$#A+1 = 302 n = 0
$#C+1 = 302 : ~/encyclopedia/old_files/data/A110/A.1100010 A priori and a posteriori bounds in matrix computations
30 KB (4,205 words) - 18:47, 5 April 2020
...lpha = 0$ 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 = 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
...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
''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
...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
''operation $ {\mathcal A} $''
A set-theoretical operation, discovered by P.S. Aleksandrov [[#References|[1]
3 KB (432 words) - 16:59, 1 July 2020
$#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
...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]].
328 bytes (49 words) - 17:22, 7 February 2011
...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
...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
...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
...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
...[Geodesic line|Geodesic line]]) of $M^n$ are also geodesic lines in $V^N$. A totally-geodesic submanifold is characterized by the fact that for every no
The existence of totally-geodesic submanifolds in a general Riemannian manifold is exceptional. Conversely, the existence of ma
1 KB (161 words) - 16:07, 10 July 2014
...T}_{<}$ on a [[linearly ordered set]] $X$ with linear order $<$, which has a [[base]] consisting of all possible [[Interval, open|open interval]]s of $X
Here "open interval" means a set of the form
2 KB (364 words) - 16:45, 23 November 2023
...to individual domains (faces). Such a subdivision of the plane is known as a planar map. Euler's formula
...aph]]) which has three vertices in each of its parts, are not planar (Fig. a).
5 KB (774 words) - 16:51, 16 March 2023
...ic to some subgroup of the group of real numbers, or else is isomorphic to a unique subring of the field of real numbers, taken with the usual order. An
712 bytes (113 words) - 17:56, 11 April 2014
...e number of different partitions of $n$, so that $p(4) = 5$. L. Euler gave a non-trivial recurrence relation for $p(n)$ (see [[#References|[a1]]]) and R
2 KB (231 words) - 15:33, 11 November 2023
...hich one is required to inscribe a triangle of least possible perimeter in a given acute-angled triangle. The solution is the [[orthocentric triangle]],
513 bytes (74 words) - 16:32, 14 August 2023
...mal processes. It shows the intensity of energy dissipation in the flow of a liquid or gas:
791 bytes (126 words) - 17:46, 14 November 2023
...ensity of a point mass $m$ located at a point $a$ is written as $m\delta(x-a)$. The delta-function may be formally defined by the relation
\[\int\limits_{\mathbf R^n}\delta(x-a)f(x)\,dx=f(a)\]
2 KB (390 words) - 22:12, 31 December 2018
''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
$#A+1 = 31 n = 0
A subset of a complete separable metric space, defined by induction as follows. The Luzin
3 KB (535 words) - 04:11, 6 June 2020
...se the concept of a proof, so that these latter could become the object of a mathematical theory — [[Proof theory|proof theory]].
...ents from those already proved. Thus, a mathematical theory is turned into a [[Formal system|formal system]].
3 KB (387 words) - 17:16, 7 February 2011
$#A+1 = 74 n = 1
be a [[Meromorphic function|meromorphic function]] on a disc
6 KB (966 words) - 08:02, 6 June 2020
$#A+1 = 133 n = 0
...tive rings and algebras]]). The latter starts out from a given algebra $ A $
12 KB (1,483 words) - 16:43, 4 June 2020
...with real coefficients and let $\{ \mu _ { i } \} _ { i = 0 } ^ { N }$ be a set of positive Borel measures supported in the real line (cf. also [[Borel
One says that (a1) is a Sobolev inner product in $\mathcal{P}$.
10 KB (1,461 words) - 08:27, 16 March 2024
$#A+1 = 28 n = 0
A semi-linear elliptic system of two fourth-order partial differential equati
8 KB (1,209 words) - 08:28, 6 June 2020
#REDIRECT[[Category of a set]]
30 bytes (5 words) - 10:07, 17 August 2012
A binary [[Algebraic operation|algebraic operation]].
...S$ on $A \times C$ defined by $a T c \Leftrightarrow \exists b \in B \,:\, a R b, b S c$.
810 bytes (132 words) - 14:29, 3 September 2017
...gma$ of the tangent space $T_pM$ are the same. The [[curvature tensor]] of a space of constant curvature is expressed in terms of the curvature $k$ and
A space of a constant curvature is a locally symmetric space.
4 KB (626 words) - 05:24, 29 May 2016
which provides a relationship between the Catalan constant $G$ and the Digamma function $\ps
and the [[Hurwitz zeta function]] $\zeta ( s , a )$, which is defined, when $\operatorname { Re } s > 1$, by
4 KB (577 words) - 14:03, 11 November 2023
$#A+1 = 35 n = 0
...m-mechanical description of a physical system is achieved by assigning: i) a [[Hilbert space|Hilbert space]] $ {\mathcal H} $,
8 KB (1,210 words) - 08:23, 6 June 2020