$#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 _
6 KB (877 words) - 19:38, 5 June 2020
...act that the set of [[prime number]]s is infinite. The partial sums of the Euler series satisfy the asymptotic relation
659 bytes (104 words) - 15:21, 10 April 2023
#REDIRECT [[Euler straight line]]
33 bytes (4 words) - 19:04, 6 November 2016
This formula was established by L. Euler (1760).
803 bytes (112 words) - 17:23, 30 July 2014
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
431 bytes (67 words) - 21:21, 29 April 2012
-
31 bytes (3 words) - 19:11, 6 November 2016
...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]]].
1 KB (171 words) - 12:50, 10 August 2014
...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
2 KB (279 words) - 19:13, 14 December 2015
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^
2 KB (366 words) - 06:13, 10 April 2023
''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
2 KB (318 words) - 09:27, 10 November 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
See also [[Euler identity|Euler identity]] and [[Zeta-function|Zeta-function]].
557 bytes (85 words) - 18:50, 18 October 2014
...oriented $n$-dimensional [[Manifold|manifold]] may be calculated from the Euler class of the [[Tangent bundle|tangent bundle]] [[#References|[a1]]], p. 348
794 bytes (108 words) - 18:57, 17 April 2014
often for $q=1$, used in the [[Euler summation method|Euler summation method]].
188 bytes (34 words) - 14:11, 23 July 2014
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
2 KB (297 words) - 11:53, 23 November 2023
...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
1 KB (217 words) - 07:30, 19 December 2014
$#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
3 KB (477 words) - 08:36, 6 January 2024
...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).
2 KB (331 words) - 14:12, 13 November 2014
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).
2 KB (328 words) - 11:50, 23 November 2023
$#C+1 = 61 : ~/encyclopedia/old_files/data/E036/E.0306620 Euler transformation
The Euler transformation of series. Given a series
6 KB (972 words) - 12:59, 6 January 2024
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).
418 bytes (68 words) - 17:34, 14 February 2020
''Euler circuit, Euler cycle, Eulerian cycle''
155 bytes (17 words) - 17:04, 7 February 2011
$#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 $
4 KB (560 words) - 08:59, 4 November 2023
$#C+1 = 89 : ~/encyclopedia/old_files/data/E036/E.0306440 Euler equation
This equation was studied in detail by L. Euler, starting from 1740.
13 KB (1,798 words) - 19:06, 26 January 2022
$#C+1 = 34 : ~/encyclopedia/old_files/data/E036/E.0306520 Euler\ANDMacLaurin formula
then the Euler–MacLaurin formula becomes
5 KB (745 words) - 09:18, 6 January 2024
#REDIRECT [[Euler function]]
28 bytes (3 words) - 21:24, 23 December 2015
...s or right-angled, or both right-angled and isosceles. The segments of the Euler line satisfy the relation
This line was first considered by L. Euler (1765).
730 bytes (113 words) - 20:16, 16 January 2016
A generalization of the Goldbach–Euler problem (cf. [[Goldbach problem|Goldbach problem]]) according to which any
...of the distribution of prime numbers allows one to solve both the Goldbach–Euler problem and the more general problem of solvability of the linear Diophanti
1 KB (201 words) - 20:18, 14 October 2014
...(x,y)$ is zero. Particular integrals for it were obtained by G. Monge. The Euler–Lagrange equation was systematically investigated by S.N. Bernshtein, who
The Euler–Lagrange equation can be generalized with respect to the dimension: The e
3 KB (501 words) - 18:46, 13 November 2014
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
-
54 bytes (8 words) - 18:01, 2 May 2012
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
2 KB (358 words) - 17:36, 14 February 2020
-
38 bytes (3 words) - 11:24, 26 December 2013
...3]]], the colliding of gravitational waves [[#References|[a6]]], etc.. The Euler–Poisson–Darboux equation has rather interesting properties, e.g. in rel
A formal solution to the Euler–Poisson–Darboux equation has the form [[#References|[a8]]]
9 KB (1,341 words) - 20:49, 23 January 2024
...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
...ars the name of Euler–Knopp summation method, see [[Euler summation method|Euler summation method]].
246 bytes (39 words) - 15:19, 1 May 2014
...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
521 bytes (82 words) - 12:45, 23 November 2014
...oriented $n$-dimensional [[Manifold|manifold]] may be calculated from the Euler class of the [[Tangent bundle|tangent bundle]] [[#References|[a1]]], p. 348
794 bytes (108 words) - 18:57, 17 April 2014
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).
418 bytes (68 words) - 17:34, 14 February 2020
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
431 bytes (67 words) - 21:21, 29 April 2012
''Euler circuit, Euler cycle, Eulerian cycle''
155 bytes (17 words) - 17:04, 7 February 2011
#REDIRECT [[Euler function]]
28 bytes (3 words) - 21:24, 23 December 2015
#REDIRECT [[Euler constant]]
28 bytes (3 words) - 19:25, 29 December 2014
...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).
730 bytes (113 words) - 20:16, 16 January 2016
#REDIRECT [[Euler straight line]]
33 bytes (4 words) - 19:04, 6 November 2016
often for $q=1$, used in the [[Euler summation method|Euler summation method]].
188 bytes (34 words) - 14:11, 23 July 2014
...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]]].
1 KB (171 words) - 12:50, 10 August 2014
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
2 KB (297 words) - 11:53, 23 November 2023
''Euler totient function, Euler totient''
Another frequently used named for the [[Euler function]] $\phi(n)$, which counts a [[reduced system of residues]] modulo
3 KB (519 words) - 10:04, 14 December 2014
''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
2 KB (318 words) - 09:27, 10 November 2023
...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
1 KB (217 words) - 07:30, 19 December 2014
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^
2 KB (366 words) - 06:13, 10 April 2023
$#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
3 KB (477 words) - 08:36, 6 January 2024
...act that the set of [[prime number]]s is infinite. The partial sums of the Euler series satisfy the asymptotic relation
659 bytes (104 words) - 15:21, 10 April 2023
$#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 $
4 KB (560 words) - 08:59, 4 November 2023
See also [[Euler identity|Euler identity]] and [[Zeta-function|Zeta-function]].
557 bytes (85 words) - 18:50, 18 October 2014
...that starts at the vertex. The centroid lies on the [[Euler straight line|Euler line]].
809 bytes (127 words) - 14:54, 2 May 2023
...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
2 KB (279 words) - 19:13, 14 December 2015
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).
2 KB (328 words) - 11:50, 23 November 2023
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
7 KB (1,043 words) - 19:38, 5 June 2020
$#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 _
6 KB (877 words) - 19:38, 5 June 2020
...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
1 KB (234 words) - 10:24, 10 December 2012
...$10^n-1$. Thus, the period length divides $\phi(q)$, the [[Euler function|Euler function]].
823 bytes (124 words) - 10:22, 27 September 2014
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
2 KB (358 words) - 17:36, 14 February 2020
''$B$-function, Euler $B$-function, Euler integral of the first kind''
1,009 bytes (161 words) - 17:31, 11 November 2023
is equal to its [[Euler characteristic|Euler characteristic]]. Betti numbers were introduced by E. Betti [[#References|[
1 KB (172 words) - 13:05, 14 February 2020
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)
1 KB (205 words) - 10:58, 26 March 2023
where $c=0.5772\ldots$ is the [[Euler constant|Euler constant]] and $\operatorname{Ci}(x)$ is the [[Integral cosine|integral cos
1 KB (161 words) - 14:30, 14 February 2020
$#C+1 = 61 : ~/encyclopedia/old_files/data/E036/E.0306620 Euler transformation
The Euler transformation of series. Given a series
6 KB (972 words) - 12:59, 6 January 2024
...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
4 KB (632 words) - 20:10, 15 March 2023
...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
1 KB (184 words) - 13:35, 14 September 2014
...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]].
1 KB (206 words) - 11:54, 2 January 2021
...|coprime]] $(k + 1)$-tuple together with $n$. This is a generalisation of Euler's [[totient function]], which is $J_1$.
1 KB (172 words) - 13:07, 19 March 2023
of the Euler $ \phi $-function (cf. [[Euler function|Euler function]]).
4 KB (652 words) - 05:18, 7 March 2022
L. Euler [[#References|[1]]] was the first to study Bernoulli polynomials for arbitr
and are closely connected with the [[Euler polynomials|Euler polynomials]]
6 KB (828 words) - 10:58, 29 May 2020
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
2 KB (411 words) - 16:38, 24 November 2018
...of Venn diagrams goes back to L. Euler and they are sometimes also called Euler diagrams.
3 KB (416 words) - 12:22, 14 February 2020
...Irregularity|irregularity]] $q=5$, with topological [[Euler characteristic|Euler characteristic]] (in case $k=\mathbf C$) equal to 27. From the Fano surface
1 KB (212 words) - 15:54, 17 July 2014
...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).
1 KB (215 words) - 20:43, 5 December 2023
...[[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).
2 KB (331 words) - 14:12, 13 November 2014
...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]].
6 KB (893 words) - 17:39, 20 January 2022
...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="
15 KB (2,038 words) - 17:21, 7 February 2011
''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
2 KB (345 words) - 19:10, 6 November 2016
...come in pairs. Two important early results were the minmax theorem and the Euler equation.
==Euler equation.==
5 KB (717 words) - 18:47, 5 April 2020
A non-orientable surface with [[Euler characteristic]] zero whose boundary is a closed curve. The Möbius strip c
662 bytes (105 words) - 08:20, 4 November 2023
...$\phi(2|d|)/2$ 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
2 KB (348 words) - 05:59, 20 August 2014
is Euler's [[beta-function]]
5 KB (641 words) - 08:58, 8 April 2023
is the [[Euler characteristic|Euler characteristic]] of $ V $.
2 KB (308 words) - 06:03, 19 March 2022
This formula was established by L. Euler (1760).
803 bytes (112 words) - 17:23, 30 July 2014
$#C+1 = 89 : ~/encyclopedia/old_files/data/E036/E.0306440 Euler equation
This equation was studied in detail by L. Euler, starting from 1740.
13 KB (1,798 words) - 19:06, 26 January 2022
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.
4 KB (531 words) - 08:28, 6 June 2020
...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
3 KB (349 words) - 21:31, 29 December 2020
...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
920 bytes (144 words) - 19:18, 6 November 2016
...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
5 KB (753 words) - 19:43, 5 June 2020
...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
4 KB (608 words) - 08:18, 4 November 2023
...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 $
5 KB (717 words) - 08:27, 6 June 2020
$#C+1 = 34 : ~/encyclopedia/old_files/data/E036/E.0306520 Euler\ANDMacLaurin formula
then the Euler–MacLaurin formula becomes
5 KB (745 words) - 09:18, 6 January 2024
[[Euler product|Euler product]] with characters. They are the basic instrument for studying by an
2 KB (347 words) - 21:23, 9 January 2015
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
4 KB (610 words) - 18:39, 26 January 2022
Thus, one has absolute Cesàro summability, absolute Euler summability, absolute Nörlund summability, absolute Riesz summability, abs
968 bytes (128 words) - 17:18, 7 February 2011
...ular extremal is, by definition, a smooth solution of the [[Euler equation|Euler equation]]
is continuous. For a regular extremal the Euler equation
4 KB (657 words) - 08:10, 6 June 2020
...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]]:
3 KB (419 words) - 20:15, 19 November 2017
...[[Singular point, index of a|Singular point, index of a]]) is equal to the Euler characteristic of $M$.
867 bytes (138 words) - 19:43, 14 August 2014
...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>
5 KB (668 words) - 17:45, 4 June 2020
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
951 bytes (154 words) - 08:27, 30 December 2015
...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} $
6 KB (955 words) - 19:40, 1 November 2023
...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) $
3 KB (494 words) - 08:31, 6 January 2024
<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
5 KB (762 words) - 19:27, 12 January 2024
...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 $
7 KB (1,026 words) - 14:54, 7 June 2020
...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''' :
4 KB (553 words) - 09:38, 22 August 2014
...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]]
10 KB (1,504 words) - 08:04, 6 June 2020
...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