...the number $V$ of its vertices plus the number $F$ of its faces minus the number $E$ of its edges is equal to 2:
Euler's theorem hold for polyhedrons of genus $0$; for polyhedrons of genus $p$ t
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If an integer $a$ is not divisible by a prime number $p>2$, then the congruence
...be a [[quadratic residue]] or non-residue modulo $p$. It was proved by L. Euler in 1761 (see [[#References|[1]]]).
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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
[[Category:Number theory]]
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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
2 KB (318 words) - 09:27, 10 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
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where $s$ is a real number and $p$ runs through all prime numbers. This product converges absolutely f
See also [[Euler identity|Euler identity]] and [[Zeta-function|Zeta-function]].
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...n the above examples these are: 09 for $1/11$ and 0 or 9 for $7/4$. If the number is irrational, the infinite decimal fraction cannot be recurrent (e.g. $\sq
...$10^n-1$. Thus, the period length divides $\phi(q)$, the [[Euler function|Euler function]].
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$#C+1 = 37 : ~/encyclopedia/old_files/data/E036/E.0306400 Euler characteristic
is the number of $ k $-
4 KB (560 words) - 08:59, 4 November 2023
...Namely, he proved that for every number $a$ relatively prime to the given number $m>1$ there is the congruence
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
The number $\gamma$ defined by
considered by L. Euler (1740). Its existence follows from the fact that the sequence
2 KB (328 words) - 11:50, 23 November 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 $\sigma
The Euler identity can be generalized in the form
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''$r$-dimensional Betti number $p^r$ of a complex $K$''
...ant of the polyhedron which realizes the complex $K$, and it indicates the number of pairwise non-homological (over the rational numbers) cycles in it. For i
1 KB (172 words) - 13:05, 14 February 2020
$#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
...|coprime]] $(k + 1)$-tuple together with $n$. This is a generalisation of Euler's [[totient function]], which is $J_1$.
* Ram Murty, M. Problems in Analytic Number Theory, ser. Graduate Texts in Mathematics '''206''' Springer-Verlag (2001)
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...artitions of an integer $n$ into parts equal to $a_1,\ldots,a_m$, i.e. the number of solutions in non-negative integers $x_1,\ldots,x_m$ of the equation
The simplest method of computing a denumerant is by Euler's recurrence relation:
1 KB (185 words) - 16:46, 23 November 2023
of the number $ n $,
is the number of integers $ k $
4 KB (652 words) - 05:18, 7 March 2022
3) The
number of terms in the Farey series of order $n$ is equal to
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Bukhshtab, "
Number theory" , Moscow (1966) (In Russian)</TD></TR><TR><TD valign="top">[2]</T
1 KB (206 words) - 11:54, 2 January 2021
...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
...modulo $p$. It follows from the criterion, in particular, that for a prime number $p$, the sequence of numbers $1,\dots,p-1$ contains exactly $(q-1)(p-1)/q$
1 KB (215 words) - 20:43, 5 December 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
...D valign="top">[1]</TD> <TD valign="top"> I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)</TD></TR></tabl
1 KB (184 words) - 13:35, 14 September 2014
...ce the factorization into prime factors in $\mathbf Q(\sqrt d)$ of a prime number that does not divide $d$ depends on whether or not $x^2-d$ is reducible mod
...gn="top">[2]</TD> <TD valign="top"> Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translati
2 KB (304 words) - 19:26, 14 August 2014
$#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
...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.
...and quadruply-connected; the graph has no loops or multiple edges, and the number $ n $
4 KB (632 words) - 20:10, 15 March 2023
...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
...length of a circle to its diameter; it is an infinite non-periodic decimal number
One frequently arrives at the number $\pi$ as the limit of certain arithmetic sequences involving simple laws. A
2 KB (352 words) - 14:43, 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
...intersections some part of the $(n+1)$-dimensional space. Here $n$ is the number of unknown functions $y_i(x)$, $i=1,\dots,n$, on which the functional to be
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
''number-theoretic function''
...er than $x$ — describes the distribution of primes; $\pi(x,q,l)$ gives the number of primes not larger than $x$ in the arithmetic progression $p\equiv l\pmod
4 KB (608 words) - 08:18, 4 November 2023
is the base of the natural logarithm, which is also known as the Napier number. This function is defined for any value of $ z $ (real or complex) by
i.e. for any natural number $ b > 0 $,
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''for the number of divisors''
...Obtained by P. Dirichlet in 1849; he noted that this sum is equal to the number of points $(x,y)$ with positive integer coordinates in the domain bounded b
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...number of all primitive roots of order $m$ is equal to the value of the [[Euler function]] $\phi(m)$ if $\mathrm{hcf}(m,\mathrm{char}(K)) = 1$.
In the field of [[complex number]]s, there are primitive roots of unity of every order: those of order $m$ t
3 KB (496 words) - 07:46, 20 December 2014
...ist, after which it is demonstrated that at least one of them is a complex number. C.F. Gauss was the first to prove the fundamental theorem of algebra witho
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$#C+1 = 34 : ~/encyclopedia/old_files/data/P074/P.0704530 Prime number
A natural number (positive [[Integer|integer]]) $ p > 1 $
6 KB (955 words) - 19:40, 1 November 2023
$#C+1 = 33 : ~/encyclopedia/old_files/data/L057/L.0507990 Lefschetz number
By definition, the Lefschetz number of $ f $
3 KB (494 words) - 08:31, 6 January 2024
is a natural number. If the congruence (1) is solvable, $ a $
...dulus power congruence problem there is a solvability criterion, due to L. Euler: For the congruence
5 KB (717 words) - 08:27, 6 June 2020
...apping|Degree of a mapping]]). It is related to the [[Euler characteristic|Euler characteristic]]. See also [[Poincaré theorem|Poincaré theorem]]; [[Krone
Cf. also [[Rotation number|Rotation number]] of a curve, which is the rotation of the unit tangent vector field of the
2 KB (376 words) - 18:01, 4 November 2014
...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(
A [[Carmichael number]] is a composite number with the property that $\lambda(n)$ divides $n-1$.
1 KB (186 words) - 16:57, 25 November 2023
''of a number $ a $
is a prime number; consequently, the notion of an index is only defined for these moduli.
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Let $f$, $g$ be functions on the [[natural number]]s. We say that $f$ has average order $g$ if the [[asymptotic equality]]
* The average order of $d(n)$, the [[number of divisors]] of $n$, is $\log n$;
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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]]:
3 KB (419 words) - 20:15, 19 November 2017
...problems of a much more general nature. The problem of writing every even number as a sum of two prime numbers has not yet (1989) been solved.
...oved that every sufficiently large even number is a sum of a prime and a number composed of at most two primes. For a proof see {{Cite|HaRi|Chapt. 11}}.
3 KB (514 words) - 21:14, 9 January 2015
Traditionally, a composite natural number $n$ is called a pseudo-prime if $2^{n-1} \equiv 1$ modulo $n$, for it has l
More recently, the concept has been extended to include any composite number that acts like a prime in some realization of a [[probabilistic primality t
3 KB (516 words) - 17:58, 8 November 2014
...field on a manifold|Vector field on a manifold]]) and let it have a finite number of isolated singular points $ A _ {1}, \dots, A _ {k} $.
is the [[Euler characteristic|Euler characteristic]] of $ V $.
2 KB (308 words) - 06:03, 19 March 2022
The formula expressing the rule for raising a [[Complex number|complex number]], expressed in trigonometric form
...power. According to de Moivre's formula the modulus $\rho$ of the complex number is raised to that power and the argument $\varphi$ is multiplied by the exp
1 KB (241 words) - 10:03, 4 June 2013
...or]] of two integers $a \ge b > 0$ is quite fast. It can be shown that the number of steps required is at most
...rs $(a,b)$ yields the [[continued fraction]] development of the [[rational number]] $a/b$.
2 KB (351 words) - 20:40, 16 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
is the number of degrees of freedom of the system, and he defined the function
3 KB (478 words) - 19:43, 5 June 2020
A field with a finite number of elements. First considered by E. Galois [[#References|[1]]].
...d|characteristic]] of this field. For any prime number $p$ and any natural number $n$ there exists a (unique up to an isomorphism) field of $p^n$ elements. I
4 KB (749 words) - 18:32, 2 November 2014
is the [[Euler constant|Euler constant]]. As a function of the complex variable $ z $,
the number of primes smaller than $ x $
4 KB (515 words) - 18:28, 25 February 2021
This function occurs in number theory as the limit
where $\Phi(x,y)$ denotes the number of positive integers not exceeding $x$ that are free of prime factors small
2 KB (243 words) - 15:57, 22 September 2017
An $n$-th root of a number $a$ is a number $x=a^{1/n}$ whose $n$-th power $x^n$ is equal to $a$.
...U_n$ is given by the [[Euler function|Euler function]] $\phi(n)$, i.e. the number of residues $\bmod\,n$ which are relatively prime to $n$. In a field of cha
4 KB (680 words) - 13:40, 30 December 2018
should pass are given. Since the general solution of the [[Euler equation|Euler equation]] of the simplest problem depends on two arbitrary constants, $
...ven manifolds. For instance, if in the [[Bolza problem|Bolza problem]] the number of boundary conditions to be satisfied by the sought curve $ x = ( x _ {1
4 KB (531 words) - 08:28, 6 June 2020
A formula which expresses the number $ \pi /2 $
Formula (1) is a direct consequence of Euler's product formula
3 KB (349 words) - 21:31, 29 December 2020
...$ 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]]]
...composition being distinct (see [[#References|[a3]]]). See also [[Additive number theory]]; [[Additive problems]].
2 KB (231 words) - 15:33, 11 November 2023
Any number $ a $,
Already in the Middle Ages a number of algorithms for constructing magic squares of odd order $ n $
5 KB (762 words) - 19:27, 12 January 2024
...the [[aliquot divisor]]s of the other, i.e. of the divisors other than the number itself. This definition is found already in Euclid's Elements and in the wo
...als of the amicable numbers converges {{Cite|Po1}}. If $A(x)$ denotes the number of integers $\le x$ that belong to an amicable pair, then it is known that
2 KB (281 words) - 14:01, 12 November 2023
[[Euler product|Euler product]] with characters. They are the basic instrument for studying by an
...$\rho : G \to {\textrm{GL}}(n,\C)$ of the Galois group $G$ of an algebraic number field $K$ (cf.
2 KB (347 words) - 21:23, 9 January 2015
...rational number $-r''$: $-r'>-r''$. The absolute value $|r|$ of a rational number $r$ is defined in the usual way: $|r|=r$ if $r\geq0$ and $|r|=-r$ if $r<0$.
...om its equivalence class, i.e. $a/b\in r$. Thus, one and the same rational number can be written as distinct, but equivalent, rational fractions.
6 KB (1,000 words) - 15:36, 14 February 2020
...ely from the formula given in [[Classical combinatorial problems]] for the
number of ''[[derangement]]s'': permutations $\pi$ such that $\pi(i)\neq i$ for al
<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
$#C+1 = 22 : ~/encyclopedia/old_files/data/N067/N.0607940 Number theory
...trical figures, were the first and the most ancient mathematical concepts. Number theory arose from problems in [[Arithmetic|arithmetic]] connected with the
10 KB (1,503 words) - 08:03, 6 June 2020
can be summed by the Abel method ($A$-method) to the number $S$ if, for any real $x$, $0<x<1$, the series
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
$#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
...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
The exponents $n$ such that the Mersenne number $M_n$ is prime that were known before 1970 are
2 KB (344 words) - 06:44, 22 March 2024
$#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
...at is, a [[Complex number|complex number]] which is not real; an imaginary number of the form $iy$ is called purely imaginary (sometimes only these numbers a
Usually the phrases "imaginary number" and "complex number" just mean the same, "imaginary" being the historically chosen word and
3 KB (437 words) - 07:46, 4 November 2023
Inequalities following from Morse theory and relating the number of critical points (cf. [[Critical point|Critical point]]) of a Morse funct
...th $ n $-dimensional manifold $ M $ (without boundary) having a finite number of critical points. Then the [[Homology group|homology group]] $ H _ \la
6 KB (845 words) - 01:33, 5 March 2022
...domains (faces). Such a subdivision of the plane is known as a planar map. Euler's formula
...e $n$ is the number of vertices, $m$ is the number of edges and $r$ is the number of faces of the map (including the exterior face) applies to any planar map
5 KB (774 words) - 16:51, 16 March 2023
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
...d) have been an important driving force for the development of [[algebraic number theory]] and [[class field theory]]. A far-reaching generalization of the q
2 KB (295 words) - 17:43, 19 December 2014
..., the [[Reynolds number|Reynolds number]] and the [[Prandtl number|Prandtl number]].
...with the mean velocity $u$ divided by the viscosity $\nu$, and the Prandlt number is the ratio of the viscosity and the thermometric conductivity $\kappa$:
9 KB (1,338 words) - 15:30, 1 July 2020
is the Euler function.
is an algebraic number field, the only prime divisors that may be ramified in $ k ( \zeta _ {n}
3 KB (435 words) - 18:23, 2 January 2021
...those values of $\alpha$ that could not be well approximated by a rational number with a small denominator. Vinogradov's estimate used the sieve of Eratosthe
...}^\infty n^{-s}$ for complex numbers $s$ with real part exceeding $1$. The Euler product formula states that
10 KB (1,642 words) - 12:01, 13 February 2024
...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.
...ong to the segment of the natural number series $[1,n^2+n]$, but not every number of this segment can be an order of the complete group of motions.
6 KB (940 words) - 09:43, 11 August 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
where $d$ is a positive integer, $\sqrt d$ is an irrational number, $c$ is an integer, and the unknowns $x$ and $y$ are integers.
where $n$ is any natural number such that $kn$ is even.
2 KB (360 words) - 09:21, 27 April 2014
For any real number $\alpha$ and any natural number $Q$ there exist integers $a$ and $q$ which satisfy the condition
...onstrated: For any real numbers $\alpha_1,\ldots,\alpha_n$ and any natural number $Q$ there exist integers $a_1,\ldots,a_n$ and $q$ such that
7 KB (1,065 words) - 14:05, 17 March 2020
...ing formula|Stirling formula]], which gives an asymptotic expansion of the Euler [[Gamma-function|gamma-function]],
upon division by a natural number $ b $
2 KB (344 words) - 09:03, 6 January 2024
is the number of series of $ \beta $(
The number of words obtainable from an arbitrary word $ \beta $
3 KB (507 words) - 13:23, 8 February 2020
where $\nu$ can be any complex number.
For $p=n$ natural number and small $x$ we have the asymptotic formulas
2 KB (361 words) - 11:28, 14 June 2019
...n. In this sense a convex polyhedron is the closed convex hull of a finite number of points and rays. The dimension of a convex polyhedron is the minimal dim
A convex polyhedron has a finite number of faces (intersections of the convex polyhedron with the supporting hyperp
11 KB (1,564 words) - 17:11, 7 February 2011
...ES) described below. Let $K$ be a number field. Fix a [[Prime number|prime number]] $p$ and consider a set $\mathcal{S}$ of square-free ideals $L$ in $\mathc
...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
...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 $,
5 KB (764 words) - 08:12, 6 June 2020
...re known (see, for example, [[#References|[1]]]). The determination of the number of magic squares of order $ n $
...ler. One of them, the problem of the 36 officers, which is to arrange this number of officers of 6 different ranks and 6 different regiments in the cells of
12 KB (1,892 words) - 19:25, 13 January 2024
is solvable for a given integer $n > 1$. The number $a$ is called a residue of degree $n$ modulo $m$. If this congruence is not
...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
...3208012.png" />. This fact, as well as the necessity for computing a large number of partial derivatives, strongly restricts the scope of applicability of th
One of the simplest numerical methods — the [[Euler method|Euler method]] — is based on an approximate computation of the integral term in
32 KB (4,604 words) - 16:55, 7 February 2011
integers containing one number from each residue class is called a complete residue system modulo $ m $.
Congruences modulo one and the same number can be added, subtracted and multiplied in the same way as normal equalitie
20 KB (3,011 words) - 09:59, 26 March 2023
occurs in the problem of estimating the number $ \Psi ( x,y ) $
where $\gamma$ is the [[Euler constant]] and $\xi(u)$
2 KB (376 words) - 08:55, 10 November 2023
is an arbitrary non-integral real number and
then the number of roots in these two sectors is equal to the even number nearest to $ \nu - 1 / 2 $,
4 KB (523 words) - 08:23, 20 January 2024
is Euler's function. Further, let $ \gamma , \gamma _ {0} \dots \gamma _ {r} $
The smallest positive number $ \nu $
10 KB (1,462 words) - 11:49, 26 March 2023
if it is "negative" . If the total number of independent trials $ n $
...a scheme is known as [[Bernoulli trials|Bernoulli trials]], and the total number of positive results,
5 KB (776 words) - 10:59, 29 May 2020
...y]]; [[Prime number]]; [[Dirichlet theorem]]). Let $\pi(x;q,a)$ denote the number of primes $p \le x$ satisfying $p \equiv a \pmod q$. One looks for inequali
...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
The union (in a three- or higher-dimensional space) of a finite number of plane polygons such that: 1) each side of any polygon is simultaneously
...as a geometric body where, for these bodies, "holes" bounded by a finite number of plane faces are allowed.
9 KB (1,472 words) - 18:56, 25 April 2012
...Dirichlet [[#References|[1]]] in 1837 in the context of the proof that the number of primes in an arithmetic progression $ \{ {dm + l } : {m = 0 , 1 ,\dots
to an arithmetic progression and are a powerful tool in analytic number theory [[#References|[2]]]–[[#References|[4]]].
15 KB (2,181 words) - 11:50, 26 March 2023
is a real number; here it is assumed that for every point $ ( x _ {1} \dots x _ {n} ) $
in its domain of definition it satisfies the Euler formula
2 KB (385 words) - 16:12, 1 August 2021
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
It did not take long before people noticed a large number of interesting congruence properties of Apéry numbers. For example, $a _ {
...valign="top"> F. Beukers, "Some congruences for the Apéry numbers" ''J. Number Theory'' , '''21''' (1985) pp. 141–155</td></tr>
3 KB (525 words) - 20:51, 23 January 2024
...he algebraic and geometric analogues of such problems concerning algebraic number fields and sets of lattice points. These problems are said to be additive,
...various branches of number theory — analytic, algebraic and probabilistic number theory.
10 KB (1,609 words) - 13:03, 8 February 2020
...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
...gn="top"> O.M. Fomenko, "Applications of the theory of modular forms to number theory" ''J. Soviet Math.'' , '''14''' : 4 (1980) pp. 1307–1362 {{ZB
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
A formula that expresses the number of fixed points of an endomorphism of a topological space in terms of the t
be the [[Lefschetz number|Lefschetz number]] of $ f $.
10 KB (1,491 words) - 19:04, 26 March 2023
...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
An oval lying inside an odd number of other ovals of $ A $
The number of components of a plane real algebraic curve of order $ m _ {1} $
16 KB (2,200 words) - 08:30, 14 January 2024
...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
...come tedious. A computation by the method of characteristics consists of a number of elementary problems: the calculation of an interior point, of a point on
6 KB (1,008 words) - 15:54, 14 February 2020
...subsequent discussion of his principle gave rise to strong objections by a number of his contemporaries. In addition to the laws of propagation of light, Mau
...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
5 KB (820 words) - 08:00, 6 June 2020
...ula for the multiplicity of a weight, and also Steinberg's formula for the number $ m _ \Lambda $
is the number of distinct presentations of an element $ \mu $
7 KB (992 words) - 09:26, 26 March 2023
is the $ k $-th Bernoulli number (cf. [[Bernoulli numbers|Bernoulli numbers]]).
is the integer equal to the number of solutions of the Diophantine equation $ F ( x _ {1} \dots x _ {m} ) =
8 KB (1,125 words) - 07:04, 29 March 2024
The first-mentioned notation is due to L. Euler; the notation $ C _ {N} ^ { n } $
as the number of distinguishable non-ordered combinations (cf. [[Combination|Combination]
5 KB (729 words) - 10:59, 29 May 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.
...]</TD> <TD valign="top"> E.V. Novoselov, "A new method in probabilistic number theory" ''Transl. Amer. Math. Soc.'' , '''52''' (1966) pp. 217–275 ''
3 KB (410 words) - 11:03, 21 October 2017
...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
...$y$, or $z$ is some prime power $q^r$, then the exponent $n$ must be prime number, the prime power must be $x=q$, $z$ differs from $y$ by $1$, and $n$ divide
8 KB (1,419 words) - 15:20, 17 March 2023
...onvergent inside $D$ if for any compact set $K \subseteq D$ there exists a
number $N = N(K)$ such that all the factors $\left({ 1 + u_k(z) }\right) \neq 0$ f
...te (1593) in his study of the quadrature of the circle. He represented the
number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
5 KB (741 words) - 17:20, 24 April 2016
The Hausdorff summation method contains as special cases a number of other well-known summation methods. Thus, for $ \mu _ {n} = 1/( q + 1)
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]]]).
The number $ 2 p + 1 $
9 KB (1,342 words) - 19:08, 9 January 2024
...$p(n)$ 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]]]
...composition being distinct (see [[#References|[a3]]]). See also [[Additive number theory]]; [[Additive problems]].
4 KB (718 words) - 15:32, 11 November 2023
A complex number is a number of the form $z=x+iy$, where $x$ and $y$ are real numbers (cf.
[[Real number|Real number]]) and $i=\def\i{\sqrt{-1}}\i$ is the so-called imaginary unit,
14 KB (2,281 words) - 09:55, 26 March 2023
Problems in number theory concerned with the asymptotic behaviour of the summation functions
...)$ is the [[number of divisors]] of $n$ and $\tau_k(n)$, $k \ge 2$, is the number of representations of $n$ as the product of $k$ natural numbers, as well as
11 KB (1,578 words) - 11:58, 26 March 2023
...s also lead to abstract generalizations and analogues of ordinary analytic number theory, which may then be applied in a unified way to further enumeration q
iii) the total number of elements $a$ with $|a| < x$ is finite, for each $x > 0$.
24 KB (3,738 words) - 07:41, 7 February 2024
...the number of primes in the prime factorization of $n/m$. The classical or number-theoretic Möbius inversion formula states:
...exceeding $n$ that are relatively prime to $n$ (cf. also [[Euler function|Euler function]]):
11 KB (1,758 words) - 09:24, 10 November 2023
...al notions and reasoning. For example, the notion "the square root of the number equal to the ratio of the length of the circumference of a circle to its di
...a, squared number; ru — from rupa, a rupee coin — free term; a dot above a number denotes subtraction).
18 KB (2,697 words) - 13:11, 13 December 2013
and the number $ a $.
...re studied by [[Gauss, Carl Friedrich|C.F. Gauss]] (1811) for an odd prime number $ q = p $
6 KB (921 words) - 19:40, 29 March 2024
is a natural number. The term (local) cyclotomic field is also sometimes applied to the fields
...rovide convenient experimental material in formulating general concepts in number theory. For example, the concept of an algebraic integer and a divisor firs
12 KB (1,769 words) - 11:19, 26 March 2023
it is also the most widely used. This method is known as Euler's method. From (2) for $ q = 2, 3 $
...rder of accuracy of an explicit Runge–Kutta method by one. To increase the number of parameters to be chosen in (2) one can consider, for example, the follow
9 KB (1,315 words) - 20:04, 15 January 2024
...he matrix $[a_{nk}]$, or simply summable by the matrix $[a_{nk}]$, and the number $s$ is referred to as its limit in the sense of this summation method. If $
...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
...g–Mills functional $\mathcal{L}$ on $\mathcal{A} ( \xi )$ assigns the real number $\mathcal{L} ( A ) = \int _ { M } \langle F _ { A } \wedge * F _ { A } \ran
the [[Euler–Lagrange equation|Euler–Lagrange equation]] of the corresponding variational principle or Yang–
7 KB (1,013 words) - 16:45, 1 July 2020
L. Euler in 1740 pointed out the connection between Bernoulli numbers and the values
Bernoulli numbers have found many applications in mathematical analysis, number theory and approximate calculations.
4 KB (684 words) - 18:44, 5 October 2023
The branch of [[Number theory|number theory]] that investigates properties of the integers by elementary methods
...refers to elementary number theory the problems that arise in branches of number theory such as the theory of divisibility, of congruences, of arithmetic fu
13 KB (2,043 words) - 20:28, 13 October 2014
...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
...rocedure is to subdivide finite magnitudes into a without limit increasing number of components which decrease without limit.
Of a similar character is the completion of the real number system by two "improper" numbers $ + \infty $
10 KB (1,463 words) - 22:12, 5 June 2020
...]], [[#References|[a4]]] on the Arnol'd conjecture (concerning the minimal number of fixed points of a symplectic diffeomorphism), which can be reformulated
...efschetz number|Lefschetz number]] of $\phi$ and the standard intersection number of $L _ { + }$ and $L_{-}$ in $M$, respectively (cf. [[Intersection theory|
15 KB (2,211 words) - 07:18, 15 February 2024
...which could not be solved previously. Methods were developed for solving a number of extremal problems in graph theory; one such problem is the construction
...f non-isomorphic graphs with the same number of vertices and/or edges. The number of non-isomorphic trees with $ n $
13 KB (1,976 words) - 20:08, 15 March 2023
...eals with the problems of distribution of primes, studies the behaviour of number-theoretic functions, and the theory of algebraic and transcendental numbers
...obtained by Euclid: There are infinitely many primes. Let $\pi(x)$ be the number of primes not exceeding $x$. Euclid's theorem can then be formulated as fol
27 KB (4,516 words) - 18:38, 18 October 2014
A theorem expressing the [[Euler characteristic|Euler characteristic]] $ \chi ( {\mathcal E} ) $
There are analogues of the Riemann–Roch theorem in algebraic number theory and arithmetic algebraic geometry.
10 KB (1,385 words) - 03:10, 2 March 2022
the [[Euler characteristic|Euler characteristic]] of the structure sheaf, $ \chi ( {\mathcal O} ) = 2 $,
...f complex numbers) the topological Betti numbers (cf. [[Betti number|Betti number]]) $ b _ {0} = b _ {4} = 1 $,
10 KB (1,443 words) - 15:43, 1 March 2022
is the [[Euler function|Euler function]].) Choose a number $ e > 1 $
and a number $ d $
12 KB (1,852 words) - 17:31, 5 June 2020
hold. The number $ L $
...','Zeta-function','../z/z099260.htm')" style="background-color:yellow;">L. Euler. One of the simplest methods for solving the Cauchy problem is named after
16 KB (2,467 words) - 15:35, 4 June 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} $,
10 KB (1,411 words) - 10:58, 29 May 2020
being considered, with the exception of a finite number of moments of time, at which they can have discontinuities of the first kin
...hich is a necessary condition that generalizes the necessary conditions of Euler, Clebsch and Weierstrass, used in classical variational calculus.
7 KB (997 words) - 08:25, 6 June 2020
...i \in Q _ { 0 } } \in \mathbf{Z} ^ { Q _ { 0 } }$ and $d _ { i j}$ is the number of arrows from $i$ to $j$ in $Q _ { 1 }$.
...e shows that the Tits form $q_Q$ coincides with the [[Euler characteristic|Euler characteristic]] $\chi _ { Q } : K _ { 0 } ( Q ) \rightarrow \mathbf{Z}$, $
18 KB (2,636 words) - 06:50, 15 February 2024
A branch of number theory studying distribution laws of prime numbers among natural numbers. T
which is the number of prime numbers not exceeding $ x $,
36 KB (5,530 words) - 08:29, 14 January 2024
...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
...a function defined on the positive integers and let $\varphi ( q )$ be the Euler [[Totient function|totient function]]. The Duffin–Schaeffer conjecture sa
...1 / q ^ { 2 }$ by a smaller function $f ( q )$ for which every irrational number $x$ can be approximated by infinitely many fractions $p / q$ such that (a1)
10 KB (1,454 words) - 07:42, 4 February 2024
are zero or negative integers, the series (2) terminates after a finite number of terms, and the hypergeometric function is a polynomial in $ z $.
Euler's formula
12 KB (1,576 words) - 01:14, 21 January 2022
...$th Fourier coefficient $r ( P , m )$ of $\theta _ { n } ( P , z )$ is the number of representations of $2 m$ as $\square ^ { t } a P a$, for some integer ve
...tion function (number theory)|partition function]] $p ( n )$, which is the number of ways of writing $n$ as a sum of positive integers. The discriminant func
13 KB (1,907 words) - 07:36, 22 March 2023
...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
...divergent series, one takes some operator or rule which assigns a specific number to a divergent series, called its sum (in this definition). Such a rule is
4 KB (679 words) - 19:36, 5 June 2020
and the number $ d ( S) $
Its [[Euler characteristic|Euler characteristic]] is 1.
10 KB (1,510 words) - 18:38, 13 January 2024
A number of methods have evolved to find solutions of general functional equations o
...solutions of problems that reduce to (2). For example, the [[Euler method|Euler method]] with irregular step $ h _ {k} $
23 KB (3,277 words) - 19:16, 16 January 2024
the number
for some natural number $ k $ (Kodaira's projective imbedding theorem [[#References|[1]]], [[#Ref
12 KB (1,750 words) - 19:20, 27 October 2023
...olution of problems of enumeration, in particular the determination of the number of configurations of a given class. The simplest examples of combinatorial
The number of combinations of size $ m $
28 KB (4,167 words) - 17:45, 4 June 2020
is called a Greco–Latin or Euler square; its elements are all the $ n ^ {2} $
An example of a pair of orthogonal Latin squares and their Euler square for $ n = 3 $
17 KB (2,485 words) - 12:08, 21 March 2022
...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
...ged during the motion. These basic models of dynamics yield solutions to a number of specific problems involving the motion of real bodies.
11 KB (1,677 words) - 20:41, 21 December 2016
...me limit $l$, then the continued fraction (1) is called convergent and the number $l$ is its value. If $\o < \infty$, that is, the continued fraction is fini
...n is not always equal to the number from which it is expanded. There are a number of criteria for the convergence of continued fractions (see, for example,
11 KB (1,649 words) - 11:41, 4 September 2013
...ed that the number of unknowns in Diophantine equations is larger than the number of equations; thus, they are also known as indefinite equations. In modern
The study of Diophantine equations is on the border-line between number theory and algebraic geometry (cf. [[Diophantine geometry|Diophantine geome
18 KB (2,685 words) - 08:57, 18 August 2022
...[Riemannian manifold|Riemannian manifold]] $( M , g )$ one can associate a number of natural elliptic differential operators which arise from the geometric s
...$ increases, but at least the leading terms (i.e. the terms with a maximal number of derivatives) in the $a_k$ for all $k$ can be described. General referenc
14 KB (2,008 words) - 16:55, 1 July 2020
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
...ory is the Iwasawa main conjecture, which has been proved for totally real number fields [[#References|[a19]]]. The conjecture is considered as an analogue o
==$\mathbf{Z} _ { p }$-extension of a number field.==
19 KB (2,876 words) - 05:38, 15 February 2024
...tained from any closed surface by removing the interior points of a finite number of non-intersecting discs. Their boundaries form the boundary of the two-di
...old, may have a very complicated structure. Thus, if one takes an infinite number of discs in $ \mathbf R ^ {2} $
20 KB (3,075 words) - 08:26, 6 June 2020
is an algebraic number field, the structure of the group $ {\mathop{\amalg\kern-0.30em\amalg}} (
7 KB (1,109 words) - 16:59, 1 July 2020
...</td></tr><tr><td valign="top">[a6]</td> <td valign="top"> G. Shimura, "Euler products and Eisenstein series" , Amer. Math. Soc. (1997) pp. Sect. 11</t
6 KB (1,050 words) - 00:38, 15 February 2024
...e-measurable functions. In these cases the optimal control has an infinite number of points of discontinuity, approaching a certain point, for example, an en
...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 style of the young author exhibited a number of characteristics
[[Bernoulli, Daniel|Daniel Bernoulli]], Clairaut and Euler, who gave as good as they
11 KB (1,657 words) - 20:48, 21 December 2023
...eld $ \mathbf Q $ of rational numbers (cf. [[Algebraic number|Algebraic number]]).
...Pell equation|Pell equation]] $ x ^{2} - D y ^{2} = 1 $ has an infinite number of solutions $ ( x ,\ y ) \in \mathbf Z \times \mathbf Z $ . Any of them
28 KB (4,440 words) - 22:00, 11 December 2019
1) The system has a finite number of periodic trajectories (including fixed points in the case of a cascade)
be the number of periodic points of index $ i $,
10 KB (1,474 words) - 20:08, 12 January 2024
defined by the standard formula but with local values of the density of the number of particles $ n ( t , r ) $,
defines the hydrodynamic equations of an ideal fluid (the Euler equations). These are the existence conditions for a first approximation to
6 KB (917 words) - 16:43, 4 June 2020
The Yang–Mills equation is the Euler–Lagrange variational equation for the functional $ L ( \nabla ) $
is the first [[Pontryagin number|Pontryagin number]] of the bundle $ \mathfrak g ( M) $
14 KB (2,063 words) - 02:45, 18 July 2022
...ied treatment of the subject [[#References|[a1]]], [[#References|[a3]]]. A number of interesting ordinary and partial differential equations admit Bäcklund
...d></tr><tr><td valign="top">[a5]</td> <td valign="top"> W.-H. Steeb, N. Euler, "Nonlinear evolution equations and Painlevé test" , World Sci. (1988)<
8 KB (1,072 words) - 17:02, 1 July 2020
...eaning of the term a dynamical system is a mechanical system with a finite number of degrees of freedom. The state of such a system is usually characterized
In a number of cases it is not possible to establish a correspondence between all the s
27 KB (4,058 words) - 19:36, 5 June 2020
is connected with the number of sheets $ n $
and with the number of branch points $ \nu $(
10 KB (1,594 words) - 06:20, 17 April 2024
...metry is due to the studies of G. Leibniz, I. Newton and, particularly, L. Euler. The tools of analytic geometry were utilized by J.L. Lagrange in his const
...een a straight line and a circle is reduced to the analytic problem of the number of solutions of the set of equations of the line and of the circle.
8 KB (1,255 words) - 18:47, 5 April 2020
...ng: "The general concept of a function requires that a function of x is a number which is given for each x and gradually changes with x. The value of a func
One says that the number of elements of a set $ A $
34 KB (5,509 words) - 22:06, 28 January 2020
...brief table of the contents of one of the best textbooks of that time, L. Euler's Introduction to algebra includes integers, ordinary and decimal fractions
...in the context of studies on complex numbers (cf. [[Complex number|Complex number]]). There appeared G. Boole's [[Algebra of logic|algebra of logic]], H. Gra
17 KB (2,478 words) - 16:09, 1 April 2020
...umber, which gives the topological type of the solution $A$. The instanton number is minus the second Chern class $c_2(F) \in H^2(M; \Z)$ of the bundle on wh
The number $b_2^+$ is defined to be the rank of the positive part of the intersection
16 KB (2,663 words) - 10:57, 13 February 2024
be the number of poles (counted with multiplicities) of an elliptic function $ f ( z) $
times, counted with multiplicities. The number $ r $
9 KB (1,292 words) - 19:08, 20 January 2022
The topological [[Euler characteristic|Euler characteristic]] satisfies the formula
...ing. They are elliptic fibrations over a rational curve. Their first Betti number is 1.
15 KB (2,046 words) - 05:46, 13 June 2022
$#C+1 = 87 : ~/encyclopedia/old_files/data/N067/N.0607900 Number
...hematics on the other, have determined the development of the concept of a number.
23 KB (3,482 words) - 08:03, 6 June 2020
...es (such as certain cell mutual exclusions), which results in reducing the number of cells to less than $ 2 ^ {n} $.
...ial insight to many algebraic concepts. The [[Venn diagram|Venn diagram]] (Euler diagram) offers the visual help of the Karnaugh map. However, it is essenti
9 KB (1,380 words) - 22:14, 5 June 2020
...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
and any number $ \epsilon > 0 $,
18 KB (2,612 words) - 13:13, 18 February 2022
...yclopediaofmath.org/legacyimages/m/m064/m064430/m064430115.png" /> and the
Euler characteristic <img align="absmiddle" border="0" src="https://www.encyclope
...hat is used in the proof of the [[Picard theorem|Picard theorem]] and in a
number of geometric questions.
39 KB (5,287 words) - 17:07, 7 February 2011
...ic functions $\operatorname { cos } nt$, $\operatorname { sin } nt$. The [[Euler formula]] $\operatorname { exp } ( i \alpha ) = \operatorname { cos } \alph
...g elastic shaft one has (with suitable boundary conditions) a self-adjoint Euler–Bernoulli equation:
9 KB (1,326 words) - 07:37, 24 November 2023
...se subject is the approximation of zero by values of functions of a finite number of integer arguments. The original problems of Diophantine approximations c
....encyclopediaofmath.org/legacyimages/d/d032/d032600/d03260013.png" />-adic number fields and Diophantine approximations in the field of power series).
54 KB (7,359 words) - 18:32, 31 March 2017
lines (points), and the number of points of the plane, which is equal to the number of lines, is $ n ^ {2} + n + 1 $.
...e existence of a finite projective plane whose order is a power of a prime number has been proved (cf. [[#References|[4]]]). The non-existence of $ P ( 2,
11 KB (1,726 words) - 06:13, 23 April 2023
...t L-function|Dirichlet $L$-function]]) form the basis of modern [[analytic number theory]]. In addition to Riemann's zeta-function one also distinguishes the
..., $\delta>0$. If $\sigma>1$, a valid representation is the [[Euler product|Euler product]]
45 KB (7,251 words) - 02:20, 29 June 2022
...ain of physical phenomena is connected with the names of J.L. Lagrange, L. Euler, J. Fourier, C.F. Gauss, B. Riemann, M.V. Ostrogradski, and many other scho
...mathematical problems in|Hydrodynamics, mathematical problems in]]), and a number of other research directions in the physics of continuous media. The mathem
13 KB (1,857 words) - 21:26, 8 November 2014
...The concepts and the results of algebraic geometry are extensively used in number theory (Diophantine equations and the evaluation of trigonometric sums), in
L. Euler studied an arbitrary polynomial $ f(x) $
29 KB (4,414 words) - 17:20, 17 December 2019
...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
...onditions of this theorem are called Chebyshev projections. They possess a number of useful properties (for them $ ( \sup \mu | _ \Delta )/( \inf \mu |
28 KB (4,001 words) - 11:10, 4 June 2020
...e considered that are specific for polyhedra, with the introduction of the number of edges or the sum of their lengths, etc. (see [[#References|[6]]]).
...odies, many isoperimetric inequalities (including the classical ones and a number of inequalities between integrals of symmetric functions of principal curva
22 KB (3,071 words) - 22:13, 5 June 2020
The assertion that for any natural number $n>2$ the equation $x^n+y^n=z^n$ (the Fermat equation) has no solution in n
For $n=3$ the theorem was proved by L. Euler (1770), for $n=5$ by P.G.L. Dirichlet and A. Legendre (1825), and for $n=7$
16 KB (2,596 words) - 09:27, 13 February 2024
is a natural number, called the order or the weight of the theta-function, and $ q $
is a number. Convergence is ensured, for example, by using coefficients of the form
14 KB (1,941 words) - 05:01, 23 February 2022
...the range. The dimension of a simplex in an a-complex is one less than the number of its elements. Each element of the set $ \overline{A}\; $
...model for this procedure was the definition of the [[Euler characteristic|Euler characteristic]]). However, this method has not achieved much popularity; f
16 KB (2,377 words) - 08:06, 6 June 2020
where the coefficients $b _ { k }$ are functions of the Courant number
...) and WAF (bottom) extensions of the Lax–Wendroff method as applied to the Euler equations of gas dynamics for ideal gases. Numerical (symbol) and exact (li
16 KB (2,341 words) - 19:00, 26 March 2023
The number $\nu$ is called the order of the Bessel equation; in the general case $x$ a
...ch equations are encountered even earlier in the works of D. Bernoulli, L. Euler and J.L. Lagrange.
7 KB (1,130 words) - 20:00, 22 March 2024
Geodesic lines were first studied by J. Bernoulli and L. Euler, who
motion of a conservative mechanical system with a finite number of
11 KB (1,778 words) - 21:59, 5 March 2012
lecturing on higher algebra and number theory in September, 1847. Buniakovsky
Euler's ''Theory of Numbers''. Buniakovsky lectured on probability theory at
12 KB (1,774 words) - 20:00, 7 March 2024
th [[Betti number|Betti number]] for Čech homology with $ \mathbf Z _ {2} $
the [[Euler characteristic]] of $ M $.
24 KB (3,546 words) - 19:41, 1 May 2023
above is differentiable; if the number of connected components of $ G $
has non-zero [[Euler characteristic|Euler characteristic]] (this is equivalent to $ G $
20 KB (2,996 words) - 08:42, 16 December 2019
and its derivative are unknown values to be observed. However, in a number of works, adaptive procedures have been constructed in which $ a = a( n)
...hich is based on interpreting suitably rescaled versions of (1) and (2) as Euler approximations to the solution of an ordinary or stochastic differential eq
9 KB (1,349 words) - 08:23, 6 June 2020
society', and that he 'was familiar with the life expectancy tables of Euler,
to the United States, Latin America and a number of European countries.
9 KB (1,317 words) - 12:34, 25 March 2023
...cf. [[Elementary number theory|Elementary number theory]]; [[Number theory|Number theory]]).
...sum of aliquot fractions. Division was carried out by subtracting from the number to be divided the numbers obtained by successive doubling of the divisor. T
20 KB (3,091 words) - 18:48, 5 April 2020
denote the number of $ a $-
(the defect relation). The constant 2 figuring in (1) is the Euler characteristic of the extended complex plane $ \mathbf C \cup \{ \infty \
20 KB (2,984 words) - 08:27, 6 June 2020
Both methods were first treated in the works of L. Euler on the basis of the classical calculus of variations (1744–1757). He also
is the length of the beam. There is a large number of concrete results for beams, beam-like systems, plates, shells, as well a
9 KB (1,409 words) - 08:22, 6 June 2020
The number
such that the normal curvature in any direction can be expressed by Euler's formula:
26 KB (4,098 words) - 17:31, 5 June 2020
The origins of the idea of a group are encountered in a number of disciplines, the principal one being the theory of solving algebraic equ
...ory is the third source of the concept of a group. As early as 1761 (cf. [[Euler function]], [[Primitive root]], [[Series]], [[Stability of an elastic syste
21 KB (3,246 words) - 17:24, 9 October 2016
...r instance, in the special example above the boundary consists of a finite number $h$ of circles and
...</tr><tr><td valign="top">[a7]</td> <td valign="top"> J. Schwermer, "On Euler products and residual Eisenstein cohomology classes for Siegel modular vari
12 KB (1,817 words) - 15:30, 1 July 2020
...etry first appeared in the 18th century and is linked with the names of L. Euler and G. Monge. The first synoptic treatise on the theory of surfaces was wri
A number of fundamental concepts in the theory of curves is based on the concept of
33 KB (5,039 words) - 11:46, 26 March 2023
...given in \ref{eq3}. These formulas had been obtained already by L. Euler (1777) by term-by-term integration.
...d minima over the period and is everywhere continuous, except at a finite number of points where it may have discontinuities of the first kind, then the Fo
21 KB (3,473 words) - 19:50, 26 April 2012
or itself or cross one of the points already existing. The number
A Sprouts game starting with $n$ points has a total number of $3n$ lives
19 KB (3,145 words) - 10:05, 18 February 2021
of an infinitely-large number of infinitely-small quantities. These problems ultimately gave rise to [[In
...he disc, etc. Thus, the area of the disc is gradually "exhausted" as the number of sides of the inscribed polygons increases. Since the ratio between the a
22 KB (3,357 words) - 17:34, 1 January 2021
which contain a number of free parameters equal to the number of degrees of freedom minus one; the $ 6 $-
...n leads to the following three cases of integrable rotating bodies: i) the Euler top, for which the fixed point and the centre of mass coincide; ii) the Lag
13 KB (1,808 words) - 22:15, 5 June 2020
3) A number of fundamental results concerned the computation of the algebraic number of fixed points of a mapping of a manifold into itself. Many fundamental fa
...e geometrical structure of manifolds and in obtaining lower bounds for the number of extremals.
19 KB (2,788 words) - 09:43, 28 October 2023
and allow one to find in a sufficiently large number of iterations a solution of (3) within prescribed accuracy $ \epsilon $.
A large number of iteration methods (the so-called descent methods) (see [[#References|[1]
20 KB (2,830 words) - 19:25, 9 January 2024
...wn function in the equation, and the class of a functional equation is the number of given functions to which the unknown function is applied. Thus, (3) is a
...tially-different variables than does the unknown function with the maximum number of variables.
19 KB (2,803 words) - 19:40, 5 June 2020
where $\nu$ is an arbitrary real or complex number. The cylinder functions are functions of one complex variable.
...|algebraic number]], $J_n (z)$ is a [[Transcendental number|transcendental number]], and $J_n (z)\neq J_m (z)$ whenever $n\geq m$.
22 KB (3,786 words) - 06:49, 24 February 2024
...aces. Differential calculus is based on the concepts of [[Real number|real number]]; [[Function|function]]; [[Limit|limit]] and [[Continuity|continuity]] —
and let there exist a number $ A $
37 KB (5,252 words) - 11:45, 26 March 2023
...bed below. The method is used in many problems of analytic [[
Number theory|
number theory]], analysis and applied mathematics.
..._ { j } | \geq \sqrt { n }$, and that if $n + 1$ is a [[Prime
number|prime
number]], then this <img align="absmiddle" border="0" src="https://www.encyclopedi
40 KB (5,729 words) - 17:51, 5 May 2024
...in the sense of Fréchet is the space of all real functions defined on the number axis $\R$, with [[pointwise convergence]]: the convergence of a sequence $f
..."rate" at which a given sequence converges to its limit. For example, the number $\pi$ can be represented in the form of a sum of series in the following tw
22 KB (3,726 words) - 10:31, 2 September 2017
largest of any given number of independent life-times,
expected number of surviving children in the context
25 KB (4,000 words) - 07:43, 14 August 2023
...[#References|[a6]]]. The situation changed radically in the 1930s due to a number of technical and scientific reasons. The basic elements of the modern theor
15 KB (2,182 words) - 12:41, 23 February 2024
strings and rods. He was involved in quarrels with his friend Euler on the
$w(t)$ of individuals at time $t$ after birth, with the number of susceptibles
10 KB (1,551 words) - 07:42, 9 March 2024
is an irrational number, it is possible to select $ ( n _ {1} \omega _ {1} + n _ {2} \omega _ {2}
is a small positive parameter. A large number of problems in physics and technology can in fact be reduced to this form.
35 KB (5,181 words) - 07:24, 8 June 2020
must satisfy the so-called Euler–Lagrange equation
...ing domains of mathematics. The first general method of integration of the Euler–Lagrange equations was suggested by Monge (1784) and A. Legendre (1787) i
29 KB (4,187 words) - 08:00, 6 June 2020
...ote skew-symmetric or exterior differential forms, which have the greatest number of applications.
The calculus of exterior forms on a complex analytic manifold has a number of special features [[#References|[6]]]. In this situation it is usual to c
27 KB (4,062 words) - 01:31, 7 May 2022
with H. v. Euler before turning single mindedly to mathematics,
20 papers in analytic number theory over the next seven years.
13 KB (2,000 words) - 18:58, 5 March 2024
the number of eigen values of $ h _ {p} ( \xi ) $
has Euler characteristic zero and a compact parallelizable covering manifold if $ K
25 KB (3,789 words) - 19:41, 5 June 2020
without becoming confused, a principle which Euler had already
statistical analysis " always to collect a large number of elements,
13 KB (1,947 words) - 18:43, 4 March 2024
...comfort of a present security system, etc. It is usually assumed that the number of ingredients is finite and the space of products is $ \mathbf R ^ {l}
is the number of ingredients. A point $ z $
37 KB (5,562 words) - 11:11, 7 January 2024
Cantor's problem on the [[Continuum, cardinality of the|cardinal number of the continuum]].
...l'fond-Schneider method|Gel'fond–Schneider theorem]], 1934; see [[Analytic number theory]]). For the general method, the [[Gel'fond–Baker method]], see e.g
29 KB (4,109 words) - 19:54, 18 March 2018
:the source domain of analysis was geometry; that of number theory was arithmetic.<ref>Ueno p. 73. Even at the beginning of the 20th ce
*the lack of any definition of the phrase "real number"
91 KB (14,186 words) - 16:58, 23 November 2023
...e, but one has to talk about a system with a variable, generally infinite, number of degrees of freedom. Quantum field theory unifies the description of fiel
...ase space. The field is represented as a canonical system with an infinite number of degrees of freedom, as the potential at each point is an independent coo
35 KB (5,270 words) - 23:26, 6 December 2016
...\mathbb C$. If to each point $z\in D$ there has been assigned some complex number $w$, then one says that on $D$ a (single-valued) function $f$ of the comple
...equations occurred already in the 18th century in J.L. d'Alembert's and L. Euler's studies on functions of a complex variable.
61 KB (9,850 words) - 19:04, 20 January 2022
The regularization method. For a number of applied problems leading to \ref{eq1} a typical situation is that the s
...various methods on a computer (for example, by solving the corresponding Euler equation for $M^\alpha[z,u_\delta]$). The element $z_\alpha$ minimizing $
33 KB (5,246 words) - 00:23, 25 April 2012
...are possible: either to introduce from the outset a larger than necessary number of parameters, and to find then the actual space implicitly using a system
...ntrinsic geometry, and thus that of manifold, not depending on the ambient number space, and in fact the notion of a structure on a manifold. This idea was c
30 KB (4,462 words) - 07:59, 6 June 2020
A term denoting a number of problems related to research on extremals and global minima of functiona
is a closed locally-minimal submanifold (i.e. one that annihilates the Euler operator) in $ S ^ {n-1} $
29 KB (4,259 words) - 11:43, 3 March 2022
...algebraic methods, arises. The use of analytical methods in geometry by L. Euler (1707–1783) and G. Monge (1746–1818) in the 18th century laid the found
...use the theory of convex bodies, or convex geometry, to solve problems in number theory led Minkowski to the [[Geometry of numbers|geometry of numbers]]. It
20 KB (2,829 words) - 18:41, 11 December 2020
...respect to the Fuchsian filtration such that the minimal polynomial of the Euler mapping on $ \oplus V _ {k} ( M)/V _ {k-1} ( M) $
...s. The work [[#References|[a13]]] contains results which predict the grade number of graded modules over $ A _ {1} ( S) $
28 KB (4,182 words) - 19:30, 19 January 2024
...int union of $K ^ { 2 }$ (respectively, $L^{2}$) with a sufficiently large number of standard complexes of ${\bf Z} _ { 2 } \times {\bf Z} _ { 4 }$, see [[#R
31 KB (4,667 words) - 17:46, 1 July 2020
...s (ringed spaces) rather than over smooth manifolds. This requires quite a number of changes in the construction, see [[Vector bundle, algebraic]].
...main types of them, -- [[Stiefel-Whitney class]], [[Pontryagin class]], [[Euler class]], [[Chern class]]).
32 KB (5,299 words) - 12:27, 12 December 2020