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 $
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...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).
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\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