$x =~
s/\n*\<table .*?\<\/table\>\n*\
s*/\n\$\$•\$\$\n/sg;
$x =~
s/<img align[^>]*>/\$•\$/sg;
804 bytes (123 words) - 13:38, 20 June 2014
''of a semi-group $S$''
...x of a semi-group $S$ if and only if $N$ is a class of some congruence on $S$ (cf. [[Congruence (in algebra)|Congruence (in algebra)]]).
683 bytes (121 words) - 20:27, 14 April 2014
...armonic function|harmonic function]] in a bounded simply-connected domain $S^+$ which, on the boundary $L$ of the domain, satisfies the condition
$$A(s)\frac{du}{dn}+B(s)\frac{du}{ds}+c(s)u=f(s),$$
734 bytes (129 words) - 19:41, 14 August 2014
...properties 1) $U(s,s) = I$; 2) $U(t,x)U(x,s) = U(t,s)$; and 3) $U(t,s) = U(s,t)^{-1}$.
...satisfied automatically. Under some circumstances, the restriction $t \ge s$ is a natural one, and the inverse need not exist at all.
740 bytes (119 words) - 18:43, 16 October 2017
...They are situated in the tangent plane to $S$ and have the same radius as $S$.
953 bytes (148 words) - 17:28, 1 August 2014
$$\prod_p\left(1-\frac{1}{p^s}\right)^{-1},$$
...erges absolutely for all $s>1$. The analogous product for complex numbers $s=\sigma+it$ converges absolutely for $\sigma>1$ and defines in this domain t
557 bytes (85 words) - 18:50, 18 October 2014
...[S]$ consists precisely of those rows of $I$ corresponding to vertices in $S$.
330 bytes (65 words) - 14:49, 10 January 2016
\def\S{\mathcal S} % subfamily
(in analogy to [[Helly's theorem]]) the smallest natural number $k$
549 bytes (88 words) - 12:15, 12 December 2013
[[Algebra|algebra]] $\Phi(S)$ over a field $\Phi$ with a basis $S$ that is at the same time a multiplicative
[[Semi-group|semi-group]]. In particular, if $S$ is a group, one obtains a
2 KB (344 words) - 19:41, 8 December 2015
...either $K(s,t)\equiv0$ if $a\leq s<t\leq b$ or $K(s,t)\equiv0$ if $a\leq t<s\leq b$. If such a function is the kernel of a linear integral operator, act
520 bytes (95 words) - 19:03, 27 April 2014
...[#References|[1]]], and if $S$ is a normal scheme, $A$ is projective over $S$, [[#References|[2]]].
1 KB (194 words) - 11:29, 27 January 2024
...which may be used to exhibit properties of various [[Dirichlet L-function]]s.
The Hurwitz zeta function $\zeta(\alpha,s)$ is defined for real $\alpha$, $0 < \alpha \le 1$ as
722 bytes (101 words) - 09:28, 19 March 2023
''of a semi-group $S$''
...complete inverse image of the [[unit element]] under some homomorphism of $S$ onto a [[Monoid|semi-group with unit element]].
644 bytes (108 words) - 18:31, 13 December 2014
..., method of]]). In Bateman's method, the degenerate kernel $ K _ {N} (x, s) $
K _ {N} (x, s) =
2 KB (269 words) - 10:33, 29 May 2020
\int\limits _ { a } ^ { b } \int\limits _ { a } ^ { b } K(x, s)
\phi (x) \overline{ {\psi (s) }}\; dx ds ,
839 bytes (120 words) - 10:59, 29 May 2020
See [[S-duality|$S$-duality]].
64 bytes (8 words) - 17:07, 3 November 2014
An integral equation in which the unknown function $\phi(s)$ occurs in a linear fashion:
$$A(x)\phi(x)+\int\limits_DK(x,s)\phi(s)ds=f(x),\quad x\in D.$$
378 bytes (72 words) - 16:46, 7 July 2014
A [[Symmetric operator|symmetric operator]] $S$ on a Hilbert space $H$ for which there exists a number $c$ such that
...n $A$ with the same lower bound $c$ (Friedrichs' theorem). In particular, $S$ and its extension have the same deficiency indices (cf. [[Defective value|
652 bytes (99 words) - 17:01, 2 July 2014
'' $ S $ with kernel $ A $''
with an epimorphism $ \phi : G \rightarrow S $
3 KB (416 words) - 12:53, 19 March 2023
Let $ S = \{ p _ {1} \dots p _ {n} \} $
...ight-line dual of the [[Voronoi diagram|Voronoi diagram]] generated by $ S $
2 KB (247 words) - 17:32, 5 June 2020
$$x=\Delta s-\frac{k_1^2\Delta s^3}{6}+o(\Delta s^3),$$
$$y=\frac{k_1\Delta s^2}{2}+\frac{k_1'\Delta s^3}{6}+o(\Delta s^3),$$
1 KB (245 words) - 16:53, 30 July 2014
algebra of Borel sets of an $ s $-
dimensional Euclidean space $ \mathbf R ^ {s} $.
6 KB (799 words) - 08:01, 6 June 2020
The direct product of automata $ \mathfrak A _ {i} = ( A _ {i} , S _ {i} , B _ {i} , \phi _ {i} , \psi _ {i} ) $,
is the automaton $ \mathfrak A = (A, S, B, \phi , \psi ) $
4 KB (613 words) - 18:49, 5 April 2020
...\setminus G(S)$ is a sub-semi-group (being, clearly, the largest ideal in $S$); such a semi-group is called a semi-group with isolated group part. The f
.... Soc. (1961)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russia
2 KB (266 words) - 15:55, 22 July 2014
An algebra $ S $
in which there are distinguished subspaces $ S _ \alpha $,
3 KB (378 words) - 19:39, 5 June 2020
\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s},\quad s=\sigma+it,
...f the zeta-function $\zeta(s)$ lie on the straight line $\operatorname{Re} s = 1/2$.
1 KB (160 words) - 20:17, 18 October 2014
''of valency $ s \geq 1 $''
A tensor of type $ ( 0, s) $,
3 KB (408 words) - 17:31, 5 June 2020
...gnals. Suppose that functions $ q ( y, s ^ \prime ; \widetilde{y} , s ^ {\prime\prime} ) $,
$ s ^ \prime , s ^ {\prime\prime} \in S $,
4 KB (540 words) - 16:43, 4 June 2020
...[[Clifford semi-group]]. The set $S^*$ of all characters of a semi-group $S$ forms a commutative semi-group with identity (the character semi-group) un
(\chi*\psi)(a) = \chi(a)\cdot\psi(a)\,,\ \ a\in S\,,\ \ \chi,\psi\in S^* \ .
3 KB (459 words) - 19:54, 1 January 2018
\int\limits _ { a } ^ { b } K ( x , s ) \phi ( s) d s ,
\int\limits _ { a } ^ { b } K ( x , s ) \phi ( s) d s = f ( s)
2 KB (335 words) - 16:32, 13 July 2021
...the power is in the interval $1\le p < \infty$, then the $L^p$ space $L^p(S, F, \mu)$ contains the equivalence classes of complex measurable functions
\int_S |f(s)|^p \; d\mu(s) < \infty
1 KB (217 words) - 14:53, 11 November 2023
...generating series for $\Lambda(n)$ is the logarithmic derivative of $\zeta(s)$:
-\frac{\zeta'(s)}{\zeta(s)} = \sum_n \Lambda(n) n^{-s}\ \ \ (\Re s > 1)
1 KB (164 words) - 18:33, 18 October 2014
...
s mapping $f : X \to Q^n$, $n = 1, 2, \dots ,$ is inessential (Aleksandrov'
s theorem).
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.
S. Aleksandrov, "Topologie" , '''1''' , Springer (1974)</TD></TR><TR><TD v
1 KB (230 words) - 06:26, 14 January 2017
...imes$ denotes the tensor product over the ring of integers, and $(r\otimes s)b = rbs$. For every left $R$-module $M$ one has the situation ${}_R M_E$, w
...}_R C_S$ that is left $R$-linear and right $S$-linear. The category of $(R,S)$-bimodules with bimodule morphisms is a [[Grothendieck category]].
2 KB (277 words) - 17:01, 23 November 2023
$$\frac h6(S+S'+4S''),$$
...e $h$ is the distance between the bases, $S$ and $S'$ are their areas and $S''$ is the area of the intersection that has equal distance to both bases.
967 bytes (161 words) - 16:41, 8 May 2024
P_n(F) = \{ s \in \mathrm{GL}_n(F) : \text{last row}\,(s) = (0,0,\ldots,1) \} \ .
...ean (cf. also [[Archimedean axiom]]), both conjectures are true (Bernstein's theorems).
1,010 bytes (146 words) - 08:10, 23 March 2018
...ng $A$ with an identity, then the symmetric algebra of $M$ is the algebra $S(M) = T(M)/I$, where $T(M)$ is the [[tensor algebra]] of $M$ and $I$ is the
S(M) = \bigoplus_{p \ge 0} S^p(M)
3 KB (487 words) - 18:21, 11 April 2017
''Ore's condition''
...]]s intersect is ''left reversible'': $\forall a,b, \in S\ \exists x,y \in S \ :\ ax = by$. A commutative semi-group is reversible, as $ab=ba$. A semi
588 bytes (88 words) - 11:41, 2 October 2016
For each $ s> 0 $,
let $ \tau _ {s} $
2 KB (273 words) - 16:15, 13 January 2021
$$\phi(x)+\int\limits_a^bK(x,s)f[s,\phi(s)]ds=0,\quad a\leq x\leq b,$$
...i.e. all its eigen values are positive. If, in addition, the function $f(x,s)$ is continuous and satisfies the condition
2 KB (338 words) - 11:07, 24 August 2014
Let $ S $
be a symbol with $ \omega \notin S $.
3 KB (455 words) - 08:11, 6 June 2020
...ays regular (cf. [[Maximal torus|Maximal torus]]). In general, a torus $ S \subset G $
is regular if and only if its centralizer $ C _{G} (S) $
2 KB (286 words) - 15:07, 17 December 2019
...s the same as the [[Voronoi diagram|Voronoi diagram]] of that set $S$. If $S$ is a [[Lattice|lattice]], it is also called the Dirichlet–Voronoi tiling
509 bytes (70 words) - 19:53, 29 April 2014
...special conditions: a right translation of a [[Semi-group|semi-group]] $ S $
for any $ x, y \in S $;
4 KB (612 words) - 08:26, 6 June 2020
''of a non-cooperative game $ \Gamma = \langle J , \{ S _ {i} \} _ {i \in J } , \{ H _ {i} \} _ {i \in J } \rangle $''
...on-cooperative game]] $ \widetilde \Gamma = \langle J , \{ \widetilde{S} {} _ {i} \} _ {i \in J } , \{ \widetilde{H} {} _ {i} \} _ {i \in J } \
4 KB (563 words) - 08:09, 6 June 2020
...$ there exist antipodes with a common image; 2) Any mapping of the sphere $
S^n$ into itself in which the images of antipodes are antipodes is an [[essen
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> K. Borsuk, "Drei
Sätze über die $n$-dimensionale euklidische Sphäre" ''Fund. Math.'' , '''
1 KB (163 words) - 06:23, 28 October 2017
...om mappings are those for which $\mathsf{P}\{\sigma=s\} = n^{-n}$ for all $s \in\Sigma_n$. A realization of such a random mapping is the result of a sim
782 bytes (121 words) - 17:57, 22 November 2016
In order that a surface $ S $
with metric $ d s ^ {2} $
1 KB (165 words) - 08:11, 6 June 2020
A measurable, in general complex-valued, function $ K (x, s) $
satisfying the conditions: 1) $ \overline{ {K (x, s) }}\; = K (s, x) $
873 bytes (124 words) - 11:32, 30 May 2020
$#C+1 = 15 : ~/encyclopedia/old_files/data/S083/S.0803210 Sample variance,
S _ {n} ( x) = \
2 KB (247 words) - 08:12, 6 June 2020