$#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
$#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
...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
$#A+1 = 34 n = 0
...cyclopedia/old_files/data/P075/P.0705030 Product of a family of objects in a category
3 KB (545 words) - 08:07, 6 June 2020
...{ r }$ 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]].
638 bytes (101 words) - 12:04, 27 October 2014
...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
...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 $u \in U$.
769 bytes (149 words) - 18:13, 15 November 2014
''$M_0=(x_0,y_0)$ 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)
805 bytes (131 words) - 18:26, 5 December 2014
''$G$ 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
$#C+1 = 29 : ~/encyclopedia/old_files/data/R077/R.0707470 Rank of a module
3 KB (439 words) - 08:09, 6 June 2020
...icient|asymmetry coefficient]] is positive (negative). If the asymmetry of a distribution is positive (negative), then the "longer" part of the densit
In Western literature one commonly calls this notion the skewness of a distribution, and accordingly one speaks of the skewness coefficient.
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''$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
$#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]]).
797 bytes (132 words) - 20:04, 29 October 2016
$#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
918 bytes (149 words) - 20:46, 18 October 2014
...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
829 bytes (132 words) - 05:43, 9 April 2023
$#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
$#A+1 = 187 n = 0
...ies are given, a converse group-theoretical construction is presented, and a few small examples are given.
11 KB (1,611 words) - 08:03, 6 June 2020
...cup B ) - m ( B )$ depends on $B$ and can be interpreted as the effect of $A$ joining $B$.
...zy measure approach) the measure that a given unlocated element belongs to a set.
7 KB (1,001 words) - 16:57, 1 July 2020
varieties are supplied by algebraic tori; for example, a three-dimensional torus $ T = R _{L/K} ^{(1)} ( G _{m} ) $,
corresponding to the biquadratic extension $ L = K ( \sqrt a ,\ \sqrt b ) $
8 KB (1,072 words) - 20:22, 21 December 2019
...$L(s,\chi)$ be a [[Dirichlet L-function]], $s = \sigma + i t$, with $\chi$ a [[Dirichlet character]] modulo $d$, $d \ge 3$. There are absolute positive
a) $L(s,\chi) \ne 0$ for $\sigma > 1 - c_1/\log(dt)$, $t \ge 3$;
3 KB (508 words) - 15:13, 3 July 2020
''over a ring $A$ in commuting variables $T_1,\ldots,T_N$''
where $F_k$ is a form of degree $k$ in $T_1,\ldots,T_N$ with coefficients in $A$. The minimal value of $k$ for which $F_k \ne 0$ is called the order of the
6 KB (1,093 words) - 08:26, 16 March 2023
...s/t/t093/t093470/t09347013.png" /> is a semi-group of unitary operators on a Hilbert space.
...g" />. Cf. also [[Irreducible representation|Irreducible representation]]. A totally-irreducible set is also called completely irreducible.
8 KB (1,059 words) - 17:18, 7 February 2011
...ase of an [[Operator ring|operator ring]], when the domain of operators is a commutative associative ring with unit.
358 bytes (56 words) - 17:03, 7 February 2011
'''Summary.''' A major developer of the foundations of statistics and an important
with a B.A. in Journalism in 1919,
13 KB (1,905 words) - 13:07, 18 March 2023
$#A+1 = 132 n = 0
of a random variable $ Y $,
10 KB (1,524 words) - 08:10, 6 June 2020
A [[Locally convex space|locally convex]] [[topological vector space]] displa
...if for each $x \in E$ there exists an $\alpha > 0$ such that $\alpha x \in A$.
3 KB (507 words) - 13:44, 17 March 2023
$#A+1 = 4 n = 0
A square [[matrix]] with non-zero [[determinant]]. For a square matrix $A$
967 bytes (138 words) - 06:01, 23 April 2023
The same as an eigen value or characteristic value of a matrix (see [[Characteristic polynomial]]).
...e tangent bundle (or another interesting bundle) of a (closed) manifold on a fundamental cycle of this manifold.
471 bytes (67 words) - 06:02, 18 April 2023
$#A+1 = 29 n = 0
is a decreasing function of the population size $ N ( t ) $.
5 KB (698 words) - 08:05, 6 June 2020
$#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
...nd translative (see [[Translativity of a summation method|Translativity of a summation method]]).
1 KB (200 words) - 14:17, 30 December 2018
...rom an automorphism $S$ of a [[Measure space|measure space]] $(X,\nu)$ and a function $f$ (defined on $X$ and taking positive integral values)''
...if $A\subset X$ and $f(x)>n$ for all $x\in A$, then $\mu(A\times\{n\})=\nu(A)$. If $\mu(M)=\int_Xfd\nu<\infty$, then one usually normalizes this measure
2 KB (399 words) - 14:54, 19 August 2014
...of incommensurate segments. They knew, e.g., that the diagonal and side of a square are incommensurate, which is equivalent to $\sqrt2$ being irrational
...cond category and has type $G_\delta$ (cf. [[Category of a set|Category of a set]]; [[Set of type F sigma(G delta)|Set of type $F_\sigma$ ($G_\delta$)]]
2 KB (331 words) - 10:10, 13 April 2014
$#A+1 = 79 n = 0
...e of curves (see [[#References|[1]]], [[#References|[2]]]) and states that a curve is defined up to an isomorphism by its periods. Let $ X $
6 KB (967 words) - 08:26, 6 June 2020
$#A+1 = 16 n = 3
By the Pontryagin–Thom theorem, there is a ring spectrum $ MU $(
4 KB (633 words) - 08:47, 26 March 2023
...ule $\mathrm{Hom}_R(U,A)$. Among the properties preserved by transition to a Morita-equivalent ring are the properties of being: Artinian, Noetherian, p
...n the rings $R$ and $S$. In particular, for $R=S$ this implies that $R$ is a [[quasi-Frobenius ring]].
3 KB (499 words) - 20:17, 26 November 2016
...differentiable function $f: \mathbb R^m \to \mathbb R^n$ with $n\leq m$''. A value $c\in \mathbb R^n$ such that the
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$#A+1 = 38 n = 0
A metric in the space $ \mathfrak M $
5 KB (670 words) - 04:11, 6 June 2020
$#A+1 = 55 n = 0
...by Yu.V. Sokhotskii [[#References|[1]]], expressing the boundary values of a Cauchy-type integral. With more complete proofs but significantly later, th
9 KB (1,249 words) - 14:55, 7 June 2020
...be deduced from other formal expressions $B_1,\dots,B_n$. If $n=0$, then $A$ is called an axiom. Derivations are either sequences or tree diagrams made
...concept of a derivation must satisfy the same requirement. The concept of a deducible expression in effective formal systems is, generally speaking, no
3 KB (392 words) - 12:21, 19 August 2014
$#A+1 = 144 n = 0
...l methods. They find extensive use in solving problems of the mechanics of a continuous medium, in particular for the equations of gas dynamics (cf. [[G
24 KB (3,317 words) - 22:11, 5 June 2020
...onents of the invariant (covariant) differential of a tensor (vector). For a tensor field $T$ of type $( 1,1 )$, the invariant differential $DT$ and the
...ma$ and $P$ determine $\square ^ { \prime } \Gamma$. T. Otsuki calls these a general connection. For $P ^ { i } _ { r } = \delta ^ { i }_r$ one obtains
5 KB (772 words) - 17:43, 1 July 2020
$#A+1 = 23 n = 0
have limits from the left and are adapted to a (given) non-decreasing family $ \mathbf F = ( F _ {t} ) _ {t \geq 0 } $
2 KB (284 words) - 08:04, 6 June 2020
$#A+1 = 19 n = 0
...ft dynamical system|shift dynamical system]]. An equivalent definition is: A function $ \phi : \mathbf R \rightarrow S $,
3 KB (414 words) - 08:10, 6 June 2020
$#A+1 = 30 n = 0
defined on the (closed) [[Linear closure|linear span]] of a basis $ \{ e _ {k} \} _ {k \geq 1 } $
3 KB (457 words) - 19:55, 27 February 2021
A ''linear algebraic group'' is
subgroup of a
14 KB (2,165 words) - 22:05, 5 March 2012
$#A+1 = 127 n = 0
A process for obtaining a sequence of interpolation functions $ \{ f _ {n} ( z) \} $
12 KB (1,759 words) - 19:07, 11 January 2024
$#A+1 = 54 n = 0
The usual equation to describe the conversion of a substrate in an enzymatic reaction. Let $ S ( t ) $
4 KB (605 words) - 08:00, 6 June 2020
The problem of the compatibility of a system of [[Diophantine equations]] of [[Waring problem|Waring type]]:
...[#References|[3]]] are imposed on the numbers $N_n,\ldots,N_1$, and $s$ is a sufficiently-large number which depends only on the natural number $4n$ whi
2 KB (254 words) - 19:09, 10 October 2017
$#A+1 = 47 n = 0
An invariant of framed [[Surgery|surgery]] of a closed 6- or 14-dimensional framed manifold.
3 KB (435 words) - 17:53, 20 January 2022
''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
...maximal one, and it is called the Freudenthal compactification (sometimes a Freudenthal compactification just means any compactification with zero-dime
2 KB (247 words) - 17:10, 7 February 2011
Enlightenment - a
a universal language useful to science and commerce alike and though his
15 KB (2,346 words) - 20:38, 22 September 2016
$#A+1 = 291 n = 2
...involving the representation of solutions by way of analytic functions of a complex variable.
30 KB (4,345 words) - 09:02, 21 January 2024
$#A+1 = 56 n = 0
be integrable functions in a domain $ X \subset \mathbf R ^ {n} $
5 KB (782 words) - 19:43, 16 January 2024
...or $AB$. The expressions $A$ and $B$ are called the conjunctive terms of $A\&B$. The normal usage of the conjunction in mathematical logic corresponds
|$A$||$B$||$A\&B$
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$#A+1 = 6 n = 0
A method to construct an approximating equation for approximate (and numerica
3 KB (401 words) - 17:32, 5 June 2020
$#A+1 = 31 n = 0
and of a rank lower than $ t $,
4 KB (563 words) - 17:28, 26 December 2020
...equation of thermal conductance and the equation of elastic equilibrium in a gravity field.
...at the moment of time $t<t_0$ from the given temperature distribution over a depth at the moment of time $t=t_0$. The solution of the thermal conductanc
4 KB (563 words) - 07:56, 15 July 2014
1960s. He founded a school of applied mathematics in Berlin and wrote the
reputation. His father, Arthur von Mises, was a prominent railroad engineer
17 KB (2,521 words) - 14:28, 15 August 2023
$#A+1 = 117 n = 1
A [[Homogeneous space|homogeneous space]] $ G/H $
9 KB (1,280 words) - 08:10, 6 June 2020
...ely few cases for which this information is explicitly known (as of 1998). A major impact of the work [[#References|[a1]]] of J. Lannes on unstable modu
...ommutes with the action of $\mathcal{A} _ { p }$. $\mathcal{A} _ { p }$ is a connected graded [[Hopf algebra|Hopf algebra]] acting on the graded algebra
8 KB (1,265 words) - 06:16, 15 February 2024
$#A+1 = 32 n = 0
A function $ f $
4 KB (569 words) - 08:09, 6 June 2020
''simple function (of a complex variable)''
Other, and older, names for a [[Univalent function|univalent function]] of a complex variable.
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...the "spins" $S _ { i } = \pm 1$ on the $N$ sites of a regular lattice in a space of dimension $d$)
Here, $J _ {i j }$ are "exchange constants" , $H$ is a (normalized) magnetic field, involving an interpretation of the model to de
16 KB (2,407 words) - 19:47, 17 February 2024
...permit a [[Projective deformation|projective deformation]]. Introduced by A. Demoulin .
...7) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad
2 KB (302 words) - 10:12, 12 April 2014
$#A+1 = 50 n = 0
...theorem was obtained by B. Mandelbrot in 1972 [[#References|[a2]]]. It is a key fact for understanding the link between random coverings and [[Potentia
5 KB (667 words) - 19:39, 5 June 2020
''of a curve''
A system of equations
2 KB (304 words) - 19:04, 26 January 2024
$#A+1 = 40 n = 0
A notion of solutions of fully non-linear second-order partial differential e
9 KB (1,306 words) - 08:28, 6 June 2020
A graded module $ A $
...cture of an associative graded algebra $ \mu : \ A \otimes A \rightarrow A $
11 KB (1,576 words) - 08:51, 16 December 2019
A synonym for [[Bit|bit]].
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$#A+1 = 46 n = 0
...en used: control through an alternative characteristic and control through a quantitative characteristic. In the former, items are classified as accepta
9 KB (1,409 words) - 08:23, 6 June 2020
...in which order is significant; $(a,b)$ is not the same as $(b,a)$ unless $a=b$. Equality between ordered pairs is defined by
(a,b) = (c,d) \ \Leftrightarrow \ a=c \wedge b=d \ .
662 bytes (112 words) - 11:56, 23 November 2023
A mapping $f$ of a
...open in $X$. This topology is the unique topology on $Y$ such that $f$ is a quotient mapping. Therefore $\mathcal{T}_f$ is called the quotient topology
7 KB (1,119 words) - 10:25, 1 January 2018
A method of solving problems in the theory of univalent functions by means of
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...s a substructure of $B$ (cf. [[Structure(2)|Structure]]), it follows that $A$ is [[existentially closed]] in $B$.
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$#A+1 = 27 n = 0
of outcomes such that a [[Domination|domination]] $ s \succ _ {K} c $
4 KB (665 words) - 01:22, 8 May 2022
$#A+1 = 46 n = 0
is a [[Riemannian metric|Riemannian metric]] on $ M $
4 KB (538 words) - 06:29, 30 May 2020
...th $ B $; and the fibre space over a point, where $ X $ is identified with a (unique) space $ F $.
A '''section''' of a fibration (fibre space) is a continuous mapping $ s: B \to X $ such that $ \pi \circ s = \operatorname{i
5 KB (754 words) - 01:34, 10 December 2016
...ximation of a given function $f(t)$, integrable on an interval $[a,b]$, by a function $\phi(t)$, where the quantity
...nd if $\sigma(t)$ is a step function with jumps $c_k$ at points $t_k$ in $[a,b]$, one has weighted mean-power approximation with respect to the system o
2 KB (331 words) - 08:01, 23 August 2014
$#A+1 = 69 n = 2
A particular case of the surjective application $ \mathbf R ^ {4} \rightar
8 KB (1,111 words) - 22:15, 5 June 2020
...ond category in itself (cf. also [[Category of a set|Category of a set]]). A space $X$ is Baire if and only if the intersection of each countable family
...)$ is a Baire space, and this is the case if and only if $( X , \tau )$ is a Baire space.
4 KB (610 words) - 08:48, 18 February 2024
$#A+1 = 58 n = 0
$#C+1 = 58 : ~/encyclopedia/old_files/data/A110/A.1100170 Acceleration wave
5 KB (817 words) - 16:08, 1 April 2020
is a closed subgroup of $ G $ ,
is a reductive group. This criterion was first found by Y. Matsushima [[#Referen
2 KB (309 words) - 10:25, 16 December 2019
The same as a [[Stochastic process|stochastic process]].
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...dule]]). All left $R$-modules are semi-chain modules if and only if $R$ is a generalized [[Uniserial ring|uniserial ring]].
...a$-uniserial are used for a direct sum of uniserial modules (respectively, a finite direct sum of uniserial modules).
751 bytes (111 words) - 16:40, 1 May 2014
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
$#A+1 = 86 n = 0
A [[Lattice|lattice]] $ \mathbf L = ( L , \leq ) $
10 KB (1,402 words) - 08:08, 6 June 2020
$#A+1 = 20 n = 1
A problem that arises when studying physical processes related to phase trans
12 KB (1,620 words) - 08:23, 6 June 2020
...ely often. It is sufficient for a one-sided infinite word (an element of $A^{\mathbf{N}}$) to be recurrent that every prefix occurs at least once again
A word is uniformly recurrent if for every factor $f$ there is an $N = N(f)$
889 bytes (131 words) - 16:11, 15 September 2023
...real number for each set of values of the parameters ($x$ in this term is a bound variable). Syntactically, terms are characterized by the fact that th
In a [[Formalized language|formalized language]] there exist formal rules, indep
1 KB (225 words) - 22:30, 2 November 2014
...ed function $f$ of $N$ variables. That is, to find a local minimizer, i.e. a point $x ^ { * }$ such that
...bed in terms of the transition from a current approximation $x _ { c }$ to a local minimizer $x ^ { * }$, to an update (and hopefully better) approximat
13 KB (1,868 words) - 07:21, 13 February 2024
...w.encyclopediaofmath.org/legacyimages/b/b015/b015170/b0151703.png" /> into
a computable sequence of elements of <img align="absmiddle" border="0" src="h
...opediaofmath.org/legacyimages/b/b015/b015170/b0151708.png" /> there exists
a general recursive function <img align="absmiddle" border="0" src="https://w
7 KB (942 words) - 18:50, 24 March 2012
Let $M$ be a smooth manifold (cf. also [[Differentiable manifold|Differentiable manifold
...\Omega ^ { l } ( M ; T M ) \rightarrow \Omega ^ { k + l } ( M ; T M )$ is a $\bf Z$-graded [[Lie bracket|Lie bracket]]:
14 KB (2,077 words) - 12:23, 12 December 2020
..., since Zak was indeed the first to systematically study that transform in a more general setting and recognize its usefulness.
The Zak transform $Z _ { a } ( f )$ of a function $f$ is defined by
10 KB (1,621 words) - 14:48, 3 February 2024
$#A+1 = 19 n = 0
A doubly-connected surface $ S $
4 KB (575 words) - 08:26, 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
$#A+1 = 21 n = 0
reduces to a Meijer transform.
3 KB (490 words) - 13:26, 13 January 2024
$#A+1 = 20 n = 0
Consider a regular [[Martingale|martingale]] $ ( X _ {n} , {\mathcal F} _ {n} ) $,
3 KB (442 words) - 06:29, 30 May 2020
$#A+1 = 29 n = 0
...ion|Multiplicative arithmetic function]]), that guarantee the existence of a mean value
6 KB (858 words) - 19:43, 5 June 2020
$#A+1 = 18 n = 0
...ication by an arbitrary function (cf. [[Tensor on a vector space|Tensor on a vector space]]).
3 KB (354 words) - 17:42, 3 January 2021
$#A+1 = 99 n = 4
...is the inductive limit of a directed set of Banach spaces and its dual is a space of type FS. Spaces of types FS and DFS are complete, separable, refle
12 KB (1,594 words) - 17:37, 1 February 2022
$#A+1 = 171 n = 0
A quiver $ Q = ( Q _ {0} , Q _ {1} , s, e) $
12 KB (1,755 words) - 19:50, 21 March 2023
$#A+1 = 40 n = 0
A smooth [[Flow (continuous-time dynamical system)|flow (continuous-time dyna
10 KB (1,474 words) - 20:08, 12 January 2024
...volution of a [[catenary]] $y=c\cosh x/c$ about the $x$-axis, where $c$ is a positive constant. The parametric equations for the catenoid are then
...for the special choice of parameters $a = b$ the corresponding surface is a minimal surface and usually the word catenoid is used only for this particu
2 KB (303 words) - 20:07, 12 November 2023
$#A+1 = 68 n = 0
7 KB (987 words) - 09:14, 6 January 2024
...-dense sets in a given [[Complete metric space|complete metric space]] has a non-empty, and in fact everywhere-dense, intersection
...the following: A non-empty complete metric space cannot be represented as a countable union of nowhere-dense subsets (i.e.
3 KB (517 words) - 17:25, 31 December 2016
$#A+1 = 115 n = 0
$#C+1 = 115 : ~/encyclopedia/old_files/data/A011/A.0101690 Algebraic systems, variety of
10 KB (1,438 words) - 16:10, 1 April 2020
.... A ''Z-number'' is a real number $x$ such that the [[Fractional part of a number|fractional part]]s
More generally, for a real number $\alpha$, define $\Omega(\alpha)$ as
1 KB (144 words) - 13:38, 25 November 2023
1) There is a sound and complete proof calculus (completeness theorem). The decidability
...er logical axioms for [[Set theory|set theory]] (e.g., ZFC) that serves as a basis for mathematics.
4 KB (514 words) - 16:59, 1 July 2020
If a [[Power series|power series]]
Moreover, if $z \notin 1 / 3 \cdot D ^ { \circ }$, then there exists a series of the form (a3) converging in $D ^ { \circ }$ and such that the est
7 KB (1,065 words) - 09:52, 11 November 2023
$#A+1 = 39 n = 0
A graph with an orientation assigned to each of its edges. An oriented graph
5 KB (790 words) - 08:14, 26 November 2023
...ems|Classical combinatorial problems]]). Each city $C_i$ is represented by a point $( x _ { i 1 } , \ldots , x _ { i r } )$ in $r$-dimensional space, an
...ed only to decide whether a tour exists whose total length does not exceed a given number rather than to find an optimal tour [[#References|[a2]]]. Ther
4 KB (704 words) - 12:30, 17 February 2021
...ct to time $t$ and phase vector $x$) can be reduced to the minimization of a function of the type
...d $\Delta x$. To each set of vectors $\{x_0,\dots,x_N\}$ there corresponds a polygonal line passing through the nodes, which is an approximate represent
3 KB (447 words) - 10:39, 31 August 2014
...ity) with respect to a [[binary operation]] $*$ defined on $M$ if for any $a \in M$ the following equation holds:
e * a = a \ \ \ (\ a * e = a\ ) \ .
2 KB (403 words) - 19:25, 3 April 2016
$#A+1 = 80 n = 0
A [[Group|group]] $ G $
5 KB (738 words) - 08:45, 8 October 2023