...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 \}$.
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$#C+1 = 101 : ~/encyclopedia/old_files/data/S083/S.0803000 \BMI S\EMI\AAhduality,
...ry homotopy and cohomotopy groups in the suspension category — for the $ S $-
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#REDIRECT [[Fermat's last theorem]]
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* Szpiro, L.; ''Séminaire sur les pinceaux des courbes de genre au moins deux'', Astérisqu
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...oup]] of residue classes modulo $p$ divides the order of the group. Fermat's little theorem was generalized by L. Euler to the case modulo an arbitrary
...is the [[Euler function|Euler function]]. Another generalization of Fermat's little theorem is the equation $x^q=x$, which is valid for all elements of
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Helly's theorem on the intersection of convex sets with a common point: Let $ K $
...eory of convex bodies (cf. [[Convex body|Convex body]]). Frequently, Helly's theorem figures in proofs of combinatorial propositions of the following ty
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Fermat's last theorem surely was mathematics' most celebrated and notorious open pro
Fermat's last theorem is the claim that $x^n+y^n=z^n$ has no solutions in non-zero i
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$x =~
s/\n*\<table .*?\<\/table\>\n*\
s*/\n\$\$•\$\$\n/sg;
$x =~
s/<img align[^>]*>/\$•\$/sg;
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''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)]]).
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...armonic function|harmonic function]] in a bounded simply-connected domain $S^+$ which, on the boundary $L$ of the domain, satisfies the condition
$$A(s)\frac{du}{dn}+B(s)\frac{du}{ds}+c(s)u=f(s),$$
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...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.
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...They are situated in the tangent plane to $S$ and have the same radius as $S$.
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$$\prod_p\left(1-\frac{1}{p^s}\right)^{-1},$$
...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
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...[S]$ consists precisely of those rows of $I$ corresponding to vertices in $S$.
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\def\S{\mathcal S} % subfamily
(in analogy to [[Helly's theorem]]) the smallest natural number $k$
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[[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
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...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
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...[#References|[1]]], and if $S$ is a normal scheme, $A$ is projective over $S$, [[#References|[2]]].
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...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
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''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]].
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..., method of]]). In Bateman's method, the degenerate kernel $ K _ {N} (x, s) $
K _ {N} (x, s) =
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\int\limits _ { a } ^ { b } \int\limits _ { a } ^ { b } K(x, s)
\phi (x) \overline{ {\psi (s) }}\; dx ds ,
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See [[S-duality|$S$-duality]].
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An integral equation in which the unknown function $\phi(s)$ occurs in a linear fashion:
$$A(x)\phi(x)+\int\limits_DK(x,s)\phi(s)ds=f(x),\quad x\in D.$$
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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|
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'' $ S $ with kernel $ A $''
with an epimorphism $ \phi : G \rightarrow S $
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Let $ S = \{ p _ {1} \dots p _ {n} \} $
...ight-line dual of the [[Voronoi diagram|Voronoi diagram]] generated by $ S $
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