$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
be independent random variables, let $ S _ {k} = \sum_{i=1}^ {k} X _ {i} $
{\mathsf P} \left \{ \max _ {1\leq k \leq n } ( S _ {k} - m ( S _ {k} -
2 KB (317 words) - 10:43, 20 January 2024
...ow \mathcal{P}S$ is a ''closure operation'' if for all $A,B \in \mathcal{P}S$:
A pair $(S,\mathrm{Cl})$ is a ''closure space'' if and only if $\mathrm{Cl}$ satisfies
2 KB (285 words) - 17:42, 13 October 2023
\sum_{n=1}^\infty \frac{1}{n^s} = \prod_p \left({1 - \frac{1}{p^s} }\right)^{-1}
...prime numbers $p$. The Euler identity also holds for all complex numbers $s = \sigma + it$ with $\sigma > 1$.
2 KB (279 words) - 19:13, 14 December 2015
...sitive integers with greatest common divisor $1$. The Frobenius number of $S$ is the largest natural number that cannot be written as a linear integer c
314 bytes (50 words) - 15:24, 10 August 2014
...Hilb}_{X/S}(S^*)=\Hom_S(S^*,\operatorname{Hilb}(X/S))$. In particular, if $S$ is the spectrum of a field $k$ (cf. [[Spectrum of a ring|Spectrum of a rin
...$. For any connected ground scheme $S$ the scheme $\operatorname{Hilb}^P(X/S)$ is also connected [[#References|[2]]].
3 KB (484 words) - 05:48, 17 April 2024
...ng to $S$. The Aitken $\Delta ^ { 2 }$ process consists of transforming $( S _ { n } )$ into the new sequence $( T _ { n } )$ defined, for $n = 0,1 , \d
...} + S _ { n } } = S _ { n } - \frac { \Delta S _ { n } } { \Delta ^ { 2 } S _ { n } }, \end{equation*}
9 KB (1,390 words) - 20:20, 25 January 2024
...ut its mean surface by a [[Conjugate net|conjugate net]] of lines. Let $ S $
...aucour congruence. Then there is a family of surfaces corresponding to $ S $
2 KB (305 words) - 08:11, 6 June 2020
...rity) $\phi_i(v+u)=\phi_i(v)+\phi_i(u)$. These axioms were introduced by L.S. Shapley [[#References|[1]]] for an axiomatic definition of the expected pa
$$\phi_i(v)=\sum_{i\in S}\frac{(|S|-1)!(n-|S|)!}{n!}[v(S)-v(S\setminus\{i\})].$$
2 KB (304 words) - 12:43, 19 August 2014
..., is an extension of the Darboux pencil. A (non-developable) hypersurface $S$ degenerates into a hyper-quadric if and only if its generalized [[Darboux
<TR><TD valign="top">[1]</TD> <TD valign="top"> S.P. Finikov, "Projective-differential geometry" , Moscow-Leningrad (1937)
2 KB (240 words) - 08:52, 8 April 2023
...eory of canonical correlations the random variables $ X _ {1} \dots X _ {s} $
and $ X _ {s+1} \dots X _ {s + t } $,
2 KB (363 words) - 06:29, 30 May 2020
$#C+1 = 50 : ~/encyclopedia/old_files/data/S087/S.0807200 Star body
An open set $ \mathfrak S $
3 KB (521 words) - 20:42, 16 January 2024
Two functions $ K ( x, s) $
and $ K _ {1} ( x, s) $
3 KB (440 words) - 17:48, 13 January 2024
...p in the ring of all endomorphisms of this group is a division ring (Schur's lemma).
587 bytes (95 words) - 19:54, 11 April 2014
\int\limits _ \Omega K ( x , s , \phi ( s) ) d s + f ( x) ,\ \
...bounded closed set in a finite-dimensional Euclidean space and $ K ( x , s , t ) $
3 KB (455 words) - 08:27, 6 June 2020
...tioned into left ideals which are (necessarily isomorphic) groups; and d) $S$ is the direct product of a group and a right zero semi-group (cf. [[Idempo
1 KB (167 words) - 17:09, 8 January 2021
...{a_s\}$ and $\{b_s\}$ be certain sets of complex numbers, $s\in S$, where $S$ is a finite or an infinite set of indices. The following inequality of Hö
...q \Bigl(\sum\limits_{s\in S}|a_s|^p\Bigr)^{\frac1p}\Bigl(\sum\limits_{s\in S}|b_s|^q\Bigr)^{\frac1q},
4 KB (729 words) - 16:13, 29 November 2012
If $ \mathbf r = \mathbf r ( s) $(
where $ s $
3 KB (450 words) - 19:38, 5 June 2020
L(a,s) = \sum_{n=1}^\infty a(n) n^{-s}
L(a,s) + L(b,s) = \sum_{n=1}^\infty (a+b)(n) n^{-s}
2 KB (358 words) - 17:25, 11 November 2023
A function $(x,s)\mapsto K_n(x,s)$ that is formed from the given kernel $K$ of an [[Integral operator|integr
$$K_1(x,s)=K(x,s),\quad K_n(x,s)=\int\limits_a^bK_{n-1}(x,t)K(t,s)dt.$$
2 KB (292 words) - 19:24, 9 October 2014
...phere]]. This group is presented by generators $R$, $S$ and relations $R^3=S^3=(RS)^2$.
* {{Ref|a1}} H.S.M. Coxeter, "Regular complex polytopes", Cambridge Univ. Press (1991) pp. 7
365 bytes (50 words) - 13:54, 8 April 2023
...ally assumed that the effect of the noise on the signal is additive: $x(t)=s(t)+n(t)$. In such a situation the problems of signal extraction are as outl
...More involved varieties of the initial hypothesis are also studied: $x(t)=s(t)+n(t)$ starting from some moment, possibly random, of time $\tau$, which
2 KB (383 words) - 14:54, 13 August 2014
...ring of fractions (cf. [[Fractions, ring of|Fractions, ring of]]) $ A [ S ^ {-1} ] $,
where $ S $
4 KB (713 words) - 21:35, 4 January 2021
''of a convex surface $S$''
...ting from some point $O\in S$ and belonging to the convex body bounded by $S$. The limit cone is defined uniquely, up to a parallel displacement dependi
1 KB (161 words) - 05:57, 18 October 2014
defined by G.W. Whitehead. In $ S ^ {k} $
Then the product of spheres $ S ^ {m} \times S ^ {n} $
4 KB (505 words) - 15:48, 29 March 2021
...$ is introduced on the bicharacteristic strip, then its equations $x_i=x_i(s)$, $i=1,\dots,n$, are defined by solving a system of $2n$ ordinary differen
\begin{equation}\dot x_i(s)=Q_{\xi_i},\quad\dot\xi_i=-Q_{x_i},\quad i=1,\dots,n.\label{*}\end{equation
3 KB (419 words) - 22:32, 10 December 2018
with at most a countable set of states $ S $:
i, j \in S.
2 KB (239 words) - 09:50, 29 October 2023
A morphism of schemes $ f: X \rightarrow S $
such that the pre-image of any open affine subscheme in $ S $
3 KB (464 words) - 05:59, 19 March 2022
U = \alpha _ {1} X _ {1} + \dots + \alpha _ {s} X _ {s} ,\ \
V = \beta _ {1} X _ {s+1} + \dots + \beta _ {t} X _ {s+t} $$
3 KB (346 words) - 08:00, 25 April 2022
A model of the manifold of lines in a three-dimensional elliptic space $ S _ {3} $
on a pair of two-dimensional elliptic planes $ S _ {2} $.
3 KB (461 words) - 19:40, 5 June 2020
...$. The shortest-path problem is to find a least cost (shortest) path from $s$ to every other vertex (node) $i$.
...eeds by constructing, one node at a time, a subtree $T$ rooted at $s$ (an $s$-[[arborescence]]). If $\Gamma$ is connected, $T$ will be a spanning subtre
2 KB (405 words) - 19:42, 6 September 2017
$$k_1=\phi(s),\quad k_2=\psi(s),$$
...pace, with [[Curvature|curvature]] $\phi(s)$ and [[Torsion|torsion]] $\psi(s)$. A necessary and sufficient condition for a curve to be in a plane is tha
2 KB (304 words) - 19:04, 26 January 2024
The number $s$ equal to the square root of the [[Arithmetic mean|arithmetic mean]] of the
$$s=\sqrt\frac{a_1^2+\dots+a_n^2}n.$$
259 bytes (45 words) - 13:55, 30 December 2018
...or]] of any element is a principal left ideal on an idempotent element of $S$.
Examples include the monoid of [[binary relation]]s on a set $A$ under composition of relations, with the empty relation as zer
714 bytes (106 words) - 16:43, 23 November 2023
...ntinuously on $t \in \mathbf{R}$. For $s \in \mathbf{R}$, denote by $C _ { S } : \mathbf{R} \rightarrow \mathcal{L} ( V )$ the solution of the initial v
\begin{equation*} X ( s ) = 0 , X ^ { \prime } ( s ) = I. \end{equation*}
2 KB (315 words) - 16:46, 1 July 2020
is called the remainder (in Peano's form). Given the asymptotic expansion
\Gamma ( s + 1 ) = \sqrt {2 \pi s } \left (
2 KB (344 words) - 09:03, 6 January 2024
...There are various generalizations of a self-perimeter for the unit sphere $S$ in a normed space of dimension greater than two (see [[#References|[5]]],
...69) pp. 431–443</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S. Gołab, "Sur la longuer d'indicatrice dans la géometrie plane de Minkow
2 KB (271 words) - 14:01, 1 October 2014
A locally trivial [[Fibration|fibration]] $ f: S ^ {2n - 1 } \rightarrow S ^ {n} $
Here, for example, $ X \star S ^ {0} = SX $,
3 KB (436 words) - 22:11, 5 June 2020
...ecisely, a homomorphism of an automaton $ \mathfrak A _ {1} = (A _ {1} , S _ {1} , B _ {1} , \phi _ {1} , \psi _ {1} ) $
into an automaton $ \mathfrak A _ {2} = (A _ {2} , S _ {2} , B _ {2} , \phi _ {2} , \psi _ {2} ) $(
3 KB (366 words) - 18:49, 5 April 2020
...onvergence of a sequence. What is needed is convergence of nets. A net $ S : D \rightarrow X $
converges to a point $ s \in X $
1 KB (190 words) - 08:02, 6 June 2020
...taining $R$. If $\bar R = R$, then $R$ is said to be integrally closed in $S$ (cf. also [[Integral ring]]).
...ty $R$ is called normal if it is reduced (i.e. has no [[nilpotent element]]s $\neq 0$) and is integrally closed in its complete ring of fractions (cf. [
2 KB (305 words) - 16:09, 11 September 2016
A \prod _ {S} B &\ \mathop \rightarrow \limits ^ { {p _ A}} \ & A \\
B &\ \mathop \rightarrow \limits _ \beta \ &S . \\
2 KB (209 words) - 11:42, 8 February 2020
An element of an algebraic structure $S$, often denoted $1$, $I$ or $e$, with a specific property with respect to a
\forall x \in S \ \ 1 * x = x * 1 = x\ .
375 bytes (55 words) - 18:27, 13 December 2014
\omega \wedge ( d \omega ) ^ {s} ( x) \neq 0 ,\ \
( d \omega ) ^ {s+1} ( x) = 0 ;
2 KB (331 words) - 07:53, 9 January 2024
...s the [[evolute]] of $\bar\gamma$. If $\mathbf{r} = \mathbf{r}(s)$ (where $s$ is the arc length parameter of $\gamma$) is the equation of $\gamma$, then
\bar{\mathbf{r}} = \mathbf{r}(s) + (c-s)\tau(s) \,,
2 KB (317 words) - 21:09, 17 December 2017
...ernel of an integral operator|Kernel of an integral operator]]) $ K ( x, s) $,
$ a \leq x, s \leq b $,
1 KB (223 words) - 22:10, 5 June 2020
...s therefore a consequence of the Riemann conjecture on the zeros of $\zeta(s)$ (cf. [[Riemann hypotheses|Riemann hypotheses]]). It is known (1982) that
...r la croissance de la fonction zêta(s)", Bull. des sciences mathématiques, série 2, vol. 32, 1908. Claude Henri Picard
2 KB (299 words) - 18:59, 7 December 2014
$$ \int \limits_0^1 \dots \int \limits_0^1 |S|^{2k} d \alpha_1 \dots d \alpha_n, $$
$$ S = \sum_{1 \leq x \leq P} e^{2 \pi i (\alpha_1 x + \dots + \alpha_n x^n)}, $
580 bytes (91 words) - 21:35, 14 January 2017
A mapping $s : Y \rightarrow X$ for which $p \circ s = \mathrm{id}_Y$. In a wider sense, a section of any morphism in an arbitra
...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
A formula expressing the surface area $S$ of a triangle in terms of its sides $a$, $b$ and $c$:
$$S=\sqrt{p(p-a)(p-b)(p-c)},$$
527 bytes (83 words) - 18:02, 17 April 2023
''least upper bound, on a set $ S $''
which is the finest of all topologies on $ S $
3 KB (490 words) - 08:27, 6 June 2020
A(x) = \int_{a}^{b} K(t,s) ~ x(s) ~ \mathrm{d}{s};
A(x) = \int_{a}^{b} K(t,s) ~ g(s,x(s)) ~ \mathrm{d}{s};
2 KB (399 words) - 05:44, 7 December 2016
...[[Dirichlet series]]. It is used in the study of [[arithmetic function]]s and yields a proof of the [[Prime number theorem]]. It was proved by [[No
...function|analytic]] for $\Re s \ge 1$, except for a simple [[pole]] at $s=1$ with residue 1. Then the [[Limit of a function|limit]] as $x$ goes to i
2 KB (298 words) - 20:23, 15 November 2023
...) it can be represented as the union of non-intersecting components (Luzin's criterion).
356 bytes (55 words) - 21:57, 19 December 2014
...p $S$ is separable if and only if its characters separate the elements of $S$.
947 bytes (144 words) - 17:58, 22 December 2023
...age of the particle was zero, then the generating function $F(t,s)={\rm E}s^{\mu(t)}$ of the number of particles $\mu(t)$ satisfies the non-linear inte
$$F(t,s) = \int_0^th(F(t-u,s))dG(u) + s(1-G(t)).$$
1 KB (204 words) - 19:33, 28 October 2014
''constructed from an automorphism $S$ of a [[Measure space|measure space]] $(X,\nu)$ and a measurable function $
...re. The motion takes place in such a way that the second coordinate of $(x,s)$ increases with unit speed until it reaches the value $f(x)$, and then the
2 KB (367 words) - 14:04, 30 August 2014
...the property that no element of $S$ is equal to a sum of two elements of $S$. It is known that in general a subset of $M$ of size $n$ contains a sum-f
The [[Cameron–Erdős conjecture]], proved by Ben Green in 2003, asserts that the set $\{1,\ldots
660 bytes (110 words) - 16:47, 23 November 2023
associating to each [[Semi-group|semi-group]] $ S $
a congruence $ \rho ( S) $(
5 KB (767 words) - 14:54, 7 June 2020
...dius function for a one-parameter family of spheres with centres in $\zeta(s)$, then the canal surface enveloping this family is characterized by the fo
\langle r - \zeta(s), r - \zeta(s) \rangle = R^2(s) \,,
3 KB (414 words) - 09:10, 26 March 2023
f ( s ) = \int\limits _ { 0 } ^ \infty {e ^ {- st } } {dG ( t ) } = {\lim\lim
If the integral converges for some complex number $ s _ {0} $,
2 KB (240 words) - 22:15, 5 June 2020
...dimensional sphere $S^2$; also, every cactoid is a monotone open image of $S^2$.
460 bytes (78 words) - 22:20, 1 November 2014
...o describe the conversion of a substrate in an enzymatic reaction. Let $ S ( t ) $
The reaction rate is proportional to $ S ( t ) $
4 KB (605 words) - 08:00, 6 June 2020
\sum_{\mathfrak{p}\in A} N(\mathfrak{p})^{-s} = a \log\frac{1}{1-s} + g(s)
holds, where $g(s)$ is regular in the closed half-plane $\mathrm{Re}(s) \ge 1$, then $A$ is a regular set of prime ideals and $a$ is called its Di
2 KB (310 words) - 09:04, 26 November 2023
f ( s) \sim \sum _ { n } A _ {n} e ^ {\Lambda _ {n} \tau }
\sum _ { n } A _ {n} e ^ {\Lambda _ {n} s } ,\ \alpha < \tau < \beta ,
3 KB (422 words) - 19:35, 5 June 2020
...this means that with each integer (or natural number) a transformation $ S _ {n} : W \rightarrow W $
S _ {n+m} (w) = \
3 KB (418 words) - 05:47, 18 May 2022
$ {\mathcal F} _ {s} \subseteq {\mathcal F} _ {t} \subseteq {\mathcal F} $,
$ s \leq t $,
1 KB (210 words) - 19:42, 5 June 2020
\frac{y ^ {s + 2k } ds }{s ( s + 1) \dots ( s + 2k) }
1 KB (147 words) - 19:35, 5 June 2020
...Lyapunov's first method for investigating stability is based) and Lyapunov's second method (see [[Lyapunov function|Lyapunov function]]). The results an
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* H. S. M. Coxeter, S. L. Greitzer, "Geometry Revisited", Mathematical Association of America (19
513 bytes (74 words) - 16:32, 14 August 2023
...ional manifold which is homotopy equivalent to the $n$-dimensional sphere $S^n$ is homeomorphic to it; at present (1991) it has been proved for all $n\g
...align="top">[a1]</TD> <TD valign="top"> S. Smale, "Generalized Poincaré's conjecture in dimensions greater than four" ''Ann. of Math. (2)'' , '''74'
1 KB (197 words) - 17:53, 10 April 2023
of analytic spaces such that for certain analytic sets $ S \subset X $
f : X \setminus S \rightarrow Y \setminus T \ \
1 KB (175 words) - 08:01, 6 June 2020
...then, for every chart $(U, \phi)$ in the atlas defining $N$, $\phi (U\cap S)$ is a set of (Lebesgue) measure zero.
...ger set|meager set]]. The latter statement is also sometimes called ''Sard's theorem'': however it is not equivalent to the one above, since closed meag
1 KB (203 words) - 16:38, 17 November 2012
\frac{x - s }{2}
0 \leq x, s \leq 2 \pi .
902 bytes (131 words) - 20:27, 18 March 2024
\xi(s) = \frac12 s(s-1) \pi^{-s/2} \Gamma(s/2) \zeta(s)
with $\zeta(s)$ the Riemann zeta function. The Riemann hypothesis is equivalent to the a
2 KB (247 words) - 20:25, 20 November 2023
...ly transitively on the set of all chambers and is generated by the set $ S $
moreover, the pair $ ( W , S ) $
4 KB (631 words) - 16:43, 4 June 2020
...ose of $\pi$. Eilenberg–MacLane spaces were introduced by S. Eilenberg and S. MacLane .
.../TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> S. Eilenberg, S. MacLane, "Relations between homology and homotopy groups of spaces. II"
2 KB (272 words) - 13:05, 24 December 2020
...$E\subset \mathbb{R}$ is a non-empty bounded set, $E_a^b=\{s \in E: a \leq s \leq b \}$ for $a,b \in E$ ($a \leq b$), $X$ is a metric space with a metri
...rem.''' Let ${E_t}^-=\{s \in E: s \leq t\}$ and ${E_t}^+=\{s \in E: t \leq s\}$. The following conditions are equivalent:
2 KB (413 words) - 17:07, 21 April 2016
$$\def\s{\sigma}\a(cx)=c^\s\a(x)$$
...g of vector spaces over the field $\C$ relative to complex conjugation $c^\s = \bar c$ is also known as an anti-linear mapping. A semi-linear mapping of
2 KB (264 words) - 22:23, 2 November 2013
$#C+1 = 12 : ~/encyclopedia/old_files/data/S086/S.0806710 Spherical indicatrix
under a mapping from the points of the curve onto the unit sphere $ S ^ {2} $
2 KB (378 words) - 08:22, 6 June 2020
...ies $\sum_{n=0}^\infty a_n$ can be summed by Riemann's method to a number $S$ if
...o0}\left[a_0+\sum_{n=1}^\infty a_n\left(\frac{\sin nh}{nh}\right)^2\right]=S.$$
2 KB (300 words) - 16:17, 4 October 2014
...always a result of an earlier calculation once one inputs $ \pi _ {1} ( S ^ {1} ) $.
It has been used to calculate $ \pi _ {n + i } ( S ^ {n} ) $
6 KB (927 words) - 19:36, 5 June 2020
( a b ) ^ {n} = a ^ {n} b ^ {n} s _ {1} ^ {n} \dots s _ {t} ^ {n}
holds, where $ s _ {1} \dots s _ {t} $
2 KB (283 words) - 08:10, 6 June 2020
\frac{\partial f }{\partial s }
= - A _ {s} f
5 KB (778 words) - 22:14, 5 June 2020
is a submonoid $ S $
of $ R $ (i.e. $ 1 \in S $
4 KB (674 words) - 07:14, 12 July 2022
A semi-group $ S $
as an ideal). To each element $ s $
4 KB (623 words) - 19:38, 5 June 2020
...ur sides $a$, $b$, $c$, $d$ intersecting in the six points $P$, $Q$, $R$, $S$, $T$, $U$. Then the three circles that have the three diagonals $PS$, $QT$
...n="top">[a1]</TD> <TD valign="top"> R. Fritsch, "Remarks on Bodenmiller's theorem" ''J. of Geometry'' , '''47''' (1993) pp. 23–31</TD></TR><TR><
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that occur. Set $ S _ {0} = S _ {0,n} = 1 $
S _ {j} = S _ { j,n} = \sum {\mathsf P} ( A _ {i _ 1 } \cap \dots \cap A _ { i _ j } )
4 KB (657 words) - 12:53, 6 May 2023
A function $ K( x, s) $,
$ a \leq x, s \leq b $,
1,003 bytes (146 words) - 08:04, 6 June 2020
...the alphabet. From an alphabet we may obtain '''strings''' (or '''[[word]]s''') which are finite [[Sequence|sequences]] of letters over $\Sigma$. The e
...''' of $\Sigma$ (or '''monoid closure''') after the American mathematician S. C. Kleene. The set $\Sigma^* \setminus \{ \lambda\}$ is denoted $\Sigma^+$
2 KB (266 words) - 07:58, 18 November 2023
In the above, an almost-periodic trajectory in a [[dynamical system]] $\{ S^t \}$ on a metric space $(X,\rho)$ is the trajectory of any point $x \in X$
\left\lbrace{ \tau \in \mathbb{R} : \rho(S^t(x), S^{t+\tau}(x)) < \epsilon \ \text{for}\ -\infty < t < +\infty }\right\rbrace
1,020 bytes (154 words) - 19:16, 25 October 2014
...three-dimensional [[Lobachevskii space|Lobachevskii space]] $ {} ^ {1} S _ {3} $
on the complex plane $ S _ {2} ( i) $(
3 KB (352 words) - 22:15, 5 June 2020
...ical constant, having the dimension of action (energy$\times$time). Planck's constant $h$ is
\[6.626\,070\,15\times10^{-34}\,\mathrm J\,\mathrm s.\]
2 KB (248 words) - 14:10, 21 May 2019
A commutative group scheme is a group scheme $G$ over a basis scheme $S$, the value of which on any
$S$-scheme is an Abelian group. Examples of commutative group schemes are [[Ab
4 KB (629 words) - 20:08, 15 December 2020
e ^ {A ( s _ {n} ) ( s _ {n} - s _ {n-1} ) } e ^ {A
( s _ {n-1} ) ( s _ {n-1} - s _ {n-2} ) }
5 KB (676 words) - 02:25, 1 March 2022
(-1)^{s+t} \det A_{i_1\ldots i_k}^{j_1\ldots j_k}
...olumns of $M$; $s = i_1 + \cdots + i_k$, $t = j_1 + \cdots + j_k$. Laplace's theorem is valid: If any $r$ rows are fixed in a determinant of order $n$,
1 KB (182 words) - 11:43, 9 February 2021
$#C+1 = 19 : ~/encyclopedia/old_files/data/S090/S.0900510 Strip method (integral equations)
\phi ( x) - \lambda \int\limits _ { a } ^ { b } K ( x , s ) \phi ( s) d s = f ( x) .
4 KB (616 words) - 18:31, 7 June 2020
\partial g ( \phi _ {k} ( s ) , z ) }{\partial n ^ {(k)}}
\partial g ( \phi _ {k} ( s ) , \zeta ) }{\partial n
3 KB (402 words) - 20:37, 16 January 2024
$#C+1 = 23 : ~/encyclopedia/old_files/data/S083/S.0803420 Schr\AGodinger representation
...tum field theory. In the Schrödinger representation the operators $ A _ {S} $
3 KB (423 words) - 08:12, 6 June 2020
...Semi-group|semi-group]] $S$ with zero for which there is an $n$ such that $S^n = 0$; this is equivalent to the identity
...Nil semi-group]]); or 3) there is a $k$ such that every sub-semi-group of $S$ can be imbedded as an ideal series of length $\leq k$.
2 KB (363 words) - 13:07, 12 December 2013
Riemann's first theorem: Let the series (*) converge at a point $ x _ {0} $
to a number $ S $.
3 KB (442 words) - 07:55, 6 January 2024
p ( s ) = a _ {0} + a _ {1} s + \dots + a _ {n} s ^ {n} = { \mathop{\rm det} } ( sI - A ) .
...open left half of the complex plane. In this case, the polynomial $ p ( s ) $
5 KB (695 words) - 22:14, 5 June 2020
An analytic function $f(s)$, $s=\sigma+i\tau$, regular in a strip $-\infty\leqslant\alpha<\sigma<\beta\leqs
...mbers. A real number $\tau$ is called an $\varepsilon$-almost-period of $f(s)$ if for all points of the strip $(\alpha, \beta)$ the inequality
2 KB (348 words) - 06:21, 16 October 2013
...its _ {t _ {0} } ^ { t } A( s) ds + \int\limits _ {t _ {0} } ^ { t } A( s) \int\limits
_ {t _ {0} } ^ { s } A( r) dr ds + \dots ,
5 KB (722 words) - 19:40, 5 June 2020
on a surface $ S $
...re conjugate diameters of the [[Dupin indicatrix|Dupin indicatrix]] of $ S $
1 KB (207 words) - 06:12, 16 April 2023
The exterior product of $ s $
vectors is said to be a decomposable $ s $-
3 KB (535 words) - 19:38, 5 June 2020
and sequences of spheres $ S _ {n} ^ \prime $,
$ S _ {n} ^ {\prime\prime} $
2 KB (343 words) - 08:28, 6 June 2020
S = \{ a _ {1} , b _ {1} \dots a _ {g} ,\
along the curves of $ S $,
3 KB (443 words) - 06:29, 30 May 2020
\frac{\phi _ {m} ( s) \overline{ {\phi _ {m} ( t) }}\; }{
of a Hermitian positive-definite continuous kernel $ K( s, t) $
3 KB (405 words) - 08:00, 6 June 2020
...roup]] into a [[group]]. A semi-group $S$ is imbeddable in a group $G$ if $S$ is isomorphic to a sub-semi-group of $G$. Necessary and sufficient conditi
([[cancellation law]]s);
3 KB (456 words) - 19:17, 23 December 2014
...n of the [[Conormal|conormal]] of some elliptic operator at the points of $S$. Oblique derivatives may figure in the boundary conditions of boundary val
...e form $l=(l_1,\ldots,l_n)$, where $l_i$ are functions of the points $P\in S$ such that $\sum_{i=1}^n(l_i)^2=1$, then the oblique derivative of a functi
1 KB (187 words) - 17:57, 30 July 2014
S _ {n} ^ {2} = \
{\mathsf E} \{ S _ {n} ^ {2} \} = \
3 KB (400 words) - 10:59, 29 May 2020
F_n(x) = \sum_{s=1}^\infty f(x^s) s^{-n} \ .
f(x) = \sum_{s=1}^\infty \mu(s) F_n(x^s) s^{-n} \ ,
2 KB (340 words) - 17:15, 30 April 2016
in a time interval $ [ s, t] $:
p _ {ij} ( s, t) = {\mathsf P} \{ \xi ( t) = j \mid \xi ( s) = i \} ,\ s< t.
5 KB (751 words) - 02:57, 4 March 2022
A surface $ S ^ \prime $
is superposed on a surface $ S $(
3 KB (529 words) - 08:08, 6 June 2020
H ^ {s} ( F, A) \times H ^ {s + 1 } ( B, A)
where $ \delta : H ^ {*} ( F, A) \rightarrow H ^ {s + 1 } ( E, F, A) $
3 KB (536 words) - 09:05, 8 April 2023
$#C+1 = 30 : ~/encyclopedia/old_files/data/S090/S.0900490 Strip (generalized),
by an equation $ \mathbf r = \mathbf r ( s) $,
2 KB (389 words) - 08:23, 6 June 2020
1) if $ S _ {p} \subset S _ {q} $,
then $ F ( S _ {p} ) \subset F ( S _ {q} ) $;
3 KB (417 words) - 08:08, 6 June 2020
\zeta ( s) = \sum_{n=1} ^ \infty \frac{1}{n ^ {s} }
L( s, \chi ) = \sum_{n=1} ^ \infty \frac{\chi ( n, k) }{n ^ {s} },
4 KB (642 words) - 13:03, 6 January 2024
...ubmodule $S$ is '''superfluous''' if it is not a summand of $M$: that is, $S + L = M$ implies $L = M$.
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...with an identity, equipped with an automorphism $\def\s{\sigma}a\mapsto a^\s$, is a mapping $q:E\times E\to A$, $(x,y)\mapsto q(x,y)$, linear in $x$ for
...E\times F\to G$, where $E$, $F$, $G$ are $A$-modules. In the case when $a^\s = a$ ($a\in A$), one obtains the notion of a
3 KB (572 words) - 13:14, 7 April 2023
be a commutative ring, let $ S $
modules. A homomorphism of $ S $-
5 KB (785 words) - 19:35, 5 June 2020
...$ m = 2 ^ \alpha p _ {1} ^ {\alpha _ {1} } \dots p _ {s} ^ {\alpha _ {s} } $
and $ g _ {1} \dots g _ {s} $
5 KB (642 words) - 22:12, 5 June 2020
...An equivalent definition is: A function $ \phi : \mathbf R \rightarrow S $,
where $ S $
3 KB (414 words) - 08:10, 6 June 2020
...quations consists of an approximation which replaces the kernel $ K ( x, s) $
K ( x, s) \phi ( s) ds = f ( x)
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...modynamical system, one describes any one of the pairs of parameters $ ( s, v) $,
$ ( s, p) $,
4 KB (556 words) - 08:25, 6 June 2020
...araboloid and the surface are all zero at $(x_0,y_0)$, where $P=p(x_0,y_0)=
s(x_0,y_0)$.
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.
S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-H
2 KB (287 words) - 13:35, 29 April 2014
\int\limits _ { a } ^ { b } | f ( x) - S _ {n} ( f , x ) | ^ {2} d x
S _ {n} ( f , x ) = \
2 KB (317 words) - 12:37, 6 January 2024
Abel's theorem on algebraic equations: Formulas expressing the solution of an arbi
The theorem was proved by N.H. Abel in 1824 . Abel's theorem may also be obtained as a corollary of [[Galois theory|Galois theor
6 KB (894 words) - 06:14, 26 March 2023
\sum _ {s = 1 } ^ { {T } - t }
( x _ {s} - \overline{x}\; )
4 KB (586 words) - 17:31, 5 June 2020
...They are block linear codes which are a special case of [[alternant code]]s.
...' of length $n$ and parameter $s$ over $GF(q)$ is defined by an $n \times s$ [[parity check matrix]] $H$ of [[alternant matrix|alternant]] form
851 bytes (132 words) - 07:25, 14 November 2023
corresponds a set $ S _ {i} ^ {(x)} $
at this point, and hence, also, the set $ S ^ {(x)} = \prod _ {i} S _ {i} ^ {(x)} $
4 KB (716 words) - 19:39, 16 January 2024
The numbers $ N _ {s} ( \beta ) $
by deletion of $ s $
3 KB (507 words) - 13:23, 8 February 2020
A concise proof was later found by S. Fisk [[#References|[a2]]]. See also [[Art gallery theorems|Art gallery the
...gn="top">[a2]</TD> <TD valign="top"> S. Fisk, "A short proof of Chvátal's watchman theorem" ''J. Combin. Th. B'' , '''24''' (1978) pp. 374</TD></T
905 bytes (142 words) - 12:23, 27 August 2014
$$s=\int(y'')^{1/3}dx.$$
...t $M_0$ of the curve: Let $M$ be a point on the curve close to $M_0$, let $s$ be the affine length of the arc $M_0M$ and let $\sigma$ be the affine leng
1 KB (204 words) - 18:12, 30 July 2014
The existence of $S$, called the Levi factor (if $S$ is a semi-simple subalgebra, respectively a semi-simple subgroup, also cal
3 KB (461 words) - 14:11, 14 August 2014
A name introduced by
S. MacLane [[#References|[a1]]] (and now largely obsolete) for a special kind
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">
S. MacLane, "Duality for groups" ''Bull. Amer. Math. Soc.'' , '''56''' (1
602 bytes (94 words) - 16:55, 7 February 2011
...B$ such that for any $x,y \in A$ it follows from $x\, R\, y$ that $f(x)\, S\, f(y)$.
493 bytes (85 words) - 18:37, 6 October 2016
$#C+1 = 40 : ~/encyclopedia/old_files/data/S090/S.0900790 Sturm theorem
f _ {0} ( x), \ldots, f _ {s} ( x)
3 KB (516 words) - 10:42, 19 February 2021
there acts the [[Symmetric group|symmetric group]] $ S _ {n} $:
f( x _ {s(} 1) \dots x _ {s(} n) ),
4 KB (590 words) - 07:45, 7 January 2024
A spectral set of an operator $A$ on a normed space is a subset $S\subset\mathbf C$ such that
$$\|p(A)\|\leq\sup\{|p(z)|:z\in S\}$$
2 KB (295 words) - 15:46, 29 December 2018
which satisfies on the boundary $S$ of this domain (or on a part of it) the linear boundary conditions
(Bu)(y) = \phi(y), \quad y\in S.
471 bytes (78 words) - 21:19, 2 May 2012
A surface $ S $
...llowing three conditions (Lyapunov's conditions): 1) at every point of $ S $
4 KB (507 words) - 04:11, 6 June 2020
...(the apothem for the base of the pyramid is the projection of the pyramid's apothem on the plane of the base).
$$V=\frac13h(s+S+\sqrt{sS}),$$
2 KB (400 words) - 21:32, 11 April 2014
One also encounters Ptolemy's theorem. Ptolemy's theorem is equivalent to the formula for $\sin(\alpha+\beta)$.
The converse to Ptolemy's theorem is also true. If in a convex quadrangle the product of the lengths
1 KB (176 words) - 21:35, 30 April 2014
B _ {n} ^ {p} ( a _ {1} \dots a _ {r} ;b _ {1} \dots b _ {s} ;x ) =
...F _ {s} } [ \Delta ( p; - n ) , a _ {1} \dots a _ {r} , b _ {1} \dots b _ {s} ;x ] ,
2 KB (357 words) - 06:29, 30 May 2020
P ( s , x ; u , \Gamma ) = \
P ( s , x ; t , d y )
2 KB (354 words) - 22:14, 5 June 2020
$#C+1 = 45 : ~/encyclopedia/old_files/data/S084/S.0804190 Semi\AAhinvariant
can be represented by Taylor's formula as
5 KB (661 words) - 12:22, 17 January 2021
Let $ \pi ^ {n} ( X) = [ X , S ^ {n} ] $
since $ S ^ {n} $
7 KB (1,086 words) - 17:45, 4 June 2020
...$. A related theorem is the sphere theorem which, in conjunction with Dehn's lemma and the loop theorem, is one of the principal tools in the topology o
...points from $L$. In particular, if $L$ is a knot (i.e. $n=1$), then $\pi_2(S^3\setminus L)=0$ (the theorem on asphericity of knots).
3 KB (466 words) - 08:36, 1 May 2023
#REDIRECT [[Fermat's last theorem]]
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==Fabry's gap theorem==
==Fabry's quotient theorem==
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c(a) = \Vert a \Vert^s = e^{s \log \Vert a \Vert} \ .
...$ is a complex number and $c_1$ is a character of $K^*$. The real part of $s$ is uniquely determined by the quasi-character $c$ and is called the real p
2 KB (320 words) - 20:07, 7 September 2017
...components. First, a [[Signature (Computer Science)|signature]] $\Sigma=(S,F)$ defining its syntactical (input/output) behaviour, and second, a corr
...rrier sets $s^{T(\Sigma)}$ of the ground term $T(\Sigma)$ are defined as $s^{T(\Sigma)} := T_s(\Sigma)$. The functions are given by:
3 KB (426 words) - 17:09, 8 February 2013
...zed formal arithmetic is often used as the essentially-undecidable theory $S$.
<TR><TD valign="top">[3]</TD> <TD valign="top"> S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951)</TD>
2 KB (273 words) - 20:42, 19 October 2017
Given an interval $ [ s,t ] $,
$ s < t $,
6 KB (972 words) - 21:59, 29 January 2020
and $ S $
place recursive relations, then the relations $ R \cup S $,
1 KB (187 words) - 14:55, 7 June 2020
Wilcoxon's test is based on the [[Rank statistic|rank statistic]]
W = s ( r _ {1} ) + \dots + s ( r _ {m} ),
3 KB (399 words) - 10:34, 7 February 2021
...8) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S.M. Nikol'skii, "A course of mathematical analysis" , '''1''' , MIR (1977
...b$ such that for all $x\in S$ one has $x\leq b$. The least upper bound of $S$ is then defined as an upper bound $B$ such that for every upper bound $b$
4 KB (722 words) - 19:45, 7 January 2015
\sum _ { s=0 } ^ { n }
s
6 KB (828 words) - 10:58, 29 May 2020
\xi _ {0} = i _ {0} \dots \xi _ {n-} s = i _ {n-} s \dots
{\mathsf P} \{ \xi _ {n} = i _ {n} \mid \xi _ {n-} s
3 KB (325 words) - 07:59, 6 June 2020
Minkowski's theorem on convex bodies is the most important theorem in the geometry of n
An equivalent formulation of Minkowski's theorem is:
3 KB (468 words) - 19:26, 15 November 2014
Mahler's problem was solved affirmatively in 1964 by V.G. Sprindzhuk [[#References|[
...>[1]</TD> <TD valign="top"> K. Mahler, "Ueber das Mass der Menge aller $S$-Zahlen" ''Math. Ann.'' , '''106''' (1932) pp. 131–139 {{ZBL|0003.2460
2 KB (260 words) - 20:25, 3 April 2016
where $a$ is a constant. Chebyshev's equation is a special case of the [[Hypergeometric equation|hypergeometric
...integrated in closed form. A fundamental system of solutions to Chebyshev's equation on the interval $-1<x<1$ with $a=n^2$, where $n$ is a natural numb
1 KB (219 words) - 19:57, 21 November 2018
...ion]] of $K$ contained in $E$. Then $S/K$ is a separable extension and $E/S$ is a purely inseparable extension.
991 bytes (163 words) - 09:02, 26 November 2023
( p) = \mathfrak P _ {1} ^ {e _ {1} } \dots \mathfrak P _ {s} ^ {e _ {s} } ,
\mathfrak P _ {2} \dots \mathfrak P _ {s} \neq \mathfrak P ,
3 KB (388 words) - 08:27, 6 June 2020
F _ {s} (t, \mathbf r _ {1} \dots \mathbf r _ {s} ,\
\mathbf p _ {1} \dots \mathbf p _ {s} ) =
5 KB (743 words) - 10:59, 29 May 2020
$#C+1 = 22 : ~/encyclopedia/old_files/data/S091/S.0901620 Symmetric derived number
\frac{\Phi ( S ( x, r _ {k} )) }{| S ( x, r _ {k} ) | }
2 KB (283 words) - 08:24, 6 June 2020
...ffine transformations: $ \overline{x}\; {} ^ {i} = A _ {s} ^ {i} x ^ {s} $.
684 bytes (92 words) - 16:43, 4 June 2020
...on-zero idempotent that is not an identity for any non-zero idempotent of $S$. If a zero is added to a completely-simple semi-group it becomes a complet
...somorphic groups (cf. [[Band of semi-groups|Band of semi-groups]]); or 2) $S$ is regular and all its idempotents are primitive. A special kind of comple
4 KB (545 words) - 20:25, 14 April 2014
...equence which converges uniformly to $f(x)$ on $(-\infty,\infty)$. Bochner's definition is extensively employed in the theory of almost-periodic functio
<TR><TD valign="top">[1]</TD> <TD valign="top"> S. Bochner, "Beiträge zur Theorie der fastperiodischen Funktionen I, Funktio
1 KB (204 words) - 18:58, 15 April 2024
...asure 0 (which may depend on $t$ and $s$); in other words, $T^t T^s = T^{t+s} \bmod 0$;
...odulo 0 of the space $(M, \mu)$ with the usual identification: if $T$ and $S$ coincide almost-everywhere, then they determine the same element of $\math
3 KB (480 words) - 02:51, 15 February 2024
...ic to $S$. The left $R$-module $A$ is put in correspondence with the left $S$-module $\mathrm{Hom}_R(U,A)$. Among the properties preserved by transition
...aces definite restrictions on 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
\begin{equation*} \phi ( \phi ( s , u ) , v ) = \phi ( s , u ^ { * } v ), \end{equation*}
\begin{equation*} s \in S , u , v \in H , \phi : S \times H \rightarrow S, \end{equation*}
2 KB (377 words) - 16:58, 1 July 2020
(Abel's differential equation of the first kind) or
...studies of N.H. Abel {{Cite|Ab}} on the theory of elliptic functions. Abel's differential equations of the first kind represent a natural
2 KB (338 words) - 22:19, 4 July 2012
...f finite groups. Sometimes the union of all three theorems is called Sylow's theorem.
...a finite group of order $p^ms$, where $p$ is a prime number not dividing $s$. Then the following theorems hold.
2 KB (372 words) - 19:17, 4 April 2023
...\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
$#C+1 = 15 : ~/encyclopedia/old_files/data/S110/S.1100180 Skorokhod stochastic differential equation
...ma ( s,X _ {s} ) } {dW _ {s} } + \int\limits _ { 0 } ^ { t } {b ( s,X _ {s} ) } {ds } ,
4 KB (501 words) - 08:14, 6 June 2020
''$S$-matrix''
...itudes forms a table with two inputs, and is called the scattering matrix $S$.
2 KB (387 words) - 22:42, 31 December 2017
...oints (cf. [[Branch point|Branch point]]) of a compact Riemann surface $ S $,
...here, extended over all finite and infinitely-distant branch points of $ S $.
1 KB (165 words) - 06:29, 30 May 2020
...ntaining group. The concept may be generalized to [[essential submodule]]s.
...'' if whenever $H$ is a non-trivial subgroup of $G$, the intersection of $S$ and $H$ is non-trivial: here "non-trivial" means "containing an element o
594 bytes (85 words) - 14:23, 12 November 2023
In 1850 the Revd T.P. Kirkman published in ''The Lady's and Gentleman's Diary'' for 1850 the problem ''Fifteen young ladies in a school walk out th
635 bytes (91 words) - 11:09, 17 March 2023
...of the stable $ N $-dimensional [[Normal bundle|normal bundle]]. Let $ S _ {i} ^ {3} $
of tubular neighbourhoods of the spheres $ S _ {i} ^ {3} $
3 KB (435 words) - 17:53, 20 January 2022
the concept of a polarized family of algebraic varieties. Let $f:X\to S$ be a
family of varieties with base $S$, that is, $f$ is a smooth projective
4 KB (644 words) - 13:06, 17 April 2023
...all tau metric]] and Spearman's rho). For an historical review of Spearman's rho and related coefficients, see [[#References|[a2]]].
...ing the sample (Pearson) correlation coefficient $r$ for $\{ ( R _ { i } , S _ { i } ) \} _ { i = 1 } ^ { n }$ yields
7 KB (1,033 words) - 18:56, 22 January 2024
''to a surface $S$ at a point $M$''
...se to $M$ is infinitesimally small as compared to the distance $MM_1$. If $S$ is given by an equation $z=f(x,y)$, then the equation of the tangent plane
740 bytes (135 words) - 10:22, 12 April 2014
A set $ S $
S \equiv \
2 KB (333 words) - 17:32, 5 June 2020
A Howell design of side $ s $
or more briefly an $ H ( s,2n ) $,
8 KB (1,138 words) - 09:31, 19 January 2021
\int_0^\infty x^{s-1}e^{-x} \rd x,
called the Euler integral of the second kind. (The latter converges for $s>0$ and is a representation of the [[Gamma-function|gamma-function]].)
431 bytes (67 words) - 21:21, 29 April 2012
\frac{\partial F (t; s) }{\partial t }
f (F (t; s))
2 KB (356 words) - 06:29, 30 May 2020
...by water in a one-dimensional porous medium (e.g., sand). If $ S _ {w} = S _ {w} ( x,t ) $
...e., the fraction of pore volume occupied by water, and $ \lambda _ {w} ( S _ {w} ) $
4 KB (526 words) - 06:29, 30 May 2020
$#C+1 = 45 : ~/encyclopedia/old_files/data/S087/S.0807260 Stationary distribution
be a homogeneous Markov chain with set of states $ S $
4 KB (609 words) - 08:23, 6 June 2020
u ( x) - \int\limits _ \Omega K ( x , s ) u ( s) d s =
...K _ {0} ( x) v ( x) + \int\limits _ \Omega K _ {1} ( x , s ) v ( s) d s ,
4 KB (524 words) - 04:11, 6 June 2020
everywhere on its boundary surface $ S $.
in the cross-section $ S ^ \prime $
2 KB (385 words) - 08:28, 6 June 2020
{\mathsf P} \{ \xi (t) = i \mid \xi (s) = i \} = 1
\ \textrm{ for } \textrm{ any } t \geq s.
3 KB (415 words) - 16:08, 1 April 2020
a _ {ij} r _ {i} ( x) s _ {j} ( y).
...o a finite two-person zero-sum [[Convex game|convex game]] $ \langle R, S, K\rangle $,
3 KB (435 words) - 17:32, 5 June 2020
...grightarrow \bigcup_{s\in S} s^A$ with $v(x)\in s^A$ for $x\in X_s$, $s\in S$ {{Cite|EM85}}. The set of all assignements is typically designated as $B(
For an assignement $v\in B(A,X)$ and for a value $a\in s^A$ the <i>assignement changed at $x$ to $a$</i>, designated as $v[x\leftarr
3 KB (467 words) - 12:21, 3 March 2013
The conjecture, stated by S. Ramanujan [[#References|[1]]], that the Fourier coefficients $\tau(n)$ of
...eta-function]]), then proved the latter (1974). This also proved Ramanujan's hypothesis.
2 KB (263 words) - 11:46, 8 April 2023
To find, in a vertical plane $(s,\tau)$, a curve such that a material point moving along it under gravity fr
$$\frac{ds}{d\tau}=-\sqrt{2g(x-s)}\sin\omega.$$
2 KB (331 words) - 14:27, 14 February 2020
in which a [[unary operation]] $ S $
is defined (i.e. $ S $
2 KB (354 words) - 08:02, 6 June 2020
A class of isotopic imbeddings of the two-dimensional sphere $ S ^ {2} $
in the four-dimensional one $ S ^ {4} $.
4 KB (597 words) - 08:26, 6 June 2020
...the past. The idea of an abstract Volterra operator appears in V. Volterra's work quite clearly, even though a formal definition and results are missing
...TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> G. Gripenberg, S.O. Londen, O. Staffans, "Volterra integral and functional equations" ,
6 KB (863 words) - 15:34, 13 July 2014
...idence assertions for mappings defined on the $n$-dimensional unit sphere $S^n\subset\mathbf R^{n+1}$. Obviously, the following three versions of this r
...very continuous mapping $f\colon S^n\to\mathbf R^n$, there exists an $x\in S^n$ with $f(x)=f(-x)$.
3 KB (471 words) - 21:50, 31 December 2018
The unboundedness of an integral kernel $ K ( x, s) $(
...al operator]]) when the product $ M( x, s)= | x - s | ^ \alpha K ( x, s) $,
4 KB (565 words) - 08:28, 6 June 2020
A [[Random field|random field]] $ X ( s) $
defined on a [[Homogeneous space|homogeneous space]] $ S = \{ s \} $
5 KB (684 words) - 08:09, 6 June 2020
it is denoted by the symbol $ S _ {n} ^ {m} $,
of the quasi-elliptic three-space $ S _ {3} ^ {1} $
6 KB (872 words) - 08:09, 6 June 2020
s ( x ) = h ( x ) g ( x )
the linear convolution $ s ( x ) $
3 KB (414 words) - 08:29, 6 June 2020
algebras $ R, S $
and $ T = R \otimes _ {K} S $
5 KB (636 words) - 22:11, 5 June 2020
is defined as follows: Let $ \sigma _ {1} \dots \sigma _ {s} $
be all real and let $ \sigma _ {s+1} \dots \sigma _ {s+t} $
3 KB (430 words) - 10:48, 20 January 2024
A semi-group theoretical construction defined as follows. Let $ S $
matrix over $ S $,
5 KB (791 words) - 08:10, 6 June 2020
\int\limits _ { a } ^ { b } K ( x, s) \phi ( s) ds -
K _ {1} ( x, s _ {j} )
3 KB (503 words) - 08:01, 6 June 2020
...ble to define the relative de Rham complex $\sum_{p\ge 0} \Gamma(\Omega_{X/S}^p)$ (see
...ed the relative de Rham cohomology sheaves. These sheaves are coherent on $S$ if $f$ is a
2 KB (380 words) - 01:16, 23 July 2018
...the restrictions of two elements $s$ and $s'$ in $F(U)$ coincide, then $s'=s$;
...l$ and $s_\mu$ to $U_\l\cap U_\mu$ coincide, then there exists an element $s\in F(U)$ which on each $U_\l$ has restriction coinciding with $s_\l$.
2 KB (273 words) - 19:52, 22 October 2016
A function $ P( s, x; t, B) $
given for $ s, t \in T $,
6 KB (935 words) - 08:26, 6 June 2020
$#C+1 = 21 : ~/encyclopedia/old_files/data/S085/S.0805120 Similar operators
Operators $ S $
1 KB (188 words) - 08:13, 6 June 2020
A system $ \Phi _{k} (S,\ G) $
split torus $ S $
6 KB (934 words) - 16:30, 17 December 2019
$#C+1 = 15 : ~/encyclopedia/old_files/data/S084/S.0804700 Serre subcategory
A locally small [[Full subcategory|full subcategory]] $ \mathfrak S $
3 KB (469 words) - 16:39, 17 March 2023
...ced by a measure distributed on a surface $S$ with density $\mu(y)$, $y\in S$; and the [[Double-layer potential|double-layer potential]]
produced by a measure distributed on $S$ with density $\nu(y)$. Physically, a simple-layer potential is interpreted
1 KB (159 words) - 14:25, 14 February 2020
...the plane bounding the half-space. The fundamental group $\pi(E^4\setminus S^2)$ of a knotted sphere is not a knot group (cf. [[Knot and link groups|Kno
583 bytes (90 words) - 08:37, 12 April 2014
* The ''Stirling numbers of the first kind'' $S(n,k)$ count the number of ways $n$ labelled objects can be arranged into $k
For example, 4 objects can be arranged into 2 cycles in eleven ways, so $S(4,2) = 11$.
2 KB (292 words) - 07:23, 7 November 2023
...heorem|Hadamard's gap theorem]]). A strengthening of this theorem is Fabry's gap theorem ([[Fabry theorem|Fabry theorem]]). If the lower density
...-valued analytic function with simply-connected domain of existence (Pólya's theorem). See also [[Over-convergence|Over-convergence]].
1 KB (203 words) - 06:24, 22 April 2012
$#C+1 = 53 : ~/encyclopedia/old_files/data/S083/S.0803440 Schur index
is the minimal degree of a field $ S $
3 KB (533 words) - 08:12, 6 June 2020
my @refs = ($f =~ /\=+References\=+\
s+(\<table\>.*?\<\/table\>)/sg);
if ($f =~ /\>\[$key\]\<(.*?),\
s*["\']/sg) {
4 KB (654 words) - 13:10, 12 November 2014
...(see [[Zermelo axiom|Zermelo axiom]]). Later it became clear that Zermelo's theorem is equivalent to the axiom of choice (in the usual system of axioms
This result is also commonly known as the well-ordering theorem or Zermelo's well-ordering theorem.
695 bytes (105 words) - 20:26, 19 December 2020
if $t \ne s$ and $\ne 0$ if $t=s$ (here, $\langle {\cdot},{\cdot} \rangle$ is the canonical bilinear form co
where $\delta_{st} = 1$ if $s = t$ and $\delta_{st} = 0$ if $s \ne t$, is also a basis; it is said to be the basis dual to $\{a_t\}$ and,
1 KB (228 words) - 17:02, 12 January 2014
\Gamma _ {ki} ^ {s} b _ {sj} -
{\widetilde \Gamma } _ {kj} ^ {s} b _ {is} = \
3 KB (368 words) - 16:09, 1 April 2020
A homomorphism $ \phi : S \rightarrow R $
algebra $ S $
2 KB (359 words) - 19:38, 5 June 2020
...y unstable and has no [[Saddle at infinity|saddle at infinity]] (Nemytskii's theorem [[#References|[1]]]).
...all points move with speed 1 along the lines $\{x\}\times\mathbf R$ ($x\in S$).
2 KB (291 words) - 08:45, 29 August 2014
where $r$ is the radius of curvature, $a=\text{const}$ and $s$ is the arc length. It can be parametrized by the [[Fresnel integrals|Fresn
$$x=\int\limits_0^t\cos\frac{s^2}{2a}\,ds,\quad y=\int\limits_0^t\sin\frac{s^2}{2a}\,ds,$$
1 KB (205 words) - 10:58, 26 March 2023
...ius $r=d(n/2(n+1))^{1/2}$. There are analogues and generalizations of Jung's theorem (e.g. replacing the Euclidean distance by other metrics) (cf. [[#Re
...2]</TD> <TD valign="top"> L. Danzer, B. Grünbaum, V.L. Klee, "Helly's theorem and its relatives" V.L. Klee (ed.) , ''Convexity'' , ''Proc. Symp.
1 KB (191 words) - 19:34, 29 April 2014
...x)$ of degree $k$ in $n$ variables $x=(x_1,\dots,x_n)$ to the unit sphere $S^{n-1}$ of the Euclidean space $E^n$, $n\geq3$. In particular, when $n=3$, t
Let $x\in E^n$, $x\neq0$, $r=|x|$, $x'=x/r\in S^{n-1}$. The basic property of spherical harmonics is the property of orthog
3 KB (519 words) - 19:48, 22 December 2018
...The [[Commutator subgroup|commutator subgroup]] $G'$ of the group $G=\pi_1(S^3\setminus k^1)$ of a Neuwirth knot is a free group of rank $2g$, where $g$
1 KB (166 words) - 20:34, 11 April 2014
be a Markov chain with discrete time and set of states $ S $,
i, j \in S.
2 KB (309 words) - 08:26, 6 June 2020
$ S _ {p} $-
is an $ S _ {p} $-
3 KB (452 words) - 12:33, 6 January 2024
$$n\alpha+t\alpha+\alpha s+\sum _{ i }{ [(k_{ i }\alpha)l_{ i }+k_{ i }^\prime (\alpha l_{ i }^\prime
where $k,t,s,k_{i},k_{i}^{\prime},l_{i},l_{i}^{\prime}$ are arbitrary elements of $K$ an
3 KB (484 words) - 20:54, 28 November 2014
and the increments $ X ( t ) - X ( s ) $,
$ 0 \leq s < t $.
5 KB (662 words) - 16:09, 1 April 2020
These are [[multiplicative arithmetic function]]s with [[Dirichlet series]]
...{-s} = \prod_p \left({(1-p^{-s})(1-p^{k-s}) }\right)^{-1} = \zeta(s) \zeta(s-k)\ .
2 KB (268 words) - 19:41, 17 November 2023
...tomorphisms of the space such that: 1) $T^t(T^s(x))=T^{t+s}(x)$ for all $t,s\in\mathbf R$, $x\in M$; and 2) the mapping $M\times\mathbf R\to M$ taking $
956 bytes (174 words) - 16:21, 19 August 2014
If two surfaces $ S $
and $ S ^ {*} $
4 KB (660 words) - 19:40, 3 January 2021
corresponds a transformation $ S _ {t} : W \rightarrow W $
S _ {0} ( w) = w \ \textrm{ and } \ S _ {t+} s ( w) = S _ {t} ( S _ {s} ( w) ) .
3 KB (541 words) - 19:39, 5 June 2020
...fixed point (i.e., there is a point $ x \in X $ such that $ x \in f(x) $). S. Kakutani showed in [[#References|[1]]] that from his theorem, the [[Minima
S. Kakutani, “A generalization of Brouwer's fixed point theorem”, ''Duke Math. J.'', '''8''': 3 (1941), pp. 457–459
1 KB (195 words) - 16:18, 15 December 2016
...ry]]. A standard construction is a functor $T : \mathfrak{S} \to \mathfrak{S}$ equipped with natural transformations $\eta : \operatorname{Id} \to T$ an
...absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087140/s0871405.png" /></td> </tr></table>
2 KB (317 words) - 09:03, 10 April 2023
...on $f(z)$ of the complex variable $z$, then there exists at least one ray $S=\{z:\arg(z-a)=\theta_0\}$ issuing from $a$ such that in every angle
...e point $e^{i\theta_0}$ is called a Julia point for $f(z)$ if every chord $S$ with end point $e^{i\theta_0}$ is a Julia chord for $f(z)$. There exist me
2 KB (343 words) - 21:49, 22 December 2018
...adsorbent and the relative concentration $\theta=c/c_s$, where the index $s$ stands for the limit value at a constant temperature, is known as the adso
Langmuir's equation of mono-molecular adsorption has the form
3 KB (421 words) - 15:48, 4 November 2014
object]] in the category of algebraic varieties or schemes. If $ S $
is a [[Group scheme|group scheme]] over $ S $,
5 KB (854 words) - 10:51, 20 December 2019
...s are called properties. Two-place relations are called [[binary relation]]s, three-place relations are called ternary, etc.
If $R$ and $S$ are $n$-place relations on $A$, then the following subsets of $A^n$ will a
2 KB (284 words) - 19:27, 13 April 2024
is summable by the method of arithmetical averages to the sum $s$ if
$$\lim_{n\to\infty}\frac{s_0+\dots+s_n}{n+1}=s,$$
1 KB (200 words) - 14:17, 30 December 2018
...$ for a curve $\gamma$ with parametric representation $\mathbf r=\mathbf r(s)$ such that the arc length on the curve between two points $\mathbf r(s_1)$
582 bytes (89 words) - 16:54, 30 July 2014
...nant]], the same [[characteristic polynomial]], and the same [[eigenvalue]]s. It is often important to select a matrix similar to a given one but having
...to two different bases, the change of basis being expressed by the matrix $S$. The rank, determinant, trace, characteristic polynomial and so forth are
1 KB (191 words) - 18:12, 7 February 2017
$#C+1 = 11 : ~/encyclopedia/old_files/data/S083/S.0803250 Scalar curvature
The scalar curvature $ s ( p) $
2 KB (248 words) - 08:12, 6 June 2020
...my matrix]] and [[Monodromy operator|Monodromy operator]]. If $ \gamma ( s) $
then $ Y ( s) = X ( \gamma ( s) ) $
3 KB (379 words) - 08:01, 6 June 2020
to an $n$-th power. According to de Moivre's formula the modulus $\rho$ of the complex number is raised to that power an
De Moivre's formula can be used to express $ \cos n \phi $ and $ \sin n \phi $ in power
1 KB (241 words) - 10:03, 4 June 2013
S ( x) = {
...rightarrow + \infty } C ( x) = \lim\limits _ {x \rightarrow + \infty } S ( x) = {
5 KB (634 words) - 19:40, 5 June 2020
Page's theorem on the zeros of Dirichlet $L$-functions.
Let $L(s,\chi)$ be a [[Dirichlet L-function]], $s = \sigma + i t$, with $\chi$ a [[Dirichlet character]] modulo $d$, $d \ge 3
3 KB (508 words) - 15:13, 3 July 2020
f ( x) e ^ {\lambda S ( x) } d x ,
is a finite interval, $ S $
4 KB (635 words) - 22:15, 5 June 2020
$#C+1 = 24 : ~/encyclopedia/old_files/data/S084/S.0804930 Shnirel\AApman method
Shnirel'man's theorem 1): If $ \alpha , \beta $
3 KB (458 words) - 19:05, 20 November 2023
$#C+1 = 21 : ~/encyclopedia/old_files/data/S091/S.0901110 Summability, strong,
''of a complex sequence $ \{ S _ {n} \} $
4 KB (521 words) - 12:35, 6 January 2024
...is called enumerable relative to a given [[Formal system|formal system]] $S$ of arithmetic if it has the following property: There is a formula $F(x_1,
...All recursive predicates, and only they, are enumerability predicates in $S$.
2 KB (311 words) - 05:34, 12 October 2014
...function|distribution function]] of a life-time on the applied stress $ S $
F ( t \mid S ) = 1 - { \mathop{\rm exp} } \left [ - {
6 KB (853 words) - 21:12, 4 April 2020
...e same distribution (cf. [[Random variable|Random variable]]), and let $ S _ {0} = 0 $,
$ S _ {k} = X _ {1} + \dots + X _ {k} $,
4 KB (549 words) - 18:47, 5 April 2020
...xcess $\epsilon$ of a spherical triangle is proportional to its area $S$: $S=R^2\epsilon$, where $R$ is the radius of the sphere containing the spherica
...987) pp. 278</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1957)</TD></
867 bytes (128 words) - 11:56, 5 July 2014
...act that isometries of real Banach spaces are affine is due to
S. Ulam and
S. Mazur [[#References|[a1]]].
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">
S. Mazur,
S. Ulam, "Sur les transformations isométriques d'espaces vectoriels" ''C.
2 KB (277 words) - 06:49, 14 January 2017
...other words, a probabilistic automaton may be defined as a system $ (A, S, B, \phi , \psi ) $
where $ A, S, B $
5 KB (760 words) - 10:01, 25 April 2020
...llection $(G,B,N,S)$, where $G$ is a group, $B$ and $N$ are subgroups and $S$ is a subset of $N/(B\cap N)$, satisfying the following conditions:
3) the set $S$ generates the group $W = N/T$ and consists of elements of order 2;
4 KB (647 words) - 09:57, 30 January 2014
to the stable homotopy group of the spectrum of the sphere $ S ^ {0} $,
...of the Whitehead group is by the Hopf construction: A mapping $ \phi : S ^ {m} \rightarrow \mathop{\rm SO} ( q) $
9 KB (1,201 words) - 20:31, 16 January 2024
...predicate calculus, see [[Logical calculus|Logical calculus]]). Aristotle's syllogistics have been formalized in an adequate fashion by J. Lukasiewicz
...on Press (1951)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> S. Yanovskaya, "The logic of classes" , ''Philosophical Encyclopaedia'' , '
2 KB (287 words) - 17:04, 30 December 2018
such that, whenever $ S $
with $ \sup S \geq a $,
2 KB (322 words) - 17:46, 4 June 2020
label("$A$",A,S); label("$B$",B,S); label("$C$",C,S);
739 bytes (100 words) - 11:51, 13 December 2014
$$f(e_i,e_j)=f(e_{s+i},e_{s+j})=0,$$
$$f(e_i,e_{s+j})=\delta_{ij},$$
3 KB (511 words) - 13:16, 7 April 2023
...to a point, i.e. if there is a continuous map $\Gamma: [0,1]\times \mathbb S^1\to \Omega$ and an element $p\in \Omega$ such that
...$n$-dimensional tori $\underbrace{\mathbb S^1 \times \ldots \times \mathbb S^1}_n$ and the annuli $\{x\in \mathbb R^2 : r<|x|<R\}$ are not simply connec
2 KB (314 words) - 09:19, 27 December 2020
$#C+1 = 46 : ~/encyclopedia/old_files/data/S090/S.0900430 Stratification
is a locally finite collection of subspaces $ S _ {i} $
5 KB (665 words) - 08:23, 6 June 2020
...nt having an isometric imbedding into the Euclidean space $ \mathbf R ^ {s _ {n} } $
of dimension $ s _ {n} = n ( n + 1 ) / 2 $.
3 KB (425 words) - 22:14, 5 June 2020
...event depends on the trial number (the so-called Poisson scheme). Poisson's theorem states that: If in a sequence of independent trials an event $ A
Bernoulli's theorem follows from Poisson's theorem when $ p _ {1} = \dots = p _ {n} $.
4 KB (651 words) - 08:06, 6 June 2020
...gebraic point of view, a finite or infinite automata $ \mathfrak A = (A, S, B, \phi , \psi ) $
is a tribasic algebra, i.e. an algebra with three sets of elements $ A, S $
7 KB (1,008 words) - 18:48, 5 April 2020
S _ {p} ^ {n + p } ( c ) = \{ {x \in \mathbf R _ {p} ^ {n + p + 1 } } :
Thus, $ S _ {p} ^ {n + p } ( c ) $
5 KB (698 words) - 13:01, 31 January 2022
...fferential equation $Lu=f$ on a domain $D$ in $\mathbf R^n$ with boundary $S$ (cf. also [[Differential equation, partial, of the second order|Differenti
$$u(x)=\phi(x),\quad x\in S,$$
810 bytes (116 words) - 14:55, 16 October 2014
( x ^ {2} + p _ {s} x + q _ {s} ) ^ {\beta _ {s} } ,
$ j = 1, \dots, s $,
3 KB (482 words) - 15:56, 2 March 2022
...y-simple semi-group|completely-simple semi-group]] of idempotents; and 4) $S$ is isomorphic to a direct product $L\times R$, where $L$ is a left-singula
4 KB (534 words) - 16:11, 12 April 2014
The former case occurs if and only if $S$ is a [[semi-group with cancellation]].
...manner to non-commutative semi-groups [[#References|[a1]]]. A semi-group $S$ with an idempotent is Archimedean (right Archimedean) if and only if it ha
3 KB (413 words) - 09:17, 2 April 2023
split torus $ S \subset G $ ;
$ D = [ {Z _{G}} (S),\ {Z _{G}} (S)] $ .
1 KB (187 words) - 10:43, 17 December 2019
associated with the $ ( s \times n ) $-
the point evaluation at $ 0 \in \mathbf R ^ {s} $.
2 KB (339 words) - 06:29, 30 May 2020
...-one mappings (permutations, cf. [[Permutation|Permutation]]) of a set $ S $
onto itself, for which there exists a partition of $ S $
6 KB (896 words) - 07:41, 26 February 2022
''$S$ in a class of semi-groups $\Sigma$''
A [[homomorphism]] of $S$ into a semi-group from $\Sigma$ (in the case of an injective homomorphism
4 KB (637 words) - 19:07, 20 March 2018
A quiver $ Q = ( Q _ {0} , Q _ {1} , s, e) $
and two mappings $ s, e: Q _ {1} \rightarrow Q _ {0} $;
12 KB (1,755 words) - 19:50, 21 March 2023
...a chord of it. The area of a segment of a circle (a circular segment) is $S=r^2(\theta-\sin\theta)/2$, where $r$ is the radius of the circle and $r\the
...nt. The area $S$ of the curved surface of a segment of a ball is given by $S=2\pi Rh$.
1 KB (221 words) - 13:34, 9 April 2014
''of a smooth dynamical system $ \{ S ^ {t} \} $''
...ddle]]. More exactly, a hyperbolic set in a smooth dynamical system $ \{ S _ {t} \} $
5 KB (745 words) - 22:11, 5 June 2020
on the Borel sets of a metric space $ S $
\lim\limits _ { n } \int\limits _ { S } f dP _ {n} = \int\limits _ { S } f dP
6 KB (936 words) - 19:36, 5 June 2020
A logical system $ S $
called provable in $ S $.
1 KB (212 words) - 04:11, 6 June 2020
....g., [[#References|[a1]]], Thm. 6.4. There a simplified version of Hermite's original proof is given.
667 bytes (110 words) - 08:23, 28 April 2023
...Gamma ( 1 - b _ {j} + s ) \prod _ {j = n + 1 } ^ { p } \Gamma ( a _ {j} - s ) }
} x ^ {s} } {ds } ,
4 KB (590 words) - 11:05, 7 June 2020
$#C+1 = 22 : ~/encyclopedia/old_files/data/S110/S.1100190 Skorokhod theorem,
...a complete and separable [[Metric space|metric space]] $ ( S, {\mathcal S} ) $
3 KB (375 words) - 08:14, 6 June 2020
is summable at a point $x_0$ by the Lebesgue summation method to the sum $s$ if in some neighbourhood $(x_0-h,x_0+h)$ of this point the integrated seri
converges and its sum $F(x)$ has symmetric derivative at $x_0$ equal to $s$:
1 KB (243 words) - 15:20, 14 February 2020
$#C+1 = 23 : ~/encyclopedia/old_files/data/S110/S.1100050 Saxon\ANDHutner theorem
...as follows. If $ A ^ {r _ {1} } B ^ {s _ {1} } \dots A ^ {r _ {k} } B ^ {s _ {k} } $
3 KB (501 words) - 08:12, 6 June 2020
An $ s $-
i = 2 \dots s,
4 KB (508 words) - 19:57, 8 February 2024
...systems. The Appell equations in independent Lagrange coordinates $ q _ {s} $(
$ s =1 \dots n $)
4 KB (661 words) - 14:38, 10 March 2024
''of a semi-group $S$''
$$A_1\subset A_2\subset\dotsb\subset A_m=S\label{*}\tag{*}$$
2 KB (370 words) - 17:11, 14 February 2020
s ^ \prime ( x) = c( x) r( x),\ c ^ \prime ( x) = - s( x) r( x),
r( x) = s ^ {2} ( x) b _ {1} ( x) + s( x) c( x)( b _ {2} ( x)- a _ {1} ( x)) +
7 KB (1,087 words) - 08:04, 6 June 2020
Approximation of a continuous function $f$ defined on a set $M$ by functions $S$ from a given class of functions, where the measure of approximation is the
$$\rho(f,S)=\sup_{x\in M}|f(x)-S(x)|.$$
2 KB (230 words) - 08:42, 19 April 2014
[[Algebraic group|algebraic group]]. Let ${\rm Sch}/S$ be the category of
schemes over a ground scheme $S$; a
5 KB (831 words) - 21:59, 5 March 2012
be a [[Locally convex space|locally convex space]] and let $ \mathfrak S $
M( S, V) = \{ {f } : {f \in L( E, F ), f( S) \subset V } \}
5 KB (737 words) - 08:04, 6 June 2020
...dorff [[#References|[1]]]; it is defined as follows. A sequence $ s = \{ s _ {n} \} $
t = \delta s,\ \
3 KB (472 words) - 18:52, 29 October 2023
...receiver), is defined by a pair of measurable spaces $ ( {\mathcal Y} , S _ {\mathcal Y} ) $,
$ ( \widetilde {\mathcal Y} , S _ {\widetilde {\mathcal Y} } ) $(
10 KB (1,309 words) - 17:45, 4 June 2020
is a finite group scheme over a scheme $ ( G, {\mathcal O} _ {S} ) $,
is a finite flat quasi-coherent sheaf of algebras over $ {\mathcal O} _ {S} $.
5 KB (753 words) - 09:02, 8 April 2023
...circle $| x | < 1$. The set of these numbers is traditionally denoted by $S$. Every positive integer $n > 1$ is a Pisot number, but a more interesting
...\infty } \| \lambda \theta ^ { n } \| ^ { 2 } < \infty$, then $\theta \in S$ and $\lambda \in \mathbf{Q} ( \theta )$ [[#References|[a1]]].
5 KB (723 words) - 19:03, 23 January 2024
...to the sum $s$ by the summation method $P$, or is $P$-summable to the sum $s$, and this fact is denoted by one of the symbols
$$\sum_{k=0}^\infty u_k=s(P),\quad\lim s_n=s(P),\quad P-\lim s_n=s,$$
4 KB (617 words) - 17:36, 14 February 2020
$#C+1 = 46 : ~/encyclopedia/old_files/data/S086/S.0806250 Spearman coefficient of rank correlation
s and $ Y _ {i} $'
4 KB (638 words) - 09:15, 6 January 2024
...an ideal extension of a group by a [[Nil semi-group|nil semi-group]], or $S$ is a subdirect product of a group and a nil semi-group.
...ns of a unipotent semi-group by $B_2$; in this case, the decomposition of $S$ into torsion classes is not necessarily a [[Band of semi-groups|band of se
5 KB (775 words) - 17:22, 17 October 2014
\Gamma = < J, \{ S _ {i} \} _ {i \in J } ,\
is the set of players, $ S _ {i} $
5 KB (790 words) - 08:03, 6 June 2020
$$X(t)=U(t,t_0)X_0V(t,t_0)+\int\limits_{t_0}^tU(t,s)C(s)V(s,t)ds,$$
...ion of the matrix differential equation $X'=B(t)X$ with the condition $X(s,s)=I$.
4 KB (628 words) - 17:38, 14 February 2020
H_n^{(s)} = \sum_{k=1}^n \frac{1}{k^s} \ .
* Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, edd. Handbook of number theory I. Dordrecht: Springer-
2 KB (222 words) - 14:19, 12 November 2023
...and equipped with a [[Proper morphism|proper morphism]] $ X \rightarrow S $
onto some base scheme $ S $.
3 KB (393 words) - 08:10, 6 June 2020
has finite area $ S ( f, D _ {0} ) $
S ( f, D _ {0} ) \geq \
4 KB (530 words) - 08:25, 6 June 2020
...f $S^n$ that corresponds to it. Since the isometric mapping of $M^n$ into $S^n$ is not single-valued, Weierstrass coordinates are defined up to sign. A
...pp. 114–119</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1965) pp. 1
1 KB (197 words) - 12:50, 14 February 2020
#REDIRECT [[Hilbert theorem#Hilbert's Nullstellen Satz]]
56 bytes (6 words) - 21:55, 29 December 2015
\phi _ {s} ( x _ {1} \dots x _ {3N} ,\
s = 1 \dots m,\ \phi _ {s} ( x, \dot{x} , t) \in C ^ {1} ,
4 KB (611 words) - 08:03, 6 June 2020
Viète's theorem asserts that the following relations (Viète's formulas) hold:
but for positive roots only; the general form of Viète's theorem was established by A. Girard [[#References|[2]]].
3 KB (355 words) - 14:04, 20 March 2023
in the hyperbolic space $ {} ^ {3} S _ {5} $.
one gets the five-dimensional hyperbolic space $ {} ^ {3} S _ {5} $.
6 KB (820 words) - 08:06, 6 June 2020
\tilde x^i = A^i_s x^s + a^i \ .
150 bytes (25 words) - 18:23, 16 October 2016
...ry and sufficient that $L$ is not confined to some open half-sphere (Krein's theorem).
775 bytes (120 words) - 09:29, 27 June 2014
\sum_{n=1}^\infty f(n) n^{-s} = \prod_p \left({1 + f(p) p^{-s} + f(p^2) p^{-2s} + \cdots }\right) \ ,
\sum_{n=1}^\infty f(n) n^{-s} = \prod_p \left({1 + f(p) p^{-s} (1 - p^{-s})^{-1}} \right) \ ,
3 KB (419 words) - 20:15, 19 November 2017
can be summed by the Abel method ($A$-method) to the number $S$ if, for any real $x$, $0<x<1$, the series
\lim_{x\rightarrow 1-0} \sum_{k=0}^\infty a_k x^k = S.
2 KB (258 words) - 12:23, 10 January 2015
\sum _ {| s |, | t | \leq m } \
\int\limits _ { G } a _ {st} D ^ {s} u \overline{ {D ^ {t} u }}\; dx
2 KB (284 words) - 14:02, 14 April 2024
a (continuous) mapping $s:Y\to X$ such that $p\circ s={\rm id}_Y$.
704 bytes (110 words) - 22:27, 24 November 2013
$#C+1 = 84 : ~/encyclopedia/old_files/data/S090/S.0900450 Stratonovich integral
\int\limits _ { 0 } ^ { t } Y( s) \circ dX( s) = \int\limits Y( s) dX( s)+
10 KB (1,470 words) - 08:23, 6 June 2020
\langle R,S \ |\ R^3=S^3=(RS)^2 \rangle \ .
This group may be realised as the group of invertible [[Hurwitz number]]s:
764 bytes (115 words) - 20:46, 23 November 2023
...urhood of $P$. One introduces a coordinate system on the tangent plane to $S$ at $P$, taking $P$ as the coordinate origin, and the vectors $\mathbf r_u$
...d $L$, $M$ and $N$ are the coefficients of the second fundamental form of $S$ calculated at $P$. The Dupin indicatrix is: a) an ellipse if $P$ is an [[E
2 KB (414 words) - 12:02, 26 March 2023
$#C+1 = 66 : ~/encyclopedia/old_files/data/S091/S.0901080 Sum function of a function
F( s) = \sum _ { n= 1} ^ \infty f( n) n ^ {- s}
6 KB (821 words) - 01:34, 26 April 2022
and let $ S $
lines of $ S $.
3 KB (443 words) - 08:48, 26 March 2023
...ns similar to terms but representing a truth value instead of a type $s\in S$, see the entry [[Formula|formulas]].
...\bigcup_{s\in S} X_s$. Then the set $T_s(\Sigma,X)$ of <i>terms of sort $s$</i> is defined inductively as the smallest set containing all
5 KB (955 words) - 16:09, 28 January 2013
$#C+1 = 28 : ~/encyclopedia/old_files/data/S086/S.0806920 Square\AAhroot method
A = S ^ {*} DS,
5 KB (756 words) - 18:15, 25 June 2020
...roup|monogenic]], i.e. is generated by a single element. The [[idempotent]]s of the bicyclic semi-group form a chain, which is ordered with respect to t
...; a similar theorem is also valid for right multiplying elements, so that $S$ has left multiplying elements if and only if it also has right multiplying
2 KB (332 words) - 18:22, 16 January 2018
Conversely, if a submodule $ S \subset A $,
and a homomorphism $ \beta : S \rightarrow B/L $
2 KB (377 words) - 20:50, 4 April 2020
on a complex manifold $ {\mathcal S} $
...cally constant sheaf of finitely-generated Abelian groups on $ {\mathcal S} $,
6 KB (883 words) - 18:39, 2 January 2021
such that for any $ s \in T $
$ t \neq s $,
3 KB (399 words) - 19:40, 5 June 2020
Davenport's constant $ D ( G ) $
such that for any sequence $ S = ( g _ {1} \dots g _ {d} ) $
4 KB (600 words) - 17:32, 5 June 2020
\sin ( x s ) \phi ( s) d s = 0
of (1), that is, for equation (1) Fredholm's theorem that a homogeneous equation has finitely many linearly independent
3 KB (465 words) - 15:24, 28 February 2022
...their product $uv$ has derivatives of the same orders, and for $n=0,\dots,s$,
612 bytes (111 words) - 14:16, 16 September 2014
$ A _ {s} $,
$ s = 1 \dots k $,
7 KB (976 words) - 19:30, 11 January 2024
into the sphere $ S ^ {2} $;
\widetilde{F} = F \circ T: M \rightarrow S ^ {2} ,
2 KB (327 words) - 22:13, 5 June 2020
...fferential equation $Lu=f$ on a domain $D$ in $\mathbf R^n$ with boundary $S$ (cf. also [[Differential equation, partial, of the second order|Differenti
$$\frac{\partial u(x)}{\partial n}=\phi(x),\quad x\in S,$$
885 bytes (129 words) - 07:13, 23 August 2014
...s $x$ in the prime decomposition of which only prime numbers from the set $S$ appear.
...\{ x \in \mathbf{Q} : | x | _ { v } = 1 , \forall | \cdot | _ { v } \notin S \}$.
5 KB (751 words) - 13:28, 25 November 2023
A finite set $ S $
of $ S $(
4 KB (651 words) - 17:45, 4 June 2020
...5]]] (see also [[#References|[a1]]], p. 299). In this masterpiece, Riemann's primary goal was to estimate the number of primes less than a given number
...{ 1 } { n ^ { s } } = \prod _ { p } \frac { 1 } { 1 - \frac { 1 } { p ^ { s } } } \end{equation}
5 KB (666 words) - 18:50, 26 January 2024
...ble at all by this method or may turn out to be summable to a number $a\ne s$ (cf [[Summation methods]]).
604 bytes (107 words) - 20:41, 14 October 2014
of the symmetric group $ S _ {m} $.
on the [[conjugacy class]] of $ S _ {m} $
4 KB (519 words) - 16:18, 5 February 2022
let $ S _ {n} $
be its partial sums and let $ S $
3 KB (401 words) - 06:28, 30 May 2020
...heses]]) on the non-trivial zeros of the Riemann zeta-function $ \zeta ( s) $.
Let $ S $
4 KB (517 words) - 10:52, 21 March 2022
...n)$. A sequence $\{s_n\}$ is summable by the method $(W,p_n)$ to a number $S$ if
$$\frac{s_0p_n+s_1p_{n-1}+\dotsb+s_np_0}{p_0+\dotsb+p_n}\to S$$
2 KB (306 words) - 13:40, 14 February 2020
...]]], IV, 495. Given a real positive-definite $(n\times n)$-matrix $T$ and $s \in \mathbf{C}$, the Epstein zeta-function is defined by
\zeta(T;s) = \sum_{\mathbf{0} \ne g \in \mathbf{Z}^n} (g^\top T g)^{-s}
5 KB (734 words) - 20:12, 5 October 2017
$ S ^ {n} \star Y $
In particular, $ S ^ {n} \star S ^ {m} \approx S ^ {n + m + 1 } $.
3 KB (443 words) - 22:14, 5 June 2020
...*)\to C^\infty(B),\qquad (s,s^*)\mapsto \left< s,s^*\right>(b)=\left< s(b),s^*(b)\right>.
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...$s$ thus formed, each coalition of interests $K$ receives the pay-off $H_K(s)$.
...sts $K$ is not interested in the departure from its coalition strategy in $s$, even if some coalition of action $K'$ departs from its strategy. Equilibr
2 KB (301 words) - 16:35, 9 April 2016
x = s + \dots ,\ \
s ^ {2} + \dots ,\ \
1 KB (210 words) - 08:02, 6 June 2020
#REDIRECT [[Riemann theorem#Riemann's theorem on conformal mappings]]
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\frac{\partial ^ {2} u }{\partial s ^ {2} }
and distance $ s $
2 KB (306 words) - 08:25, 6 June 2020
in $ S ^ {n+2} $,
in $ S ^ {n+2} $.
4 KB (529 words) - 17:40, 5 April 2023
in which the arc element $ d s $
d s = F
6 KB (908 words) - 19:48, 12 January 2024
...nts of $S$, and where the domain and codomain of an element $s$ are $s^{-1}s$ and $ss^{-1}$, respectively; here the objects have a natural meet semi-lat
...D valign="top"> Ch. Ehresmann, "Structures locales et catégories ordonnés" , ''Oeuvres complètes et commentées'' , ''Supplément aux Cahiers de Top
3 KB (429 words) - 20:52, 16 March 2023
...ple and a nilpotent endomorphism that commute with each other: $ g = g _{s} + g _{n} $.
The endomorphisms $ g _{s} $
10 KB (1,474 words) - 02:51, 25 February 2022
...its definition being: $\mathrm{res} A = 2\mathrm{res}_{s=0} \tr(A\Delta^{-s})$, with $\Delta$ the [[Laplace operator]]. It satisfies the trace conditio
\mathrm{res} A = \int_{S^*M} \tr a_{-n}(x,\xi) d\xi
2 KB (395 words) - 18:13, 25 September 2017
s _ \tau = \sum _ {i = 1 } ^ { k } m _ {i} \Delta x _ {i} \
S _ \tau = \sum _ {i = 1 } ^ { k } M _ {i} \Delta x _ {i} $$
8 KB (1,150 words) - 11:22, 26 March 2023
...s theorem which is often referred to as "Burnside's Lemma" or "Burnside's Theorem" states that when a [[Finite group|finite group]] $G$ acts on a se
...the Cauchy–Frobenius theorem and also, on rare occasions, as Not Burnside's Lemma.
3 KB (516 words) - 22:29, 1 December 2014
$$s^2=c^2(t_2-t_1)^2-(x_2-x_1)^2-(y_2-y_1)^2-(z_2-z_1)^2,$$
...rval with $s^2>0$ is called a time-like space-time interval, and one with $s^2<0$ is called a space-like space-time interval.
1 KB (224 words) - 14:51, 30 December 2018
from its values on (a part of) the boundary $ S $
( Bu ) ( y) = \phi ( y) ,\ y \in S ,
928 bytes (133 words) - 08:03, 6 June 2020
...ew, in the sense that inside the model there is no enumeration of the set $
S$.
<table><TR><TD valign="top">[1]</TD> <TD valign="top">
S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951)</TD>
2 KB (258 words) - 14:04, 10 April 2014
...in the representation space of an irreducible unitary representations $D^{(s)}$ of the unitary-unimodular group $\mathrm{SU}(2)$. The dimension of the r
<TR><TD valign="top">[a2]</TD> <TD valign="top"> A.S. Wightman, "L'invariance dans la mécanique relativiste" , ''Relations de
2 KB (347 words) - 19:55, 12 January 2017
\frac{c _ {k} ( q n ) }{n ^ {s} }
\frac{c _ {k} ( q n ) }{k ^ {s} }
3 KB (502 words) - 14:27, 17 March 2023
If the kernel $ K ( s , x ) $
\overline{u}\; ( s) = \int\limits _ \alpha ^ \beta K ( s , x ) u ( x) dx
4 KB (526 words) - 22:12, 5 June 2020
$#C+1 = 100 : ~/encyclopedia/old_files/data/S091/S.0901670 Symmetric group
is denoted by $ S ( X) $.
7 KB (1,003 words) - 08:24, 6 June 2020
...( \mathfrak X \times \widetilde{\mathfrak X} , S _ {\mathfrak X } \times S _ {\widetilde{\mathfrak X} } ) $,
where $ ( \mathfrak X , S _ {\mathfrak X } ) $
4 KB (536 words) - 22:12, 5 June 2020
M = \left({ \begin{array}{cc} P & Q \\ R & S \end{array} }\right) \ ,
M/P = S - R P^{-1} Q \ .
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$#C+1 = 40 : ~/encyclopedia/old_files/data/S090/S.0900230 Stochastic process, renewable,
are the mean-square closed linear hulls of all the values $ \xi _ {s} $(
5 KB (659 words) - 08:23, 6 June 2020
...\alpha m :\Longleftrightarrow \alpha(n)=\alpha(m)$ for $n,m\in C_s$, $s\in S$ {{Cite|W90}}.
...e</i> resp. <i>co-semicomputable</i>, if $C$ (and $\equiv_s$ for all $s\in S$) is recursive resp. recursively enumerable resp. co-recursively enumerable
3 KB (404 words) - 09:05, 21 April 2013
...a category]]) and satisfying the following axiom: There exist [[coproduct]]s (sums) of arbitrary families of objects, and for each directed family of su
...as a ''localizing subcategory'' if it is closed with respect to [[colimit]]s and if, in an exact sequence
2 KB (366 words) - 19:42, 30 October 2016
...ctorial $s$-mappings preserve a pointwise-countable base, and a factorial $s$-mapping of a space with a pointwise-countable base onto a space of pointwi
756 bytes (124 words) - 22:05, 11 April 2014
$#C+1 = 45 : ~/encyclopedia/old_files/data/S086/S.0806820 Spline interpolation
...tisfy further conditions at the end points. E.g., for the cubic spline $ S _ {3} ( \Delta _ {n} , x ) $,
5 KB (751 words) - 08:22, 6 June 2020
...terms involve a [[Stieltjes integral]], i.e., $ds$ gets replaced by $dg(s)$ for appropriate functions $g$.
237 bytes (33 words) - 16:59, 10 January 2016
$#C+1 = 5 : ~/encyclopedia/old_files/data/S090/S.0900180 Stochastic point process with limited memory
in which the intervals $ s _ {i} = t _ {i+} 1 - t _ {i} $
822 bytes (110 words) - 08:23, 6 June 2020
==Brouwer's fixed-point theorem==
...fs of theorems on the existence of solutions of various equations. Brouwer's theorem can be generalized to infinite-dimensional topological vector space
4 KB (586 words) - 11:27, 17 March 2023
...cup_{A \in \mathcal{E}} A$ ($\mathcal{E}$ need not be a $\sigma$-ring and $S$ need not be in $\mathcal{E}$: cf. [[Ring of sets]]). Let a function $\gamm
Given a content $\gamma$ on $\mathcal{E}$ and any $A \subset S$, $A \neq \emptyset$, one defines
2 KB (378 words) - 15:57, 8 October 2017
...verges for all $n=1,2,\ldots,$ and if the sequence $\sigma_n$ has a limit $s$ for $n \rightarrow \infty$:
\lim_{n\rightarrow\infty} \sigma_n = s,
3 KB (519 words) - 22:33, 8 May 2012
$ f(f^{-1}(S)) = S $ for all subsets $S$ of the range $B$.
...compatible with the underlying structure are often called ''[[projection]]s''.
1 KB (230 words) - 21:19, 18 December 2014
The function $N(y,f)$ was defined by S. Banach [[#References|[1]]] (see also [[#References|[2]]]).
...(In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.S. Kronrod, "On functions of two variables" ''Uspekhi Mat. Nauk'' , '''5''
3 KB (485 words) - 12:32, 16 May 2015
...he theory of [[scheme]]s. To define the Picard scheme for an arbitrary $ S $-
one considers the relative Picard functor $ \mathop{\rm Pic}\nolimits _{X/S} $
7 KB (1,008 words) - 07:42, 20 March 2024
If $ S $
denotes the state space of the system and $ {T ( t ) } : S \rightarrow S $
5 KB (668 words) - 17:45, 4 June 2020
...e use in [[Commutative algebra|commutative algebra]]. One says that Hensel's lemma is valid for a local ring $ A $
Hensel's lemma is fulfilled, for example, for a complete local ring. Hensel's lemma makes it possible to reduce the solution of an algebraic equation ove
3 KB (386 words) - 22:10, 5 June 2020
$#C+1 = 13 : ~/encyclopedia/old_files/data/S087/S.0807120 Stable homotopy group,
'' $ k $-stable homotopy group $ \pi _ {k} ^ {s\star} ( X) $, of a topological space $ X $''
2 KB (275 words) - 16:00, 2 March 2022
$$\sum_{n=2}^\infty\left(\pi(n)-\pi(n-1)-\frac1{\ln n}\right)\frac{\ln^mn}{n^s}$$
has a finite limit as $s\to1+$.
3 KB (542 words) - 16:58, 14 February 2020
...h discrete time obeys the following arcsine law: Let $ S _ {0} = 0 \dots S _ {n} \dots $
S _ {n} = \sum _ {u = 1 } ^ { n }
4 KB (576 words) - 18:48, 5 April 2020
cf. [[#References|[2]]]) given on a metric space $ S $,
If $ S = \mathbf R ^ {n} $,
3 KB (380 words) - 22:15, 5 June 2020
$$V\phi(x)=\int\limits_a^x K(x,s)\phi(s)ds.$$
$$V\phi(x)=\int\limits_a^x K(x,s,\phi(s))ds.$$
2 KB (370 words) - 19:05, 27 April 2014
...o [[Knot theory|Knot theory]]). Alexander's theorem has its roots in Brunn's result (1897) that every knot has a projection with only one multiple point
946 bytes (146 words) - 08:32, 17 March 2023
$#C+1 = 15 : ~/encyclopedia/old_files/data/S084/S.0804970 Shot effect
$ W ( s) = 0 $,
3 KB (388 words) - 09:42, 20 July 2021
algebra $ S $
such that $ f ^ { - 1 } ( E) \in S $,
8 KB (1,244 words) - 08:05, 6 June 2020
label("$x_0$",(0.7,0),S);
label("$a$",(0.4,0),S);
1 KB (134 words) - 11:41, 13 December 2014
...in the unit of $G$ and if for any $x,y\in S(g)$ the intersection $S(x)\cap S(y)$ is non-empty.
2 KB (365 words) - 20:04, 12 July 2014
of outcomes such that a [[Domination|domination]] $ s \succ _ {K} c $
cannot hold for any outcomes $ s \in S $,
4 KB (665 words) - 01:22, 8 May 2022
$#C+1 = 40 : ~/encyclopedia/old_files/data/S084/S.0804030 Semi\AAhcontinuous summation method
are used to convert a given sequence $ \{ s _ {n} \} $
4 KB (580 words) - 08:21, 6 January 2024
A finite sum $ S $
S = \sum _ {x = 1 } ^ { P } e ^ {2 \pi iF ( x) } ,
4 KB (621 words) - 08:26, 6 June 2020
...o null-systems are absolute null-systems of the quasi-symplectic space $ S _ {P _ {2n-} 1 } ^ {2m-} 1 $.
Collineations of $ S _ {P _ {2n-} 1 } ^ {2m-} 1 $
3 KB (441 words) - 08:09, 6 June 2020
...$C ( S )$ consisting of functions that are constant on $S$. For $f \in C ( S \times T )$, one defines
..._ { s } f ( s , t ) + \frac { 1 } { 2 } \operatorname { inf } _ { s } f ( s , t ) \end{equation*}
8 KB (1,276 words) - 07:35, 8 February 2024
A table of the [[binomial coefficients]]. In this table, there are 1's at the lateral sides of an equilateral triangle and each of the remaining n
...Western literature the triangle occurs printed on the title page of Stifel's Arithmetica integra, published in 1544, almost a century before the birth o
3 KB (379 words) - 07:44, 4 November 2023
...the set of all points on the boundary of these components which belong to $S$. The definition quoted is an attempt to give a precise meaning to a concep
...cally, discrete analogues of this region exist. In particular, such a set $S$ then refers to a part of the complex plane; often a model problem like $\d
2 KB (315 words) - 16:58, 15 September 2014
One of the [[cardinal characteristic]]s of a [[topological space]] $X$. The least upper bound of the cardinalities
d(X) \le 2^{s(X)} \, ;
861 bytes (124 words) - 18:14, 14 November 2023
is the unit matrix), lying in a linear subspace $ \Pi ^ {s} \subset \mathbf R ^ {n} $
of dimension $ s < n $,
2 KB (319 words) - 22:17, 5 June 2020
\frac{\partial \psi _ {S} ( t) }{\partial t }
H \psi _ {S} ( t) = \
4 KB (623 words) - 11:53, 25 January 2022
...e symbols given by the [[Signature (Computer Science)|signature]] $\Sigma=(S,F)$ and a set $X$ of variables, whereas the semantics is defined by a [[Sig
...ch extends $v$, i.e. which fulfills $v^\ast_s (x) = v_s(x)$ for all $s \in S$, $x \in X$ {{Cite|ST99}} {{Cite|W90}}. This extension is called an <i>inte
6 KB (991 words) - 21:29, 24 March 2013
...gamma _ {r} ( x) \gamma _ {s} ( x) = ( {} _ { s } ^ {r+ s } ) \gamma _ {r+s} ( x) $;
4) $ \gamma _ {s} ( \gamma _ {r} ( x))= \epsilon _ {s,r} \gamma _ {rs} ( x) $;
4 KB (663 words) - 08:28, 20 January 2024
...order of Lamé's curve is $pq$ if $m>0$ and $2pq$ if $m<0$. If $m=1$, Lamé's curve is a straight line, if $m=2$ it is an ellipse, and if $m=2/3$ and $a=
931 bytes (151 words) - 18:29, 4 May 2023
$#C+1 = 49 : ~/encyclopedia/old_files/data/S090/S.0900710 Student distribution
{\mathsf P} \{ t _ {f} \leq x \} = S _ {f} ( x) =
5 KB (724 words) - 14:55, 7 June 2020
contains the class $ S _ {r} $
s ( z, t) \
5 KB (677 words) - 08:27, 6 June 2020
induces a mapping $ S _ {n} : ST _ {n} \rightarrow T _ {n + 1 } $,
where $ S $
3 KB (438 words) - 08:25, 6 June 2020
...not\in H$, $a\in H$ the intersection $S(g)\cap S(ga)$ is non-empty, where $S(x)$ is the minimal invariant sub-semi-group of $G$ containing $x$. A convex
2 KB (249 words) - 16:16, 12 April 2014
===Tolsa's solution===
...it is equal to zero if and only if the three points lie on one line. Tolsa's solution is:
2 KB (356 words) - 17:55, 24 September 2012
'' $ s $-
Each term of the table is indexed by $ s \geq 2 $
998 bytes (149 words) - 08:02, 6 June 2020
$ f^{-1}(f(S)) = S $ for all subsets $S$ of the domain $A$.
...e compatible with the underlying structure are often called ''[[embedding]]s''.
1 KB (234 words) - 12:11, 12 December 2013
...eprint (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I.S. Sokolnikoff, "Tensor analysis" , Wiley (1964)</TD></TR></table>
...and commuting with contraction. Regarding a tensor field of type $ ( r, s) $
11 KB (1,607 words) - 08:25, 6 June 2020
\begin{equation}\int\limits_0^x\frac{\phi(s)}{\sqrt{x-s}}ds=f(x),\label{1}\end{equation}
\begin{equation}\int\limits_a^x\frac{\phi(s)}{(x-s)^\alpha}ds=f(x),\quad a\leq x\leq b,\label{2}\end{equation}
4 KB (633 words) - 19:23, 22 December 2018
...case of \eqref{a1} where $A$ and $B$ are $(n\times n)$-matrices is due to S. Lie, and is simply known as the product formula for matrix exponentials.
$$\exp(it(A+B))=s-\lim_{n\to\infty}(\exp(itA/n)\exp(itB/n))^n,$$
2 KB (344 words) - 21:49, 3 December 2017
For any Riemann surface $ S $
...f a finite number of triangles in a [[Triangulation|triangulation]] of $ S $;
2 KB (348 words) - 19:20, 17 January 2024
$#C+1 = 44 : ~/encyclopedia/old_files/data/S086/S.0806730 Spherical trigonometry
\frac{\sin ( s- b) \sin ( s- c) }{\sin b \sin c }
6 KB (898 words) - 08:22, 6 June 2020
A region $ S $
there are at least two points of $ S $;
2 KB (222 words) - 08:01, 6 June 2020
...ty]$). It is usually assumed that the empty set is an element of $\mathcal{S}$ and that $\mu (\emptyset) =0$.
for every finite collection $\{E_i\}$ of ''disjoint'' elements of $\mathcal{S}$.
3 KB (459 words) - 14:45, 19 September 2012
\frac{\mu ( \alpha S ) }{\mu ( S) }
and $ S $
3 KB (413 words) - 08:01, 6 June 2020
( z - z _ {0} ) ^ {-s} [ \mathop{\rm ln} ( z - z _ {0} ) ] ^ {k}
where $ s $
940 bytes (141 words) - 16:10, 1 April 2020
...group scheme over a field of characteristic 0 is reduced is called Cartier's theorem, cf. also [[#References|[a1]]].
...e scheme $S$ is reduced but that $X \times_S T$ is not reduced (even with $S$ and $T$ reduced). The classical objects of study in [[Algebraic geometry|a
2 KB (310 words) - 07:40, 19 March 2023
\leq 1 } | S _ {n} ( f ) | ,\ \
S _ {n} ( f ) ( t) = \sum _ { k=1 } ^ { n }
1 KB (217 words) - 19:29, 28 February 2021
...e $(n-1)$-dimensional simplex $OA_1\ldots A_{i-1}A_{i+1}\ldots A_n$. Then $s^2=\sum_{i=1}^ns_i^2$.
1 KB (160 words) - 14:17, 14 August 2014
..._ { s } ) M$ is of finite length; in that case $y _ { 1 } , \dots , y _ { s }$ is called a system of parameters of $M$.
The ideal $( y _ { 1 } , \dots , y _ { s } )$ is called a parameter ideal.
3 KB (406 words) - 17:45, 1 July 2020
Binet's formula for the $n$-th number is given by W. Spickerman in [[#References|[a
...>[a1]</TD> <TD valign="top"> A. Shannon, "Tribonacci numbers and Pascal's pyramid" ''The Fibonacci Quart.'' , '''15''' : 3 (1977) pp. 268–275
1 KB (166 words) - 20:08, 21 March 2018
real s=sin(a);
if (n*abs(s)>=1) { continue; }
1 KB (203 words) - 11:38, 13 December 2014
under conditions when it is not the variables $ X ^ {(} s) $
\widetilde{X} {} _ {i} ^ {(} s) = \
5 KB (715 words) - 17:46, 4 June 2020
S _ {mn} = \sum _ {i = 1 } ^ { m }
...ar partial sums. They also form a double sequence. If this sequence $ \{ S _ {mn} \} $
7 KB (1,082 words) - 19:36, 5 June 2020
...rface of the body and that of the gas (fluid) outside the boundary layer, $S$ is the area of the surface, $l$ is the characteristic length, and $\lambda
1 KB (169 words) - 15:07, 10 April 2023
$#C+1 = 47 : ~/encyclopedia/old_files/data/S083/S.0803070 Saddle point method
F( \lambda ) = \int\limits _ \gamma f( z) e ^ {\lambda S( z) } dz,
5 KB (774 words) - 08:12, 6 June 2020
...= 0$ and $\int _ { 0 } ^ { t } I _ { ( 0 ) } ( Y _ { s } ) d \text{l} _ { s } = \text{l} _ { t }$.
...Y _ { t } = B _ { t } - \operatorname { min } _ { 0 \leq s \leq t } B _ { s } \bigwedge 0, \end{equation*}
4 KB (565 words) - 16:56, 1 July 2020
$#C+1 = 26 : ~/encyclopedia/old_files/data/S091/S.0901610 Symmetric derivative
\frac{\Phi ( S ( x; r)) }{| S ( x; r) | }
3 KB (376 words) - 08:24, 6 June 2020
...it contains all purely-logical laws of set-theoretic mathematics. K. Gödel's proof [[#References|[1]]] yields a means of constructing a countermodel (i.
...i.e. by arithmetical means. One obtains thus a constructive form of Gödel's completeness theorem (Bernays' lemma): For each predicate formula $ A $
4 KB (578 words) - 14:27, 19 March 2023
===Dirichlet's theorem in the theory of Diophantine approximations===
<TR><TD valign="top">[1]</TD> <TD valign="top"> J.W.S. Cassels, "An introduction to diophantine approximation" , Cambridge Univ
7 KB (1,065 words) - 14:05, 17 March 2020
and some semi-simple subalgebra $ S $:
A = N \oplus S,
3 KB (465 words) - 08:28, 6 June 2020
i.e. $ f \in S $),
Theorem 3) If $ f \in S $,
5 KB (681 words) - 17:31, 5 June 2020
\sum _ { s=0} ^ { k } \alpha _ {s} f ^ { 0 } ( t, x, u _ {s} ) dt,
\dot{x} = \sum _ { s=0} ^ { k } \alpha _ {s} f( t, x, u _ {s} ),
13 KB (1,801 words) - 19:58, 15 January 2024
...{\int\limits \int\limits } _ {\partial V } ( \mathbf n , \mathbf a ) d s ,
and $ s $
2 KB (251 words) - 08:28, 6 June 2020
and $ S(0, 0, -1) $
and $ S $
2 KB (322 words) - 10:58, 29 May 2020
...$T$ is substructure complete, then it is model complete. To see this, set $S=A$ for every model $A$ of $T$ which is a substructure of a model $B$ of $T$
...lign="top">[a2]</TD> <TD valign="top"> L. van den Dries, "Alfred Tarski's elimination theory for real closed fields" ''J. Symb. Logic'' , '''53'''
3 KB (435 words) - 09:40, 26 November 2016
and $ S = S ( X ) $
...} ( S ^ {p} ) \leq {\mathsf E} ( X ^ {*p } ) \leq C _ {p} {\mathsf E} ( S ^ {p} ) ,
3 KB (442 words) - 06:29, 30 May 2020
...orems|Selection theorems]]). There is also a widespread statement of König's theorem in terms of graphs: In a bipartite graph (cf. [[Graph, bipartite|Gr
König's theorem is often used in various combinatorial questions relating to select
2 KB (415 words) - 22:37, 26 October 2014
given on the hypersphere $ S $,
\int\limits _ { S } \xi
4 KB (575 words) - 08:00, 6 June 2020
\forall s \in [ \xi , x ] \Rightarrow G ( s ; \xi ) \geq 0 } \} ,
The theorem was obtained by S.A. Chaplygin in 1919.
3 KB (360 words) - 16:43, 4 June 2020
the ring of skew polynomials $ F( x, S) $
with an automorphism $ S: F \rightarrow F $(
5 KB (880 words) - 19:00, 9 January 2024
...wing property: All prime ideals $ \mathfrak P _ {1} \dots \mathfrak P _ {s} $
associated with the primary ideals $ \mathfrak Q _ {1} \dots \mathfrak Q _ {s} $
2 KB (246 words) - 19:37, 5 June 2020