$#A+1 = 302 n = 0
$#C+1 = 302 : ~/encyclopedia/old_files/data/A110/A.1100010 A priori and a posteriori bounds in matrix computations
30 KB (4,205 words) - 18:47, 5 April 2020
$#A+1 = 76 n = 1
...1 = 76 : ~/encyclopedia/old_files/data/A010/A.0100550 Action of a group on a manifold
7 KB (1,066 words) - 06:29, 26 March 2023
...lpha = 0$ can be uniquely represented as $\phi = \psi \, \mu$. A kernel of a morphism $\alpha$ is denoted by $\ker \alpha$.
...of $\alpha$. Thus, the kernels of a morphism $\alpha$ form a subobject of $A$, denoted by $\operatorname{Ker} \alpha$.
3 KB (482 words) - 13:57, 26 December 2017
...following classical assertion is well known. Let $D \subset \mathbf{C}$ be a simply connected bounded domain with smooth boundary $\partial D$, and $f \
if and only if $f ( z )$ extends into the domain $D$ as a [[Holomorphic function|holomorphic function]] of the class $H ( D ) \cap C
6 KB (978 words) - 17:44, 1 July 2020
$#A+1 = 26 n = 0
...data/C021/C.0201550 Character of a finite\AAhdimensional representation of a semi\AAhsimple Lie algebra
2 KB (326 words) - 07:50, 21 January 2022
$#A+1 = 34 n = 0
...cyclopedia/old_files/data/P075/P.0705030 Product of a family of objects in a category
3 KB (545 words) - 08:07, 6 June 2020
...{ r }$ generate $\mathfrak{m}$, and in that case $( A , \mathfrak m )$ is a regular local ring.
...s of finite length; in that case $y _ { 1 } , \dots , y _ { s }$ is called a system of parameters of $M$.
3 KB (406 words) - 17:45, 1 July 2020
$#A+1 = 95 n = 0
...~/encyclopedia/old_files/data/W097/W.0907540 Weight of a representation of a Lie algebra
5 KB (778 words) - 02:00, 4 January 2022
$#A+1 = 62 n = 1
...ncyclopedia/old_files/data/C021/C.0201570 Character of a representation of a group
6 KB (828 words) - 09:30, 26 March 2023
''operation $ {\mathcal A} $''
A set-theoretical operation, discovered by P.S. Aleksandrov [[#References|[1]
3 KB (432 words) - 16:59, 1 July 2020
...ummable ones. A measurable function $f(x)$ is called $A$-integrable over $[a,b]$ if
The number $I$ is called the $A$-integral. It is denoted by
940 bytes (157 words) - 11:01, 31 March 2017
$#A+1 = 27 n = 0
$#C+1 = 27 : ~/encyclopedia/old_files/data/A010/A.0100080 \BMI A\EMI\AAhsystem
3 KB (440 words) - 18:48, 5 April 2020
''analytic set, in a complete separable metric space''
A continuous image of a [[Borel set]]. Since any Borel set is a continuous
2 KB (327 words) - 14:21, 15 August 2023
...n is actually observed. The a posteriori probability is connected with the a priori probability by the [[Bayes formula|Bayes formula]].
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...sumption on its existence is not sufficiently founded). For the use of the a priori distribution, see [[Bayesian approach]].
1 KB (204 words) - 19:22, 21 April 2024
...posteriori probability|a posteriori probability]]. This terminology is, as a rule, used in connection with the [[Bayes formula|Bayes formula]].
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...id $R$ if for any $\lambda \in R$ and $a \in A$ the product $\lambda a \in A$ is defined, such that
\lambda (\mu a) = (\lambda\mu) a
6 KB (1,055 words) - 05:59, 22 April 2023
...riable, to be contrasted with its unconditional or [[A priori distribution|a priori distribution]].
...$ when $\Theta=\theta$; then the a posteriori distribution of $\Theta$ for a given $X=x$, according to the [[Bayes formula|Bayes formula]], has the dens
2 KB (248 words) - 21:34, 1 January 2019
$#A+1 = 98 n = 0
$#C+1 = 98 : ~/encyclopedia/old_files/data/D031/D.0301210 Derivation in a ring
6 KB (945 words) - 17:32, 5 June 2020
...one means the density $d(A)$ (introduced in 1930 by L.G. Shnirel'man) of $A$, namely
$$d(A)=\inf\frac{A(n)}{n},$$
3 KB (461 words) - 11:41, 14 February 2020
$#A+1 = 27 n = 0
$#C+1 = 27 : ~/encyclopedia/old_files/data/O070/O.0700510 Oscillation of a function
2 KB (281 words) - 08:04, 6 June 2020
...Grundy function]]: the function $g$ that associates to every vertex $k$ of a graph $\Gamma$ the number
...and external stability, and in this regard it is similar to the concept of a [[von Neumann–Morgenstern solution]] in [[cooperative game]]s.
2 KB (361 words) - 16:41, 4 June 2023
.... Then the full subcategory of $\mathfrak K$ generated by these objects is a skeleton of $\mathfrak K$.
...rits many properties of the category itself: local smallness, existence of a bicategory structure, various forms of completeness, etc.
1 KB (223 words) - 07:36, 28 November 2017
...erves as the base of two different deformations $F'$ and $F''$, then it is a principal base of deformation.
...ns, then the cross ratio $t=(\kappa^2,\kappa'^2,\kappa''^2,\kappa_t^2)$ is a constant quantity for all positions of $M$ on $F$.
5 KB (766 words) - 12:40, 2 November 2014
...h is either a [[prime number]] or the number zero, uniquely determined for a given field in the following way. If for some $n > 0$,
...of all rational numbers, and a field of finite characteristic $p$ contains a subfield isomorphic to the field of residue classes modulo $p$: in each cas
885 bytes (152 words) - 13:43, 12 December 2013
$#A+1 = 27 n = 0
$#C+1 = 27 : ~/encyclopedia/old_files/data/V096/V.0906280 Variety in a category
3 KB (532 words) - 08:28, 6 June 2020
...and that the set of orders of all its elements is bounded). The period of a group is also called the exponent of the group.
The order of an element $x$ of a group $G$ is the least positive integer $n$ such that $x^n=e$ (where $e$ is
444 bytes (83 words) - 12:02, 23 August 2014
$#A+1 = 77 n = 0
$#C+1 = 77 : ~/encyclopedia/old_files/data/V096/V.0906140 Variation of a set
7 KB (1,016 words) - 08:27, 6 June 2020
$#A+1 = 15 n = 0
$#C+1 = 15 : ~/encyclopedia/old_files/data/G044/G.0404000 Genus of a surface
2 KB (319 words) - 14:59, 30 March 2023
-
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$#A+1 = 74 n = 0
$#C+1 = 74 : ~/encyclopedia/old_files/data/E036/E.0306980 Extension of a group
6 KB (917 words) - 20:29, 17 January 2024
...orphism]] of the group into the group of all invertible transformations of a set $V$.
...ation representation'' is a homomorphism to the [[symmetric group]] $S_V$: a [[group action]] of $G$ on $V$: cf. [[Permutation group]].
5 KB (653 words) - 19:41, 1 December 2014
$#A+1 = 9 n = 0
$#C+1 = 9 : ~/encyclopedia/old_files/data/S091/S.0901020 Subrepresentation of a representation
812 bytes (117 words) - 08:24, 6 June 2020
$#A+1 = 215 n = 0
$#C+1 = 215 : ~/encyclopedia/old_files/data/G043/G.0403270 Game with a hierarchy structure
20 KB (2,700 words) - 19:41, 5 June 2020
...the local structure of the space, while the properties of the covering as a whole (in particular, the number of elements, the multiplicity and the comb
...tary relation between coverings. A family $\gamma$ of sets is inscribed in a family $\lambda$ of sets if each element of the family $\gamma$ is containe
10 KB (1,606 words) - 07:49, 17 May 2017
A number $T \ne 0$ such that for any $x \in X \subset \mathbf{R}$ (or $x \in
...function $f=\text{const.}$ on an axis or on a plane, any number $T\ne0$ is a period; for the [[Dirichlet-function|Dirichlet function]]
1 KB (227 words) - 21:30, 18 November 2017
$#A+1 = 14 n = 0
$#C+1 = 14 : ~/encyclopedia/old_files/data/H047/H.0407160 Hessian of a function
2 KB (372 words) - 01:47, 17 June 2022
...nt of <math> Y </math> if and only if <math> \mathit{f}^{-1}(C) </math> is a component of <math> X </math>.
990 bytes (165 words) - 22:41, 25 March 2011
$#A+1 = 32 n = 0
$#C+1 = 32 : ~/encyclopedia/old_files/data/I052/I.0502450 Inversion of a series
4 KB (554 words) - 20:40, 16 January 2024
$#A+1 = 62 n = 0
$#C+1 = 62 : ~/encyclopedia/old_files/data/M065/M.0605510 Multiplicity of a weight
6 KB (885 words) - 08:02, 6 June 2020
''$X$ in a compactification $bX$''
...tion]] is simply called a bordering of $X$. In the language of borderings, a series of theorems has been formulated on the dimensions of the remainder o
2 KB (294 words) - 16:43, 24 September 2017
$#A+1 = 56 n = 0
$#C+1 = 56 : ~/encyclopedia/old_files/data/S091/S.0901930 System (in a category),
4 KB (556 words) - 08:24, 6 June 2020
Let $X$ be a topological space and $f:X\to \mathbb R$ a function. The support of $f$, denoted by ${\rm supp}\, (f)$ is the smallest
The same concept can be readily extended to maps taking values in a vector space or more generally in an additive group.
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...Locally finite group]]), finiteness of rank (cf. [[Rank of a group|Rank of a group]]), and finiteness of conjugacy classes (cf. [[Conjugate elements|Con
...valent to the maximum condition for Abelian subgroups [[#References|[4]]]. A similar result was also established for the minimum condition in locally so
5 KB (720 words) - 17:15, 7 February 2011
...al subgroup]] of a group $G$ belonging to a given radical class of groups. A class of groups is called radical if it is closed under homomorphic images
...adical normal subgroup — the radical. The quotient group by the radical is a semi-simple group, that is, it has trivial radical.
3 KB (517 words) - 11:55, 23 November 2013
...ty ($IP$-loop) coincide, and for [[Moufang loop]]s they form, in addition, a normal subloop (see [[Loop]]).
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$#A+1 = 122 n = 0
$#C+1 = 122 : ~/encyclopedia/old_files/data/C025/C.0205180 Connections on a manifold
9 KB (1,365 words) - 18:18, 19 January 2021
$#A+1 = 33 n = 0
$#C+1 = 33 : ~/encyclopedia/old_files/data/V096/V.0906130 Variation of a mapping
3 KB (406 words) - 13:07, 6 January 2024
...a triangle is proportional to its area. The maximum value of the defect of a triangle equals two right angles if all the vertices of the triangle are po
For a triangle in [[spherical geometry]] one has the opposite effect that the sum
1 KB (162 words) - 11:52, 13 December 2015
...a subgroup of $G$. The core of $H$ is the maximal subgroup of $H$ that is a [[normal subgroup]] of $G$. It follows that
...resentation of $G$ on the set of right cosets of $H$ in $G$ (cf [[Coset in a group]]). Then its kernel is the core of $H$ in $G$.
802 bytes (136 words) - 21:48, 30 November 2016
$#A+1 = 41 n = 0
$#C+1 = 41 : ~/encyclopedia/old_files/data/D031/D.0301180 Depth of a module
3 KB (524 words) - 17:32, 5 June 2020
...peration of tensor algebra that associates with a tensor with components $ a^{i_{1} \ldots i_{p}}_{j_{1} \ldots j_{q}} $, $ p,q \geq 1 $, the tensor
...q - 1} 1} + a^{2 i_{2} \ldots i_{p}}_{j_{1} \ldots j_{q - 1} 2} + \cdots + a^{n i_{2} \ldots i_{p}}_{j_{1} \ldots j_{q - 1} n} \\
1 KB (237 words) - 02:26, 9 September 2015
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''$M$ over a commutative ring $R$''
...ctrum of a ring]]). For example, for a finite Abelian group$M$ regarded as a module over the ring of integers, $\mathrm{Supp}(M)$ consists of all prime
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...ing $[a]\leq[b]$, where $[a]$ is the class containing the element $a$, if $a\triangleleft b$. For examples and references see [[Partially ordered set]].
...ters use the notation $a||b$ to indicate that neither $a\leq b$ nor $b\leq a$ holds.
3 KB (573 words) - 12:32, 6 February 2021
...pact; etc. If every connected compact subset of $Y\setminus X$ consists of a single point (for example, if $\operatorname{ind}(Y\setminus X)=0$), the re
...by all remainders of $X$ or by none. When the remainders of $X$ have such a property, $X$ is said to have the property "at infinity" .
1 KB (228 words) - 14:25, 10 April 2014
$#A+1 = 22 n = 0
$#C+1 = 22 : ~/encyclopedia/old_files/data/V096/V.0906120 Variation of a functional,
5 KB (687 words) - 08:27, 6 June 2020
A permutation is
...-one mapping of a set onto itself. The term permutation is used mainly for a finite set $X$. In that case it is convenient to assume that $X=\{1,\dots,n
7 KB (1,262 words) - 20:15, 27 September 2016
$#A+1 = 37 n = 0
$#C+1 = 37 : ~/encyclopedia/old_files/data/P072/P.0702750 Place of a field
2 KB (326 words) - 08:06, 6 June 2020
...rve from $F$ to its distance from a given line (the directrix) is equal to a constant (the eccentricity). See also [[Conic sections]].
The foci of a second-order curve can be defined as the points of intersection of the tang
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$#A+1 = 106 n = 0
$#C+1 = 106 : ~/encyclopedia/old_files/data/I050/I.0500250 Immersion of a manifold
6 KB (923 words) - 22:11, 5 June 2020
$#A+1 = 55 n = 0
$#C+1 = 55 : ~/encyclopedia/old_files/data/H047/H.0407830 Homology of a polyhedron
6 KB (897 words) - 08:54, 25 April 2022
A field extension $K$ is a field containing a given field $k$ as a subfield. The notation $K/k$
Let $K/k$ and $L/k$ be two extensions of a field $k$. An isomorphism of
6 KB (1,030 words) - 17:29, 12 November 2023
''$x$ in a topological space $X$''
...line{\{y\}} \subseteq \overline{\{x\}}$; every [[neighbourhood]] of $x$ is a [[neighbourhood]] of $y$.
3 KB (403 words) - 08:08, 26 November 2023
$#A+1 = 13 n = 0
$#C+1 = 13 : ~/encyclopedia/old_files/data/K110/K.1100090 Kernel of a matrix
2 KB (237 words) - 22:14, 5 June 2020
$#A+1 = 48 n = 0
$#C+1 = 48 : ~/encyclopedia/old_files/data/M064/M.0604110 Minor of a graph
4 KB (617 words) - 08:00, 6 June 2020
...showed that a non-null subalgebra of a free algebra over a field $\Phi$ is a free algebra.
* Kurosh, A. "Nonassociative free algebras and free products of algebras". Mat. Sb., N
1 KB (163 words) - 08:20, 12 November 2023
$#A+1 = 49 n = 0
$#C+1 = 49 : ~/encyclopedia/old_files/data/M065/M.0605490 Multiplicity of a module
4 KB (548 words) - 06:47, 16 June 2022
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$#A+1 = 25 n = 0
$#C+1 = 25 : ~/encyclopedia/old_files/data/H047/H.0407520 Holomorph of a group
2 KB (302 words) - 22:10, 5 June 2020
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$#A+1 = 22 n = 0
$#C+1 = 22 : ~/encyclopedia/old_files/data/C021/C.0201440 Channel with a finite memory
3 KB (419 words) - 16:43, 4 June 2020
$#A+1 = 97 n = 0
...= 97 : ~/encyclopedia/old_files/data/G044/G.0404910 Graph, connectivity of a
7 KB (1,143 words) - 07:00, 6 July 2023
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''of a set $A$ in a [[topological space]] $X$''
...$x$. The interior may be denoted $A^\circ$, $\mathrm{Int} A$ or $\langle A \rangle$.
1 KB (244 words) - 16:55, 25 November 2023
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...,b \in A$, $aRb$ implies $b R a$, i.e. $R \subseteq R^{-1}$. An example of a symmetric relation is an [[Equivalence relation]].
...n $R$ such that $R \cap R^{-1} \subseteq \Delta = \{ (x,x) : \forall x \in A \}$.
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$#A+1 = 43 n = 0
$#C+1 = 43 : ~/encyclopedia/old_files/data/F040/F.0400660 Flow in a network
7 KB (1,074 words) - 20:15, 15 March 2023
$#A+1 = 29 n = 0
$#C+1 = 29 : ~/encyclopedia/old_files/data/C024/C.0204560 Conductor of a character
3 KB (411 words) - 18:32, 14 June 2020
$#A+1 = 20 n = 0
$#C+1 = 20 : ~/encyclopedia/old_files/data/E036/E.0306850 Exhaustion of a domain,
2 KB (348 words) - 19:20, 17 January 2024
...e also the [[Borel set|Borel sets]]) and $\mu: \mathcal{B}\to [0, \infty]$ a ($\sigma$-additive) measure (cp. with [[Borel measure]]). The support of $\
The existence of a countable base guarantees that
3 KB (393 words) - 16:15, 29 November 2012
''$A$ in a topological space $X$''
The intersection of all [[closed set]]s of $X$ containing the set $A$.
1 KB (198 words) - 16:57, 9 November 2014
$#A+1 = 5 n = 0
$#C+1 = 5 : ~/encyclopedia/old_files/data/O068/O.0608030 Object in a category
2 KB (270 words) - 08:03, 6 June 2020
''(general and special rank of a group)''
...is the minimal number such that any finitely-generated subgroup of $G$ has a system of generators of at most $r$ elements. If no such finite number exis
1 KB (182 words) - 16:24, 19 April 2014
...the triangle with the midpoint of the opposite side. The three medians of a triangle intersect at one point, called the centre of gravity, the [[centro
...] (cf. [[Lobachevskii geometry]]) the meridians of a triangle intersect at a point.
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...e $X$ is said to be of the first category in $X$ if it can be expressed as a finite or countable union of [[nowhere-dense set|nowhere dense sets]] in $X
...h sets is residual (or comeager), cp. again with Chapter 9 of {{Cite|Ox}}. A non-empty closed set of real numbers, in particular an interval, is not of
2 KB (291 words) - 19:06, 7 December 2023
#REDIRECT [[Interior of a set]]
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...and its deformation $F^*$ outside their points of congruence. The base of a deformation is characterized by the fact that the bend — the relation bet
...<TD valign="top"> V.F. Kagan, "Foundations of the theory of surfaces in a tensor setting" , '''2''' , Moscow-Leningrad (1948) (In Russian)</TD></TR
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A concept of classical measure theory generalized further in [[Geometric meas
===Lebesgue density of a set===
10 KB (1,662 words) - 09:58, 16 August 2013
...ly one fixes a quotient module $X/A$, that is, an extension of the module $A$ by the module $B$ is an [[exact sequence]]
0 \rightarrow A \rightarrow X \rightarrow B \rightarrow 0 \ .
2 KB (354 words) - 21:55, 30 October 2016
A side of a face of a [[Polyhedron|polyhedron]].
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''of a [[trapezium]]''
A segment joining the mid-points of the two lateral sides of the trapezium. T
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#REDIRECT[[Density of a set]]
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$#A+1 = 71 n = 0
$#C+1 = 71 : ~/encyclopedia/old_files/data/S086/S.0806590 Spectrum of a ring
5 KB (778 words) - 13:13, 6 January 2024
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$#A+1 = 101 n = 0
$#C+1 = 101 : ~/encyclopedia/old_files/data/N067/N.0607360 Norm on a field
6 KB (1,003 words) - 21:35, 13 January 2021
The [[equivalence relation]] on the domain of a function expressing the property that equivalent elements have the same im
...onversely, every equivalence relation $\sim\,$ on a set $X$ gives rise to a function of which it is the kernel. Consider the ''quotient set'' $X/\sim
1 KB (189 words) - 17:00, 23 November 2023
$#A+1 = 16 n = 0
$#C+1 = 16 : ~/encyclopedia/old_files/data/G043/G.0403930 Generator of a category,
3 KB (545 words) - 19:41, 5 June 2020
$#A+1 = 29 n = 0
$#C+1 = 29 : ~/encyclopedia/old_files/data/N067/N.0607740 Normalizer of a subset
2 KB (277 words) - 08:03, 6 June 2020
The inclusion of any finite number of zeros between adjacent terms of a series. For the series
a diluted series has the form
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...$A$. Usually, $A$ is taken to be either the multiplicative group $k^*$ of a field $k$ or the subgroup
...character of a group was originally introduced for finite groups $G$ with $A = T$ (in this case every character $G → ℂ^*$ takes values in $T$).
3 KB (438 words) - 13:36, 23 July 2015
$#A+1 = 33 n = 0
$#C+1 = 33 : ~/encyclopedia/old_files/data/I050/I.0500150 Image of a morphism
4 KB (576 words) - 22:11, 5 June 2020
...of points $a$, $b$, $c$, $d$ are said to be congruent if and only if $Q ( a - b ) = Q ( c - d )$.
...L$ be the set of all lines of $( P , \equiv )$. Such a structure is called a Euclidean plane if:
12 KB (1,960 words) - 17:44, 1 July 2020
...metry, and topology. Thus, in topology, each topological space $X$ defines a chain complex in the category $\textsf{Ab}$ of Abelian groups: $(C_n(X),\pa
The concept of the cohomology of a cochain complex is defined in a dual manner.
2 KB (265 words) - 21:51, 22 October 2016
$#A+1 = 61 n = 0
$#C+1 = 61 : ~/encyclopedia/old_files/data/D032/D.0302330 Differentiation of a mapping
6 KB (918 words) - 19:35, 5 June 2020
The ''pole of a function'' is
...of the point at infinity $a=\infty$, the function $f(z)$ can be written as a [[Laurent series]] of special form:
3 KB (549 words) - 09:10, 18 January 2014
$#A+1 = 54 n = 0
$#C+1 = 54 : ~/encyclopedia/old_files/data/G044/G.0404970 Graph of a mapping
3 KB (568 words) - 19:42, 5 June 2020
...$. In a wider sense, a section of any morphism in an arbitrary category is a right-inverse morphism.
If $U \subset Y$ is a subset of $Y$, a section over $U$ of $p$ is a mapping $s : U \rightarrow X$ such that $p(s(u)) = u$ for all $u \in U$.
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''$M_0=(x_0,y_0)$ relative to a circle''
$$(x-a)^2+(y-b)^2=R^2$$
2 KB (298 words) - 06:25, 12 October 2023
.... The quotient group $G/Z$ of
a group $G$ by its centre is not necessarily
a group without centre.
<table><TR><TD valign="top">[1]</TD> <TD valign="top">
A.G. Kurosh, "The theory of groups" , '''1–2''' , Chelsea (1955–1956)
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''$G$ by a subgroup $H$ (from the left)''
A set of elements of $G$ of the form
1 KB (215 words) - 20:26, 4 December 2014
Also called ''total variation''. A numerical characteristic of functions of one or more real variables which
where $N$ is an arbitrary natural number. The total variation of a function $f: I\to \mathbb R$ is given by
16 KB (2,511 words) - 16:07, 12 October 2012
A non-negative real-valued function $d$ defined on the set of all pairs of el
...(relative to $d$) if and only if $d(x,A) > 0$ for each $x \in X \setminus A$. Here
2 KB (339 words) - 20:16, 11 December 2016
...ages/c/c025/c025860/c02586012.png" /> is
a continuous representation, then
a contraction of <img align="absmiddle" border="0" src="https://www.encyclope
...w.encyclopediaofmath.org/legacyimages/c/c025/c025860/c02586053.png" /> and
a representation <img align="absmiddle" border="0" src="https://www.encyclope
8 KB (1,091 words) - 17:21, 7 February 2011
$#A+1 = 76 n = 0
$#C+1 = 76 : ~/encyclopedia/old_files/data/I052/I.0502440 Inversion of a matrix
11 KB (1,584 words) - 22:13, 5 June 2020
#REDIRECT [[Homology of a complex]]
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The centre of a ring $R$ is
$$Z=\{z: az = za \textrm{ for all }a \in R\}.$$
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''in a group $G$''
...$G$ is usually denoted by $\left|G:H\right|$. The product of the order of a subgroup $H$ by its index $\left|G:H\right|$ is equal to the order of $G$ (
1 KB (186 words) - 11:18, 20 April 2012
$#A+1 = 29 n = 1
$#C+1 = 29 : ~/encyclopedia/old_files/data/R077/R.0707470 Rank of a module
3 KB (439 words) - 08:09, 6 June 2020
...icient|asymmetry coefficient]] is positive (negative). If the asymmetry of a distribution is positive (negative), then the "longer" part of the densit
In Western literature one commonly calls this notion the skewness of a distribution, and accordingly one speaks of the skewness coefficient.
589 bytes (87 words) - 17:11, 7 February 2011
''$N$ to a base $a$''
...very positive number $N$ and given base $a>0$, $a\neq1$, there corresponds a unique real logarithm (logarithms of negative numbers are complex numbers).
3 KB (453 words) - 15:11, 19 August 2014
$#A+1 = 38 n = 0
$#C+1 = 38 : ~/encyclopedia/old_files/data/G043/G.0403990 Genus of a curve
3 KB (483 words) - 06:28, 31 March 2023
$#A+1 = 39 n = 1
$#C+1 = 39 : ~/encyclopedia/old_files/data/D030/D.0300900 Degree of a mapping,
4 KB (562 words) - 17:50, 13 January 2024
$#A+1 = 14 n = 0
$#C+1 = 14 : ~/encyclopedia/old_files/data/B017/B.0107320 Boundary (of a manifold)
1 KB (204 words) - 06:28, 30 May 2020
$#A+1 = 6 n = 0
$#C+1 = 6 : ~/encyclopedia/old_files/data/S091/S.0901530 Swerve of a curve,
1 KB (206 words) - 05:43, 9 April 2023
$#A+1 = 51 n = 0
$#C+1 = 51 : ~/encyclopedia/old_files/data/T092/T.0902630 Thinness of a set
4 KB (696 words) - 08:25, 6 June 2020
...les of a spherical triangle and two right angles. The excess $\epsilon$ of a spherical triangle is proportional to its area $S$: $S=R^2\epsilon$, where
...he defect of the triangle (cf. [[Triangle, defect of a|Triangle, defect of a]]).
867 bytes (128 words) - 11:56, 5 July 2014
...y, is the most important [[Cardinal characteristic|cardinal invariant]] of a topological space.
A space satisfies the [[second axiom of countability]] if and only if it has
873 bytes (124 words) - 20:48, 5 December 2023
...s closed. Every [[Proximate point|proximate point]] of a set $M$ is either a limit point or an [[Isolated point|isolated point]] of it.
A limit point of a set is usually called an [[Accumulation point|accumulation point]] of that
2 KB (248 words) - 17:14, 11 April 2014
The ''complexification of a Lie group $G$ over $\R$'' is
...subgroup such that the Lie algebra $\def\fg{ {\mathfrak g}}\fg$ of $G$ is a real form of the Lie algebra $\fg_\C$ of $G_\C$ (see
3 KB (501 words) - 15:05, 13 April 2024
A system of equations
obtained as the limit as $h\to0$, $z(m,h)\to z$ of a system of partially solved equations
4 KB (651 words) - 17:23, 14 February 2020
#REDIRECT [[Generator of a category]]
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...nite [[word]] over a finite alphabet describes the largest number of times a contiguous subsequence can be repeated. For example, the critical exponent
...s a factor $y$ of $w$ with $y = x^a x_p$ where $x_p$ is a prefix of $x$, $a$ is the integer part of $\alpha$, and the length $|y| \ge \alpha |x|$|: we
4 KB (533 words) - 20:21, 7 November 2023
''minimum polynomial of a matrix''
...] of $A$ and, more generally, it divides every polynomial $f$ such that $f(A)=0$.
679 bytes (100 words) - 15:17, 1 May 2014
''$A$ in a topological space $X$''
...in $X$ are closed. The pseudo-character $\psi(x,X)$ of a point $x\in X$ in a topological space $X$ is understood to be the pseudo-character $\psi(\{x\},
2 KB (346 words) - 13:09, 27 September 2014
''in a Euclidean space $E^n$''
...ty, it is perfect and has the cardinality of the continuum. The concept of a condensation point can be generalized to arbitrary topological spaces.
861 bytes (136 words) - 09:56, 26 March 2023
$#A+1 = 37 n = 0
$#C+1 = 37 : ~/encyclopedia/old_files/data/I051/I.0501550 Integral of a differential equation
5 KB (803 words) - 12:07, 31 March 2024
...mposition of a '''signed''' measure we refer to [[Jordan decomposition (of a signed measure)]].
...e $p\in [0,1]$, $\mu_a$ is an atomic probability measure and $\mu_{na}$ is a nonatomic probability measure. The decomposition into atomic and nonatomic
532 bytes (81 words) - 10:44, 17 September 2012
The collection of points at which $K$ is a [[Smooth continuum|smooth continuum]].
103 bytes (18 words) - 13:42, 29 April 2014
$#A+1 = 17 n = 0
$#C+1 = 17 : ~/encyclopedia/old_files/data/P072/P.0702420 Perturbation of a linear system
3 KB (437 words) - 07:39, 26 February 2022
''$S$ in a class of semi-groups $\Sigma$''
...rtain simple representations by union, multiple repetition, restriction to a subset, and by the operation of [[imbedding of semi-groups]].
4 KB (637 words) - 19:07, 20 March 2018
$#A+1 = 59 n = 0
$#C+1 = 59 : ~/encyclopedia/old_files/data/P074/P.0704120 Potential of a mass distribution
6 KB (915 words) - 05:56, 13 June 2022
$#A+1 = 25 n = 0
...1 = 25 : ~/encyclopedia/old_files/data/P075/P.0705300 Projective object of a category
4 KB (597 words) - 08:08, 6 June 2020
$#A+1 = 41 n = 0
...= 41 : ~/encyclopedia/old_files/data/P075/P.0705360 Projective spectrum of a ring
3 KB (468 words) - 09:09, 6 January 2024
The same as the [[Variation of a function|variation of a function]].
See also [[Oscillation of a function|Oscillation of a function]].
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...the remaining nodes of $G$ as (isolated) components. A matching is called a $k$-matching if it contains exactly $k$ edges. If $G$ contains $p$ nodes, t
...en and $k=p/2$; in this case, all the nodes of $G$ are covered with edges (a [[perfect matching]]); and
6 KB (1,005 words) - 20:14, 15 March 2023
$#A+1 = 37 n = 0
...= 37 : ~/encyclopedia/old_files/data/B017/B.0107560 Branching process with a random medium,
5 KB (646 words) - 06:29, 30 May 2020
$#A+1 = 158 n = 0
$#C+1 = 158 : ~/encyclopedia/old_files/data/T092/T.0902400 Tensor on a vector space
13 KB (1,934 words) - 08:25, 6 June 2020
The linear subspace of the [[domain of definition]] of a [[linear operator]] that consists of all vectors that are mapped to zero.
...of the domain onto the range) if and only if the [[adjoint operator]] has a weakly-dense range.
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$#A+1 = 85 n = 0
$#C+1 = 85 : ~/encyclopedia/old_files/data/F042/F.0402080 Functional of a Markov process
7 KB (1,069 words) - 19:40, 5 June 2020
$#A+1 = 14 n = 0
...4 : ~/encyclopedia/old_files/data/M062/M.0602310 Mapping, principal net of a,
1 KB (229 words) - 07:59, 6 June 2020
A [[Differential field|differential field]] $ F \supset F _{0} $
with a set $ \Delta $
13 KB (1,981 words) - 20:02, 17 December 2019
$#A+1 = 27 n = 0
...: ~/encyclopedia/old_files/data/C021/C.0201660 Characteristic function of a set
2 KB (248 words) - 18:24, 24 December 2020
$#A+1 = 13 n = 0
...13 : ~/encyclopedia/old_files/data/V096/V.0906440 Vector field, source of a
2 KB (251 words) - 08:28, 6 June 2020
$#A+1 = 112 n = 0
...encyclopedia/old_files/data/F040/F.0400280 Finite group, representation of a
10 KB (1,488 words) - 19:39, 5 June 2020
$#A+1 = 55 n = 0
$#C+1 = 55 : ~/encyclopedia/old_files/data/M064/M.0604490 Moduli of a Riemann surface
4 KB (668 words) - 06:30, 31 March 2023
...to the focal surface. Every conjugate net on a two-dimensional surface is a focal net of the congruence of tangents to the lines of one of the families
816 bytes (137 words) - 18:17, 24 December 2020
-
27 bytes (3 words) - 16:03, 8 August 2012
Also called ''negative increment of a function''
...te increment or [[Variation of a function|variation of a function]] $f$ on a given interval.
2 KB (271 words) - 08:48, 16 September 2012
A curve $\bar\gamma$ assigned to the plane curve $\gamma$ such that $\gamma$
...and $k'(s_1) \ne 0$ (the point corresponding to $s=s_1$ on the evolvent is a cusp of the second kind).
2 KB (317 words) - 21:09, 17 December 2017
...ge\aleph_0$ such that if $x$ is in the closure $\bar A$, then $A$ contains a subset $B$ of cardinality $\le \mathfrak{t}$ with $x \in\bar B$ . The tigh
566 bytes (91 words) - 20:48, 5 December 2023
$#A+1 = 88 n = 0
$#C+1 = 88 : ~/encyclopedia/old_files/data/R081/R.0801490 Representation with a highest weight vector
6 KB (805 words) - 08:11, 6 June 2020
$#A+1 = 43 n = 0
$#C+1 = 43 : ~/encyclopedia/old_files/data/C024/C.0204900 Congruence modulo a prime number
7 KB (1,033 words) - 17:46, 4 June 2020
The ''complexification of a Lie algebra
is a complex Lie algebra $\fg_\C$ that is the
3 KB (431 words) - 17:08, 22 November 2013
$#A+1 = 122 n = 0
$#C+1 = 122 : ~/encyclopedia/old_files/data/D110/D.1100090 Defect group of a block
6 KB (919 words) - 17:32, 5 June 2020
$#A+1 = 40 n = 0
$#C+1 = 40 : ~/encyclopedia/old_files/data/S087/S.0807230 Star of a function element,
5 KB (749 words) - 19:11, 19 June 2020
$#A+1 = 72 n = 0
$#C+1 = 72 : ~/encyclopedia/old_files/data/L060/L.0600280 Localization in a commutative algebra
4 KB (713 words) - 21:35, 4 January 2021
...register sequence|Shift register sequence]]) capable of producing $\mathbf{a}$. The linear complexity also equals the maximum number of linearly indepen
\begin{equation*} {\bf a}^ { ( t ) } = ( a _ { t } , a _ { t + 1} , \dots , a _ { n + t - 1 }) ( t \geq 0 ) \end{equation*}
4 KB (622 words) - 19:04, 17 February 2024
...screte subgroup]] $\Gamma$ of a [[Lie group]] $G$ such that $G/\Gamma$ has a finite volume relative to the (induced) $G$-invariant measure.
...nsional vector space $V$ over $\mathbf R$ is discrete if and only if it is a lattice [[#References|[1]]].
874 bytes (134 words) - 18:43, 11 April 2023
''zero (object) of a category''
...it is the zero module, etc. Not every category contains a null object, but a null object can always be formally adjoined to any given category.
2 KB (327 words) - 18:38, 13 November 2023
$#A+1 = 47 n = 0
$#C+1 = 47 : ~/encyclopedia/old_files/data/A012/A.0102340 Analytic model of a language,
9 KB (1,377 words) - 18:47, 5 April 2020
$#A+1 = 39 n = 0
...clopedia/old_files/data/G044/G.0404960 Graph, numerical characteristics of a
5 KB (780 words) - 20:06, 15 March 2023
''$\{f^tx\}$ of a dynamical system $f^t$''
A point
1 KB (254 words) - 17:08, 14 February 2020
A non-zero lower semi-continuous semi-finite trace $ f $
on a [[C*-algebra| $ C ^ {*} $-
2 KB (358 words) - 14:43, 27 January 2024
...led a compact extension, and if $Y$ is a [[Hausdorff space]], it is called a Hausdorff extension.
536 bytes (75 words) - 22:21, 7 November 2014
$#A+1 = 15 n = 0
...encyclopedia/old_files/data/M063/M.0603170 Mean\AAhsquare approximation of a function
2 KB (334 words) - 20:15, 10 January 2024
...hen $x-y\in K$. A proper cone $K$ in a space $X$ is called reproducing (or a domain of positivity) if $K-K=X$.
...[[topological vector space]] $k$, let $V$ be the intersection of $K$ with a bounded open ball with centre at $0\in k$, and let $U\neq\emptyset$ be an o
1 KB (235 words) - 10:40, 26 May 2016
...ed with this notion is that of the uniqueness of a trigonometric series of a function (see [[Uniqueness set|Uniqueness set]]).
...<TR><TD valign="top">[2]</TD> <TD valign="top"> N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russ
1,018 bytes (145 words) - 17:01, 7 February 2011
$#A+1 = 32 n = 0
$#C+1 = 32 : ~/encyclopedia/old_files/data/K055/K.0505240 Kernel of a complex sequence
3 KB (439 words) - 22:14, 5 June 2020
$#A+1 = 31 n = 0
$#C+1 = 31 : ~/encyclopedia/old_files/data/P074/P.0704210 Power function of a test
3 KB (374 words) - 08:07, 6 June 2020
''$X$ with respect to a coefficient group $G$''
...uality|Pontryagin duality]], the homological approach with coefficients in a compact group $G$ is equivalent to the cohomological approach with coeffici
5 KB (741 words) - 06:05, 22 November 2018
$#A+1 = 12 n = 0
...12 : ~/encyclopedia/old_files/data/S085/S.0805600 Singular point, index of a
2 KB (300 words) - 08:21, 21 March 2022
-
32 bytes (3 words) - 14:13, 11 July 2018
A canonical decomposition theorem (due to Jordan) for [[Function of bounded
...e pair is unique up to addition of a constant, i.e. if $g^+$ and $g^-$ is a second pair with the same property, then $g^+-f^+=g^--f^-\equiv {\rm const
2 KB (301 words) - 14:05, 10 December 2012
...rd semi-group]]. The set $S^*$ of all characters of a semi-group $S$ forms a commutative semi-group with identity (the character semi-group) under [[poi
(\chi*\psi)(a) = \chi(a)\cdot\psi(a)\,,\ \ a\in S\,,\ \ \chi,\psi\in S^* \ .
3 KB (459 words) - 19:54, 1 January 2018
$#A+1 = 17 n = 0
...= 17 : ~/encyclopedia/old_files/data/I052/I.0502330 Invariant subspace of a representation
1 KB (174 words) - 22:13, 5 June 2020
$#A+1 = 62 n = 0
...62 : ~/encyclopedia/old_files/data/D032/D.0302260 Differential operator on a module
5 KB (785 words) - 19:35, 5 June 2020
...l character'' $\chi(x,X)$ at a point $x \in X$ is the least cardinality of a [[local base]] at $x$. The character $\chi(X)$ is the least upper bound of
A space satisfies the [[first axiom of countability]] if and only if it has c
542 bytes (83 words) - 12:05, 23 November 2023
...he trace of a matrix $A = [a_{ij}]$ is denoted by $\tr A$, $\Tr A$ or $\Sp A$:
\tr A = \sum_{i=0}^n a_{ii}
2 KB (389 words) - 19:21, 11 November 2023
$#A+1 = 26 n = 0
$#C+1 = 26 : ~/encyclopedia/old_files/data/R082/R.0802490 Risk of a statistical procedure
3 KB (411 words) - 08:11, 6 June 2020
...clopediaofmath.org/legacyimages/c/c025/c025850/c02585023.png" /> is called
a contraction of the group <img align="absmiddle" border="0" src="https://www
...cyclopediaofmath.org/legacyimages/c/c025/c025850/c02585037.png" /> becomes
a commutative ideal of <img align="absmiddle" border="0" src="https://www.enc
10 KB (1,419 words) - 17:01, 7 February 2011
A [[Normal series|normal series]] all factors of which are central, that is, a series of subgroups
...upper central series have the same length, which is the minimal length of a central series of the group. This length is called the nilpotency class (or
1 KB (185 words) - 12:34, 14 February 2020
-
106 bytes (15 words) - 17:08, 7 February 2011
A function defined for all real numbers $x$ and equal to the difference betwe
275 bytes (39 words) - 13:07, 16 December 2012
...rable ring $B$ can be uniquely extended to a continuous homomorphism $\hat A \rightarrow B$.
...f the discrete rings $A/\mathfrak{a}_\lambda$. The separable completion of a module is achieved in the same way.
1 KB (174 words) - 15:55, 21 December 2014
...eria can be stated as follows: 1) its complement must also be an $\mathcal{A}$-set (Suslin's criterion); and 2) it can be represented as the union of no
356 bytes (55 words) - 21:57, 19 December 2014
...ct to the grid parameters (see [[#References|[9]]], [[#References|[10]]]). A differential approximation includes some of the terms of the expansion; the
...s possible, using the apparatus of differential approximations, to produce a group classification of difference schemes (see [[#References|[9]]]).
7 KB (987 words) - 14:03, 24 December 2020
$#A+1 = 73 n = 0
...= 73 : ~/encyclopedia/old_files/data/I051/I.0501450 Integral extension of a ring
3 KB (562 words) - 22:12, 5 June 2020
$#A+1 = 33 n = 0
$#C+1 = 33 : ~/encyclopedia/old_files/data/M065/M.0605500 Multiplicity of a singular point
3 KB (388 words) - 17:12, 7 February 2021
$#A+1 = 17 n = 0
$#C+1 = 17 : ~/encyclopedia/old_files/data/H110/H.1100170 Henselization of a valued field
2 KB (285 words) - 22:10, 5 June 2020
$#A+1 = 55 n = 0
$#C+1 = 55 : ~/encyclopedia/old_files/data/I052/I.0502180 Invariance of a statistical procedure
5 KB (735 words) - 07:44, 13 May 2022
Let $z=a+ib$ in its geometric representation.
...corresponding to this number, i.e. the point with Cartesian coordinates $(a,b)$. The affix is sometimes identified with the complex number itself.
232 bytes (39 words) - 09:35, 17 October 2013
The smallest two-sided [[ideal]] in the semi-group. Not every semi-group has a kernel.
<TR><TD valign="top">[1]</TD> <TD valign="top"> A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , Amer. Ma
485 bytes (62 words) - 18:11, 7 May 2016
...exactly, an extension $L$ of a field $K$ is called the splitting field of a polynomial $f$ over the field $K$ if $f$ decomposes over $L$ into linear fa
...identity on $K$. It follows from the definition that a splitting field is a finite [[algebraic extension]] of $K$.
1 KB (237 words) - 14:06, 20 March 2023
''$\xi$ of a space with a normalized measure $(X,\mu)$''
...ned as follows. If the elements of $\xi$ having measure zero form in total a set of positive measure, then the entropy of $\xi$ is $H(\xi)=\infty$; othe
768 bytes (122 words) - 08:36, 29 August 2014
A partially resolving operator $L_m^h$, uniformly bounded in $h$ and $m$, des
...[[#References|[6]]]). There are also other definitions of the stability of a computational algorithm (cf. e.g. [[#References|[1]]], [[#References|[3]]])
2 KB (295 words) - 14:06, 14 August 2014
...esentation|continuous representation]] of the compact topological group in a topological vector space (cf. [[Compact group|Compact group]]).
159 bytes (20 words) - 17:05, 7 February 2011
$#A+1 = 22 n = 0
$#C+1 = 22 : ~/encyclopedia/old_files/data/C024/C.0204090 Completion of a uniform space
2 KB (257 words) - 17:46, 4 June 2020
$#A+1 = 91 n = 0
...: ~/encyclopedia/old_files/data/P075/P.0705220 Projective determination of a metric
7 KB (1,055 words) - 10:04, 21 March 2022
...it a finite number of terms. More precisely, a summation method $\mathcal{A}$ is said to be translative if the summability of the series
...method $\mathcal{A}$ defined by transformation of the sequence $S_n$ into a sequence or function, the property of translativity consists of the equival
2 KB (319 words) - 19:29, 23 December 2015
$#A+1 = 16 n = 0
$#C+1 = 16 : ~/encyclopedia/old_files/data/C024/C.0204220 Complexification of a vector space
1 KB (197 words) - 17:46, 4 June 2020
$#A+1 = 39 n = 0
$#C+1 = 39 : ~/encyclopedia/old_files/data/E036/E.0306990 Extension of a Lie algebra
3 KB (416 words) - 12:53, 19 March 2023
'' $ A $''
on the set $ A ^ {+} $
3 KB (490 words) - 13:58, 21 January 2024
The term of a convergent [[Series|series]] of numbers or functions with positive terms th
$$ \sum_{k=0}^{\infty}c_{k}(z-a)^{k} $$
1 KB (256 words) - 13:07, 12 December 2013
...ich is valid for all finite semi-groups (a property of this kind is called a finiteness condition). The definition of the property $\theta$ may be phras
...emi-group can be imbedded in a semi-group with two generators, and also in a semi-group with three idempotent generators [[#References|[8]]].
6 KB (870 words) - 11:58, 1 August 2014
...racteristic]]s of a [[topological space]]. The minimal [[cardinality]] of a [[dense set]].
A [[separable space]] is one with countable density.
381 bytes (48 words) - 11:57, 23 November 2023
...ical test|statistical test]] for testing a simple hypothesis $H_0$ against a simple hypothesis $H_1$ rejects $H_0$ when in fact $H_1$ is true. In the ca
...wer function of the test (see [[Power function of a test|Power function of a test]]).
2 KB (246 words) - 11:23, 13 August 2014
$#A+1 = 242 n = 0
$#C+1 = 242 : ~/encyclopedia/old_files/data/D032/D.0302240 Differential on a Riemann surface
18 KB (2,560 words) - 19:35, 5 June 2020
...If $\Sigma\subset\Omega$ is a $C^1$ $n-1$-dimensional surface oriented by a (continuous) unit normal $\nu$, the flux of the vector field $v$ through th
...computed using the [[Area formula]]. If $\Sigma$ is given by the graph of a function $f: \mathbb R^{n-1} \supset V \to \mathbb R$ with its natural orie
4 KB (589 words) - 15:27, 4 February 2014
$#A+1 = 57 n = 1
...7 : ~/encyclopedia/old_files/data/D032/D.0302070 Differential equations on a torus,
12 KB (1,841 words) - 18:33, 5 June 2020
$#A+1 = 64 n = 0
$#C+1 = 64 : ~/encyclopedia/old_files/data/C024/C.0204890 Congruence modulo a double modulus
4 KB (700 words) - 18:53, 18 January 2024
$#A+1 = 72 n = 0
$#C+1 = 72 : ~/encyclopedia/old_files/data/V096/V.0906430 Vector field on a manifold
6 KB (884 words) - 08:28, 6 June 2020
$#A+1 = 49 n = 0
$#C+1 = 49 : ~/encyclopedia/old_files/data/R081/R.0801690 Retract of a topological space
5 KB (871 words) - 19:55, 3 February 2021
...resentation theory of Lie groups and Lie algebras (cf. [[Representation of a Lie algebra]]).
<TR><TD valign="top">[a1]</TD> <TD valign="top"> T.A. Springer, "Invariant theory" , ''Lect. notes in math.'' , '''585''' , Spri
2 KB (267 words) - 19:16, 27 October 2023
$#A+1 = 50 n = 0
...1 = 50 : ~/encyclopedia/old_files/data/T093/T.0903890 Transmission rate of a channel
6 KB (879 words) - 08:26, 6 June 2020
An argument $x$ of a function $f$ that satisfies the condition
...ly speaking, the converse is not true. If a one-sided derivative exists at a smooth point, an ordinary derivative exists as well.
569 bytes (102 words) - 06:04, 22 August 2014
...et of points at which $f$ vanishes (cf. [[Support of a function|Support of a function]]). For example, $\mathop{\mathrm{supp}} x = \mathbf{R}^1$, $\math
...that in any neighbourhood of them the generalized function is not equal to a $C^\infty$-function. For example, $\mathop{\mathrm{sing}} \mathop{\mathrm{s
3 KB (450 words) - 09:18, 12 August 2022
$#A+1 = 97 n = 1
.../encyclopedia/old_files/data/A011/A.0101980 Almost\AAhperiodic function on a group
12 KB (1,716 words) - 11:05, 10 May 2020
$#A+1 = 69 n = 0
$#C+1 = 69 : ~/encyclopedia/old_files/data/L058/L.0508910 Limit set of a trajectory
5 KB (812 words) - 04:31, 14 September 2022
$#A+1 = 54 n = 0
$#C+1 = 54 : ~/encyclopedia/old_files/data/E037/E.0307010 Extension of a semi\AAhgroup
4 KB (623 words) - 19:38, 5 June 2020
$#A+1 = 25 n = 1
$#C+1 = 25 : ~/encyclopedia/old_files/data/V096/V.0906150 Variation of a univalent function
4 KB (590 words) - 08:27, 6 June 2020
$#A+1 = 15 n = 0
$#C+1 = 15 : ~/encyclopedia/old_files/data/K055/K.0505310 Kernel of a summation method
3 KB (389 words) - 22:14, 5 June 2020
#REDIRECT[[Variation of a function]]
36 bytes (5 words) - 14:36, 18 September 2012
$#A+1 = 50 n = 0
$#C+1 = 50 : ~/encyclopedia/old_files/data/D033/D.0303920 Double of a Riemann surface
4 KB (635 words) - 19:36, 5 June 2020
...]). If a Lie group $G$ coincides with the set of real or complex points of a [[linear algebraic group]] $\widehat G$, then the rank of $G$ coincides wit
<TR><TD valign="top">[a3]</TD> <TD valign="top"> A.W. Knapp, "Representation theory of semisimple groups" , Princeton Univ. Pr
1 KB (177 words) - 19:00, 7 April 2023
$#A+1 = 37 n = 0
...1 = 37 : ~/encyclopedia/old_files/data/F042/F.0402200 Fundamental forms of a surface
6 KB (990 words) - 19:40, 5 June 2020
...ith an optimal one. In mathematical statistics the notion of optimality of a statistical procedure is usually expressed in terms of the risk (function)
...istical procedure is a concept not altogether clearly defined. It acquires a more precise meaning in specific problems of mathematical statistics such a
2 KB (252 words) - 05:48, 15 August 2013
Also called ''positive increment of a function''
...e increment or [[Variation of a function|variation of a function]] $f$ on a given interval.
2 KB (269 words) - 08:46, 16 September 2012
$#A+1 = 11 n = 0
...11 : ~/encyclopedia/old_files/data/A110/A.1100780 Asymptotic invariant of a group
3 KB (362 words) - 09:44, 14 April 2024
$#A+1 = 32 n = 0
...clopedia/old_files/data/G043/G.0403820 Generalized function, derivative of a
4 KB (499 words) - 08:03, 25 April 2022
$#A+1 = 87 n = 2
$#C+1 = 87 : ~/encyclopedia/old_files/data/R081/R.0801390 Representation of a Lie algebra
10 KB (1,344 words) - 08:11, 6 June 2020
$#A+1 = 114 n = 2
$#C+1 = 114 : ~/encyclopedia/old_files/data/R081/R.0801420 Representation of a topological group
20 KB (2,775 words) - 16:40, 31 March 2020
$#A+1 = 91 n = 0
$#C+1 = 91 : ~/encyclopedia/old_files/data/D031/D.0301110 Density of a probability distribution,
6 KB (898 words) - 17:32, 5 June 2020
$#A+1 = 27 n = 0
$#C+1 = 27 : ~/encyclopedia/old_files/data/S087/S.0807000 Stability of a computational process
3 KB (507 words) - 08:22, 6 June 2020
...of a category|generator of a category]]. Explicitly, an integral object is a generator which is "minimal" in the sense that it has no non-trivial idem
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''in a topological space''
...sense. The [[interior]] of a set $A$ consists of the interior points of $A$.
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$#A+1 = 77 n = 0
...1 = 77 : ~/encyclopedia/old_files/data/S085/S.0805380 Simplicial object in a category
7 KB (966 words) - 21:39, 10 June 2020
$#A+1 = 79 n = 0
$#C+1 = 79 : ~/encyclopedia/old_files/data/C024/C.0204800 Conformal radius of a domain
5 KB (759 words) - 16:51, 5 June 2020
$#A+1 = 14 n = 1
$#C+1 = 14 : ~/encyclopedia/old_files/data/H047/H.0407820 Homology of a dynamical system,
4 KB (528 words) - 22:10, 5 June 2020
A numerical invariant of a [[field]] describing the structure of [[quadratic form]]s over the field.
...uadratic form|isotropic]], or that every form of dimension at least $u$ is a [[universal quadratic form]].
3 KB (488 words) - 20:22, 15 November 2023
...set of axioms which generalise the notion of median, or majority vote, as a [[Boolean function]].
...wedge (z \vee x)$ satisfies these axioms, so that every Boolean algebra is a median algebra.
2 KB (261 words) - 18:11, 14 October 2023
$#A+1 = 50 n = 0
...al based on [[Richardson extrapolation|Richardson extrapolation]]. Suppose a value $ I $
5 KB (738 words) - 19:50, 16 January 2024
$#A+1 = 46 n = 0
A programmed grammar (introduced in [[#References|[a3]]]) is a construct
4 KB (620 words) - 08:10, 6 June 2020
...ferences|[a5]]], B. Eckmann and A. Schöpf defined an injective envelope of a module to be an embedding $M \subset E$ with $E$ injective and with $M$ ess
...ope. If $R$ is a [[Ring|ring]], Bass proved that every left $R$-module has a projective cover if and only if every flat left $R$-module is projective (c
6 KB (1,031 words) - 16:55, 1 July 2020
A [[Complex number|complex number]] $i$ whose square is equal to minus one: $
98 bytes (17 words) - 22:37, 16 March 2014
A finite-dimensional [[Lie algebra|Lie algebra]] over a field $ k $
...ion of a Lie algebra|Representation of a Lie algebra]]). The property that a Lie algebra $ \mathfrak g $
3 KB (380 words) - 21:59, 16 December 2019
...ce as special cases. A convex cone is sometimes meant to be the surface of a convex cone.
391 bytes (67 words) - 13:27, 9 June 2016
...lelopipedon all faces of which are squares is called a cube. The volume of a parallelopipedon is equal to the product of the area of its base and its he
...opipedon is a special case of a [[Parallelohedron|parallelohedron]] and of a [[Parallelotope|parallelotope]]. Two special parallelopipeda, namely the go
1 KB (188 words) - 16:21, 11 April 2014
A one-dimensional manifold properly embedded in a $3$-ball, $D^3$.
...term "arcbody" is used for a one-dimensional manifold properly embedded in a $3$-dimensional manifold.
2 KB (336 words) - 09:53, 16 March 2023
where $D$ is a given natural number. Principal characters serve to define the concepts of
340 bytes (45 words) - 20:04, 9 January 2015
''of a ring''
...with respect to the addition operation in the ring. The additive group of a ring is always Abelian.
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...the set of husbands and $B$ the set of wives.) Now, take a random pairing (a bijection again). What is the chance that this random pairing gives at leas
...mort, Pierre Rémond de|P.R. de Montmort]] (around 1700) in connection with a card game known as the "jeu du treize", "jeu de rencontre" or simply "r
1 KB (177 words) - 07:34, 2 December 2016
A sequence $x=(\xi_i)_1^\infty$ is said to be summable to $A(x)$ by a method $A$ given by an infinite matrix $(a_{i,k})$, if
$$A(x)=\lim_i\sum_{k=1}^\infty a_{i,k}\xi_i.$$
2 KB (413 words) - 14:16, 3 August 2014
...then the statement "X is subject to the Poisson law with parameter 1" is
a simple hypothesis. See also [[Composite hypothesis|Composite hypothesis]].
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">
A.M. Mood, F.
A. Graybill, "Introduction to the theory of statistics" , McGraw-Hill (196
675 bytes (94 words) - 09:30, 27 April 2014
...our which is described by a point $M$ of a circle of radius $r$ rolling on a circle with the same radius $r$; an [[Epicycloid|epicycloid]] with modulus
...] of the circle, a special case of a [[Pascal limaçon|Pascal limaçon]] and a [[Sinusoidal spiral|sinusoidal spiral]].
1 KB (192 words) - 09:12, 26 March 2023
...tors of $n$ must be greater than $\mathcal P(n)$, where $\mathcal P(n)$ is a function that increases more slowly than $n$. For example,
...TD valign="top"> Diamond, Harold G.; Halberstam, H.; Galway, William F. "A higher-dimensional sieve method. With procedures for computing sieve functi
823 bytes (114 words) - 18:31, 5 May 2023
...n of the current $\psi(z,r)$ of a flow generated in an unbounded liquid by a vortical ring as integrals containing the Bessel functions of order zero an
...it itself produces in the ring, moves in the direction of the $z$-axis at a constant velocity $c$, which is given by the following approximate formula:
1 KB (238 words) - 06:39, 15 August 2014
...example, a central source of electric power. One is seeking the minimum of a function
...n (of a real variable)]]) and one shows that, with some exceptions, it has a unique minimizer. These assertions remain valid when $\rho$ is allowed to b
5 KB (888 words) - 14:13, 21 August 2014
...or its minimum value. The usual practice is to find the extremal values of a certain characteristic by imposing restrictions on other numerical characte
where $n=k[n/k]+r$. A complete $k$-partite graph, the cardinalities of the parts of which differ
4 KB (567 words) - 20:10, 15 March 2023
A sequence or series is said to be absolutely summable by a given [[Matrix summation method|matrix summation method]] if the sequence o
968 bytes (128 words) - 17:18, 7 February 2011
For a compact [[Riemannian manifold|Riemannian manifold]] $M = M ^ { n }$, let $\
Note that a lower bound for the first eigenvalue, without any curvature assumptions, is
2 KB (297 words) - 16:59, 1 July 2020
$#A+1 = 14 n = 1
$#C+1 = 14 : ~/encyclopedia/old_files/data/H047/H.0407820 Homology of a dynamical system,
4 KB (528 words) - 22:10, 5 June 2020
$#A+1 = 72 n = 0
From a purely mathematical point of view, the Golay codes are the most interesting
5 KB (772 words) - 11:10, 23 March 2023
$#A+1 = 10 n = 0
A system
2 KB (224 words) - 08:10, 6 June 2020
...decor'' was adapted from an original article by Alicia Carriquiry, Herbert A. David, which appeared in ''StatProb: The Encyclopedia Sponsored by Statist
George W. Snedecor is best remembered as a pioneer in making
14 KB (2,125 words) - 19:20, 24 March 2023
.... The quotient group $G/Z$ of
a group $G$ by its centre is not necessarily
a group without centre.
<table><TR><TD valign="top">[1]</TD> <TD valign="top">
A.G. Kurosh, "The theory of groups" , '''1–2''' , Chelsea (1955–1956)
805 bytes (131 words) - 18:26, 5 December 2014
A discrete-time [[Stochastic process|stochastic process]] $ ( X _ {n} ) _ {
a function of $ m + 1 $
2 KB (320 words) - 12:34, 18 February 2022
...egins at the moment when the concept of convergence has been introduced on a set of elements.
...ove, given specific conditions, the existence of a solution and to provide a method that makes it possible to calculate this solution with the required
22 KB (3,726 words) - 10:31, 2 September 2017
...project onto the $(p,T)$-plane (pressure-temperature plane) in the form of a curve, the general form of its equation — equality of the chemical potent
...d the second phases. A three-phase state is represented by a point, called a triple point.
1 KB (222 words) - 17:50, 5 August 2014
$#A+1 = 10 n = 3
A statement according to which for real linear forms
2 KB (294 words) - 20:27, 11 November 2023
...ightly narrower sense, by the semantics of a formalized language one means a system of agreements that determine the understanding of the formulas of th
...and intuitionistic logic has an extensional nature: that is, the truth of a complex statement is determined only by the truth character of the expressi
5 KB (771 words) - 16:59, 7 February 2011
A property of a [[binary operation]]. For the [[logical operation]] of [[conjunction]] ($\w
a \wedge a = a\ \ \text{and}\ \ a \vee a = a \ .
682 bytes (101 words) - 19:48, 13 November 2016
...left primitive ideals. The set $ \mathfrak{P} $ of all primitive ideals of a ring, endowed with some topology, is useful in the study of various classes
\forall A \subseteq \mathfrak{P}: \qquad
960 bytes (147 words) - 04:53, 24 April 2017
$#A+1 = 42 n = 0
''theorem on the completeness of a system of powers $ \{ x ^ {\lambda _ {k} } \} $
4 KB (588 words) - 12:46, 6 January 2024
In an algebraic structure $A$ with a [[binary operation]] $\cdot$, the left and right cancellation laws respecti
Such a structure is termed "cancellative".
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$#A+1 = 134 n = 0
A locally Euclidean space with a differentiable structure. Let $ X $
17 KB (2,329 words) - 01:38, 30 December 2021
''for measurability of a function of a real variable''
...ient that for any $\epsilon>0$ there is a function $\phi$, continuous on $[a,b]$, such that the measure of the set
3 KB (436 words) - 19:43, 28 December 2017
$#A+1 = 69 n = 0
which is defined by a non-degenerate $ 2 $-
6 KB (837 words) - 08:24, 6 June 2020
$#A+1 = 77 n = 3
...a11]]] solved the following problem: find all rigid bodies, rotating about a fixed point in the presence of gravity, such that the equations of motion a
13 KB (1,808 words) - 22:15, 5 June 2020
..., $(2,3)$ and $(5,6)$, $(3,4)$ and $(6,1)$ — say, 9, 7 and 8 — also lie on a single line for every choice of three points each on any two straight lines
...ean geometry]].) The possibility of proving Pascal's theorem is similar in a certain sense to that of proving the [[Desargues assumption|Desargues assum
5 KB (796 words) - 21:08, 11 April 2014
$#A+1 = 27 n = 0
...with multiplication having a two-sided homotopy identity. More precisely, a pointed topological space $ ( X , e) $
3 KB (410 words) - 19:42, 5 June 2020
$#A+1 = 6 n = 0
A [[Riemannian space|Riemannian space]] $ M $
3 KB (397 words) - 08:10, 6 June 2020
$#A+1 = 141 n = 0
...ii geometry]]: In the plane, through a given point that is not incident to a given straight line there are at least two straight lines that do not inter
19 KB (3,123 words) - 23:52, 14 December 2020
...coordinate frames defines a unique [[affine transformation]] of the space $A^n$ which converts the first frame into the second (see also [[Affine coordi
...s $\mathbf{e}_i$ in the definition should be understood as independence in a corresponding vector space.
1 KB (211 words) - 19:23, 21 January 2016
$#A+1 = 50 n = 0
A [[Topological space|topological space]] $ | A | $
3 KB (513 words) - 08:14, 6 June 2020
...emi-classical approximation for $\sum | e | ^ { \gamma }$, which serves as a heuristic motivation for (a1). It is (cf. [[#References|[a14]]]):
...n = 1$ in [[#References|[a14]]] and for $n > 1$ in [[#References|[a2]]] by A. Laptev and Weidl. (See also [[#References|[a1]]].)
15 KB (2,262 words) - 19:26, 26 March 2023
$#A+1 = 31 n = 0
\dot{x} dot = - a \sin x,
5 KB (785 words) - 08:05, 6 June 2020
$#A+1 = 47 n = 0
be a prime number. The $ p $-
4 KB (568 words) - 14:10, 31 December 2020
...n R$. Thus, $R$ determines a function (perhaps not defined everywhere) on $A$. When it satisfies $R^{-1}\circ R=\Delta$ this function is well-defined ev
...$R\subset A\times B$ between sets $A$ and $B$ such that $(a,b)\in R$ and $(a,c)\in R$ imply $b=c$.
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...puting numerically the [[conformal mapping]] $g$ of the unit disc $D$ onto a star-like region $\Delta$ given by the polar coordinates $\tau$, $\rho ( \t
...torname { log } ( g ( w ) / w )$ is analytic in $D$ and can be extended to a [[Homeomorphism|homeomorphism]] of the closure $\overline{ D }$ onto the cl
7 KB (1,076 words) - 07:40, 7 February 2024
$#A+1 = 79 n = 1
respectively. A linear subspace $ E = E ( X, \mu ) $
6 KB (916 words) - 16:51, 19 February 2022
$#A+1 = 13 n = 0
A statistical test for testing the hypothesis $ H _ {0} $
2 KB (288 words) - 18:11, 3 March 2021
$#A+1 = 180 n = 0
...by K. Menger in 1942 and has since been developed by a number of authors. A treatment, comprehensive up to 1983, may be found in {{Cite|SSk}}.
12 KB (1,757 words) - 08:07, 6 June 2020
A ''field'' is a commutative, associative ring containing a unit in which the set of
non-zero elements is not empty and forms a group under multiplication
6 KB (929 words) - 00:28, 18 May 2013
...orm distribution of integral points on surfaces; for example, for $n=3$ on a sphere or on an ellipsoid. The strongest results are obtained by the method
...ice points, since the set $\mathbf Z^n\subset\mathbf R^n$ is an example of a [[lattice of points]].
2 KB (239 words) - 19:03, 2 October 2016
...m $a^b$ is called a power (that of the form $x\to a^x$ a power function); $a$ is called the base and $b$ is called the exponent.
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$#A+1 = 7 n = 0
A [[Propositional calculus|propositional calculus]] using only the primitive
2 KB (329 words) - 22:11, 5 June 2020
...s in the plane for which the sum of the distances from two given points is a constant magnitude.
..."geometric locus" is somewhat obsolete, and nowadays usually simply means a point set defined by some property.
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Surreal numbers are a creation of the British mathematician J.H. Conway [[#References|[a2]]]. The
...never terminates, and therefore the surreal numbers do not form a set but a proper class (cf. [[Types, theory of|Types, theory of]]).
8 KB (1,226 words) - 16:11, 3 July 2016
$#A+1 = 80 n = 1
A [[Semi-group|semi-group]] in which every element is regular (see [[Regular
10 KB (1,515 words) - 08:10, 6 June 2020
$#A+1 = 65 n = 0
grows at a rate $ n ^ {\alpha + 1/2 } $
8 KB (1,016 words) - 22:14, 5 June 2020
which little is known, at home and in a local Franciscan school. In
He worked as a lawyer in the local parliament in Toulouse, becoming
7 KB (1,004 words) - 19:40, 8 March 2024
An orthonormal system of vectors is a system $(x_\alpha)$ of vectors in a Euclidean (Hilbert) space with inner product $(\cdot,\cdot)$ such that $(x_
...system $(\phi_k)$ of functions which is both an [[orthogonal system]] and a [[normalized system]] , i.e.
2 KB (321 words) - 18:58, 7 September 2017
...on $P$ admits (in the convolution semi-group of probability distributions) a factorization
...bution that is either degenerate or is representable as the convolution of a finite or countable set of indecomposable distributions (cf. [[Indecomposab
2 KB (326 words) - 16:26, 9 April 2016
.... It satisfies the trace condition: $\mathrm{res}(AB) = \mathrm{res}(BA)$. A very important property is that it can be expressed as an integral (local f
\mathrm{res} A = \int_{S^*M} \tr a_{-n}(x,\xi) d\xi
2 KB (395 words) - 18:13, 25 September 2017
$#A+1 = 39 n = 0
The vector field generated by the gradients of a scalar function $ f $
4 KB (527 words) - 08:07, 6 June 2020
$#A+1 = 55 n = 0
''over a ring $ k $''
5 KB (790 words) - 22:16, 5 June 2020
...p, and of a circle with respect to a point at infinity; the [[evolute]] of a nephroid is another nephroid.
...Freeth''' is the [[strophoid]] of a circle with respect to its centre and a point on the circumference. It has equation
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...y straight line lying in $\alpha$. See [[Orthogonality|Orthogonality]] for a generalization of the concept of perpendicularity.
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A trigonalizable element of the algebra $ \mathop{\rm End}\nolimits \ V $
of endomorphisms of a finite-dimensional vector space $ V $
4 KB (574 words) - 11:56, 22 December 2019
...llows. Start with any node and connect it to the node nearest to it. Given a connected set of nodes, take the unconnected node that is nearest to one in
...a [[set function]], the greedy algorithm works as follows. Let $\nu(Q)$ be a real-valued function defined on all subsets $Q$ of $\{1,2,\ldots,n\}$ and c
2 KB (387 words) - 15:16, 20 November 2014
''of a ring $R$''
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...ubcomplex of some simplex of sufficiently high dimension. The dimension of a finite geometric complex is the largest dimension of its constituent simpli
...ds of the vectors of some orthonormal basis. Any geometric complex defines a topological space, constituted by all the points of its simplices and known
3 KB (419 words) - 17:16, 7 February 2011
$#A+1 = 40 n = 0
$#C+1 = 40 : ~/encyclopedia/old_files/data/A012/A.0102100 Alternion
3 KB (462 words) - 16:35, 1 April 2020
...ces|[5]]], and in the theory of formal groups [[#References|[6]]]. Let $ A $
...e, commutative ring with unit element. Witt vectors with components in $ A $
17 KB (2,502 words) - 17:25, 22 December 2019
$#A+1 = 69 n = 0
...crimination, S. Kullback and R.A. Leibler [[#References|[a13]]] introduced a measure of the "distance" or "divergence" between statistical populatio
9 KB (1,370 words) - 22:15, 5 June 2020
''entier, integer part of a (real) number $x$''
...the integers. Using the integral part one defines the [[fractional part of a number]] $x$, denoted by the symbol $\{x\}$ and given by
2 KB (239 words) - 16:50, 23 November 2023
...theta )$ for the sample size required to attain with a level-$\alpha$ test a prescribed power $\beta$ at an alternative $\theta$. If one has two tests w
...both the computation and interpretation, thus hoping that the limit gives a sufficiently good approximation of the far more complicated finite-sample c
15 KB (2,246 words) - 16:10, 11 February 2024
...two opposite sides $AB$ and $CD$ of a rectangle $ABCD$ so that the points $A$ and $B$ are matched with the points $C$ and $D$, respectively (see Fig.).
In the Euclidean space $E^3$ the Möbius strip is a one-sided surface (see [[One-sided and two-sided surfaces]]).
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$#A+1 = 15 n = 0
A variable whose values can be predicates (cf. [[Predicate|Predicate]]). In t
2 KB (229 words) - 08:07, 6 June 2020
A subalgebra $ A $
of bounded linear operators on a Hilbert space $ H $
17 KB (2,631 words) - 19:37, 19 January 2024
...mega )$ such that for every compact subset $K \subset \Omega$ there exists a $C = C _ { f , K} > 0$ satisfying
...d, if possible, to find the critical value(s) of $s$, i.e. those for which a change of behaviour occurs. In particular, all weak solutions to the [[Heat
21 KB (3,112 words) - 08:01, 6 February 2024
...the zero (for instance, a formula for the number of these zeros). Consider a mapping
..., \dots , z _ { n } )$, $f = ( f _ { 1 } , \dots , f _ { n } )$. Consider a function $\varphi$ holomorphic in $D$ and continuous on $\overline{ D }$.
7 KB (1,090 words) - 10:25, 16 March 2024
A [[Trigonometric sum|trigonometric sum]] of the form
637 bytes (102 words) - 15:20, 14 February 2020
$#A+1 = 101 n = 0
A [[Stochastic process|stochastic process]] that changes its state only at ra
9 KB (1,334 words) - 22:14, 5 June 2020
...satisfied), but not every Tikhonov space is normal (cf. [[Normal space]]). A.N. Tikhonov (1929) characterized Tikhonov spaces as subspaces of compact Ha
<TR><TD valign="top">[2]</TD> <TD valign="top"> A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: p
1 KB (177 words) - 12:35, 4 November 2023
$#A+1 = 117 n = 1
A [[Homogeneous space|homogeneous space]] $ G/H $
9 KB (1,280 words) - 08:10, 6 June 2020
...er, in general the poly-nilpotent length is less than the solvable length. A poly-nilpotent group of length 2 is called meta-nilpotent.
...tent classes not exceeding the numbers $c_1,\dots,c_l$, respectively, form a variety $\mathfrak M$, which is the product of nilpotent varieties:
2 KB (229 words) - 16:12, 4 October 2014
$#A+1 = 20 n = 0
A general strategy for computing linear and cyclic convolutions by applying t
3 KB (414 words) - 08:29, 6 June 2020
$#A+1 = 37 n = 1
For a positive square-free integer $ d > 0 $,
5 KB (775 words) - 07:59, 26 March 2023
...asing positive function $\epsilon(r)$, $0\leq r$ (as $r\to\infty$), having a positive lower bound on any finite interval, an entire function $g(z)$ such
...uch that any point $z$ of the complement $CE$ can be joined to $\infty$ by a Jordan curve situated outside $E$ and outside the disc $|\zeta|<\eta(|z|)$.
2 KB (243 words) - 18:27, 13 August 2015
...ecture: For each language $L$ over a finite alphabet $\Sigma$ there exists a finite subset $F$ of $L$ such that for any two morphisms $g$ and $h$ on $\S
Such an $F$ is called a test set for $L$. In general, its existence is non-effective; however, it d
7 KB (1,052 words) - 16:57, 1 July 2020
$#A+1 = 101 n = 0
fold integral over a domain $ D $
13 KB (1,868 words) - 19:42, 5 June 2020
$#A+1 = 29 n = 0
A pseudo-complemented [[Distributive lattice|distributive lattice]] $ L $(
4 KB (569 words) - 08:23, 6 June 2020
...ralization of the narrow [[Denjoy integral|Denjoy integral]] introduced by A.Ya. Khinchin in [[#References|[1]]].
A function $f$ is said to be integrable in the sense of Khinchin on $[a,b]$ if it is Denjoy-integrable in the wide sense and if its indefinite inte
1 KB (157 words) - 08:39, 23 July 2014
$#A+1 = 13 n = 0
\eta ( x,t ) = a \cos ( \zeta ) + {
7 KB (969 words) - 10:58, 29 May 2020
$\newcommand{\A}{\mathcal A}
...formally, it is a pair $(X,\A)$ consisting of a set $X$ and a σ-algebra $\A$ of subsets of $X$.
15 KB (2,605 words) - 07:10, 23 September 2012
$$h=c+ab-\frac{\partial a}{\partial u},\quad k=c+ab-\frac{\partial b}{\partial v}, $$
$$\frac{\partial^2\theta}{\partial u\partial v} = a\frac{\partial\theta}{\partial u}+b\frac{\partial\theta}{\partial v}+c\theta
2 KB (242 words) - 13:06, 12 December 2013
$#A+1 = 30 n = 3
A [[Cohomology|cohomology]] that takes the action of some group into account.
5 KB (772 words) - 11:12, 20 January 2021
A sequence $a_0,a_1,\dots,$ defined over a [[field]] $K$ that satisfies a relation
...succession, if the first $p$ terms are known. A classical example of such a sequence is the sequence of [[Fibonacci numbers]] $1,1,2,3,5,8$ defined by
2 KB (283 words) - 16:38, 30 December 2018
...ty \}$, where $<_1,\,<_2$ stand for $<$ or $\le$. For $n \ge 1$, a subset $A$ of the Cartesian product $M^n$ is called parametrically definable if there
A = \{ (a_1,\ldots,a_n) \in M^n : \phi(a_1,\ldots,a_n,b_1,\ldots,b_k)\ \text{
5 KB (799 words) - 09:27, 26 November 2016
...ysohn space; \(X\) is hereditarily separable; \(X\) is a quotient space of a metric space.
...ringer (1975)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> B.A. Efimov, "Dyadic bicompacta" ''Trans. Moscow Math. Soc.'' , '''14''' (1
3 KB (415 words) - 20:38, 2 January 2021
$#A+1 = 14 n = 0
of points $ \mathbf a = g _ {1} \mathbf e _ {1} + \dots + g _ {n} \mathbf e _ {n} $,
3 KB (421 words) - 22:15, 5 June 2020
''to a curve in space at a point $M$''
946 bytes (164 words) - 17:20, 30 July 2014
...to the central limit theorem (''CLT''), the distribution function $F_n$ of a normalized sum
...nce $npq (q=1-p)$, the normal approximation is given , for integers $0\leq a \leq b\leq n$, by
11 KB (1,719 words) - 20:18, 12 March 2016
...l{S}$ and its properties can be used as an alternative characterization of a Sasakian manifold.
...he collection of tensors $\{ \xi ^ { a } , \eta ^ { a } , \Phi ^ { a } \}_{a = 1,2,3}$ is traditionally called the $3$-Sasakian structure on $( \mathcal
17 KB (2,475 words) - 12:23, 12 December 2020
$#A+1 = 166 n = 0
A complex-valued function $ f: R \rightarrow \overline{\mathbf C}\; $
13 KB (1,964 words) - 08:11, 6 June 2020
...rmalization and far-reaching generalization of the concept of direction on a curve. The orientation of special classes of spaces is defined (cf.
...dinate systems belong to the same class if they are positively related (in a specific sense).
18 KB (2,980 words) - 17:05, 13 June 2020
$#A+1 = 43 n = 0
...uished by the fact that in such a game the players control the "motion of a point" in the state space $ X $.
4 KB (716 words) - 19:39, 16 January 2024
A [[triple system]] closely related to [[Jordan algebra]]s.
...th a $K$-[[trilinear mapping]] $V \times V \times V \rightarrow V$, called a ''triple product'' and usually denoted by $\{ \cdot , \cdot , \cdot \}$ (so
3 KB (525 words) - 10:13, 30 April 2018
...higher power unto infinity into two like powers, is impossible; and I have a marvelous proof for this. But the margin is too narrow to contain it" .)
...amuel proceeded to collect Fermat's mathematical correspondence. That, and a reprint of the Arithmetika, together with Fermat's marginal notes, was publ
8 KB (1,419 words) - 15:20, 17 March 2023
...pings a [[Brownian motion|Brownian motion]]. Then a Brownian functional is a measurable mapping defined on the [[Probability space|probability space]] $
a) they are met in a great number of applications, including filtering and mathematical finance;
7 KB (1,053 words) - 22:49, 7 March 2019
$#A+1 = 10 n = 1
...exist, and the synthetic methods cannot be made mathematically rigorous in a direct way.
5 KB (681 words) - 08:24, 6 June 2020
Let $f(q)$ be a non-negative function defined on the positive integers. Gallagher's ergodic
...mensional generalization is due to V.T. Vil'chinskii [[#References|[a5]]]. A complex version is given in [[#References|[a3]]].
2 KB (270 words) - 18:14, 11 April 2016
...ef\a{\alpha}X_\a$ with respect to a system of continuous mappings $f_\a:X_\a\to X_0$,
...\cA$ is the subset $X_\cA$ of the [[Tikhonov product]] $\prod_{\a\in\cA}X_\a$, which is considered in the [[induced topology]] and which consists of the
1 KB (238 words) - 15:27, 20 November 2014
$#A+1 = 19 n = 0
An inequality [[#References|[a3]]] that began a series of [[correlation inequalities]] for finite partially ordered sets. L
4 KB (565 words) - 19:26, 13 April 2024
...s, as well as a more precise definition of procedure bodies in the case of a description of them in other languages.
2 KB (340 words) - 17:06, 7 February 2011
A collection of methods for the summation of series of numbers and functions.
A series
3 KB (519 words) - 15:44, 7 May 2012
...ngle assumption that a polynomial of odd degree with real coefficients has a real root.
...TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.G. Bashmakova, "On a proof of the fundamental theorem of algebra" ''Istor. Mat. Issled.'' : 10
2 KB (348 words) - 05:59, 20 August 2014
...re: If $J(X)$ has constant eigenvalues, then $(M,g)$ is flat or is locally a rank-$1$ symmetric space.
...algebraic problem. Let $R(X,Y,Z,W)$ be a $4$-tensor on $R^m$ which defines a corresponding curvature operator $R(X,Y)$. If $R$ satisfies the identities,
5 KB (875 words) - 18:06, 5 February 2021
$#A+1 = 51 n = 0
be a [[Field|field]] and $ G $
5 KB (726 words) - 08:01, 6 June 2020
...than the number of unknowns. In the linear case, such
a system is given by
a rectangular <img align="absmiddle" border="0" src="https://www.encyclopedia
...0.png" /> is the symbol of differentiation, the compatibility condition is
a homogeneous system of equations with constant coefficients:
5 KB (715 words) - 08:35, 19 October 2014
...$R$ in $S$ is the set of all $s \in S$ which are integral over $R$. It is a subring $\bar R$ of $S$ containing $R$. If $\bar R = R$, then $R$ is said t
...omain]] and is closed in its field of fractions. In some of the literature a normal ring is also required to be an integral domain.
2 KB (305 words) - 16:09, 11 September 2016
...of the social and economic life of the people there arose a need to create a more modern notation for numbers than the verbal writing and to work out th
...h century B.C.; the number system was used in Attica up to the 1st century A.D., although in other Greek countries it was displaced long before by the m
3 KB (570 words) - 20:06, 11 April 2014
...if for any $\beta'<\beta$ there exists an $x'\in X$ such that $x'>\beta'$. A number $\alpha$ is said to be the greatest lower bound of $X$, denoted by $
$$\inf[a,b]=a,\quad\sup[a,b]=b;$$
4 KB (722 words) - 19:45, 7 January 2015
...rphism of $F$ onto the direct product of these groups). As an operation on a class of groups, the verbal product is associative and, within the correspo
<TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations in ter
883 bytes (142 words) - 19:33, 11 December 2015
...mple colon. If only one one-point subset in $F$ is open, $F$ is said to be a connected colon. Finally, if only the empty subset and all of $F$ in $F$ ar
...every compactum can be obtained as the continuous image of a closed set of a Cantor cube of the same weight. These facts generalize well-known results o
2 KB (246 words) - 17:35, 27 February 2018
$#A+1 = 29 n = 0
A real-valued function $ F ( x) $
3 KB (400 words) - 08:10, 6 June 2020
$#A+1 = 18 n = 0
that depends on a parameter $ t $
3 KB (392 words) - 08:13, 6 June 2020
...p of a distribution function (probability density) of a random variable in a certain set of distribution functions containing more than one element. See
504 bytes (64 words) - 16:57, 7 February 2011
$#A+1 = 191 n = 1
A problem generated by the following equation, where $ x $
22 KB (3,082 words) - 19:17, 17 January 2024
$#A+1 = 71 n = 0
''Grassmann algebra, of a vector space $ V $
7 KB (1,013 words) - 19:38, 5 June 2020
...stem written in the language of restricted predicate calculus (that is, in a first-order language). Examples of elementary axiom systems are those of fo
407 bytes (61 words) - 17:28, 7 February 2011
A formula for computing
...$ u $ and $ v $, and $ \widehat{L} $ is the closure of $ L $ in $ \widehat{A} $. Then the mapping
6 KB (1,020 words) - 17:41, 4 May 2017
...chosen in each factor $G_i$, then the subgroups $H_i$, $i\in I$, generate
a subgroup $H$ of $G$which is the same as the product of the $H_i$:
<table><TR><TD valign="top">[1]</TD> <TD valign="top">
A.G. Kurosh, "The theory of groups" , '''1''' , Chelsea (1955) (Translate
1 KB (191 words) - 00:54, 24 December 2018
$#A+1 = 18 n = 0
A trigonometric series
3 KB (394 words) - 10:58, 29 May 2020
...es a baseline to determine whether or not gene frequencies have changed in a population, and thus whether evolution has occurred.
....H. Hardy, an English mathematician [[#References|[a1]]], and W. Weinberg, a German physician [[#References|[a3]]], concluded that gene pool frequencies
6 KB (948 words) - 13:11, 4 October 2014
$#A+1 = 75 n = 0
and not containing derivatives of higher orders. For instance, a linear equation of the second order has the form
7 KB (980 words) - 17:33, 5 June 2020
$#A+1 = 34 n = 0
...onnection]] (without torsion), at each point of which the tangent space is a [[Pseudo-Euclidean space|pseudo-Euclidean space]].
6 KB (888 words) - 08:08, 6 June 2020
...ent of $K$ is the root of a non-zero polynomial with coefficients in $k$. A finite degree extension is necessarily algebraic, but the converse does not
Algebraic extensions form a ''[[Distinguished class of extensions|distinguished class]]'' : that is, th
1 KB (190 words) - 14:18, 12 November 2023
$#A+1 = 81 n = 0
A projective $ n $-
6 KB (872 words) - 08:09, 6 June 2020
...n H$ can be approximated with arbitrary accuracy (in the metric of $H$) by a finite linear combination of elements of the system; in other words: For an
...e in the $p$-th degree $(p\geq1)$ (or $p$-summable) on a finite interval $[a,b]$ with (integral) weight $\mu(x)$. The inequality \eqref{1} then reads
7 KB (1,066 words) - 15:42, 14 February 2020
''(general and special rank of a group)''
...is the minimal number such that any finitely-generated subgroup of $G$ has a system of generators of at most $r$ elements. If no such finite number exis
1 KB (182 words) - 16:24, 19 April 2014
$#A+1 = 4 n = 0
902 bytes (131 words) - 20:27, 18 March 2024
A procedure for solving an equation of the form
...ith designed distance $2 t + 1$ consist of all codeword polynomials having a common set of $2 t$ consecutive powers of $\alpha$ as zeros, the zero set o
8 KB (1,155 words) - 18:48, 26 January 2024
...e group|solvable group]] (cf. also [[Finite group|Finite group]]) provides a measure of how far the group is from being nilpotent. For any finite solvab
As one measure of the complexity of a solvable group, the Fitting height is related to many other such measures.
5 KB (855 words) - 17:03, 1 July 2020
''of a topological space $X$, base of a topology, basis of a topology, open base''
...ual concept of a closed base, formed by the complements of the elements of a base, is used in [[compactification]] theory.
6 KB (1,071 words) - 14:02, 17 March 2020
...ctral analysis]] of the [[Schrödinger equation|Schrödinger equation]] with a periodic potential. In time, this method has been very well-developed and a
...sense, and more precisely, they are families of functions which depend on a vector parameter, say $\eta \in \mathbf{R} ^ { N }$, that have the form
16 KB (2,451 words) - 17:45, 1 July 2020
...$(
a,b)$. For any real numbers $x_0$, $x_0'$, and $t_0 \in (
a,b)$, there is
a unique solution <img align="absmiddle" border="0" src="https://www.encyclop
...ww.encyclopediaofmath.org/legacyimages/l/l059/l059360/l05936017.png" /> is
a particular solution of the inhomogeneous equation (1), then the [[General s
16 KB (2,227 words) - 17:26, 28 August 2018
...the representation of arbitrary functions in $L ^ { 2 } ( \mathbf{R} )$ as a sum of time-frequency atoms, or Gabor functions (cf. also [[Gabor transform
\begin{equation*} \{ e ^ { 2 \pi i m b x } g ( x - n a ) : n , m \in \mathbf{Z} \} = \{ g _ { n , m} : n , m \in \mathbf{Z} \}, \e
8 KB (1,207 words) - 07:48, 12 February 2024
...d $\chi$ is called the characteristic mapping of the cell $e^n$. If $X$ is a [[cellular space]], then the cells of $X$ are defined as those cells of $X$
485 bytes (89 words) - 21:46, 11 November 2017
$#A+1 = 24 n = 0
is a set in the space $ \mathbf R ^ {n} $,
4 KB (565 words) - 08:28, 6 June 2020
...ountable set" can mean either "a finite or countably infinite set" or "a countably infinite set" .
715 bytes (107 words) - 12:12, 12 April 2014
...]], [[#References|[a10]]]). These coincide for non-relativistic systems of a finite number of particles with the (Dirac) canonical quantization.
...perators $\hat\phi^i(x)$ corresponding to the local components $\phi^i$ of a field $\phi$, "localized" at the point $x$ of the space-time $M$:
16 KB (2,356 words) - 19:53, 13 January 2018
...g a given curve. The trajectory has to be found of the uniform movement of a point $N$, such that the tangent drawn towards the trajectory at any moment
In a plane, the system of equations which the curve of pursuit must satisfy take
1 KB (199 words) - 11:15, 26 March 2023
$#A+1 = 5 n = 0
...]]). A simplex can be ascribed one of two orientations, which then induces a specific orientation on each of its $ ( n - 1) $-
2 KB (249 words) - 08:14, 6 June 2020
A sequence $\{x_n\}$ such that for all $n=1,2,\dots,$ either
$x_n<x_{n+1}$ (a strictly-increasing sequence);
387 bytes (55 words) - 22:33, 30 November 2018
$#A+1 = 184 n = 0
...fibres (or of their homotopy invariants) of a fibre space corresponding to a path in the base. More precisely, let $ p : E \rightarrow B $
14 KB (1,947 words) - 08:01, 6 June 2020
$#A+1 = 25 n = 0
A subset of points of a projective space $ P ^ {n} $
3 KB (364 words) - 08:09, 13 July 2022
$#A+1 = 37 n = 0
$#C+1 = 37 : ~/encyclopedia/old_files/data/P072/P.0702750 Place of a field
2 KB (326 words) - 08:06, 6 June 2020
$#A+1 = 90 n = 0
...rhood of the identity, see [[#References|[1]]]). The precise definition of a local topological group is as follows.
6 KB (894 words) - 22:17, 5 June 2020
...Slavic numerals in adjacent positions. Thousands were denoted by prefixing a certain sign to the number to express the number of thousands.
...D valign="top">[a3]</TD> <TD valign="top"> G. Ifrah, "From one to zero: a universal history of numbers" , Penguin (1987) (Translated from French)</
1,012 bytes (155 words) - 17:18, 7 February 2011
$#A+1 = 179 n = 2
...ying type (elliptic, hyperbolic or parabolic) in its domain of definition. A linear (or quasi-linear) differential equation of the second order with two
19 KB (2,773 words) - 06:19, 24 February 2022
$#A+1 = 66 n = 0
A $ \mathbf Z / 2 $-
8 KB (1,143 words) - 16:30, 31 March 2020
$#A+1 = 30 n = 0
on a manifold $ M $.
3 KB (502 words) - 17:42, 1 February 2022
''in a partially ordered set''
...lement $x\in P$ such that $a<x<b$: the [[Interval and segment|interval]] $[a,b]$ is an ''atomic'' or ''[[elementary interval]]''.
351 bytes (59 words) - 07:37, 24 January 2016
...ue?, i.e. does $\operatorname{det} JF \in \mathbf{C}^*$ imply that $F$ has a polynomial inverse?, [[#References|[a4]]]. This problem is now known as Kel
...t each $H _ { i }$ is of the form $l _ { i } ^ { 3 }$, where each $l_i$ is a linear form (L. Drużkowski). The cubic homogeneous case has been verified
7 KB (1,112 words) - 05:20, 15 February 2024
...$ is a sector of $K(x)$ other than a semi-disc; or 4) $K_G(x)$ consists of a finite number of sectors $u_i(x)$ of $K(x)$ with no common points except $x
...region is either a closed surface or a surface with boundary consisting of a finite number of pairwise non-intersecting Jordan polygons.
3 KB (509 words) - 11:10, 16 April 2023
$#A+1 = 114 n = 0
An [[Algebraic system|algebraic system]] $ \{ A; \mathbf P , +, \cdot, \cle \} $
5 KB (833 words) - 22:15, 5 June 2020
$#A+1 = 54 n = 0
under a univalent conformal mapping of the disc $ \{ \zeta : {| \zeta | < 1 } \
6 KB (783 words) - 18:43, 11 October 2023
...riable, to be contrasted with its unconditional or [[A priori distribution|a priori distribution]].
...$ when $\Theta=\theta$; then the a posteriori distribution of $\Theta$ for a given $X=x$, according to the [[Bayes formula|Bayes formula]], has the dens
2 KB (248 words) - 21:34, 1 January 2019
$#A+1 = 49 n = 0
...c equation by the decomposition of the phase density of the particles into a finite sum of [[Spherical functions|spherical functions]] in arguments that
9 KB (1,280 words) - 20:00, 16 January 2024
...(u)$ is an even [[Convex function (of
a real variable)|convex function (of
a real variable)]] which is increasing for $u$ positive, and
...M(u)=u^p/p$, $1<p<\infty$, then $N(u)=n^q/q$, where $p^{-1}+q^{-1}=1$. For
a pair of complementary functions, the Young inequality
2 KB (377 words) - 14:21, 11 August 2014
#REDIRECT [[Order (on a set)]]
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$#A+1 = 47 n = 0
...ons depends on the type of the discontinuous variational problem. Thus, in a first-order discontinuous variational problem the polygonal extremal is fou
7 KB (1,027 words) - 19:35, 5 June 2020
...register sequence|Shift register sequence]]) capable of producing $\mathbf{a}$. The linear complexity also equals the maximum number of linearly indepen
\begin{equation*} {\bf a}^ { ( t ) } = ( a _ { t } , a _ { t + 1} , \dots , a _ { n + t - 1 }) ( t \geq 0 ) \end{equation*}
4 KB (622 words) - 19:04, 17 February 2024
$#A+1 = 69 n = 0
The problem of finding a [[Minimal surface|minimal surface]] with a given boundary $ \Gamma $.
10 KB (1,540 words) - 11:46, 3 March 2022
...o [[Algebraic number|Algebraic number]]; [[Number field|Number field]]) or a completion of such an $F$ with regard to an (infinite or finite) prime.
...ups and groups with opposite multiplication shows that $S ( F )$ is indeed a subgroup of $B ( F )$. The celebrated Brauer–Witt theorem implies that th
4 KB (692 words) - 17:46, 1 July 2020
...It is therefore possible to write the [[Dirac equation|Dirac equation]] in a form that is covariant with respect to the Lorentz group of transformations
2 KB (328 words) - 16:06, 15 December 2016
$#A+1 = 85 n = 0
$#C+1 = 85 : ~/encyclopedia/old_files/data/A110/A.1100100 Abstract approximation theory
9 KB (1,258 words) - 07:37, 4 November 2023
$#A+1 = 28 n = 0
The graded algebra $ A _ {p} $
4 KB (522 words) - 08:23, 6 June 2020
$#A+1 = 88 n = 0
A [[Correlation|correlation]] $ \pi $
6 KB (909 words) - 08:06, 6 June 2020
''with respect to a fixed point $O$''
A domain $D$ in the complex space $\mathbf C^n$, $n\geq1$, such that, for any
2 KB (305 words) - 14:28, 30 July 2014
$#A+1 = 31 n = 1
be a positive real number. The Zygmund class $ Z _ {M} $
3 KB (453 words) - 04:01, 15 June 2022
...ommutes with each operator of a given set operating on these systems, i.e. a homomorphism of operator groups, operator rings, etc.
246 bytes (36 words) - 15:52, 1 May 2014
...is an extremely overloaded notion used in different contexts. Here follows a partial list of possible disambiguations.
350 bytes (40 words) - 09:15, 5 May 2012
$#A+1 = 16 n = 0
...n effective criterion for the existence of Euler cycles (Euler's theorem): A connected graph has an Euler cycle if and only if each of its vertices (exc
4 KB (632 words) - 20:10, 15 March 2023
$#A+1 = 28 n = 0
...s to the concept of a regular mapping (a submersion) of complex manifolds. A finitely-represented (local) morphism of schemes $ f: X \rightarrow Y $
3 KB (548 words) - 06:26, 21 April 2024
$#A+1 = 95 n = 0
An element of the reduced [[Whitehead group]] $ \overline{K}_{1} A $,
6 KB (926 words) - 19:41, 16 January 2024
...yclopediaofmath.org/legacyimages/l/l059/l059620/l0596208.png" />. There is a conjecture that
The most recent result, obtained by a method of I.M. Vinogradov, has the form
3 KB (384 words) - 11:53, 17 January 2020
An integral of a function with values in a Banach space with respect to a scalar-valued measure. It belongs to the family of so-called [[Strong integ
* $ X $ is a Banach space (resp. real or complex).
7 KB (1,028 words) - 21:30, 26 April 2016
...the one side stands a [[Modular function|modular function]]; on the other, a finite [[Sporadic simple group|sporadic simple group]]. Moonshine is the ex
...( \mathbf{R} )$. The group $G _ { g }$ contains $\Gamma _ { 0 } ( N )$ as a [[Normal subgroup|normal subgroup]], where $N$ divides $o ( g ) \operatorna
12 KB (1,765 words) - 09:46, 10 November 2023
...to a reconsideration of the views on logic, which he stated in the form of a ramified theory of types (cf. [[Types, theory of|Types, theory of]]). Howev
...w-Hill (1963)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.A. Fraenkel, Y. Bar-Hillel, "Foundations of set theory" , North-Holland
3 KB (445 words) - 17:13, 7 February 2011
$#A+1 = 35 n = 0
''of a morphism in a category''
3 KB (434 words) - 22:14, 5 June 2020
$#A+1 = 30 n = 0
A property of an object in a [[Category|category]] which characterizes it as a representing object for some (covariant or contravariant) set-valued [[Func
3 KB (396 words) - 16:44, 4 November 2023
$#A+1 = 16 n = 0
...in complex of operations (designs, programs, topics, etc.) on the basis of a [[Network model|network model]] of the complex; it is also known as PERT (s
5 KB (745 words) - 08:02, 6 June 2020
[[Algebraic number theory|Algebraic number theory]]) and $G$ is a connected
[[Linear algebraic group|linear algebraic group]] defined over $k$, a
4 KB (613 words) - 21:48, 5 March 2012
$#A+1 = 13 n = 0
$#C+1 = 13 : ~/encyclopedia/old_files/data/K110/K.1100090 Kernel of a matrix
2 KB (237 words) - 22:14, 5 June 2020
The approximate representation of a function $f$ in the form of a [[Trigonometric polynomial|trigonometric polynomial]]
$$T(x)=A+\sum_{k=1}^n(a_k\cos kx+b_k\sin kx)$$
2 KB (269 words) - 15:19, 14 February 2020
$#A+1 = 14 n = 0
is the ring of regular functions on a certain affine variety $ X $,
2 KB (246 words) - 19:37, 5 June 2020
$#A+1 = 29 n = 0
''of a point $ P ( t _ {0} , x _ {0} ) $
3 KB (466 words) - 22:12, 5 June 2020
$#A+1 = 34 n = 0
...somorphic entities thereof (cf. also [[Boolean algebra|Boolean algebra]]). A classical Karnaugh map of $ n $
9 KB (1,380 words) - 22:14, 5 June 2020
...the theory of continuous mappings, in the theory of transformation groups (a connection with quotient spaces), in the theory of generalized manifolds (i
4 KB (561 words) - 11:33, 22 August 2014
$#A+1 = 263 n = 0
A class of special functors from the category of pairs of spaces into the cat
23 KB (3,297 words) - 19:41, 5 June 2020
$#A+1 = 5 n = 0
...g of one of the axioms: instead of the condition $ w ( a \cdot b ) = w ( a) w ( b) $
1 KB (165 words) - 08:08, 6 June 2020
An argument $x$ of a function $f$ that satisfies the condition
...ly speaking, the converse is not true. If a one-sided derivative exists at a smooth point, an ordinary derivative exists as well.
569 bytes (102 words) - 06:04, 22 August 2014
$#A+1 = 41 n = 0
...on 2 can be expanded. They act on two-component spin functions $ \psi _ {A} $,
6 KB (785 words) - 01:11, 19 March 2022
$#A+1 = 276 n = 0
...ween one magnitude and another, of the same kind, and which is accepted as a unit" . Rigorous theories of real numbers were constructed at the end of th
26 KB (4,086 words) - 09:51, 4 April 2020
A calculating frame used for arithmetical calculations in ancient Greece, anc
In nomography an abacus is a special sketch (the so-called net chart nomogram).
674 bytes (100 words) - 17:27, 7 February 2011
measuring the ''Bayesian Probability of a Model''.
== Measuring the Evidence in Favor of a Model ==
10 KB (1,679 words) - 18:34, 29 February 2024
$#A+1 = 14 n = 0
A [[Stochastic process|stochastic process]] $ X ( t) $
2 KB (331 words) - 08:13, 6 June 2020
$#A+1 = 193 n = 0
....) can be described by boundary value problems for differential equations. A very wide class of models is reducible to such boundary value problems.
27 KB (3,927 words) - 19:13, 11 January 2024
$#A+1 = 72 n = 7
A smooth projective [[Algebraic surface|algebraic surface]] $ X $
10 KB (1,443 words) - 15:43, 1 March 2022
...] $\mathfrak C$ of a [[category]] $\mathfrak K$ such that for any objects $A$ and $B$ from $\mathfrak C$ one has the equality
$$H_\mathfrak C(A,B)=H_\mathfrak K(A,B).$$
1 KB (160 words) - 17:51, 15 November 2014
$#A+1 = 16 n = 0
...problem, partial differential equations]]). Suppose, for example, that in a bounded domain $ \Omega $
2 KB (350 words) - 19:19, 11 September 2020
...er formulation Bertrand's postulate states that for any $x>1$ there exists a prime number in the interval $(x, 2x)$. The postulate was advanced by [[Jos
806 bytes (127 words) - 15:15, 10 April 2023
$#A+1 = 3 n = 0
A [[Net (in differential geometry)|net (in differential geometry)]] on a surface $ V _ {2} $
882 bytes (119 words) - 08:28, 6 June 2020
$#A+1 = 29 n = 0
...ro-sum game|two-person zero-sum game]] in which each player possesses only a finite number of pure strategies. If player I possesses $ m $
5 KB (689 words) - 08:00, 6 June 2020
...oncept of a [[group]] to the case of an $n$-ary operation. An $n$-group is a [[universal algebra]] with one $n$-ary associative operation that is unique
Let $\Gamma(\circ)$ be a group with multiplication operation $\circ$; let $n\geq 3$ be an arbitrary
2 KB (307 words) - 21:09, 21 November 2014
$#A+1 = 61 n = 0
...ematically. Below one such a formalization will be considered, followed by a discussion of some examples.
8 KB (1,192 words) - 16:43, 4 June 2020
The area formula is a tool to compute the $n$-dimensional volume of (sufficiently regular) subset
...Rademacher theorem|Rademacher's theorem]], $f$ is differentiable $\lambda$-a.e.. At any point $y\in \mathbb R^n$ of differentiability we denote by $J f
6 KB (1,027 words) - 15:56, 12 November 2013
$#A+1 = 15 n = 0
$#C+1 = 15 : ~/encyclopedia/old_files/data/G044/G.0404000 Genus of a surface
2 KB (319 words) - 14:59, 30 March 2023
...x+3=2$. If one says "12 is divisible by 2", then the term "12" is used in a non-autonymous manner (as the designation of an integer), while if one says
...fer from the same term used without the quotation marks. Autonymous use of a term merges the meaning of the term with the term itself: it is both the ob
2 KB (318 words) - 14:25, 30 December 2018
$#A+1 = 19 n = 3
A flow on a [[Solv manifold|solv manifold]] $ M = G/H $
3 KB (490 words) - 08:14, 6 June 2020
$#A+1 = 46 n = 1
...hod|balayage method]]. O. Perron [[#References|[1]]] was the first to give a full exposition of the method of upper and lower functions for this last ca
6 KB (849 words) - 19:25, 11 January 2024
...n [[Linear-algebra(2)|linear algebra]]. Let $V$ be a [[vector space]] over a field $K$; the vectors $a_1,\ldots,a_n$ are said to be linearly independent
...lled the rank, or dimension, of the space, and the subset itself is called a [[basis]] (or base).
2 KB (325 words) - 19:52, 20 November 2014
Given a [[Polynomial|polynomial]] $P ( x _ { 1 } , \ldots , x _ { n } )$ with compl
...|geometric mean]] of $| P |$ over the $n$-torus. If $n = 1$ and $P ( x ) = a _ { 0 } \prod _ { k = 1 } ^ { d } ( x - \alpha _ { k } )$, Jensen's formula
7 KB (1,101 words) - 14:50, 27 January 2024
==Upper and lower limit of a real sequence==
The upper and lower limit of a sequence of real numbers $\{x_n\}$ (called also limes superior and limes in
8 KB (1,481 words) - 09:55, 16 August 2013
$#A+1 = 142 n = 0
...ends to become equal to its probability as the number of trials increases (a phenomenon which was probably first noted for games of chance) may serve as
18 KB (2,728 words) - 20:05, 12 January 2024
$#A+1 = 221 n = 0
A function $ f : E \rightarrow Y $
12 KB (1,760 words) - 08:00, 21 January 2024
$#A+1 = 38 n = 0
A derivative defined by Ch.J. de la Vallée-Poussin [[#References|[1]]]. Let
3 KB (405 words) - 17:32, 5 June 2020
$#A+1 = 23 n = 0
...ion channel]] for which the statistical properties of the output signal at a time $ t $
2 KB (344 words) - 08:00, 6 June 2020
$#A+1 = 18 n = 0
A generalization of the notion of an associative ring (cf. [[Associative ring
3 KB (408 words) - 08:11, 6 June 2020
...s a [[Universal space|universal space]] in the class of metric spaces with a countable base (Urysohn's metrization theorem).
...y (cf. [[Infinite-dimensional space|Infinite-dimensional space]]). This is a rich and fruitful area of investigation.
1,008 bytes (144 words) - 19:09, 12 April 2014
$#A+1 = 13 n = 0
A family of operators $ T ( t) $,
4 KB (555 words) - 08:04, 6 June 2020
...le $x$ is defined in a neighborhood of a point $x_0$ and that there exists a finite or infinite limit
...er, there are continuous functions that have no derivative at any point of a given interval (see [[Non-differentiable function]]).
4 KB (596 words) - 11:47, 5 July 2016
...tive ring $R$ is said to be non-full if it can be represented in the form $A=BC$ where $B$, $C$ are matrices of orders $n\times r$ and $r\times n$, resp
$$A=(a,a_2,\ldots,a_n),\quad B=(b,a_2,\ldots,a_n)$$
5 KB (851 words) - 14:57, 30 July 2014
$#A+1 = 127 n = 11
Let $ A _ {0} $
14 KB (2,017 words) - 09:05, 26 March 2023
$#A+1 = 32 n = 0
...e, of realizing a controlled dynamical process. This dynamical process, as a rule, can be described using differential, integral, functional, and finite
21 KB (3,161 words) - 08:04, 6 June 2020
$#A+1 = 18 n = 0
...vative) derivative of a functional or a mapping. The Fréchet derivative of a mapping $ f: X \rightarrow Y $
2 KB (279 words) - 14:29, 15 April 2023
...Halin graphs. Indeed, the construction just described yields an example of a class of edge-minimal planar $3$-connected graphs. The graph shown below is
...attribute is relevant since it is known that Halin graphs are contained in a so-called $3$-terminal recursive class. As shown in [[#References|[a3]]], t
5 KB (722 words) - 20:45, 16 March 2023
...hen a stopping time (relative to this family of subalgebras) is a mapping (a [[random variable]]) $\tau : \Omega \rightarrow T \cup \{\infty\}$ such tha
...stance, the first time of entry of a stochastic process $X(t)$ into a set $A$ (hitting time). In the (translated) Russian literature the phrase [[Markov
2 KB (318 words) - 19:06, 11 October 2017
...of the circle in the Möbius strip. Thus, the cycle $\alpha$ (see Fig.) is a one-sided curve, while the cycle $\beta$ is two-sided (in general, any [[De
...hbf RP^2$ the projective plane, $\mathbf RP^3$ projective space, and $S^1$ a desorienting path on $\mathbf RP^2$).
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''in a group $G$''
...$G$ is usually denoted by $\left|G:H\right|$. The product of the order of a subgroup $H$ by its index $\left|G:H\right|$ is equal to the order of $G$ (
1 KB (186 words) - 11:18, 20 April 2012
A finite $2$-group defined by generators $x,y$ and defining relations
...he latter for any value of $m$. A quasi-dihedral group is sometimes called a semi-dihedral group.
663 bytes (104 words) - 09:07, 19 October 2014
'''Summary.''' Markov, with Liapunov a disciple of [[Chebyshev, Pafnutii Lvovich|Chebyshev]], gave rigorous proofs
revolution, and is now again - called St. Petersburg). He was a poor student
13 KB (1,874 words) - 18:35, 8 March 2024
...roup is zero, which is convenient in any kind of definition of acyclicity. A [[Proper cycle|proper cycle]] $z^0=\{z_1^0,\dots,z_k^0,\dots\}$ is called n
537 bytes (95 words) - 10:42, 31 August 2014
$#A+1 = 45 n = 0
A numerical characteristic of random variables related to the concept of a [[Moment|moment]] of higher order. If $ \xi = ( \xi _ {1} \dots \xi _ {k}
5 KB (661 words) - 12:22, 17 January 2021
$#A+1 = 42 n = 0
''of a real stochastic process $ \{ {X ( t) } : {t \in T } \} $''
6 KB (906 words) - 17:31, 5 June 2020
''of a module $M$ over a ring $R$''
...ly-reducible module]]). If $\phi$ is a homomorphism from a module $A$ into a module $B$, then the set
2 KB (361 words) - 19:20, 2 October 2016