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103-396: Hermite polynomials , Hermite interpolation , Hermite normal form , Hermitian operators , and cubic Hermite splines are named in his honor. One of his students was Henri Poincaré . He was the first to prove that e , the base of natural logarithms , is a transcendental number . His methods were used later by Ferdinand von Lindemann to prove that π is transcendental. Hermite

206-578: A 0,0 = 1 , a 1,0 = 0 , a 1,1 = 1 . For the physicist's polynomials, assuming H n ( x ) = ∑ k = 0 n a n , k x k , {\displaystyle H_{n}(x)=\sum _{k=0}^{n}a_{n,k}x^{k},} we have H n + 1 ( x ) = 2 x H n ( x ) − H n ′ ( x ) . {\displaystyle H_{n+1}(x)=2xH_{n}(x)-H_{n}'(x).} Individual coefficients are related by

309-658: A 0,0 = 1 , a 1,0 = 0 , a 1,1 = 2 . The Hermite polynomials constitute an Appell sequence , i.e., they are a polynomial sequence satisfying the identity He n ′ ⁡ ( x ) = n He n − 1 ⁡ ( x ) , H n ′ ( x ) = 2 n H n − 1 ( x ) . {\displaystyle {\begin{aligned}\operatorname {He} _{n}'(x)&=n\operatorname {He} _{n-1}(x),\\H_{n}'(x)&=2nH_{n-1}(x).\end{aligned}}} An integral recurrence that

412-564: A ; b ; z ) {\displaystyle {}_{1}F_{1}(a;b;z)} are Confluent hypergeometric functions of the first kind . The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric functions, see below. With more general boundary conditions , the Hermite polynomials can be generalized to obtain more general analytic functions for complex-valued λ . An explicit formula of Hermite polynomials in terms of contour integrals ( Courant & Hilbert 1989 )

515-458: A constant. Rewriting the differential equation as an eigenvalue problem L [ u ] = u ″ − x u ′ = − λ u , {\displaystyle L[u]=u''-xu'=-\lambda u,} the Hermite polynomials He λ ⁡ ( x ) {\displaystyle \operatorname {He} _{\lambda }(x)} may be understood as eigenfunctions of

618-775: A courtesy. American and British veterans who served in either World War on French soil, or during the 1944 campaigns to liberate France, may be eligible for appointment as Chevalier of the Legion of Honour, provided they were still living when the honour was approved. Collective appointments can be made to cities, institutions or companies. A total of 64 settlements in France have been decorated, as well as six foreign cities: Liège in 1914, Belgrade in 1920, Luxembourg City in 1957, Volgograd (the World War II 'Stalingrad') in 1984, Algiers in 2004, and London in 2020. French towns display

721-479: A façade to give political favours, gifts, and concessions. The Légion d'honneur was loosely patterned after a Roman legion , with legionaries , officers , commanders, regional " cohorts " and a grand council. The highest rank was not a Grand Cross but a Grand aigle (Grand Eagle), a rank that wore the insignia common to a Grand Cross. The members were paid, the highest of them extremely generously: Napoleon famously declared, "You call these baubles, well, it

824-464: A gesture of the hand the figures he studies. Plainly he sees and he is eager to paint, this is why he calls gesture to his aid. With M. Hermite, it is just the opposite, his eyes seem to shun contact with the world; it is not without, it is within he seeks the vision of truth. Reading one of [Poincare's] great discoveries, I should fancy (evidently a delusion) that, however magnificent, one ought to have found it long before, while such memoirs of Hermite as

927-543: A new system of nobility. However, the Légion d'honneur did use the organization of the old French orders of chivalry, for example, the Ordre de Saint-Louis . The insignia of the Légion d'honneur bear a resemblance to those of the Ordre de Saint-Louis , which also used a red ribbon. Napoleon originally created this award to ensure political loyalty. The organization would be used as

1030-556: A publication now in the public domain :  Herbermann, Charles, ed. (1913). "Charles Hermite". Catholic Encyclopedia . New York: Robert Appleton Company. Hermite polynomials In mathematics , the Hermite polynomials are a classical orthogonal polynomial sequence . The polynomials arise in: Hermite polynomials were defined by Pierre-Simon Laplace in 1810, though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work

1133-428: A simple proof of Niels Abel 's proposition concerning the impossibility of an algebraic solution to equations of the fifth degree . A correspondence with Carl Jacobi , begun in 1843 and continued the next year, resulted in the insertion, in the complete edition of Jacobi's works, of two articles by Hermite, one concerning the extension to Abelian functions of one of the theorems of Abel on elliptic functions , and

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1236-1362: A wider range of evaluation, it is necessary to include a factor for changing amplitude: e − x 2 2 ⋅ H n ( x ) ∼ 2 n π Γ ( n + 1 2 ) cos ⁡ ( x 2 n − n π 2 ) ( 1 − x 2 2 n + 1 ) − 1 4 = 2 Γ ( n ) Γ ( n 2 ) cos ⁡ ( x 2 n − n π 2 ) ( 1 − x 2 2 n + 1 ) − 1 4 , {\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)\sim {\frac {2^{n}}{\sqrt {\pi }}}\Gamma \left({\frac {n+1}{2}}\right)\cos \left(x{\sqrt {2n}}-{\frac {n\pi }{2}}\right)\left(1-{\frac {x^{2}}{2n+1}}\right)^{-{\frac {1}{4}}}={\frac {2\Gamma (n)}{\Gamma \left({\frac {n}{2}}\right)}}\cos \left(x{\sqrt {2n}}-{\frac {n\pi }{2}}\right)\left(1-{\frac {x^{2}}{2n+1}}\right)^{-{\frac {1}{4}}},} which, using Stirling's approximation , can be further simplified, in

1339-447: A world of physical reality exists, the one like the other independent of ourselves, both of divine creation. I shall risk nothing on an attempt to prove the transcendence of π . If others undertake this enterprise, no one will be happier than I in their success. But believe me, it will not fail to cost them some effort. While speaking, M. Bertrand is always in motion; now he seems in combat with some outside enemy, now he outlines with

1442-532: A year to preparing for the notoriously difficult entrance examination . In 1842 he was admitted to the school. However, after one year the school would not allow Hermite to continue his studies there because of his deformed foot. He struggled to regain his admission to the school, but the administration imposed strict conditions. Hermite did not accept this, and he quit the École Polytechnique without graduating. In 1842, Nouvelles Annales de Mathématiques published Hermite's first original contribution to mathematics,

1545-820: Is Honneur et Patrie ("Honour and Fatherland"); its seat is the Palais de la Légion d'Honneur next to the Musée d'Orsay , on the left bank of the Seine in Paris . Since 1 February 2023, the Order's grand chancellor has been retired General François Lecointre , who succeeded fellow retired General Benoît Puga in office. The order is divided into five degrees of increasing distinction: Chevalier ( Knight ), Officier (Officer), Commandeur ( Commander ), Grand officier (Grand Officer) and Grand-croix ( Grand Cross ). During

1648-462: Is a constant. Imposing the boundary condition that u should be polynomially bounded at infinity, the equation has solutions only if λ is a non-negative integer, and the solution is uniquely given by u ( x ) = C 1 He λ ⁡ ( x ) {\displaystyle u(x)=C_{1}\operatorname {He} _{\lambda }(x)} , where C 1 {\displaystyle C_{1}} denotes

1751-602: Is a rescaling of the other: H n ( x ) = 2 n 2 He n ⁡ ( 2 x ) , He n ⁡ ( x ) = 2 − n 2 H n ( x 2 ) . {\displaystyle H_{n}(x)=2^{\frac {n}{2}}\operatorname {He} _{n}\left({\sqrt {2}}\,x\right),\quad \operatorname {He} _{n}(x)=2^{-{\frac {n}{2}}}H_{n}\left({\frac {x}{\sqrt {2}}}\right).} These are Hermite polynomial sequences of different variances; see

1854-464: Is also possible. The sequence of probabilist's Hermite polynomials also satisfies the recurrence relation He n + 1 ⁡ ( x ) = x He n ⁡ ( x ) − He n ′ ⁡ ( x ) . {\displaystyle \operatorname {He} _{n+1}(x)=x\operatorname {He} _{n}(x)-\operatorname {He} _{n}'(x).} Individual coefficients are related by

1957-611: Is as follows: These equations have the form of a Rodrigues' formula and can also be written as, He n ⁡ ( x ) = ( x − d d x ) n ⋅ 1 , H n ( x ) = ( 2 x − d d x ) n ⋅ 1. {\displaystyle \operatorname {He} _{n}(x)=\left(x-{\frac {d}{dx}}\right)^{n}\cdot 1,\quad H_{n}(x)=\left(2x-{\frac {d}{dx}}\right)^{n}\cdot 1.} The two definitions are not exactly identical; each

2060-2430: Is deduced and demonstrated in is as follows: He n + 1 ⁡ ( x ) = ( n + 1 ) ∫ 0 x He n ⁡ ( t ) d t − H e n ′ ( 0 ) , {\displaystyle \operatorname {He} _{n+1}(x)=(n+1)\int _{0}^{x}\operatorname {He} _{n}(t)dt-He'_{n}(0),} H n + 1 ( x ) = 2 ( n + 1 ) ∫ 0 x H n ( t ) d t − H n ′ ( 0 ) . {\displaystyle H_{n+1}(x)=2(n+1)\int _{0}^{x}H_{n}(t)dt-H'_{n}(0).} Equivalently, by Taylor-expanding , He n ⁡ ( x + y ) = ∑ k = 0 n ( n k ) x n − k He k ⁡ ( y ) = 2 − n 2 ∑ k = 0 n ( n k ) He n − k ⁡ ( x 2 ) He k ⁡ ( y 2 ) , H n ( x + y ) = ∑ k = 0 n ( n k ) H k ( x ) ( 2 y ) n − k = 2 − n 2 ⋅ ∑ k = 0 n ( n k ) H n − k ( x 2 ) H k ( y 2 ) . {\displaystyle {\begin{aligned}\operatorname {He} _{n}(x+y)&=\sum _{k=0}^{n}{\binom {n}{k}}x^{n-k}\operatorname {He} _{k}(y)&&=2^{-{\frac {n}{2}}}\sum _{k=0}^{n}{\binom {n}{k}}\operatorname {He} _{n-k}\left(x{\sqrt {2}}\right)\operatorname {He} _{k}\left(y{\sqrt {2}}\right),\\H_{n}(x+y)&=\sum _{k=0}^{n}{\binom {n}{k}}H_{k}(x)(2y)^{n-k}&&=2^{-{\frac {n}{2}}}\cdot \sum _{k=0}^{n}{\binom {n}{k}}H_{n-k}\left(x{\sqrt {2}}\right)H_{k}\left(y{\sqrt {2}}\right).\end{aligned}}} These umbral identities are self-evident and included in

2163-754: Is headed by the Grand Chancellor ( grand chancelier ), usually a retired general, as well as the Secretary-General ( secrétaire général ), a civilian administrator. The Grand Chancery also regulates the National Order of Merit and the médaille militaire (Military Medal). There are several structures funded by and operated under the authority of the Grand Chancery, like the Legion of Honour Schools ( Maisons d'éducation de la Légion d'honneur ) and

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2266-1057: Is needed to resolve the wavefunction of a quantum harmonic oscillator such that it agrees with the classical approximation in the limit of the correspondence principle . A better approximation, which accounts for the variation in frequency, is given by e − x 2 2 ⋅ H n ( x ) ∼ ( 2 n e ) n 2 2 cos ⁡ ( x 2 n + 1 − x 2 3 − n π 2 ) ( 1 − x 2 2 n + 1 ) − 1 4 . {\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)\sim \left({\frac {2n}{e}}\right)^{\frac {n}{2}}{\sqrt {2}}\cos \left(x{\sqrt {2n+1-{\frac {x^{2}}{3}}}}-{\frac {n\pi }{2}}\right)\left(1-{\frac {x^{2}}{2n+1}}\right)^{-{\frac {1}{4}}}.} A finer approximation, which takes into account

2369-706: Is the Airy function of the first kind. Legion of Honour The National Order of the Legion of Honour ( French : Ordre national de la Légion d'honneur [ɔʁdʁ nɑsjɔnal də la leʒjɔ̃ dɔnœʁ] ), formerly the Imperial Order of the Legion of Honour ( Ordre impérial de la Légion d'honneur ), is the highest French order of merit , both military and civil , and currently comprises five classes. Established in 1802 by Napoleon Bonaparte , it has been retained (with occasional slight alterations) by all later French governments and regimes. The order's motto

2472-677: Is the Kronecker delta . The probabilist polynomials are thus orthogonal with respect to the standard normal probability density function. The Hermite polynomials (probabilist's or physicist's) form an orthogonal basis of the Hilbert space of functions satisfying ∫ − ∞ ∞ | f ( x ) | 2 w ( x ) d x < ∞ , {\displaystyle \int _{-\infty }^{\infty }{\bigl |}f(x){\bigr |}^{2}\,w(x)\,dx<\infty ,} in which

2575-606: Is the double factorial . Note that the above expression is a special case of the representation of the probabilist's Hermite polynomials as moments: He n ⁡ ( x ) = 1 2 π ∫ − ∞ ∞ ( x + i y ) n e − y 2 2 d y . {\displaystyle \operatorname {He} _{n}(x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }(x+iy)^{n}e^{-{\frac {y^{2}}{2}}}\,dy.} Asymptotically, as n → ∞ ,

2678-681: Is to appreciate that the entire function F ( z ) = ∫ − ∞ ∞ f ( x ) e z x − x 2 d x = ∑ n = 0 ∞ z n n ! ∫ f ( x ) x n e − x 2 d x = 0 {\displaystyle F(z)=\int _{-\infty }^{\infty }f(x)e^{zx-x^{2}}\,dx=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}\int f(x)x^{n}e^{-x^{2}}\,dx=0} vanishes identically. The fact then that F ( it ) = 0 for every real t means that

2781-430: Is uniquely given in terms of physicist's Hermite polynomials in the form u ( x ) = C 1 H λ ( x ) {\displaystyle u(x)=C_{1}H_{\lambda }(x)} , where C 1 {\displaystyle C_{1}} denotes a constant, after imposing the boundary condition that u should be polynomially bounded at infinity. The general solutions to

2884-1054: Is valid for all complex values of x and t , and can be obtained by writing the Taylor expansion at x of the entire function z → e (in the physicist's case). One can also derive the (physicist's) generating function by using Cauchy's integral formula to write the Hermite polynomials as H n ( x ) = ( − 1 ) n e x 2 d n d x n e − x 2 = ( − 1 ) n e x 2 n ! 2 π i ∮ γ e − z 2 ( z − x ) n + 1 d z . {\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}}=(-1)^{n}e^{x^{2}}{\frac {n!}{2\pi i}}\oint _{\gamma }{\frac {e^{-z^{2}}}{(z-x)^{n+1}}}\,dz.} Using this in

2987-412: Is with baubles that men are led... Do you think that you would be able to make men fight by reasoning? Never. That is good only for the scholar in his study. The soldier needs glory, distinctions, rewards." This has been often quoted as "It is with such baubles that men are led." Napoleon was also occasionally noted after a battle to ask who the bravest man in a regiment was, and upon the regiment declaring

3090-531: Is worn by the Grand Cross (in gilt on the left chest) and the Grand Officer (in silver on the right chest) respectively; it is similar to the badge, but without enamel, and with the wreath replaced by a cluster of rays in between each arm. The central disc features the head of Marianne , surrounded by the legend République Française ('French Republic') and the motto Honneur et Patrie . The ribbon for

3193-498: The Hôtel de Salm , headquarters of the Légion d'honneur , was burned to the ground in fierce street combats; the archives of the order were lost. In the second term of President Jules Grévy , which started in 1885, newspaper journalists brought to light the trafficking of Grévy's son-in-law, Daniel Wilson, in the awarding of decorations of the Légion d'honneur . Grévy was not accused of personal participation in this scandal, but he

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3296-572: The Légion , for "eminent merit" ( mérites éminents ) in military or civil life. In practice, in current usage, the order is conferred on entrepreneurs, high-level civil servants , scientists, artists, including famous actors and actresses, sport champions , and others with connections in the executive . Members of the French Parliament cannot receive the order, except for valour in war, and ministers are not allowed to nominate their accountants. Until 2008, French nationals could only enter

3399-505: The Légion d'honneur among his family and his senior ministers. This collar was abolished in 1815. Although research is made difficult by the loss of the archives, it is rumoured that three women who fought with the army were decorated with the order: Virginie Ghesquière , Marie-Jeanne Schelling and a nun , Sister Anne Biget . The Légion d'honneur was prominent and visible in the French Empire. The Emperor always wore it, and

3502-410: The Légion d'honneur . All previous orders were Christian, or shared a clear Christian background, whereas the Légion d'honneur is a secular institution. The badge of the Légion d'honneur has five arms. In a decree issued on the 10 Pluviôse XIII (30 January 1805), a grand decoration was instituted. This decoration, a cross on a large sash and a silver star with an eagle, symbol of

3605-624: The Fourier transform of f ( x ) e is 0, hence f is 0 almost everywhere . Variants of the above completeness proof apply to other weights with exponential decay . In the Hermite case, it is also possible to prove an explicit identity that implies completeness (see section on the Completeness relation below). An equivalent formulation of the fact that Hermite polynomials are an orthogonal basis for L ( R , w ( x ) dx ) consists in introducing Hermite functions (see below), and in saying that

3708-536: The Fourth Republic in 1946 brought about the latest change in the design of the Legion of Honour. The date "1870" on the obverse was replaced by a single star. No changes were made after the establishment of the Fifth Republic in 1958. The Legion of Honour is a national order of France, meaning a public incorporated body. The Legion is regulated by a civil law code , the "Code of the Legion of Honour and of

3811-579: The French Revolution , all of the French orders of chivalry were abolished and replaced with weapons of honour . It was the wish of Napoleon Bonaparte , the First Consul , to create a reward to commend civilians and soldiers. From this wish was instituted a Légion d'honneur , a body of men that was not an order of chivalry , for Napoleon believed that France wanted a recognition of merit rather than

3914-466: The Legion of Honour Museum ( Musée de la Légion d'honneur ). The Legion of Honour Schools are élite boarding schools in Saint-Denis and Camp des Loges in the forest of Saint-Germain-en-Laye . Study there is restricted to daughters, granddaughters, and great-granddaughters of members of the order, the médaille militaire or the ordre national du Mérite . There are five classes in

4017-569: The differential operator representation detailed below, He n ⁡ ( x ) = e − D 2 2 x n , H n ( x ) = 2 n e − D 2 4 x n . {\displaystyle {\begin{aligned}\operatorname {He} _{n}(x)&=e^{-{\frac {D^{2}}{2}}}x^{n},\\H_{n}(x)&=2^{n}e^{-{\frac {D^{2}}{4}}}x^{n}.\end{aligned}}} In consequence, for

4120-748: The exponential generating function e x t − 1 2 t 2 = ∑ n = 0 ∞ He n ⁡ ( x ) t n n ! , e 2 x t − t 2 = ∑ n = 0 ∞ H n ( x ) t n n ! . {\displaystyle {\begin{aligned}e^{xt-{\frac {1}{2}}t^{2}}&=\sum _{n=0}^{\infty }\operatorname {He} _{n}(x){\frac {t^{n}}{n!}},\\e^{2xt-t^{2}}&=\sum _{n=0}^{\infty }H_{n}(x){\frac {t^{n}}{n!}}.\end{aligned}}} This equality

4223-591: The floor function : H n ( x ) = n ! ∑ m = 0 ⌊ n 2 ⌋ ( − 1 ) m m ! ( n − 2 m ) ! ( 2 x ) n − 2 m . {\displaystyle H_{n}(x)=n!\sum _{m=0}^{\left\lfloor {\tfrac {n}{2}}\right\rfloor }{\frac {(-1)^{m}}{m!(n-2m)!}}(2x)^{n-2m}.} The probabilist's Hermite polynomials He have similar formulas, which may be obtained from these by replacing

Charles Hermite - Misplaced Pages Continue

4326-1086: The m th derivatives the following relations hold: He n ( m ) ⁡ ( x ) = n ! ( n − m ) ! He n − m ⁡ ( x ) = m ! ( n m ) He n − m ⁡ ( x ) , H n ( m ) ( x ) = 2 m n ! ( n − m ) ! H n − m ( x ) = 2 m m ! ( n m ) H n − m ( x ) . {\displaystyle {\begin{aligned}\operatorname {He} _{n}^{(m)}(x)&={\frac {n!}{(n-m)!}}\operatorname {He} _{n-m}(x)&&=m!{\binom {n}{m}}\operatorname {He} _{n-m}(x),\\H_{n}^{(m)}(x)&=2^{m}{\frac {n!}{(n-m)!}}H_{n-m}(x)&&=2^{m}m!{\binom {n}{m}}H_{n-m}(x).\end{aligned}}} It follows that

4429-1043: The normal distribution with expected value 0 and standard deviation 1. The n th-order Hermite polynomial is a polynomial of degree n . The probabilist's version He n has leading coefficient 1, while the physicist's version H n has leading coefficient 2 . From the Rodrigues formulae given above, we can see that H n ( x ) and He n ( x ) are even or odd functions depending on n : H n ( − x ) = ( − 1 ) n H n ( x ) , He n ⁡ ( − x ) = ( − 1 ) n He n ⁡ ( x ) . {\displaystyle H_{n}(-x)=(-1)^{n}H_{n}(x),\quad \operatorname {He} _{n}(-x)=(-1)^{n}\operatorname {He} _{n}(x).} H n ( x ) and He n ( x ) are n th-degree polynomials for n = 0, 1, 2, 3,... . These polynomials are orthogonal with respect to

4532-1818: The weight function ( measure ) w ( x ) = e − x 2 2 ( for  He ) {\displaystyle w(x)=e^{-{\frac {x^{2}}{2}}}\quad ({\text{for }}\operatorname {He} )} or w ( x ) = e − x 2 ( for  H ) , {\displaystyle w(x)=e^{-x^{2}}\quad ({\text{for }}H),} i.e., we have ∫ − ∞ ∞ H m ( x ) H n ( x ) w ( x ) d x = 0 for all  m ≠ n . {\displaystyle \int _{-\infty }^{\infty }H_{m}(x)H_{n}(x)\,w(x)\,dx=0\quad {\text{for all }}m\neq n.} Furthermore, ∫ − ∞ ∞ H m ( x ) H n ( x ) e − x 2 d x = π 2 n n ! δ n m , {\displaystyle \int _{-\infty }^{\infty }H_{m}(x)H_{n}(x)\,e^{-x^{2}}\,dx={\sqrt {\pi }}\,2^{n}n!\,\delta _{nm},} and ∫ − ∞ ∞ He m ⁡ ( x ) He n ⁡ ( x ) e − x 2 2 d x = 2 π n ! δ n m , {\displaystyle \int _{-\infty }^{\infty }\operatorname {He} _{m}(x)\operatorname {He} _{n}(x)\,e^{-{\frac {x^{2}}{2}}}\,dx={\sqrt {2\pi }}\,n!\,\delta _{nm},} where δ n m {\displaystyle \delta _{nm}}

4635-460: The 35,000 to 38,000 members. The images of Napoleon and his eagle were removed and replaced by the image of King Henry IV , the popular first king of the Bourbon line. Three Bourbon fleurs-de-lys replaced the eagle on the reverse of the order. A king's crown replaced the imperial crown. In 1816, the grand cordons were renamed grand crosses and the legionnaires became knights. The king decreed that

4738-517: The Cross of a Knight of the Legion of Honour: The order has had five levels since the reign of King Louis XVIII, who restored the order in 1815. Since the reform, the following distinctions have existed: Due to the order's long history, and the remarkable fact that it has been retained by all subsequent governments and regimes since the First Empire, the order's design has undergone many changes. Although

4841-463: The French nation. The insignia were drastically altered; the cross now displayed tricolour flags . In 1847, there were 47,000 members. Yet another revolution in Paris (in 1848) brought a new republic (the second) and a new design to the Légion d'honneur . A nephew of the founder, Louis-Napoléon Bonaparte , was elected president and he restored the image of his uncle on the crosses of the order. In 1852,

4944-490: The Hermite functions are an orthonormal basis for L ( R ) . The probabilist's Hermite polynomials are solutions of the differential equation ( e − 1 2 x 2 u ′ ) ′ + λ e − 1 2 x 2 u = 0 , {\displaystyle \left(e^{-{\frac {1}{2}}x^{2}}u'\right)'+\lambda e^{-{\frac {1}{2}}x^{2}}u=0,} where λ

5047-715: The Hermite polynomials also satisfy the recurrence relation He n + 1 ⁡ ( x ) = x He n ⁡ ( x ) − n He n − 1 ⁡ ( x ) , H n + 1 ( x ) = 2 x H n ( x ) − 2 n H n − 1 ( x ) . {\displaystyle {\begin{aligned}\operatorname {He} _{n+1}(x)&=x\operatorname {He} _{n}(x)-n\operatorname {He} _{n-1}(x),\\H_{n+1}(x)&=2xH_{n}(x)-2nH_{n-1}(x).\end{aligned}}} These last relations, together with

5150-449: The Legion of Honour at the class of Chevalier (Knight). To be promoted to a higher class, one had to perform new eminent services in the interest of France and a set number of years had to pass between appointment and promotion. This was however amended in 2008 when entry became possible at Officer, Commander and Grand Officer levels, as a recognition of "extraordinary careers" ( carrières hors du commun ). In 2009, Simone Veil became

5253-555: The Legion of Honour award for service is achieved after 20 years of meritorious service, having been awarded the rank of Chevalier of the Ordre National du Mérite . Bravery awards lessen the time needed for the award—in fact decorated servicemen become directly chevaliers of the Légion d'Honneur , skipping the Ordre du Mérite . NCOs almost never achieve that award, except for the most heavily decorated service members. Collective appointments can be made to military units. In

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5356-412: The Legion of Honour: The "eminent merits" required to be awarded the order require the flawless performance of one's trade as well as doing more than ordinarily expected, such as being creative, zealous and contributing to the growth and well-being of others. The order has a maximum quota of 75 Grand Cross, 250 Grand Officers, 1,250 Commanders, 10,000 Officers, and 113,425 (ordinary) Knights. As of 2010,

5459-614: The Legion on 14 May 2017. The Grand Master appoints all other members of the order, on the advice of the French government. The Grand Master's insignia is the Grand Collar of the Legion. The President of the Republic, as Grand Master of the order, receives the Collar as part of his investiture, but the Grand Masters have not worn the Collar since Valéry Giscard d'Estaing . The Grand Chancery

5562-673: The Military Medal". While the President of the French Republic is the Grand Master of the order, day-to-day running is entrusted to the Grand Chancery ( Grande Chancellerie de la Légion d'honneur ). Since the establishment of the Legion, the Grand Master of the order has always been the Emperor, King or President of France. President Emmanuel Macron therefore became the Grand Master of

5665-701: The Napoleonic Empire, became known as the Grand aigle (Grand Eagle), and later in 1814 as the Grand cordon (big sash, literally "big ribbon"). After Napoleon crowned himself Emperor of the French in 1804 and established the Napoleonic nobility in 1808, award of the Légion d'honneur gave right to the title of "Knight of the Empire" ( Chevalier de l'Empire ). The title was made hereditary after three generations of grantees. Napoleon had dispensed 15 golden collars of

5768-597: The above explicit expressions, that is, those for monomials in terms of probabilist's Hermite polynomials He are x n = n ! ∑ m = 0 ⌊ n 2 ⌋ 1 2 m m ! ( n − 2 m ) ! He n − 2 m ⁡ ( x ) . {\displaystyle x^{n}=n!\sum _{m=0}^{\left\lfloor {\tfrac {n}{2}}\right\rfloor }{\frac {1}{2^{m}m!(n-2m)!}}\operatorname {He} _{n-2m}(x).} The corresponding expressions for

5871-409: The above second-order differential equations are in fact linear combinations of both Hermite polynomials and confluent hypergeometric functions of the first kind. For example, for the physicist's Hermite equation u ″ − 2 x u ′ + 2 λ u = 0 , {\displaystyle u''-2xu'+2\lambda u=0,} the general solution takes

5974-539: The actual membership was 67 Grand Cross, 314 Grand Officers, 3,009 Commanders, 17,032 Officers and 74,384 Knights. Appointments of veterans of World War II , French military personnel involved in the North African Campaign and other foreign French military operations, as well as wounded soldiers, are made independently of the quota. Members convicted of a felony ( crime in French) are automatically dismissed from

6077-755: The approximation is e − x 2 2 ⋅ H n ( x ) = π 1 4 2 n 2 + 1 4 n ! n − 1 12 ( Ai ⁡ ( 2 1 2 n 1 6 t ) + O ( n − 2 3 ) ) , {\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)=\pi ^{\frac {1}{4}}2^{{\frac {n}{2}}+{\frac {1}{4}}}{\sqrt {n!}}\,n^{-{\frac {1}{12}}}\left(\operatorname {Ai} \left(2^{\frac {1}{2}}n^{\frac {1}{6}}t\right)+O\left(n^{-{\frac {2}{3}}}\right)\right),} where Ai

6180-502: The award when it was offered to them. While membership in the Légion is technically restricted to French nationals, foreign nationals who have served France or the ideals it upholds may receive the honour. Foreign nationals who live in France are subject to the same requirements as the French. Foreign nationals who live abroad may be awarded a distinction of any rank or dignity in the Légion . Foreign heads of state and their spouses or consorts of monarchs are made Grand Cross as

6283-424: The base of the natural system of logarithms , is transcendental . Techniques similar to those used in Hermite's proof of e 's transcendence were used by Ferdinand von Lindemann in 1882 to show that π is transcendental. The following is a list of his works: There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our mind, just as

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6386-399: The basic shape and structure of the insignia has remained generally the same, the hanging device changed back and forth and France itself swung back and forth between republic and monarchy . The central disc in the centre has also changed to reflect the political system and leadership of France at the time. As each new regime came along the design was altered to become politically correct for

6489-516: The case of a military unit, its flag is decorated with the insignia of a knight, which is a different award from the fourragère . Twenty-one schools, mainly schools providing reserve officers during the World Wars, were awarded the Légion d'Honneur. Foreign military units can be decorated with the order, such as the U.S. Military Academy . The Flag or Standard of the following units was decorated with

6592-663: The commandants were now commanders. The Légion d'honneur became the second-ranking order of knighthood of the French monarchy, after the Order of the Holy Spirit . Following the overthrow of the Bourbons in favour of King Louis Philippe I of the House of Orléans , the Bourbon monarchy's orders were once again abolished and the Légion d'honneur was restored in 1830 as the paramount decoration of

6695-913: The decoration in their municipal coat of arms . Organisations to receive the honour include the French Red Cross ( Croix-Rouge Française ), the Abbaye de Nôtre-Dame des Dombes ( Abbey of Notre-Dame des Dombes ), the French National Railway Company ( SNCF , Société Nationale des Chemins de fer Français ), the Préfecture de Police de la Ville de Paris ( Prefecture of Police of Paris ), and various Grandes Écoles (National (Elite) Colleges) and other educational establishments. The military distinctions ( Légion d'honneur à titre militaire ) are awarded for bravery ( actions de guerre ) or for service. For active-duty commissioned officers,

6798-400: The differential operator L [ u ] {\displaystyle L[u]} . This eigenvalue problem is called the Hermite equation , although the term is also used for the closely related equation u ″ − 2 x u ′ = − 2 λ u . {\displaystyle u''-2xu'=-2\lambda u.} whose solution

6901-579: The expansion e − x 2 2 ⋅ H n ( x ) ∼ 2 n π Γ ( n + 1 2 ) cos ⁡ ( x 2 n − n π 2 ) {\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)\sim {\frac {2^{n}}{\sqrt {\pi }}}\Gamma \left({\frac {n+1}{2}}\right)\cos \left(x{\sqrt {2n}}-{\frac {n\pi }{2}}\right)} holds true. For certain cases concerning

7004-595: The family. Hermite obtained his secondary education at Collège de Nancy and then, in Paris, at Collège Henri IV and at the Lycée Louis-le-Grand . He read some of Joseph-Louis Lagrange 's writings on the solution of numerical equations and Carl Friedrich Gauss 's publications on number theory . Hermite wanted to take his higher education at École Polytechnique , a military academy renowned for excellence in mathematics, science, and engineering. Tutored by mathematician Eugène Charles Catalan , Hermite devoted

7107-628: The fashion of the time allowed for decorations to be worn most of the time. The king of Sweden therefore declined the order; it was too common in his eyes. Napoleon's own decorations were captured by the Prussians and were displayed in the Zeughaus (armoury) in Berlin until 1945. Today, they are in Moscow. Louis XVIII changed the appearance of the order, but it was not abolished. To have done so would have angered

7210-643: The first American was admitted: Thomas Wiltberger Evans , dentist of Napoleon III. In 1870, the defeat of the French Imperial Army in the Franco-Prussian War brought the end of the Empire and the creation of the Third Republic (1871–1940). As France changed, the Légion d'honneur changed as well. The crown was replaced by a laurel and oak wreath. In 1871, during the Paris Commune uprising,

7313-576: The first kind), and h λ ( x ) {\displaystyle h_{\lambda }(x)} are physicist's Hermite functions (of the second kind). The latter functions are compactly represented as h λ ( x ) = 1 F 1 ( − λ 2 ; 1 2 ; x 2 ) {\displaystyle h_{\lambda }(x)={}_{1}F_{1}(-{\tfrac {\lambda }{2}};{\tfrac {1}{2}};x^{2})} where 1 F 1 (

7416-597: The first person to enter the Order at Grand Officer level. Veil was a member of the Académie française , a former Health Minister and President of the European Parliament , as well as an Auschwitz survivor. She was promoted to Grand Cross in 2012. Every year at least five recipients decline the award. Even if they refuse to accept it, they are still included in the order's official membership. The composers Maurice Ravel and Charles Koechlin , for example, declined

7519-440: The first recorded woman, Angélique Duchemin , an old revolutionary of the 1789 uprising against the absolute monarchy, was admitted into the order. On 2 December 1851, President Louis-Napoléon Bonaparte staged a coup d'état with the help of the armed forces. He made himself Emperor of the French exactly one year later on 2 December 1852, after a successful plebiscite . An Imperial crown was added. During Napoleon III 's reign,

7622-2702: The following multiplication theorem holds: H n ( γ x ) = ∑ i = 0 ⌊ n 2 ⌋ γ n − 2 i ( γ 2 − 1 ) i ( n 2 i ) ( 2 i ) ! i ! H n − 2 i ( x ) , He n ⁡ ( γ x ) = ∑ i = 0 ⌊ n 2 ⌋ γ n − 2 i ( γ 2 − 1 ) i ( n 2 i ) ( 2 i ) ! i ! 2 − i He n − 2 i ⁡ ( x ) . {\displaystyle {\begin{aligned}H_{n}(\gamma x)&=\sum _{i=0}^{\left\lfloor {\tfrac {n}{2}}\right\rfloor }\gamma ^{n-2i}(\gamma ^{2}-1)^{i}{\binom {n}{2i}}{\frac {(2i)!}{i!}}H_{n-2i}(x),\\\operatorname {He} _{n}(\gamma x)&=\sum _{i=0}^{\left\lfloor {\tfrac {n}{2}}\right\rfloor }\gamma ^{n-2i}(\gamma ^{2}-1)^{i}{\binom {n}{2i}}{\frac {(2i)!}{i!}}2^{-i}\operatorname {He} _{n-2i}(x).\end{aligned}}} The physicist's Hermite polynomials can be written explicitly as H n ( x ) = { n ! ∑ l = 0 n 2 ( − 1 ) n 2 − l ( 2 l ) ! ( n 2 − l ) ! ( 2 x ) 2 l for even  n , n ! ∑ l = 0 n − 1 2 ( − 1 ) n − 1 2 − l ( 2 l + 1 ) ! ( n − 1 2 − l ) ! ( 2 x ) 2 l + 1 for odd  n . {\displaystyle H_{n}(x)={\begin{cases}\displaystyle n!\sum _{l=0}^{\frac {n}{2}}{\frac {(-1)^{{\tfrac {n}{2}}-l}}{(2l)!\left({\tfrac {n}{2}}-l\right)!}}(2x)^{2l}&{\text{for even }}n,\\\displaystyle n!\sum _{l=0}^{\frac {n-1}{2}}{\frac {(-1)^{{\frac {n-1}{2}}-l}}{(2l+1)!\left({\frac {n-1}{2}}-l\right)!}}(2x)^{2l+1}&{\text{for odd }}n.\end{cases}}} These two equations may be combined into one using

7725-478: The following recursion formula: a n + 1 , k = { − a n , k + 1 k = 0 , 2 a n , k − 1 − ( k + 1 ) a n , k + 1 k > 0 , {\displaystyle a_{n+1,k}={\begin{cases}-a_{n,k+1}&k=0,\\2a_{n,k-1}-(k+1)a_{n,k+1}&k>0,\end{cases}}} and

7828-503: The following recursion formula: a n + 1 , k = { − ( k + 1 ) a n , k + 1 k = 0 , a n , k − 1 − ( k + 1 ) a n , k + 1 k > 0 , {\displaystyle a_{n+1,k}={\begin{cases}-(k+1)a_{n,k+1}&k=0,\\a_{n,k-1}-(k+1)a_{n,k+1}&k>0,\end{cases}}} and

7931-534: The form u ( x ) = C 1 H λ ( x ) + C 2 h λ ( x ) , {\displaystyle u(x)=C_{1}H_{\lambda }(x)+C_{2}h_{\lambda }(x),} where C 1 {\displaystyle C_{1}} and C 2 {\displaystyle C_{2}} are constants, H λ ( x ) {\displaystyle H_{\lambda }(x)} are physicist's Hermite polynomials (of

8034-455: The great aid he gave those beginning scientific life. His published courses of lectures have exercised a great influence. His important original contributions to pure mathematics , published in the major mathematical journals of the world, dealt chiefly with Abelian and elliptic functions and the theory of numbers . In 1858, Hermite showed that equations of the fifth degree could be solved by elliptic functions. In 1873, he proved that e ,

8137-400: The head of Marianne , surrounded by the legend République Française on a blue enamel ring. The reverse central disc is also in gilt, with a set of crossed tricolores , surrounded by the Légion's motto Honneur et Patrie ('Honour and Fatherland') and its foundation date on a blue enamel ring. The badge is suspended by an enameled laurel and oak wreath. The star (or plaque )

8240-481: The individual, the Emperor would take the Legion d'Honneur from his own coat and pin it on the chest of the man. The order was the first modern order of merit . Under the monarchy, such orders were often limited to Roman Catholics, all knights had to be noblemen, and military decorations were restricted to officers . The Légion d'honneur , however, was open to men of all ranks and professions; only merit or bravery counted. The new legionnaire had to be sworn into

8343-753: The initial polynomials H 0 ( x ) and H 1 ( x ) , can be used in practice to compute the polynomials quickly. Turán's inequalities are H n ( x ) 2 − H n − 1 ( x ) H n + 1 ( x ) = ( n − 1 ) ! ∑ i = 0 n − 1 2 n − i i ! H i ( x ) 2 > 0. {\displaystyle {\mathit {H}}_{n}(x)^{2}-{\mathit {H}}_{n-1}(x){\mathit {H}}_{n+1}(x)=(n-1)!\sum _{i=0}^{n-1}{\frac {2^{n-i}}{i!}}{\mathit {H}}_{i}(x)^{2}>0.} Moreover,

8446-481: The inner product is given by the integral ⟨ f , g ⟩ = ∫ − ∞ ∞ f ( x ) g ( x ) ¯ w ( x ) d x {\displaystyle \langle f,g\rangle =\int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,w(x)\,dx} including the Gaussian weight function w ( x ) defined in

8549-701: The limit, to e − x 2 2 ⋅ H n ( x ) ∼ ( 2 n e ) n 2 2 cos ⁡ ( x 2 n − n π 2 ) ( 1 − x 2 2 n + 1 ) − 1 4 . {\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)\sim \left({\frac {2n}{e}}\right)^{\frac {n}{2}}{\sqrt {2}}\cos \left(x{\sqrt {2n}}-{\frac {n\pi }{2}}\right)\left(1-{\frac {x^{2}}{2n+1}}\right)^{-{\frac {1}{4}}}.} This expansion

8652-427: The material on variances below. The notation He and H is that used in the standard references. The polynomials He n are sometimes denoted by H n , especially in probability theory, because 1 2 π e − x 2 2 {\displaystyle {\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}} is the probability density function for

8755-403: The medal is plain red. The badge or star is not usually worn, except at the time of the decoration ceremony or on a dress uniform or formal wear . Instead, one normally wears the ribbon or rosette on their suit. For less formal occasions, recipients wear a simple stripe of thread sewn onto the lapel (red for chevaliers and officiers , silver for commandeurs ). Except when wearing

8858-405: The medal, whatever the Légion d'honneur rank and the military rank of the bearer. This is not mandatory with the ribbon. In practice, however, this is rarely done. There is not a single, complete list of all the members of the Legion in chronological order. The number is estimated at one million, including about 2,900 Knights Grand Cross. French nationals, men and women, can be received into

8961-1432: The monotonic and transition regions. Specifically, if x = 2 n + 1 cosh ⁡ ( φ ) , 0 < ε ≤ φ ≤ ω < ∞ , {\displaystyle x={\sqrt {2n+1}}\cosh(\varphi ),\quad 0<\varepsilon \leq \varphi \leq \omega <\infty ,} then e − x 2 2 ⋅ H n ( x ) = 2 n 2 − 3 4 n ! ( π n ) − 1 4 ( sinh ⁡ φ ) − 1 2 ⋅ e ( n 2 + 1 4 ) ( 2 φ − sinh ⁡ 2 φ ) ( 1 + O ( n − 1 ) ) , {\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)=2^{{\frac {n}{2}}-{\frac {3}{4}}}{\sqrt {n!}}(\pi n)^{-{\frac {1}{4}}}(\sinh \varphi )^{-{\frac {1}{2}}}\cdot e^{\left({\frac {n}{2}}+{\frac {1}{4}}\right)\left(2\varphi -\sinh 2\varphi \right)}\left(1+O\left(n^{-1}\right)\right),} while for x = 2 n + 1 + t {\displaystyle x={\sqrt {2n+1}}+t} with t complex and bounded,

9064-466: The one referred to in the text arouse in me the idea: “What magnificent results! How could he dream of such a thing?” I turn with terror and horror from this lamentable scourge of continuous functions with no derivatives. In addition to the mathematics properties named in his honor, the Hermite crater near the Moon 's north pole is named after Hermite. [REDACTED]  This article incorporates text from

9167-464: The order. Members convicted of a misdemeanour ( délit in French) can be dismissed as well, although this is not automatic. Wearing the decoration of the Légion d'honneur without having the right to do so is a serious offence. Wearing the ribbon or rosette of a foreign order is prohibited if that ribbon is mainly red, like the ribbon of the Legion of Honour. French military personnel in uniform must salute other military members in uniform wearing

9270-427: The other concerning the transformation of elliptic functions. After spending five years working privately towards his degree, in which he befriended eminent mathematicians Joseph Bertrand , Carl Gustav Jacob Jacobi, and Joseph Liouville , he took and passed the examinations for the baccalauréat , which he was awarded in 1847. He married Joseph Bertrand's sister, Louise Bertrand, in 1848. In 1848, Hermite returned to

9373-559: The physicist's Hermite polynomials H follow directly by properly scaling this: x n = n ! 2 n ∑ m = 0 ⌊ n 2 ⌋ 1 m ! ( n − 2 m ) ! H n − 2 m ( x ) . {\displaystyle x^{n}={\frac {n!}{2^{n}}}\sum _{m=0}^{\left\lfloor {\tfrac {n}{2}}\right\rfloor }{\frac {1}{m!(n-2m)!}}H_{n-2m}(x).} The Hermite polynomials are given by

9476-639: The power of 2 x with the corresponding power of √ 2   x and multiplying the entire sum by 2 : He n ⁡ ( x ) = n ! ∑ m = 0 ⌊ n 2 ⌋ ( − 1 ) m m ! ( n − 2 m ) ! x n − 2 m 2 m . {\displaystyle \operatorname {He} _{n}(x)=n!\sum _{m=0}^{\left\lfloor {\tfrac {n}{2}}\right\rfloor }{\frac {(-1)^{m}}{m!(n-2m)!}}{\frac {x^{n-2m}}{2^{m}}}.} The inverse of

9579-764: The preceding section An orthogonal basis for L ( R , w ( x ) dx ) is a complete orthogonal system . For an orthogonal system, completeness is equivalent to the fact that the 0 function is the only function f ∈ L ( R , w ( x ) dx ) orthogonal to all functions in the system. Since the linear span of Hermite polynomials is the space of all polynomials, one has to show (in physicist case) that if f satisfies ∫ − ∞ ∞ f ( x ) x n e − x 2 d x = 0 {\displaystyle \int _{-\infty }^{\infty }f(x)x^{n}e^{-x^{2}}\,dx=0} for every n ≥ 0 , then f = 0 . One possible way to do this

9682-485: The standard normal (with expected value zero) may be read off directly from the relation for even indices: E ⁡ [ X 2 n ] = ( − 1 ) n He 2 n ⁡ ( 0 ) = ( 2 n − 1 ) ! ! , {\displaystyle \operatorname {\mathbb {E} } \left[X^{2n}\right]=(-1)^{n}\operatorname {He} _{2n}(0)=(2n-1)!!,} where (2 n − 1)!!

9785-723: The sum ∑ n = 0 ∞ H n ( x ) t n n ! , {\displaystyle \sum _{n=0}^{\infty }H_{n}(x){\frac {t^{n}}{n!}},} one can evaluate the remaining integral using the calculus of residues and arrive at the desired generating function. If X is a random variable with a normal distribution with standard deviation 1 and expected value μ , then E ⁡ [ He n ⁡ ( X ) ] = μ n . {\displaystyle \operatorname {\mathbb {E} } \left[\operatorname {He} _{n}(X)\right]=\mu ^{n}.} The moments of

9888-472: The time, sometimes even changed multiple times during one historical era. The badge of the Légion is shaped as a five-armed "Maltese Asterisk ", using five distinctive "arrowhead" shaped arms inspired by the Maltese Cross . The badge is rendered in gilt (in silver for chevalier) enameled white, with an enameled laurel and oak wreath between the arms. The obverse central disc is in gilt, featuring

9991-424: The uneven spacing of the zeros near the edges, makes use of the substitution x = 2 n + 1 cos ⁡ ( φ ) , 0 < ε ≤ φ ≤ π − ε , {\displaystyle x={\sqrt {2n+1}}\cos(\varphi ),\quad 0<\varepsilon \leq \varphi \leq \pi -\varepsilon ,} with which one has

10094-1040: The uniform approximation e − x 2 2 ⋅ H n ( x ) = 2 n 2 + 1 4 n ! ( π n ) − 1 4 ( sin ⁡ φ ) − 1 2 ⋅ ( sin ⁡ ( 3 π 4 + ( n 2 + 1 4 ) ( sin ⁡ 2 φ − 2 φ ) ) + O ( n − 1 ) ) . {\displaystyle e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x)=2^{{\frac {n}{2}}+{\frac {1}{4}}}{\sqrt {n!}}(\pi n)^{-{\frac {1}{4}}}(\sin \varphi )^{-{\frac {1}{2}}}\cdot \left(\sin \left({\frac {3\pi }{4}}+\left({\frac {n}{2}}+{\frac {1}{4}}\right)\left(\sin 2\varphi -2\varphi \right)\right)+O\left(n^{-1}\right)\right).} Similar approximations hold for

10197-498: The École Polytechnique as répétiteur and examinateur d'admission . In July 1848, he was elected to the French Academy of Sciences . In 1856 he contracted smallpox. Through the influence of Augustin-Louis Cauchy and of a nun who nursed him, he resumed the practice of his Catholic faith. From 1862 to 1873 he was lecturer at the École Normale Supérieure . In 1869, he succeeded Jean-Marie Duhamel as professor of mathematics, both at

10300-548: The École Polytechnique, where he remained until 1876, and at the University of Paris , where he remained until his death. Upon his 70th birthday, he was promoted to grand officer in the French Legion of Honour . Hermite died in Paris on 14 January 1901, aged 78. An inspiring teacher, Hermite strove to cultivate admiration for simple beauty and discourage rigorous minutiae. His correspondence with Thomas Stieltjes testifies to

10403-451: Was born in Dieuze , Moselle , on 24 December 1822, with a deformity in his right foot that would impair his gait throughout his life. He was the sixth of seven children of Ferdinand Hermite and his wife, Madeleine née Lallemand. Ferdinand worked in the drapery business of Madeleine's family while also pursuing a career as an artist. The drapery business relocated to Nancy in 1828, and so did

10506-474: Was overlooked, and they were named later after Charles Hermite , who wrote on the polynomials in 1864, describing them as new. They were consequently not new, although Hermite was the first to define the multidimensional polynomials. Like the other classical orthogonal polynomials , the Hermite polynomials can be defined from several different starting points. Noting from the outset that there are two different standardizations in common use, one convenient method

10609-420: Was slow to accept his indirect political responsibility, which caused his eventual resignation on 2 December 1887. During World War I , some 55,000 decorations were conferred, 20,000 of which went to foreigners. The large number of decorations resulted from the new posthumous awards authorised in 1918. Traditionally, membership in the Légion d'honneur could not be awarded posthumously. The establishment of

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