\in \R$ ,
\item a smooth, non-vanishing, function $h_{\ups}^{(0)}$ on $ \intoo{ - \mf{x}_0 }{ \mf{x}_0 } \times V^{(0)}_{[\ell,n_{\ell}]} \times I_{\ell}^{(0)}$,
\item a smooth function $\mc{V}_{\ups}^{(0)}$ on $\intoo{ - \mf{x}_0 }{ \mf{x}_0 } \times V^{(0)}_{[\ell,n_{\ell}]}$,
\end{itemize}
such that
\beq
\mf{x} + \tilde{z}_{\ups}(\bs{x}) \, = \, \vsg_{\ups;\ell} \cdot \Big[ x_{n_{\ell}}^{(\ell)} - \mc{V}_{\ups}^{(0)}\big( \bs{x}_{[\ell, n_{\ell}]} \big) \Big] \cdot h_{\ups}^{(0)}(\mf{x}; \bs{x})
\qquad \e{with} \quad \vsg_{\ups;\ell} \, =\, -\zeta_{\ell} \, \e{sgn}\big( \mf{u}_{1}^{\prime}(k_0) + \ups \op{v} \big) \;.
\label{ecriture factorisation x plus z pm modele effectif}
\enq
Here, again, I kept the $\mf{x}$-dependence of $ \mc{V}_{\ups}^{(0)}$ implicit.
Note that here, $\vsg_{\ups;\ell}$ is exactly as defined in \eqref{ecriture factorisation x plus z pm} owing to the very choice of the parameters
$\veps_a$, $\xi_a$, $\op{u}$, defining the effective function $\tilde{z}_{\ups}(\bs{x})$. The function $h_{\ups}^{(0)}$ enjoys the identity
\beq
h_{\ups}^{(0)}\big( \mf{x}; \bs{x}_{\ups} \big) \; = \; \Big| \ups \op{v} + \mf{h}_{\ell}^{\prime}\Big( \mc{V}_{\ups}^{(0)}\big( \bs{x}_{[\ell, n_{\ell}]} \big) \Big) \Big| \, > \, 0
\quad \e{with} \quad
\bs{x}_{\ups} \, = \, \Big( \bs{x}_{[\ell, n_{\ell}]} , \mc{V}_{\ups}^{(0)}\big( \bs{x}_{[\ell, n_{\ell}]} \big) \Big) \;.
\label{ecriture simplification locale de hups effectif}
\enq
Furthermore, there exist two smooth functions on $V^{(0)}_{[\ell,n_{\ell}]}$ such that
\beq
\mc{V}_{-}^{(0)}\big( \bs{x}_{[\ell, n_{\ell}]} \big)\, - \, \mc{V}_{+}^{(0)}\big( \bs{x}_{[\ell, n_{\ell}]} \big) \; = \; \mf{x} \, \mc{U}_1^{(0)}\big( \bs{x}_{[\ell, n_{\ell}]} \big) \, + \, \mc{U}_2^{(0)}\big( \bs{x}_{[\ell, n_{\ell}]} \big)
\enq
and satisfying
\beqa
\mc{U}_1^{(0)}\big( \bs{x}_{[\ell, n_{\ell}]} \big) & = & \f{ -2\op{v} \zeta_{\ell} }{ \op{v} ^2 \, - \, \big( \mf{u}_1^{\prime}(k_0) \big)^2 } \; + \;\e{O}\big( |\mf{x}|+\norm{ \bs{x}_{[\ell, n_{\ell}]} } \big) \label{ecriture des dvpmnts de U1 0 et U20 en 0} \\
\mc{U}_2^{(0)}\big(\bs{x}_{[\ell, n_{\ell}]} \big) & = & \f{ - \zeta_{\ell} \, \op{v} }{ \op{v}^2 - \big( \mf{u}_1^{\prime}(k_0) \big)^2 } \, \big( \bs{x}_{[\ell, n_{\ell}]} ,\op{M} \, \bs{x}_{[\ell, n_{\ell}]} \big)
\; + \;\e{O}\big( \norm{\bs{x}_{[\ell, n_{\ell}]}}^3 \big) \;.
\eeqa
Following the above reasoning, and re-adjusting the domains $W_{\bs{\phi}}, V^{\prime\prime}_{[\ell, n_{\ell}]}$ appearing above if necessary, one eventually concludes that there exists
a smooth diffeormorphism $\bs{\phi}^{(0)}_{\mf{x}}: \, W_{\bs{\phi}} \; \tend \; V^{(0)}_{[\ell, n_{\ell}]} $ such that
\beq
\f{ \mc{U}_2^{(0)}\big( \bs{\phi}^{(0)}_{\mf{x}}(\bs{v}) \big) }{ \mc{U}_1^{(0)}\big( \bs{\phi}^{(0)}_{\mf{x}}(\bs{v}) \big) } \; = \; \big(\bs{v}, \op{M} \bs{v} \big) \quad \e{and} \quad
\left\{ \ba{ccc} D_{\bs{0}}\bs{\phi}^{(0)}_{\mf{x}} & = & D_{\bs{0}}\bs{\phi}_{\mf{x}} \\
\bs{\phi}^{(0)}_{\mf{x}}\big( \bs{0} \big) & = & \bs{0} \ea \right. \;.
\enq
Likewise to the previous situation, $(\mf{x}, \bs{v}) \mapsto \bs{\phi}^{(0)}_{\mf{x}}(\bs{v})$ is smooth on $\intoo{ - \mf{x}_0 }{ \mf{x}_0 } \times W_{\bs{\phi}}$.
I stress that the open neighbourhood $W_{\bs{\phi}}$ appearing above coincides exactly with the domain of the diffeomorphism $\bs{\phi}_{\mf{x}}$ introduces earlier on.
Also, I should comment relatively to the possibility of choosing $\bs{\phi}^{(0)}_{\mf{x}}$ such that $D_{\bs{0}}\bs{\phi}^{(0)}_{\mf{x}} \, = \, D_{\bs{0}}\bs{\phi}_{\mf{x}}$.
Just as for the case of $\bs{\phi}_{\mf{x}}$, one deduces that any Morse function $\bs{\phi}^{(0)}_{\mf{x}}$ rectifying $\tf{ \mc{U}_2^{(0)} }{ \mc{U}_1^{(0)} }$ has to satisfy
$ \Big(D_{\bs{0}}\bs{\phi}^{(0)}_{\mf{x}} \Big)^{\op{t}} \cdot \op{M} \cdot D_{\bs{0}}\bs{\phi}^{(0)}_{\mf{x}} \, = \, 2 \op{M}$.
\textit{A priori} this equation has a space of solutions that is isomorphic to $SO(p,q)$ where
$(p,q)$ is the signature of $\op{M}$. However, upon looking at the proof of the Morse Lemma, one constructs
a Morse function from a given choice of a solution to this equation. Thus, when constructing $\bs{\phi}^{(0)}_{\mf{x}}$, the latter can always be chosen
so that $D_{\bs{0}}\bs{\phi}^{(0)}_{\mf{x}} \, = \, D_{ \bs{0} }\bs{\phi}_{\mf{x}}$.
\subsubsection*{$\bullet$ The \textit{per-se} rectification}
With all the ingredients being introduced, one may define the smooth diffeomorphism
\beq
\Phi_{\mf{x}} \; : \; \left\{ \ba{ccc} V^{(0)}_{[\ell,n_{\ell}]} & \tend & V^{\prime\prime}_{[\ell,n_{\ell}]} \vspace{3mm} \\
\bs{x}_{[\ell,n_{\ell}]} & \mapsto & \bs{\phi}_{\mf{x}} \circ \Big( \bs{\phi}^{(0)}_{\mf{x}} \Big)^{-1} \big( \bs{x}_{[\ell,n_{\ell}]} \big) \ea \right.
\quad \e{which} \; \e{satisfies} \quad
\f{\mc{U}_2}{\mc{U}_1}\circ \Phi_{\mf{x}} \, = \, \f{ \mc{U}_2^{(0)} }{ \mc{U}_1^{(0)} } \;.
\enq
One is now in position to introduce the smooth map
\beq
\Psi_{\mf{x}} \; : \; \left\{ \ba{ccc} V^{(0)}_{[\ell,n_{\ell}]} \times I_{\ell}^{(0)} & \tend & \Psi_{\mf{x}}\Big( V^{(0)}_{[\ell,n_{\ell}]} \times I_{\ell}^{(0)} \Big) \\
\bs{x} & \mapsto & \Psi_{\mf{x}} (\bs{x}) \, = \, \bigg( \Phi_{\mf{x}}(\bs{x}_{[\ell,n_{\ell}]} ),
\mc{V}_{-}\circ\Phi_{\mf{x}} (\bs{x}_{[\ell,n_{\ell}]} ) \, + \, \Big( x_{n_{\ell}}^{(\ell)} - \mc{V}_{-}^{(0)}(\bs{x}_{[\ell,n_{\ell}]} ) \Big)
\cdot \f{\mc{U}_1\circ \Phi_{\mf{x}} (\bs{x}_{[\ell,n_{\ell}]} ) }{\mc{U}_1^{(0)}(\bs{x}_{[\ell,n_{\ell}]} ) }
\bigg) \ea \right. \;.
\enq
I first establish that $\Psi_{\mf{x}}$ is a diffeomorphism.
Indeed, from its very construction, one has that $\Phi_{\mf{x}}(\bs{0})=\bs{t}_{[\ell,n_{\ell}]}(k_0)$ and that $D_{ \bs{0} } \Phi_{\mf{x}} \, = \, \op{I}_{ \ov{\bs{n}}_{\ell}-1}$, with $\ov{\bs{n}}_{\ell}=\sul{r=1}{\ell}n_r$.
Thence, the expansions \eqref{ecriture des dvpmnts de U1 et U2 en t de k0} and \eqref{ecriture des dvpmnts de U1 0 et U20 en 0} ensure that
\beq
\left. \f{\mc{U}_1\circ \Phi_{\mf{x}} (\bs{x}_{[\ell,n_{\ell}]} ) }{\mc{U}_1^{(0)}(\bs{x}_{[\ell,n_{\ell}]} ) } \right|_{ \bs{x}_{[\ell,n_{\ell}]} = \bs{0} } \; = \;
1+\e{O}\big( \mf{x} \big) \;.
\label{ecriture valeur en zero ration U1 sur U1 effectif}
\enq
Furthermore, since $\mf{z}_{\ups}^{([\ell,n_{\ell}])} \big( \bs{t}_{[\ell,n_{\ell}]}(k_0) \big)=0$, one has
\beq
\mc{V}_{\ups} \Big( \bs{t}_{[\ell,n_{\ell}]}(k_0) \Big) \; = \; \big( \mf{w}^{(\ell)}_{\ups} \big)^{-1}\Big( \f{ \mf{x} }{ \zeta_{\ell}} ; t_{\ell}(k_0) \Big) \, = \, t_{\ell}(k_0)+\e{O}(\mf{x})
\label{ecriture dvpmt local zero V ups}
\enq
and, similarly, $\mc{V}_{\ups}^{(0)} ( \bs{0} ) \; = \; \e{O}(\mf{x})$. All of the above put together entails that
\beq
{\Psi_{\mf{x}}(\bs{x})}_{\mid_{\bs{x}=0}} \; = \; \Big( \bs{t}_{[\ell,n_{\ell}]}(k_0) , t_{\ell}(k_0) + \e{O}(\mf{x}) \Big) \, = \, \bs{t}(k_0) \, + \, \big( \underbrace{ \bs{0} }_{ \in \R^{ \ov{\bs{n}}_{\ell}-1 } } , \e{O}(\mf{x}) \big) \;.
\label{ecriture valeur Psi en origine}
\enq
Furthermore, denote by $\big[\Psi_{\mf{x}}(\bs{x})\big]_{n_{\ell}}^{(\ell)} $ the ultimate scalar entry of $\Psi_{\mf{x}}(\bs{x})$. Then, the expansion \eqref{ecriture developpement nu ups}
and an analogous one for $\mc{V}_{-}^{(0)}(\bs{x}_{[\ell,n_{\ell}]})$, yields for $\bs{u}\in \R^{\ov{\bs{n}}_{\ell}-1}$ and $s\in \R$
\beq
D_{\bs{0}} \Big( \big[\Psi_{\mf{x}}(\bs{x})\big]_{n_{\ell}}^{(\ell)} \Big) \cdot \big(\bs{u},s \big) \, = \, \zeta_{\ell}^{-1}\, \mc{V}_{-;1} \, \big(D_{\bs{0}}\Phi_{\mf{x}} \cdot \bs{u},\bs{y} \big)
+\Big(s-\zeta_{\ell}^{-1}\, \mc{V}_{-;1}^{(0)} \, \big( \bs{u},\bs{y} \big) \Big) \, \f{\mc{U}_1\circ \Phi_{\mf{x}} (\bs{0} ) }{\mc{U}_1^{(0)}(\bs{0} ) } \; + \;
\underbrace{ \mc{V}_{-;0}^{(0)} }_{=\e{O}( \mf{x} ) } \,
D_{\bs{0}} \bigg( \f{\mc{U}_1\circ \Phi_{\mf{x}} }{\mc{U}_1^{(0)} } \bigg) \cdot \bs{u} \;.
\enq
Since $\mc{V}_{-;0}^{(0)} = \e{O}( \mf{x} ) $ and $\mc{V}_{-;1} -\mc{V}_{-;1}^{(0)} = \e{O}(\mf{x}^2)$, it holds that there exists a linear form $\mc{L}$ on $\R^{ \ov{\bs{n}}_{\ell} }$
such that
\beq
D_{\bs{0}} \Big( \big[\Psi_{\mf{x}}(\bs{x})\big]_{n_{\ell}}^{(\ell)} \Big) \cdot \big(\bs{u},s \big) \, = \, s \, + \, \mf{x} \, \mc{L}\cdot \big(\bs{u},s \big) \;.
\enq
Thence, there exists an endomorphism $\op{N}_{\Psi}$ on $\R^{\ov{\bs{n}}_{\ell} }$, with $ \norm{ \op{N}_{\Psi} } \leq C $ for some $\mf{x}$-independent constant,
such that $D_{\bs{0}} \Psi_{\mf{x}} \, = \, \e{id} + \mf{x} \op{N}_{\Psi} $ \;.
Thus, $\Psi_{\mf{x}}$ is invertible in some open neighbourhood of $\bs{0}$ which, upon reducing $V^{(0)}_{[\ell,n_{\ell}]}$ and $I_{\ell}^{(0)}$ if necessary,
may be taken to be $V^{(0)}_{[\ell,n_{\ell}]} \times I_{\ell}^{(0)}$. Note that the estimates on the differential $D_{\bs{0}} \Psi_{\mf{x}} \, = \, \e{id} + \mf{x} \op{N}_{\Psi} $
and \eqref{ecriture valeur Psi en origine} ensures that $ \bs{t}(k_0)\in \Psi_{\mf{x}}\Big( V^{(0)}_{[\ell,n_{\ell}]} \times I_{\ell}^{(0)} \Big)$
and that the latter set contains a $\mf{x}$-independent open neighbourhood of $ \bs{t}(k_0)$ in $\R^{ \ov{\bs{n}}_{\ell} }$.
All is now in place so as to establish the rectification relation. Observe that there exists a direct relation between the zeroes $\mc{V}_{\pm} $ and $\mc{V}_{\pm}^{(0)}$:
\bem
\mc{V}_{-}\circ\Phi_{\mf{x}} (\bs{x}_{[\ell,n_{\ell}]} ) \, - \, \mc{V}_{+}\circ\Phi_{\mf{x}} (\bs{x}_{[\ell,n_{\ell}]} )
\, = \, \mc{U}_1\circ \Phi_{\mf{x}} (\bs{x}_{[\ell,n_{\ell}]} ) \Bigg[ \mf{x} + \f{\mc{U}_2\circ \Phi_{\mf{x}} (\bs{x}_{[\ell,n_{\ell}]} )}{\mc{U}_1\circ \Phi_{\mf{x}} (\bs{x}_{[\ell,n_{\ell}]} )} \Bigg] \\
\, = \, \mc{U}_1\circ \Phi_{\mf{x}} (\bs{x}_{[\ell,n_{\ell}]} ) \cdot \bigg[ \mf{x} + \Big( \big(\bs{\phi}^{(0)}_{\mf{x}}\big)^{-1}(\bs{x}_{[\ell,n_{\ell}]} ), \op{M} \, \big(\bs{\phi}^{(0)}_{\mf{x}}\big)^{-1}(\bs{x}_{[\ell,n_{\ell}]} ) \Big) \bigg] \\
\, = \, \mc{U}_1\circ \Phi_{\mf{x}} (\bs{x}_{[\ell,n_{\ell}]} ) \cdot \Bigg[ \mf{x} + \f{ \mc{U}_2^{(0)}\big( \bs{x}_{[\ell,n_{\ell}]} \big) }{ \mc{U}_1^{(0)} \big( \bs{x}_{[\ell,n_{\ell}]} \big) } \Bigg]
\; = \; \f{ \mc{U}_1\circ \Phi_{\mf{x}} (\bs{x}_{[\ell,n_{\ell}]} ) }{ \mc{U}_1^{(0)} \big( \bs{x}_{[\ell,n_{\ell}]} \big) } \cdot
\Big[ \mc{V}_{-}^{(0)} (\bs{x}_{[\ell,n_{\ell}]} ) \, - \, \mc{V}_{+}^{(0)} (\bs{x}_{[\ell,n_{\ell}]} ) \Big] \;.
\end{multline}
This identity entails that, for any $\ups \in \{\pm 1\}$,
\beq
\big[\Psi_{\mf{x}}(\bs{x})\big]_{n_{\ell}}^{(\ell)} - \mc{V}_{\ups}\circ\Phi_{\mf{x}} (\bs{x}_{[\ell,n_{\ell}]} ) \; = \;
\Big[ x_{n_{\ell}}^{(\ell)} - \mc{V}_{\ups}^{(0)} (\bs{x}_{[\ell,n_{\ell}]} ) \Big] \cdot \f{\mc{U}_1\circ \Phi_{\mf{x}} (\bs{x}_{[\ell,n_{\ell}]} ) }{\mc{U}_1^{(0)}(\bs{x}_{[\ell,n_{\ell}]} ) } \;.
\label{ecriture trivialisation coordonnel ell nelle de Psi}
\enq
Thus, starting from the factorisation \eqref{ecriture factorisation x plus z pm} and applying the equality \eqref{ecriture trivialisation coordonnel ell nelle de Psi} followed by
an application of the factorisation \eqref{ecriture factorisation x plus z pm modele effectif} backwards, one gets that
\beq
\mf{x} + \mf{z}_{\ups}\circ\Psi_{\mf{x}}(\bs{x}) \, = \, \mf{f}_{\ups}(\mf{x};\bs{x}) \cdot \Big( \mf{x} + \tilde{z}_{\ups}(\bs{x}) \Big)
\qquad \e{with} \qquad
\mf{f}_{\ups}(\mf{x}; \bs{x}) \, = \, \f{\mc{U}_1\circ \Phi_{\mf{x}} (\bs{x}_{[\ell,n_{\ell}]} ) }{\mc{U}_1^{(0)}(\bs{x}_{[\ell,n_{\ell}]} ) } \cdot
\f{ h_{\ups}\Big(\mf{x}; \Psi (\bs{x} ) \Big) }{ h_{\ups}^{(0)}(\mf{x};\bs{x} ) } \;.
\enq
Clearly, $\mf{f}_{\ups}$ is smooth. I now establish that $\mf{f}_{\ups}$ has the claimed form of the expansion.
Putting \eqref{ecriture dvpmt local zero V ups} and \eqref{ecriture valeur Psi en origine} together, one infers that, for any $\ups\in \{ \pm \}$,
\beq
\Psi_{\mf{x}}(\bs{0}) \, = \,\Big( \bs{t}_{[\ell, n_{\ell}]}(k_0), \mc{V}_{\ups}\big(\bs{t}_{[\ell, n_{\ell}]}(k_0) \big) \Big) + \e{O}\big( \mf{x} \big) \;.
\enq
This, along with $\mc{V}^{(0)}_{\ups}(0)=\e{O}(\mf{x})$ and the smoothness of $h_{\ups}$ and $h_{\ups}^{(0)}$,
allows one to use the expressions \eqref{ecriture simplification locale de hups} and \eqref{ecriture simplification locale de hups effectif} so as to
deduce that
\bem
\left. \f{ h_{\ups}\Big(\mf{x};\Psi_{\mf{x}} (\bs{x}) \Big) }{ h_{\ups}^{(0)}(\mf{x}; \bs{x}) } \right|_{\bs{x}=0} \; = \;
\f{ h_{\ups}(\mf{x};\bs{p}_{\ups})\mid_{\bs{p}_{[\ell,n_{\ell}]}= \bs{t}_{[\ell,n_{\ell}]}(k_0) }+ \e{O}(\mf{x}) }
{ h_{\ups}^{(0)}(\mf{x};\bs{x}_{\ups})\mid_{\bs{x}_{[\ell,n_{\ell}]}= \bs{0} } + \e{O}(\mf{x} ) } \, = \,
\f{ \big| \ups \op{v}+\mf{u}^{\prime}_{\ell}\big( t_{\ell}(k_0)\big) + \e{O}(\mf{x}) \big| + \e{O}(\mf{x} ) }
{ \big| \ups \op{v}+\mf{h}^{\prime}_{\ell}\big( 0 \big) + \e{O}(\mf{x}) \big| + \e{O}(\mf{x}) } \\
\, = \, \f{ \ups \op{v}+\mf{u}^{\prime}_{1}\big( k_0 \big) + \e{O}(\mf{x}) }
{ \ups \op{v}+ \op{u} + \e{O}(\mf{x}) }
\, = \, 1 + \e{O}(\mf{x}) \;,
\end{multline}
where $\bs{p}_{\ups}$, resp. $\bs{x}_{\ups}$, are as defined through \eqref{definition vecteur p ups}, resp. \eqref{ecriture simplification locale de hups effectif}.
Furthermore, I used that $\mf{u}^{\prime}_1(k_0)=\op{u}$.
Thence, a similar result for the ratio of $\mc{U}_1$'s established in \eqref{ecriture valeur en zero ration U1 sur U1 effectif}, and smoothness in $\bs{x}$ all together,
entail that one has $ \mf{f}_{\ups}(\mf{x};\bs{x}) = 1+ \e{O}\big( \norm{\bs{x}}+|\mf{x}| \big) $.
Recall the definition \eqref{ecriture domaine D eff eta prime the rectification des z ups} of the domain $\mc{D}^{(\e{eff})}_{\eta^{\prime}}$.
To complete the proof, it remains to establish that
\beq
\mc{D}^{(\e{eff})}_{\eta^{\prime}} \subset V^{(0)}_{[\ell,n_{\ell}]} \times I_{\ell}^{(0)} \qquad \e{provided}\; \e{that} \qquad 0<\eta^{\prime}<\eta_0
\enq
for some $\eta_0>0$ small enough. The map
\beq
G \; : \; \R^{ \ov{\bs{n}}_{\ell}} \tend \R^{\ov{\bs{n}}_{\ell}} \quad \e{such}\, \e{that} \quad
\big[G(\bs{x})\big]_{a}^{(r)} = t_{r}^{(0)}\big( x_1^{(1)} \big) - x_a^{(r)} (1-\de_{a,1}\de_{r,1}) \;,
\enq
is obviously continuous, and thus, upon agreeing to denote $B_{\eps}(0)= \big\{x \in \R \; : \; |x| < \eps \big\}$ the open ball around $0$ in $\R$ of radius $\eps$, one gets that
\beq
\mc{D}^{(\e{eff})}_{\eta^{\prime}} \; = \; G^{-1} \bigg( B_{C \eta^{\prime}}(0) \times \pl{r=1}{\ell} \Big( B_{ \xi_r^2 \eta^{\prime} }(0) \Big)^{ n_r- \de_{r,1} } \bigg)
\enq
is open as a pre-image of an open set by a continuous function. Since $\bs{0} \in \mc{D}^{(\e{eff})}_{\eta^{\prime}} $, it is an open neighbourhood of that point in $\R^{ \ov{\bs{n}}_{\ell} }$. Since its diameter
shrinks to $0$ as $\eta^{\prime}\tend 0$, and since $V^{(0)}_{[\ell,n_{\ell}]} \times I_{\ell}^{(0)}$ is also an open neighbourhood of $\bs{0}$ in $\R^{ \ov{\bs{n}}_{\ell} }$, the claim follows. \qed
\subsection{Factorisation of the maps $\tilde{z}_{\ups}$}
\begin{prop}
\label{Proposition factorisation jolie en zeroes de tilde z ups}
Let $\tilde{z}_{\ups}$ correspond to the below multivariate polynomial on $\pl{r=1}{\ell}\R^{n_r}$:
\beq
\tilde{z}_{\ups}(\bs{x}) \; = \;- \sul{ (r,a)\in \mc{M} }{} \zeta_{r} \bigg\{ \mf{h}_r\big( x_a^{(r)} \big) +\ups \op{v} x_a^{(r)} \bigg\} \qquad where \qquad
\mf{h}_r\big( x \big) \; = \; - \zeta_r \veps_r \f{ x^2 }{ 2 \xi_r^2 } \, + \, \op{u}\, x \;,
\enq
$\veps_{r}\in \{ \pm \}$, $\xi_r \in \R^*$ and $(\op{u}, \op{v}) \in \R \times \R^+$.
Then, there exists a linear map $\op{M} $ on $\pl{r=1}{\ell}\R^{n_r} $ such that:
\begin{itemize}
\item $\op{M}$ is invertible;
\item there exist integers $m_{\pm} \in \mathbb{N}$ satisfying $m_++m_-+1=\sul{r=1}{\ell}n_r$ such that
it holds
\beq
\tilde{z}_{\ups}\Big( \op{M} (y,\bs{z}) \Big) \, = \, \wt{P}_{\ups}(y,\bs{z}) \quad with \quad \bs{z}=\big( \bs{z}^{(+)}, \bs{z}^{(-)} \big) \in \R^{m_+}\times \R^{m_-}
\enq
and
\beq
\wt{P}_{\ups}(y,\bs{z}) \; = \; \f{ y^2 }{2 \mc{P}_{\e{eff}} } -\big( \op{u}+\ups \op{v} \big) y \, + \, \sul{s=1}{m_+} \big( z_s^{(+)} \big)^2 \, - \; \sul{s=1}{m_-} \big( z_s^{(-)} \big)^2 \;,
\quad with \quad \mc{P}_{\e{eff}}=\sul{r=1}{\ell} \veps_r n_r \xi_r^2 \;.
\enq
\end{itemize}
\end{prop}
\Proof
One may recast the polynomial $\tilde{z}_{\ups}$ in the form
\beq
\tilde{z}_{\ups}(\bs{x}) \; = \; - \big( \op{u}+\ups \op{v} \big) \, \ov{\bs{x}}_{\zeta} \, + \, \sul{ (r,a)\in \mc{M} }{} \f{\veps_{r}}{2 \xi_r^2} \big( x_a^{(r)} \big)^2 \qquad \e{with} \qquad
\ov{\bs{x}}_{\zeta} \; = \; \sul{ (r,a)\in \mc{M} }{} \zeta_{r} x_a^{(r)} \;.
\enq
Then, let $\wt{\op{M}} \, = \, \wt{\op{D}} \, + \, \bs{g}\cdot \bs{e}^{\op{t}}$ with $\bs{e}^{\op{t}} = \big(1, \dots, 1 \big) \in \R^{ \ov{\bs{n}}_{\ell} }$,
$\bs{g}^{\op{t}} = \big( \big(\bs{g}^{(1)}\big)^{\op{t}} , \dots, \big( \bs{g}^{(\ell)} \big)^{\op{t}} \big) $ and where
\beq
\Big( \bs{g}^{(r)} \Big)^{\op{t}} = \f{ \zeta_1 \zeta_r \veps_r }{ \mc{P}_{\e{eff}} } \cdot \xi_r^2 \cdot \big(1, \dots, 1 \big) \in \R^{ n_r }
\qquad \e{and} \qquad \wt{\op{D}} \, = \, \left( \ba{ccccc} 0 & \cdots & \cdots & \cdots & 0 \\
\vdots & -I_{n_1-1} & 0 & \cdots \\
\vdots & 0 & -\zeta_1 \zeta_2 I_{n_2} & 0 & \vdots \\
\vdots & & 0 & \ddots & 0 \\
0 & \cdots & 0 & 0 & -\zeta_1 \zeta_{\ell} I_{n_{\ell}} \\ \ea \right) \;.
\enq
It is straightforward to see that $|\det[\wt{\op{M}}]|=| \mc{P}_{\e{eff}} |^{-1} \not=0$.
Then, a straightforward calculation shows that, given $\bs{y} \in \pl{r=1}{\ell} \R^{n_{r} }$
\beq
\tilde{z}_{\ups}\big( \, \wt{\op{M}} \bs{y} \, \big) \, = \, + \f{ \big( y_1^{(1)} \big)^2 }{2 \mc{P}_{\e{eff}} }
- \zeta_1 \big( \op{u} + \ups \op{v} \big) y_1^{(1)} + \mc{Q} \big( \bs{y}_{[1,1] } \big)
\enq
in which I employed the convention introduced in \eqref{definition vecteur p avec composantes omises}, while the quadratic form $\mc{Q}$ reads
\beq
\mc{Q}\big( \bs{y}_{[1,1] } \big) \, = \, \sul{ \substack{ (r,a) \in \\ \mc{M}_{[1,1]} } }{} \f{ \zeta_r }{ 2 \xi_r^2 } \big( y_a^{(r)} \big)^2
\, - \; \f{1}{2 \mc{P}_{\e{eff}} } \bigg( \sul{ \substack{ (r,a) \in \\ \mc{M}_{[1,1]} } }{} y_a^{(r)} \bigg)^2 \;.
\enq
Here $\mc{M}_{[1,1]}$ is as defined in \eqref{definition ensemble M elle n ell enleve}.
Representing the quadratic form as $ \mc{Q}\big( \bs{y}_{[1,1] } \big) \, = \, \Big( \bs{y}_{[1,1] } , \op{M}_{\mc{Q}} \bs{y}_{[1,1] } \Big) $, one gets that
the matrix $\op{M}_{\mc{Q}}$ is a rank one perturbation of a diagonal matrix:
\beq
\op{M}_{\mc{Q}}\; = \; \op{D}_{\mc{Q}} \, - \, \f{ 1 }{ 2 \mc{P}_{ \e{eff} } } \bs{e} \cdot \bs{e}^{\op{t}} \quad \e{ with} \quad \bs{e}^{\op{t}} = \big(1, \dots, 1 \big) \in \R^{ \ov{\bs{n}}_{\ell} -1 }
\enq
and where I denoted
\beq
\op{D}_{\mc{Q}}\, = \, \left( \ba{ccccc} \\
\tfrac{1}{2} \veps_1 \cdot \xi_1^{-2} \cdot I_{n_1-1} & 0 & \cdots &0 \\
0 & \tfrac{1}{2} \veps_2 \cdot \xi_2^{-2} I_{n_2} & 0 & \vdots \\
& 0 & \ddots & 0 \\
0 & \cdots & \cdots & \tfrac{1}{2} \veps_{\ell} \cdot \xi_{\ell}^{-2} I_{n_{\ell}} \\ \ea \right) \;.
\enq
The determinant of $\mc{M}_{\mc{Q}}$ can thus be computed in a closed form
\beq
\det\big[ \op{M}_{\mc{Q}} \big] \, = \, \f{ \veps_1 \, \xi_1^2 }{ \mc{P}_{\e{eff}} } \pl{r=1 }{\ell} \bigg\{ \f{ \veps_r }{ 2 \xi_r^2} \bigg\}^{n_r- \de_{r,1} } \; \not= \; 0 \;.
\enq
$ \op{M}_{\mc{Q}}$ being invertible and symmetric, there exists an orthogonal linear map $\op{N}$ such that
\beq
\op{M}_{\mc{Q}} \; = \; \op{N} \left( \ba{cc} \op{I}_{m_+} & 0 \\ 0 & - \op{I}_{m_-}\ea \right) \op{N}^{\op{t}}
\enq
in which $(m_{+},m_{-})$ is the signature of $\op{M}_{\mc{Q}} $. Thus the map
\beq
\op{M}\, = \, \wt{\op{M}} \cdot \left( \ba{cc} \zeta_1 & 0 \\ 0 & \op{N} \ea \right)
\enq
does the job. \qed
\subsection{Local expansion of a Vandermonde determinant}
Recall the notations for norms and partial order on vectors of integers \eqref{notation norme et ordre partiel sur entiers vecteurs} and the one for exponents $\bs{x}^{\bs{\a}}$ \eqref{notation exposant polynomial vectoriel}
with $\bs{x}\in \mathbb{R}^{\ov{\bs{n}}_{\ell} } \, = \, \pl{r=1}{\ell} \R^{ n_{r}} $ and $\bs{\a}\in \mathbb{N}^{\ov{\bs{n}}_{\ell} }$.
$\ov{\bs{n}}_{\ell}$ is as defined in \eqref{definition bs ne ell}.
\begin{lemme}
\label{Lemme VdM local expansion}
Let $\Psi: U \tend \Psi(U)$ be a smooth diffeomorphism on a open neighbourhood $U$ of $\bs{0}\in \mathbb{R}^{\ov{\bs{n}}_{\ell} }$ such that
\begin{itemize}
\item $\Psi(\bs{0})=\bs{v} \in \mathbb{R}^{\ov{\bs{n}}_{\ell} }$ with $\bs{v}=\big(\bs{v}^{(1)},\dots, \bs{v}^{(\ell)} \big)$, each entry $\bs{v}^{(r)}\, = \, \big(v^{(r)},\dots, v^{(r)} \big) \in \R^{n_r}$ having equal coordinates;
\item $D_{\bs{x}}\Psi = \e{id}+\mf{x} \op{N}_{\Psi}$, with $\op{N}_{\Psi} \in \mc{L}\big( \mathbb{R}^{\ov{\bs{n}}_{\ell} } \big)$ such that $\norm{ \op{N}_{\Psi} } \leq C$, for a $|\mf{x}|$-independent constant $C$.
\end{itemize}
Let
\beq
V(\bs{x})=\pl{r=1}{\ell} \pl{a<b}{n_r} \big(x_a^{(r)}-x_b^{(r)} \big)^2
\enq
be a product of Vandermonde determinants relative to each of the $r$-coordinates.
Then, there exists a smooth map $Q:U \tend \R$ such that $V\big( \Psi( \bs{x} ) \big) \, = \, V(\bs{x}) \, + \, Q(\bs{x})$. The map $Q$ has the expansion around $\bs{x}=\bs{0}$ of the form
\beq
Q(\bs{x}) \, = \, \sul{ \substack{ \bs{\a}, |\bs{\a}|\geq m \\ \bs{\a}_0 \geq \bs{\a} } }{} \mf{x} C_{\bs{\a}} \bs{x}^{ \bs{\a} }
\, + \, \sul{ \substack{ \bs{\a} , |\bs{\a}|\geq m+1 \\ \bs{\a}_1 \geq \bs{\a} } }{} D_{ \bs{\a} } \bs{x}^{ \bs{\a} }
\, + \, \e{O}\Big( |\mf{x}| \bs{x}^{\bs{\a}_{0}} + \bs{x}^{\bs{\a}_1} \Big) \qquad with \qquad m=\sul{r=1}{\ell} n_r(n_r-1) \;.
\enq
In the above expansion, the even integer coordinate vectors $\bs{\a}_0, \bs{\a}_{1} \in \big( 2 \mathbb{N}\big)^{\ov{\bs{n}}_{\ell} }$ can be taken arbitrary provided that $|\bs{\a}_a|\geq m+a$ , and
$C_{\bs{\a}}, D_{\bs{\a}} \in \R$ are coefficients that are bounded uniformly in $\mf{x}$.
\end{lemme}
\Proof
The hypotheses on $\Psi$ entail that $\Psi(\bs{x}) \, =\, \bs{v} \, + \, \bs{x} \, + \, \mf{x} \op{N}_{\Psi} \cdot \bs{x} \, + \, \de\Psi(\bs{x})$, with $\de \Psi(\bs{x}) = \e{O}\big( \norm{\bs{x}}^2 \big)$.
Then, one can write
\beq
\Big( \Psi(\bs{x}) \Big)^{(r)}_{a} \; = \; x_a^{(r)}+y_a^{(r)} \qquad \e{with} \quad y_a^{(r)} \; = \; \e{O}\Big( \mf{x} \norm{ \bs{x} } + \norm{ \bs{x} }^2 \Big)
\enq
and smooth in $\bs{x}$.
Then, one has
\beq
V\big( \Psi( \bs{x} ) \big) \, = \, \pl{r=1}{\ell} \det^2_{ n_r } \Big[ \big(x_a^{(r)}+y_a^{(r)} \big)^{b-1} \Big] \;.
\enq
Upon expanding the power-law, one gets that
\beq
\big(x_a^{(r)}+y_a^{(r)} \big)^{b-1} \, = \, \big(x_a^{(r)} \big)^{b-1} \, + \, \sul{k=1}{b-1} C_{b-1}^{k} \, \big(x_a^{(r)} \big)^{b-1-k} \, \big(y_a^{(r)} \big)^{k}
\, = \, \big(x_a^{(r)} \big)^{b-1} \, + \, P_{a,b}^{(r)}(\bs{x}) \;,
\enq
where $ C_{b-1}^{k}$ are binomial coefficients. The smoothness of $y_a^{(r)}$ and the estimates in $\mf{x}$ ensure that $P_{a,b}^{(r)}(\bs{x})$ takes the form:
\beq
P_{a,b}^{(r)}(\bs{x}) \; = \; \sul{ \substack{ \bs{\a}, |\bs{\a}| \geq b-1 \\ \bs{\a} \leq \bs{\a}_0 } }{} \mf{x} \, C_{\bs{\a},a,b}^{(r)} \, \bs{x}^{ \bs{\a} }
\, + \, \sul{ \substack{ \bs{\a} , |\bs{\a}|\geq b \\ \bs{\a} \leq \bs{\a}_1 } }{} D_{\bs{\a},a,b}^{(r)} \, \bs{x}^{ \bs{\a} }
\, + \, \e{O}\Big( |\mf{x}| \, \bs{x}^{ \bs{\a}_{0} } + \bs{x}^{ \bs{\a}_1 } \Big)
\enq
for some coefficients $C_{\bs{\a},a,b}^{(r)}, D_{\bs{\a},a,b}^{(r)}$ and $\bs{\a}_0,\bs{\a}_1 \in (2\mathbb{N})^{ \ov{\bs{n}}_{\ell} }$. Developing the determinant in respect to the sum appearing in each column yields
\beq
\det_{ n_r } \Big[ \big(x_a^{(r)}+y_a^{(r)} \big)^{b-1} \Big] \; = \; \det_{ n_r } \Big[ \big(x_a^{(r)} \big)^{b-1} \Big] \; + \; R^{(r)}(\bs{x})
\label{ecriture dvpment binomial du determinant}
\enq
with
\beq
R^{(r)}(\bs{x}) \; = \; \sul{k=1}{n_r} \sul{ \substack{ \intn{1}{n_r} = L\sqcup \ov{L} \\ |\ov{L} |=k } }{} \det_{n_r}\big[ \op{M}_{ L,\ov{L} }\big] \quad ,
\qquad \Big( \op{M}_{ L,\ov{L} } \Big)_{ab} \; = \; \left\{ \ba{cc} \big(x_a^{(r)} \big)^{b-1} & \e{if} \, a \in L \vspace{2mm} \\
P_{a,b}^{(r)}(\bs{x}) & \e{if} \, a \in \ov{L} \ea \right. \;.
\enq
Above, the sum runs through all partitions $L\sqcup \ov{L}$ of $\intn{1}{n_r}$ such that $\ov{L}$ has fixed cardinality $k$.
Upon using the expansion
\beq
\det_{n_r}\big[ \op{M}_{ L,\ov{L} }\big] \, = \, \sul{ \sg \in \mf{S}_{N} }{} (-1)^{\sg} \pl{a \in L}{} \big(x_a^{(r)} \big)^{ \sg(a)-1} \cdot \pl{a \in \ov{L} }{} P_{a,\sg(a) }^{(r)}(\bs{x})
\enq
a direct exponent counting argument entails that there exist constants $ C_{\bs{\a},L, \ov{L} }^{(r)}, D_{\bs{\a},L, \ov{L} }^{(r)} \in \R$ such that
\beq
\det_{n_r}\big[ \op{M}_{ L,\ov{L} }\big] \, = \, \sul{ \substack{ \bs{\a}, |\bs{\a}|\geq \tfrac{n_r(n_r-1)}{2} \\ \bs{\a} \leq \bs{\a}_0 } }{} \mf{x} \, C_{\bs{\a},L, \ov{L} }^{(r)} \, \bs{x}^{ \bs{\a} }
\quad + \, \sul{ \substack{ \bs{\a} , |\bs{\a}|\geq \tfrac{n_r(n_r-1)}{2} +1 \\ \bs{\a} \leq \bs{\a}_1 } }{} D_{\bs{\a},L, \ov{L} }^{(r)} \, \bs{x}^{ \bs{\a} }
\, + \, \e{O}\Big( |\mf{x}| \, \bs{x}^{\bs{\a}_{0}} + \bs{x}^{\bs{\a}_1} \Big) \;.
\enq
All of this being established, it remains to take the square of the expression in \eqref{ecriture dvpment binomial du determinant} and then the product over $r$ so as to get the claim. \qed
\subsection{Asymptotic behaviour of a local integral}
\subsubsection{The integral associated with the $|\mf{u}_1^{\prime}(k_0)|< \op{v}$ regime}
Given $\de_{\pm}>0$ and $m_{\pm} \in \mathbb{N}^*$ define
\beq
I_{\de_{\ups},m_{\ups}} \; = \; \big[0 ; \sqrt{\de}_+ \, \big]^{m_+}\times \big[0 ; \sqrt{\de}_- \, \big]^{m_-} \;.
\label{definition intervalle I de pm m pm}
\enq
\begin{lemme}
\label{Lemme integrale beta multi-dim auxiliaire locale cas u less than v}
Let $\de_{\pm}, \eta>0$ be fixed and small enough, $m_{\pm} \in \mathbb{N}$. Let $a,b$ be two smooth functions on $\intff{ - 2 m_+ \de_+ - 2 m_- \de_- }{ 2 m_+ \de_+ + 2 m_- \de_- } $
depending, possibly, on an auxiliary parameter $\mf{x}$ and such that
\beq
a(0) \, = \, b(0) \, = \, 0 \qquad \e{and} \qquad |a(s)| \, + \, |b(s)| \, \leq \, \eta
\label{proprietes fcts a et b}
\enq
uniformly in $s \in \intff{ - 2 m_+ \de_+ - 2 m_- \de_- }{ 2 m_+ \de_+ + 2 m_- \de_- }$. Let $ A, B > -1 $ be smooth function on $I_{\de_{\ups},m_{\ups}} \times \intff{-\eta}{ \eta}$ and
let $\mc{G}$ be a smooth function on $I_{\de_{\ups},m_{\ups}} \times \intff{-\eta}{ \eta} \times \R^+ \times \R^+$
such that
\beq
\mc{G}\big( \bs{u}, x, y \big) \; = \; G(\bs{u}) +\e{O}\big( |x|^{1-\tau} \, + \, |y|^{1-\tau} \big) \qquad with \qquad
0< \tau <1 \;,
\label{ecriture propriete fct G etape 1}
\enq
$G$ being a smooth function on $I_{\de_{\ups},m_{\ups}} \times \intff{-\eta}{ \eta}$ and the remainder being differentiable in the sense of Definition \ref{Defintion reste differentiable}.
Further, assume that $\mc{G}(\bs{u},x,y)=0$ whenever
\beq
\bs{u}=\big( \bs{u}^{(+)}, \bs{u}^{(-)}, s \big) \qquad with \qquad \sul{a=1}{m_{+}} \big( u^{(+)}_a \big)^{2} \; > \; \de_{+} \quad or \quad
\sul{a=1}{m_{-}} \big( u^{(-)}_a \big)^{2} \; > \; \de_{-} \;.
\label{ecriture propriete fct G etape 2}
\enq
Let $\mc{W}$ be smooth on $I_{\de_{\ups},m_{\ups}} \times \intff{-\eta}{ \eta} \times \R^+ $
and admit the expansion around the origin
\beq
\mc{W}\big( \bs{u}, \kappa \big) \; = \; \sul{ \substack{ \bs{\a}, | \bs{\a} |\geq m_0 \\ \bs{\a}=(\a_0,\bs{\be}) } }{} c_{\bs{\a}} \cdot \kappa^{\a_0} \, \bs{u}^{\bs{\be} }
\qquad with \qquad m_0 \in 2 \mathbb{N} \;.
\label{ecriture propriete fct W}
\enq
Consider the integral
\bem
\mc{J}[\mc{G}_{\e{tot}},A,B]( \mf{x} ) \;= \; \pl{\ups= \pm }{}\Bigg\{ \Int{ 0 }{ \de_{\ups} } \f{ \dd^{ m_{\ups} } w^{(\ups)} }{ \pl{a=1}{m_{\ups}} \sqrt{ w_a^{(\ups)} } } \Bigg\} \Int{0}{1}\dd t \;
\bigg[ \mc{G}_{\e{tot}}\Big( \bs{u} , ( 1- t )\vp_{\mf{x}}\big(\bs{u}_{\bs{w}}\big), t \vp_{\mf{x}}\big(\bs{u}_{\bs{w}}\big) \Big) \\
\times ( 1- t )^{ A(\bs{u}) } \cdot t^{ B(\bs{u}) }\cdot \Xi\big[ \vp_{\mf{x}}\big(\bs{u}_{\bs{w}}\big) \big] \cdot
\big[ \vp_{\mf{x}}\big(\bs{u}_{\bs{w}}\big) \big]^{ A(\bs{u})+B(\bs{u})+1 } \bigg]_{ \bs{u}_{\bs{w}}-\e{even} }
\end{multline}
where the even part of a function is as defined in \eqref{definition even part of a function} and vectors $\bs{u}$, $\bs{u}_{\bs{w}}$ appearing under the integral sign are parameterised in terms of $\bs{w}^{(\pm)}$, $t$ as
\beq
\bs{u} \, = \, \Big( \bs{u}_{\bs{w}} , a\circ\vp_{\mf{x}}(\bs{u}_{\bs{w}}) + t b\circ\vp_{\mf{x}}(\bs{u}_{\bs{w}}) \Big) \quad with \quad \bs{u}_{\bs{w}} \, = \, \Big( \bs{u}_{\bs{w}}^{(+)}, \bs{u}_{\bs{w}}^{(-)} \Big)
\quad and \quad \bs{u}_{\bs{w}}^{(\ups)} \, = \, \Big( \sqrt{ w_1^{(\ups)} },\dots,\sqrt{ w_{m_{\ups}}^{(\ups)} } \Big) \;,
\label{definition variable u full et u index w}
\enq
while
\beq
\vp_{\mf{x}}\big(\bs{u}_{\bs{w}}\big) \; = \; \mf{x} \, + \, \sul{a=1}{m_+} w_a^{(+)} \, - \, \sul{a=1}{m_-} w_a^{(-)} \;.
\label{definition fct varphi de u index w}
\enq
Finally, the main building block of the integrand reads
\beq
\mc{G}_{\e{tot}}\big( \bs{u} ,x, y \big) \, = \, \mc{W}\big( \bs{u}, \vp_{\mf{x}}\big(\bs{u}_{\bs{w}}\big) \big)\cdot \mc{G} \big( \bs{u} , x,y \big) \;.
\enq
Then, the integrand belongs to $L^1(\intff{0}{\de_+}^{m_+}\times \intff{0}{\de_-}^{m_-} \times \intff{ 0 }{ 1 })$ and for any smooth function $F$ on $I_{\de_{\ups},m_{\ups}} \times \intff{-\eta}{ \eta}$ satisfying $F(\bs{0})=1$,
there exists a smooth function $\mc{R}$ around $0$ such that one has the $ \mf{x} \tend 0$ behaviour
\beq
\mc{J}[\mc{G}_{\e{tot}},A,B]( \mf{x} ) \;= \; G(\bs{0}) \, \mc{J}\big[ \mc{W}_0 \, F ,A(\bs{0}),B(\bs{0}) \big]( \mf{x} ) \; + \; \mc{R}(\mf{x}) \; + \;
\e{O}\big( \, |\mf{x}|^{ \varrho } \, \big)
\enq
where
\beq
\varrho \, = \, \frac{ 1 }{2}( m_+ + m_0 + m_- ) + 1+ (A+B)(\bs{0}) + 1-\tau \;,
\enq
and
\beq
\mc{W}_0\big( \bs{u}, \kappa \big) \; = \; \sul{ \substack{ \bs{\a}, |\bs{\a}|= m_0 \\ \bs{\a}=(\a_0,\bs{\be}) } }{} c_{\bs{\a}} \cdot \kappa^{\a_0} \, \bs{u}^{\bs{\be} }
\qquad with \qquad m_0 \in 2 \mathbb{N} \;.
\enq
\end{lemme}
\Proof
One first implements the change of the $\bs{w}$-integration variables $\bs{w}^{(\ups)} \hookrightarrow \bs{s}^{(\ups)}$ with
\beq
s_k^{(\ups)}\,=\, \sul{ a = k }{ m_{\ups} } w_a^{(\ups)} \qquad i.e. \qquad w_k^{(\ups)}=s_k^{(\ups)} - s_{k+1}^{(\ups)} \; \; \e{for} \; \;
k=1,\dots, m_{\ups}-1 \;\; \e{and} \;\; w_{m_{\ups}}^{(\ups)}=s_{m_{\ups}}^{(\ups)}
\enq
whose Jacobian equals to $1$. The inequalities $0 \leq w_k^{(\ups)} \leq \de_{\ups} $ defining the integration domain in the original variables can be recast in terms of an equivalent set of encased
inequalities defining the integration domain in the $\bs{s}^{(\ups)}$ variables:
\beq
\left\{ \ba{c} 0\leq s_1^{(\ups)} \leq m_{\ups} \de_{\ups} \vspace{2mm} \\
\mf{s}_{k-1}^{(\ups;-)} \, \leq \, s_k^{(\ups)} \, \leq \, \mf{s}_{k-1}^{(\ups;+)} \quad \e{for} \; k=2,\dots, m_{\ups} \ea \right.
\qquad \e{with} \qquad \left\{ \ba{ccl} \mf{s}_{k-1}^{(\ups;-)} &=& \e{max}\big\{ 0 , s_{k-1}^{(\ups)}-\de_{\ups} \big\} \vspace{1mm} \\
\mf{s}_{k-1}^{(\ups;+)} &=& \e{min}\big\{ s_{k-1}^{(\ups)}, ( m_{\ups} +1-k) \de_{\ups} \big\} \ea \right.
\;.
\label{ecriture domaine integration variables s ups}
\enq
This recasts the original integral in the form
\bem
\mc{J}[\mc{G}_{\e{tot}},A,B]( \mf{x} ) \;= \; \pl{\ups= \pm }{} \Bigg\{ \Int{ 0 }{ m_{\ups} \de_{\ups} }\dd s^{(\ups)}_1 \pl{k=2}{m_{\ups}} \Int{ \mf{s}_{k-1}^{(\ups;-)} }{ \mf{s}_{k-1}^{(\ups;+)} } \dd s_{k}^{(\ups)} \Bigg\}
\Int{0}{1}\dd t \; \pl{\ups=\pm}{} \pl{a=1}{m_{\ups}} \Bigg\{ \f{ 1 }{ \sqrt{ s_a^{(\ups)} - s_{a+1}^{(\ups)} } } \Bigg\} \\
\times \bigg[ ( 1- t )^{ A(\bs{x}) } \cdot t^{ B(\bs{x}) } \mc{G}_{\e{tot}}\Big( \bs{x} , ( 1- t ) \vp_{\mf{x}}\big(\bs{x}_{\bs{s}}\big), t \vp_{\mf{x}}\big(\bs{x}_{\bs{s}}\big) \Big)
\cdot \Xi\big[ \vp_{\mf{x}}\big(\bs{x}_{\bs{s}}\big) \big] \cdot
\big[ \vp_{\mf{x}}\big(\bs{x}_{\bs{s}}\big) \big]^{ A(\bs{x})+B(\bs{x})+1 } \bigg]_{ \bs{x}_s-\e{even} }
\end{multline}
where it is understood that $ s_{m_{\ups}+1}^{(\ups)} \equiv 0$, while $\bs{x}\, =\, \big( \bs{x}_{\bs{s}}, a\circ\vp_{\mf{x}}(\bs{x}_{\bs{s}})+t b\circ\vp_{\mf{x}}(\bs{x}_{\bs{s}}) \big)$, in which
\beq
\bs{x}_{\bs{s}} \, = \, \Big( \bs{x}_{\bs{s}}^{(+)}, \bs{x}_{\bs{s}}^{(-)} \Big) \qquad \e{where} \qquad
\bs{x}_{\bs{s}}^{(\ups)} \, = \, \bigg( \sqrt{s_1^{(\ups)}-s_2^{(\ups)}}, \dots, \sqrt{s_{m_{\ups}-1}^{(\ups)}-s_{m_{\ups}}^{(\ups)} }, \sqrt{ s_{m_{\ups}}^{(\ups)} } \bigg) \; .
\enq
Finally, one has $\vp_{\mf{x}}\big(\bs{x}_{\bs{s}}\big) \; = \; \mf{x} \, + \, s_1^{(+)}\, - \, s_1^{(-)}$.
Recall that the integrand vanishes when, either, $s_1^{(+)}\geq \de_{+}$ or $s_1^{(-)}\geq \de_{-}$. Thus, one may reduce the $\big( s_1^{(+)} , s_1^{(-)} \big)$ integration from
$\intff{0}{m_+ \de_{+} } \times \intff{0}{m_- \de_{-} }$ to
the rectangle $\intff{0}{\de_{+}}\times \intff{0}{\de_{-}}$. However, as soon as it holds $0 \leq s_1^{(\ups)} \leq \de_{\ups} $, one can readily check that the endpoints of integration
$ \mf{s}_{k-1}^{(\ups;-)}, \mf{s}_{k-1}^{(\ups;+)}$ in \eqref{ecriture domaine integration variables s ups} for the variable $s_{k}^{(\ups)} $ reduce to
\beq
\mf{s}_{k-1}^{(\ups;-)}=0 \quad \e{and} \quad \mf{s}_{k-1}^{(\ups;+)}=s_{k-1}^{(\ups)} \quad \e{for} \quad k=2,\dots,m_{\ups} \; .
\enq
Upon this reduction, one may implement another change of variables $\bs{s}^{(\ups)} \; \hookrightarrow \; \bs{u}^{(\ups)}$ with
\beq
s_{a}^{(\ups)}=u_1^{(\ups)}\dots u_a^{(\ups)} \qquad \e{and} \qquad
\det\Big[ D_{\bs{u}^{(\ups)}}\bs{s}^{(\ups)} \Big] \, = \, \pl{ a=2 }{ m_{\ups} }\big\{ u_1^{(\ups)} \cdots u_{a-1}^{(\ups)} \big\} = \pl{a=1}{ m_{\ups} } \big\{ u^{(\ups)}_a \big\}^{m_{\ups}-a} \;.
\enq
Since
\beq
\pl{ a = 1 }{ m_{\ups} } \sqrt{ s_a^{(\ups)}-s_{a+1}^{(\ups)} } \; = \; \pl{a=2}{m_{\ups}}\sqrt{1-u_a^{(\ups)}} \cdot \pl{a=1}{m_{\ups}} \big\{ u_a^{(\ups)} \big\}^{ \tfrac{m_{\ups}+1-a }{2} }
\enq
the integral takes the form
\bem
\mc{J}[\mc{G}_{\e{tot}},A,B]( \mf{x} ) \;= \; \pl{\ups= \pm }{} \Bigg\{ \Int{ 0 }{ \de_{\ups} }\dd u^{(\ups)}_1 \big[ u_1^{(\ups)} \big]^{ \tfrac{m_{\ups}-2 }{2} }
\pl{k=2}{m_{\ups}} \Int{ 0 }{ 1 } \dd u_{k}^{(\ups)} \f{ \big( u_k^{(\ups)} \big)^{ \tfrac{m_{\ups}-1-k }{2} } }{ \sqrt{ 1-u_{k}^{(\ups)} } } \Bigg\} \\
\times \Int{0}{1}\dd t \; \bigg[ \mc{G}_{\e{tot}}\Big( \bs{y} , ( 1- t )\vp_{\mf{x}}\big(\bs{y}_{\bs{u}}\big), t \vp_{\mf{x}}\big(\bs{y}_{\bs{u}}\big) \Big)\cdot
( 1- t )^{ A(\bs{y}) } \cdot t^{ B(\bs{y}) } \cdot \Xi\big[ \vp_{\mf{x}}\big(\bs{y}_{\bs{u}}\big) \big] \cdot
\big[ \vp_{\mf{x}}\big(\bs{y}_{\bs{u}}\big) \big]^{ A(\bs{y})+B(\bs{y})+1 } \bigg]_{\bs{y}_{\bs{u}}-\e{even} }\;.
\end{multline}
There, one should identify $\bs{y}=\big( \bs{y}_{\bs{u}}, a\circ\vp_{\mf{x}}(\bs{y}_{\bs{u}}) + t b\circ\vp_{\mf{x}}(\bs{y}_{\bs{u}}) \big)$ where $\bs{y}_{\bs{u}} \, = \, \big( \bs{y}_{\bs{u}}^{(+)}, \bs{y}_{\bs{u}}^{(-)} \big) $ and
\beq
\bs{y}_{\bs{u}}^{(\ups)} \, = \, \bigg( \sqrt{u_1^{(\ups)} \Big(1-u_2^{(\ups)} \Big) }, \dots, \sqrt{u_1^{(\ups)}\cdots u_{m_{\ups}-1}^{(\ups)} \Big(1-u_{m_{\ups}}^{(\ups)} \Big) } ,
\sqrt{u_1^{(\ups)}\cdots u_{m_{\ups} }^{(\ups)} } \bigg) \; .
\enq
Finally, $ \vp_{\mf{x}}\big(\bs{y}_{\bs{u}}\big) \; = \; \mf{x} \, + \, u_1^{(+)}\, - \, u_1^{(-)}$.
At this stage, one may perform explicitly the reduction of the integration domain due to the presence of the Heaviside function. One has $ \vp_{\mf{x}}\big(\bs{y}_{\bs{u}}\big) \geq 0 $ on
\beq
\bigg\{ \Big( u_{1}^{(+)}, u_{1}^{(-)} \Big) \in \intff{ 0 }{ \de_{+} }\times \intff{ 0 }{ \de_{-} }\; , \; -\e{min}(0,\mf{x}) \leq u_{1}^{(+)} \leq \de_+ \;\;
\e{and} \;\; 0 \leq u_{1}^{(-)} \leq \e{min}\big(u_{1}^{(+)} + \mf{x}, \de_{-} \big) \bigg\} \;.
\enq
Since the integrand vanishes when $u_{1}^{(-)}\geq \de_-$, one may just as well, for fixed $u_1^{(+)}$, extend the $u_{1}^{(-)}$ integration to the segment $ \intff{0}{ u_{1}^{(+)} + \mf{x}}$.
This form of the integration domain takes explicitly into account the constraints implied by the presence of the Heaviside function. This reduced form of the
integration domain leads naturally to the last change of variables
\beq
u_a^{(+)} =z_{a}^{(+)}\;\;, \;\; a=1,\dots, m_{+} \; , \quad u_1^{(-)}=z_{1}^{(-)} \cdot \Big( z_{1}^{(+)} + \mf{x}\Big) \quad \e{and} \quad u_a^{(-)} =z_{a}^{(-)}\;\;, \;\; a=2,\dots, m_{-} \;.
\enq
The integration variables are then gathered in $\bs{r}=\big( \bs{r}_{\bs{z}}, a\circ\vp_{\mf{x}}(\bs{r}_{\bs{z}}) + t b\circ\vp_{\mf{x}}(\bs{r}_{\bs{z}}) \big)$, with
\beq
\bs{r}_{\bs{z}} \, = \, \Big( \bs{r}_{\bs{z}}^{(+)}, \big[ z_{1}^{(+)} + \mf{x}\big]^{\f{1}{2}} \cdot \bs{r}_{\bs{z}}^{(-)} \Big)
\label{definition variable r index z}
\enq
and
\beq
\bs{r}_{\bs{z}}^{(\ups)} \, = \, \bigg( \sqrt{ z_1^{(\ups)} \Big( 1 - z_2^{(\ups)} \Big) }, \dots, \sqrt{z_1^{(\ups)}\cdots z_{m_{\ups}-1}^{(\ups)} \Big( 1 - z_{m_{\ups}}^{(\ups)} \Big) } ,
\sqrt{z_1^{(\ups)}\cdots z_{m_{\ups} }^{(\ups)} } \bigg) \; .
\label{definition variable r full}
\enq
All of this recasts the integral in the form
\bem
\mc{J}[\mc{G}_{\e{tot}},A,B]( \mf{x} ) \;= \hspace{-4mm} \Int{ -\e{min}(0,\mf{x}) }{ \de_{+} } \hspace{-2mm} \dd z^{(+)}_1 \Int{ 0 }{ 1 } \dd z^{(-)}_1
\pl{\ups= \pm }{} \Bigg\{ \pl{k=2}{m_{\ups}} \Int{ 0 }{ 1 } \dd z_{k}^{(\ups)} \f{ \big( z_k^{(\ups)} \big)^{ \tfrac{m_{\ups}-1-k }{2} } }{ \sqrt{ 1-z_{k}^{(\ups)} } } \Bigg\}\cdot
\big[z_1^{(+)}+\mf{x} \big]^{ \frac{m_-}{2} } \cdot \pl{\ups = \pm }{} \big[ z_1^{(\ups)} \big]^{ \tfrac{m_{\ups} }{2}-1 } \\
\times \Int{0}{1}\dd t \; \bigg[ ( 1- t )^{ A(\bs{r}) } \cdot t^{ B(\bs{r}) } \cdot
\Big[ (z_1^{(+)}+\mf{x})(1-z_1^{(-)}) \Big]^{ A(\bs{r})+B(\bs{r}) +1 }
\cdot \mc{G}_{\e{tot}}\Big( \bs{r} , \, ( 1- t ) \vp_{\mf{x}}\big(\bs{r}_{\bs{z}}\big) , \, t \vp_{\mf{x}}\big(\bs{r}_{\bs{z}}\big)\Big) \bigg]_{ \bs{r}_{\bs{z}}-\e{even} } \;.
\label{ecriture representation integrale quasi 1D pour integrale modele u prime lower than v}
\end{multline}
in which $\vp_{\mf{x}}\big(\bs{r}_{\bs{z}}\big) \; = \; \big( \mf{x} \, + \, z_1^{(+)} \big) \cdot \big( 1\, - \, z_1^{(-)} \big)$.
The $L^1$ nature of the integrand is manifest on the level of \eqref{ecriture representation integrale quasi 1D pour integrale modele u prime lower than v}.
By composing the various expansions at $\bs{0}$, it is easy to see that
\bem
\mc{G}_{\e{tot}}\Big( \bs{r} , (1-t) \vp_{\mf{x}}\big(\bs{r}_{\bs{z}}\big) , t \vp_{\mf{x}}\big(\bs{r}_{\bs{z}}\big) \Big) \, = \, \sul{ \substack{ k, \ell \\ 2k+2\ell=m_0} }{} C_{k,\ell}\Big(\bs{z}^{(+)}_2,\bs{z}^{(-)},t \Big)
\cdot \big( z_1^{(+)} \big)^{ \ell }\cdot \big( z_1^{(+)}+\mf{x} \big)^{ k + 1 + A(\bs{0}) + B(\bs{0}) } \\
\; + \; \e{O}\bigg( \sul{ \substack{ k, \ell \\ 2k+2\ell = m_0 + 1 } }{} D_{k,\ell,p}\Big(\bs{z}^{(+)}_2,\bs{z}^{(-)},t \Big)
\cdot \big( z_1^{(+)} \big)^{ \ell } \cdot \big( z_1^{(+)}+\mf{x} \big)^{ k + 1 + A(\bs{0}) + B(\bs{0})} \Big\{ 1 \, + \, \big| \ln \big( z_1^{(+)}+\mf{x} \big) \, \big| \Big\} \bigg) \\
\; + \;
\e{O}\bigg( \sul{ \substack{ k, \ell \\ 2k+2\ell = m_0 } }{} \wt{D}_{k,\ell,p}\Big(\bs{z}^{(+)}_2,\bs{z}^{(-)},t \Big)
\cdot \big( z_1^{(+)} \big)^{ \ell } \cdot \big( z_1^{(+)}+\mf{x} \big)^{ k + 2-\tau + A(\bs{0}) + B(\bs{0})} \bigg) \;.
\end{multline}
There, $\bs{z}^{(+)}_2 \, = \, \big( z_{2}^{(+)}, \dots, z_{m_+}^{(+)} \big)$ and the functions $C_{k,\ell,p}\Big(\bs{z}^{(+)}_2,\bs{z}^{(-)},t \Big)$,
$ D_{k,\ell,p}\Big(\bs{z}^{(+)}_2,\bs{z}^{(-)},t \Big)$, $\wt{D}_{k,\ell,p}\Big(\bs{z}^{(+)}_2,\bs{z}^{(-)},t \Big)$ are all continuous on $\intff{0}{1}^{m_++m_-}$.
Finally, the remainders are differentiable.
An application of Lemma \ref{Lemme integrale beta auxiliaire} relatively to the $z_1^{(+)}$ integration then leads to the claim. \qed
\subsubsection{The integral associated with the $|\mf{u}_1^{\prime}(k_0)| > \op{v}$ regime}
\begin{lemme}
\label{Lemme integrale beta multi-dim auxiliaire locale cas u greater than v}
Assume the notations and hypotheses outlined in Lemma \ref{Lemme integrale beta multi-dim auxiliaire locale cas u less than v}, with the exception that $b$ does not have to vanish at $0$.
Pick $\de_0$ small enough and such that $\de_0>2 \de_+$ and assume further that
\beq
\mc{G}(\bs{u},x,y)=0 \quad if \quad s>\eta^{\prime} \qquad where \qquad \bs{u}=\big( \bs{u}^{(+)}, \bs{u}^{(-)}, s\big) \;,
\label{ecriture condition de support compact par rapport a variable t de G cal}
\enq
such that
\beq
a\circ\vp_{\mf{x}}(\bs{v}) \;+\; t \, b\circ\vp_{\mf{x}}(\bs{v}) > 2 \eta^{\prime} \qquad uniformly \; in \quad t> \f{\de_0 }{ 2 }\; , \quad \bs{v} \in \intff{ 0 }{ \de_+ }^{ m_{+} } \times \intff{ 0 }{ \de_- }^{ m_{-} } \;.
\enq
Consider the integral
\bem
\chi[\mc{G}_{\e{tot}},A,B]( \mf{x} ) \;= \; \pl{\ups= \pm }{}\Bigg\{ \Int{ 0 }{ \de_{\ups} } \f{ \dd^{ m_{\ups} } w^{(\ups)} }{ \pl{ a = 1 }{ m_{\ups} } \sqrt{ w_a^{(\ups)} } } \Bigg\} \Int{0}{\de_0}\dd t \;
\\
\times
\bigg [\mc{G}_{\e{tot}}\Big( \bs{u} , t , t \, + \, \vp_{\mf{x}}\big(\bs{u}_{\bs{w}}\big) \Big) t^{ A(\bs{u}) } \cdot \big[ t \, + \, \vp_{\mf{x}}\big(\bs{u}_{\bs{w}}\big) \big]^{ B(\bs{u}) }
\cdot \Xi\big[ \vp_{\mf{x}}\big(\bs{u}_{\bs{w}}\big) \big] \bigg]_{ \bs{u}_{\bs{w}}-\e{even} }
\end{multline}
where the vectors $\bs{u}$, $\bs{u}_{\bs{w}}$ appearing under the integral sign are as defined in
\eqref{definition variable u full et u index w}, the even part of a function is as defined in \eqref{definition even part of a function},
while $\vp_{\mf{x}}\big(\bs{u}_{\bs{w}}\big)$
has been defined in \eqref{definition fct varphi de u index w}.
Then, the integrand belongs to $L^1(\intff{0}{\de_+}^{m_+}\times \intff{0}{\de_-}^{m_-} \times \intff{ 0 }{ \de_0 })$ and, for any smooth function $F$ on $I_{\de_{\ups},m_{\ups}} \times \intff{-\eta}{ \eta}$ satisfying $F(\bs{0})=1$,
there exists a smooth function $\mc{R}$ around $0$ such that one has the $ \mf{x} \tend 0$ behaviour
\beq
\mc{J}[\mc{G}_{\e{tot}},A,B]( \mf{x} ) \;= \; G(\bs{0}) \, \mc{J}\big[ \mc{W}_0 \, F ,A(\bs{0}),B(\bs{0}) \big]( \mf{x} ) \; + \; \mc{R}(\mf{x}) \; + \;
\e{O}\big( \, |\mf{x}|^{ \varrho } \, \big)
\enq
where
\beq
\varrho \, = \, \frac{ 1 }{2}( m_+ + m_0 + m_- ) + 1+ (A+B)(\bs{0}) + 1-\tau \;,
\enq
and
\beq
\mc{W}_0\big( \bs{u}, \kappa \big) \; = \; \sul{ \substack{ \bs{\a}, |\bs{\a}|= m_0 \\ \bs{\a}=(\a_0,\bs{\be}) } }{} c_{\bs{\a}} \cdot \kappa^{\a_0} \, \bs{u}^{\bs{\be} }
\qquad with \qquad m_0 \in \mathbb{N} \;.
\enq
\end{lemme}
\Proof
The very same chain of transformation that was implemented in the proof of Lemma \ref{Lemme integrale beta multi-dim auxiliaire locale cas u less than v}
and the vanishing condition \eqref{ecriture condition de support compact par rapport a variable t de G cal}
allows one to recast the original integral as
\bem
\mc{J}[\mc{G}_{\e{tot}},A,B]( \mf{x} ) \;=
\Int{ -\e{min}(0,\mf{x}) }{ \de_{+} } \hspace{-2mm} \dd z^{(+)}_1 \Int{ 0 }{ \de_- } \dd z^{(-)}_1 \Int{ 0 }{ \de_0 }\dd t \;
\pl{\ups= \pm }{} \Bigg\{ \pl{k=2}{m_{\ups}} \Int{ 0 }{ \de_{\ups} } \dd z_{k}^{(\ups)} \f{ \big( z_k^{(\ups)} \big)^{ \tfrac{m_{\ups}-1-k }{2} } }{ \sqrt{ 1-z_{k}^{(\ups)} } } \Bigg\} \\
\times \; \big[z_1^{(+)}+\mf{x} \big]^{ \frac{m_-}{2} } \pl{\ups=\pm}{} \Big\{ \big[ z_1^{(\ups)} \big]^{ \tfrac{m_{\ups} }{2}-1 } \Big\} \cdot \phi(t)
\cdot \bigg[ t^{ A(\bs{r}) } \cdot \Big[t\, + \, (z_1^{(+)}+\mf{x})(1-z_1^{(-)} ) \Big]^{ B(\bs{r}) } \mc{G}_{\e{tot}}\Big( \bs{r} , t , t \, + \, (z_1^{(+)}+\mf{x})(1-z_1^{(-)} ) \Big) \bigg]_{ \bs{r}_{\bs{z}}-\e{even} }
\label{ecriture simplification vers integrales quasi 1D de integrale modele a u prime bigger than v}
\end{multline}
in which $\bs{r}$, $\bs{r}_{\bs{z}}$ are as defined in \eqref{definition variable r full}, \eqref{definition variable r index z} and
$\phi \geq 0$ is smooth with compact support on $\intff{0}{\de_0}$ and such that $\phi_{\mid \intff{ 0 }{ \tf{\de_0}{2} } } = 1$.
The $L^{1}$ nature of the integrand is already manifest on the level of \eqref{ecriture simplification vers integrales quasi 1D de integrale modele a u prime bigger than v}.
Then a direct application of Lemma \ref{Lemme integrale triple type beta auxiliaire} leads to the claim. \qed
\begin{lemme}
\label{Lemme integrale triple type beta auxiliaire}
Let $\de_0> 2 \de_+ >0$, $m_{\pm} \in \mathbb{N}^*$ and $\mf{x} \in \R^{*}$. Consider the integral
\bem
\mc{L}(\mf{x}) \, = \, \Int{ -\e{min}(0,\mf{x}) }{ \de_+ } \dd z_+ \Int{ 0 }{ 1 } \dd z_- \Int{0}{\de_0} \dd t \; \phi(t) \cdot
\big[ z_+ + \mf{x} \big]^{ \f{m_-}{2} } \cdot \big[ z_+ \big]^{ \f{m_+}{2} -1 } \\
\times \bigg[ t^{ A(\bs{z}) } \cdot \Big[t\, + \, (z_++\mf{x})(1-z_- ) \Big]^{ B(\bs{z}) }
\mc{F}\Big( \bs{z}_0 ; t, t + (z_++\mf{x})(1-z_- ) \Big) \bigg]_{\bs{z}_0-\e{even} }
\end{multline}
in which
\beq
\bs{z} \, = \, \Big(\bs{z}_{0}, g_1\big( (z_++\mf{x})(1-z_- ) \big) \, + \, t g_{2}\big( (z_++\mf{x})(1-z_- ) \big) \Big)
\quad \bs{z}_0\, = \, \big( \sqrt{z_+}, \big[ z_-(z_+ + \mf{x}) \big]^{\f{1}{2}} \big) \; ,
\enq
$g_1, g_2$ are smooth, such that $g_1(0)=0$, $g_2(0)=1$ while the even part of a function is as given in \eqref{definition even part of a function}.
Furthermore, the function $\mc{F}$ is assumed smooth and has the small argument expansion, with a differentiable remainder:
\beq
\mc{F}\big( x, y ; u, v \big) \; = \;\, x^{2 s_+} \, y^{2 s_-}\Big( F_0 \, + \, \e{O}\big( u^{1-\a} + v^{1-\a} + x + y \big) \, \Big) \quad for \, some \quad 0 < \tau < 1 \;,
\enq
and integers $s_{\pm}$. The functions $A,B$ are smooth, $A,B>-1$, while $\phi \geq 0$ is smooth with compact support on $\intff{0}{\de_0}$ and such that $\phi_{\mid \intff{ 0 }{ \tf{\de_0}{2} } } = 1$
Then, for any function of two variables $G$ such that $G(\bs{0})=1$, there exists a smooth
function $\mf{r}$ around $\mf{x}=0$ such that
\bem
\mc{L}(\mf{x}) \, = \, F_0 \hspace{-3mm} \Int{ -\e{min}(0,\mf{x}) }{ \de_+ } \hspace{-3mm} \dd z_+ \Int{ 0 }{ 1 } \dd z_- \Int{0}{\de_0} \dd t \; \phi(t) \cdot
\big[ z_+ + \mf{x} \big]^{ \f{m_-}{2} } \cdot \big[ z_+ \big]^{ \f{m_+}{2} -1 }
\bigg[ t^{ A(\bs{0}) } \cdot \Big[t\, + \, (z_++\mf{x})(1-z_- ) \Big]^{ B(\bs{0}) }
\cdot G\big( \bs{z}_0 \big) \bigg]_{\bs{z}_0-\e{even} }\; \\
\; + \; \mf{r}(\mf{x}) \; + \; \e{O}\Big( \mf{x}^{3 + a_0 + b_0 + \a_+ + \a_- } |\ln \mf{x} | \, + \, \mf{x}^{3 + a_0 + b_0 + \a_+ + \a_- -\tau} \Big) \;,
\end{multline}
with
\beq
\a_+ \, = \, s_+\, + \, \frac{m_+}{2} -1 \;, \quad \a_-\, = \, s_- \, + \, \frac{ m_- }{2 } \;.
\enq
\end{lemme}
\Proof
We only discuss the proof in the case of $\mf{x}>0$ and small enough in that the case $\mf{x}<0$ can be dealt with in much the same way.
For further convenience, set $a_0 = A(\bs{0})$ and $b_0 = B(\bs{0})$. Let $n$ be such that
\beq
-2 \, < \, 1 + a_0 + b_0 + \a_- + \a_+ - n \, < \, -1 \; .
\label{ecriture bon choix parametres pour DA L}
\enq
It is obvious from the form of the integrand that $\mc{L}(\mf{x})$ defines a smooth function of $\mf{x}$ on $\R^+$ and that its derivative can be obtained directly by carrying the derivative under the integral sign.
Then, it holds
\bem
\Dp{\mf{x}}^n\bigg[ t^{ A(\bs{z}) } \cdot \Big[t\, + \, (z_++\mf{x})(1-z_- ) \Big]^{ B(\bs{z}) } \cdot \big[ z_+ + \mf{x} \big]^{ \f{m_-}{2} } \cdot \big[ z_+ \big]^{ \f{m_+}{2} -1 } \cdot
\mc{F}\Big( \sqrt{z_+}, \big[ z_-(z_+ + \mf{x}) \big]^{\f{1}{2}} ; t + (z_++\mf{x})(1-z_- ) \Big) \bigg]_{\bs{z}_0-\e{even} } \\
\; = \; \sul{p=0}{n} C_{n}^{p} \cdot \big(\a_- \big)_{n-p} \big( b_0 \big)_{p} \, z_+^{\a_+} \cdot z_-^{\a_-} \big(z_+ + \mf{x} \big)^{\a_--n+p}
\Big[t\, + \, (z_++\mf{x})(1-z_- ) \Big]^{ b_0 -p} (1-z_-)^p \cdot t^{a_0}\\
\times \bigg\{ F_0 +\e{O}\bigg( t^{ 1 - \tau }+\big[t\, + \, (z_++\mf{x})(1-z_- ) \big]^{1-\tau} \\
\, + \; \Big\{ z_+ + z_-(z_++\mf{x}) + t \Big\} \cdot \Big\{ 1 + |\ln t| + \ln\big[t\, + \, (z_++\mf{x})(1-z_- )\; \big] \Big\} \bigg) \; \bigg\}
\end{multline}
where $(x)_p=x (x-1) \cdots (x-p+1)$ refers to the descending Pochhammer symbol.
Thus, upon setting
\beq
\mc{T}(a,b,c,d,e,f) \; = \; \Int{0}{\de_+} \dd z_+ \Int{0}{1} \dd z_- \Int{0}{\de_0} \dd t \; \phi(t) \cdot t^a \big[t\, + \, (z_++\mf{x})(1-z_- ) \big]^{b} \, z_+^c \, z_-^{d} \, (\mf{x}+z_+)^e (1-z_-)^{f}
\enq
all together, one gets that
\bem
\Dp{\mf{x}}^n\mc{L}(\mf{x}) \, = \, \sul{p=0}{n} C_{n}^{p} F_0 (b_0)_{p}(\a_-)_{n-p} \mc{T}\Big( \, a_0 , b_0-p , \a_+ , \a_-, \a_--n+p , p \Big) \\
\, + \, \sul{ \ups\in \{ \pm, 0 \} }{}\e{O}\bigg( \Big[|\Dp{a}\mc{T}| \, + \, |\Dp{b}\mc{T}| \, + \, |\mc{T}| \Big]\Big( \, a_0 + \de_{\ups,0} , b_0-p , \a_+ + \de_{\ups, +} ,\a_- + \de_{\ups, -}, \a_--n+p + \de_{\ups, -} ,p \, \Big) \bigg) \\
\, + \,\sul{\ups= \pm }{} \e{O}\bigg( \, \big|\mc{T}\Big( \, a_0 + \a \de_{\ups, +} , b_0-p + \a \de_{\ups, -} ,\a_+,\a_-,\a_--n+p,p \, \Big) \big| \, \bigg) \;.
\label{ecriture derivee neme integraleL}
\end{multline}
At this stage, it remains to focus on $\mc{T}$. There, one implements the change of variables
\beq
t= \f{ \zeta \, \nu }{ 1- \nu} \qquad \e{with} \qquad \zeta= (z_++\mf{x})(1-z_- ) \quad \e{i.e.} \quad \nu \, = \, \f{t}{t+\zeta} \; ,
\enq
leading to
\beq
\mc{T}(a,b,c,d,e,f) \; = \; \Int{0}{\de_+} \dd z_+ \Int{0}{1} \dd z_- z_+^c \; z_-^{d} \, (1-z_-)^{f+a+b+1} (z_++\mf{x})^{e+a+b+1} h\big( (z_++\mf{x})(1-z_- ) \big)
\enq
where
\beq
h(\zeta) \;= \; \Int{0}{ \frac{\de_0}{\zeta + \de_0 } }
\dd \nu \; \f{ \nu^a }{ (1-\nu)^{2+a+b} } \phi\Big( \f{ \zeta \, \nu }{ 1- \nu} \Big) \;.
\enq
The fact that $\phi$ is smooth with compact support on $\intff{0}{\de_0}$ entails that for $\Re(a)>-1$, and irrespectively of the value of $b$, $h$ is smooth in the neighbourhood of the
origin and $h(0)= \tf{\Ga(a+1)\Ga(-1-a-b) }{ \Ga(-b) }$, so that upon making use of hypergeometric like notations, an application of Lemma \ref{Lemme integrale beta auxiliaire} yields
\bem
\mc{T}(a,b,c,d,e,f) \; = \; \Ga\left( \ba{c} -2-a-b-c-e , \, -1-a-b, \, f+a+b+2 \\ -b , \, f+a+b+d+3 , \, -1-a-b-e \ea \right) \\
\times \Ga( a+1, \, d+1, \, 1+c ) \cdot \mf{x} ^{2+e+a+b+c} \; + \; \mc{R}(\mf{x})
\; + \; \e{O}\big( \mf{x}^{3+e+a+b+c} \big) \;,
\end{multline}
in which $\mc{R}(\mf{x})$ is smooth in $\mf{x}$. Upon inserting this expansion into \eqref{ecriture derivee neme integraleL}, one gets
\bem
\Dp{\mf{x}}^n\mc{L}(\mf{x}) \, = \, \sul{p=0}{n} C_{n}^{p} F_0 (b_0)_{p}(\a_-)_{n-p} \cdot \mf{x}^{ 2+ a_0+b_0 + \a_- + \a_+ - n } \\
\times \Ga\left( \ba{c} a_0+1, \, \a_+ + 1, \, \a_- + 1 \, , p-1-b_0-a_0, \, a_0+b_0+2 , \, n - a_0 - b_0 - \a_- - \a_{+} - 2 \\
p-b_0 , \, a_0+b_0 + \a_- + 3 , \, n-1-a_0-b_0 - \a_- \ea \right) \\
\; + \; \wt{\mc{R}}(\mf{x}) \; + \e{O}\bigg(\mf{x}^{ 3+ a_0+b_0 + \a_- + \a_+ - n } |\ln \mf{x} | \, + \, \mf{x}^{ 3+ a_0 + b_0 + \a_- + \a_+ - n + \tau }\bigg) \;.
\end{multline}
Then, the bounds \eqref{ecriture bon choix parametres pour DA L} followed by a direct integration of the above expansion lead to the claim.
\qed
\section{Asymptotic behaviour of a model integral}
\label{Appendix Section Asymptotics of model integral}
\subsection{Reduction of the model integral into regular and singular parts}
Recall that
\beq
V(\bs{x}) \; = \; \pl{r=1}{\ell} \pl{a<b}{n_r} \Big( x_a^{(r)}-x_b^{(r)}\Big)^2 \;.
\label{rappel definition V et P alpha}
\enq
\begin{prop}
\label{Proposition reduction vers voisinage singularite integrale modele}
Let $\de_{\ups}>0$, $\sum_{r=1}^{\ell}n_r\geq 2$, $\xi_{r} \in \R^{*}$ and $\veps_r \in \{\pm 1\}$ be such that
\beq
\sum_{r=1}^{\ell} \veps_{r}\, \xi_r^2 \, n_r \not=0 \;.
\label{hypothese positivite contrainte sur nr epsr zetar squared}
\enq
Consider the integral
\beq
\mc{J}(\mf{x}) \, = \, \Int{ \mc{D}_{\bs{n}} }{} \ex{-(\bs{x},\bs{x}) } V(\bs{x}) \pl{\ups=\pm }{} \Big\{ \Xi\big(\, \mf{x} + z_{\ups}(\bs{x}) \big)\cdot \big[ \, \mf{x} + z_{\ups}(\bs{x}) \big]^{\de_{\ups}-1} \Big\} \cdot \dd \bs{x}
\quad over \quad
\mc{D}_{\bs{n}}=\pl{r=1}{\ell} \R^{n_r}
\enq
where the functions $ z_{\ups}(\bs{x})$, $\ups=\pm$, are quadratic forms
\beq
z_{\ups}(\bs{x}) \; = \; \f{1}{2} \big( \bs{x},\op{E} \bs{x} \big) \, + \, \big( \bs{x}, \bs{e}) \cdot \f{ \op{u}+\ups \op{v} }{ 2\op{v} }
\quad with \quad
\op{E}=\left( \ba{ccc} \veps_1 \e{I}_{n_1} & 0& \ddots \\
0 & \ddots & 0 \\
\ddots & 0 & \veps_{\ell} \e{I}_{n_{\ell} } \ea \right)\;.
\label{definition z ups etape 1}
\enq
The parameters $(\op{u} , \op{v})\in \R \times \R^+$ are such that $\op{u} \not= \pm \op{v}$ while, one has
\beq
\bs{e} \, = \, \big( \bs{e}^{(1)},\cdots, \bs{e}^{(\ell)} \big) \quad with \quad \bs{e}^{(r)}\, = \,
- 2\op{v} \xi_{r}\cdot \big(1,\cdots, 1 \big) \in \R^{n_r} \;.
\label{definition z ups etape 2}
\enq
\vspace{2mm}
\noindent Then, for $|\mf{x}|\not=0$ and small enough,
\beq
\bs{x} \; \mapsto \; \ex{-(\bs{x},\bs{x}) } V(\bs{x}) \pl{\ups=\pm }{} \Big\{ \Xi\big(\, \mf{x} + z_{\ups}(\bs{x}) \big)\cdot \big[ \, \mf{x} + z_{\ups}(\bs{x}) \big]^{\de_{\ups}-1} \Big\} \, \in \, L^{1}\big( \mc{D}_{\bs{n}} \big) \;,
\enq
ensuring that $\mc{J}(\mf{x})$ is well-defined.
\vspace{2mm} \noindent Furthermore, there exists a smooth function $\mc{S}(\mf{x})$ in an open neighbourhood of $\mf{x}=0$ such that it holds
\beq
\mc{J}(\mf{x}) \, = \, \mc{J}_{\e{eff}}(\mf{x}) \, + \, \mc{S}(\mf{x})
\enq
where
\beq
\mc{J}_{\e{eff}}(\mf{x}) \, = \, \pl{r=1}{\ell} |\xi_r|^{-n_r^{2} } \cdot \Int{ \mc{D}^{(\e{eff})}_{ 2 \eta^{\prime} } }{} \ex{-(\bs{x},\op{M} \bs{x}) } \Big( \vp^{(\e{\sslash})}_{\e{eff}}\vp^{(\e{sg})}_{\e{eff}}V \Big)(\bs{x})
\pl{\ups=\pm }{} \Big\{ \Xi\big(\, \mf{x} + \tilde{z}_{\ups}(\bs{x}) \big)\cdot \big[ \, \mf{x} + \tilde{z}_{\ups}(\bs{x}) \big]^{\de_{\ups}-1} \Big\} \cdot \dd \bs{x} \;.
\label{definition integrale effective J eff modele}
\enq
The integral above runs through the domain
\beq
\mc{D}^{(\e{eff})}_{ \eta^{\prime} } \, = \, \bigg\{ \bs{x} \in \mc{D}_{\bs{n}} \; : \; |x_1^{(1)}| \, \leq \, C \eta^{\prime} \; , \;
\forall (r,a )\in \mc{M} \; : \; \big| t^{(0)}_r(x_1^{(1)})-x_a^{(r)} \big| \, \leq \, \xi_{r}^2 \eta^{\prime} \bigg\}
\label{definition du domaine D eff}
\enq
where $C>0$ is some constant while $t^{(0)}_r(x) = \tfrac{ \veps_1 \zeta_1}{ \veps_r \zeta_r } \big( \tfrac{\xi_r^2}{ \xi_1^2 } \big) \, x $. Furthermore, $\op{M}$ stands for the positive definite matrix
\beq
\op{M}=\left( \ba{ccc} \xi_1^{-2} \e{I}_{n_1} & 0& \ddots \\
0 & \ddots & 0 \\
\ddots & 0 & \xi_{\ell}^{-2} \e{I}_{n_{\ell} } \ea \right)\;,
\label{definition matrice positive definie M}
\enq
and the function $\tilde{z}_{\ups}$ takes the explicit form
\beq
\tilde{z}_{\ups}(\bs{x}) \; = \;- \sul{ (r,a)\in \mc{M} }{} \zeta_{r} \Bigg\{ \mf{h}_r\big( x_a^{(r)} \big) +\ups \op{v} x_a^{(r)} \Bigg\} \qquad where \qquad
\mf{h}_r\big( x \big) \; = \; - \zeta_r \veps_r \f{ x^2 }{ 2 \xi_r^2 } \, + \, \op{u} x \;.
\label{definition fonction z tilde effective}
\enq
The variables $\zeta_r\in \{ \pm \}$ are arbitrary.
Finally, one has that $\vp^{(\e{sg})}_{\e{eff}}$, $\vp^{(\sslash)}_{\e{eff}}$ are arbitrary smooth functions on $\mc{D}_{\bs{n}}$, such that
$0\leq\vp^{(*)}_{\e{eff}} \leq 1 $
\beq
\left\{ \ba{ccc} \vp^{(\e{sg})}_{\e{eff}} \; = \; 1 & on & \ov{\mc{D}^{(\e{eff})}_{ \eta^{\prime} }} \vspace{2mm} \\
\vp^{(\sslash)}_{\e{eff}} \; = \; 1 & o n& \ov{ \mc{D}^{(\sslash;\e{eff})}_{ \frac{1}{2}\eta^{\prime} } } \ea \right. \qquad and \qquad
\left\{ \ba{ccc} \vp^{(\e{sg})}_{\e{eff}} \; = \; 0 & on & \mc{D}_{\bs{n}} \setminus \mc{D}^{(\e{eff})}_{ 2\eta^{\prime} } \vspace{2mm} \\
\vp^{(\sslash)}_{\e{eff}} \; = \; 0 & on & \mc{D}_{\bs{n}} \setminus\mc{D}^{(\sslash;\e{eff})}_{ \eta^{\prime} } \ea \right.
\label{definition fcts approx unite sg et sslash integrale modele}
\enq
with
\beq
\mc{D}^{(\sslash;\e{eff})}_{ \eta^{\prime} } \; = \; \bigg\{ \bs{x} \in \mc{D}_{\bs{n}} \; : \; \; \forall (r,a )\in \mc{M} \; : \; \big| t^{(0)}_r(x_1^{(1)})-x_a^{(r)} \big| \, \leq \, \xi_{r}^2 \eta^{\prime} \bigg\} \;.
\label{definition domaine De eff sslash}
\enq
\end{prop}
\Proof
To start with, I take for granted that
\beq
\bs{x} \; \mapsto \, \ex{-(\bs{x},\bs{x}) } V(\bs{x}) \pl{\ups=\pm }{} \Big\{ \Xi\big(\, \mf{x} + z_{\ups}(\bs{x}) \big)\cdot \big[ \, \mf{x} + z_{\ups}(\bs{x}) \big]^{\de_{\ups}-1} \Big\} \; \in \; L^{1}\big( \mc{D}_{\bs{n}} \big) \;.
\enq
This issue will be dealt with at the end of the proof.
It is convenient to introduce $\check{\xi}_r \, = \, \zeta_r \xi_r$ with $\zeta_r\in \{ \pm \}$ and change variables through a rescaling
\beq
\bs{y}^{(r)} \; = \; \check{\xi}_r \bs{x}^{(r)} \;.
\enq
This yields
\beq
\mc{J}(\mf{x}) \, = \, \Int{ \mc{D}_{\bs{n}} }{} \mc{F}_{\e{tot}}(\bs{x})\cdot \dd \bs{x}
\enq
with
\beq
\mc{F}_{\e{tot}}(\bs{x}) \; = \; \pl{r=1}{\ell} |\xi_r|^{-n_r^{2} } \cdot \ex{-(\bs{x},\op{M} \bs{x}) } V(\bs{x})
\cdot \pl{\ups=\pm }{} \Big\{ \Xi\big(\, \mf{x} + \tilde{z}_{\ups}(\bs{x}) \big)\cdot \big[ \, \mf{x} + \tilde{z}_{\ups}(\bs{x}) \big]^{\de_{\ups}-1} \Big\}
\enq
and the positive definite matrix $\op{M}$ is as given in \eqref{definition matrice positive definie M}. Finally, the
functions $\tilde{z}_{\ups}$ have been introduced in \eqref{definition fonction z tilde effective}.
Observe that the functions $\mf{h}_r$ appearing as building blocks of $\tilde{z}_{\ups}$ are such that $\mf{h}_r^{\prime}$ is strictly monotonous.
Furthermore, it is readily checked that $t_r^{(0)}$, as given above of \eqref{definition du domaine D eff}, satisfies
$\mf{h}_r^{\prime}\big( t_r^{(0)}(x) \big) \, = \, \mf{h}_1^{\prime}\big(x \big)$ on $\R$.
One has the decomposition $\mc{D}_{\bs{n}}=\mc{D}^{(\perp ;\e{eff})}_{\eta^{\prime}}\sqcup \mc{D}^{(\sslash ;\e{eff})}_{\eta^{\prime}}$, with
\beq
\mc{D}^{(\perp ;\e{eff})}_{\eta^{\prime}} \, = \, \Big\{ \bs{x} \in \mc{D}_{\bs{n}} \; : \; \exists (r,a) \in \mc{M} \;\; , \;\; \big| \mf{h}_1^{\prime}( x_1^{(1)} ) \, - \, \mf{h}_r^{\prime}( x_a^{(r)} ) \big| > \eta^{\prime} \Big\}
\enq
and
\beq
\mc{D}^{(\sslash ;\e{eff})}_{\eta^{\prime}} \, = \, \Big\{ \bs{x} \in \mc{D}_{\bs{n}} \; : \; \forall (r,a) \in \mc{M} \;\; , \;\; \big| \mf{h}_1^{\prime}( x_1^{(1)} ) \, - \, \mf{h}_r^{\prime}( x_a^{(r)} ) \big| \, \leq \, \eta^{\prime} \Big\} \;.
\enq
Let $\vp^{(\sslash)}_{\e{eff}} $ be smooth and such that $0 \leq \vp^{(\sslash)}_{\e{eff}} \leq 1$
\beq
\vp^{(\sslash)}_{\e{eff}} \, = \, 1 \quad \e{on} \quad \ov{ \mc{D}^{(\sslash;\e{eff})}_{ \frac{ 1 }{ 2 } \eta^{\prime} } } \qquad \e{and} \qquad
\vp^{(\sslash)}_{\e{eff}} \, = \, 0 \quad \e{on} \quad \mc{D}^{(\perp;\e{eff})}_{ \eta^{\prime} } \;.
\enq
This allows one to split the original integral as $\mc{J}(\mf{x}) \, = \, \mc{J}^{(\perp)}(\mf{x})\, + \, \mc{J}^{(\sslash)}(\mf{x})$, with
\beq
\mc{J}^{(\perp)}(\mf{x}) \, = \, \Int{ \mc{D}^{(\perp ;\e{eff})}_{ \frac{ 1 }{ 2 } \eta^{\prime} } }{} \mc{F}_{\e{tot}}^{(\perp)}(\bs{x})\cdot \dd \bs{x}
\qquad \e{and} \qquad
\mc{J}^{(\sslash)}(\mf{x}) \, = \, \Int{ \mc{D}^{(\sslash ;\e{eff})}_{ \eta^{\prime} } }{} \mc{F}_{\e{tot}}^{(\sslash)}(\bs{x})\cdot \dd \bs{x} \;,
\enq
where it is understood that
\beq
\mc{F}_{\e{tot}}^{(\perp)}(\bs{x}) \; = \; \Big(1-\vp^{(\sslash)}_{\e{eff}}(\bs{x})\Big) \cdot \mc{F}_{\e{tot}} (\bs{x}) \qquad \e{and} \qquad
\mc{F}_{\e{tot}}^{(\sslash)}(\bs{x}) \; = \; \vp^{(\sslash)}_{\e{eff}}(\bs{x}) \cdot \mc{F}_{\e{tot}} (\bs{x}) \;.
\enq
\subsubsection*{$\bullet$ The integral $\mc{J}^{(\perp)}(\mf{x})$}
Pick $R>0$ large enough and let $\vp^{(\perp)}_{\e{eff}}$ be a smooth function on $\R$ satisfying
\beq
0 \, \leq \, \vp^{(\perp)}_{\e{eff}} \, \leq \, 1 \; , \quad
\vp^{(\perp)}_{\e{eff}}(x) = 1 \quad \e{for} \quad |x| \, \leq \, R \qquad \e{and} \qquad \vp^{(\perp)}_{\e{eff}}(x) = 0 \quad \e{for} \quad |x| \, \geq \, R+1 \;.
\enq
Thus, by writing $1=1\, - \, \vp^{(\perp)}_{\e{eff}}\big(x_a^{(r)} \big) \, + \, \vp^{(\perp)}_{\e{eff}}\big(x_a^{(r)} \big)$, $\vp^{(\perp)}_{\e{eff}}$ allows one
to build a partition of unity on $\mc{D}_{ \bs{n} }$ which separates, in each variable, the pieces containing $\infty$ in some of the variables and those being bounded
\beq
1\, = \hspace{-3mm}
\sul{ \substack{ \mc{M} = \\ \mc{M}_{\e{in}} \sqcup \mc{M}_{\e{out}} } }{} \hspace{-3mm} \Phi_{ \mc{M}_{\e{in}} ; \mc{M}_{\e{out}} }(\bs{x})
\qquad \e{with} \qquad
\Phi_{ \mc{M}_{\e{in}} ; \mc{M}_{\e{out}} }(\bs{x}) \, = \, \pl{(a,r) \in \mc{M}_{\e{in}} }{} \Big\{ \vp^{(\perp)}_{\e{eff}}\big(x_a^{(r)} \big) \Big\} \; \cdot \hspace{-3mm}
\pl{(a,r) \in \mc{M}_{\e{out}} }{} \Big\{ 1-\vp^{(\perp)}_{\e{eff}}\big(x_a^{(r)} \big) \Big\}
\enq
where the sum runs through all partitions of $\mc{M}$ into two disjoint sets $\mc{M}_{\e{in}}$ and $\mc{M}_{\e{out}}$.
This partition of unity leads to the decomposition
\beq
\mc{J}^{(\perp)}(\mf{x}) \; = \; \sul{ \mc{M} = \mc{M}_{\e{in}} \sqcup \mc{M}_{\e{out}} }{} \mc{J}^{(\perp)}_{ \mc{M}_{\e{in}} ; \mc{M}_{\e{out}} }(\mf{x})
\enq
where, upon making the change of variables
\beq
y_a^{(r)}\, = \, (x_a^{(r)})^{-1} \quad \e{if} \quad (r,a) \in \mc{M}_{\e{out}} \; , \qquad
y_a^{(r)}\, = \, x_a^{(r)} \quad \e{if} \quad (r,a) \in \mc{M}_{\e{in}}
\enq
and denoting $\bs{x}(\bs{y}; \mc{M}_{\e{out}} )$ the obtained vector, one has
\beq
\mc{J}^{(\perp)}_{ \mc{M}_{\e{in}} ; \mc{M}_{\e{out}} }(\mf{x}) \; = \;
\Int{ \mc{D}^{(\perp)}_{R; \mc{M}_{\e{out}} } }{} \f{ \mc{F}_{\e{tot}}^{(\perp)}\Big( \bs{x}(\bs{y}; \mc{M}_{\e{out}} ) \Big) }{ \pl{(a,r) \in\mc{M}_{\e{out}} }{} \big( y_a^{(r)} \big)^2 } \cdot \dd \bs{y}
\label{ecriture integrale J perp ramenee a un interval compact}
\enq
with
\beq
\mc{D}^{(\perp)}_{R; \mc{M}_{\e{out}} } \; = \; \Bigg\{ \bs{y} \; : \; \bs{x}(\bs{y}; \mc{M}_{\e{out}} ) \in \mc{D}^{(\perp)}_{\eta^{\prime}} \quad \e{and} \quad
\ba{cc} |y_a^{(r)}| \, \leq \, R +1 & \forall (r,a) \in \mc{M}_{\e{in}} \vspace{2mm} \\
|y_a^{(r)}| \, \leq \, R^{-1} & \forall (r,a) \in \mc{M}_{\e{out}} \ea \Bigg\} \;.
\enq
Note that $\bs{x}(\bs{y}; \mc{M}_{\e{out}} )$ does depend on the given choice of the partition.
It is easy to see that the integrand in \eqref{ecriture integrale J perp ramenee a un interval compact} is smooth and vanishes on $\Dp{} \mc{D}^{(\perp)}_{R; \mc{M}_{\e{out}} } $. The smoothness at the origin
follows from the Gaussian decay of $\mc{F}_{\e{tot}}^{(\perp)}$ at infinity.
Furthermore, for any $\bs{k}\in \mc{D}^{(\perp)}_{R; \mc{M}_{\e{out}} } $, there exists $(r,a)\not= (1,1)$ such that
\beq
\bigg| \mf{h}^{\prime}_1\Big(x_1^{(1)}(\bs{k}; \mc{M}_{\e{out}} ) \Big) - \mf{h}^{\prime}_r\Big(x_a^{(r)}(\bs{k}; \mc{M}_{\e{out}} ) \Big) \bigg| \, > \, \eta^{\prime} \;.
\enq
Then, set
\beq
f_{[r,a]}(\bs{y}) \, = \, \Big( \bs{y}_1^{(1)},\bs{y}^{(2)},\cdots, \bs{y}^{(r-1)}, \bs{y}_{[a]}^{(r)}, \bs{y}^{(r+1)}, \cdots, \bs{y}^{(\ell)},
\tilde{z}_{+}\big( \bs{x}(\bs{y}; \mc{M}_{\e{out}} ) \big), \tilde{z}_{-}\big( \bs{x}(\bs{y}; \mc{M}_{\e{out}} ) \big) \Big)
\enq
so that it holds
\beq
\det\Big[ D_{\bs{k}} f_{[r,a]} \Big] \; = \; (-1)^{m_{r,a}} \zeta_1 \zeta_r 2 \op{v} \cdot \bigg( \f{ -1 }{ (k_1^{(1)})^2 } \bigg)^{\bs{1}_{ \mc{M}_{\e{out}}(1,1) }} \cdot
\bigg( \f{ -1 }{ (k_a^{(r)})^2 } \bigg)^{\bs{1}_{ \mc{M}_{\e{out}}(a,r) }}
\cdot \Big( \mf{h}^{\prime}_r \big( x_a^{(r)}(\bs{k}; \mc{M}_{\e{out}} ) \big) \, - \, \mf{h}^{\prime}_1 \big( x_1^{(1)}(\bs{k}; \mc{M}_{\e{out}} ) \big) \Big) \;,
\enq
where $m_{r,a}=a+\sul{b=1}{r-1}n_b$. The latter ensures the local invertibility of $f_{[r,a]}(\bs{y})$ around $\bs{k}$ and thus the applicability of Lemma \ref{Lemme integrale multidimensionnelle auxiliaire reguliere} to the integral of interest.
Hence, for any given partition $ \mc{M}_{\e{in}}\sqcup \mc{M}_{\e{out}} $ of $\mc{M}$, $ \mc{J}^{(\perp)}_{ \mc{M}_{\e{in}} ; \mc{M}_{\e{out}} }(\mf{x}) $ is smooth in $\mf{x}$ and thus so is $\mc{J}^{(\perp)}(\mf{x}) $
as a finite sum of smooth functions.
\subsubsection*{$\bullet$ The integral $\mc{J}^{(\sslash)}(\mf{x})$}
For the purpose of further reasoning, it is convenient to define
\beq
\msc{P}(x) \; = \; \f{ x }{ \veps_1 \zeta_1 \xi_1^2 } \sul{r=1}{\ell} n_r \veps_r \xi_r^2 \qquad \e{and} \qquad
\msc{E}(x) \; = \; \sul{r=1}{\ell} n_r \, \zeta_r \cdot \mf{h}_r\Big( t_r^{(0)} (x ) \Big)
\label{definition impulsion et energie effectives locales}
\enq
so that, by assumption \eqref{hypothese positivite contrainte sur nr epsr zetar squared}, $\big| \msc{P}^{ \prime }(x) \big|\geq c>0$ for any $x \in \R$ and some $c>0$.
To start with, one observes that $\tilde{z}_{\ups}(\bs{x}) \; = \; \mc{Z}_{\ups}(x_1^{(1)}) \, + \, \de \tilde{z}_{\ups}(\bs{x}) $ with
\beq
\mc{Z}_{\ups}(x) \; = \; - \Big\{ \msc{E}(x) + \ups \op{v} \msc{P}(x) \Big\} \qquad \e{and} \qquad
\de \tilde{z}_{\ups}(\bs{x}) \; = \; - \sul{ (r,a)\in \mc{M} }{} \zeta_{r} \mf{w}_{\ups}^{(r)}\Big( x_a^{(r)}, t_r^{(0)}(x_1^{(1)}) \Big) \;,
\enq
in which $\mf{w}_{\ups}^{(r)}\big( x, y \big) \, = \, \mf{h}_{r}(x) - \mf{h}_{r}(y) \, + \, \ups \op{v} (x-y) $.
Observe that the bound
\beq
\big| \mf{h}_1^{\prime}( x_1^{(1)} ) \, - \, \mf{h}_r^{\prime}( x_a^{(r)} ) \big| \, \leq \, \eta^{\prime}
\qquad \e{is} \; \e{equivalent} \; \e{to} \qquad
| t_r^{(0)}(x_1^{(1)})-x_a^{(r)} |\, \leq \, \xi_r^2 \eta^{\prime} \;,
\enq
what ensures that $ \mf{w}_{\ups}^{(r)}\Big( x_a^{(r)}, t_r^{(0)}(x_1^{(1)}) \Big) = \e{O}(\eta^{\prime})$
under such bounds and thus, it holds uniformly in $\bs{x}\in \mc{D}^{(\sslash)}_{\eta^{\prime}}$ that $ \de \tilde{z}_{\ups}(\bs{x}) \, = \, \e{O}(\eta^{\prime})$.
It is readily seen that
\beq
\mc{Z}_{\ups}^{\prime}(x) \; = \;- \msc{P}^{\prime}(x) \Big( \mf{h}_{1}^{\prime}(x)+ \ups \op{v} \Big)
\enq
so that $\mc{Z}_{\ups}^{\prime}$ vanishes at the points $s_{ \ups}= \zeta_1 \veps_1 \xi^2_1 \big( \op{u}+\ups \op{v} \big)$.
However, one has that for $\ups, \ups^{\prime} \in \{\pm 1\}$,
\beq
\mc{Z}_{\ups}(s_{\ups^{\prime}}) \; = \; - \f{ 1 }{ 2 } \big( \op{u} + \ups^{\prime} \op{v} \big) \big( \op{u}+(2\ups-\ups^{\prime}) \op{v} \big) \cdot \sul{r=1}{\ell} \veps_r n_r \xi_r^2 \; \not= \; 0
\label{ecriture evalutation Z ups sur s ups prime}
\enq
owing to the hypotheses of the proposition.
Thus, it follows from the above that $\mc{Z}_{\ups}$ is strictly monotonous on $\intoo{-\infty}{ s_{\ups} }$ and on $\intoo{ s_{\ups} }{+\infty}$. Furthermore, one has $s_{+}\not= s_{-}$.
Thus one may introduce the three intervals
\beq
I^{(-)} \, = \, \intoo{-\infty}{ \min_{\ups= \pm} \{ s_{\ups} \} }\; \; , \qquad
I^{(c)} \, = \, \intoo{ \min_{\ups= \pm} \{ s_{\ups} \} }{ \max_{\ups= \pm} \{ s_{\ups} \} } \; \; , \qquad
I^{(+)} \, = \, \intoo{ \max_{\ups= \pm} \{ s_{\ups} \} }{ + \infty } \;.
\enq
In each of these intervals $x \mapsto \mc{Z}_{\pm}(x)$ are both strictly monotonous. First assume that $\op{u} \not= \pm 3 \op{v}$. Then, the previous calculations ensure that $\mc{Z}_{\pm} \not=0$
on $\Dp{} I^{(\tau)}$, for any $\tau \in \{c, \pm \}$.
Let $\tau_0\in \{c, \pm \}$ be such that $0 \in I^{(\tau_0)}$. Note that $\tau_{0}$ is well defind since $s_{\pm} \not=0$. This entails that $0 \not \in \Dp{} I^{(\tau)}$ for any $\tau \in \{c, \pm \}$.
Finally, one decomposes
\beq
\mc{D}^{(\sslash;\e{eff})}_{\eta^{\prime}} \, = \, \bigcup_{ \tau \in \{c, \pm \} } \mc{D}^{(\sslash;\e{eff})}_{_{\eta^{\prime}}; \tau} \qquad \e{with} \qquad
\mc{D}^{(\sslash;\e{eff})}_{_{\eta^{\prime}};\tau}\, = \, \Big\{ \bs{x} \in \mc{D}^{(\sslash;\e{eff})}_{\eta^{\prime}} \; : \; x_1^{(1)} \in \ov{I^{(\tau)}} \Big\}
\enq
what induces the decomposition of the original integral $\mc{J}^{(\sslash)}(\mf{x})= \sul{ \tau \in \{c, \pm \} }{} \mc{J}^{(\sslash)}_{\tau} (\mf{x})$
with
\beq
\mc{J}^{(\sslash)}_{\tau}(\mf{x}) \, = \, \Int{ \mc{D}^{(\sslash)}_{_{\eta^{\prime}};\tau} }{} \mc{F}_{\e{tot}}^{(\sslash)}(\bs{x})\cdot \dd \bs{x} \;.
\enq
\subsubsection*{$\bullet$ The integral $\mc{J}^{(\sslash)}_{\tau_0}(\mf{x})$}
The analysis depends on whether $|\op{u}|>\op{v}$ or $|\op{u}|<\op{v}$.
\vspace{2mm}
{\bf i)} The $|\op{u}|<\op{v}$ case.
\vspace{2mm}
\noindent Let $\sg=\e{sgn}\big( \msc{P}^{\prime}(0) \, (\op{u}+\op{v}) \big)$. Since $x \mapsto \mc{Z}_{\ups}(x)$ is strictly monotonous on $I^{(\tau_0)}$, $0 \in I^{(\tau_0)}$
it follows that
\beq
\e{sgn}\Big( \mc{Z}_{\ups}^{\prime}(x) \Big) = \e{sgn}\Big( \mc{Z}_{\ups}^{\prime}(0) \Big) = -\ups \sg \quad \e{for} \quad x \in I^{(\tau_0)} \; .
\enq
By using that $\mc{Z}_{\ups}(0)=0$, since $x \mapsto \mc{Z}_{\ups}(x)$ is strictly monotonous and since $\mc{Z}_{\ups}^{\prime}(x)$ vanishes, at most, on $\Dp{}I^{(\tau_0)}$,
one infers that
\beq
\ba{ccclccc} \sg \ups \tilde{z}_{\ups}(\bs{x}) & < & - \eta^{\prime} & \e{if} & x_1^{(1)} & > & C \eta^{\prime} \vspace{3mm} \\
\sg \ups \tilde{z}_{\ups}(\bs{x}) & > & \eta^{\prime} & \e{if} & x_1^{(1)} & < & - C \eta^{\prime}
\ea \;,
\label{region positivite tilde z ups}
\enq
this uniformly in $\bs{x} \in \mc{D}^{(\sslash)}_{\eta^{\prime};\tau_0}$ and for some constant $C>0$.
Therefore \eqref{region positivite tilde z ups} entails that, at least for $|\mf{x}|<\tf{ \eta^{\prime} }{2}$,
$\mf{x}+\tilde{z}_{\ups}(\bs{x})$ have both opposite signs on $\mc{D}^{(\sslash)}_{\eta^{\prime};\tau_0}$ as soon as $ |x_1^{(1)}| \, > \, C \eta^{\prime} $.
The presence of Heaviside functions in the integrand $\mc{F}_{\e{tot}}(\bs{x})$ entails that one has the reduction of the integration domain so that it holds
\beq
\mc{J}^{(\sslash)}_{\tau_0}(\mf{x}) \, = \, \Int{ \mc{D}^{(\e{eff})}_{ \eta^{\prime} } }{} \mc{F}_{\e{tot}}^{(\sslash)}(\bs{x}) \cdot \dd \bs{x}
\qquad \e{for} \;\; |\mf{x}|<\tf{ \eta^{\prime} }{2}.
\enq
The domain $\mc{D}^{(\e{eff})}_{ \eta^{\prime} }$ appearing above is as defined in \eqref{definition du domaine D eff}.
Finally, let $\vp^{(\e{sg})}_{\e{eff}}$ be smooth on $\pl{r=1}{\ell} \R^{n_r}$ and such that
\beq
0 \leq \vp^{(\e{sg})}_{\e{eff}} \leq 1 \quad, \quad \vp^{(\e{sg})}_{\e{eff}}=1 \quad \e{on} \quad \ov{\mc{D}^{(\e{eff})}_{ \eta^{\prime} }} \quad \e{and} \quad
\vp^{(\e{sg})}_{\e{eff}}=0 \quad \e{on} \quad \mc{D}_{\bs{n}} \setminus \mc{D}^{(\e{eff})}_{ 2 \eta^{\prime} } \;.
\label{definition fct charact lisse ensemble D eff eta prime}
\enq
Since the integrand vansihes anyway outside of $\mc{D}^{(\e{eff})}_{ \eta^{\prime} } $, it holds
\beq
\mc{J}^{(\sslash)}_{\tau_0}(\mf{x}) \, = \, \Int{ \mc{D}^{(\e{eff})}_{ 2 \eta^{\prime} } }{} \vp^{(\e{sg})}_{\e{eff}}(\bs{x}) \mc{F}_{\e{tot}}^{(\sslash)}(\bs{x}) \cdot \dd \bs{x} \; ,
\enq
what corresponds exactly to $\mc{J}^{(\sslash)}_{\e{eff}}(\mf{x})$ as given in \eqref{definition integrale effective J eff modele}.
\vspace{2mm}
{\bf ii)} The $|\op{u}|>\op{v}$ case.
\vspace{2mm}
Keeping the definition for $\sg$ as above, the same reasonings ensure that, now, $\e{sgn}\Big( \mc{Z}_{\ups}^{\prime}(x) \Big) = - \sg $.
This then leads to
\beq
\ba{ccc} \sg \tilde{z}_{\ups}(\bs{x}) \, < \, - \eta^{\prime} & \e{if} & x_1^{(1)} > C \eta^{\prime} \vspace{3mm} \\
\sg \tilde{z}_{\ups}(\bs{x}) \, > \, \eta^{\prime} & \e{if} & x_1^{(1)} < - C \eta^{\prime}
\ea \;,
\enq
this uniformly in $\bs{x} \in \mc{D}^{(\sslash)}_{\eta^{\prime};\tau_0}$ and for some constant $C>0$. One then introduces $\vp^{(\e{sg})}_{\e{eff}}$
as in \eqref{definition fct charact lisse ensemble D eff eta prime} and using that
\beq
\mc{J}^{(\sslash)}_{\tau_0}(\mf{x}) \, = \, \Int{ \mc{D}^{(\sslash;\e{eff})}_{ 2\eta^{\prime} ;\tau_0} }{} \mc{F}_{\e{tot}}^{(\sslash)}(\bs{x})\cdot \dd \bs{x}
\enq
since $ \mc{F}_{\e{tot}}^{(\sslash)}$ vanishes on $ \mc{D}^{(\sslash;\e{eff})}_{2\eta^{\prime};\tau_0} \setminus \mc{D}^{(\sslash;\e{eff})}_{\eta^{\prime};\tau_0} $, one may
decompose the integral as
\beq
\mc{J}^{(\sslash)}_{\tau_0}(\mf{x}) \, = \, \mc{J}_{\e{eff}}(\mf{x}) \; + \; \mc{J}_{\tau_0; \e{out}}(\mf{x})
\qquad \e{with} \qquad
\mc{J}_{\tau_0; \e{out}}(\mf{x}) \, = \hspace{-2mm} \Int{ \mc{D}^{(\e{out})}_{ \eta^{\prime} ; \tau_0 } }{} \hspace{-2mm} \big( 1 - \vp^{(\e{sg})}(\bs{x}) \big) \cdot \mc{F}_{\e{tot}}^{(\sslash)}(\bs{x}) \cdot \dd \bs{x}
\enq
in which $\mc{J}_{\e{eff}}(\mf{x})$ is as given in \eqref{definition integrale effective J eff modele}, while
\beq
\mc{D}^{(\e{out})}_{ \eta^{\prime}; \tau_0 } \; = \; \bigg\{ \bs{x} \in \mc{D}^{(\sslash)}_{\eta^{\prime};\tau_0} \; : \; |x_1^{(1)}| \, > \, C \eta^{\prime} \; , \; \forall (r,a )\in \mc{M} \; : \;
\big| t_r^{(0)}(x_1^{(1)})-x_a^{(r)} \big| \, < \, 2 \xi_{r}^2 \eta^{\prime} \bigg\} \;.
\enq
The above bounds ensure that if $|\mf{x}|< \tf{\eta^{\prime}}{2}$, then $\big| \mf{x}+\tilde{z}_{\ups}(\bs{x}) \big| \geq \tf{\eta^{\prime}}{2}$ uniformly on $ \mc{D}^{(\e{out})}_{ \eta^{\prime}; \tau_0 }$.
Then, derivation under the integral theorems ensure that $\mf{x} \mapsto \mc{J}_{\tau_0; \e{out}}(\mf{x}) $ is a smooth function of $\mf{x}$
in an open neighbourhood of $\mf{x}=0$.
\subsubsection*{$\bullet$ The integral $\mc{J}^{(\sslash)}_{ \tau }(\mf{x})$ with $\tau \not= \tau_0$}
When $\tau \not= \tau_0$, by construction, it holds that $\e{d}(I^{(\tau)},0)>0$, in which $\e{d}(A,x)$ stands for the Euclidian distance from the set $A$ to the point $x$.
This property, along with the explicit form for $\msc{P}$ and $\msc{E}$ given in \eqref{definition impulsion et energie effectives locales} both ensure that it holds
\beq
(0,0) \not\in \Big\{ (\msc{P}(x),\msc{E}(x)) \; : \; x \in I^{(\tau)} \Big\} \;.
\enq
Furthermore, the previous arguments ensure that
\beq
\min_{ \substack{ x \in \Dp{} I^{(\tau)} \\ \ups = \pm } } \Big| \msc{E}(x) + \ups \msc{P}(x) \Big| >0 \;.
\enq
The above properties guarantee that the integral $\mc{J}^{(\sslash)}_{\tau}(\mf{x})$ is a particular example of the general class
of integrals considered in the section "Behaviour of $\mc{I}^{(\sslash)}$ in the regular case"
of the proof of Theorem \ref{Theorem principal caractere lisse et non lisse des integrals multi particules}.
One should simply make the identification $\mf{u}_r \hookrightarrow \mf{h}_r$. Then, that very same analysis ensures that
$\mf{x} \mapsto \mc{J}^{(\sslash)}_{\tau}(\mf{x})$ is smooth in $\mf{x}$.
\vspace{2mm}
It remains to comment on the case when $\op{u} = 3 \ups^{\prime} \op{v}$, for some $\ups^{\prime} \in \{\pm 1\}$. Then, by \eqref{ecriture evalutation Z ups sur s ups prime},
for any $\ups \in \{ \pm \}$ it holds that $\mc{Z}_{\ups}(s_{\ups})\not=0$. However, one has that $\mc{Z}_{- \ups^{\prime}}(s_{\ups^{\prime}}) =0$
and $\mc{Z}_{ \ups^{\prime}}(s_{-\ups^{\prime}}) \not=0$.
One should then split the integration domain as
\beq
\intoo{ -\infty }{ s_{-\ups^{\prime}} } \cup \intoo{ s_{-\ups^{\prime}} }{ s_{ \ups^{\prime}} -\eps } \cup \intoo{ s_{ \ups^{\prime}} -\eps }{ s_{ \ups^{\prime}} +\eps } \cup
\intoo{ s_{ \ups^{\prime}} +\eps }{+\infty}
\enq
where $\eps>0$ is taken small enough and, for simplicity, I assumed that $s_{\ups^{\prime}}>s_{-\ups^{\prime}}$, the other situation being tractable in a similar way.
The analysis on all the intervals other that $\intoo{ s_{ \ups^{\prime}} -\eps }{ s_{ \ups^{\prime}} +\eps } $ goes along the lies described above, while on
$\intoo{ s_{ \ups^{\prime}} -\eps }{ s_{ \ups^{\prime}} +\eps } $, one should proceed by implementing a change of variables analogous to \eqref{ecriture chngement vars f ups 11}.
Reasonings as in \eqref{ecriture integrale I ups sslash}-\eqref{ecriture integrale I ups k} then allow one to conclude on the smoothness of such contributions.
\subsubsection*{$\bullet$ The $L^1(\mc{D}_{\bs{n}})$ character}
It remains to prove the $L^{1}(\mc{D}_{\bs{n}})$ nature of the integrand. Since
\beq
V(\bs{x}) \pl{\ups=\pm }{} \Big\{ \Xi\big(\, \mf{x} + z_{\ups}(\bs{x}) \big)\cdot \big[ \, \mf{x} + z_{\ups}(\bs{x}) \big]^{\de_{\ups}-1} \Big\}
\enq
grows algebraically in $\norm{\bs{x}}$ at infinity, the Gaussian prefactor $ \ex{-(\bs{x},\bs{x}) } $ ensures integrability at $\infty$.
Furthermore, if both $\de_{\pm}\geq 1$, then the integrand is bounded on compact subsets of $\mc{D}_{ n }$ what entails its
$L^{1}(\mc{D}_{\bs{n}})$ nature. If at least one inequality $0<\de_{\ups} <1$ holds, then integrability issues may arise
from a neighbourhood of the points where $\mf{x} + z_{\ups}(\bs{x}) = 0$. Moreover, since the integrand is strictly positive, it is enought to prove
local integrability. By following the integral reduction steps that are outlined in the last part of the proof of Theorem \ref{Theorem principal caractere lisse et non lisse des integrals multi particules},
one eventually ends up with one-dimensional integrals whose direct inspection shows that the local $L^1$-character boils down to the condition $0<\de_{\ups} <1$.
\qed
\subsection{Asymptotic behaviour of the model integral}
\begin{prop}
\label{Proposition asymptotique integrale modele}
Let $V$ be as given in \eqref{rappel definition V et P alpha}. Consider the integral
\beq
\mc{J}(\mf{x}) \, = \, \Int{ \mc{D}_{\bs{n}} }{} \ex{-(\bs{x},\bs{x}) } V(\bs{x}) \pl{\ups=\pm }{} \Big\{ \Xi\big(\, \mf{x} + z_{\ups}(\bs{x}) \big)\cdot \big[ \, \mf{x} + z_{\ups}(\bs{x}) \big]^{\de_{\ups}-1} \Big\} \cdot \dd \bs{x}
\qquad over \qquad
\mc{D}_{\bs{n}}=\pl{r=1}{\ell} \R^{n_r} \;.
\label{definition integrale modele pour extraire le comportement singulier}
\enq
Here $\de_{\ups}>0$ and it is assumed that $\sum_{r=1}^{\ell}n_r\geq 2$. Further, the functions $ z_{\ups}(\bs{x}) $ are quadratic forms
\beq
z_{\ups}(\bs{x}) \; = \; \f{1}{2} \big( \bs{x},\op{E} \bs{x} \big) \, + \, \big( \bs{x}, \bs{e}) \cdot \f{ \op{u}+\ups \op{v} }{ 2\op{v} }
\qquad with \qquad
\op{E}=\left( \ba{ccc} \veps_1 \e{I}_{n_1} & 0& \ddots \\
0 & \ddots & 0 \\
\ddots & 0 & \veps_{\ell} \e{I}_{n_{\ell} } \ea \right)\;.
\enq
There $\veps_r \in \{\pm 1\}$, $(\cdot, \cdot)$ is the canonical scalar product on $\mc{D}_{\bs{n}}$, $(\op{u} , \op{v})\in \R \times \R^+$ are such that $\op{u}\not= \pm \op{v}$ and, given $\xi_{r} \in \R$ one has
\beq
\bs{e} \, = \, \big( \bs{e}^{(1)},\cdots, \bs{e}^{(\ell)} \big) \quad with \quad \bs{e}^{(r)}\, = \, - 2\op{v} \xi_{r}\cdot \big(1,\cdots, 1 \big) \in \R^{n_r} \;.
\enq
The parameters in play are such that
\beq
\sum_{r=1}^{\ell} \veps_{r} \xi_r^2 n_r \not=0 \;.
\label{ecriture hypothese sur les nr veps et zetar}
\enq
Then, there exists a smooth function $\mc{S}$ in a neighbourhood of $0$ such that the $\mf{x}\tend 0$ asymptotic expansion holds:
\bem
\mc{J}(\mf{x}) \, = \, | \mf{x} |^{ \vth } \Ga(\de_+) \Ga(\de_-) \Ga(-\vth ) \cdot \f{ [2\op{v}]^{\de_++\de_--1} }{ \pl{\ups= \pm }{} \big| \op{v}-\ups \op{u} \big|^{\de_{\ups}} }
\cdot \f{ \pl{r=1}{\ell} \Big\{ G(2+n_r) \cdot \big( 2\pi \big)^{ \tfrac{ n_r-\de_{r,1} }{2} } \Big\} }{ \sqrt{ \big| \sum_{r=1}^{\ell} \veps_r \xi_r^2 n_r \big| } } \\
\times \bigg\{ \Xi(\mf{x}) \f{ \sin[\pi \nu_+] }{ \pi } \, + \, \Xi(-\mf{x}) \f{ \sin[\pi \nu_-] }{ \pi } \bigg\} \Big( 1 + \e{O}(\mf{x}) \Big) \; + \; \mc{ S }( \mf{x} ) \;.
\label{ecriture DA kappa zero de integrale modele}
\end{multline}
where
\beq
\vth \; = \; \f{1}{2}\sul{r=1}{\ell}n_r^2 \; - \; \f{3}{2} \, + \, \de_+ \, + \, \de_- \;
\label{definition de cal theta}
\enq
and given $\eps \in \{ \pm \}$
\beq
\nu_{ \eps } \; = \; \f{1}{2}\sul{ \substack{ r=1 \, : \, \\ \eps \veps_r= -1} }{ \ell } n_r^2 \; - \; \f{ 1 +\eps \vsg }{ 4 } \, + \, \sul{ \substack{ \ups=\pm \\ \eps \sg_{\ups}>0 } }{} \de_{\ups}
\qquad with \qquad \left\{ \ba{ccc}
\vsg & =& \e{sgn}\Big( - \sul{r=1}{\ell} \veps_r n_r \xi_r^2 \Big) \vspace{2mm} \\
\sg_{\ups} &= & 1-\ups \f{ \op{u} }{ \op{v} } \ea \right. \;.
\label{definition du signe xi}
\enq
\end{prop}
\Proof
Recall the notation \eqref{notation vecteur dans produit cartesien} for the vector $\bs{x}$.
Recall that, for $\om \in \R^{*}$, one has the integral representation
\beq
\f{ \Ga(\a) }{ 2\pi } \Int{ \msc{C}_{\om} }{} \dd t \f{ \ex{-\i \om t} }{ [-\i t]^{\a} } \, = \, \om^{\a-1} \Xi(\om)
\label{ecriture rep int fonction saut fois puissance}
\enq
where the contour $\msc{C}_{\om}$ passes slightly above $0$ and then goes to infinity in either of the two directions $\Re(t)\tend \pm \infty$
along two rays $\mc{R}_{\om}$ which enjoy the property that
$\Im(\om t) \tend -\infty$, linearly in $|t|$, when $ t \tend \infty$ along $\mc{R}_{\om}$.
The maps $\wh{z}_{\ups}=\mf{x}+z_{\ups}$ are such that
\beq
D_{\bs{x}} \wh{z}_{\ups}\cdot \bs{h} = \big( \bs{x},\op{E}\bs{h}) \, + \, \f{\op{u}+\ups \op{v} }{2\op{v} } \big( \bs{e},\bs{h} \big)
\enq
so that $D_{\bs{x}} \wh{z}_{\ups}$ is surjective with the exception of the point $\bs{x}_{\ups}=-\op{E}^{-1}\bs{e}\tfrac{\op{u}+\ups \op{v} }{2\op{v} }$.
Still, as ensured by the assumption \eqref{ecriture hypothese sur les nr veps et zetar}, one has, for $|\mf{x}|$ small enough,
\beq
\wh{z}_{\ups}(\bs{x}_{\ups}) \, = \, \mf{x} -\f{1}{2} \Big( \f{\op{u}+\ups \op{v} }{2\op{v} } \Big)^2 \Big( \bs{e}, \op{E}^{-1} \bs{e} \Big) \not=0\;.
\enq
Therefore, $\wh{z}_{\ups}$ is a submersion in an open neighbourhood of $\mc{M}_{\ups}=(\,\wh{z}_{\ups})^{-1}(0)$. Hence, $\mc{M}_{\ups}$
is a $\sum_{r=1}^{\ell}n_r-1$ dimensional sub-manifold of $\mc{D}_{\bs{n}}$
and, as such, has Lebesgue measure zero. It follows that upon setting $\mc{M}=\mc{M}_{+}\cup\mc{M}_-$, one has the representation
\beq
\f{ \mc{J}(\mf{x}) }{ \Ga(\de_+) \, \Ga( \de_-) } \; = \;
\Int{ \mc{D}_{\bs{n}} \setminus \mc{M} }{} \hspace{-3mm} \dd \bs{x}\; \ex{-(\bs{x},\bs{x}) } V(\bs{x}) \Int{ \msc{C}_{\,\wh{z}_+(\bs{x})} }{} \! \f{ \dd \la }{2 \pi} \, \f{ \ex{-\i \la \, \wh{z}_+(\bs{x}) } }{ [-\i \la]^{\de_{+}} }
\Int{ \msc{C}_{\,\wh{z}_-(\bs{x})} }{} \! \f{ \dd \mu }{2 \pi} \, \f{ \ex{-\i \mu \, \wh{z}_-(\bs{x}) } }{ [-\i \mu]^{\de_{-}} } \;.
\enq
This representation follows by, first, replacing $\mc{D}_{\bs{n}}$ by $\mc{D}_{\bs{n}}\setminus\mc{M}$ in $\mc{J}(\mf{x})$ as given by \eqref{definition integrale modele pour extraire le comportement singulier}
and then using the integral representation \eqref{ecriture rep int fonction saut fois puissance} for the products involving the $\wh{z}_{\ups}$ functions.
The form of the $\la,\mu$ contours ensures exponential decay at infinity of these integrals. Furthermore, it is easy to check that
\beq
\bigg| \Int{ \msc{C}_{\,\wh{z}_{\pm}(\bs{x})} }{} \f{ \dd \mu }{2 \pi} \f{ \ex{-\i \mu \, \wh{z}_{\pm}(\bs{x}) } }{ [-\i \mu]^{\de_{\pm}} } \bigg| \; \leq \; C \, | \wh{z}_{\pm}(\bs{x}) |^{\de_{\pm}-1}
\enq
for some constant $C$. By using Proposition \ref{Proposition reduction vers voisinage singularite integrale modele}, it is easy to see that
\beq
\bs{x} \; \mapsto \; \ex{-(\bs{x},\bs{x}) } V(\bs{x}) \pl{\ups=\pm }{} \Big\{| \wh{z}_{\pm}(\bs{x}) |^{\de_{\pm}-1} \Big\}
\enq
is in $L^{1}( \mc{D}_{\bs{n}} )$. Thence, one can apply dominated convergence to get
\beq
\f{ \mc{J}(\mf{x}) }{\Ga(\de_+) \, \Ga( \de_-) } \; = \; \lim_{\tau_t, \tau_s\tend 0^+} \Int{ \mc{D}_{\bs{n}} \setminus \mc{M} }{} \hspace{-3mm} \dd \bs{x}\; \ex{-(\bs{x},\bs{x}) } V(\bs{x}) \hspace{-4mm}
\Int{ \msc{C}_{\,\wh{z}_+(\bs{x})} \times \msc{C}_{\,\wh{z}_-(\bs{x})} }{} \hspace{-5mm} \f{ \dd \la \, \dd \mu }{(2 \pi)^2} \f{ \ex{-\i \la \, \wh{z}_+(\bs{x}) } }{ [-\i \la]^{\de_{+}} }
\cdot \f{ \ex{-\i \mu \, \wh{z}_-(\bs{x}) } }{ [-\i \mu]^{\de_{-}} } \cdot \exp\bigg\{ -\tau_t\Big(\f{\la+\mu}{2}\Big)^2-\tau_s\Big(\f{\la-\mu}{2}\Big)^2 \bigg\} \;.
\enq
Since the integrand under the limit has now Gaussian convergence in $\la, \mu \tend \infty$, one can deform the contours $\msc{C}_{\,\wh{z}_{\ups}(\bs{x})} \hookrightarrow \R+\i \a$
for some $\a>0$ and small enough. Since the new $\la, \mu$ contours become $\bs{x}$ independent and one has a rapid convergence of the integrand at infinity, by Fubbini's theorem, one can swap the order of integration and take the
$\bs{x}$ integration first. Then, using again that $\mc{M}$ has Lebesgue measure $0$, yields
\beq
\f{ \mc{J}(\mf{x}) }{ \Ga(\de_+) \, \Ga( \de_-) } \; = \; \lim_{\tau_t, \tau_s\tend 0^+} \Int{ (\R+\i\a)^2 }{} \hspace{-3mm} \f{ \dd \la \, \dd \mu }{(2 \pi)^2}
\Int{ \mc{D}_{\bs{n}} }{} \dd \bs{x}\; \ex{-(\bs{x},\bs{x}) } V(\bs{x})
\f{ \ex{-\i \la \, \wh{z}_+(\bs{x}) } }{ [-\i \la]^{\de_{+}} }
\cdot \f{ \ex{-\i \mu \, \wh{z}_-(\bs{x}) } }{ [-\i \mu]^{\de_{-}} } \cdot \exp\bigg\{ -\tau_t\Big(\f{\la+\mu}{2}\Big)^2-\tau_s\Big(\f{\la-\mu}{2}\Big)^2 \bigg\} \;.
\enq
Then the change of variables
\beq
\la\, = \, t\Big( 1 - \tfrac{ \op{u} }{ \op{v} }\Big) \, + \, s \quad \e{and} \quad \mu \, = \, t\Big( 1 + \tfrac{ \op{u} }{ \op{v} }\Big) \, - \, s
\enq
recasts the integral in the form
\beq
\mc{J}(\mf{x}) \; = \; \lim_{\tau_t, \tau_s\tend 0^+}
\Int{ \R+\i\a }{} \hspace{-2mm} \dd t \Int{ \R +\i \a_{\op{u}} }{} \hspace{-1mm} \dd s \ex{ -\tau_t t^2-\tau_s (s-\f{\op{u}}{\op{v}}t) ^2 }
\chi(t,s) \; \mc{I}_{V}(t,s)
\enq
where $\a_{\op{u}}=\f{ \op{u} }{ \op{v} } \a$,
\beq
\chi(t,s) \, = \, \pl{\ups=\pm}{} \bigg\{ \f{ \i }{ t \cdot \sg_{\ups}+ \ups s } \bigg\}^{ \de_{\ups} } \quad \e{and} \quad
\sg_{\ups}=1-\ups \f{ \op{u} }{ \op{v} } \;.
\label{definition chi et sigma ups}
\enq
while
\beq
\mc{I}_{V}(t,s) \, = \, \f{ \Ga(\de_+) \, \Ga( \de_-) }{ 2\pi^2 } \Int{ \mc{D}_{\bs{n}} }{} \dd \bs{x}\; V(\bs{x}) \cdot
\ex{- \i s ( \bs{x}, \bs{e}) } \cdot \ex{-(\bs{x},[\e{id}+\i t \op{E}]\bs{x}) -2 \i t \mf{x} }
\enq
Above, $\e{id}$ refers to the identity matrix acting on $\mc{D}_{\bs{n}}$.
\vspace{2mm}
The $\bs{x}$ integral can already be taken explicitly. Indeed, for $t \in \R+\i\a$ with $|\a|<1$, upon dilating the variables and then shifting them, \textit{viz}. leading to the substitution
\beq
\bs{x}^{(r)} \; = \; \f{ \bs{y}^{(r)} - \i s \, \bs{e}^{(r)} }{ 2 \sqrt{1+\i \veps_r t } } \quad \bs{y}^{(r)} \in \R^{n_r} \, ,
\enq
one obtains
\beq
\mc{I}_{V}(t,s) \, = \, \f{ \ex{ - \be_{t} s^2 } \Ga(\de_+) \, \Ga( \de_-) }{ 2\pi^2 \pl{r=1}{\ell} \big[ 1+\i \veps_r t \big]^{ \frac{ 1 }{2} n_r^2 } } \cdot \ex{-2\i t \mf{x} }
\cdot \Int{ \mc{D}_{\bs{n}} }{} \dd \bs{y} \; V(\bs{y}) \ex{ - ( \bs{y} , \bs{y} ) } \;,
\enq
where
\beq
\be_t\;= \; \sul{r=1}{\ell} \f{\op{v}^2 \xi_r^2 n_r }{ 1+\i\veps_r t } \;.
\label{definition constante beta de t et suite mr}
\enq
The remaining integrals can be taken by means of the Gaudin-Mehta formula \eqref{formule integrale Gaudin-Mehta}, leading to
\beq
\mc{I}_{V}(t,s) \, = \, \f{ \Ga(\de_+) \, \Ga( \de_-) }{ 2\pi^2 \cdot \ex{ \be_{t} s^2 } }
\pl{r=1}{\ell} \Bigg\{ \f{ \big( 2\pi \big)^{ \f{n_r}{2} } G(2+n_r) }{ \big[ 2(1+\i \veps_r t) \big]^{ \f{1}{2} n_r^2 } } \Bigg\} \cdot \ex{-2\i t \mf{x} }\;.
\enq
Thus, all-in-all, one gets
\beq
\mc{J}(\mf{x}) \; = \; \f{ \Ga(\de_+) \, \Ga( \de_-) }{ 2\pi^2 }
\pl{r=1}{\ell} \Bigg\{ \f{ \big( 2\pi \big)^{ \f{n_r}{2} } G(2+n_r) }{ 2 ^{ \f{1}{2} n_r^2 } } \Bigg\} \cdot \mc{K}(\mf{x}) \;,
\enq
where
\beq
\mc{K}(\mf{x}) \, = \, \lim_{\tau_t, \tau_s\tend 0^+}
\Int{ \R+\i\a }{} \hspace{-1mm} \dd t \Int{ \R +\i \a_{\op{u}} }{} \hspace{-1mm} \dd s \; \ex{ -\tau_t t^2-\tau_s (s-\f{\op{u}}{\op{v}}t) ^2 }
\f{ \chi(t,s) \, \ex{-\be_{t}s^2-2 \i t \mf{x} } }{ \prod_{r=1}^{\ell} \big[ 1+\i \veps_r t \big]^{ \frac{1}{2}n_r^2 } } \;.
\label{formule intermediaire pour J kappa modele}
\enq
Since the double limit $ \lim_{\tau_t, \tau_s\tend 0^+}$ exists, it can be computed in any way, in particular, by taking the successive limits
$ \lim_{\tau_t \tend 0^+} \lim_{\tau_s\tend 0^+}$. For $(t,s)\in (\R+\i \a)\times (\R+\i\a_{\op{u}})$ one has the lower bound
$|t \sg_{\ups}+\ups s| \geq \a >0$ and thus
\beq
\big| \chi(t,s) \big| \, \leq \, \pl{\ups= \pm }{} \a^{-\de_{\ups}} \;.
\enq
For such $t,s$, owing to $|\be_{t}|\leq C$ for some $C>0$, one thus has the bound
\beq
\Big| \ex{ -\tau_t t^2-\tau_s (s-\f{\op{u}}{\op{v}}t) ^2 }
\f{ \chi(t,s) \, \ex{-\be_{t}s^2-2 \i t \mf{x} } }{ \prod_{r=1}^{\ell} \big[ 1+\i \veps_r t \big]^{ \frac{1}{2} n_r^2 } } \Big| \; \leq \; W(t,s)
\label{ecriture borne pour dominantion limite taus vers zero dans s integrale}
\enq
with
\beq
W(t,s)
=C \cdot
\ex{-\tau_t (\Re(t))^2 } \ex{-\Re(\be_{t}) (\Re(s))^2 +2 \a_{\op{u}} \Re(s) \Im(\be_{t}) } \;.
\enq
Observe that $\Re(\be_t)>0$ for finite $t$ and
\beq
\be_t= -\f{\i}{t}\op{v}^2 \sul{r=1}{\ell} \veps_r \xi_r^2 n_r + \f{1}{t^2}\sul{r=1}{\ell} \op{v}^2 n_r \xi_r^2 + \e{O}(t^{-3}) \quad \e{when} \quad \Re(t) \tend \pm \infty\, .
\enq
Thus, for $\a$ small enough, one has the bound
\beq
\Int{\R+\i\a }{} \hspace{-2mm} \dd t \Int{ \R + \i\a_{\op{u}} }{} \hspace{-2mm} \dd s \; W(t,s) \; \leq \;
C \Int{\R+\i\a }{} \hspace{-2mm} \dd t \ex{-\tau_t (\Re(t))^2 } \ex{ \a_{\op{u}}^2 \f{ (\Im(\be_{t}))^2 }{ \Re(\be_{t}) } } \cdot \Int{ \R + \i\a_{\op{u}} }{} \hspace{-2mm} \dd s \; \ex{-\Re(\be_{t}) (\Re(s))^2 } \;.
\enq
Then, owing to the asymptotics at large $\Re(t)$ of $\be_t$, observe that for some $C_0$
\beq
\Big| \tfrac{ \Im(\be_{t}) }{ \sqrt{ |\Re(\be_{t})| } } \Big| \, \leq \, C_0 \, .
\enq
Upon the rescaling $\Re(s) \hookrightarrow |\Re(\be_{t})|^{-\tfrac{1}{2} } \Re(s)$, the above bounds entail that, for some constant $C^{\prime}$
\beq
\Int{\R+\i\a }{} \hspace{-2mm} \dd t \Int{ \R + \i\a_{\op{u}} }{} \hspace{-2mm} \dd s \; W(t,s) \; \leq \;
C^{\prime} \Int{\R }{} \dd t \ex{-\tau_t t^2 } \cdot \big( 1+|t| \big) \cdot \Int{ \R }{} \dd s \; \ex{- s^2 }
\; < \; +\infty\;.
\enq
Hence, since the bounding function in \eqref{ecriture borne pour dominantion limite taus vers zero dans s integrale} is positive, by Fubbini's theorem,
the above estimate ensures that it is in $L^1\Big( (\R+\i \a)\times (\R+\i\a_{\op{u}}) \Big)$, so that one can apply dominated convergence so as to take the $\tau_{s}\tend 0^+$ limit in \eqref{formule intermediaire pour J kappa modele},
hence yielding
\beq
\mc{K}(\mf{x}) \; = \; \lim_{\tau_t \tend 0^+}
\Int{ \R+\i\a }{} \hspace{-2mm} \dd t
\f{ \ex{ -\tau_t t^2 } \ex{ -2 \i t \mf{x} } }{ \prod_{r=1}^{\ell} \big[ 1+\i \veps_r t \big]^{ \frac{1}{2}n_r^2 } } \cdot \mc{F}(t)
\qquad \e{where} \qquad
\mc{F}(t) \, = \, \Int{ \R +\i \a_{\op{u}} }{} \hspace{-2mm} \dd s \; \chi(t,s) \ex{-\be_{t}s^2} \;.
\enq
The function $\mc{F} (t)$ is analysed in Lemma \ref{Lemme cpmyt asymptotique de F}, which ensures that there exist three functions $\mc{F}^{(a)}(t)$
such that $\mc{F}=\mc{F}^{(1)}+\mc{F}^{(2)}+\mc{F}^{(3)}$ where
\begin{itemize}
\item $\mc{F}^{(1)}$ and $\mc{F}^{(2)}$ are holomorphic on the set
\beq
\mc{S}_{\th_0;A} \, = \, \Big\{ t \in \Cx : |\Re(t)|\, > \, A \; \; \e{and} \; \; t=\rho \ex{\i \th} \quad \e{with} \quad \rho \in \R \quad \e{and} \quad |\th|<\th_0 \Big\}
\enq
where $A$ is large enough while $\th_0$ is small enough;
\item when $ \Re(t) \tend + \infty $
\beq
\mc{F}^{(1)}(t) \; = \; \sqrt{\pi} \cdot \chi(t,0) \cdot \Big( \tfrac{1}{\be_t} \Big)^{ \f{1}{2} }
\cdot \Big( 1+ \e{O}\Big( \f{1}{ t }\Big) \Big)
\label{ecriture DA partie algebrique fct F varrho un}
\enq
with a remainder that is uniform and holomorphic on $\mc{S}_{\th_0;A}$;
\item there exists $C_1,C_2>0$ such that, for $\rho \tend \pm \infty$, $\th>0$ and $\vsg=-\e{sgn}\big( \sum_{r=1}^{\ell} \veps_r \xi_r^2 n_r \big) $,
\beq
\Big| \mc{F}^{(2)}\Big( \rho \ex{-\i\th \vsg \e{sgn}(\rho) } \Big) \Big| \, \leq \, C_1 \ex{ - C_2 |\rho| \th } \quad \e{and} \quad
\big| \mc{F}^{(3)}( t ) \big| \, \leq \, C_1 \ex{ - C_2 |\Re(t)| } \quad \e{for} \quad t\in \R+\i\a \; .
\enq
\end{itemize}
Define the contours $\msc{C}^{(1)} \, = \, \mc{R}_{0}^{(\mf{x})}\cup_{\ups=\pm} \mc{R}_{\ups}^{(\mf{x})} $, where $\mc{R}_{0}^{(\mf{x})}$ is a curve joining $-A$ to $A$ in the upper half-plane but having a sufficiently small imaginary part,
while $\mc{R}_{\pm}^{(\mf{x})}$ are two rays
\beq
\mc{R}_{\pm}^{(\mf{x})} \; = \; \bigg\{ z\; = \; \pm A \pm \rho \ex{ \mp \i \th \e{sgn}(\mf{x}) } \;, \quad \rho \in \R^+ \bigg\}
\enq
going to $\infty$ in the direction $\Re(z)\tend \pm \infty$ with a slight angle $\th>0$ small enough, so that $\Im( t \mf{x}) \tend +\infty$ linearly in $|t|$ along these rays.
The contour $\msc{C}^{(2)}$ has a similar structure: $\msc{C}^{(2)} \, = \, \mc{R}_{0}^{(\mf{x})}\cup_{\ups=\pm} \mc{R}_{\ups;\th} $ with the
two rays $ \mc{R}_{\ups;\th}$ given as
\beq
\mc{R}_{\ups;\th} \; = \; \bigg\{ z\; = \; \ups A + \ups \rho \ex{ -\ups \vsg \i \th } \;, \quad \rho \in \R^+ \bigg\}
\enq
for $\th >0$ and small enough. Finally, take $\msc{C}^{(3)}\, = \, \R + \i\a$.
Upon
\begin{itemize}
\item[i)] inserting the decomposition $\mc{F} \, = \, \sum_{a=1}^{3}\mc{F}^{(a)}$ into $\mc{K}(\mf{x})$;
\item[ii)] splitting the integrations for each piece ;
\item[iii)] deforming the $t$-integration contour to $\msc{C}^{(a)}$ in the integrals associated with $\mc{F}^{(a)}$;
\end{itemize}
one obtains three integrals whose respective integrands decay, uniformly in $\tau_t$ small enough, exponentially fast to $0$ along $\msc{C}^{(a)}$.
Thus, one can invoke dominated convergence so as to send $\tau_t\tend 0^+$ and obtain that
\beq
\mc{K}(\mf{x})=\sul{a=1}{3} \mc{K}^{(a)}(\mf{x}) \qquad \e{with} \qquad
\mc{K}^{(a)}(\mf{x}) \; = \; \Int{ \msc{C}^{(a)} }{} \dd t
\f{ \ex{ -2 \i t \mf{x} } \, \mc{F}^{(a)}(t) }{ \prod_{r=1}^{\ell} \big[ 1+\i \veps_r t \big]^{ \frac{1}{2} n_r^2 } } \;.
\enq
Since, the integrands in $ \mc{K}^{(2)}(\mf{x})$ and $ \mc{K}^{(3)}(\mf{x})$ are bounded and decay exponentially fast to $0$ at $\infty$, this uniformly in $|\mf{x}|$ small enough,
one can apply derivation under the integral theorems so as to infer that $ \mc{K}^{(2)}+ \mc{K}^{(3)}$ is smooth in $\mf{x}$ around $0$.
Hence, it remains to focus on the $\mf{x} \tend 0^+ $ behaviour of $ \mc{K}^{(1)}(\mf{x})$ which can be decomposed as
\beq
\mc{K}^{(1)}(\mf{x}) \, = \, \sul{ c \in \{ \pm , 0 \} }{} \mc{K}^{(1)}_{c}(\mf{x}) \quad \e{with} \quad
\mc{K}^{(1)}_{ c}(\mf{x}) \, = \, \Int{ \mc{R}_{ c }^{(\mf{x})} }{} \hspace{-2mm} \dd t \; \f{ \ex{ -2 \i t \mf{x} } \, \mc{F}^{(1)}(t) }{ \prod_{r=1}^{\ell} \big[ 1+\i \veps_r t \big]^{ \frac{1}{2} n_r^2 } } \;.
\enq
Since the integration in $\mc{K}^{(1)}_{0}$ runs through a compact set and since the integrand in bounded, it follows that $\mc{K}^{(1)}_{0}(\mf{x})$ is a smooth function of $\mf{x}$ by derivation under the integral theorems.
It thus remains to estimate $\mc{K}^{(1)}_{\pm}(\mf{x})$. The properties of $\mc{F}^{(1)}(t)$ ensure that
\beq
\f{ \mc{F}^{(1)}(t) }{ \prod_{r=1}^{\ell} \big[ 1+\i \veps_r t \big]^{ \frac{1}{2}n_r^2 } } \; = \; \vp_{\mf{s}_t}(t) \; + \; \psi_{\mf{s}_t}(t) \qquad \e{with} \qquad \psi_{\mf{s}_t}(t)=\e{O}\Big( \f{ \vp_{\mf{s}_t}(t) }{ t } \Big)
\;\; , \; \mf{s}_t \,=\, \e{sgn}\big[ \Re(t) \big]
\enq
and $\vp_{\mf{s}_t }(t)\, = \,\ga_{\mf{s}_t} \cdot \big(\mf{s}_t \cdot t)^{ -\vth - 1 }$ with $\vth$ as defined in \eqref{definition de cal theta}.
The constant prefactor takes the form
\beq
\ga_{\mf{s}_t} \; = \; \f{ \sqrt{\pi} \cdot \ex{ -\i \vsg \mf{s}_t \f{\pi}{4} } }
{ \Big| \op{v}^2 \sum_{r=1}^{\ell} n_r \veps_r \xi_r^2 \Big|^{\frac{1}{2} } }
\cdot \pl{\ups = \pm }{} \bigg\{ \f{ \ex{ \i \mf{s}_t \mf{s}_{\sg_{\ups}} \f{\pi}{2} } }{ |\sg_{\ups}| } \bigg\}^{\de_{\ups}}
\cdot \pl{r=1}{\ell} \Big\{ \ex{-\i \veps_r \mf{s}_t n_r^2 \f{\pi}{4} } \Big\} \;.
\enq
$\vsg$ has been introduced in \eqref{definition du signe xi}, $\sg_{\ups}$ is given by \eqref{definition chi et sigma ups},
while $\mf{s}_{\sg_{\ups}}=\e{sgn}\big( \sg_{\ups} \big)$.
This being settled, one splits the integrals as
\beq
\mc{K}^{(1)}_{\pm}(\mf{x}) \; = \; \msc{J}^{(1)}_{\pm}(\mf{x}) \, + \, \de \! \msc{J}^{(1)}_{\pm}(\mf{x})
\enq
where
\beq
\msc{J}^{(1)}_{ \ups}(\mf{x}) \, = \, \Int{ \mc{R}_{ \ups}^{(\mf{x}) } }{} \ex{ -2 \i t \mf{x} } \, \vp_{\ups }(t) \cdot \dd t \qquad \e{and} \qquad
\de \! \msc{J}^{(1)}_{ \ups}(\mf{x}) \, = \, \Int{ \mc{R}_{ \ups}^{(\mf{x}) } }{} \ex{ -2 \i t \mf{x} } \, \psi_{\ups}(t)\cdot \dd t \;.
\enq
The rest depends on how large $\vth$ is. Let $n \in \mathbb{N}$ and $0\leq \a <1$ be such that $\vth+2=\a+n$.
First, focus on $\de \! \msc{J}^{(1)}_{ \ups}(\mf{x})$ and introduce $\psi_{\ups;0} \, = \, \psi_{\ups}$, and, for $p\geq 1$,
\beq
\psi_{\ups;p}(t)\; = \; \Int{ \mc{R}_{ \ups}^{(\mf{x}) } }{ t } \psi_{\ups;p-1}(s) \cdot \dd s
\enq
where the integration runs from $\infty$, along $\mc{R}_{ \ups}^{(\mf{x}) }$, up to $t \in \mc{R}_{ \ups}^{(\mf{x}) }$.
Then, integrating by parts $n$ times, one has
\beq
\de \! \msc{J}^{(1)}_{ \ups}(\mf{x}) \, = \, -\ups \sul{p=0}{n-1} (2\i \mf{x} )^{p} \, \psi_{ \ups ; p +1}( \ups A ) \, \ex{-2\i \ups \mf{x} A t }
\, + \, (2\i \mf{x} )^{n} \Int{ \mc{R}_{ \ups}^{(\mf{x}) } }{} \ex{ -2 \i t \mf{x} } \, \psi_{ \ups ; n }( t ) \cdot \dd t \;.
\label{ecriture dvpmt de J 1 ups}
\enq
One has $ \psi_{ \ups ; n }( t )= \e{O}\big( |t|^{-\a} \big)$. Upon setting $\mf{s}_{\mf{x}}=\e{sgn}(\mf{x})$ and taking $\th>0$ and small enough, one gets
\bem
\bigg| \Int{ \mc{R}_{ \ups}^{(\mf{x}) } }{} \ex{ -2 \i t \mf{x} } \psi_{ \ups ; n }( t ) \cdot \dd t \bigg| \; = \;
\bigg| \Int{ 0 }{ + \infty} \ex{ -\ups 2 \i \rho \mf{x} \ex{ -\ups \i \mf{s}_{\mf{x}}\th} } \psi_{ \ups ; n }\Big( \ups A + \ups \rho \ex{ -\ups \i \mf{s}_{\mf{x}}\th} \Big) \cdot \dd \rho \bigg| \\
\; \leq \; C \Int{ 0 }{ + \infty} \f{ \ex{ - 2 \rho |\mf{x}| \sin(\th) } }{ \big[ A + \cos(\th) \rho \big]^{\a} } \cdot \dd \rho
\; = \; C \, |\mf{x}|^{ \a - 1 } \Int{ 0 }{ + \infty} u_{\mf{x}}( \rho ) \cdot \dd \rho \;,
\end{multline}
for some constant $C>0$. In the last integral, I have set
\beq
u_{\mf{x}}(\rho ) = \ex{ - 2 \rho \sin(\th) } \cdot \big[ |\mf{x}| A + \cos(\th) \rho \big]^{-\a} \; \limit{ \mf{x} }{ 0 } \, u_{0}(\rho ) \in L^1(\R^+)
\enq
point-wise on $\R^{+}\setminus\{ 0 \}$. Since $u_{\mf{x}}(\rho) \leq u_{0}(\rho)$, by dominated convergence, the expansion \eqref{ecriture dvpmt de J 1 ups} ensures that there exist smooth functions $g_{\pm}$ such that
\beq
\de \! \msc{J}^{(1)}_{ \ups }(\mf{x}) \, = \, g_{\ups}(\mf{x}) \, + \, \e{O}\Big( |\mf{x}|^{ \vth +1} \Big) \;.
\enq
The integral $ \msc{J}^{(1)}_{ \ups }(\mf{x})$ can be dealt with analogously by doing $n-1$ integration by parts, where $\vth + 1 = n - 1 + \a$. Namely, one has
\bem
\Int{ \mc{R}_{ \ups }^{(\mf{x})} }{} \f{ \ex{ -2 \i \mf{x} t } }{ (\ups t)^{\vth+1} } \cdot \dd t
\; = \; \Ga\bigg( \ba{c} \vth \\ \vth+1 \ea \bigg) \f{ 1 }{ A^{\vth } } \ex{ -2 \i \ups \mf{x} A }
\, -\, 2\i \mf{x} \ups \Ga\bigg( \ba{c} \vth \\ \vth +1 \ea \bigg) \Int{ \mc{R}_{ \ups }^{(\mf{x})} }{} \f{ \ex{ -2 \i \mf{x} t } }{ (\ups t)^{\vth} } \cdot \dd t \\
\; = \; \ex{ -2 \i \ups \mf{x} A } \sul{p=0}{n-2} \f{ \big(-2\i \mf{x} \ups\big)^{p } }{ A^{\vth-p} } \Ga\bigg( \ba{c} \vth-p \\ \vth+1 \ea \bigg)
\, +\, \big(-2\i \mf{x} \ups\big)^{n-1 } \Ga\bigg( \ba{c} \vth -n +2 \\ \vth +1 \ea \bigg) \Int{ \mc{R}_{ \ups }^{(\mf{x})} }{} \f{ \ex{ -2 \i \mf{x} t } }{ (\ups t)^{\a} } \cdot \dd t \;.
\end{multline}
All this leads to the representation
\beq
\msc{J}^{(1)}(\mf{x})= \sul{\ups=\pm}{} \msc{J}^{(1)}_{\ups}(\mf{x}) \, = \, \msc{J}^{(1)}_{\e{reg}}(\mf{x}) \, + \, \msc{J}^{(1)}_{\e{sing}}(\mf{x})
\enq
where $\msc{J}^{(1)}_{\e{reg}}$ is smooth and given by
\beq
\msc{J}^{(1)}_{\e{reg}}(\mf{x}) \; = \; \sul{p=0}{n-2} \sul{\ups=\pm}{} \ga_{ \ups } \cdot \Ga\bigg( \ba{c} \vth - p \\ \vth + 1 \ea \bigg) \, \cdot \,
\ex{ -2 \i \ups \mf{x} A } \f{ \big(-2\i \mf{x} \ups\big)^{p } }{ A^{ \vth - p } }
\enq
while, upon deforming the contours $\mc{R}_{ \ups }^{(\mf{x})}$ to $\ups A -\i \mf{s}_{\mf{x}} \R^+_{\ups}$, where $\R^+_{\ups}$ corresponds to $\R^+$ oriented with the sign $\ups$,
\bem
\msc{J}^{(1)}_{\e{sing}}(\mf{x}) \, = \, \Ga\bigg( \ba{c} \vth -n +2 \\ \vth +1 \ea \bigg) \cdot |2 \mf{x}|^{n-1} \Bigg\{
\ga_{ - } \cdot \big( \i \mf{s}_{\mf{x}} \big)^{n } \ex{2\i \mf{x} A } \Int{ 0 }{ + \infty } \f{ \ex{ -2 |\mf{x}| t } }{ (A + \i t \mf{s}_{\mf{x}} )^{\a} } \cdot \dd t \\
\hspace{7cm} \, + \, \ga_{ + } \cdot \big(- \i \mf{s}_{\mf{x}} \big)^{n } \ex{-2\i \mf{x} A } \Int{ 0 }{ + \infty } \f{ \ex{ -2 |\mf{x}| t } }{ (A -\i t \mf{s}_{\mf{x}} )^{\a} } \cdot \dd t
\Bigg\} \\
\, = \, (-1)^n |2 \mf{x}|^{ \vth } \cdot \Ga\bigg( \ba{c} \vth -n +2 \\ \vth + 1 \ea \bigg) \sul{\ups=\pm}{}
\ga_{\ups} \cdot \big( \ups\i \mf{s}_{\mf{x}} \big)^{n+\a } \ex{-2\i \mf{x} \ups A } \Int{ 0 }{ + \infty } \f{ \ex{ - t } }{ (t +\ups \i 2\mf{x} A )^{\a} } \cdot \dd t \\
\noindent
\end{multline}
By dominated convergence, one has that
\beq
\Int{ 0 }{ + \infty } \f{ \ex{ - t } }{ (t +\ups \i 2\mf{x} A )^{\a} } \cdot \dd t \; = \; \Ga(1-\a) \cdot (1+\e{o}(1) )
\enq
when $\mf{x} \tend 0$. With some more efforts, one can even establish that it is a
$\e{O}(\mf{x})$. Since $\Ga(1-\a)=\Ga(n-1-\vth )$, straightforward calculation lead to
\beq
\msc{J}^{(1)}_{\e{sing}} (\mf{x}) \, = \, 2 | 2 \mf{x} |^{ \vth }\cdot \f{ \sqrt{\pi} \cdot \Ga( -\vth ) \cdot \prod_{\ups= \pm }^{} | \sg_{\ups} |^{-\de_{\ups}} }
{ \Big| \op{v}^2 \sum_{r=1}^{\ell} n_r \veps_r \xi_r^2 \Big|^{\frac{1}{2} } }
\sul{\ups=\pm}{} \bigg\{ \Xi( \ups \mf{x}) \cdot \sin[\pi \nu_{\ups} ]
\bigg\} \cdot \Big( 1 + \e{O}(\mf{x}) \Big) \;.
\enq
Above, it is understood that
\beq
\nu_{ \eps } \; = \; \frac{1}{2}\sul{ \substack{ r=1 \, : \, \\ \eps \veps_r= -1} }{ \ell } n_r^2 \; - \; \f{ 1 +\eps \vsg }{ 4 } \; + \sul{ \substack{ \ups=\pm \\ \eps \sg_{\ups}>0 } }{} \de_{\ups} \;.
\enq
Finally, the obtained estimates are readily seen to be differentiable. Thus, upon putting all the results together, the asymptotic expansion given in \eqref{ecriture DA kappa zero de integrale modele} follows.
\subsection{Asymptotics of auxiliary functions}
\begin{lemme}
\label{Lemme cpmyt asymptotique de F}
Let $\a>0$, $t \in \R + \i\a $, $(\op{u},\op{v}) \in \R \times \R^+$ be such that $\op{u}\not= \pm \op{v}$ and set $\a_{\op{u}} \, = \, \f{ \op{u} }{ \op{v} } \a $. Let
\beq
\mc{F}_n(t) \, = \, \Int{ \R +\i \a_{\op{u}} }{} \hspace{-2mm} \dd s \; \chi(t,s) \, s^{n} \, \ex{-\be_{t}s^2}
\enq
where $\be_t$ is as defined in \eqref{definition constante beta de t et suite mr} and $\chi(t,s)$ has been introduced in \eqref{definition chi et sigma ups}.
Then, there exist functions $\mc{F}_n^{(a)}(t)$, $a=1,2,3$ such that $\mc{F}_n=\mc{F}_n^{(1)} + \mc{F}_n^{(2)} + \mc{F}_n^{(3)}$, and enjoying the properties:
\begin{itemize}
\item $\mc{F}_n^{(1)}$ and $\mc{F}_n^{(2)}$ are holomorphic on the set
\beq
\mc{S}_{\th_0;A} \, = \, \Big\{ t \in \Cx : |\Re(t)| \, > \, A \; \; and \; \; t=\rho \ex{\i \th} \quad with \quad \rho \in \R \quad and \quad |\th|<\th_0 \Big\}
\enq
where $A$ is large enough while $\th_0$ is small enough;
\item when $ \Re(t) \tend + \infty $
\beq
\mc{F}_{n}^{(1)}(t) \; = \; \chi(t,0) \cdot \Big( \tfrac{1}{\be_t} \Big)^{ \f{1+n}{2} } \cdot u_n \cdot \Big( \tfrac{1}{ t \sqrt{\be_{t}} }\Big)^{w_n}
\cdot \Big( 1+ \e{O}\Big( \f{1}{ t }\Big) \Big)
\enq
for some constant
\beq
\left\{ \ba{ccc} u_n = \Ga\Big( \tfrac{n+1}{2} \Big) & \e{if} & n \in 2\mathbb{N} \vspace{1mm} \\
u_n \not=0 & \e{if} & n \in 2\mathbb{N} +1 \ea \right. \; ,
\quad \; an \; integer \quad \left\{ \ba{ccc} w_n = 0 & \e{if} & n \in 2\mathbb{N} \vspace{1mm} \\
w_n \geq 1 & \e{if} & n \in 2\mathbb{N} +1 \ea \right. \;,
\label{definition proprietes un et wn}
\enq
and with a remainder that is uniform and holomorphic on $\mc{S}_{\th_0;A}$;
\item there exists $C_1,C_2>0$ such that, for $\rho \tend \pm \infty$ and $\th>0$,
\beq
\Big| \mc{F}_n^{(2)}\Big( \rho \ex{-\i \th \vsg \e{sgn}(\rho) } \Big) \Big| \, \leq \, C_1 \ex{ - C_2 |\rho| \th } \quad and \quad
\big| \mc{F}_n^{(3)}( t ) \big| \, \leq \, C_1 \ex{ - C_2 |\Re(t)| } \quad for \quad t\in \R+\i\a \; ,
\enq
where $\vsg = - \e{sgn} \Big( \sul{r=1}{\ell} \veps_r \xi_r^2 n_r \Big)$.
\end{itemize}
\end{lemme}
\Proof
One has to distinguish between the two cases: $|\op{u}| \, > \, \op{v}$ or $|\op{u}| \, < \, \op{v}$. Also, recall the notation $\sg_{\ups} \, = \, 1-\ups \tfrac{ \op{u} }{ \op{v} }$
\subsection*{A) The regime $|\op{u}| \, > \, \op{v}$ }
Let $\mf{s}_{\op{u}}= \e{sgn}(\op{u})$, so that, using that $\e{sgn}(\sg_{\ups}) \, = \, -\ups \mf{s}_{\op{u}} $ one can recast
\beq
\chi(t,s) \, = \, \pl{ \ups= \pm }{} \bigg\{ \f{ \i \ups }{ s \, - \, t \mf{s}_{ \op{u} } |\sg_{\ups}| } \bigg\}^{ \de_{\ups} }
\enq
meaning that, for fixed $t$, the map $s \mapsto \chi(t,s)$ has cuts along $ \mf{s}_{\op{u}} t |\sg_{\ups}| -\i \R^{\ups}$.
Since the integration runs through $\R+\i \a_{\op{u}}$, and, for $t \in \R + \i \a $,
\beq
\Im\Big( \i \a_{\op{u}} \, - \, \mf{s}_{\op{u}}|\sg_{\ups}| t \Big) \; = \; \a \ups
\enq
the cut along $ \mf{s}_{\op{u}} t |\sg_{+}| -\i \R^{+}$ lies below the integration line $\R+\i \a_{\op{u}}$ and the one along
$ \mf{s}_{\op{u}} t |\sg_{-}| - \i \R^{-}$ lies above the integration line $\R+\i \a_{\op{u}}$.
One can represent $\be_{t} \, = \, \ex{ \i \th_{\be_t} } |\be_{t}|$. Since, for $t$ large,
\beq
\be_{t}\; = \; \i \f{ \vsg C_{\be} }{ t } \; + \; \e{O}\big( t^{-2} \big)
\quad \e{one} \, \e{has} \quad
\th_{\be_t} \, \sim \, \f{\pi}{2} \mf{s}_t \vsg \qquad \e{with} \qquad
\left\{ \ba{ccc} \vsg & = & - \e{sgn} \Big( \sul{r=1}{\ell} \veps_r \xi_r^2 n_r \Big) \\
C_{\be} & = & \Big| \sul{r=1}{\ell} \veps_{r} \xi_r^2 \op{v}^2 n_r \Big| \ea \right. \;.
\label{definition de xiu et C beta}
\enq
Here, I introduced the shorthand notation $\mf{s}_{t}=\e{sgn}\Big( \Re(t) \Big)$. Also, for further convenience, it is useful to set
\beq
\vsg_{\op{u}} \, = \, \mf{s}_{\op{u}} \cdot \vsg \;.
\enq
It is then convenient to deform the integration contour towards the curve depicted in Figure \ref{Figure contour deformes pour u bigger than v re(t) su negatif} in the case
when $\Re(t) \mf{s}_{\op{u}}<0$ and the one depicted in Figure \ref{Figure contour deformes pour u bigger than v re(t) su positif} in the case when $\Re(t) \mf{s}_{\op{u}}>0$.
\begin{figure}[ht]
\begin{center}
\begin{pspicture}(12,5)
\psline[linestyle=dashed, dash=3pt 2pt](0,2.5)(11.8,2.5)
\psline[linewidth=2pt]{->}(11.8,2.5)(11.9,2.5)
\rput(11,2.2){$\R + \i \a_{\op{u}}$}
\rput(0.1,3.1){ $\mf{s}_{\op{u}} t |\sg_-|$ }
\psdots(1,3)
\psline(1,3)(1,4.6)
\psdot(1,4.6)
\pscurve(1,4.6)(1.2,3.5)(1.3,2.9)(1,2.8)(0.7,2.9)(0.8,3.5)(1,4.6)
\rput(1.3,4.7){$ r_- $}
\rput(1.7 , 3.1){$\Ga_-^{\be_t}$}
\rput(2.8,2){ $\mf{s}_{\op{u}} t |\sg_+|$ }
\psdots(2,2)
\psline(2,2)(2,0.8)
\psdots(2,0.85)
\pscurve(2,0.85)(2.2,1.6)(2,2.1)(1.8,1.6)(2,0.85)
\rput(2.35,0.7){$ r_+ $}
\rput(1.5,1.5){$\Ga_+^{\be_t}$}
\psline(0,0)(12,5)
\psline[linewidth=2pt]{->}(8,3.35)(8.1,3.39)
\psline[linewidth=2pt]{->}(1,2.8)(1.1,2.8)
\rput(11,0){$ \vsg_{\op{u}}<0$}
\rput(11.5,0.8){ $ \ex{ -\tfrac{\i}{2} \th_{\be_t} } \R $ }
\psline(0,5)(12,0)
\psline[linewidth=2pt]{->}(8,1.65)(8.1,1.60)
\psline[linewidth=2pt]{->}(1.8,1.6)(1.8,1.65)
\rput(11,5){$ \vsg_{\op{u}}>0$}
\rput(11,4.2){ $\ex{-\tfrac{\i}{2} \th_{\be_t} } \R$ }
\end{pspicture}
\caption{ Deformed contours in the case $|\op{u}|>\op{v}$ and for $\Re(t) \mf{s}_{\op{u}}<0$
\label{Figure contour deformes pour u bigger than v re(t) su negatif} }
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\begin{pspicture}(12,5)
\psline[linestyle=dashed, dash=3pt 2pt](0,2.5)(11.8,2.5)
\psline[linewidth=2pt]{->}(11.8,2.5)(11.9,2.5)
\rput(11,2.2){$\R + \i \a_{\op{u}}$}
\rput(11.9,3.1){ $\mf{s}_{\op{u}} t |\sg_-|$ }
\psdots(11,3)
\psline(11,3)(11,4.6)
\psdot(11,4.6)
\pscurve(11,4.6)(10.8,3.5)(10.7,2.9)(11,2.8)(11.3,2.9)(11.2,3.5)(11,4.6)
\rput(10.7,4.7){$ r_- $}
\rput(10.3, 3.1){$\Ga_-^{\be_t}$}
\rput(9.2,2){ $\mf{s}_{\op{u}} t |\sg_+|$ }
\psdots(10,2)
\psline(10,2)(10,0.8)
\psdots(10,0.85)
\pscurve(10,0.85)(9.8,1.6)(10,2.1)(10.2,1.6)(10,0.85)
\rput(9.65,0.7){$ r_+ $}
\rput(10.5,1.5){$\Ga_+^{\be_t}$}
\psline(0,0)(12,5)
\psline[linewidth=2pt]{->}(8,3.35)(8.1,3.39)
\psline[linewidth=2pt]{->}(11,2.8)(11.1,2.8)
\rput(1,0){$ \vsg_{\op{u}}<0$}
\rput(0.5,0.8){ $ \ex{ -\tfrac{\i}{2} \th_{\be_t} } \R $ }
\psline(0,5)(12,0)
\psline[linewidth=2pt]{->}(8,1.65)(8.1,1.60)
\psline[linewidth=2pt]{->}(10.2,1.65)(10.2,1.6)
\rput(1,5){$ \vsg_{\op{u}}>0$}
\rput(1,4.3){ $\ex{-\tfrac{\i}{2} \th_{\be_t} } \R$ }
\end{pspicture}
\caption{ Deformed contours in the case $|\op{u}|>\op{v}$ and for $\Re(t) \mf{s}_{\op{u}} > 0$
\label{Figure contour deformes pour u bigger than v re(t) su positif} }
\end{center}
\end{figure}
Upon the change of variables $s=\be_t^{-\tf{1}{2}} s^{\prime}$ in the integration along $\ex{-\tfrac{\i}{2}\th_{\be_t} } \R$, one decomposes $\mc{F}_n$ as
\beq
\mc{F}_n(t) \, = \, \Big( \f{ 1 }{ \be_t } \Big)^{ \frac{1}{2}(n+1) } \Int{ \R }{} \! \dd s \, \ex{-s^2}\, s^{n} \, \chi\Big(t, \tfrac{ s }{ \sqrt{\be_t} } \Big) \; + \;
\Int{ \Ga^{\be_t}_{ \vsg_{\op{u}}} }{}\, \dd s\, s^n \, \chi(t,s) \ex{-\be_t s^2 } \;.
\enq
\subsubsection*{ $\bullet$ The Gaussian integral }
The first integral appearing in this decomposition can be analysed by observing that
\beq
\chi\Big(t, \tfrac{ s }{ \sqrt{\be_t} } \Big) \; = \; \chi(t,0) \cdot \Bigg\{ 1\, - \, \f{ s }{ t \sqrt{\be_t} } \sul{\ups=\pm}{} \f{ \ups \de_{\ups} }{ \sg_{\ups} }
\, + \, \e{O}\Big( \f{ s^2 }{ t^2 \cdot \be_t } \Big) \Bigg\}
\enq
uniformly in $|s| \leq |\Re(t)|^{\f{1}{4}}$. Thus, it is convenient to introduce the intervals
\beq
I_{\e{in}} \, = \, \intoo{ - \tau }{ \tau } \; \quad \e{and} \quad
I_{ \e{out} } \, = \, \intoo{ - \infty }{ - \tau } \cup \intoo{ \tau }{ + \infty } \qquad \e{with} \quad \tau=|\Re(t)|^{\f{1}{4}} \;.
\enq
Then, one has
\beq
\Big( \f{ 1 }{ \be_t } \Big)^{ \frac{1}{2}(n+1) } \Int{ \R }{} \! \dd s \, \ex{-s^2} s^n \chi\Big(t, \tfrac{ s }{ \sqrt{\be_t} } \Big) \; = \; \mc{H}_{ I_{\e{in}} }(t) + \mc{H}_{ I_{\e{out}} }(t)
\label{ecriture decomposition integrale Gaussienne sur H in et H out}
\enq
where, for even $n$, one has
\beq
\mc{H}_{ I_{\e{in}} }(t) \; = \; \chi(t,0) \cdot \Big( \f{ 1 }{ \be_t } \Big)^{ \frac{1}{2}(n+1) } \Bigg\{ \Int{ I_{\e{in}} }{} \ex{-s^{2}} s^n \cdot \dd s \; + \;
\e{O}\bigg( \Int{ I_{\e{in}} }{} \f{ s^{2+n} \ex{-s^{2}} }{ \be_t \, t^2 } \cdot \dd s \bigg) \Bigg\}
\enq
The integrations can be extended to infinity upon adding some $\e{O}\Big( \ex{-\frac{1}{2} \tau^2} \Big)$ corrections, so that, for even $n$, one has
\beq
\mc{H}_{ I_{\e{in}} }(t) \; = \; \chi(t,0) \cdot \Big( \f{ 1 }{ \be_t } \Big)^{ \frac{1}{2}(n+1) } \Bigg\{ \Ga\Big( \tfrac{n+1}{2} \Big)
\; + \; \e{O}\bigg( \f{ 1 }{ \be_t \, t^2 } \bigg) \, + \, \e{O}\bigg( \ex{ - \tfrac{1}{2}\sqrt{|\Re(t)|} } \bigg) \Bigg\} \;.
\enq
Similar handlings in the case of odd $n$ entail that in such a case
\beq
\mc{H}_{ I_{\e{in}} }(t) \; = \; \chi(t,0) \cdot \Big( \f{ 1 }{ \be_t } \Big)^{ \frac{1}{2}(n+1) } \cdot \Big( \tfrac{1}{ t \sqrt{\be_{t}} }\Big)^{w_n}
\cdot \Bigg\{ u_n \; + \; \e{O}\bigg( \f{ 1 }{ \be_t \, t^2 } \bigg) \, + \, \e{O}\bigg( \ex{ - \tfrac{1}{2}\sqrt{|\Re(t)|} } \bigg) \Bigg\}
\enq
for some integer $w_n \geq 1$ and a coefficient $u_n \not=0$.
Regarding to the second contribution in \eqref{ecriture decomposition integrale Gaussienne sur H in et H out}, it can be presented as
\beq
\mc{H}_{ I_{\e{out}} }(t) \; = \; \Big( \f{ 1 }{ \be_t } \Big)^{ \frac{1}{2}(n+1) } \Int{0}{+\infty} \dd s \, \ex{-(s+\tau)^2} \, (s+\tau )^{n} \bigg\{ \chi\Big(t, \tfrac{ s+\tau }{ \sqrt{\be_t} } \Big)
\; + \; (-1)^n \chi\Big(t, - \tfrac{ s+\tau }{ \sqrt{\be_t} } \Big) \bigg\} \;.
\enq
Since, for $s\geq 0$,
\beq
\Im\Big( t \sg_{\ups} \pm \ups \tfrac{ s+\tau }{ \sqrt{\be_t} } \Big) \, = \, \a \sg_{\ups} \mp \ups \frac{ s+\tau }{ \sqrt{|\be_t|} } \sin\big[ \tfrac{\th_{\be_t} }{2} \big]
\enq
which obviously does not vanish for $|t|$ large enough, with $t \in \R +\i\a $, and is dominated in this regime by the second term, it follows that
\beq
\Big| \chi\Big(t, \pm \tfrac{ s+\tau }{ \sqrt{\be_t} } \Big) \Big| \, \leq \, C \cdot
\big| \be_t \big|^{ \tfrac{ \de_{+} + \de_{-} }{2} }
\enq
so that, upon inserting the large-$t$ behaviour of $\be_t$, one has
\beq
\Big| \mc{H}_{ I_{\e{out}} }(t) \Big| \; \leq \; C^{\prime} |t|^{ \tfrac{ 1}{ 2 }( n+1-\de_{+} - \de_{-}) } \cdot \ex{ - \sqrt{ |\Re(t)| } }
\Int{0}{+\infty} \dd s \ex{-s^2-2s\tau} \Big( \, |t|^{ \tfrac{ 1}{ 4 }( |\de_{+}| + |\de_{-}| +n)} + |s|^{ |\de_{+}| + |\de_{-}| + n } \Big) \bigg\}
=\e{O}\bigg( \ex{ - \tfrac{3}{4}\sqrt{|\Re(t)|} } \bigg) \;.
\enq
Thence, all in all, since $\chi(t,0)\cdot \be_t^{-\tfrac{1}{2}(n+1) }$ has at most an algebraic growth in $t$,
\beq
\Int{ \R }{} \hspace{-2mm} \f{ \dd s }{ \sqrt{\be_t} } \ex{-s^2} \chi\Big(t, \tfrac{ s }{ \sqrt{\be_t} } \Big) \; = \;
\chi(t,0) \cdot \Big( \f{ 1 }{ \be_t } \Big)^{ \frac{1}{2}(n+1) } \cdot \Big( \tfrac{1}{ t \sqrt{\be_{t}} }\Big)^{w_n}\cdot \Bigg\{ u_n \; + \; \e{O}\bigg( \f{ 1 }{ t } \bigg) \Bigg\}
\, + \, \e{O}\bigg( \ex{ - \tfrac{1}{2}\sqrt{|\Re(t)|} } \bigg) \;.
\enq
Above, $u_n$ and $w_n$ are as appearing in \eqref{definition proprietes un et wn}. Finally, it is readily seen that both remainders are
holomorphic in $t \in \mc{S}_{\th_0,A}$ for some $\th_0$ small enough and $A$ large enough.
\subsubsection*{ $\bullet$ The loop integral contribution}
It now remains to focus on the integral along $\Ga_{ \vsg_{\op{u}} }^{ \be_t }$. The contour $\Ga_{\ups}^{\be_t}$ can be deformed as
\beq
\Ga_{\ups}^{\be_t} \; \hookrightarrow \; \Big\{ \mf{s}_{\op{u}} t |\sg_{\ups}| \, - \, \ups \Dp{}\mc{D}_{0,\eps} \Big\} \cup
\Big\{ \mf{s}_{\op{u}} t |\sg_{\ups}| \, +\, \i \intff{ -\ups\eps }{ -\ups T_{\ups} } \Big\} \;.
\enq
Here, $ - \, \ups \Dp{}\mc{D}_{0,\eps}$ stands for the circle of radius $\eps$ centred at $0$ and oriented $-\ups$ counterclockwise.
However, in doing so, one has to take into account the discontinuity of the integrand along the line $\mf{s}_{\op{u}} t |\sg_{\ups}| \, - \, \i \ups \intff{ \eps }{ T_{\ups} }$, where
I have set
\beq
T_{\ups} \; = \; - \ups \Big( \Im\big( r_{\ups} \big) - \mf{s}_{\op{u}} |\sg_{\ups}| \a \Big) \; \sim \; C \cdot |\Re(t)|
\label{ecriture T ups et estimation de sa croissance à l'infini}
\enq
for some $C>0$ and when $ |\Re(t)| \tend +\infty$. This yields that
\beq
\Int{ \Ga^{\be_t}_{\ups} }{} \! \dd s\; \chi(t,s) s^n\, \ex{-\be_t s^2 } \; = \; \sul{a=1}{3}\Psi_{a;\ups }(t)
\enq
where, given $\eta>0$ small enough,
\beq
\Psi_{1;\ups }(t) \; = \; - \ups \eps \hspace{-1mm} \Int{ \Dp{} \mc{D}_{0,1} }{} \hspace{-2mm} \dd z \; \f{ \ex{- \be_t \big(\eps z + \mf{s}_{\op{u}} t |\sg_{\ups}| \big)^2 } }{ \big[ -\i \eps \ups z \big]^{\de_{\ups}} }
\cdot \big(\eps z + \mf{s}_{\op{u}} t |\sg_{\ups}| \big)^n \cdot \bigg\{ \f{ \i }{ 2t - \ups \eps z } \bigg\}^{ \de_{-\ups} } \;,
\enq
\beq
\Psi_{2;\ups }(t) \; = \; (-\i \ups)^n 2 \sin[\pi \de_{\ups} ] \hspace{-1mm} \Int{ \eps }{ \eta(t-\i\a) \mf{s}_{t} } \hspace{-3mm} \dd w \, h_{\ups}(w,t)
\qquad \e{and} \qquad
\Psi_{3;\ups }(t) \; = \; (-\i \ups)^n 2 \sin[\pi \de_{\ups} ] \hspace{-1mm} \Int{ \eta |\Re(t)| }{T_{\ups} } \hspace{-3mm} \dd w \, h_{\ups}(w,t) \;.
\enq
Above, I have introduced
\beq
h_{\ups}(w,t) \; = \; \big(w + \i \ups \mf{s}_{\op{u}} |\sg_{\ups}| \, t \big)^n \cdot \f{ \ex{ \, \be_t \big(w + \i \ups \mf{s}_{\op{u}} |\sg_{\ups}| \, t \big)^2 } }
{ w^{ \de_{\ups} } \cdot \big[ w - 2 \i t \big]^{ \de_{-\ups} } } \;.
\enq
\subsubsection*{ $\bullet$ The bound on $\Psi_{3; \vsg_{\op{u}} } (t) $}
Using the expansion \eqref{definition de xiu et C beta}, one gets, uniformly in $w \in \Big[ \eta |\Re(t)| ; T_{\ups} \Big] $ and in respect to $|\Re(t)| \tend + \infty$
\bem
\be_t \big(w + \i \ups \mf{s}_{\op{u}} |\sg_{\ups}| \, t \big)^2 \; = \; \f{ \i \vsg C_{\be} }{t}
\Big( w^2 \, + \, 2 w \i \ups \mf{s}_{\op{u}} |\sg_{\ups}| \, t \, - \, \sg_{\ups}^{2} \, t^2 \Big) \; + \; \e{O}(1) \\
\; = \; \i \vsg C_{\be} \Big\{ \tfrac{ w^2 }{ t } \, - \, \sg_{\ups}^2 t \Big\} \, - \, 2 C_{\be} \ups \vsg_{\op{u}} |\sg_{\ups}| w \, + \, \e{O}\big( 1 \big) \;.
\end{multline}
Also, given $t\in \R + \i \a$, one has that for $|\Re(t)| \tend +\infty$ and $w \in \Big[ \eta |\Re(t)| ; T_{\ups} \Big]$
\beq
\big| w + \i \ups \mf{s}_{\op{u}} |\sg_{\ups}| \, t \big|^n \, \leq \, C |\Re(t) |^{n} \qquad \e{and} \qquad
\Big|w^{ -\de_{\ups} } \cdot \big[ w - 2 \i t \big]^{ -\de_{-\ups} } \Big| \, \leq \, C |\Re(t) |^{ |\de_{+}| + |\de_{-}| }
\enq
for some constant $C>0$ and where one uses in the intermediate steps that both $|w|$ and $|w - 2 \i t|$ are bounded from below.
Hence, by putting these bounds together, one gets for some constants $C, C^{\prime}, C^{\prime \prime}$,
\beq
\Psi_{3; \vsg_{\op{u}} }(t) \; \leq \; C |\Re(t)|^{ n-\de_+ - \de_- } \Int{ \eta |\Re(t)| }{ T_{\ups} } \ex{-2 C_{\be} |\sg_{ \vsg_{\op{u}}} | w } \cdot \dd w
\; \leq \; C^{\prime} \ex{ -C^{\prime \prime}|\Re(t)| }
\enq
where the last bound follows from \eqref{ecriture T ups et estimation de sa croissance à l'infini} and holds for $t\in \R+\i\a$ with $|t|$-large enough.
\subsubsection*{ $\bullet$ The bound on $\Psi_{2; \vsg_{\op{u}} }(t) $}
Assume that $t = \rho \ex{\i \th} $ with $\rho \in \R$, $\rho \tend \pm \infty$ and $|\th|<\th_0$ with $\th_0$ small enough.
Then, it is convenient to rescale the integration variable as
\beq
w_{x}\, = \; (1-x) \eps \, + \, x \eta \mf{s}_{t} (t-\i\a)
\enq
so that
\beq
\Psi_{2;\ups }(t) \; = \; 2 (-\i \ups)^{n} \sin[\pi \de_{\ups} ]\, \Big( \eta \mf{s}_{t} (t-\i\a) \, -\, \eps \Big) \Int{ 0 }{1 } \hspace{-1mm} \dd x \, h_{\ups}(w_x,t) \;.
\enq
Furthermore, one has that
\beq
\be_t \big(w_x + \i \ups \mf{s}_{\op{u}} |\sg_{\ups}| \, t \big)^2 \; = \; - \i \vsg C_{\be} t \Big( |\sg_{\ups}|-\i \ups \mf{s}_{\op{u}} \mf{s}_{t}\eta x \Big)^2 \; + \; \e{O}\big( 1 \big)
\enq
and, for $\eta$ small enough, it holds
\beq
|\sg_{\ups}|-\i \ups \mf{s}_{\op{u}} \mf{s}_{t}\eta x \, = \, g_{x,\ups} \ex{-\tfrac{\i}{2} \vp_{x} \ups \mf{s}_{\op{u}} \mf{s}_{t} }
\qquad \e{for} \, \e{some} \quad
0\leq \vp_{x} < C \eta \quad \e{and} \quad g_{x,\ups} \, > \, C^{\prime} \, > \, 0
\enq
uniformly in $x$. Thus, for $\ups = \vsg_{\op{u}}$,
\beq
\be_t \big(w_x + \i \vsg_{\op{u}} \mf{s}_{\op{u}} |\sg_{ \vsg_{\op{u}} }| \, t \big)^2 \; = \; |\rho| \, C_{\be} \, g_{x,\vsg_{\op{u}} }^2 \exp\Big\{ \i \big[\th-(\vp_{x}+\tfrac{\pi}{2})\mf{s}_{t} \vsg \big] \Big\} \; + \; \e{O}\big( 1 \big) \;.
\enq
In particular, for $\th=-\mf{s}_t \vsg \psi$ with $\psi>0$, one gets that
\beq
\Re\Big[ \be_t \big(w_x + \i \ups \mf{s}_{\op{u}} |\sg_{\ups}| \, t \big)^2 \Big] \; \leq \; - C^{\prime\prime} \, |\rho| \, \sin\Big( \psi +\vp_{x} \Big) \, \leq \, - C^{(3)} \, |\rho| \, \psi
\enq
uniformly in $x \in \intff{0}{1}$. Furthermore,
\beq
\Re\big( w_{x} \big) \, = \, (1-x) \eps + x \eta |\rho| \cos(\th) \quad \e{so}\; \e{that} \quad
\big| w_x^{-\de_{\ups} } \big| \; \leq \; C
\enq
since $|w_x|$ is bounded from below away from $0$. Finally, one also has that
\beq
\Im\Big( w_x -2 \i t \Big) \; = \; -\a x \eta \mf{s}_{t} \, - \, 2 \rho \cos(\th) + x \eta \rho \mf{s}_{t} \sin(\th)
\enq
so that uniformly in $x \in \intff{0}{1}$ and for $|\rho|$ large enough and $\th_0$ small enough $\big| \Im\big( w_x -2 \i t \big) \big| \; \geq \; C |\rho|$
hence ensuring that $| w_x -2 \i t |$ is bounded from below and thus
\beq
\Big| \big( w_x -2 \i t \big)^{-\de_{-\ups} } \Big| \; \leq \; C \;.
\enq
The
numerator in $h_{\vsg_{\op{u}} }(w_x,t)$ generates a power-law bound in $\rho$ so that, all-in-all,
\beq
\big| h_{\vsg_{\op{u}} }(w_x,t) \big| \, \leq \, C^{\prime} \ex{- C \psi |\rho|} \qquad \e{for} \qquad t = |\rho| \ex{-\i \vsg \mf{s}_t \psi } \;.
\enq
This entails that
\beq
\Big| \Psi_{2; \vsg_{\op{u}} }(t) \Big| \; \leq \; C^{\prime} \ex{- C \psi |\rho|} \qquad \e{for} \qquad t = |\rho| \ex{-\i \psi \vsg \mf{s}_t} \;.
\enq
Also, the above reasonings and estimates ensure that $\Psi_{2; \vsg_{\op{u}} }$ is holomorphic on $\mc{S}_{\th_0;A}$ with $A$ large enough and $\th_0$ small enough.
\subsubsection*{ $\bullet$ The bound on $\Psi_{1;\ups }(t) $}
Similar handlings to what has been exposed show that, for $z=\ex{\i\psi}$,
\beq
-\Re\Big( \be_t \, (\eps z + \mf{s}_{\op{u}} t |\sg_{\ups} | \big)^2 \Big) \, = \, - \sg_{\ups}^2 C_{\be} |\rho| \sin(\th) \, + \, \e{O}(1) \qquad \e{with} \quad
t \, = \, \rho \ex{-\i \vsg \mf{s}_{t} \th } \;.
\enq
Furthermore, for $z=\ex{\i \psi}$ one has
\beq
\Big| \Im\Big( \eps z \,- \,2 \i t \Big)\Big| \, = \, \Big| \eps \sin(\psi) \, - \, 2 \rho \cos(\th) \Big| \geq C |\rho|
\enq
provided that $\th$ is small enough and $|\rho|$ large enough, one readily gets that, for some constants $C, C^{\prime}$,
\beq
\Big| \Psi_{1;\ups }(t) \Big| \; \leq \; C^{\prime} \ex{- C \th |\rho|} \qquad \e{for} \qquad t=\rho \ex{-\i \th \vsg \mf{s}_t} \;.
\enq
Again, the above also ensures that $\Psi_{1;\ups }$ is holomorphic on $\mc{S}_{\th_0;A}$ with $A$ large enough and $\th_0$ small enough.
Thus, upon putting all the intermediate bounds together, the claim follows.
\subsubsection*{B) The regime $|\op{u}|< \op{v}$}
The analysis is quite similar to the previous regime. I thus only highlight the main steps.
In the present case, since $\e{sgn}( \sg_{\pm} )=+$, it is convenient to represent
\beq
\chi(t,s) \, = \, \pl{ \ups= \pm }{} \bigg\{ \f{ \i \ups }{ s \,+ \, t \ups \sg_{\ups} } \bigg\}^{ \de_{\ups} }
\enq
meaning that, for fixed $t$, the map $s \mapsto \chi(t,s)$ has cuts along the lines $ -\ups t \sg_{\ups} - \ups \i \R^{+}$.
Since $|\a_{\op{u}}| <\a$, the cut along $ -t \sg_{+} - \i \R^{+}$ is located below the original integration line $\R + \i \a_{ \op{u} }$
while the one along $ t \sg_{-} + \i \R^{+}$ is located above $\R + \i \a_{ \op{u} }$.
\begin{figure}[ht]
\begin{center}
\begin{pspicture}(12,5)
\psline[linestyle=dashed, dash=3pt 2pt](0,2.5)(11.8,2.5)
\psline[linewidth=2pt]{->}(11.8,2.5)(11.9,2.5)
\rput(12,2.2){$\R + \i \a_{\op{u}}$}
\rput(0.4,2.9){ $ t \sg_-$ }
\psdots(1,3)
\psline(1,3)(1,4.6)
\psdot(1,4.6)
\pscurve(1,4.6)(1.2,3.5)(1.3,2.9)(1,2.8)(0.7,2.9)(0.8,3.5)(1,4.6)
\rput(1.3,4.7){$ r_- $}
\rput(1.7 , 3.1){$\Ga_-^{\be_t}$}
\rput(9.4,2){ $- t \sg_+ $ }
\psdots(10,2)
\psline(10,2)(10,0.8)
\psdots(10,0.85)
\pscurve(10,0.85)(9.8,1.6)(10,2.1)(10.2,1.6)(10,0.85)
\rput(9.65,0.7){$ r_+ $}
\rput(10.5,1.5){$\Ga_+^{\be_t}$}
\psline(0,0)(12,5)
\psline[linewidth=2pt]{->}(8,3.35)(8.1,3.39)
\psline[linewidth=2pt]{->}(1,2.8)(1.1,2.8)
\rput(11,0){$ \vsg <0$}
\rput(11.5,0.8){ $ \ex{ -\tfrac{\i}{2} \th_{\be_t} } \R $ }
\psline(0,5)(12,0)
\psline[linewidth=2pt]{->}(8,1.65)(8.1,1.60)
\psline[linewidth=2pt]{->}(9.8,1.6)(9.8,1.65)
\rput(11,5){$ \vsg >0$}
\rput(11,4.2){ $\ex{-\tfrac{\i}{2} \th_{\be_t} } \R$ }
\end{pspicture}
\caption{ Deformed contours in the case $|\op{u}|<\op{v}$ and for $\Re(t) < 0$
\label{Figure contour deformes pour u bigger than v re(t) negatif} }
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\begin{pspicture}(12,5)
\psline[linestyle=dashed, dash=3pt 2pt](0,2.5)(11.8,2.5)
\psline[linewidth=2pt]{->}(11.8,2.5)(11.9,2.5)
\rput(11,2.2){$\R + \i \a_{\op{u}}$}
\rput(11.8,3.1){ $ t \sg_-$ }
\psdots(11,3)
\psline(11,3)(11,4.6)
\psdot(11,4.6)
\pscurve(11,4.6)(10.8,3.5)(10.7,2.9)(11,2.8)(11.3,2.9)(11.2,3.5)(11,4.6)
\rput(10.7,4.7){$ r_- $}
\rput(10.3, 3.1){$\Ga_-^{\be_t}$}
\rput(2.6,2){ $ -t \sg_+$ }
\psdots(2,2)
\psline(2,2)(2,0.8)
\psdots(2,0.85)
\pscurve(2,0.85)(2.2,1.6)(2,2.1)(1.8,1.6)(2,0.85)
\rput(2.35,0.7){$ r_+ $}
\rput(1.5,1.5){$\Ga_+^{\be_t}$}
\psline(0,0)(12,5)
\psline[linewidth=2pt]{->}(8,3.35)(8.1,3.39)
\psline[linewidth=2pt]{->}(11,2.8)(11.1,2.8)
\rput(1,0){$ \vsg < 0$}
\rput(0.5,0.8){ $ \ex{ -\tfrac{\i}{2} \th_{\be_t} } \R $ }
\psline(0,5)(12,0)
\psline[linewidth=2pt]{->}(8,1.65)(8.1,1.60)
\psline[linewidth=2pt]{->}(2.2,1.65)(2.2,1.6)
\rput(1,5){$ \vsg > 0$}
\rput(1,4.3){ $\ex{-\tfrac{\i}{2} \th_{\be_t} } \R$ }
\end{pspicture}
\caption{ Deformed contours in the case $|\op{u}|<\op{v}$ and for $\Re(t) >0$
\label{Figure contour deformes pour u bigger than v re(t) positif} }
\end{center}
\end{figure}
The analysis then depends on the sign of $ \vsg$
\subsubsection*{$\bullet$ $ \vsg >0$ }
In this case, one can deform the contour as in Figures \ref{Figure contour deformes pour u bigger than v re(t) negatif} or \ref{Figure contour deformes pour u bigger than v re(t) positif},
depending on the sign of $\Re(t)$, without having to deal with the cuts of $\chi(t,s)$. This yields
\beq
\mc{F}_n(t) \, = \, \Big( \f{ 1 }{ \be_t } \Big)^{ \frac{1}{2}(n+1) } \Int{ \R }{} \dd s \, s^n \ex{-s^2} \, \chi\Big(t, \tfrac{ s }{ \sqrt{\be_t} } \Big)
\enq
and one can conclude by the previous analysis.
\subsubsection*{$\bullet$ $ \vsg < 0$ }
In this case, when deforming the contour as in Figures \ref{Figure contour deformes pour u bigger than v re(t) negatif} or \ref{Figure contour deformes pour u bigger than v re(t) positif},
according to the sign of $\Re(t)$, one observes that one has to take into account both cuts stemming from $\chi(t,s)$. This yields
\beq
\mc{F}_n(t) \, = \, \Big( \f{ 1 }{ \be_t } \Big)^{ \frac{1}{2}(n+1) } \Int{ \R }{} \dd s \, s^n \ex{-s^2} \, \chi\Big(t, \tfrac{ s }{ \sqrt{\be_t} } \Big) \; + \;
\sul{\ups= \pm }{} \Int{ \Ga^{\be_t}_{ \ups } }{} \dd s\, s^n \, \chi(t,s) \ex{-\be_t s^2 } \;.
\enq
The two cut-issued integrals are very similar in structure to those studied previously, and, eventually,
one ends up with the same conclusions. \qed | 68,096 |
Interruption to Supply Eton Irrigation Water Supply Scheme Eton Irrigation will be chemically treating for aquatic weed control in the channel system during the following period when water WILL NOT be available.
Closedown Schedule for Period Monday 7th December to Saturday 12th December 2020
- Mt Alice Reticulation is off line for a longer period, as the storage has to be treated prior to dewatering the channel. The channel must then be refilled in order to re-instate the pump station.
Recent Comments | 381,082 |
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Finally, avoid eating junk food as much as possible. Many folks who want a flat stomach are attracted to the concept of doing the”oxidative detoxification diet,” which contains drinking highly-filled drinks will want to focus! | 361,140 |
TITLE: Finite additive measure
QUESTION [1 upvotes]: Problem: Let $[0,1]\cap\mathbb{Q} $ denote
the set of all rational number inside the interval $\left[0,1\right]$,
let $\mathcal{A}$ be the algebra of sets that can be expressed as
finite unions of non-intersecting sets $A$ of the form $\left\{ r:a<r<b\right\} ,\left\{ r:a\leq r<b\right\} ,\left\{ r:a<r\leq b\right\} ,\left\{ r:a\leq r\leq b\right\} $,
and let $\mathbb{P}\left(A\right)=b-a$. Prove that the set function $\mathbb{P}\left(A\right),A\in\mathcal{A}$,
is finitely additive but not countably additive.
Attempt I have managed to show that $\mathbb{P}$ is not countably additive. However, I don't know how to show the finite additivity. Namely, if we have two disjoint sets $(a,b)$ and $(c,d)$ where $b\neq c$, how to show that
$$\mathbb{P}((a,b)\cup(c,d))=\mathbb{P}(a,b)+\mathbb{P}(c,d)$$ I think if $b=c$ then everything works out fine. But this is not the case. My idea is that we have
\begin{align*}
\mathbb{P}((a,b)\cup[b,c]\cup(c,d))& =\mathbb{P}(a,d)\\
& = d-a\\
& = (d-c)+(c-b)+(b-a)\\
& = \mathbb{P}(a,b)+\mathbb{P}[b,c]+\mathbb{P}(c,d)
\end{align*}
subtract both sides by $\mathbb{P}([b,c])$, we have
\begin{align*}
\mathbb{P}((a,b)\cup[b,c]\cup(c,d))-\mathbb{P}([b,c])=\mathbb{P}(a,b)+\mathbb{P}(c,d)
\end{align*}
So it suffices to show that the left hand side is $\mathbb{P}((a,b)\cup(c,d))$.This is where I got stuck.
REPLY [0 votes]: The function $\mathbb{P}$ is finitely additive by definition. It is defined by $\mathbb{P}(A) = b-a$ for any interval with endpoints $a\leq b$, and $\mathbb{P}(A_1 \cup \cdots \cup A_n) = \sum_{i=1}^n \mathbb{P}(A_i)$ for any collection of disjoint intervals $A_1,\ldots,A_n$.
What is slightly less clear is whether $\mathbb{P}$ is well-defined. For example, as you mention, it could happen that a single set can be written as the finite disjoint union of intervals in several ways, for example $[a,b] = [a,c) \cup [c,b]$. In this particular case, there is no problem, since using the decomposition on the left we get a measure of $b-a$, and using the decomposition on the right we get a measure of $(c-a) + (b-c) = b-a$. But how do we know that this is always the case? This is exactly the point of the exercise.
You can probably prove that $\mathbb{P}$ is well-defined from first principles, but a shorter route is to show that $\mathbb{P}$ agrees with the standard Lebesgue measure on its domain of definition. | 62,728 |
TITLE: Prove $\sqrt[5]{2} > \ln(\pi)$
QUESTION [3 upvotes]: For any positive rational numbers $p/q$ and $m/n$, I can decide which is bigger by comparing $pqn$ and $mqn$.
For irrational numbers, we can't use this test. Furthermore, almost all irrational numbers can't be specified by a finite formula in a given symbolic system (i.e. computed to arbitrary precision with an algorithm whose definition is finite). Some can: $\sqrt[5]{2}$ and $\ln{\pi}$ are two examples. There are algorithms available to compute either of these to arbitrary precision, and at some point, their decimal expansions diverge and we can conclude which one is bigger. This is equivalent to finding a rational number which is bigger than one but less than the other.
However, is it possible to prove a fact like $\sqrt[5]{2} > \ln{\pi}$ without obtaining a rational number between them?
More generally, say we can describe two expressions for real numbers, $r$ and $s$, in terms of $+,-,\times,\div$, some transcendental functions, infinite series, etc., such that we could approximate each of their decimal expansions. Is it possible in general to compute/prove which of the expressions $r$ and $s$ is bigger without finding a decimal expansion or otherwise finding a rational number in between them?
REPLY [4 votes]: The set of numbers for which there is a formula to approximate their decimal expansions is called the computable numbers. The order relation on the computable numbers is non-computable. If the two numbers are known to be unequal, then it is computable, but the method is successive approximation. | 172,916 |
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Most homemade granola bars are packed with dried fruit and nuts, which is all well and good unless you are allergic (like me). Not only is this recipe for granola bars nut free but it’s dairy-free and gluten-free as well! If you aren’t allergic to nuts, you can certainly customize this recipe and sub in… Read More >
Popcorn Crusted Macaroni and Cheese
Who doesn’t love mac and cheese? It’s a dinner that will bring a smile to anyone’s face – after all, it is the ultimate comfort food. This macaroni and cheese recipe takes a unique turn by using popcorn as a crunchy topping instead of the more typical bread crumbs. While I enjoyed the uniqueness, the… Read More >
Cheddar Popcorn Bread
Popcorn is a healthy whole grain snack, but its uses far exceed snack time cravings. It’s a versatile ingredient that you can use in everything from dessert to breakfast and beyond. I had never thought to use it in cornbread and the result was fantastic – very light and airy yet at the same time,… Read More > | 127,795 |
\begin{document}
\title{The number of edge-disjoint transitive triples in a tournament}
\author{Raphael Yuster
\thanks{
e-mail: [email protected] \qquad
World Wide Web: http:$\backslash\backslash$research.haifa.ac.il$\backslash$\symbol{126}raphy}
\\ Department of Mathematics\\ University of
Haifa at Oranim\\ Tivon 36006, Israel}
\date{}
\maketitle
\setcounter{page}{1}
\begin{abstract}
We prove that a tournament with $n$ vertices has more than $0.13n^2(1+o(1))$ edge-disjoint transitive triples.
We also prove some results on the existence of large packings of $k$-vertex transitive
tournaments in an $n$-vertex tournament. Our proofs combine probabilistic arguments and some powerful packing results
due to Wilson and to Frankl and R\"odl.
\end{abstract}
\section{Introduction}
All graphs and digraphs considered here are finite and have no loops or multiple edges.
For the standard terminology used the reader is referred to \cite{Bo}.
A {\em tournament} on $n$ vertices is an orientation of $K_n$.
Thus, for every two distinct vertices $x$ and $y$, either $(x,y)$ or $(y,x)$ is an edge,
but not both.
Let $TT_k$ denote the unique transitive tournament on $k$ vertices. $TT_3$
is also called a {\em transitive triple} as it consists of some triple $\{(x,y), (x,z), (y,z)\}$.
A {\em $TT_k$-packing} of a directed graph $D$ is a set of edge-disjoint copies of $TT_k$
subgraphs of $D$. The {\em $TT_k$-packing number} of $D$, denoted $P_k(D)$,
is the maximum size of a $TT_k$-packing of $D$.
The $TT_3$-packing number of $D_n$, the complete digraph with $n$ vertices and $n(n-1)$ edges,
has been extensively studied. See, e.g., \cite{Ga,PhLi,Sk}.
In this paper we consider only $TT_k$-packings of tournaments.
Let $f_k(n)$ denote the minimum possible value of $P_k(T_n)$, where $T_n$ ranges over all possible
$n$-vertex tournaments. For simplicity, put $f(n)=f_3(n)$ and $P(T_n)=P_3(T_n)$.
Trivially, $P_k(T_n) \leq n(n-1)/(k(k-1))$, and in particular $f(n) \leq n(n-1)/6 < 0.167n^2(1+o(1))$.
In fact, it is not difficult to show that $f(n) \leq \lceil n(n-1)/6-n/3 \rceil$ (see Section 4
for this and also for a general way to construct an upper bound for $f_k(n)$).
We conjecture the following:
\begin{conj}
\label{c1}
$f(n)= \lceil n(n-1)/6-n/3 \rceil$.
\end{conj}
This conjecture was verified for all $n \leq 8$.
Our main result is the following lower bound for $f(n)$.
\begin{theo}
\label{t1}
$f(n) > 0.13n^2(1+o(1))$.
\end{theo}
We prove Theorem \ref{t1} in Section 2. In section 3 we show that if $T_n$ is
the random tournament on $n$ vertices then $P(T_n) \geq \frac{1}{6}n^2(1-o(1))$ almost surely.
In fact, we show that $P_k(T_n) \geq \frac{1}{k(k-1)}n^2(1-o(1))$ almost surely.
The final section contains some concluding remarks.
\section{Proof of the main result}
From here on we assume that the vertex set of a tournament with $k$
vertices is $[k]=\{1,\ldots,k\}$. Let $T_k$ be any $k$-vertex tournament.
For $v \in [k]$, let $d^+(v)$ denote the out-degree of $v$ in $T_k$.
Let $a(T_k)$ denote the total number of transitive triples in $T_k$,
and let $t(T_k)$ denote the total number of directed triangles in $T_k$.
Clearly, $a(T_k)+t(T_k)={k \choose 3}$. We shall also make use of the obvious inequality,
which follows from the fact that in a transitive triple there is one source and one sink.
\begin{equation}
\label{e1}
a(T_k) = \sum_{i=1}^k \frac{1}{2}\left({{d^+(v)} \choose 2} + {{k-1-d^+(v)} \choose 2}\right)
\geq \frac{k(k-1)(k-3)}{8}.
\end{equation}
In the proof of Theorem \ref{t1} we need the following (special) case
of Wilson's Theorem \cite{Wi}.
\begin{lemma}
\label{l21}
There exists a positive integer $N$ such that for all $n > N$, if $n \equiv 1 \bmod 42$
then $K_n$ decomposes into ${n \choose 2}/21$ edge-disjoint copies of $K_7$.
\endpf
\end{lemma}
The next lemma quantifies the fact that if $t(T_7)$ is relatively small then $P(T_7)$ is relatively large.
\begin{lemma}
\label{l22}
If $t(T_7) \leq 4$ then $P(T_7)=7$. If $t(T_7) \leq 11$ then $P(T_7) \geq 6$.
If $t(T_7) \geq 12$ then $P(T_7) \geq 5$.
\end{lemma}
{\bf Proof}\,
Clearly, the expected number of directed triangles in a random Steiner triple system
of $T_7$ is $7 \cdot \frac{t(T_7)}{t(T_7)+a(T_7)}= \frac{7}{35}t(T_7)$.
Hence, if $t(T_7) < 5$ then this expectation is less than $1$. Thus, there is a Steiner
triple system with no directed triangle. Namely, $P(T_7)=7$ in this case.
Similarly, by \ref{e1}, we always have $a(T_7) \geq 21$ and so $t(T_7) \leq 14$.
Therefore, the expectation above is always at most $14 \cdot \frac{7}{35} \leq 2.8$.
Thus, there is always a Steiner triple system with at most two directed triangles.
Namely, $P(T_7) \geq 5$ always.
We remain with the case where $t(T_7) \leq 11$. Notice that we may assume $t(T_7)=11$
or $t(T_7)=10$ since otherwise the above expectation argument yields $P(T_7) \geq 6$.
Assume first that $t(T_7)=11$. Hence $a(T_7)=24$ and by (\ref{e1}) the only possible scores
(sorted out-degree sequence) of such a $T_7$ are $(4,4,4,3,2,2,2)$,
$(5,3,3,3,3,2,2)$ and $(4,4,3,3,3,3,1)$. The last two scores are complementary (namely,
reversing the edges of a $T_7$ with one of these scores yields a tournament with the other score)
and the first score is self-complementary. Hence, one needs only to check the first two scores.
There are precisely 18 non-isomorphic tournaments with the score $(4,4,4,3,2,2,2)$,
and each can be checked to have at least 6 edge-disjoint transitive triples.
A convenient way to enumerate these 18 non-isomorphic tournaments is as follows.
Let $A_i$ be the set of vertices with out-degree $i$, $i=2,3,4$. $|A_2|=|A_4|=3$, $|A_3|=1$.
First case: The subgraph induced by $A_2$ is a directed triangle and the subgraph
induced by $A_4$ is also a directed triangle. There are four non-isomorphic tournaments
with this restriction. Second case: The subgraph induced by $A_2$ is a directed triangle
and the subgraph induced by $A_4$ is a transitive triple. There are four non-isomorphic tournaments
with this restriction. Third case: The subgraph induced by $A_2$ is a transitive triple
and the subgraph induced by $A_4$ is a directed triangle. There are four non-isomorphic tournaments
with this restriction. Fourth case: Both $A_2$ and $A_4$ induce a transitive triple. There are six
non-isomorphic tournaments with this restriction. Altogether there are $4+4+4+6=18$ possibilities.
There are precisely 15 non-isomorphic tournaments with the score $(5,3,3,3,3,2,2)$,
and each can be checked to have at least 6 edge-disjoint transitive triples.
A convenient way to enumerate these 18 non-isomorphic tournaments is as follows.
Let $A_i$ be the set of three vertices with out-degree $i$, $i=2,3,5$. $A_2=\{a,b\}$
$A_5=\{c\}$, $A_3=\{d,e,f,g\}$. We may assume the edge inside $A_2$ is $(a,b)$.
First case: $(a,c)$ is an edge. There is a unique tournament with this restriction.
Second case: $(c,a)$, $(a,d)$ and $(d,c)$ are edges. There are four non-isomorphic
tournaments. Third case: $(c,a)$, $(a,d)$ $(c,d)$ and $(b,d)$ are edges.
There are three non-isomorphic tournaments. Fourth case: $(c,a)$, $(a,d)$ $(c,d)$ and $(d,b)$
are edges. There are 7 non-isomorphic tournaments. Altogether there are $1+4+3+7=15$ possibilities.
In case $t(T_7)=10$ the expected number of directed triangles in a random Steiner triple
system is precisely 2. However, the distribution is easily seen to be non-constant
(e.g., the variance is positive). Thus, there is a Steiner triple system with less than two directed triangles.
Namely, $P(T_7) \geq 6$ in this case. \endpf
Fix $T_n$, and let $3 \leq m \leq n$.
Let $T_m$ be a randomly chosen $m$-vertex induced subgraph of $T_n$.
Let $X=a(T_m)$ denote the random variable corresponding to the
number of transitive triples of $T_m$, and let $E[X]$ denote the expectation of $X$.
\begin{prop}
\label{p23}
$E[X] \geq \frac{3}{4}\frac{n-3}{n-2}{m \choose 3}$.
\end{prop}
{\bf Proof}\,
A specific triple of $T_n$ belongs to precisely ${{n-3} \choose {m-3}}$ induced subgraphs
on $m$ vertices. Thus, by (\ref{e1}),
$$
E[X] =\frac{a(T_n) { {n-3} \choose {m-3}} }{ {n \choose m}}= a(T_n)\frac{m(m-1)(m-2)}{n(n-1)(n-2)}
\geq \frac{3}{4}\frac{n-3}{n-2}{m \choose 3}.
$$
\endpf
{\bf Proof of Theorem \ref{t1}:}\,
Let $n > N+41$ where $N$ is the constant from lemma \ref{l21}.
Let $T_n$ be a fixed $n$-vertex tournament.
We may assume that $n \equiv 1 \bmod 42$, since otherwise we may
delete at most $41$ vertices, apply the theorem on the smaller graph,
and this will not affect the claimed asymptotic number of transitive triples in
the original graph. By Proposition \ref{p23}, the expected number of transitive triples in a random $T_7$
of $T_n$ is at least $26.25(n-3)/(n-2)=26.25(1-o_n(1))$. Hence, the expected number of directed triangles
is at most $8.75(1+o_n(1))$.
Let $p_1$ denote the probability that a random $T_7$ has $t(T_7) \leq 4$. Let $p_2$ denote the
probability that a random $T_7$ has $5 \leq t(T_7) \leq 11$. Let$p_3$ denote the probability that
a random $T_7$ has $t(T_7) \geq 12$. Clearly, $p_1+p_2+p3=1$ and
$$
5p_2+12p_3 \leq 8.75(1+o_n(1)).
$$
Let $Y$ denote the random variable corresponding to $P(T_7)$.
By definition of $p_1,p_2,p_3$ and by Lemma \ref{l22} we have
$$
E[Y] \geq 7p_1+6p_2+5p_3.
$$
Minimizing $E[Y]$ subject to $p_1+p_2+p_3=1$, $p_i \geq 0$ and $5p_2+12p_3 \leq 8.75(1+o_n(1))$.
yields $p_1=0$, $p_2=13/28(1-o_n(1))$, $p_3=15/28(1+o_n(1))$ and $E[Y] \geq \frac{153}{28}(1-o_n(1))$.
Let $S$ be a fixed $K_7$-decomposition of $K_n$ into ${n \choose 2}/21$ edge-disjoint copies of $K_7$.
By Lemma \ref{l21} such an $S$ exists. Each $s \in S$ corresponds to a $7-set$ of $[n]$.
Let $\sigma$ be a random permutation of $[n]$ and let $S_\sigma$ denote the $T_7$-decomposition
of $T_n$ corresponding to $S$ and $\sigma$. Namely, for each $s \in S$ the corresponding $T_7$-subgraph
of $T_n$, denoted $s_\sigma$, consists of the 7 vertices $\{\sigma(i) ~: ~ i \in s\}$.
Notice that since $\sigma$ is a random permutation, $s_\sigma$ is a random $T_7$ of $T_n$.
Thus, the expected number of edge-disjoint transitive triples of $s_\sigma$ is at least $\frac{153}{28}(1-o_n(1))$.
By linearity of expectation we get that
$$
P(T_n) \geq \frac{{n \choose 2}}{21}\frac{153}{28}(1-o_n(1))=\frac{51}{392}n^2(1+o_n(1)) > 0.13n^2(1+o_n(1)).
$$
\endpf
\section{Edge-disjoint transitive triples in a random tournament}
A random tournament with $n$ vertices is obtained by selecting the orientation of each edge
by flipping an unbiased coin, where all ${n \choose 2}$ choices are independent.
Assume, therefore, that $T_n$ is a random tournament.
\begin{prop}
\label{p31}
$$
{\rm Prob} \left[P_k(T_n) \geq \frac{1}{k(k-1)}n^2(1-o_n(1)) \right] \geq 1-o_n(1).
$$
\end{prop}
{\bf Proof}\,
Let $(x,y)$ be any edge of $T_n$. Clearly, each $K_k$ containing $(x,y)$
induces a $TT_k$ with probability $k!/2^{k \choose 2}$.
Hence, letting $n(x,y)$ denote the number of
transitive $k$-vertex tournaments containing $(x,y)$, we have $E[n(x,y)]={{n-2} \choose {k-2}}k!/2^{k \choose 2}$.
As any two $k$-vertex tournaments containing $(x,y)$ share at most $k-3$ vertices (other than $x$ and $y$) there is
limited dependence between the tournaments containing $(x,y)$ (in fact, for $k=3$ there is complete independence).
Hence, standard large deviation arguments for limited dependence yield that for every $0.5 > \epsilon > 0$,
$$
{\rm Prob}\left[ \left| n(x,y)-{{n-2} \choose {k-2}}\frac{k!}{2^{k \choose 2}}\right| > n^{k-2-\epsilon}\right] =o(n^{-2}).
$$
Thus, with probability $1-o_n(1)$, all edges of $T_n$ lie on at least
$(k(k-1)/2^{k \choose 2})n^{k-2}(1-o_n(1))$
copies of $TT_k$ and at most $(k(k-1)/2^{k \choose 2})n^{k-2}(1+o_n(1))$
copies of $TT_k$.
Consider the ${k \choose 2}$-uniform hypergraph $H$ whose $N={n \choose 2}$ vertices are the edges of $T_n$
and whose edges are the (edge sets of) $TT_k$ copies of $T_n$.
The degree of all the vertices in this hypergraph
is $(k(k-1)/2^{k \choose 2})n^{k-2}(1 \pm o_n(1))=2^{k/2-1-k(k-1)/2}k(k-1)N^{k/2-1}(1 \pm o_n(1))$,
(i.e. the hypergraph is almost regular).
Furthermore, the co-degree of any two vertices in this hypergraph is at most $O(n^{k-3})=O(N^{k/2-1.5})=o(N^{k/2-1})$.
By the result of Frankl and R\"odl \cite{FrRo},
this hypergraph has a matching that covers all but at most $N(1-o_N(1))$ vertices.
Such a matching corresponds to a set of $\frac{1}{k(k-1)}n^2(1-o_n(1))$ edge-disjoint
copies of $TT_k$ in $T_n$. \endpf
\section{Concluding remarks}
\begin{itemize}
\item
Whenever $P(T_n) = n(n-1)/6$ we say that $T_n$ has a {\em transitive} Steiner
triple system. Clearly, this may occur only if $K_n$ has a Steiner triple system,
namely, when $n \equiv 1,3 \bmod 6$. It would be interesting to characterize the
tournaments that have a transitive Steiner triple system.
\item
Conjecture \ref{c1}, if true, would be best possible. We show $f(n) \leq \lceil n(n-1)/6-n/3 \rceil$.
Let $T_3(n)$ be the complete $3$-partite Tur\'an graph with $n$ vertices. It is well-known that
$T_3(n)$ has ${n \choose 2}-\lceil n(n-1)/6-n/3 \rceil$ edges. Denote the partite classes
by $V_1,V_2,V_3$. Orient all edges between $V_1$ and $V_2$ from $V_1$ to $V_2$.
Orient all edges between $V_2$ and $V_3$ from $V_1$ to $V_2$.
Orient all edges between $V_1$ and $V_3$ from $V_3$ to $V_1$.
Complete this oriented graph to a tournament $T_n$ by adding directed edges between any
two vertices in the same partite class in any arbitrary way. Notice that each transitive triple
in $T_n$ contains at least one edge with both endpoints in the same vertex class.
Hence, $P(T_n) \leq \lceil n(n-1)/6-n/3 \rceil$.
\item
Conjecture \ref{c1} has been verified for $n \leq 8$. The values $f(1)=f(2)=f(3)=0$
and $f(4)=1$ are trivial. The values $f(5) = 2$, $f(6)=3$ are easy exercises.
The value $f(7) \geq 5$ is a consequence of Lemma \ref{l22}, and thus $f(7) = 5$ by the above Tura\'n
graph argument. The value $f(8) \geq 7$ (and hence $f(8)=7$) is computer verified.
\item
Conjecture \ref{c1} claims, in particular, that one can cover almost all edges of $T_n$ with
edge-disjoint transitive triples. Proposition \ref{p31} asserts that this is true for the random tournament
and that, in fact, the random tournament can be covered almost completely with edge-disjoint copies of $TT_k$
for every fixed $k$. However, for $k \geq 4$ there are constructions showing that a significant amount
of edges must be uncovered by any set of edge-disjoint $TT_k$. Consider $TT_4$. It is well-known
(cf. \cite{RePa}) that there is a unique $T_7$ with no $TT_4$. Consider the complete $7$-partite digraph
with $n$ vertices obtained by blowing up each vertex of this unique $T_7$ with $n/7$ vertices.
Add arbitrary directed edges connecting two vertices in the same vertex class to obtain a $T_n$.
Clearly, any $TT_4$ of this $T_n$ must contain an edge with both endpoints in the same vertex class.
Hence, $f_4(n) \leq P_4(T_n) \leq 7{{n/7} \choose 2}=O(\frac{1}{14}n^2)$.
Hence at least ${n \choose 2} - 6 f_4(T_n) \geq \frac{1}{14}n^2(1+o(1))$ must be uncovered.
Similar constructions exist for all $k \geq 4$, where the fraction of covered edges tends to
zero as $k$ increases.
\item
It is possible to slightly improve the constant appearing in Theorem \ref{t1}.
Recall that the proof of Theorem \ref{t1} assumed a worst case of $p_1 \geq 0$, where $p_1$ is
the probability that a random $T_7$ has at most four directed triangles.
However, it is very easy to prove that for $n$ sufficiently large, $p_1 > c > 0$ where $c$ is some (small) absolute
constant. This follows from the fact that every $T_{54}$ contains a $TT_7$ \cite{Sa}.
Thus there exists a positive constant $c'$ such that for $n$ sufficiently large,
$T_n$ has at least $c'n^7$ copies of $TT_7$. Hence, a random induced $7$-vertex subgraph of $T_n$ is
a $TT_7$ with constant positive probability. This improvement for $p_1$ immediately implies a
(very small) improvement for the constant appearing in Theorem \ref{t1}.
\end{itemize}
\section*{Acknowledgment}
The author thanks Noga Alon for useful comments. | 84,458 |
ASHLAND, Ore., July 11, 2008 (PRIME NEWSWIRE) -- Electric Moto Corp., Inc. , a U.S.-based designer, manufacturer and retailer of highly efficient electric battery-powered motorcycles, scooters and ATVs, today announced the resignation of Ely Schless as Chief Executive Officer, effective immediately.
In conjunction with his resignation as CEO, Mr. Schless has been named Chief Technology Consultant (CTC) of Electric Moto. The new position was created specifically to optimally capitalize on his unique capabilities surrounding the development of 2nd and 3rd generation electric vehicles, which is highlighted by his successful design of the company's first-generation "Blade" motor bike.
In addition, the company's board of directors announced the creation of an Interim Management Team of corporate personnel designed to (a) launch an immediate executive search for a new CEO, and (b) continue to pursue initiatives outlined by the firm's investment bankers surrounding the establishment of a new production facility and the development of distribution channels throughout Central and South America and the Caribbean. Electric Moto's well-established European distribution channel is viewed by management as a key competitive strength and is expected to remain unchanged going forward.
Management is also pleased to announce that several "Blade XT" demo bikes have recently been shipped from the company's current production facility in Oregon to distributors in Europe. In addition, Electric Moto anticipates shipping two additional "demo" bikes to the offices of its investment banking firm in NY in the coming weeks.
About Electric Moto Corp., Inc:
Electric Moto Corp. is an Ashland, Oregon-based company that specializes in the design, manufacture, marketing and sales of vehicle chassis and components and eco-friendly, zero emissions, electric battery-powered vehicles for recreational, law enforcement, governmental and military applications. The engineering and management team behind Electric Moto and its rapidly expanding line of lithium-ion powered vehicles have decades of experience developing and implementing cutting edge technologies for powering the company's expanding line of practical, clean and economic vehicles of the future. The company, which currently has distributors in the United States and Europe, has plans to continue adding distributors wherever Earth-friendly, green vehicles will be in demand. Electric Moto is currently poised to begin production of its Blade XT off-road motorcycles and is in preparation to launch its own lines of ATVs and scooters. For more information on Electric Moto please visit:.
In addition to the historical information contained herein, this news release contains forward-looking statements that are subject to risks and uncertainties. Actual results may differ substantially from those referred to herein due to a number of factors. Forward looking statements may include, but are not limited to statements regarding EMOT's growth and profitability, growth strategy, access to technology, liquidity and access to public markets, product flow from suppliers, and trends in the industry in which EMOT operates, which could have a material effect on EMOT's current business model. The forward-looking statements contained in this press release are also subject to other risks and uncertainties, including those more fully described in EMOT's filings with the SEC. EMOT assumes no obligation to update these forward-looking statements to reflect actual results, changes in risks, uncertainties or assumptions underlying or affecting such statements, or for prospective events that may have a retroactive effect.
CONTACT: OutcastTrader.com
Investor Relations:
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Rashad Badr, MPA
There is a great deal of discussion about the need to adjust the balance between civilian agencies and the military in executing U.S. foreign policy and programs. The easiest argument to make is that the vast American defense complex overshadows US diplomatic and developmental efforts in almost every way. The Department of Defense (DoD) budget in 2010 was $691 billion, whereas the State Department’s budget for that year was just $16.4 billion. The United States Agency for International Development (USAID), the development arm of State, similarly faced a shorter stack in pursuing its goals abroad. However one wants to add it up, defense spending surpasses civilians projects by about 40 to 1.
Other State advocates point to the sprawling manpower that the military possesses when compared to its civilian counterparts. Just one example of this mismatch: the personal staff of the Central Command Combatant Commander (CCDR) – known for previously serving a famous Woodrow Wilson School graduate, General David Petraeus *85 *87 – is larger than many of the embassies that fall into Central Command’s theatre. When CCDRs travel, they normally arrive with a small army of assistants and personnel. When I saw Assistant Secretary of State Jeff Feltman travel to the Middle East last summer, he traveled with a single assistant.
Secretary of Defense Robert Gates has argued before Congress that the State Department needs more funding. Secretary of State Hillary Clinton, with the help of our very own former dean, Professor Anne-Marie Slaughter, has pioneered the Quadrennial Diplomacy and Development Review (QQDR). In it, Clinton and Slaughter aim to remap American diplomatic and developmental efforts, putting them on par with defensive operations, in accordance with President Obama’s “3D” approach to foreign policy: defense, diplomacy, and development. This document has many creative ideas and useful insights. But what have we seen of it so far?
More importantly, we’ve heard all of this before. So what should we do about it?
Unfortunately, the situation is bit more complex than a simple issue of funding parity or even mission creep. But first off, let’s get one thing straight: I’m a big fan of State and a staunch advocate for the need to elevate diplomacy as a tool of national security. That being said, the department needs to critically alter its mission and operations in three ways.
First, State has to get serious about assuming greater risks while conducting diplomacy. Current security measures, left largely in the hands of Regional Security Officers abroad, effectively keep diplomats trapped behind embassy walls. If a country is deemed “dangerous,” then diplomats have to jump through numerous hoops before they are allowed to leave compound – and when they do get permission, they have to be escorted by armed guards and in armored vehicles. There is something counter-intuitive about effectively marginalizing our Foreign Service Officers in the places that need the greatest diplomatic efforts. Of course relaxing these standards will come with attendant risks and dangers, but diplomacy is a dangerous endeavor. Unfortunately, the State Department’s allergy to potentially hostile situations – to which the military is largely immune – has ultimately led to its marginalization.
Second, the department needs to reassert control over peacekeeping, nation building, and wartime operations. The US’s two biggest engagements currently are in Iraq and Afghanistan. A study of peacekeeping operations and nation building efforts in both of these countries reveal DoD dominance in both developmental and diplomatic activities. Diplomats and aid workers argue that they simply don’t have the funding or the operational capacity to work in these environments. Fine, but let’s also not forget that the State Department’s mission has been steadily cut down since the Clinton presidency (without much of a fight might I add) and traditional State and USAID operations have been farmed out to DoD. In fact, one of the biggest complaints I hear from people in uniform (at all levels) is that State and USAID are just not stepping up to the plate. State and USAID will have to not only reassert themselves in these areas on a macro level, but take substantive steps to fund and train civilians in taking over from DoD.
Which brings me to my third point: the State Department needs to implement its goal of “engaging beyond the state,” as referenced in the QDDR. In Iraq and Afghanistan, that means picking up some of the work done by Special Operations in navigating and channeling tribal and ethnic currents. Elsewhere in the world, it means engaging outside of our “comfort zone,” to include more engagement with Islamist groups and opposition movements. American diplomatic efforts will always be limited if we (read: American policymakers) are content engaging with official, traditional government counterparts and Western, liberal thinkers. American diplomacy cannot be considered robust if it is not widened to take into account the full spectrum and picture of political actors operating in today’s complex international environment.
These criticisms may come off as a bit harsh on the State Department and USAID, but these are necessary issues to keep in mind if civilian efforts will ever near parity with military power. Because at the end of the day, American policymakers can’t just ask DoD to give up turf; they need to have a strong and aggressive civilian sector willing to pick up and take over.
And things won’t be easier in FY2011, as Congress’s 11th hour budget cut nearly $8 billion more from State and USAID.
While I definitely agree with you that a) we must do a better job rebalancing and b) State bears some of the responsibility for its diminished status, I have a reservation on your second point. Isn’t it a bit of a chicken and the egg problem? How can State take on these missions if it doesn’t have the financial resources to do so? | 193,762 |
\begin{document}
\chapter{Gradings on cluster algebra structure for coordinate algebras of matrices and Grassmannians} \label{chp:matrix_grassmannian}
In this chapter we consider gradings on cluster algebra structures for coordinate algebras of matrices and Grassmannians and prove that, for the infinite type cases, they contain variables of infinitely many different degrees. We also prove that all positive degrees occur in these cluster algebras.
While we only deal with the classical cluster algebra structures here, any results about gradings on their quantum analogues are also determined by the classical case. This follows from \cite[Theorem 6.1]{BZ} and \cite[Remark 3.6]{GL}.
At various points we use some fairly long mutation paths to prove certain results. Given such paths, it is not difficult to prove these results by hand, but the reader may wonder how such paths were found. So it is worth mentioning that specific code was written to help discover some of these paths, or other patterns that led to them. As noted previously, this code is available with the electronic version of the thesis.
\section{Graded cluster algebra structure for matrices}
\begin{dfn}
Let $M(k,l)$ be the set of $k \times l$ matrices. The coordinate algebra of the $k \times l$ matrices is $\mathcal{O}(M(k,l)) =\C[ x_{i,j}]$, for $1 \leq i \leq k$, $1 \leq j \leq l$, where $x_{ij}$ is the \emph{coordinate function} $M(k,l) \rightarrow \C$ given by $A \mapsto a_{ij}$.
\end{dfn}
$\mathcal{O}(M(k,l))$ has the structure of a cluster algebra, $\A \big(\mathcal{O}(M(k,l)) \big)$, as follows. For row set $I$ and column set $J$ with $|I|=|J|$, let $ {\scriptsize \left[\begin{matrix}J \\ I \end{matrix} \right] }$ denote the corresponding minor of the matrix
\[
\begin{pmatrix}
x_{11} & x_{12} & \dots & x_{1l} \\
\vdots & \vdots & \ddots & \vdots \\
x_{k1} & x_{k2} & \dots & x_{kl}
\end{pmatrix}.
\]
For $1 \leq r \leq k$ and $1 \leq s \leq l$, define the sets
\begin{eqnarray}
R(r,s)= &\{ k-r+1,k-r+2, \dots, k-r+s\} \cap \{1, \dots, k \},\\
C(r,s) = &\{l-s+1, l-s+2, \dots, l-s+r \} \cap \{1, \dots, l \}.
\end{eqnarray}
For a cluster algebra of this cardinality, it is more efficient to define the initial seed using a quiver. We show the initial quiver for $M(4,4)$ (which we will denote by $Q_{M(4,4)}$) in Figure \ref{fig:init_quiver_M(4,4)} below. This generalises to $M(k,l)$ by replacing the $4 \times 4$ grid by a $k \times l$ grid of the same form, again with only the bottom and rightmost vertices frozen (see also \cite[p.~715]{GL}).
The initial cluster is given by defining the entry of the cluster corresponding to the vertex in position $(i,j)$ to be equal to $ {\scriptsize \left[\begin{matrix} C(i,j) \\ R(i,j) \end{matrix} \right] }$. The initial grading vector is then given by assigning the vertex in the $(i,j)$ position degree $\min(i,j)$.
It is easy to check that this definition gives a valid degree quiver. Moreover, since $\mathcal{O}(M(k,l))$ is an $\N$-graded algebra (\cite[p.~716]{GL}) we have that the corresponding cluster algebra is also $\N$-graded (as, similarly, is the cluster algebra corresponding to $\mathcal{O}(Gr(k,k+l))$ which we will introduce in Section \ref{sec:corollaries_for_grassmannians}). For a justification that this initial structure gives rise to a cluster algebra that agrees with the coordinate ring of (quantum) matrices, see \cite[Corollary 12.10]{GLS} or \cite{GL}.
\section{Infinitely many degrees}
We first consider the case for when $k \geq 4$ and $l \geq 4$. Without loss of generality, we may consider the smallest such case: $M(4,4)$. We prove our result by considering a certain subquiver of $Q_{M(4,4)}$. Larger cases will always contain the same subquiver, and it is thus possible to show they also contain infinitely many degrees in exactly the same way.
The initial exchange quiver, $Q_{M(4,4)}$, is given below. (Recall that we denote frozen vertices by placing a square around them.)
\begin{figure}[H]
\begin{center}
\begin{tikzpicture}[node distance=1.8cm, auto, font=\footnotesize]
\node (AA) []{1};
\node (AB) [right of = AA]{2};
\node (AC) [right of = AB]{3};
\node (AD) [solid node, rectangle, fill=white, right of = AC]{16};
\node (BA) [below of = AA]{4};
\node (BB) [right of = BA]{5};
\node (BC) [right of = BB]{6};
\node (BD) [solid node, rectangle, fill=white, right of = BC]{15};
\node (CA) [below of = BA]{7};
\node (CB) [right of = CA]{8};
\node (CC) [right of = CB]{9};
\node (CD) [solid node, rectangle, fill=white, right of = CC]{14};
\node (DA) [solid node, rectangle, fill=white, below of = CA]{10};
\node (DB) [solid node, rectangle, fill=white, right of = DA]{11};
\node (DC) [solid node, rectangle, fill=white, right of = DB]{12};
\node (DD) [solid node, rectangle, fill=white, right of = DC]{13};
\draw[-latex] (AA) to node {} (AB);
\draw[-latex] (AB) to node {} (AC);
\draw[-latex] (AC) to node {} (AD);
\draw[-latex] (BA) to node {} (BB);
\draw[-latex] (BB) to node {} (BC);
\draw[-latex] (BC) to node {} (BD);
\draw[-latex] (CA) to node {} (CB);
\draw[-latex] (CB) to node {} (CC);
\draw[-latex] (CC) to node {} (CD);
\draw[-latex] (AA) to node {} (BA);
\draw[-latex] (BA) to node {} (CA);
\draw[-latex] (CA) to node {} (DA);
\draw[-latex] (AB) to node {} (BB);
\draw[-latex] (BB) to node {} (CB);
\draw[-latex] (CB) to node {} (DB);
\draw[-latex] (AC) to node {} (BC);
\draw[-latex] (BC) to node {} (CC);
\draw[-latex] (CC) to node {} (DC);
\draw[-latex] (BB) to node {} (AA);
\draw[-latex] (BC) to node {} (AB);
\draw[-latex] (BD) to node {} (AC);
\draw[-latex] (CB) to node {} (BA);
\draw[-latex] (CC) to node {} (BB);
\draw[-latex] (CD) to node {} (BC);
\draw[-latex] (DB) to node {} (CA);
\draw[-latex] (DC) to node {} (CB);
\draw[-latex] (DD) to node {} (CC);
\end{tikzpicture}
\end{center}
\captionof{figure}{Initial quiver for $M(4,4)$.}
\label{fig:init_quiver_M(4,4)}
\end{figure}
\begin{lem}
\label{lem:MA_4_4_inf_degrees}
The graded cluster algebra structure for $\mathcal{O}(M(4,4))$ has variables in infinitely many degrees.
\end{lem}
\begin{proof}
We will show that the mutation sequence $[4,8,2,9,5,6,1,5,2]$ transforms $Q$ into a quiver to which we can apply Proposition \ref{prop:degree_subquiver_tends_infty_relaxed}. Note that embedding the same sequence into larger initial quivers (we can embed the subquiver in Figure \ref{fig:subquiver_M(4,4)} at the bottom left of the larger quiver) allows us to immediately generalise any results to $\mathcal{O}(M(k,l))$ for $k,l \geq 4$. (Of course, using the exact same mutation sequence as above would not work in general, since the vertices of the corresponding subquiver are relabelled when embedded in a larger initial quiver, but it is clear how to adjust the mutation path to compensate for this.)
Consider the following subquiver of $Q(M(4,4))$.
\begin{figure}[H]
\begin{center}
\begin{tikzpicture}[node distance=1.6cm, auto, font=\footnotesize]
\node (A5) []{5};
\node (A4) [left of = A5]{4};
\node (A6) [right of = A5]{6};
\node (A2) [above of = A5]{2};
\node (A8) [below of = A5]{8};
\node (A1) [above of = A4]{1};
\node (A9) [right of = A8]{9};
\node (A11) [solid node, rectangle, fill=white,below of = A8]{11};
\draw[-latex] (A1) to node {} (A2);
\draw[-latex] (A4) to node {} (A5);
\draw[-latex] (A5) to node {} (A6);
\draw[-latex] (A8) to node {} (A9);
\draw[-latex] (A1) to node {} (A4);
\draw[-latex] (A2) to node {} (A5);
\draw[-latex] (A5) to node {} (A8);
\draw[-latex] (A8) to node {} (A11);
\draw[-latex] (A6) to node {} (A9);
\draw[-latex] (A5) to node {} (A1);
\draw[-latex] (A9) to node {} (A5);
\draw[-latex] (A8) to node {} (A4);
\draw[-latex] (A6) to node {} (A2);
\end{tikzpicture}
\end{center}
\captionof{figure}{Subquiver of $Q(M(4,4))$.}
\label{fig:subquiver_M(4,4)}
\end{figure}
We now perform the sequence of mutations to this subquiver, as shown in the following diagram. After a mutation, we will often no longer be interested in the mutated vertex, and if so we will remove it from our mutated subquiver.
\begin{center}
\Image{
\begin{tikzpicture}[node distance=1.3cm, auto, font=\footnotesize]
\node (A5) []{5};
\node (A4) [left of = A5]{4};
\node (A6) [right of = A5]{6};
\node (A2) [above of = A5]{2};
\node (A8) [below of = A5]{8};
\node (A1) [above of = A4]{1};
\node (A9) [right of = A8]{9};
\node (A11) [solid node, rectangle, fill=white,below of = A8]{11};
\draw[-latex] (A1) to node {} (A2);
\draw[-latex] (A4) to node {} (A5);
\draw[-latex] (A5) to node {} (A6);
\draw[-latex] (A8) to node {} (A9);
\draw[-latex] (A1) to node {} (A4);
\draw[-latex] (A2) to node {} (A5);
\draw[-latex] (A5) to node {} (A8);
\draw[-latex] (A8) to node {} (A11);
\draw[-latex] (A6) to node {} (A9);
\draw[-latex] (A5) to node {} (A1);
\draw[-latex] (A9) to node {} (A5);
\draw[-latex] (A8) to node {} (A4);
\draw[-latex] (A6) to node {} (A2);
\end{tikzpicture}
} \Arrow{2}
\Image{
\begin{tikzpicture}[node distance=1.3cm, auto, font=\footnotesize]
\node (A5) []{5};
\node (A4) [left of = A5]{4};
\node (A6) [right of = A5]{6};
\node (A2) [above of = A5]{2};
\node (A8) [below of = A5]{8};
\node (A1) [above of = A4]{1};
\node (A9) [right of = A8]{9};
\node (A11) [solid node, rectangle, fill=white,below of = A8]{11};
\draw[latex-] (A1) to node {} (A2);
\draw[-latex] (A4) to node {} (A5);
\draw[-latex] (A8) to node {} (A9);
\draw[-latex] (A1) to node {} (A4);
\draw[latex-] (A2) to node {} (A5);
\draw[-latex] (A5) to node {} (A8);
\draw[-latex] (A8) to node {} (A11);
\draw[-latex] (A6) to node {} (A9);
\draw[-latex] (A9) to node {} (A5);
\draw[-latex] (A8) to node {} (A4);
\draw[latex-] (A6) to node {} (A2);
\end{tikzpicture}
} \Arrow{5}
\Image{
\begin{tikzpicture}[node distance=1.3cm, auto, font=\footnotesize]
\node (A5) []{5};
\node (A4) [left of = A5]{4};
\node (A6) [right of = A5]{6};
\node (A2) [above of = A5]{2};
\node (A8) [below of = A5]{8};
\node (A1) [above of = A4]{1};
\node (A9) [right of = A8]{9};
\node (A11) [solid node, rectangle, fill=white,below of = A8]{11};
\draw[latex-] (A1) to node {} (A2);
\draw[latex-] (A4) to node {} (A5);
\draw[-latex] (A1) to node {} (A4);
\draw[-latex] (A2) to node {} (A5);
\draw[latex-] (A5) to node {} (A8);
\draw[-latex] (A8) to node {} (A11);
\draw[-latex] (A6) to node {} (A9);
\draw[latex-] (A9) to node {} (A5);
\draw[latex-] (A6) to node {} (A2);
\draw[latex-] (A2) to node {} (A4);
\draw[latex-] (A2) to node {} (A9);
\end{tikzpicture}
} \Arrow{1}
\Image{
\begin{tikzpicture}[node distance=1.3cm, auto, font=\footnotesize]
\node (A5) []{5};
\node (A4) [left of = A5]{4};
\node (A6) [right of = A5]{6};
\node (A2) [above of = A5]{2};
\node (A8) [below of = A5]{8};
\node (A9) [right of = A8]{9};
\node (A11) [solid node, rectangle, fill=white,below of = A8]{11};
\draw[latex-] (A4) to node {} (A5);
\draw[-latex] (A2) to node {} (A5);
\draw[latex-] (A5) to node {} (A8);
\draw[-latex] (A8) to node {} (A11);
\draw[-latex] (A6) to node {} (A9);
\draw[latex-] (A9) to node {} (A5);
\draw[latex-] (A6) to node {} (A2);
\draw[latex-] (A2) to node {} (A9);
\end{tikzpicture}
} \Arrow{6}
\Image{
\begin{tikzpicture}[node distance=1.3cm, auto, font=\footnotesize]
\node (A5) []{5};
\node (A4) [left of = A5]{4};
\node (A2) [above of = A5]{2};
\node (A8) [below of = A5]{8};
\node (A9) [right of = A8]{9};
\node (A11) [solid node, rectangle, fill=white,below of = A8]{11};
\draw[latex-] (A4) to node {} (A5);
\draw[-latex] (A2) to node {} (A5);
\draw[latex-] (A5) to node {} (A8);
\draw[-latex] (A8) to node {} (A11);
\draw[latex-] (A9) to node {} (A5);
\end{tikzpicture}
} \Arrow{5}
\Image{
\begin{tikzpicture}[node distance=1.3cm, auto, font=\footnotesize]
\node (A5) []{};
\node (A4) [left of = A5]{4};
\node (A2) [above of = A5]{2};
\node (A8) [below of = A5]{8};
\node (A9) [right of = A8]{9};
\node (A11) [solid node, rectangle, fill=white,below of = A8]{11};
\draw[-latex] (A8) to node {} (A11);
\draw[-latex] (A2) to node {} (A4);
\draw[-latex] (A8) to node {} (A4);
\draw[-latex] (A2) to node {} (A9);
\draw[-latex] (A8) to node {} (A9);
\end{tikzpicture}
}\Arrow{9}
\Image{
\begin{tikzpicture}[node distance=1.3cm, auto, font=\footnotesize]
\node (A5) []{};
\node (A4) [left of = A5]{4};
\node (A2) [above of = A5]{2};
\node (A8) [below of = A5]{8};
\node (A9) [right of = A8]{9};
\node (A11) [solid node, rectangle, fill=white,below of = A8]{11};
\draw[-latex] (A8) to node {} (A11);
\draw[-latex] (A2) to node {} (A4);
\draw[-latex] (A8) to node {} (A4);
\draw[latex-] (A2) to node {} (A9);
\draw[latex-] (A8) to node {} (A9);
\end{tikzpicture}
}\Arrow{2}
\Image{
\begin{tikzpicture}[node distance=1.3cm, auto, font=\footnotesize]
\node (A5) []{};
\node (A4) [left of = A5]{4};
\node (A8) [below of = A5]{8};
\node (A9) [right of = A8]{9};
\node (A11) [solid node, rectangle, fill=white,below of = A8]{11};
\draw[-latex] (A8) to node {} (A11);
\draw[-latex] (A8) to node {} (A4);
\draw[latex-] (A8) to node {} (A9);
\draw[latex-] (A4) to node {} (A9);
\end{tikzpicture}
}\Arrow{8}
\Image{
\begin{tikzpicture}[node distance=1.6cm, auto, font=\footnotesize]
\node (A9) []{9};
\node (A11) [solid node, rectangle, fill=white,below left of = A9]{11};
\node (A4) [below right of = A9]{4};
\draw[-latex] (A9) to node [] {2} (A4);
\draw[-latex] (A9) to node [swap] {} (A11);
\end{tikzpicture}
}\Arrow{4}
\Image{
\begin{tikzpicture}[node distance=1.6cm, auto, font=\footnotesize]
\node (A9) []{9};
\node (A11) [solid node, rectangle, fill=white,below left of = A9]{11};
\node (A4) [below right of = A9]{4};
\draw[latex-] (A9) to node [] {2} (A4);
\draw[-latex] (A9) to node [swap] {} (A11);
\end{tikzpicture}.}
\end{center}
Direct computation shows that this last subquiver corresponds to the degree subquiver
\begin{center}
\Image{
\begin{tikzpicture}[node distance=1.6cm, auto, font=\footnotesize]
\node (A7) []{$(5)_9$};
\node (A1) [solid node, rectangle, fill=white,below left of = A7]{$(2)_{11}$};
\node (A14) [below right of = A7]{$(9)_4$};
\draw[latex-] (A1) to node {} (A7);
\draw[latex-] (A7) to node [] {2} (A14);
\end{tikzpicture}}.
\end{center}
Since $9 \geq 5$, we can now apply Proposition \ref{prop:degree_subquiver_tends_infty_relaxed}, noting that the frozen vertex $11$ is not part of the mutation path stipulated by the proposition.
\end{proof}
\begin{cor}
For $k\geq 4$ and $l \geq 4$, the graded cluster algebra structure for $M(k,l)$ has variables in infinitely many degrees.
\end{cor}
The other infinite type cases that are not covered by the above are $M(3,6)$ and $M(6,3)$. Note that $Q_{M(3,6)} \cong Q_{M(6,3)}$, so we only need to consider $Q_{M(3,6)}$. For this, we can start with the subquiver
\begin{center}
\Image{
\begin{tikzpicture}[node distance=1.6cm, auto, font=\footnotesize]
\node (A1) []{1};
\node (A2) [right of = A1]{2};
\node (A3) [right of = A2]{3};
\node (A4) [right of = A3]{4};
\node (A6) [below of = A1]{6};
\node (A7) [right of = A6]{7};
\node (A9) [below of = A4]{9};
\node (A10) [right of = A9]{10};
\node (A11) [solid node, rectangle, fill=white,below of = A6]{11};
\draw[-latex] (A1) to node {} (A2);
\draw[-latex] (A2) to node {} (A3);
\draw[-latex] (A3) to node {} (A4);
\draw[-latex] (A6) to node {} (A7);
\draw[-latex] (A9) to node {} (A10);
\draw[-latex] (A6) to node {} (A11);
\draw[-latex] (A7) to node {} (A1);
\draw[-latex] (A9) to node {} (A3);
\draw[-latex] (A1) to node {} (A6);
\draw[-latex] (A4) to node {} (A9);
\draw[-latex] (A2) to node {} (A7);
\end{tikzpicture}}
\end{center}
and perform the mutation path $[2,4,3,7,10,6,1,2,7,6,9,3]$ to obtain a quiver containing the subquiver
\Image{
\begin{tikzpicture}[node distance=1.6cm, auto, font=\footnotesize]
\node (A7) []{6};
\node (A1) [solid node, rectangle, fill=white,below left of = A7]{11};
\node (A14) [below right of = A7]{4};
\draw[latex-] (A1) to node {} (A7);
\draw[latex-] (A7) to node [] {2} (A14);
\end{tikzpicture}}
(this time we do not write down all the intermediate computations).
The corresponding degree quiver is
\Image{
\begin{tikzpicture}[node distance=1.6cm, auto, font=\footnotesize]
\node (A7) []{$(4)_6$};
\node (A1) [solid node, rectangle, fill=white,below left of = A7]{$(1)_{11}$};
\node (A14) [below right of = A7]{$(10)_4$};
\draw[latex-] (A1) to node {} (A7);
\draw[latex-] (A7) to node [] {2} (A14);
\end{tikzpicture}},
which satisfies Proposition \ref{prop:degree_subquiver_tends_infty_relaxed}. We again note that vertex $11$ is not part of the stipulated mutation path.
We have now established the following.
\begin{prop}
If the graded cluster algebra associated to $\mathcal{O}(M(k,l))$ is of infinite type, it has cluster variables of infinitely many different degrees.
\end{prop}
\section{Occurring degrees}
A natural question to ask is whether variables of all positive degrees exist in the graded cluster algebra associated to $\mathcal{O}(M(k,l))$. We will prove that this is indeed the case for $\mathcal{O}(M(4,4))$ and $\mathcal{O}(M(3,6))$, which, as before, also proves the result for all larger cases and therefore all infinite type cases.
\begin{lem}
\label{lem:M_4_4_all_degrees_occur}
$\A \big( \mathcal{O}(M(4,4)) \big)$ has cluster variables of each degree in $\N$.
\end{lem}
\begin{proof}
We show this by writing down certain mutation sequences that result in degree subquivers of the form of \pref{form:linear_increasing_degree_quiver}. It turns out in this case that these will yield sequences of degrees increasing by $4$ at each mutation.
The initial quiver mutated by the path $[6,4,8,9,1,4,2,6]$ has the subquiver and degree subquiver
\striL{$4$}{$6$} and \striL{$(8)$}{$(12)$}, so we have variables in all degrees of the form $4k$ for $k \geq 2$ by Lemma \ref{lem:linear_increasing_degree_quiver}.
The path $[4,6,8,9,1,4,2,6]$ gives the subquiver and degree subquiver \striR{$4$}{$6$} and \striR{$(10)$}{$(6)$}, so we have variables in all degrees of the form $4k+2$ for $k \geq 1$.
Next, $[1,2,4,8,6,9,1]$ gives the subquiver and degree subquiver \striR{$1$}{$9$} and \striR{$(9)$}{$(5)$}, which leads to variables in all degrees of the form $4k+1$ for $k \geq 1$.
Finally, $[1,9,1,8,6,1,2,6,4,8]$ gives
\striR{$1$}{$9$} and \striR{$(11)$}{$(7)$} which leads to variables in all degrees of the form $4k+3$ for $k \geq 1$.
The initial cluster contains variables of degrees $1,2$ and $3$, and it is easy to find a mutation path that yields a variable of degree $4$. (For example, $\deg [ 6 ]=4$.) This covers all positive degrees.
\end{proof}
\begin{rmk}
In lifting the result of Lemma \ref{lem:M_4_4_all_degrees_occur} to larger cases, we should check that the subquiver (corresponding to $Q_{M(4,4)}$) that we deal with will not interact with the rest of the quiver it is embedded in. In fact it is easy to see that if mutation of $Q_{M(k,l)}$ is restricted to the vertices at $(i,j)$ within the rectangle $1\leq i \leq k$, $1 \leq j \leq l$, then there can be no arrow created between one of these vertices and a vertex in position $(i',j')$ where $i' \geq k+2$ or $j' \geq l+2$. So since vertices we mutate at in $Q_{M(4,4)}$ satisfy $1 \leq i,j \leq 3$ (and since this time, rather than embedding the pattern in the bottom left, we will keep vertices corresponding to the above mutation paths where they are in any larger initial quivers), we do not need to worry about vertices outside the leftmost $4 \times 4$ square in larger cases.
\end{rmk}
\begin{lem}
\label{lem:M_3_6_all_degrees_occur}
$\A \big( \mathcal{O}(M(3,6)) \big)$ has cluster variables of each degree in $\N$.
\end{lem}
\begin{proof}
In this case, we find sequences of degrees increasing by $6$ at each mutation. We might expect that the sequences in this case will increase by a larger amount than in $\mathcal{O}(M(4,4))$ because having so few vertices in which to mutate means achieving a double arrow is not possible in as few mutations as previously, which in turn means degrees have longer to grow along the way. The proof is essentially the same as for $\mathcal{O}(M(4,4))$ so we will summarise it in Table \ref{tab:M_3_6_paths_giving_all_degrees} below.
This leaves degrees $1,2,3,4,6$ and $7$ to consider, but all these degrees are contained in $\cl [ 10, 12, 4, 7, 10, 5, 8, 1, 4, 9, 4 ]$, for example.
\begin{center}
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{ | c c c c |}
\hline
Path & Quiver & Degree Quiver & Degrees \\ \hline
$[9,7,5,2,10,9,5,8,1,4,9]$ & \striL{$7$}{$9$} & \striL{$(12)$}{$(18)$} & $6k$, $k \geq 2$ \\
$[10,2,4,7,10,5,8,1,4,9,4]$ & \striR{$10$}{$2$} & \striR{$(19)$}{$(13)$} & $6k+1$, $k \geq 2$ \\
$[7,9,5,2,10,9,5,8,1,4,9]$ & \striR{$7$}{$9$} & \striR{$(14)$}{$(8)$} & $6k+2$, $k \geq 1$ \\
$[8,3,5,2,6,9,8,9,3,4,7]$ &\striR{$8$}{$3$} & \striR{$(15)$}{$(9)$} & $6k+3$, $k \geq 1$ \\
$[10,9,2,7,1,5,9,10,4,8,9]$ &\striR{$10$}{$9$} & \striR{$(16)$}{$(10)$} & $6k+4$, $k \geq 1$ \\
$[10,7,9,7,2,9,8,4,9,1,5]$ &\striR{$10$}{$7$} & \striR{$(11)$}{$(5)$} & $6k+5$, $k \geq 0$ \\ \hline
\end{tabular}
\captionof{table}{\label{tab:M_3_6_paths_giving_all_degrees} Mutation paths leading to all degrees for $M(3,6)$}
\end{center}
\end{proof}
\begin{cor}
If the graded cluster algebra associated to $\mathcal{O}(M(k,l))$ is of infinite type, it has cluster variables of each degree in $\N$.
\end{cor}
\section{Corollaries for Grassmannians} \label{sec:corollaries_for_grassmannians}
\begin{dfn}
The \emph{Grasmmannian} $Gr(k,k+l)$ is the set of $k$-dimensional subspaces of a $(k+l)$-dimensional vector space $V$ over $\C$.
After fixing a basis for $V$, such a subspace can be described by a $k \times (k+l)$ matrix whose rows are linearly independent vectors forming a basis of the subspace. Let $I$ be a subset of $\{1,\dots,(k+l)\}$ such that $| I |=k$. The \emph{Pl{\"u}cker coordinate} $x_I$ is the function that maps a $k \times (k+l)$ matrix to its minor indexed by $I$.
\sloppy The \emph{coordinate ring} $\mathcal{O}(Gr(k,k+l))$ is isomorphic to the subalgebra of ${\mathcal{O}(M(k,k+l))}$ generated by the Pl{\"u}cker coordinates.
\end{dfn}
The Grasmannian coordinate ring $\mathcal{O}(Gr(k,k+l))$ has the structure of a graded cluster algebra as follows. For the initial exchange quiver, we take the same quiver as for $\mathcal{O}(M(k,l))$, but add a frozen vertex with a single arrow to the vertex at position $(1,1)$. We then assign all vertices degree $1$. By results of \cite{GL}, this quiver gives rise to a cluster algebra for $\mathcal{O}(Gr(k,k+l))$. While an explicit expression for the initial cluster variables is not given in \cite{GL}, one can be obtained by tracing through the construction therein. References \cite{GSV} and \cite{Sc} give more explicit initial clusters but with different quivers. Since we do not need to know the cluster variables explicitly, we do not concern ourselves unduly with this.
Again, it is easy to check that the degree quiver is valid. For further background on how the associated cluster algebra has the desired structure, see \cite{Sc} (for the classical case), \cite{GL} (for the quantum case) or \cite{GSV}.
In terms of proving the existence of infinitely many degrees, the corresponding results for $\mathcal{O}(Gr(k,k+l))$ will immediately follow from the matrix algebra case. Since the grading on $\mathcal{O}(Gr(k,k+l))$ is an $\N$-grading, and since the initial exchange quiver is the same as that for $\mathcal{O}(M(k,l))$ but for an added frozen vertex, the same mutation sequences as in the matrix case will yield quivers which have the same subquivers as above, to which we can again apply Proposition \ref{prop:degree_subquiver_tends_infty_relaxed}.
Thus we have:
\begin{prop}
If the graded cluster algebra associated to $\mathcal{O}(Gr(k,k+l))$ is of infinite type, it has cluster variables of infinitely many different degrees.
\end{prop}
We also have variables in each degree and prove this in a very similar way as for the matrix case.
\begin{lem}
$\A \big( \mathcal{O}(Gr(4,8)) \big)$ has cluster variables of each degree in $\N$.
\end{lem}
\begin{proof}
In this case we find paths that give degree sequences increasing by $2$ with each mutation.
Applying $[9,1,2,4,8,6,9,1]$ gives the subquiver and degree subquiver
\striR{$9$}{$1$} and \striR{$(6)$}{$(4)$}, so by Lemma \ref{lem:linear_increasing_degree_quiver} we have variables in all degrees of the form $2k$ for $k \geq 2$.
The path $[4,6,8,9,1,4,2,6]$ gives the subquiver and degree subquiver
\striR{$4$}{$6$} and \striR{$(5)$}{$(3)$}, so we have variables in all degrees of the form $2k+1$ for $k \geq 1$.
This only leaves degree $2$, which is easy to find. For example, $\var[6]=2$.
\end{proof}
\begin{lem}
$\A \big( \mathcal{O}(Gr(3,9)) \big)$ has cluster variables of each degree in $\N$.
\end{lem}
\begin{proof}
As in the previous section, the degree sequences for this case increase by a larger amount than for the matrix case. We summarise three paths which work in Table \ref{tab:Gr_3_9_paths_giving_all_degrees} below.
\begin{center}
\begin{tabular}{ | c c c c |}
\hline
Path & Quiver & Degree Quiver & Degrees \\ \hline
$[10,7,9,7,2,9,8,4,9,1,5]$ &\striR{$10$}{$7$} & \striR{$(6)$}{$(3)$} & $3k$, $k \geq 1$ \\
$[7,4,3,1,4,9,10,6,2,9,3]$ & \striL{$4$}{$7$} & \striL{$(4)$}{$(7)$} & $3k+1$, $k \geq 1$ \\
$[10,9,2,7,1,5,9,10,4,8,9]$ &\striR{$10$}{$9$} & \striR{$(8)$}{$(5)$} & $3k+2$, $k \geq 1$ \\ \hline
\end{tabular}
\captionof{table}{ \label{tab:Gr_3_9_paths_giving_all_degrees} Mutation paths leading to all degrees for $Gr(3,9)$}
\end{center}
This leaves degrees $1$ and $2$ which are found in $\cl [7]$, for example.
\end{proof}
\begin{cor}
If the graded cluster algebra associated to $\mathcal{O}(Gr(k,k+l))$ is of infinite type, it has cluster variables of each degree in $\N$.
\end{cor}
\biblio
\end{document} | 31,033 |
First major named snowstorm of the year, Juno, hit my area.
Predictions:
Like there was only going to be one this year?
Twitter named it: #Snowmageddon2015 #Blizzard2015 #Juno
The media went wild. Because, of course it did.
"Juno?" ('cho know), says the weather folks: This could be a big one.
To be fair, the totals were anticipated in FEET, not inches.
Preparations:
I dragged out my lighting and emergency flashlights, radios, clocks, and charged up phones, batteries, etc.
And shopped with moderation.
Mostly worried about the water and Diet Coke.
Not pictured: water, Diet Coke, crackers, pb & jelly, pizzas, chocolate
Hadn't stressed out on this, until the folks started to.
Observations:
The office said to use best case judgement on travel to work on Monday, so I decided not to get possibly stranded (and honestly harangued by the folks) and just worked from home.
Monday: Coming down
Pretty as a picture
So glad I don't have to deal with it.
Monday: At Dusk
That's not a filter, that's Mother Nature.
That's not a filter, that's Mother Nature.
Just from watching from the window and news, glad not to have been cold,
whipped by snow and wind, or sloshing around.
Tuesday: 12:30a.m.
Why is someone riding their bicycle down the middle of the street?
Idiots
Tuesday morning:
My first reaction was: That's it? What? Well, okay.
The office is closed, so ... I'm going to laze about.
Really? This is it? Where are my DRIFTS?
Still glad I don't have to deal with it.
Enjoying two days of staying home, working from home, in my sweats.
I kind of wish it could be three.
The weather people seem to be a bit annoyed and disappointed that NYC didn't get the big storm.
The transit, sanitation, and government people are justifiably peeved that they are being asked to justify the closing of the roads, transit ways, schools, etc.
Better prepared than under-prepared.
Because, inevitably ...
There will be "the big one" at some point.
Maybe time to buy more Diet Coke. | 377,876 |
TITLE: Statement from Enderton's A Mathematical Introduction to Logic, Section 1.3.
QUESTION [1 upvotes]: Section 1.3, A Parsing Algorithm, presents an algorithm that given a wff, will produce its "ancestral tree". That is, a tree whose root is the complete wff, each leave is a sentence symbol, and each other node is an application of a connective symbol $\neg,\wedge,\vee,\to$, or $\leftrightarrow$ to one or two nodes (its children). This is how the relevant parts go:
If all minimal vertices (the ones at the bottom) have sentence
symbols, then the procedure is completed. (The given expression is
indeed a wff, and we have constructed its tree.) Otherwise, select a
minimal vertex having an expression that is not a sentence symbol. We
examine that expression.
The first symbol must be $($. If the second symbol is the negation symbol, jump to step 4. Otherwise, go on to step 3.
Scan the expression from the left until first reaching $(\alpha$, where
$\alpha$ is a nonempty expression having a balance between left and right parentheses. Then $\alpha$ is the first of the two constituents. The next symbol must be $\wedge$, $\vee$, $\to$, or $\leftrightarrow$. This is the principal connective. The remainder of the expression, $\beta)$, must consist of an expression $\beta$ and a right parenthesis. We extend the tree by creating two new vertices below the present one, with $\alpha$ as the expression at the “left child” vertex, and $\beta$ as the expression at the “right child” vertex. Return to step 1.
...
Enderton then gives "comments in support of the correctness" of the algorithm. In particular, he says the following:
Secondly, the choices made by the procedure could not have been made differently. For example, in step 3 we arrive at an expression $\alpha$. We could not use less than $\alpha$ for a constituent, because it would not have a balance between left and right parentheses (as required by Lemma 13A). We could not use more than $\alpha$, because that would have the proper initial segment $\alpha$ that was balanced (violating Lemma 13B). Thus $\alpha$ is forced upon us. And then the choice of the principal connective is inevitable. We conclude that this algorithm constructs the only possible tree for the given expression.
The first assertion about the impossibility of taking less than $\alpha$ is clear, although I reason that Lemma 13B is the one that guarantees it (any proper initial sequence of a wff has more left parenthesis than right parenthesis).
The second assertion is where the real confusion comes from. "We could not use more than $\alpha$, because that would have the proper initial segment $\alpha$ that was balanced (violating Lemma 13B)." What does "taking more than $\alpha$" have anything to do with a proper initial sequence of $\alpha$? Is he just recycling symbols in a confusing way? I can see intuitively why the algorithm works, but I don't see how this admittedly informal argument supports it.
REPLY [0 votes]: After a comment from Andreas Blass, it makes a lot of sense. I made the mistake of forgetting everything that was true prior to a claim and failed to see the connection as a result. An alternative, more explicit reading of that paragraph could be the following.
Secondly, the choices made by the procedure could not have been made differently. At the start of step 3, the expression must be of the form $(\gamma\square\delta)$ if it is to be a wff, where $\gamma$ and $\delta$ are wffs themselves (called the left and right constituents, respectively). Hence we could not use less than $\alpha$ (i.e. the first balanced expression) for the left constituent because Lemma 13A requires it. Furthermore, $\alpha$ must make up the entirety of the left constituent (in other words, $\alpha = \gamma$) or it would be a proper initial segment of $\gamma$ that was balanced, contradicting Lemma 13B. Thus $\alpha$ is forced upon us. And then the choice of the principal connective is inevitable. We conclude that this algorithm constructs the only possible tree for the given expression. | 136,944 |
TITLE: Where does the factor of half appear from in the Klein-Gordon Lagrangian for a real scalar field?
QUESTION [1 upvotes]: The lagangian density of a scalar field or a Klein-Gordon field has the form of
$$\begin{align}
\mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2 \phi^2.
\end{align}$$
Why is there a factor of half appearing in the lagrangian? Being a constant, when entered into the Euler-lagrange equation, should yield the same equation when compared to a lagrangian without the factor of half, so what is the reason for having that factor in the equation?
REPLY [1 votes]: This normalization is convenient when you work in Fourier space, since it implies that
$$ \langle \phi(p) \phi(q) \rangle = \frac{1}{p^2 + m^2} \delta(p+q)$$
which is easy to remember. You normally derive this using path integrals/functional integration, and in that way the factor of $1/2$ comes from a Gaussian integral -- where ultimately it's more natural to have $\int dx \exp(-x^2/2)$ than $\int dx \exp(-x^2)$. Of course, it's possible to use an arbitrary normalization, and you could write
$$ \mathcal{L} = A (\partial \phi)^2 + B \phi^2 $$
for arbitrary constants $A,B$. In perturbation theory, the Feynman rules would be a bit more complicated, and all observables would depend on the ratio $B/A$. The usual normalization is nice because the Lagrangian parameter $m^2$ is actually the mass of the one-particle state in the free theory. | 18,565 |
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ROCKFORD (WIFR) – One candidate for governor has received a vote of confidence from a former state representative.
Former State Representative of the 89th House District, Jim Sacia, announced his endorsement of Illinois State Treasurer Dan Rutherford. The Republican candidate for Governor is proud to have the support of Sacia who said it was an easy decision for who to line up behind.
“He does what is right and that’s a rare commodity in any politician and I don’t look at Dan Rutherford as a politician, I look at him as a fellow citizen that wants only the best for all of us in Illinois.
Rutherford is running with Steve Kim as his Lieutenant Governor asnd the primary election is set for March 18. | 172,870 |
Early Reader: The Ant's Pants Paperback / softback
by Lydia Ripper
Part of the Early Reader series
Description.
Information
- Format: Paperback / softback
- Pages: 64 pages
- Publisher: Hachette Children's Group
- Publication Date: 09/08/2018
- Category: Educational: English language: readers & reading schemes
- ISBN: 9781510101838
£5.11 | 288,225 |
{\bf Problem.} Find the domain of the function
\[g(x) = \frac{x^3 + 11x - 2}{|x - 3| + |x + 1|}.\]
{\bf Level.} Level 2
{\bf Type.} Intermediate Algebra
{\bf Solution.} The expression is defined as long as the denominator $|x - 3| + |x + 1|$ is not equal to 0. Since the absolute value function is always non-negative, the only way that $|x - 3| + |x + 1| = 0$ is if both $|x - 3|$ and $|x + 1|$ are equal to 0. In turn, this occurs if and only if $x = 3$ and $x = -1$. Clearly, $x$ cannot be both 3 and $-1$ at the same time, so the denominator is always non-zero. Therefore, the domain of the function is $\boxed{(-\infty,\infty)}.$ | 130,385 |
Most Gold Cards are valid for up to five years. If your Gold Card is due to expire within the next 30 days then you should renew your card.
If your Gold Card was issued because of a disability, or you've changed your name or address since you were last issued with a Gold Card please contact your district, borough or Derby City Council for advice on when and how to renew.
If you've moved out of Derbyshire or Derby city altogether you will no longer be entitled to a Derbyshire Gold Card. You'll still qualify for the English National Concessionary Travel Scheme pass in your new local authority area.
Browser does not support script. | 64,495 |
Elegant WordPress Themes provide many benefits to websites. By choosing an elegant theme for your blog or website, you make it easier for the visitors to quickly navigate through your pages. It also makes it easier to create professional looking sites that will stick out from others and bring a range lots of different themes available for the best of both worlds as well as the typical Joe-Bob web developer.
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Chatsworth, CA 11/16/06 - Noble House CD Carrying "H-3000" New 220 DVD/CD Printer Autoloader
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More information about Noble House, its history and products is available on the Internet at or by calling 1-800-260-5053 (USA and Canada). Outside of the USA and Canada, call (818)709-5053. E-mail to [email protected]. | 263,467 |
AFC Fylde manager Dave Challinor says he doesn’t anticipate having much time to rest this summer with preparations already underway for next season.’t miss the latest football transfer news and gossip from leagues in England, Scotland and Europe with footballwire.co.uk
Challinor guided the Coasters to the play-offs this season in what was their debut campaign in the National League, losing to Boreham Wood at the elimination stage.
Fylde will want to go one better next season in their bid to become an EFL club – and to achieve that the club must get their pre-season preparations spot on according to Challinor.
When asked if he was looking forward to a summer break after a long, hard season, Challinor told The Gazette: “It’s not so much a break because the next few weeks are massively important.
“We will have a break where we go away for a week but do I expect to put my phone down? Absolutely not.
“Then you’re straight back at it and it’s a quick turnaround.
“I think we’ll be back on June 25 and you’ve got to put a pre-season schedule together before the season kicks off on August 4.
“I suppose at some point you’d like to recharge your batteries but having been part of this process for eight or nine years now, I know it can be just as busy as the normal season is.
“We want this preparation to set us up for what we hope will be a successful season next year.”
Challinor also revealed the Coasters may have to play the majority of their pre-season friendlies away from home.
That’s because the Mill Farm pitch, which has caused a number of issues for the club this season, is due to be relaid having been ripped up on Thursday.
Challinor is still hopeful of bringing one or two high-profile clubs to Mill Farm, but admitted the current situation is not ideal.
“We’ve had issues with the pitch so we’re having that redone, so the work is continuing,” he said.
“Whether we get any home games, we’ll have to wait and see.
“At the minute we’ll have to put away games in and for home games it might be a bit different this year in terms of playing games against clubs higher up as behind closed doors games.
“That’s still up in the air but it’s certainly not ideal. But we will see where we are.
“Hopefully the pitch gets sorted and we can arrange a couple of high-profile games here at Mill Farm so we can show off a few players.” | 88,812 |
The Quarter After
The Quarter After
Abre iTunes para escuchar un fragmento, comprar y descargar música.
Reseña de álbum
Between them, singing brothers Dominic Campanella and Rob Campanella have been at least an adjunct member of just about every band in the axis between the Brian Jonestown Massacre, Beachwood Sparks and the Tyde. Unsurprisingly, the self-titled debut by their own band works that circa-'67 L.A. sound, with heavy echoes of the pre-David Axelrod the Electric Prunes, Buffalo Springfield, and various other half-forgotten exemplars of the sound, minimizing the country-rock inflections of Beachwood Sparks (only notable on the Neil Young-like "Mirror to You") or much of the slightly unhinged experimentalism of the the Brian Jonestown Massacre. For a little less than half of the album, the brothers, along with bassist David Koenig and drummer Nelson Bragg, do a pretty good pastiche of Sunset Strip psychedelia, kicking up a particularly lysergic head of steam on the self-explanatory "One Trip Later." The problem is that the other half of the album, nearly a full thirty minutes, consists of three endless acid-guitar jams that don't justify their overextended length; the most frustrating one is the nine-minute "Taken," which cooks up a good old-fashioned freight train momentum and then blows it on a flaccid and seemingly endless solo. At about four-and-a-half minutes, it would be the best song on the album, but at nine-minutes-and-16-seconds, it's a prime candidate for the forward skip button. With an editor and a bit more emphasis on Love than the Quicksilver Messenger Service, the Quarter After may really have something.
Top álbumes y canciones de The Quarter After
Otros usuarios también compraron
- USD 9.99
- Géneros: Rock, Música, Psicodélica, Alternativa, College rock, Pop
- Publicado: 12/07/2005
- ℗ 2005 Bird Song Recordings | 414,408 |
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Erie MetroParks returns with a walking program this year, with a simplified schedule of events to make it easier for walkers to stick with it.
The program resumes with a kickoff event 6:30 p.m. April 9 at Osborn Park, then will continue on alternate Mondays through August.
Last year's Look Who's Walking" events were held on different days and different times of day. The idea was to provide different times so everyone could attend some events, but all of the changes confused people, said Lois TerVeen, supervisor of program services for Erie MetroParks.
This year, every walk will be at 6:30 p.m., every other Monday, a regular time, which should make it easier for local walkers to remember to show up, TerVeen said.
"We want people to come out and walk," she said. "We want people to come out and exercise. We hope this will help them get started and get in the habit."
All walks will last for about one hour.
"We hope they go out on their own on the Monday in between," she said.
TerVeen explained that the parks department's walking program runs through the spring and summer. The Friends of Erie MetroParks then take over with a fall and early winter walking program, which ends in December.
Winter months are omitted because of the weather.
The walking program is free and everyone is welcome. The parks department hopes to distribute free pedometers to participants, TerVeen said.
Schedule of events (all at 6:30 p.m.)
. | 329,818 |
\begin{document}
\title[A curious differential calculus on the quantum disc and cones]
{A curious differential calculus\\ on the quantum disc and cones}
\author{Tomasz Brzezi\'nski}
\address{
Department of Mathematics, Swansea University,
Swansea SA2 8PP, U.K.\ \newline
Department of Mathematics, University of Bia{\l}ystok, K.\ Cio{\l}kowskiego 1M,
15-245 Bia\-{\l}ys\-tok, Poland. E-mail: {[email protected]} }
\author{Ludwik D\k{a}browski}
\address{SISSA, Via Bonomea 265, 34136 Trieste, Italy. E-mail: [email protected]}
\subjclass{Primary 58B32}
\keywords{non-commutative geometry; differential forms; integral forms}
\date{October 2016}
\begin{abstract}
A non-classical differential calculus on the quantum disc and cones is constructed and the associated integral is calculated.
\end{abstract}
\maketitle
\section{Introduction}
The aim of this note is to present a two-dimensional differential calculus on the quantum disc algebra, which has no counterpart in the classical limit, but admits a well-defined (albeit different from the one in \cite{BegMaj:spe}) integral, and restricts properly to the quantum cone algebras. In this way the results of \cite{Brz:dif} are extended to other classes of non-commutative surfaces and to higher forms. The presented calculus is associated to an orthogonal pair of skew-derivations, which arise as a particular example of skew-derivations on generalized Weyl algebras constructed recently in \cite{AlmBrz:ske}. It is also a fundamental ingredient in the construction of the Dirac operator on the quantum cone \cite{BrzDab:spe} that admits a twisted real structure in the sense of \cite{BrzCic:twi}.
The reader unfamiliar with non-commutative differential geometry notions is referred to \cite{Brz:abs}.
\section{A differential calculus on the quantum disc}\label{sec.diff}
Let $0<q<1$. The coordinate algebra of the quantum disc, or the quantum disc algebra $\Oo(D_q)$ \cite{KliLes:two} is a complex $*$-algebra generated by $z$ subject to
\begin{equation}\label{disc}
z^*z - q^2zz^* = 1-q^2.
\end{equation}
To describe the algebraic contents of $\Oo(D_q)$ it is convenient to introduce a self-adjoint element $x = 1-zz^*$, which $q^2$-commutes with the generator of $\Oo(D_q)$, $xz = q^2zx$. A linear basis of $\Oo(D_q)$ is given by monomials
$x^kz^l, x^k{z^*}^{l}$.
We
view $\Oo(D_q)$ as a $\ZZ$-graded algebra, setting $\deg(z)=1$, $\deg(z^*)=-1$. Associated with this grading is the degree-counting automorphism
$\sigma:\Oo(D_q) \to \Oo(D_q)$, defined on homogeneous $a\in \Oo(D_q)$ by $\sigma(a) = q^{2\deg(a)} a$. As explained in \cite{AlmBrz:ske} there is an orthogonal pair of skew-derivations $\partial,\bar\partial: \Oo(D_q)\to \Oo(D_q)$ twisted by $\sigma$ and given on the generators of $\Oo(D_q)$ by
\begin{equation}\label{partial}
\partial(z) = z^*, \quad \partial(z^*) = 0, \qquad \bar\partial(z) = 0, \quad \bar\partial(z^*) = q^2z,
\end{equation}
and extended to the whole of $\Oo(D_q)$ by the (right) $\sigma$-twisted Leibniz rule. Therefore, there is also a corresponding first-order differential calculus $\Omega^1 (D_q)$ on $\Oo(D_q)$, defined as follows.
As a left $\Oo(D_q)$-module, $\Omega^1 (D_q)$ is freely generated by one forms $\omega, \bar\omega$. The right $\Oo(D_q)$-module structure and the differential $d :\Oo(D_q) \to \Omega^1 (D_q)$ are defined by
\begin{equation}\label{diff}
\omega a = \sigma(a) \omega, \quad \bar\omega a = \sigma(a) \bar\omega, \qquad d(a) = \partial(a)\omega + \bar\partial(a)\bar\omega.
\end{equation}
In particular,
\begin{equation}\label{dz}
dz = z^*\omega = q^2\omega z^*, \qquad dz^*= q^2z \bar\omega = \bar\omega z,
\end{equation}
and so, by the commutation rules \eqref{diff},
\begin{equation}\label{omegadz}
\omega = \frac{q^{-2}}{1-q^2}\left(dz z - q^4zdz\right), \qquad \bar\omega = \frac{q^{-2}}{1-q^2}\left( z^*dz^* - q^2dz^*z^*\right).
\end{equation}
Hence $\Omega^1 (D_q) = \{\sum_i a_i db_i\; |\; a_i,b_i\in \Oo(D_q)\}$, i.e.\ $(\Omega^1 (D_q), d)$ is truly a first-order differential calculus not just a degree-one part of a differential graded algebra. The appearance of $q^2-1$ in the denominators in \eqref{omegadz} indicates that this calculus has no classical (i.e.\ $q=1$) counterpart.
The first-order calculus $(\Omega^1 (D_q), d)$ is a $*$-calculus in the sense that the $*$-structure extends to the bimodule $\Omega^1 (D_q)$ so that $(a \nu b)^* = b^*\nu^* a^*$ and $(da)^* =d(a^*)$, for all $a,b\in \Oo(D_q)$ and $\nu \in \Omega^1 (D_q)$, provided $\omega^* = \bar\omega$ (this choice of the $*$-structure justifies the appearance of $q^2$ in the definition of $\bar\partial$ in equation \eqref{partial}). From now on we view $(\Omega^1 (D_q), d)$ as a $*$-calculus, which allows us to reduce by half the number of necessary checks.
Next we aim to show that the module of 2-forms $\Omega^2 (D_q)$ obtained by the universal extension of $\Omega^1 (D_q)$ is generated by the anti-self-adjoint 2-form\footnote{One should remember that the $*$-conjugation takes into account the parity of the forms; see \cite{Wor:dif}.}
\begin{equation}\label{volume}
\mathsf{v} = \frac{q^{-6}}{q^2-1}(\omega^*\omega + q^8\omega\omega^*), \qquad \mathsf{v}^* = - \mathsf{v}
\end{equation}
and to describe the structure of $\Omega^2 (D_q)$. By \eqref{diff}, for all $a\in \Oo(D_q)$,
\begin{equation}\label{va}
\mathsf{v} a = \sigma^2(a) \mathsf{v}.
\end{equation}
Combining commutation rules \eqref{diff} with the relations \eqref{dz} we obtain
\begin{equation}\label{zdz}
z^* dz = q^2dzz^*, \qquad dzz - q^4 zdz = q^2(1-q^2)\omega,
\end{equation}
and their $*$-conjugates. The differentiation of the first of equations \eqref{zdz} together with \eqref{diff} and \eqref{disc} yield
\begin{equation}\label{omom}
\omega\omega^* =(1-x)\mathsf{v}, \qquad \omega^*\omega = q^6(q^2x -1)\mathsf{v},
\end{equation}
which means that $\omega\omega^*$ and $\omega^*\omega$ are in the module generated by $\mathsf{v}$. Next, by differentiating $\omega z^* = q^{-2}z^*\omega$ and $\omega z = q^2z\omega$ and using \eqref{dz} and \eqref{diff} one obtains
\begin{equation}\label{domegaz}
d\omega z^* = q^{-2} z^*d\omega + z(\omega^*\omega+q^4\omega\omega^*), \quad d\omega z = q^2zd\omega + (q^2+q^{-2})z^*\omega^2.
\end{equation}
The differentiation of $dz = z^* \omega$ yields
\begin{equation}\label{zdom}
z^*d\omega = -q^2z\omega^*\omega.
\end{equation}
Multiplying this relation by $z$ from left and right, and using commutation rules \eqref{disc} and \eqref{diff} one finds that
$
(1-x) d\omega = q^{-4} z^*d\omega z.
$
Developing the right hand side of this equality with the help of the second of equations \eqref{domegaz} we find
\begin{equation}\label{domega}
d\omega = \frac{1+q^{-4}}{q^2-1} {z^*}^2 \omega^2.
\end{equation}
Combining \eqref{domegaz} with \eqref{domega} we can derive
\begin{equation}\label{zomega}
{z^*}^3\omega^2 = -z\frac{q^8}{q^4 +1}\left(\omega^*\omega +q^4\omega\omega^*\right) .
\end{equation}
The multiplication of \eqref{zomega} by $z^3$ from the left and right and the usage of \eqref{disc}, \eqref{diff} give
\begin{subequations}\label{omegas}
\begin{equation}\label{omega1}
(1-x)(1-q^{-2}x)(1-q^{-4}x) \omega^2 = - \frac{q^8}{q^4 +1}z^4\left(\omega^*\omega +q^4\omega\omega^*\right) ,
\end{equation}
\begin{equation}\label{omega2}
(1-q^2x)(1-q^{4}x)(1-q^{6}x) \omega^2 = - \frac{q^8}{q^4 +1}z^4\left(\omega^*\omega +q^4\omega\omega^*\right) .
\end{equation}
\end{subequations}
Comparing the left hand sides of equations \eqref{omegas}, we conclude that
\begin{equation}\label{xomegas}
x\omega^2 = 0 = \omega^2 x \quad \mbox{and, by $*$-conjugation,} \quad x\omega^{*2} = 0 = \omega^{*2} x,
\end{equation}
and hence in view of either of \eqref{omegas}
\begin{equation}\label{omega.sq}
\omega^2 = - \frac{q^8}{q^4 +1}z^4\left(\omega^*\omega +q^4\omega\omega^*\right).
\end{equation}
By \eqref{omom}, the right hand side of \eqref{omega.sq} is in the module generated by $\mathsf{v}$, and so is $\omega^2$ and its adjoint $\omega^{*2}$. Thus, the module $\Omega^2 (D_q)$ spanned by all products of pairs of one-forms is indeed generated by $\mathsf{v}$.
Multiplying \eqref{domega} and \eqref{zdom} by $x$ and using relations \eqref{xomegas} we obtain
\begin{equation}\label{xzomegas1}
xz\omega^*\omega = 0 = \omega^*\omega xz.
\end{equation}
Following the same steps but now starting with the differentiation of
$dz^*\!=\!q^2z \omega^*$ (see \eqref{dz}), we obtain the complementary relation
\begin{equation}\label{xzomegas2}
xz\omega\omega^* = 0 = \omega\omega^* xz.
\end{equation}
In view of the definition of $\mathsf{v}$, \eqref{xzomegas1} and \eqref{xzomegas2} yield
$
xz\mathsf{v} = 0 = \mathsf{v}xz.
$
Next, the multiplication of, say, the first of these equations from the left and right by $z^*$ and the use of \eqref{disc} yield
$
x(1-x)\mathsf{v} =0$ and $x(1-q^2x)\mathsf{v} =0$.
The subtraction of one of these equations from the suitable scalar multiple of the other produces the necessary relation
\begin{equation}\label{xv}
x\mathsf{v} = 0 = \mathsf{v} x,
\end{equation}
which fully characterises the structure of $\Omega^2 (D_q)$ as an $\Oo(D_q)$-module generated by $\mathsf{v}$. In the light of \eqref{xv}, the $\CC$-basis of $\Omega^2 (D_q)$ consists of elements $\mathsf{v}z^n$, $ \mathsf{v}z^{*m}$, and hence, for all $w\in \Omega^2 (D_q)$, $wx = xw =0$, i.e., $\Omega^2 (D_q)$ is a torsion (as a left and right $\Oo(D_q)$-module). Since $\Oo(D_q)$ is a domain and $\Omega^2 (D_q)$ is a torsion, the dual of $\Omega^2 (D_q)$ is the zero module, hence, in particular $\Omega^2 (D_q)$ is not projective. Again by \eqref{xv}, the annihilator of $\Omega^2 (D_q)$,
$$
\mathrm{Ann}(\Omega^2 (D_q)) := \{ a\in \Oo(D_q) \; |\; \forall w\in \Omega^2 (D_q), \, aw=wa=0\},
$$
is the ideal of $\Oo(D_q)$ generated by $x$. The quotient $\Oo(D_q)/ \mathrm{Ann}(\Omega^2 (D_q))$ is the Laurent polynomial ring in one variable, i.e.\ the algebra $\Oo(S^1)$ of coordinate functions on the circle. When viewed as a module over $\Oo(S^1)$, $\Omega^2 (D_q)$ is free of rank one, generated by $\mathsf{v}$. Thus, although the module of 2-forms over $\Oo(D_q)$ is neither free nor projective, it can be identified with sections of a trivial line bundle once pulled back to the (classical) boundary of the quantum disc.
With \eqref{xv} at hand, equations \eqref{omom}, \eqref{omega.sq}, \eqref{domega} and their $*$-conjugates give the following relations in $\Omega^2 (D_q)$
\begin{subequations}\label{full}
\begin{equation}
d\omega = q^8 z^2\mathsf{v}, \quad d\omega^* = -z^{*2}\mathsf{v}, \quad \omega\omega^* = \mathsf{v}, \quad \omega^*\omega = -q^6\mathsf{v},
\end{equation}
\begin{equation}
\omega^2 = q^{12}\frac{q^2 -1}{q^4+1}z^4\mathsf{v}, \qquad \omega^{*2} = q^{-4}\frac{q^2 -1}{q^4+1}z^{*4}\mathsf{v}.
\end{equation}
\end{subequations}
One can easily check that \eqref{full}, \eqref{xv} and \eqref{va} are consistent with \eqref{diff} with no further restrictions on $\mathsf{v}$.
Setting $\Omega^n (D_q) =0$, for all $n>2$, we thus obtain a 2-dimensional calculus on the quantum disc.
\section{Differential calculus on the quantum cone}
The quantum cone algebra $\Oo(C^N_q)$ is a subalgebra of $\Oo(D_q)$ consisting of all elements of the $\ZZ$-degree congruent to 0 modulo a positive natural number $N$. Obviously $\Oo(C^1_q)= \Oo(D_q)$, the case we dealt with in the preceding section, so we may assume $N>1$. $\Oo(C^N_q)$ is a $*$-algebra generated by the self-adjoint $x=1-zz^*$ and by $y=z^N$, which satisfy the following commutation rules
\begin{equation}\label{cone}
xy = q^{2N} yx, \qquad yy^* = \prod_{l=0}^{N-1}\left(1-q^{-2l}x\right), \qquad y^*y = \prod_{l=1}^{N}\left(1-q^{2l}x\right).\end{equation}
The calculus $\Omega(C^N_q)$ on $\Oo(C^N_q)$ is obtained by restricting of the calculus $\Omega(D_q)$, i.e.\ $\Omega^n(C^N_q) = \{ \sum_{i}a_0^i d(a_1^i)\cdots d(a_n^i)a_{n+1}^i\; |\; a_k^i\in \Oo(C^N_q)\}$. Since $d$ is a degree-zero map $\Omega(C^N_q)$ contains only these forms in $\Omega(D_q)$, whose $\ZZ$-degree is a multiple of $N$. We will show that all such forms are in $\Omega(C^N_q)$. Since $\deg(\omega)= 2$, $\deg(\omega^*) =-2$ and $\deg(\mathsf{v})=0$,
this is equivalent to
$$
\Omega^1(C^N_q) = \Oo(D_q)_{\overline{-2}}\, \omega \oplus \Oo(D_q)_{\overline{2}}\, \omega^*, \qquad \Omega^2(C^N_q)= \Oo(C^N_q) \mathsf{v},
$$
where $\Oo(D_q)_{\overline{s}} = \{a\in \Oo(D_q)\; |\; \deg(a) \equiv s \!\! \mod\! N\}$.
As an $\Oo(C^N_q)$-module, $\Oo(D_q)_{\overline{-2}}$ is generated by $z^{N-2}$ and ${z^*}^2$, hence to show that $\Oo(D_q)_{\overline{-2}}\, \omega\subseteq \Omega^1(C^N_q)$ suffices it to prove that $z^{N-2}\omega, {z^*}^2\omega \in \Omega^1(C^N_q)$. Using the Leibniz rule one easily finds that
$$
d y = \left(\qn{N}{q^2} -q^{-2N+4} \qn{N}{q^4}x\right) z^{N-2}\omega,
$$
where
$\qn{n}{s} := \frac{ s^n -1}{s-1}$. Hence, in view of \eqref{disc} and \eqref{diff},
\begin{subequations}\label{ydy}
\begin{equation}\label{y*dy}
y^* dy = \qn{N}{q^2}\left( 1 - q^4\frac{\qn{N}{q^4}}{\qn{N}{q^2}} x\right) \prod_{l=3}^N\left(1-q^{2l}x\right) {z^*}^2\omega,
\end{equation}
\begin{equation}\label{dyy*}
dy y^* = q^{-2N}\qn{N}{q^2}\left( 1 - q^{-2N+4}\frac{\qn{N}{q^4}}{\qn{N}{q^2}} x\right) \prod_{l=0}^{N-3}\left(1-q^{-2l}x\right) {z^*}^2\omega.
\end{equation}
\end{subequations}
The polynomial in $x$ on the right hand side of \eqref{y*dy} has roots in common with the polynomial on the right hand side of \eqref{dyy*} if and only if there exists an integer
$k\in \left[-2N+2, -N-1\right] \cup \left[2, N-1\right]$
such that
\begin{equation}\label{crit}
q^{2k}(q^{2N} +1) = {q^2}+1.
\end{equation}
Equation \eqref{crit} is equivalent to
$
{q^2} \qn{N+k-1}{q^2} + \qn{k}{q^2} =0,
$
with the left hand side strictly positive if $k>0$ and strictly negative if $k\leq -N$. So, there are no solutions within the required range of values of $k$.
Hence the polynomials \eqref{y*dy}, \eqref{dyy*} are coprime, and so there exists a polynomial (in $x$) combination of the left had sides of equations \eqref{ydy} that gives ${z^*}^2\omega$. This combination is an element of $\Omega^1(C^N_q)$ and so is ${z^*}^2\omega$.
Next,
$$
{z^*}^2\omega \, y = q^{2N}(1-{q^2}x)(1-q^4 x)z^{N-2}\omega ,
$$
$$
y {z^*}^2\omega = (1-q^{-2N+4}x)(1-q^{-2N+2}x)z^{N-2}\omega,
$$
so again there is an $x$-polynomial combination of the left hand sides (which are already in $\Omega^1(C^N_q)$) giving $z^{N-2}\omega$. Therefore, $\Oo(D_q)_{\overline{-2}}\, \omega\subseteq \Omega^1(C^N_q)$. The case of $\Oo(D_q)_{\overline{2}}$ follows by the $*$-conjugation.
Since $z^2\omega^*$, ${z^*}^2\omega$
are elements of $\Omega^1(C^N_q)$,
\begin{equation}\label{o*o1}
\Omega^2(C^N_q) \ni z^2\omega^*{z^*}^2\omega = q^{-4}(1-x)(1-q^{-2}x) \omega^*\omega = -q^2 \mathsf{v},
\end{equation}
by the quantum disc relations and \eqref{full} and \eqref{xv}.
Consequently, $\mathsf{v} \in \Omega^2(C^N_q)$. Therefore, $\Omega(C^N_q)$ can be identified with the subspace of $\Omega(D_q)$, of all the elements whose $\ZZ$-degree is a multiple of $N$.
\section{The integral}\label{sec.int}
Here we construct an algebraic integral associated to the calculus constructed in Section~\ref{sec.diff}.
We start by observing that since $\sigma$ preserves the $\ZZ$-degrees of elements of $\Oo(D_q)$ and $\partial$ and $\bar\partial$ satisfy the $\sigma$-twisted Leibniz rules, the definition \eqref{partial} implies that $\partial$ lowers while $\bar\partial$ raises degrees by $2$. Hence, one can equip $\Omega^1 (D_q)$ with the $\ZZ$-grading so that $d$ is the degree zero map, provided $\deg(\omega) = 2$, $\deg(\omega^*) = -2$. Furthermore, in view of the definition of $\sigma$, one easily finds that
\begin{equation}\label{q-deriv}
\sigma^{-1}\circ\partial \circ\sigma = q^4 \partial, \qquad \sigma^{-1}\circ\bar\partial \circ\sigma = q^{-4} \bar \partial,
\end{equation}
i.e.\ $\partial$ is a $q^4$-derivation and $\bar\partial$ is a $q^{-4}$-derivation. Therefore, by \cite{BrzElK:int}, $\Omega (D_q)$ admits a divergence, for all right $\Oo(D_q)$-linear maps $f:\Omega^1 (D_q)\to \Oo(D_q)$, given by
\begin{equation}\label{div}
\nabla_0(f) = q^4\partial\left(f\left(\omega\right)\right) + q^{-4}\bar\partial\left(f\left(\omega^*\right)\right) .
\end{equation}
Since the $\Oo(D_q)$-module $\Omega^2 (D_q)$ has a trivial dual, $\nabla_0$ is flat. Recall that by the {\em integral} associated to $\nabla_0$ we understand the cokernel map of $\nabla_0$.
\begin{thm}\label{thm.int}
The integral associated to the divergence \eqref{div} is a map $\Lambda: \Oo(D_q) \to \CC$, given by
\begin{equation}\label{int}
\Lambda(x^kz^l) = \lambda \frac{\qn{k+1}{q^2}}{\qn{k+1}{q^4}} \delta_{l,0}, \qquad \mbox{for all $k\in \NN,\, l\in \ZZ$},
\end{equation}
where, for $l<0$, $z^l$ means ${z^*}^{-l}$ and $\lambda \in \CC$.
\end{thm}
\begin{proof}
First we need to calculate the image of $\nabla_0$. Using the twisted Leibniz rule and the quantum disc algebra commutation rules \eqref{disc}, one obtains
\begin{equation}\label{part1}
\partial(x^k) = -q^{-2}\qn{k}{q^4} x^{k-1}{z^*}^2.
\end{equation}
Since $\partial(z^*) =0$, \eqref{part1} means that all monomials $x^k {z^*}^{l+2}$ are in the image of $\partial$ hence in the image of $\nabla_0$. Using the $*$-conjugation we conclude the $x^k {z}^{l+2}$ are in the image of $\bar\partial$ hence in the image of $\nabla_0$. So $\Lambda$ vanishes on (linear combinations of) all such polynomials.
Next note that
\begin{equation}\label{part2}
\partial(z^2) = ({q^2}+1) - (q^4+1)x,
\end{equation}
hence
$$
\partial(z^*z^2 - q^4z^2z^*) = (1-q^4)z^*, \quad \partial(z^*z^2 - {q^2}z^2z^*) =(1-{q^2})(1+q^4)xz^*.
$$
This means that $z^*$ and $xz^*$ are in the image of $\partial$, hence of $\nabla_0$. In fact, all the $x^kz^*$ are in this image which can be shown inductively. Assume $x^kz^* \in \mathrm{Im}(\partial)$, for all $k\leq n$. Then using the twisted Leibniz rule, \eqref{part1} and \eqref{part2} one finds
\begin{equation}\label{part3}
\partial(x^nz^2) = -{q^2}\qn{N}{q^4} x^{n-1} + ({q^2}+1)\qn{n+1}{q^4} x^n - \qn{n+2}{q^4}x^{n+1}.
\end{equation}
Since $\partial(z^*) =0$, equation \eqref{part3} implies that $\partial(z^nz^2z^*)$ is a linear combination of monomials $x^{n-1}z^*$, $x^nz^*$ and $x^{n+1}z^*$. Since the first two are in the image of $\partial$ by the inductive assumption, so is the third one. Therefore, all linear combinations of $x^kz^*$ and $x^kz$ (by the $*$-conjugation) are in the image of $\nabla_0$.
Put together all this means that $\Lambda$ vanishes on all the polynomials
$\sum_{k,l=1}^n(c_{kl}x^kz^l + c'_{kl}x^k{z^*}^l).
$
The rest of the formula \eqref{int} can be proven by induction. Set $\lambda = \Lambda(1)$. Since $\Lambda$ vanishes on all elements in the image of $\nabla_0$, hence also in the image of $\partial$, the application of $\Lambda$ to the right hand side of \eqref{part1} confirms \eqref{int} for $k=1$. Now assume that \eqref{int} is true for all $k\leq n$. Then the application $\Lambda$ to the right hand side of \eqref{part3} followed by the use of the inductive assumption yields
\begin{eqnarray*}
\qn{n+2}{q^4}\Lambda\left(x^{n+1}\right) &=& {q^2}\qn{N}{q^4} \Lambda\left(x^{n-1}\right) -({q^2}+1)\qn{n+1}{q^4} \Lambda\left(x^n\right) \\
&=& \lambda\left( ({q^2}+1)\qn{n+1}{q^2}-{q^2}\qn n{q^2} \right) = \lambda\qn{n+2}{q^2}.
\end{eqnarray*}
Therefore, the formula \eqref{int} is true also for $n+1$, as required.
\end{proof}
The restriction of $\Lambda$ to the elements of $\Oo(D_q)$, whose $\ZZ$-degree is a multiple of $N$ gives an integral on the quantum cone $\Oo(C^N_q)$.
\subsection*{Acknowledgment}
The work on this project began during the first author's visit to SISSA, supported by INdAM-GNFM. He would like to thank the members of SISSA for hospitality. The second author was supported in part by the Simons Foundation grant 346300 and the Polish Government MNiSW 2015-2019 matching fund. | 138,699 |
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TITLE: Imaginary unit $i$ is not a limit of a real Cauchy sequence
QUESTION [1 upvotes]: I saw this in some book once and it has been bugging me. The book had, I think as the first exercise it mentioned, to prove that the imaginary unit $i = \sqrt{-1}$ is not a limit of any real valued Cauchy sequence.
Given all I know of complex numbers, this shouldn't seem so hard to show. Like, I know any Cauchy sequence converges in $\mathbb R$ and elements of $\mathbb R$ have total ordering and elements of $\mathbb C$ cannot and so on, but still I have great doubts of how this is supposed to be done.
REPLY [3 votes]: Hint: If $a_n$ is real, then the distance from $a_n$ to $i$ is $\sqrt{a_n^2+1}$, which is always $\ge 1$.
For an $\epsilon$-$N$ argument that the limit of the $a_n$ is not $i$, choose $\epsilon=\frac{1}{2}$. | 20,383 |
f(x)=2x-3 and g(x)=√4x+3 find (f ° g)(x) Find the domain of (f ° g)(x) Find (g ° f)(x) Find the domain of (g ° f)(x) can someone lead me in the right direction on how to do these?? Oct 7 | Erica from Seattle, WA | 2 Answers | 0 Votes Mark favorite Subscribe Comment | 45,772 |
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