''minimum polynomial of a matrix''
...] of $A$ and, more generally, it divides every polynomial $f$ such that $f(A)=0$.
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''$A$ 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.
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$#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
The collection of points at which $K$ is a [[Smooth continuum|smooth continuum]].
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$#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
...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
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...ive to $0$. A chain $C$ in $P$ is maximal if no proper superset of $C$ is a chain. If all maximal chains have the same cardinality, then $P$ is said t
If the rank of each element of the poset $(P,\le)$ is finite, and there is a finite number $p_n$ of elements of rank $n$ (in particular, if $P$ is finit
1 KB (210 words) - 19:08, 7 December 2023
A function $f$, defined on some interval, satisfying the condition
...some interval lying inside $(a, b)$, it is continuous on $(a, b)$. Thus, a discontinuous convex function is unbounded on any interior interval and is
9 KB (1,555 words) - 08:20, 25 July 2013
$#A+1 = 41 n = 0
$#C+1 = 41 : ~/encyclopedia/old_files/data/M063/M.0603250 Measure in a topological vector space
7 KB (1,017 words) - 08:00, 6 June 2020
...itself is an interior point with a unique pre-image. In a neighbourhood of a branching point $(u=0,v=0)$ the coordinates $(x,y,z)$ of the minimal surfac
where $a=c(1+i)\neq0$ and $b\neq0$ are two complex constants, $\operatorname{Im}c=0$
3 KB (495 words) - 13:00, 3 October 2014
$#A+1 = 34 n = 0
...34 : ~/encyclopedia/old_files/data/D030/D.0300410 Decimal approximation of a real number
3 KB (445 words) - 17:32, 5 June 2020
A projective system of topological spaces that is associated with a [[Topologized category|topologized category]] and which makes it possible t
provided with a [[Grothendieck topology|Grothendieck topology]] $ \tau $
6 KB (842 words) - 13:07, 15 December 2019
Let $C$ be a smooth [[Jordan curve|Jordan curve]] (of class $C ^ { 3 }$) in the complex
...a2]]]. In some sense, $\lambda$ is a measure for the deviation of $C$ from a circle. $\lambda$ is invariant under Möbius transformations (cf. also [[Fr
7 KB (989 words) - 20:48, 23 January 2024
...ice|Banach lattice]] (cf. also [[Riesz space|Riesz space]]) of measures on a [[Measurable space|measurable space]] $ ( \Omega, {\mathcal F} ) $,
is a set of probability measures on $ ( \Omega, {\mathcal F} ) $.
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-
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...f a [[compact group]] into the group of continuous linear automorphisms of a (complex) Banach space that is continuous with respect to the strong operat
be a compact group, let $ V $
5 KB (636 words) - 06:35, 19 May 2024
''$V$ of a representation $T$ of a Lie group $G$''
A vector $\xi \in V$ for which the mapping
3 KB (401 words) - 18:39, 17 October 2016
$#A+1 = 24 n = 0
...: ~/encyclopedia/old_files/data/W097/W.0907400 Weierstrass conditions (for a variational extremum)
3 KB (508 words) - 08:28, 6 June 2020
...armonic conformal mapping $X : M \rightarrow {\bf R} ^ { n }$ then defines a [[Minimal surface|minimal surface]] in ${\bf R} ^ { n }$, $n \geq 3$ (cf. a
is a holomorphic $1$-form on $M$. Hence any (branched) minimal surface in ${\bf
4 KB (595 words) - 16:57, 1 July 2020
...mes n } \rightarrow \mathbf{C}$ acts on the $( n \times n )$-matrix $A = [ a _ {i j } ]$ by
...A ) : = \sum _ { \sigma \in G } \chi ( \sigma ) \prod _ { i = 1 } ^ { n } a _ {i \sigma ( i ) }. \end{equation*}
19 KB (2,837 words) - 05:34, 15 February 2024
...$f, g$ are functions taking values in $X$ then the pointwise extension of a [[binary operation]] $\star$ is
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$#A+1 = 20 n = 0
...erences|[a1]]], [[#References|[a2]]], [[#References|[a3]]] if there exists a [[Homeomorphism|homeomorphism]] $ h : {\mathbf R ^ {n} } \rightarrow {\m
3 KB (414 words) - 19:37, 5 June 2020
$#A+1 = 16 n = 0
A smooth morphism of algebraic varieties or schemes of relative dimension $
3 KB (413 words) - 19:38, 5 June 2020
$#A+1 = 37 n = 1
...the latter) has a quasi-discrete spectrum (or is a system, flow, etc. with a quasi-discrete spectrum)" .
7 KB (978 words) - 05:56, 19 March 2022
\langle A,B \ |\ A^5=B^3=(AB)^2 \rangle \ .
It is finite of order 120. It occurs as a subgroup of the [[unit quaternion]]s.
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...: | z | = 1 \}$ and $\mu$ is a positive [[Measure|measure]] on $\bf T$, by a [[Quadrature formula|quadrature formula]] of the form
...space of dimension $2 n - 1$, which is the maximal dimension possible with a quadrature formula of this form.
3 KB (454 words) - 16:59, 1 July 2020
...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
$#A+1 = 69 n = 0
...concentric slits); it usually arises in the following manner. One chooses a one-parameter family of domains $ Q _ {t} $,
8 KB (1,168 words) - 20:13, 12 January 2024
$#A+1 = 127 n = 0
''with values in a sheaf of Abelian groups''
10 KB (1,477 words) - 22:17, 5 June 2020
$#A+1 = 91 n = 1
A group with a distinguished system of generators $ \{ {r _ {i} } : {i \in I } \} $
15 KB (2,070 words) - 14:45, 1 November 2023
$#A+1 = 209 n = 0
...ciding between two hypotheses (determined by the Neyman–Pearson lemma) for a fixed sample size and the same error probabilities.
18 KB (2,425 words) - 14:55, 7 June 2020
...the sphere $S^n$ into the Euclidean space $E^n$ there exist antipodes with a common image; 2) Any mapping of the sphere $S^n$ into itself in which the i
1 KB (163 words) - 06:23, 28 October 2017
$#A+1 = 21 n = 0
is a [[Function of bounded variation|function of bounded variation]] in every fi
3 KB (415 words) - 17:03, 13 January 2024
A unified way of looking at various geometries (for example, Euclidean, affin
...the properties of figures that are preserved under all transformations of a given group is called the geometry of this group.
3 KB (468 words) - 19:29, 17 April 2014
The curve of intersection of the surfaces of a sphere of radius $R$ and a certain circular cylinder of radius $R/2$ through the centre of the sphere.
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$#A+1 = 138 n = 0
that arise in a body occupying a region $ \Omega $
28 KB (4,176 words) - 05:30, 19 March 2022
$#A+1 = 35 n = 0
...valued under unrestricted [[Analytic continuation|analytic continuation]]. A Schwarz function is an [[Automorphic function|automorphic function]]. The c
5 KB (568 words) - 08:12, 6 June 2020
...l group]]), Engel groups (cf. [[Engel group|Engel group]]) and groups with a normalizer condition. The majority of classes of generalized nilpotent grou
...ed from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.G. Kurosh, S.N. Chernikov, "Solvable and nilpotent groups" ''Uspekhi M
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A [[category]] in which the [[subobject]]s of each object may be indexed by a set.
The dual notion is a ''co-well-powered'' category.
419 bytes (60 words) - 16:20, 18 November 2023