$$x=\Delta s-\frac{k_1^2\Delta s^3}{6}+o(\Delta s^3),$$
$$y=\frac{k_1\Delta s^2}{2}+\frac{k_1'\Delta s^3}{6}+o(\Delta s^3),$$
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