...ars the name of Euler–Knopp summation method, see [[Euler summation method|Euler summation method]].
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...numbers, where $\phi(m)$ is Euler's $\phi$-function (cf. [[Euler function|Euler function]]). One usually takes the numbers mutually prime with $m$ in the c
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...oriented $n$-dimensional [[Manifold|manifold]] may be calculated from the Euler class of the [[Tangent bundle|tangent bundle]] [[#References|[a1]]], p. 348
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Euler's theorem hold for polyhedrons of genus $0$; for polyhedrons of genus $p$ t
holds. This theorem was proved by L. Euler (1758); the relation \eqref{*} was known to R. Descartes (1620).
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called the Euler integral of the first kind, or the [[Beta-function|beta-function]], and
called the Euler integral of the second kind. (The latter converges for $s>0$ and is a repre
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''Euler circuit, Euler cycle, Eulerian cycle''
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#REDIRECT [[Euler function]]
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#REDIRECT [[Euler constant]]
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...s or right-angled, or both right-angled and isosceles. The segments of the Euler line satisfy the relation
This line was first considered by L. Euler (1765).
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#REDIRECT [[Euler straight line]]
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often for $q=1$, used in the [[Euler summation method|Euler summation method]].
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...ues of the complex variable $z$. In particular, for a real value $z=x$ the Euler formulas become
These formulas were published by L. Euler in [[#References|[1]]].
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The recurrence formula for the Euler numbers ($E^n\equiv E_n$ in symbolic notation) has the form
...\dots$; $E_2=-1$, $E_4=5$, $E_6=-61$, $E_8=1385$, and $E_{10}=-50521$. The Euler numbers are connected with the [[Bernoulli numbers|Bernoulli numbers]] $B_n
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''Euler totient function, Euler totient''
Another frequently used named for the [[Euler function]] $\phi(n)$, which counts a [[reduced system of residues]] modulo
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''Euler's totient function''
...ot exceeding $n$ and relatively prime to $n$ (the "totatives" of $n$). The Euler function is a [[multiplicative arithmetic function]], that is $\phi(1)=1$ a
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...be a [[quadratic residue]] or non-residue modulo $p$. It was proved by L. Euler in 1761 (see [[#References|[1]]]).
Euler also obtained a more general result: A number $a \not\equiv 0 \pmod p$ is a
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The first Euler substitution: If $a>0$, then
The second Euler substitution: If the roots $x_1$ and $x_2$ of the quadratic polynomial $ax^
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$#C+1 = 23 : ~/encyclopedia/old_files/data/E036/E.0306550 Euler polynomials
are the [[Euler numbers]]. The Euler polynomials can be computed successively by means of the formula
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...act that the set of [[prime number]]s is infinite. The partial sums of the Euler series satisfy the asymptotic relation
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$#C+1 = 37 : ~/encyclopedia/old_files/data/E036/E.0306400 Euler characteristic
It was given this name in honour of L. Euler, who proved in 1758 that the number $ V $
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See also [[Euler identity|Euler identity]] and [[Zeta-function|Zeta-function]].
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...that starts at the vertex. The centroid lies on the [[Euler straight line|Euler line]].
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...itrary real number and the product extends over all prime numbers $p$. The Euler identity also holds for all complex numbers $s = \sigma + it$ with $\sigma
The Euler identity can be generalized in the form
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considered by L. Euler (1740). Its existence follows from the fact that the sequence
...mber $\gamma$ is also known as the ''Euler-Mascheroni'' constant, after L. Euler (1707–1783) and L. Mascheroni (1750–1800).
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A smooth solution of the [[Euler equation|Euler equation]], which is a necessary extremum condition in the problem of varia
Euler's equation has the form
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$#C+1 = 34 : ~/encyclopedia/old_files/data/E036/E.0306530 Euler method
by a polygonal line (Euler's polygonal line) whose segments are rectilinear on the intervals $ [ x _
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...ides the order of the group. Fermat's little theorem was generalized by L. Euler to the case modulo an arbitrary $m$. Namely, he proved that for every numbe
where $\phi(m)$ is the [[Euler function|Euler function]]. Another generalization of Fermat's little theorem is the equati
2 KB (257 words) - 18:02, 8 November 2014
..., 1740). There exists a constant $\gamma>0$, known as the [[Euler constant|Euler constant]], such that $S_n = \ln n + \gamma + \varepsilon_n$, where $\lim\l
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...$10^n-1$. Thus, the period length divides $\phi(q)$, the [[Euler function|Euler function]].
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is summable by means of the Euler summation method ($(E,q)$-summable) to the sum $S$ if
...hen its terms $a_n$ satisfy the condition $a_n=o((2q+1)^n)$, $q\geq0$. The Euler summation method can also be applied for analytic continuation beyond the d
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''$B$-function, Euler $B$-function, Euler integral of the first kind''
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is equal to its [[Euler characteristic|Euler characteristic]]. Betti numbers were introduced by E. Betti [[#References|[
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The simplest method of computing a denumerant is by Euler's recurrence relation:
1 KB (185 words) - 16:46, 23 November 2023
...a is employed in the calculus of variations to derive the [[Euler equation|Euler equation]] in its integral form. In this proof it is not necessary to assum
1 KB (183 words) - 15:14, 27 August 2014
The spiral of Cornu is sometimes called the spiral of Euler after L. Euler, who mentioned it first (1744). Beginning with the works of A. Cornu (1874)
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where $c=0.5772\ldots$ is the [[Euler constant|Euler constant]] and $\operatorname{Ci}(x)$ is the [[Integral cosine|integral cos
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$#C+1 = 61 : ~/encyclopedia/old_files/data/E036/E.0306620 Euler transformation
The Euler transformation of series. Given a series
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...the existence of Euler cycles (Euler's theorem): A connected graph has an Euler cycle if and only if each of its vertices (except two) has even degree.
A graph is said to be Hamiltonian (Eulerian) if it has a Hamilton (Euler) cycle. A graph is said to be Hamilton-connected if any two of its vertices
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...ence is unsolvable, then $a$ is called a quadratic non-residue modulo $m$. Euler's criterion: Let $p>2$ be prime. Then an integer $a$ coprime with $p$ is a
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...n the $(x,y)$-plane for which the general integral of the [[Euler equation|Euler equation]] can be represented in the form
1 KB (227 words) - 19:28, 26 March 2023
Of course, in \eqref{*} $\phi$ denotes the [[Euler function|Euler function]].
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...|coprime]] $(k + 1)$-tuple together with $n$. This is a generalisation of Euler's [[totient function]], which is $J_1$.
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of the Euler $ \phi $-function (cf. [[Euler function|Euler function]]).
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L. Euler [[#References|[1]]] was the first to study Bernoulli polynomials for arbitr
and are closely connected with the [[Euler polynomials|Euler polynomials]]
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A collection of solutions of the [[Euler equation|Euler equation]], depending on $n$ arbitrary constants and filling without mutual
depends. Euler's equation is understood in the vector sense, that is, it is a system of $n
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...of Venn diagrams goes back to L. Euler and they are sometimes also called Euler diagrams.
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...Irregularity|irregularity]] $q=5$, with topological [[Euler characteristic|Euler characteristic]] (in case $k=\mathbf C$) equal to 27. From the Fano surface
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...mal [[Euler identity]] beween the Dirichlet series $L(a,s)$ and a formal [[Euler product]] over primes
and is [[Totally multiplicative function|totally multiplicative]] if the Euler product is of the form
2 KB (358 words) - 17:25, 11 November 2023
...ence $x^3\equiv a$ ($\bmod\,p$) may be checked for solvability using the [[Euler criterion]]: The congruence $x^3\equiv a$ ($\bmod\,p$), $(a,p)=1$, is solva
...the form $p=x^2+27y^2$ with integers $x$ and $y$ (a result conjectured by Euler and first proved by Gauss).
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...[[Polyhedron, abstract|Polyhedron, abstract]]) with [[Euler characteristic|Euler characteristic]] equal to 2 can be realized as a convex polyhedron. Here, a
2 KB (277 words) - 15:40, 15 April 2014
...elative to another one $0x'y'z'$ with the same origin and orientation. The Euler angles are regarded as the angles through which the former must be successi
These angles were introduced by L. Euler (1748).
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...ans of the series (2), which converges throughout the complex plane, or by Euler's formula
For Euler's formula see also [[Euler formulas|Euler formulas]].
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...o construct the formulas of an iteration method. Thus, after e.g. applying
Euler's method, (12) is replaced by the relations
Numerical integration of (15) by
Euler's method with respect to the points <img align="absmiddle" border="0" src="
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''Euler circle, Feuerbach circle''
...nine-point circle. The points $H,T,O,E$ then lie on a straight line (the [[Euler line]]), $E$ being the midpoint of the segment $HO$, and the pair of points
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...come in pairs. Two important early results were the minmax theorem and the Euler equation.
==Euler equation.==
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A non-orientable surface with [[Euler characteristic]] zero whose boundary is a closed curve. The Möbius strip c
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...$\phi(2|d|)/2$ or $\phi(4|d|)/2$, where $\phi(n)$ is the [[Euler function|Euler function]]. The quadratic reciprocity law makes it possible to establish fa
2 KB (304 words) - 19:26, 14 August 2014
...bra was first given by J. d'Alembert in 1746. There followed the proofs of Euler, P.S. Laplace, J.L. Lagrange and others in the second half of the 18th cent
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is Euler's [[beta-function]]
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is the [[Euler characteristic|Euler characteristic]] of $ V $.
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This formula was established by L. Euler (1760).
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$#C+1 = 89 : ~/encyclopedia/old_files/data/E036/E.0306440 Euler equation
This equation was studied in detail by L. Euler, starting from 1740.
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should pass are given. Since the general solution of the [[Euler equation|Euler equation]] of the simplest problem depends on two arbitrary constants, $
...em yields arbitrary constants, which appear in the general integral of the Euler equation.
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...nuation to the whole complex plane. After this, using the [[Euler formulas|Euler formulas]] one sees that (3) holds, from which the series expansions are re
9 KB (1,307 words) - 13:15, 9 December 2012
...apping|Degree of a mapping]]). It is related to the [[Euler characteristic|Euler characteristic]]. See also [[Poincaré theorem|Poincaré theorem]]; [[Krone
2 KB (376 words) - 18:01, 4 November 2014
...f $\pi$ was finally elucidated in analysis, with a decisive part played by Euler's formula:
2 KB (352 words) - 14:43, 14 February 2020
Formula (1) is a direct consequence of Euler's product formula
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...lassical [[triangle centre]]s. The orthocentre of a triangle lies on the [[Euler line]]. The mid-points of the three sides, the mid-points of the segments j
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...in the classical calculus of variations to represent the [[Euler equation|Euler equation]] in canonical form. Let $ L ( t, x, \dot{x} ) $
If the Hamilton function is used, the Euler equation
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...ithmetic functions have traditional symbolic notations: $\phi(n)$ is the [[Euler function]]; $d(n)$ or $\tau(n)$ is the [[number of divisors]]; $\mu(n)$ is
...ral argument. Thus, for instance, in an algebraic field $K$ of degree $n$, Euler's function $\phi(\mathfrak U)$ — the number of residue classes by the ide
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...dulus power congruence problem there is a solvability criterion, due to L. Euler: For the congruence
It follows immediately from Euler's criterion that the numbers $ 1 \dots p - 1 $
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$#C+1 = 34 : ~/encyclopedia/old_files/data/E036/E.0306520 Euler\ANDMacLaurin formula
then the Euler–MacLaurin formula becomes
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[[Euler product|Euler product]] with characters. They are the basic instrument for studying by an
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The Legendre condition, like the [[Euler equation|Euler equation]], is a necessary condition for a weak extremum. If the Legendre c
...r extremal. Such an extremal is twice continuously differentiable, and the Euler equation for it can be represented as an ordinary differential equation of
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Thus, one has absolute Cesàro summability, absolute Euler summability, absolute Nörlund summability, absolute Riesz summability, abs
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...ular extremal is, by definition, a smooth solution of the [[Euler equation|Euler equation]]
is continuous. For a regular extremal the Euler equation
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...nction $\sigma(m)$, the [[sum of divisors]] of a natural number $m$; the [[Euler function]] $\phi(m)$; and the [[Möbius function]] $\mu(m)$. The function $
Formally, the [[Dirichlet series]] of a multiplicative function $f$ has an [[Euler product]]:
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...[[Singular point, index of a|Singular point, index of a]]) is equal to the Euler characteristic of $M$.
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...nh z$, $\cosh z$, and the function $1/\Gamma(z)$, where $\Gamma(z)$ is the Euler [[Gamma-function|gamma-function]]. Entire transcendental functions have one
...tal functions consist of the frequently encountered special functions: the Euler [[gamma-function]] and [[beta-function]], the [[hypergeometric function]] a
3 KB (477 words) - 13:50, 9 April 2023
...avier–Stokes equations|Navier–Stokes equations]] from the [[Euler equation|Euler equation]] [[#References|[a7]]]. See [[#References|[a8]]] for an example of
...Ellis, M Pinsky, "The projection of Navier--Stokes equations from the Euler equations" ''J. Math. Pures Appl.'' , '''54''' (1975) pp. 157–182</TD>
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The Pochhammer equation has been integrated using the [[Euler transformation|Euler transformation]], and its particular integrals have the form
3 KB (479 words) - 17:43, 16 December 2020
where $\tau(n)$ is the [[number of divisors]] of $n$ and $\gamma$ is the [[Euler constant]], $\gamma \approx 0.577$. Obtained by P. Dirichlet in 1849; he n
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...d on the use of tools of mathematical analysis. The further development of Euler's analytic method proved to be fruitful (cf. [[Analytic number theory|Analy
in the Gaussian number field (cf. [[Gauss number|Gauss number]]) one obtains Euler's theorem: $ p = a ^ {2} + b ^ {2} $
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...mechanics)|Lagrange equations (in mechanics)]] (or to the [[Euler equation|Euler equation]] in the classical calculus of variations), in which the unknown m
3 KB (478 words) - 19:43, 5 June 2020
is equal to the [[Euler characteristic|Euler characteristic]] $ \chi ( X) $
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Knobloch, "Euler and the history of a problem in probability theory" ''Ganita–Bharati'' ,
1 KB (177 words) - 07:34, 2 December 2016
is the [[Euler constant|Euler constant]]. As a function of the complex variable $ z $,
4 KB (515 words) - 18:28, 25 February 2021
Necessary conditions for an extremum, additional to the [[Euler equation|Euler equation]], specified at points at which the extremal has a corner. Let
3 KB (496 words) - 08:28, 6 June 2020
..., abstract|Polyhedron, abstract]]; for example, the [[Euler characteristic|Euler characteristic]] or the homology groups, cf. [[Homology group|Homology grou
3 KB (522 words) - 14:11, 19 April 2014
...al is ensured by the first-order necessary condition: the [[Euler equation|Euler equation]], the [[Transversality condition|transversality condition]] and t
The Euler equation for $ \delta ^ {2} J ( \eta ) $
8 KB (1,142 words) - 22:14, 5 June 2020
...ogonal Latin squares]]) and magic squares, which has been studied since L. Euler (see [[#References|[a1]]] and [[#References|[a2]]]). See also [[#References
<TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Euler, "De quadratis magicis" G. Kowalewski (ed.) , ''Opera Omnia Ser. 1; oper
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...law of conservation of mass for any volume of a moving fluid (or gas). In Euler variables the continuity equation has the form
1 KB (175 words) - 12:28, 30 December 2018
is the [[Euler function|Euler function]], then $ g ^ \gamma $
5 KB (642 words) - 22:12, 5 June 2020
is the [[Euler constant|Euler constant]]. Its graph is:
4 KB (588 words) - 19:18, 11 January 2024
as their common point, the Euler line (cf. [[Euler straight line|Euler straight line]]) through $ O $,
is on the Euler line between $ G $
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...tation $\tau$ of its smooth boundary $\partial V^2$ are connected with the Euler characteristic $\chi$ of $V^2$ by the relation
...D valign="top">[4]</TD> <TD valign="top"> V.A. Sharafutdinov, "Relative Euler class and the Gauss–Bonnet theorem" ''Siberian Math. J.'' , '''14''' :
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...mplicit form was first stated by P.L.M. Maupertuis [[#References|[1]]]; L. Euler [[#References|[2]]] gave a proof of it for the case of the motion of one ma
...748) pp. 417</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L. Euler, "Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes
4 KB (544 words) - 22:15, 5 June 2020
...we have $\lambda(p^a) = \phi(p^a) = p^{a-1}(p-1)$ where $\phi$ denotes the Euler [[totient function]]. For powers of 2, we have $\lambda(2) = 1$, $\lambda(
1 KB (186 words) - 16:57, 25 November 2023
must be an extremal, i.e. it must satisfy the [[Euler equation|Euler equation]]
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...number of all primitive roots of order $m$ is equal to the value of the [[Euler function]] $\phi(m)$ if $\mathrm{hcf}(m,\mathrm{char}(K)) = 1$.
for $1 \le \gamma < \phi(m )$, where $\phi(m)$ is the Euler function. For a primitive root $g$, its powers $g^0=1,\ldots,g^{\phi(m)-1}$
3 KB (496 words) - 07:46, 20 December 2014
...(one writes $\Gamma(z) = (z-1)!$). It was introduced in 1729 by L. Euler in a letter to Ch. Goldbach, using the infinite product
...ain the integral representation (Euler integral of the second kind, cf. [[Euler integrals]])
10 KB (1,513 words) - 09:50, 5 December 2012
2) Euler base-$b$ pseudo-primes, whose [[Jacobi symbol]] with $b$ satisfies
...rmat) witness for $n$ if $n$ is composite and not a base-$b$ pseudo-prime. Euler and strong witnesses are similarly defined. If $W(n)$, the smallest strong
3 KB (516 words) - 17:58, 8 November 2014
...cit method for stiff initial-value problems is the backward [[Euler method|Euler method]]
...thereof is used, again for reasons of numerical stability. However, as the Euler method has only order of consistency $ p = 1 $,
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...e non-resonant case the Fuchsian system is locally gauge equivalent to the Euler system
;Example (Euler equation).
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taken in an arbitrary direction, by means of Euler's formula:
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...the number of which is $\phi(p^n-1)$, where $\phi$ is the [[Euler function|Euler function]]. Each primitive root of unity of degree $p^n-1$ is a primitive e
4 KB (749 words) - 18:32, 2 November 2014
Equation (4) is called the Euler–Ostrogradski equation (sometimes the Ostrogradski equation). This equatio
...(2). The Euler–Ostrogradski equation is analogous to the [[Euler equation|Euler equation]] for one-dimensional problems. In expanded form, (4) is a second-
4 KB (627 words) - 08:01, 6 June 2020
...bers. It was posed in 1742 by Ch. Goldbach in a communication to L. Euler. Euler replied by pointing out that in order to solve this problem it is sufficien
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...polygonal extremal is a piecewise-smooth solution of the [[Euler equation|Euler equation]] satisfying certain additional necessary conditions at the vertic
...ts determining the polygonal extremal — it is a particular solution of the Euler equation satisfying the boundary conditions. In fact, the equations (9) giv
7 KB (1,027 words) - 19:35, 5 June 2020
One of the simplest numerical methods — the [[
Euler method|
Euler method]] — is based on an approximate computation of the integral term in
The truncation error of
Euler's method is proportional to <img align="absmiddle" border="0" src="https://
32 KB (4,604 words) - 16:55, 7 February 2011
...e functional $W$ are called Willmore surfaces and are characterized by the Euler equation $\Delta H + 2 H ( H ^ { 2 } - K ) = 0$ corresponding to the variat
...hi ( \Sigma )$, where $\chi ( \Sigma )$ denotes the [[Euler characteristic|Euler characteristic]] of the surface, so that $W$ and $\tilde { W }$ have the sa
8 KB (1,192 words) - 16:56, 1 July 2020
L. Euler made an exceptionally important contribution to number theory. He proved Fe
...led to the creation of [[Analytic number theory|analytic number theory]]. Euler's generating functions (cf. [[Generating function|Generating function]]) ap
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Euler observed that the Euclidean algorithm applied to a pair of natural numbers
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...ents satisfy certain compatibility conditions. Generally, almost all known Euler systems satisfy the condition ES) described below. Let $K$ be a number fiel
...pping from $K ( L l )$ down to $K ( L )$. Next to condition ES), any given Euler system may have additional properties, cf. [[#References|[a4]]], [[#Referen
19 KB (2,901 words) - 17:41, 25 November 2023
...unity. The number of such roots in $U_n$ is given by the [[Euler function|Euler function]] $\phi(n)$, i.e. the number of residues $\bmod\,n$ which are rela
4 KB (680 words) - 13:40, 30 December 2018
...$H_1(K;\mathbf Z)=\mathbf Z\oplus\mathbf Z/(2)$, $H_2(K;\mathbf Z)=0$. Its Euler characteristic is $\chi(K)=0$. Together with the torus it is the only smoot
2 KB (269 words) - 19:36, 17 April 2014
.... [[Du Bois-Reymond lemma|du Bois-Reymond lemma]]) is the [[Euler equation|Euler equation]]:
5 KB (687 words) - 08:27, 6 June 2020
...so [[Convolution transform|Convolution transform]]; [[Euler transformation|Euler transformation]]; [[Fourier transform|Fourier transform]]; [[Gauss transfor
7 KB (959 words) - 19:34, 17 January 2024
...he constant $e^{-\gamma}$ as $u\rightarrow\infty$, where $\gamma$ is the [[Euler constant]]. The difference $\omega(u)-e^{-\gamma}$ behaves asymptotically l
2 KB (243 words) - 15:57, 22 September 2017
coincides with the reduction modulo 2 of the [[Euler class|Euler class]].
5 KB (692 words) - 17:18, 6 January 2024
is the [[Euler constant|Euler constant]].
6 KB (811 words) - 19:20, 11 January 2024
...was found by A. de Moivre (1707), its modern notation was suggested by L. Euler (1748).
1 KB (241 words) - 10:03, 4 June 2013
* The average order of $\phi(n)$, the [[Euler totient function]] of $n$, is $ \frac{6}{\pi^2} n$;
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...$ counts the number of different partitions of $n$, so that $p(4) = 5$. L. Euler gave a non-trivial recurrence relation for $p(n)$ (see [[#References|[a1]]]
2 KB (231 words) - 15:33, 11 November 2023
...acteristics of manifolds such as orientability, the [[Euler characteristic|Euler characteristic]], the [[Signature|signature]], etc.
The Euler class. Suppose that the base $ B $
35 KB (5,194 words) - 09:33, 26 March 2023
...domains (faces). Such a subdivision of the plane is known as a planar map. Euler's formula
5 KB (774 words) - 16:51, 16 March 2023
''Euler–Lagrange operator''
...for variational problems must satisfy (cf. also [[Euler–Lagrange equation|Euler–Lagrange equation]]).
17 KB (2,575 words) - 17:45, 1 July 2020
is the [[Euler characteristic|Euler characteristic]] of $ M $.
6 KB (845 words) - 01:33, 5 March 2022
This summation method can already be found in the works of L. Euler and G. Leibniz. The name "Abel summation method" originates from the [[Abel
2 KB (258 words) - 12:23, 10 January 2015
...approximate numerical method for solving equation (4) — the [[Euler method|Euler method]].
5 KB (793 words) - 11:49, 26 March 2023
...special) plane curves by means of a relation $k_1=\phi(s)$ goes back to L. Euler.
2 KB (304 words) - 19:04, 26 January 2024
The [[Euler line]] contains some of the classical centres: the centroid, the orthocentr
2 KB (333 words) - 08:29, 23 November 2023
...he point $O(a,b,c)$ into which the origin is mapped and the [[Euler angles|Euler angles]], i.e. this group of motions is hexaparametric.
6 KB (940 words) - 09:43, 11 August 2014
The reciprocity law for quadratic residues was first stated in 1772 by L. Euler. A. Legendre in 1785 formulated the law in modern form and proved a part of
2 KB (295 words) - 17:43, 19 December 2014
...}^\infty n^{-s}$ for complex numbers $s$ with real part exceeding $1$. The Euler product formula states that
where the product is over all prime numbers $p$. Taking the reciprocal of the Euler product, one sees that
10 KB (1,642 words) - 12:01, 13 February 2024
...nts. The arbitrary constants on which the solution of the [[Euler equation|Euler equation]] depends are determined by means of the tranversality condition.
...mined together with the $x^i(t)$, $i=1,\ldots,m$, from the solution of the Euler system of differential equations
9 KB (1,496 words) - 08:59, 13 February 2024
...{ R \}$. The uniformization theorem implies that for a surface of negative Euler characteristic, the following two determinations are equivalent: a complex
...sed to points" and each component of the nodal-complement having negative Euler characteristic). Expansions for the Weil–Petersson metric in a neighbourh
10 KB (1,540 words) - 17:43, 1 July 2020
representation introduced a century earlier by Euler, as Venn well knew.
general but more clumsy than Euler's and convex shapes can only work
6 KB (913 words) - 18:45, 4 March 2024
...[Dirichlet L-function|Dirichlet $L$-function]]) is then an [[Euler product|Euler product]].
...ts]]. The coefficients are not multiplicative, so the $L$-series is not an Euler product.
7 KB (1,047 words) - 17:38, 17 March 2024
be an admissible function satisfying the [[Euler equation|Euler equation]]
to be extremized and the Euler equation for the solution $ u $.
15 KB (2,260 words) - 08:17, 9 January 2024
...fluid without taking into account the viscosity (cf. also [[Euler equation|Euler equation]]). To introduce the viscosity, and also compressibility, this equ
...o, a first-order macroscopic approximation is provided by the compressible Euler equation. The connection with the Navier–Stokes equation turned out to be
9 KB (1,338 words) - 15:30, 1 July 2020
...quations used to model flows can also be solved (cf. also [[Euler equation|Euler equation]]).
8 KB (1,092 words) - 17:08, 7 February 2011
satisfies the finite-difference approximation of the [[Euler equation|Euler equation]] up to terms of order $ \max ( h , h / \tau , h / \tau ^ {2} )
6 KB (898 words) - 17:24, 31 December 2020
...known, for example, whether or not $e\pi$, $e+\pi$ or the [[Euler constant|Euler constant]] $\gamma$ are rational.
6 KB (1,000 words) - 15:36, 14 February 2020
L. Euler discovered 59 pairs of amicable numbers, while the use of electronic comput
2 KB (281 words) - 14:01, 12 November 2023
is Euler's [[Beta-function|beta-function]], and the integral on the right-hand side
5 KB (776 words) - 10:59, 29 May 2020
Also known as Cauchy-Riemann conditions and D'Alembert-Euler conditions, they are the partial differential equations that must be satisf
...Their first appearance as a criterion for analyticity was in a paper of L. Euler, delivered at the Petersburg Academy of Sciences in 1777 {{Cite|Eu}}. A.L.
7 KB (1,063 words) - 12:51, 22 March 2023
in its domain of definition it satisfies the Euler formula
2 KB (385 words) - 16:12, 1 August 2021
A number of combinatorial problems was investigated by L. Euler. One of them, the problem of the 36 officers, which is to arrange this numb
...to that of the existence of a pair of orthogonal Latin squares of order 6. Euler conjectured that there are no orthogonal Latin squares of order $ n = 4 k
12 KB (1,892 words) - 19:25, 13 January 2024
...le of least action, for the special case of an isolated body, is due to L. Euler. He showed (1744) that under the actions of central forces bodies describe
...s, Vol. 11.))</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> L. Euler, "Methodus inveniendi lineas curvas maximi minimive proprietate gandentes
5 KB (820 words) - 08:00, 6 June 2020
...ing formula|Stirling formula]], which gives an asymptotic expansion of the Euler [[Gamma-function|gamma-function]],
2 KB (344 words) - 09:03, 6 January 2024
where $\gamma$ is the [[Euler constant]].
2 KB (361 words) - 11:28, 14 June 2019
...uces to a commutative algebra of transformations; and the [[Euler equation|Euler equation]] for the rotation of a solid body, when the $ \eta _ {i} $
7 KB (1,014 words) - 20:21, 17 January 2024
...ory — mainly owing to the work of G. Leibniz, Jacob, and Johann Bernoulli, Euler and Lagrange.
...arked the beginning of the study of numerically solving extremum problems. Euler's method was the first representative of a large class of methods known as
28 KB (4,182 words) - 07:56, 16 April 2023
is Euler's function. Further, let $ \gamma , \gamma _ {0} \dots \gamma _ {r} $
is the [[Euler constant|Euler constant]]. This asymptotic equation shows that it is not possible, in gene
10 KB (1,462 words) - 11:49, 26 March 2023
...g concrete numbers $M_n$ it has been shown, for example, that $M_{31}$ (L. Euler, 1750) and $M_{61}$ (I.M. Pervushin, 1883) are Mersenne numbers. Computers
2 KB (344 words) - 06:44, 22 March 2024
Euler's method); $ k = 1 , u _ {0} = 3 / 2 , u _ {-1} = - 1/2 $;
6 KB (888 words) - 21:01, 4 April 2020
...mages/c/c026/c026310/c02631011.png" />-dimensional convex polyhedron; then Euler's relation
11 KB (1,564 words) - 17:11, 7 February 2011
is the [[Euler function|Euler function]], which is equal to the number of elements in the set $ 1 \dots
is explained to a significant degree by the theorems of L. Euler and P. Fermat, which show that the order of any element of the group divide
20 KB (3,011 words) - 09:59, 26 March 2023
or the topological [[Euler characteristic|Euler characteristic]]).
7 KB (986 words) - 19:41, 5 June 2020
where $\gamma$ is the [[Euler constant]] and $\xi(u)$
2 KB (376 words) - 08:55, 10 November 2023
is Euler's function. This relation may be interpreted as the law of uniform distribu
7 KB (1,065 words) - 14:05, 17 March 2020
...onstant matrix. The monodromy group has been computed for the equations of Euler and Papperitz (see [[#References|[1]]], [[#References|[2]]]).
3 KB (379 words) - 08:01, 6 June 2020
Equation (1) was examined by W. Brouncker (1657), P. Fermat and J. Wallis. L. Euler, on account of a misunderstanding, ascribed it to J. Pell.
2 KB (360 words) - 09:21, 27 April 2014
is the Euler function.
3 KB (435 words) - 18:23, 2 January 2021
is the Euler characteristic of the variety $X$ with coefficients in the structure sheaf
2 KB (416 words) - 12:41, 19 August 2014
...PIC method, cf. [[#References|[3]]]). This uses two calculation grids, the Euler and the Lagrange grid. Because of the discrete representation of continuous
3 KB (428 words) - 17:24, 7 February 2011
...eralized function]]. The concept of a generalized solution goes back to L. Euler [[#References|[9]]].
...from Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> L. Euler, "Institutionum calculi integralis" G. Kowalewski (ed.) , ''Opera Omnia
7 KB (1,006 words) - 19:41, 5 June 2020
...ce scheme thus described is of the first order of accuracy (an analogue of Euler's method of polygonal lines for solving ordinary differential equations). A
6 KB (1,008 words) - 15:54, 14 February 2020
that is, it is the left-hand side of the Euler equation, which is a necessary condition for a minimum of $ I ( y) $.
3 KB (409 words) - 19:40, 5 June 2020
...edges plus the number of faces) is equal to two; symbolically, $V-E+F=2$ (Euler's theorem). For a polyhedron of genus $p$ one has $V-E+F=2-2p$. A
9 KB (1,472 words) - 18:56, 25 April 2012
(Euler's identity);
7 KB (992 words) - 09:26, 26 March 2023
as an Euler product over prime numbers $ p $:
is Euler's function. If, on the other hand, $ \chi \neq \chi _ {0} $
15 KB (2,181 words) - 11:50, 26 March 2023
...ty of the congruence $x^n \equiv a \pmod p$ can be answered by using the [[Euler criterion]]: If $q = \mathrm{hcf}(n,p-1)$, then for the congruence $x^n \eq
3 KB (435 words) - 19:32, 19 December 2014
...ning of an intensive development of mathematical analysis. The works of L. Euler, Jacob and Johann Bernoulli and J.L. Lagrange played an essential role in i
(see [[Euler substitutions|Euler substitutions]]); certain cases of integrals of differential binomials (cf.
22 KB (3,436 words) - 22:12, 5 June 2020
...{ dim} H _ { S } ^ { i } ( D ) < \infty$ and the [[Euler characteristic|Euler characteristic]] $\chi ( D ) = \sum ( - 1 ) ^ { i } \operatorname { dim } H
8 KB (1,101 words) - 17:44, 1 July 2020
ii) all the factors $\sum_{k=0}^\infty f(p^k) p^{-ks}$ of the [[Euler product]] of $\sum_n f(n) n^{-s}$ are non-zero.
3 KB (410 words) - 11:03, 21 October 2017
the [[Euler–Lagrange equation|Euler–Lagrange equation]] of the corresponding variational principle or Yang–
...sion two is Gauss' classical theorem expressing the [[Euler characteristic|Euler characteristic]] as the integral of the [[Scalar curvature|scalar curvature
7 KB (1,013 words) - 16:45, 1 July 2020
===Euler===
...silon\iota\alpha$, i.e. circumference, 1736; the notation was borrowed by Euler from H. Jones.), and to introduce the imaginary unit $i$ (from the French
18 KB (2,697 words) - 13:11, 13 December 2013
is played by the [[Euler characteristic|Euler characteristic]] $ \chi ( B _ {+} ) $,
with positive, zero and negative Euler characteristics. In a similar manner, $ N _ {+} , N _ {0} , N _ {-} $
16 KB (2,200 words) - 08:30, 14 January 2024
the Hausdorff method turns into the Euler method $ ( E, q) $,
3 KB (472 words) - 18:52, 29 October 2023
to a canonical form. Condition (4) was first obtained by L. Euler in 1755 (see [[#References|[2]]]).
...) pp. 76–135</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L. Euler, "Institutiones calculi differentialis" G. Kowalewski (ed.) , ''Opera Om
9 KB (1,342 words) - 19:08, 9 January 2024
...gles whose sides and angles are less than $\pi$ (such triangles are called Euler triangles). The sides $a,b,c$ of a spherical triangle are measured by the p
In every spherical (Euler) triangle, each side is less than the sum of, and more than the difference
8 KB (1,389 words) - 15:53, 19 April 2014
...these numbers gained more ground, mainly owing to the contributions of L. Euler (1707–1783), who also introduced the letter $i$ to denote $\sqrt{-1}$ ($i
3 KB (437 words) - 07:46, 4 November 2023
...tions (cf. [[Variational calculus|Variational calculus]]; [[Euler equation|Euler equation]]; [[Weierstrass conditions (for a variational extremum)|Weierstra
...numerical methods of the calculus of variations appeared in the work of L. Euler. However, their most rapid development took place in the mid-20th century a
23 KB (3,462 words) - 08:27, 6 June 2020
is the Euler function; it is asymptotically maximal as $ n \rightarrow \infty $.
3 KB (507 words) - 13:23, 8 February 2020
...Finsler space. The Euler–Lagrange equations (cf. [[Euler–Lagrange equation|Euler–Lagrange equation]]) of $ ( M ^ {n} ,F ) $
by the [[Euler theorem|Euler theorem]] on homogeneous functions. Note that $ G ^ {i} $,
10 KB (1,411 words) - 10:58, 29 May 2020
is the Euler constant;
4 KB (523 words) - 08:23, 20 January 2024
...','Zeta-function','../z/z099260.htm')" style="background-color:yellow;">L. Euler. One of the simplest methods for solving the Cauchy problem is named after
This is Euler's method.
16 KB (2,467 words) - 15:35, 4 June 2020
...stimates of coefficients, analyticity properties, the functional equation, Euler product expansion) in view of the presence of a non-trivial ring of corresp
...operators are characterized by the fact that their Mellin transforms have Euler product expansions.
8 KB (1,125 words) - 07:04, 29 March 2024
it is also the most widely used. This method is known as Euler's method. From (2) for $ q = 2, 3 $
9 KB (1,315 words) - 20:04, 15 January 2024
[[Euler characteristic|Euler characteristic]] is non-negative and the rational cohomology algebra satisf
7 KB (1,128 words) - 07:52, 9 December 2023
...e shows that the Tits form $q_Q$ coincides with the [[Euler characteristic|Euler characteristic]] $\chi _ { Q } : K _ { 0 } ( Q ) \rightarrow \mathbf{Z}$, $
...$ is of global dimension at most two, the form $q_{ R}$ coincides with the Euler characteristic $\chi _ { R } : K _ { 0 } ( \operatorname { mod } R ) \right
18 KB (2,636 words) - 06:50, 15 February 2024
Trigonometric series are first encountered in the work of L. Euler (1744). He obtained the expansions
...ysts of that time, such as D. Bernoulli, J. d'Alembert, J.L. Lagrange, and Euler. These arguments related to the essence of the notion of a function. At tha
15 KB (2,280 words) - 14:56, 7 June 2020
is the [[Euler characteristic|Euler characteristic]] of $ P $.
10 KB (1,458 words) - 07:41, 10 February 2024
The properties mentioned can be combined in the Euler product expansion of the [[formal Dirichlet series]]
3 KB (483 words) - 14:20, 17 March 2023
...the [[Voronoi summation method]], the [[Cesàro summation methods]], the [[Euler summation method]], the [[Riesz summation method]] $(R,p_n)$, the [[Hausdor
3 KB (519 words) - 22:33, 8 May 2012
contains, beside a Pontryagin class, also the [[Euler class|Euler class]] $ e \in H ^ {n} ( \mathop{\rm BSO} _ {n} ) $.
be the Euler class of $ \xi \otimes \mathbf C $,
14 KB (1,987 words) - 19:15, 16 January 2024
...opediaofmath.org/legacyimages/t/t110/t110020/t11002015.png" /> one may use
Euler's notation: <img align="absmiddle" border="0" src="https://www.encyclopedia
.../www.encyclopediaofmath.org/legacyimages/t/t110/t110020/t11002049.png" /> (
Euler's notation) satisfy the plane triangle inequalities.
32 KB (4,639 words) - 17:23, 7 February 2011
Euler's formula
12 KB (1,576 words) - 01:14, 21 January 2022
...ign="top">[a4]</td> <td valign="top"> A.J. van der Poorten, "A proof that Euler missed $...$ Apéry's proof of the irrationality of $\zeta ( 3 )$" ''Math.
3 KB (525 words) - 20:51, 23 January 2024
The Laplace equation occurs in papers of L.
Euler and J. d'Alembert (see [[#References|[1]]], [[#References|[2]]]) in connect
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.
Euler, ''Novi Commentarii Acad. Sci. Petropolitanae'' , '''6''' (1761) {{MR|}
9 KB (1,300 words) - 19:02, 9 January 2024
...multipliers, are written in specific formulations as the [[Euler equation|Euler equation]], the [[Weierstrass conditions (for a variational extremum)|Weier
11 KB (1,859 words) - 23:04, 27 June 2014
The first-mentioned notation is due to L. Euler; the notation $ C _ {N} ^ { n } $
5 KB (729 words) - 10:59, 29 May 2020
Its [[Euler characteristic|Euler characteristic]] is 1.
10 KB (1,510 words) - 18:38, 13 January 2024
A theorem expressing the [[Euler characteristic|Euler characteristic]] $ \chi ( {\mathcal E} ) $
10 KB (1,385 words) - 03:10, 2 March 2022
...exceeding $n$ that are relatively prime to $n$ (cf. also [[Euler function|Euler function]]):
...red set $P$ with minimum $0$ and maximum $1$ is the [[Euler characteristic|Euler characteristic]] of the order complex of $P$, the [[Simplicial complex|simp
11 KB (1,758 words) - 09:24, 10 November 2023
...h the aid of analysis, via a certain series-production relation called the Euler product formula for $G$.
...ne may substitute the symbol $y$ for $c ^ { - z }$ and obtain the modified Euler product formula:
24 KB (3,738 words) - 07:41, 7 February 2024
is the [[Euler function|Euler function]].) Choose a number $ e > 1 $
12 KB (1,852 words) - 17:31, 5 June 2020
...solutions of problems that reduce to (2). For example, the [[Euler method|Euler method]] with irregular step $ h _ {k} $
For example, applying Euler's method with forward step $ h _ {k + 1 } $
23 KB (3,277 words) - 19:16, 16 January 2024
...ncyclopediaofmath.org/legacyimages/k/k120/k120090/k12009019.png" /> is the Euler [[Gamma-function|gamma-function]].
11 KB (1,477 words) - 13:44, 14 November 2012
is called the Finsler metric tensor. By Euler's theorem on homogeneous functions,
12 KB (1,843 words) - 19:39, 5 June 2020
...formula is the Hopf formula, which asserts that the [[Euler characteristic|Euler characteristic]] $ \chi ( X) $
10 KB (1,491 words) - 19:04, 26 March 2023
...\omega , L _ { + } , L _ { - } )$ are the standard [[Euler characteristic|Euler characteristic]] of $M$, the [[Lefschetz number|Lefschetz number]] of $\phi
15 KB (2,211 words) - 07:18, 15 February 2024
...is fulfilled. In concrete problems, this method was already applied by L. Euler and D. Bernoulli, but its complete elaboration was given by J.L. Lagrange [
4 KB (627 words) - 07:23, 16 June 2022
...ong minimum can be stated in the form of a maximum principle, in which the Euler equation (3), the transversality condition (4) and the Weierstrass conditio
5 KB (680 words) - 22:15, 5 June 2020
...t was defined by A.M. Legendre (1789), but it was considered earlier by L. Euler (1776).
...ces it to the [[Hamilton function|Hamilton function]]. Here, the system of Euler equations (in variational calculus) and the Lagrange equation (in classical
8 KB (1,082 words) - 17:09, 17 January 2024
where $\gamma$ is the [[Euler constant|Euler constant]]. The asymptotics of the sum
11 KB (1,578 words) - 11:58, 26 March 2023
...ical integration of ordinary differential equations. Applying the simplest Euler method in the direct method of variation of parameter to the Cauchy problem
10 KB (1,328 words) - 17:04, 7 February 2011
...hese proofs imply that the Weyl character gives the [[Euler characteristic|Euler characteristic]] of $ {\mathcal L} _ \chi $
10 KB (1,486 words) - 08:19, 4 March 2022
...f had dealt with the case $n=4$. It was many years later, in 1753, that L. Euler dealt with the case $n=3$. There was an alleged omission in the argument, l
8 KB (1,419 words) - 15:20, 17 March 2023
Infinite products with factors that are functions were encountered by L. Euler (1742); an example is
5 KB (741 words) - 17:20, 24 April 2016
...ed in the works of mathematicians of the 17th century and 18th century. L. Euler first came to the conclusion that the question must be posed, not what the
4 KB (679 words) - 19:36, 5 June 2020
The first systematic results in this field were obtained in 1748 by L. Euler, who employed power series to study the partition of integers into positive
...are solved by means of generating functions. This method was introduced by Euler and forms the basis of the analytic methods developed by G.H. Hardy, J.E. L
10 KB (1,609 words) - 13:03, 8 February 2020
the [[Euler characteristic|Euler characteristic]] of the structure sheaf, $ \chi ( {\mathcal O} ) = 2 $,
and the Euler–Poincaré characteristic, $ e( X)= 24 $.
10 KB (1,443 words) - 15:43, 1 March 2022
[[Euler formulas|Euler formulas]] $e^{\pm i\phi} = \cos\phi\pm i\sin\phi$ transform the trigonomet
(1722). The symbol $i=\i$ was proposed by L. Euler (1777, published
14 KB (2,281 words) - 09:55, 26 March 2023
L. Euler in 1740 pointed out the connection between Bernoulli numbers and the values
4 KB (684 words) - 18:44, 5 October 2023
...integral]] (cf. also [[Distribution of prime numbers]]) and $\phi$ is the Euler totient function (cf. [[Totient function]]).
4 KB (622 words) - 16:31, 23 December 2014
...bsequent development and improvement of the laws of dynamics are due to L. Euler, J. d'Alembert and J.L. Lagrange, who gave general methods for establishing
<TR><TD valign="top">[4]</TD> <TD valign="top"> L. Euler, "Principles of point dynamics" , Moscow-Leningrad (1938) (In Russian;
11 KB (1,677 words) - 20:41, 21 December 2016
...[[Gauss–Bonnet theorem|Gauss–Bonnet theorem]], the [[Euler characteristic|Euler characteristic]] of $M$ if $M$ is a surface). There are more results along
...\ldots , \operatorname { spec } ( M , \Delta ^ { ( \dim M ) } )$ gives the Euler characteristic of $M$ and another linear combination of the derivatives at
14 KB (2,008 words) - 16:55, 1 July 2020
to the solution of the Euler–Lagrange equation for a minimal surface. Experiments by J. Plateau (1849)
of a minimal surface having a given topological type (i.e. a given Euler characteristic and orientability character) and being bounded by a given co
10 KB (1,540 words) - 11:46, 3 March 2022
...lée-Poussin theorem|de la Vallée-Poussin theorem]]). Riemann considers the Euler zeta-function (also called the [[Riemann zeta-function|Riemann zeta-functio
5 KB (666 words) - 18:50, 26 January 2024
...lier (1737, 1749) the zeta-function with real $s$-values was studied by L. Euler, who demonstrated the identity specifying the connection between $\zeta(s)$
where $\phi(k)$ is the [[Euler function|Euler function]] and $1\leq k\leq \ln^Bx$, where $A$ and $B$ are arbitrary fixed
27 KB (4,516 words) - 18:38, 18 October 2014
...etric series]], but had been considered (together with its solution) by L. Euler at an even earlier date.
8 KB (1,050 words) - 22:11, 5 June 2020
...$p(n)$ gives the number of different partitions of $n$. So, $p(4) = 5$. L. Euler gave a non-trivial recurrence relation for $p(n)$ (see [[#References|[a1]]]
4 KB (718 words) - 15:32, 11 November 2023
is Euler's function. The field $ K _ {n} $
12 KB (1,769 words) - 11:19, 26 March 2023
The Euler–Frobenius polynomials $p _ { m } ( x )$ of degree $m - 1 \geq 0$ are char
...invariance of this kind is to look for an appropriate space with which the Euler–Frobenius polynomials $( p _ { m } ( x ) ) _ { m \geq 1 }$ are attached i
5 KB (742 words) - 07:27, 25 January 2024
L. Euler (1737, 1749, see [[#References|[1]]]) introduced the function
...in the theory of the distribution of prime numbers. Based on the identity, Euler proved that the series $ \sum 1/p $
36 KB (5,530 words) - 08:29, 14 January 2024
sometimes called the Laplace–Fourier kernel, the Mellin kernel and the Euler kernel, respectively. Various changes of variables lead to modified forms o
...at can be used throughout the entire domain of existence. For example, the Euler integral of the second kind,
21 KB (2,949 words) - 22:12, 5 June 2020
...s of semi-linear evolution equations (Navier–Stokes, Kuramoto–Sivashinksi, Euler, the [[Ginzburg–Landau equation|Ginzburg–Landau equation]]) with period
...al equations; and [[#References|[a20]]] for a generalized [[Euler equation|Euler equation]].
21 KB (3,112 words) - 08:01, 6 February 2024
The [[Euler equation|Euler equation]] of $ J ( \phi ) $
14 KB (2,005 words) - 08:29, 6 June 2020
The topological [[Euler characteristic|Euler characteristic]] satisfies the formula
15 KB (2,046 words) - 05:46, 13 June 2022
and the Euler characteristic $ \chi ( X) $(
5 KB (749 words) - 07:33, 22 December 2023
...), and entropy inequality (a5), with $f$ given by (a6). This system is the Euler system of mono-atomic perfect gas dynamics.
...tic description of hyperbolic systems of conservation laws, other than the Euler system of gas dynamics. Most of the known generalized BGK-models fit in the
13 KB (2,050 words) - 16:58, 1 July 2020
...ons allows one to define "free" functions under constraints given by the Euler equations and the Weierstrass necessary condition.
12 KB (1,716 words) - 17:18, 7 February 2011
The Calabi–Yau manifolds with non-vanishing [[Euler characteristic|Euler number]] provide the only known "compactifications" $ M ^ {4} \times K $
12 KB (1,750 words) - 19:20, 27 October 2023
...ions of functions with respect to them have appeared in the research of L. Euler, D. Bernoulli, A. Legendre, P. Laplace, F. Bessel, and others into various
...[15]</TD> <TD valign="top"> A.B. Paplauskas, "Trigonometric series from Euler to Lebesgue" , Moscow (1966) (In Russian)</TD></TR></table>
18 KB (2,612 words) - 13:13, 18 February 2022
...ings of the sphere onto the plane, the system (3) turns into the so-called Euler–Urmaev system (see [[#References|[5]]], [[#References|[11]]]), while for
...class of projections, of classical heritage, is known, namely the class of Euler projections: $ ab = 1 $,
28 KB (4,001 words) - 11:10, 4 June 2020
is the identity mapping of a simply-connected complex with Euler characteristic $ \chi $,
6 KB (926 words) - 19:41, 16 January 2024
...aph without passing through any edge more than once; it was obtained by L.
Euler in 1736 in solving the problem of the bridges of Königsberg. The [[Four-co
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.
Euler, , ''Commentationes Arithmetica Collectae'' , St. Petersburg (1766) pp.
13 KB (1,976 words) - 20:08, 15 March 2023