a+b=b+a,\quad \text{ and } \quad ab=ba.
...e law of commutativity) if in the given algebraic system the identity $a*b=b*a$ holds.
427 bytes (66 words) - 22:17, 26 October 2014
|$A$||$B$||$A\downarrow B$
...junction $A\lor B$ is equivalent to $(A\downarrow B)\downarrow(A\downarrow B)$. This arrow was introduced by C. Peirce.
1 KB (194 words) - 12:27, 12 August 2014
$$A\supset(B\supset A),\quad(A\supset(B\supset C))\supset((A\supset B)\supset(A\supset C)),$$
$$A\&B\supset A,\quad A\&B\supset B,\quad A\supset(B\supset A\&B),$$
2 KB (246 words) - 08:00, 12 August 2014
''of an arithmetical fraction $a/b$''
...raction. The denominator of an algebraic fraction $A/B$ is the expression $B$ (see [[Fraction|Fraction]]).
340 bytes (49 words) - 09:49, 15 April 2014
...$a$ and $b$ and including the end points $a$ and $b$. It is denoted by $[a,b]$. See also [[Interval and segment|Interval and segment]].
238 bytes (42 words) - 13:25, 9 April 2014
''of an arithmetic fraction $a/b$''
...is used to make up the fraction. The numerator of an algebraic fraction $A/B$ is the expression $A$ (cf. [[Fraction|Fraction]]).
238 bytes (42 words) - 09:50, 15 April 2014
...of $a$ and $b$. If this number $n$ can be chosen independently of $a$ and $b$, then $G$ is called an Engel group of finite class $n$. The class of Engel
...pringer (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> B. Huppert, "Finite groups" , '''3''' , Springer (1982)</TD></TR></table>
1 KB (200 words) - 11:46, 26 April 2014
...ment $x\in P$ such that $a<x<b$: the [[Interval and segment|interval]] $[a,b]$ is an ''atomic'' or ''[[elementary interval]]''.
351 bytes (59 words) - 07:37, 24 January 2016
''$B$-function, Euler $B$-function, Euler integral of the first kind''
B(p,q) = \int_0^1 x^{p-1}(1-x)^{q-1} \rd x.
1,009 bytes (161 words) - 17:31, 11 November 2023
|$A$||$B$||$A|B$
...A|A$; the [[Disjunction|disjunction]] $A\lor B$ of two assertions $A$ and $B$ is expressed as:
2 KB (249 words) - 17:56, 29 November 2014
''of vectors $\mathbf{a},\mathbf{b},\mathbf{c}$''
...product]] of the vector $\mathbf{a}$ by the [[vector product]] of $\mathbf{b}$ and $\mathbf{c}$:
397 bytes (60 words) - 08:09, 14 January 2018
...cture(2)|Structure]]), it follows that $A$ is [[existentially closed]] in $B$.
309 bytes (49 words) - 19:56, 9 December 2016
...tent to a subset of the other), then there is a bijection between $A$ and $B$ (they are [[equipotent sets]]).
...{b}$ and $\mathfrak{b} \le \mathfrak{a}$ implies $\mathfrak{a} = \mathfrak{b}$.
1,006 bytes (155 words) - 19:36, 17 November 2023
$$x=a\cos t,\quad y=\pm\sqrt{b^2-a^2\sin^2t},\quad z=a\sin t,\quad b\geq a,$$
...e $a$ and $b$ are the radii of the cylinders and $t$ is a parameter. If $a=b$, the bicylindrics is a pair of congruent ellipses.
379 bytes (73 words) - 09:40, 5 August 2014
''of a group $A$ by a group $B$''
...h element $a \in A$ corresponds an automorphism $\alpha_a \in \mathrm{Aut}(B)$, which is [[conjugation]] by the element $a$:
2 KB (359 words) - 16:54, 23 November 2023
...\,b\,\omega$ or $(a,b)\omega$, but more commonly in infix form as $a \star b$ where $\star$ is the operator symbol. Many arithmetic, algebraic and logi
A binary operation is ''partial'' if it is not defined on all pairs $(a,b) \in A \times A$ (as for example division by zero is not defined). Propert
1 KB (193 words) - 19:47, 13 November 2016
...er product]] of vectors (where the product $(a,b)$ of two vectors $a$ and $b$ is, in general, a complex number) that satisfies the following axioms:
1) $(a,b)=\overline{(b,a)}$;
1 KB (173 words) - 05:40, 20 April 2023
...ddition of numbers is commutative: $a+b=b+a$, and associative: $(a+b)+c=a+(b+c)$. The operation inverse to addition is called subtraction.
739 bytes (126 words) - 20:47, 16 March 2014
Two subextensions $A$ and $B$ of an extension $\def\O{\Omega}\O$ of $k$ are called linearly disjoint if
subalgebra generated by $A$ and $B$ in $\O$ is (isomorphic to) the
1 KB (259 words) - 22:07, 5 March 2012
...out remainder. In other words, a divisor of the integer $a$ is an integer $b$ such that, for a certain integer $c$, the equality $a=bc$ holds. A ''prop
A divisor of a polynomial $A(x)$ is a polynomial $B(x)$ that divides $A(x)$ without remainder (cf. [[Division|Division]]).
1 KB (209 words) - 08:06, 26 November 2023
Linear operators $B$ and $T$, of which $T$ is of general type and $B$ is bounded, and which are such that
...rator|Extension of an operator]]). The commutation relation is denoted by $B\cup T$ and satisfies the following rules:
1 KB (259 words) - 15:29, 14 February 2020
''of two elements $a$ and $b$ in a group with multiple operators''
$$-a-b+a+b.$$
515 bytes (87 words) - 14:01, 10 April 2014
...is called the consequence. The precise meaning of the expression $A\supset B$ differs in the classical, constructive and other approaches to the semanti
!$A$||$B$||$A\supset B$
1 KB (166 words) - 16:47, 30 December 2018
''of three elements $a,b,c$ of a ring $R$''
It is denoted by $(a,b,c)$.
125 bytes (27 words) - 13:41, 10 April 2014
...is an [[incidence system]] on a set $V$ of $v$ points and a family $B$ of $b$ blocks: subsets of points, each of size $k$. It is required that each poi
b \ge v \ .
563 bytes (91 words) - 19:38, 4 January 2016
...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
...ven two sets $A$ and $B$, their symmetric difference, denoted by $A \Delta B$, is given by
...\setminus A) = (A \cup B) \setminus (A \cap B) = (A \cap B') \cup (A' \cap B)
2 KB (273 words) - 08:47, 29 April 2023
...it is defined as 1 if $a = \pm 1$ and 0 otherwise. For $b \neq 0$, write $b$ as a product $\prod_i p_i$ where the $p_i$ are primes, not necessarily dis
\left({\frac{a}{b}}\right) = \prod_i \left({\frac{a}{p_i}}\right)
857 bytes (137 words) - 16:43, 23 November 2023
$$\frac{a-b}{a+b}=\frac{\tan\frac 12(A-B)}{\tan\frac 12(A+B)}.$$
759 bytes (117 words) - 18:13, 1 June 2023
of a [[Riemann surface|Riemann surface]] to which a given domain $ B $
\iint_ { B } | F ^ { \prime } ( z) | ^ {2} d \sigma
2 KB (311 words) - 20:36, 22 December 2020
''of a vector $a$ by a vector $b$ in $\mathbb{R}^3$''
The [[vector]] $c$, denoted by the symbol $a\times b$ or $[a,b]$, satisfying the following requirements:
1 KB (206 words) - 05:53, 15 April 2023
...called the lower class, while the set $B$ is called the upper class of $A|B$. Dedekind cuts of the set of rational numbers are used in the construction
...B$: a Dedekind cut is a cut in which either $A$ has a maximal element or $B$ has a minimal element. A [[continuous set]] is a totally ordered set in w
1 KB (249 words) - 20:56, 28 September 2016
...[[Fibration|fibration]] $ f ^ { * } ( \pi ) : X ^ \prime \rightarrow B ^ \prime $
induced by the mapping $ f : B ^ \prime \rightarrow B $
3 KB (430 words) - 01:57, 21 January 2022
...epresentations of the natural numbers with respect to base $b$. Thus for $b=2$ the word begins
This is a [[disjunctive word]], that is, contains each finite sequence over $B$ as a factor infinitely often.
997 bytes (135 words) - 19:33, 6 December 2023
$$a-b=c-d,$$
where $a,b,c,d$ are given numbers. An arithmetic proportion is also called a differenc
151 bytes (26 words) - 09:19, 13 April 2014
...,b)=1$. If $a$ and $b$ are coprime, there exist numbers $u$ and $v$, $|u|<|b|$, $|v|<|a|$, such that $au+bv=1$.
667 bytes (98 words) - 18:57, 18 October 2014
''skew product, $ \mathbf a \lor \mathbf b $,
and $ \mathbf b $''
925 bytes (135 words) - 08:08, 6 June 2020
of the direct sum $ A \oplus B $
and $ B $
2 KB (377 words) - 20:50, 4 April 2020
A \prod _ {S} B &\ \mathop \rightarrow \limits ^ { {p _ A}} \ & A \\
p _ {B} \downarrow \ &{} &\downarrow \ \alpha \\
2 KB (209 words) - 11:42, 8 February 2020
...elements of $M$ and such that for any pair $a,b \in M$ the subgroup $H_{a,b}$ is non-trivial, where
H_{a,b} = \{ h \in G : h(a)=a\,,\ h(b)=b \} \ ;
1 KB (166 words) - 19:15, 22 October 2017
is everywhere dense in a set $ B $
if every non-empty portion of $ B $
939 bytes (161 words) - 08:07, 6 June 2020
...\mathfrak{b}$, $\mathfrak{a} = \mathfrak{b}$, $\mathfrak{a} > \mathfrak{b}$ holds. This asserts that of any two sets, one may be put into one-to-one
720 bytes (101 words) - 09:59, 21 January 2021
Z B ( X , Y ) = B
( \nabla _ {Z} X , Y ) + B
3 KB (368 words) - 16:09, 1 April 2020
''$A$ and $B$ over a ring $R$''
...sequence of matrices each of which is either a) a [[permutation matrix]]; b) an [[elementary matrix]]; c) an invertible [[diagonal matrix]].
703 bytes (117 words) - 20:29, 18 November 2016
and $ b $
a \leq b \leq a ^ {n} ,\ &b \leq a \leq b ^ {n} ,\ \\
2 KB (231 words) - 18:48, 5 April 2020
...of matrices is an [[equivalence relation]]. Congruence arises when $A$, $B$ represent a [[bilinear form]] or [[quadratic form]] with respect to differ
504 bytes (75 words) - 18:11, 14 November 2023
...r, the dual bundle is a vector bundle $\pi^*:E^*\to B$ over the same base $B$ with the fiber $F^*$ [[bundle#dual|dual]] to the fiber $F$.
...o C^\infty(B),\qquad (s,s^*)\mapsto \left< s,s^*\right>(b)=\left< s(b),s^*(b)\right>.
585 bytes (112 words) - 13:20, 20 May 2012
A subset consisting of two elements $a\leq b$ such that there are no other elements in the [[partially ordered set]] bet
$$a\leq x\leq b\Rightarrow a=x\text{ or }a=b.$$
641 bytes (100 words) - 07:37, 24 January 2016
is defined on a bounded or unbounded interval $ [ a, b) $,
$ - \infty \leq a \leq b \leq \infty $,
1 KB (167 words) - 19:36, 5 June 2020
1) If $a/b$ and $a'/b'$ are two consecutive terms of the Farey series of order $n$, then
2) If $a/b$, $a'/b'$, $a''/b''$ are three consecutive terms of the Farey series of order $n$, then the [
1 KB (206 words) - 11:54, 2 January 2021
Quadruples $ ( X; A, B, x _ {0} ) $,
and $ B $
3 KB (461 words) - 08:26, 6 June 2020
...A \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
...It consists of the set $\{a,b\}$ with open sets $\{ \emptyset, \{a\}, \{a,b\} \}$.
331 bytes (43 words) - 19:39, 17 November 2023
and $ B $,
and $ B $
2 KB (350 words) - 22:13, 5 June 2020
...B}|f(x)-f_B|^{p^*}\,dx\Bigr)^{\frac{1}{p^*}} \leqslant C\Bigl(\int\limits_{B}|\nabla f(x)|^{p}\,dx\Bigr)^{\frac{1}{p}}
...ant $C$ depends only on $n$ and $p$. Here $f_B = \frac{1}{|B|}\int\limits_{B}f\,dx$.
1 KB (211 words) - 15:44, 1 September 2020
...es$ 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 ${}_R M_E$, whe
The centre of an $(R,R)$-bimodule (also called an $R$-bimodule) $B$ is defined to be the set
2 KB (277 words) - 17:01, 23 November 2023
A mapping $ f: ( X, A) \rightarrow ( Y, B) $
into another $ ( Y, B) $
2 KB (282 words) - 16:43, 4 June 2020
$$A\equiv B\Leftrightarrow((A\supset B)\mathbin{\&}(B\supset A)).\tag{1}$$
$$A\mathbin{\&}B\equiv\neg(\neg A\lor\neg B)\equiv\neg(A\supset\neg B),\tag{2}$$
2 KB (311 words) - 15:56, 14 February 2020
pair B=(0,0);
draw( A--B--E--cycle,currentpen+1.5 );
739 bytes (100 words) - 11:51, 13 December 2014
does not pass through any of the vertices of a triangle $ A B C $
points: the side $ A B $
2 KB (347 words) - 10:08, 4 June 2020
...information. By a deletion of a letter in a word $ \beta = b _ {1} \dots b _ {n} $
over some alphabet $ B $
3 KB (507 words) - 13:23, 8 February 2020
...n)$ matrices then the sum $A+B$ is also an $(m \times n)$ matrix with $(A+B)_{ij} = A_{ij} + B_{ij}$ for $i=1,\ldots,m$ and $j = 1,\ldots n$.
334 bytes (63 words) - 20:52, 27 October 2016
...a divisor of $a$. The divisibility of $a$ by $b$ is denoted by the symbol $b | a$.
b | a \ \text{and}\ c | b \Rightarrow c | a \ ;
2 KB (396 words) - 18:39, 25 September 2017
$#C+1 = 14 : ~/encyclopedia/old_files/data/B016/B.0106190 Bifunctor
A mapping $ T: \mathfrak A \times \mathfrak B \rightarrow \mathfrak C $,
2 KB (314 words) - 19:02, 29 March 2024
$$c^2=a^2+b^2-2ab\cos C.$$
...a,b,c$ are the sides of the triangle and $C$ is the angle between $a$ and $b$.
311 bytes (66 words) - 14:25, 19 March 2014
...\otimes (b \otimes c) \cong (a \otimes b) \otimes c$ is natural for all $a,b,c \in \mathcal{C}$ and the diagram
...s b) \otimes (c \otimes d) & \stackrel{\alpha}{\rightarrow} & ((a \otimes b) \otimes c) \otimes d \\
4 KB (612 words) - 14:59, 6 April 2023
$$||B\& C||=T \iff ||B||=T \text{ and } ||C||=T,$$
$$||B \vee C||=T \iff ||B||=T \text{ or } ||C||=T,$$
3 KB (439 words) - 22:34, 16 June 2014
...to the additive group of $B$ and for which $(ab)\phi=b\phi\cdot a\phi$ ($a,b\in A$).
624 bytes (103 words) - 19:11, 24 November 2023
...is a metric space with a metric $d$, $V(g,E_a^b)$ variation of $g$ on $E_a^b$.
(b) $V(f,{E_x}^-)=x+c$, $x \in E$, where $c=-\inf(E)$;
2 KB (413 words) - 17:07, 21 April 2016
...difference'' relation $a\sim b$ if and only if $a\leq b$ and $b\leq a$, $a,b\in M$, is an [[equivalence]] on $M$. The pre-order $\leq$ induces an [[orde
...ed from a real-valued utility function $u$ with $a \leq b$ if $u(a) \leq u(b)$.
1 KB (162 words) - 06:54, 9 April 2023
...the left and the right) by the smaller, that is, $a<b$ implies that $ax=ya=b$ for some $x,y\in H$. The positive cone of any partially ordered group (cf.
671 bytes (106 words) - 09:15, 2 April 2023
\sum _ {k = 1 } ^ \infty b _ {k} = B
\frac{a _ {n} }{b _ {n} }
1 KB (187 words) - 11:30, 16 April 2023
and $ B $
are events and $ {\mathsf P} ( B) > 0 $,
4 KB (559 words) - 17:46, 4 June 2020
...ical triangles (see [[Spherical geometry|Spherical geometry]]). Let $ A, B, C $
be the angles and let $ a, b, c $
6 KB (898 words) - 08:22, 6 June 2020
...$ but with multiplicity not counted: so $C(\{a,a,b\}) = C(\{a,b,b\}) = \{a,b\}$. A multiset over a set $C$ is a multiset for which $C$ is a carrier.
2 KB (265 words) - 19:01, 9 November 2023
...ivalence relation $A \equiv_{\mathrm{T}} B$ if $A \le_{\mathrm{T}} B$ and $B \le_{\mathrm{T}} A$ is ''Turing equivalence'' and the equivalence classes a
1 KB (177 words) - 19:32, 17 November 2023
In other words, a function $ f : A \to B $ from a set $A$ to a set $B$ is
...)=B $, i.e., for each $ b \in B $ there is an $ a \in A $ such that $ f(a)=b $.
1 KB (230 words) - 21:19, 18 December 2014
...^2-b^2$, $y=2ab$, $z=a^2+b^2$, where $a$ and $b$ are positive integers $(a>b)$. The Pythagorean numbers can be interpreted as the sides of a right-angle
761 bytes (116 words) - 07:30, 10 December 2016
...e set $A$) with respect to the set of all formulas in the theory (the set $B$).
922 bytes (149 words) - 10:24, 30 August 2014
...pr}_1 : A \times B \rightarrow A$, $\mathrm{pr}_2 : A \times B \rightarrow B$;
...: C \times A \rightarrow B$ there is a unique arrow $[f] : C \rightarrow A^B$ with $\mathrm{ev}\circ ([f]\times \mathrm{id}_A) = f$.
2 KB (374 words) - 20:31, 27 December 2017
$$(A\supset B)\supset(\neg B\supset\neg A).$$
279 bytes (46 words) - 13:49, 29 April 2014
\frac{A \ A \supset B }{B}
and $ B $
2 KB (308 words) - 08:01, 6 June 2020
...ent if they are fibrewise homotopic. No method of consistently defining $( B _ { n } , \phi _ { n } )$-structures for equivalent fibrations exists, beca
...old $M ^ { n }$ with a fixed $( B , \phi )$-structure on it is called a $( B , \phi )$-manifold.
3 KB (457 words) - 17:02, 1 July 2020
...bject variable, then $(\neg A)$, $(A\mathbin\&B)$, $(A\lor B)$, $(A\supset B)$, $(\forall y\ A)$, $(\exists y\ A)$ are formulas.
2 KB (252 words) - 16:42, 30 December 2018
...cal{P}(Y) \rightarrow \mathcal{P}(X)$ defined for $A \in \mathcal{P}(X)$, $B \in \mathcal{P}(Y)$ by
f^\dashv(B) = \{ x \in X : \exists b \in B\,,\, b = f(x) \} \ .
1 KB (194 words) - 16:56, 25 November 2023
and $ B $
and $ B $
2 KB (345 words) - 17:45, 4 June 2020
...b]$ such that $\phi_2(a)=0$, $\phi_2(b)=1$. If $n>2$, then one divides $[a,b]$ into $n-2$ parts by the points $w_1,\dots,w_{n-1}$ and one chooses the in
In the case when $a=0$, $b=1$, and $\{w_n\}$ is the sequence of all dyadic rational points in $[0,1]$,
3 KB (436 words) - 09:23, 6 September 2014
$#C+1 = 20 : ~/encyclopedia/old_files/data/B015/B.0105020 Baer multiplication
and $ B $
3 KB (321 words) - 10:26, 27 April 2020
...lation, $a R b$ for all $a,b \in A$, the equality relation, $a R b$ for $a=b \in A$ and the empty (nil) relation are transitive.
...d as $a R^* b$ if there exists a finite chain $a = a_0, a_1, \ldots, a_n = b$ such that for each $i=1,\ldots,n$ we have $a_{i-1} R a_i$.
1 KB (245 words) - 19:34, 17 November 2023
...nd replacing each with its negation) from a given theorem: "If A, then B".
474 bytes (75 words) - 18:01, 30 July 2014
and $ \mathbf b $
\frac{( \mathbf a , \mathbf b ) }{|
2 KB (283 words) - 08:56, 8 April 2023
1) for all $ a,b,c \in A $,
a \cle b \Rightarrow a + c \cle b + c,
5 KB (833 words) - 22:15, 5 June 2020
$$\frac{A\supset B,B\supset C}{A\supset C}$$
$$A\supset B,B\supset C\vdash A\supset C.$$
854 bytes (130 words) - 19:13, 17 October 2014
S = \{ {A _ \sigma , B _ \sigma } : {\sigma \in \Sigma
$ H \subset B _ \sigma $
2 KB (295 words) - 11:42, 12 January 2021
...ntable in the form $a+b\sqrt{d}$, where $a$ and $b$ are rational numbers, $b\ne 0$, and $d$ is an integer which is not a perfect square. A real number $
683 bytes (103 words) - 20:31, 1 October 2016
* $A,B\in \mathcal{A}\Rightarrow A\cup B\in\mathcal{A}$;
* $A,B\in \mathcal{A}\Rightarrow A\cap B\in\mathcal{A}$
789 bytes (133 words) - 18:36, 25 November 2012
...tient module $X/A$, that is, an extension of the module $A$ by the module $B$ is an [[exact sequence]]
0 \rightarrow A \rightarrow X \rightarrow B \rightarrow 0 \ .
2 KB (354 words) - 21:55, 30 October 2016
...\to B $, where $ \pi = \operatorname{id} $ and $ X $ is identified with $ B $; and the fibre space over a point, where $ X $ is identified with a (uniq
...inuous mapping $ s: B \to X $ such that $ \pi \circ s = \operatorname{id}_{B} $.
5 KB (754 words) - 01:34, 10 December 2016
...In this case one says that the element $a$ is incident with $B$, or that $B$ is incident with $a$. The concept of an incidence system is introduced wit
...ubsets of $A$ is taken for $\mathfrak{B}$; then $a\,I\,B$ is simply $a \in B$.
3 KB (488 words) - 19:37, 7 November 2023
\exp(A+B)=\lim_{n\to\infty}(\exp(A/n)\exp(B/n))^n.\label{a1}
...algebra over $\mathbf Q$ in the variables $A$ and $B$, where both $A$ and $B$ are given degree $1$.
2 KB (344 words) - 21:49, 3 December 2017
...onical mapping $a/\theta \mapsto \phi(a)$, where $a/\theta = \{ b \in A : (b,a) \in \theta \}$, is an isomorphism of the quotient system $A/\theta$ onto
764 bytes (124 words) - 07:39, 13 November 2016
...peration consists in addition of some (interior) point $v$ on the line $(a,b)$; this point then becomes a new vertex. A graph $G'$ is called a subdivisi
796 bytes (134 words) - 17:22, 18 October 2014
a _ {k} \dots a _ {0} . b _ {1} \dots b _ {l} ,
where $ 0 \leq a _ {i} , b _ {j} < 10 $
2 KB (331 words) - 17:32, 5 June 2020
...nts and a statement $A$ imply both a statement $B$ and the statement $\neg B$, then $\Gamma$ implies $\neg A$. The rule of reductio ad absurdum can, e.g
\frac{\Gamma, A \rightarrow B\,;\ \Gamma,A \rightarrow \neg B}{\Gamma \rightarrow \neg A}
802 bytes (131 words) - 18:28, 17 September 2017
A formula for calculating an integral over a finite interval $[a,b]$:
where $h=(b-a)/N$ and $\alpha\in[a,a+h]$. Its algebraic degree of accuracy is 1 if $\al
2 KB (295 words) - 17:35, 24 March 2018
...sure pay-off. The minimax principle holds in such a game $\Gamma=\langle A,B,H\rangle$ if the equality
$$v=\max_{a\in A}\min_{b\in B}H(a,b)=\min_{b\in B}\max_{a\in A}H(a,b)\label{*}\tag{*}$$
2 KB (344 words) - 15:57, 14 February 2020
...f probability theory: $\mu(A\cap B)=\mu(A)\mu(B)$ for $A\in B(\xi)$, $B\in B(\eta)$. Under these conditions, if a measurable partition that is a refinem
2 KB (251 words) - 18:07, 3 August 2014
by a group $ B $
as a normal subgroup and such that $ G/A \cong B $,
6 KB (917 words) - 20:29, 17 January 2024
and $ N _ {1} ( b , \sigma ^ {2} ) $,
$ b $,
2 KB (319 words) - 22:17, 5 June 2020
...ls. It was described in geometrical form in Euclid's Elements (3rd century B.C.).
...ainder $b_1$ is either 0 or a positive integer less than $b$, $0 \le b_1 < b$. Successive divisions are performed:
2 KB (351 words) - 20:40, 16 November 2023
...cdots + b_k$, $k=1,\ldots,n$, are bounded by a number $B$, i.e. $|B_k| \le B$, and if either $a_i \ge a_{i+1}$ or $a_i \le a_{i+1}$, $i = 1,2,\ldots,n-1
\left\vert{ \sum_{k=1}^n a_k b_k }\right\vert \le B(|a_1| + 2|a_n|)
651 bytes (126 words) - 12:13, 9 November 2014
...en points: If $x, y \in A$ and $f \left({x}\right), f \left({y}\right) \in B$, then
...sometric to each other. Isometric spaces are homeomorphic. If in addition $B$ is the same as $A$, then the isometric mapping is said to be an isometric
2 KB (277 words) - 06:49, 14 January 2017
...s $\def\g{\gamma}\g:X\to A$, $\def\d{\delta}\d:X\to B$ for which $\g\a =\d\b$ there exists a unique morphism $\xi:X\to D$ such that $\xi\phi = \g$, $\xi
\begin{array}{ccc} D & \ra{\psi} & B\\ \da{\phi} & & \da{\b}\\A&\ra{\a}& C\end{array}$$
3 KB (575 words) - 10:30, 23 November 2013
...eighbourhoods; here, $[A]$ and $[B]$ are the closures of the sets $A$ and $B$, while $\emptyset$ is the empty set. Totally-normal spaces and only such s
1 KB (199 words) - 16:35, 1 May 2014
...$ and $l'$, then the points of intersection of $AB'$ and $A'B$, $BC'$ and $B'C$, $AC'$ and $A'C$ are collinear.
1,003 bytes (158 words) - 12:52, 10 August 2014
$$\int\limits_a^bf(x)dx\int\limits_a^bg(x)dx\leq(b-a)\int\limits_a^bf(x)g(x)dx,$$
where $f$ and $g$ are either both increasing or both decreasing on $[a,b]$.
690 bytes (132 words) - 12:59, 14 February 2020
of elements of a [[Banach space|Banach space]] $ B $
whose norms are all equal to one, $ \| x _ {i} \| _ {B} = 1 $.
1 KB (216 words) - 08:03, 6 June 2020
...cal{P}S \rightarrow \mathcal{P}S$ is a ''closure operation'' if for all $A,B \in \mathcal{P}S$:
K3) $K(A) \subseteq K(A\cup B)$;
2 KB (285 words) - 17:42, 13 October 2023
...[[Measurable space|Measurable space]]) $ ( \mathfrak X _ {k} , \mathfrak B _ {k} ) $.
...ariables is the function $ P _ {X _ {1} \dots X _ {n} } ( B _ {1} \dots B _ {n} ) $
2 KB (319 words) - 22:14, 5 June 2020
An algebraic equation of the form $ax^n+b=0$, where $a$ and $b$ are [[complex number]]s, with $ab\neq0$. Two-term equations have $n$ disti
...term equation in the complex plane are located on the circle with radius $|b/a|^{1/n}$ and centre at the coordinate origin, at the vertices of the inscr
524 bytes (94 words) - 17:30, 23 December 2014
A subset containing with any two elements $a$ and $b$ the entire interval $[a,b]$ (cf. [[Interval and segment|Interval and segment]]).
...a partially ordered set is convex if $a\leq b\leq c$ and $a,c\in A$ imply $b\in A$.
755 bytes (123 words) - 18:15, 16 October 2014
...tisfying the following requirement: For any two elements $a$ and $b$ with $b\neq0$ one can find elements $q$ and $r$ such that
where either $r=0$ or $n(r)<n(b)$.
1 KB (169 words) - 06:52, 18 October 2014
if $a\leq b$, then $a\phi\geq b\phi$;
if $a'\leq b'$, then $a'\psi\geq b'\psi$;
1 KB (219 words) - 20:00, 23 December 2016
...$a,b \in X$, there are only finitely many $x \in X$ such that $a \le x \le b$.
414 bytes (65 words) - 19:36, 7 November 2023
...pectively, by the equations $ {x ^ {2} } / {a ^ {2} } + {y ^ {2} } / {b ^ {2} } = 1 $,
and $ {x ^ {2} } / {a ^ {2} } - {y ^ {2} } / {b ^ {2} } = 1 $,
2 KB (317 words) - 19:36, 5 June 2020
''of two propositions (formulas) $A$ and $B$''
...lent if for each admissible choice of values of the parameters of $A$ and $B$ either both are true or both are false. For example, equivalence of equati
912 bytes (130 words) - 18:24, 16 April 2014
...$ is the [[least upper bound]] of $A$ and the [[greatest lower bound]] of $B$.
632 bytes (103 words) - 20:44, 23 November 2023
...,
B}\right)$ or, using arrows, $\alpha : A \to
B$, $A \xrightarrow{\alpha}
B$, etc.
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">
B. Mitchell, "Theory of categories" , Acad. Press (1965)</TD></TR></table>
2 KB (284 words) - 13:56, 26 December 2017
...positional formulas, then so are $(A\mathbin\&B)$, $(A\lor B)$, $(A\supset B)$, and $(\neg A)$.
1 KB (139 words) - 13:57, 30 December 2018
...$\mathsf{E}(X\, |\, \mathfrak{B})$, measurable with respect to $\mathfrak{B}$ and such that
\int\limits_BX\mathsf{P}(d\,\omega)=\int\limits_B\mathsf{E}(X\, |\, \mathfrak{B})\mathsf{P}(d\,\omega)
4 KB (642 words) - 11:43, 14 June 2017
...frak C$ of a [[category]] $\mathfrak K$ such that for any objects $A$ and $B$ from $\mathfrak C$ one has the equality
$$H_\mathfrak C(A,B)=H_\mathfrak K(A,B).$$
1 KB (160 words) - 17:51, 15 November 2014
Let $A,B$ be two subsets of a group $G$, then $A^B$ denotes the set
\{ a^b : a \in A\,,\, b \in B \}
1 KB (255 words) - 21:08, 29 November 2014
...nd $a\land b=0$. If for any $a,b\in L$ with $a\leq b$ the [[interval]] $[a,b]$ is a complemented lattice, then $L$ is called a relatively complemented l
...$ with $b\nleq a$ there is an element $c\in L$ such that $a\land c=0$ and $b\land c\neq0$;
3 KB (489 words) - 08:09, 26 November 2023
...on is the kernel of a linear integral operator, acting on the space $L_2(a,b)$, and is itself square-integrable in the triangle in which it is non-zero,
520 bytes (95 words) - 19:03, 27 April 2014
...y a colon $(a:b)$, a horizontal stroke $\frac ab$ or an oblique stroke $(a/b)$.
...totally divisible (divisible without remainder) by $b$; this is noted as $b\mid a$. Division of complex numbers is defined by the formula
3 KB (464 words) - 18:40, 30 December 2018
an orthogonal system $ b _ {1}, \dots, b _ {k} $
is constructed such that every vector $ b _ {i} $ ($ i = 1, \dots, k $)
4 KB (542 words) - 08:33, 13 May 2022
C3) $C(A \cup B) = C(A) \cup C(B)$.
...ightarrow Y$ is ''continuous'' if $f(C_X(B)) \subseteq C_Y(f(B))$ for any $B \subseteq C$.
1 KB (218 words) - 14:04, 12 November 2023
...ng from $A$ to $B$ solely under the influence of gravity, would arrive at $B$ within the shortest possible time. The problem can be reduced to finding a
$$J(y)=\int\limits_a^b\sqrt{\frac{1+y'^2}{2gy}}dx,$$
2 KB (275 words) - 19:39, 29 April 2014
...dependently developed by P. de Casteljau at Citroën (about 1959) and by P. Bézier at Rénault (about 1962) for the construction of car bodies.
{} _ s^{t} B _ i^{n} ( u ) = {
6 KB (972 words) - 21:59, 29 January 2020
in a space $ L _ {p} ( a , b ) $
on $ [ a, b ] $
2 KB (259 words) - 22:17, 5 June 2020
...whereas Specker proved in 1950 that every countable subgroup of $\mathcal{B}$ is free abelian.
775 bytes (113 words) - 07:30, 24 November 2023
( \nu + a ^ {2} ) }{( b ^ {2} - a ^ {2} ) ( c ^ {2} - a ^ {2} ) }
\frac{( \lambda ^ {2} + b ^ {2} ) ( \mu ^ {2} + b ^ {2} ) ( \nu + b
3 KB (374 words) - 19:37, 5 June 2020
...t describes a largest [[Quotient object|quotient object]] of an object $ B $
that annihilates the image of a homomorphism $ \alpha : A \rightarrow B $.
3 KB (417 words) - 16:16, 5 June 2020
(- \lambda ) ^ {n} + b _ {1} (- \lambda ) ^ {n - 1 } + \dots + b _ {n} $$
the coefficient $ b _ {1} $
3 KB (434 words) - 11:43, 24 December 2020
such that for all $ a , b \in G $
( a b ) ^ {n} = a ^ {n} b ^ {n} s _ {1} ^ {n} \dots s _ {t} ^ {n}
2 KB (283 words) - 08:10, 6 June 2020
$#C+1 = 35 : ~/encyclopedia/old_files/data/B015/B.0105650 Bernoulli polynomials
B _ {n} (x) = \
6 KB (828 words) - 10:58, 29 May 2020
...leading coefficient 1, having minimal norm in the space $C[a,b]$ or $L_p[a,b]$.
$$T_n(x)=2\left(\frac{b-a}{4}\right)^n\cos n\arccos\left(\frac{2x-a-b}{b-a}\right),$$
3 KB (552 words) - 15:05, 14 February 2020
$$\inf[a,b]=a,\quad\sup[a,b]=b;$$
$$\inf(a,b)=a,\quad\sup(a,b)=b;$$
4 KB (722 words) - 19:45, 7 January 2015
A '''finite''' element $b$ of a lattice $L$ is one for which the condition
b \le \bigvee_{d \in D} d
879 bytes (153 words) - 08:06, 26 November 2023
$ B $
define $ A \lor B = \{ {a \lor b } : {a \in A, b \in B } \} $
4 KB (523 words) - 19:25, 4 April 2020
...r of elements is a semi-lattice with respect to the operation $a+b=\sup\{a,b\}$. In this case one says that the partially ordered set is an upper semi-l
<TR><TD valign="top">[a1]</TD> <TD valign="top"> A.H. Clifford, G.B. Preston, "The algebraic theory of semigroups" , '''1''' , Amer. Math. So
2 KB (272 words) - 12:05, 23 November 2023
...he points $((k\pi-c)/b,0)$. Its extrema are at the points $(((k+1/2)\pi-c)/b,(-1)^ka)$.
1 KB (238 words) - 11:36, 13 December 2014
...e of the axioms: instead of the condition $ w ( a \cdot b ) = w ( a) w ( b) $
only $ w ( a \cdot b ) \leq w ( a) w ( b) $
1 KB (165 words) - 08:08, 6 June 2020
An extension $ B $
with unit element such that every element $ x \in B $
3 KB (562 words) - 22:12, 5 June 2020
\lim\limits _ {n \rightarrow \infty } ( \underline{a _ {n} } \pm \underline{b _ {n} } ) = \
a \pm b,\ \
3 KB (445 words) - 17:32, 5 June 2020
if $ F ( x) = F _ {0} (( x - b)/a) $.
is the scale parameter and $ b $
1 KB (203 words) - 08:12, 6 June 2020
In a suitable coordinate system (see Fig. a, Fig. b) the equation of a [[One-sheet hyperboloid|one-sheet hyperboloid]] is
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1,$$
2 KB (330 words) - 09:19, 22 August 2014
...try on the Lobachevskii plane (cf. [[Lobachevskii geometry]]). Let $a$, $b$, $c$ be the lengths of the sides of a triangle on the Lobachevskii plane,
\cosh a = \cosh b \cosh c - \sinh b \sinh c \cos \alpha.
982 bytes (147 words) - 08:44, 8 April 2023
In other words, a function $ f : A \to B $ from a set $A$ to a set $B$ is
: $ f(A) = B $ and $ a_1 \ne a_2 $ implies $ f(a_1) \ne f(a_2) $ for all $ a_1, a_2 \in
1 KB (239 words) - 12:13, 12 December 2013
is pseudo-open if and only if for every $ B \subseteq Y $
the corestriction $ f _ {B} : f ^ { - 1 } [ B] \rightarrow B $
1 KB (165 words) - 08:08, 6 June 2020
...to $G$ on $[a,b]$ having $M(a)=m(a)=0$ and such that at each point $x\in[a,b]$,
...is called the Perron–Stieltjes integral of $f$ with respect to $G$ on $[a,b]$ and is denoted by
2 KB (442 words) - 08:20, 23 July 2014
...A^p$, and if $A$ and $B$ are symmetric matrices of the same order, then $A+B$ is a symmetric matrix, while $AB$ is symmetric if and only if $AB=BA$.
...is symmetric if and only if $B$ is a symmetric bilinear form, i.e. $B(u,v)=B(v,u)$.
2 KB (288 words) - 22:15, 30 November 2018
of a Euclidean space), defined on the square $ [ a, b] \times [ a, b] $
\int\limits _ { a } ^ { b } K ( x, t) K _ {1} ( t, s) dt
3 KB (440 words) - 17:48, 13 January 2024
$#C+1 = 14 : ~/encyclopedia/old_files/data/B017/B.0107000 Booth lemniscate
...called hyperbolic (it has a nodal point at the coordinate origin, cf. Fig. b, where $ n > 2 m ^ {2} $).
2 KB (323 words) - 08:33, 26 March 2023
...of functions $\{\phi_n\}_{n=1}^\infty$ is orthonormal on the interval $[a,b]$ (cf. [[Orthonormal system|Orthonormal system]]) and a sequence of numbers
(that is, $c_n\in l_2$), then there exists a function $f\in L_2[a,b]$ for which
2 KB (290 words) - 14:27, 14 February 2020
...(r) = \{ x \in \mathbf{R}^n : |x| \le r \}$, for $r > 0$, is injective in $B^n(\psi(n,K))$.
2 KB (236 words) - 19:29, 1 November 2016
\frac{( \sigma + a ^ {2} ) ( \tau + a ^ {2} ) }{a ^ {2} - b ^ {2} }
\frac{( \sigma + b ^ {2} ) ( \tau + b ^ {2} ) }{b ^ {2} - a ^ {2} }
3 KB (373 words) - 19:37, 5 June 2020
...e]] of $B$: $X \setminus [A] = \langle B \rangle$ and $X \setminus \langle B \rangle = [ A ]$.
1 KB (244 words) - 16:55, 25 November 2023
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
...^ { * } B ^ { * }$, natural in $A$ and $B$, such that for all objects $A , B , C \in \mathcal{C}$ the following diagram commutes:
where in the bottom arrow $s = s ( ( A ^ { * } ) ^ { ( B ^ { * } ) } , ( B ^ { * } ) ^ { ( C ^ { * } ) } )$.
3 KB (375 words) - 17:46, 1 July 2020
Let $A$, $B$ be sets in some universal domain $\Omega$ and $\complement$ denote complem
\complement (A \cap B) = (\complement A) \cup (\complement B)
1 KB (199 words) - 14:15, 12 November 2023
A fibre bundle (cf. [[Fibre space|Fibre space]]) $ \pi : X \rightarrow B $
such that for any point of the base $ b \in B $
2 KB (303 words) - 20:28, 16 January 2024
...rally, a module over a [[ring]] in which $2$ is invertible, then defining $b$ by
b(x,y) = \frac12 ( q(x+y) - q(x) - q(y) )
1 KB (207 words) - 15:05, 19 November 2023
$#C+1 = 22 : ~/encyclopedia/old_files/data/B110/B.1100410 Bessel polynomials
...tions]], [[#References|[a2]]], the Bessel polynomials $ \{ y _ {n} ( x,a,b ) \} _ {n = 0 } ^ \infty $
3 KB (437 words) - 10:58, 29 May 2020
(A+B)^* = A^* + B^*\,,\ \ \ (\lambda A)^* = \bar\lambda A^*
(AB)^* = B^* A^*\,,\ \ \ (A^*)^{-1} = (A^{-1})^*\,,\ \ \ (A^*)^* = A \ .
930 bytes (131 words) - 16:56, 30 November 2014
$$a-b=c,$$
$a$ is the minuend, $b$ is the subtrahend and $c$ is the difference.
565 bytes (96 words) - 06:07, 23 April 2023
\sum _ {k = 1 } ^ { n } ( a _ {k} \cos kx + b _ {k} \sin kx)
with real coefficients $ a _ {0} , a _ {k} , b _ {k} $,
1 KB (158 words) - 14:03, 28 June 2020
...1(x),\ldots,\phi_k(x)$ form a [[Chebyshev system|Chebyshev system]] on $(a,b)$.
a) $1,x,x^2,\ldots,$ on any interval $[a,b]$;
1 KB (202 words) - 06:03, 5 August 2014
...$ denote the ideal of $R$ generated by elements of the form $a+b$, with $a,b \in G_R$. It is further assumed that:
2) if $a,b \in G_R$ and $a+b \in I_R^2$, then $a+b=0$;
2 KB (318 words) - 22:48, 13 December 2014
where $ A , B $
or $ B \subseteq P $.
2 KB (393 words) - 08:07, 6 June 2020
...precisely, an additive functor $ F : \mathfrak A \rightarrow \mathfrak B $
and $ \mathfrak B $
2 KB (336 words) - 19:38, 5 June 2020
...nu(a/b) = r$ where $a,b$ are integers and $a/b = p^r \cdot a'/b'$ with $a',b'$ coprime to $p$; set $\nu_p(0) = \infty$. The $p$-adic [[Norm on a field|
1 KB (235 words) - 08:06, 26 November 2023
\langle A,B \ |\ A^5=B^3=(AB)^2 \rangle \ .
520 bytes (74 words) - 14:29, 12 November 2023
\int\limits _ { a } ^ { b } f ( x) dx \cong \
\frac{b - a }{2}
3 KB (476 words) - 16:30, 29 March 2024
...in binary relation $R$ to the element $b$. An alternative notation for $(a,b)\in R$ is $aRb$.
$$R^{-1}=\{(a,b)\colon(b,a)\in R\},$$
4 KB (625 words) - 08:47, 26 November 2023
and $ B $(
is the point $ A + B $
3 KB (482 words) - 08:29, 6 June 2020
A submodule $ B $
...be homogeneous if it can be decomposed into a direct sum of submodules $ B _ {n} $
2 KB (330 words) - 16:25, 23 November 2023
and $ b $
b = b ^ {j _ {1} \dots {j _ {q} } } e _ {j _ {1} } \otimes \dots \otimes e _ {
3 KB (535 words) - 19:38, 5 June 2020
\frac{b _ {1} \mid }{\mid a _ {1} }
\frac{b _ {n} \mid }{\mid a _ {n} }
3 KB (392 words) - 17:33, 5 June 2020
...d if and only if for any set $B$ and function $g:\mathcal{P}(B)\rightarrow B$ there is a unique function $f$ such that the following diagram commutes (c
A & \xrightarrow{f} & B\\
2 KB (250 words) - 19:23, 15 June 2017
x_{1,2} = \frac{-b \pm\sqrt{b^2-4ac}}{2a}.
...jugate) numbers, when $b^2=4ac$ the equation has the double root $x_1=x_2=-b/2a$.
2 KB (384 words) - 07:34, 18 December 2014
The theorem was demonstrated by B. Bolzano {{Cite|Bo}}; it was later also independently deduced by K. Weierst
|valign="top"|{{Ref|Bo}}|| B. Bolzano, ''Abhandlungen der königlichen böhmischen Gesellschaft der Wissenschaften. v.''
602 bytes (92 words) - 08:01, 2 May 2014
...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
The quaternion algebra $(a,b)_F$ is the four-dimensional vector space over $F$with basis $\mathbf{1}, \m
\mathbf{i}^2 = a\mathbf{1},\ \ \mathbf{j}^2 = b\mathbf{1},\ \ \mathbf{i}\mathbf{j} = -\mathbf{j}\mathbf{i} = \mathbf{k}\ .
1 KB (180 words) - 20:49, 5 December 2023
\mu_1(a) \mu_2(b) \le \mu_1(a \vee v) \mu_2(a \wedge b)
for all $a,b \in \Gamma$, then
2 KB (253 words) - 18:02, 16 December 2016
...rations. A sublattice $A$ is called convex if $a,b \in A$ and $a\leq c\leq b$ imply $c\in A$. An example of a sublattice is any one-element subset of a
1 KB (237 words) - 21:15, 20 December 2015
...f the magnetic field. Drift equations apply if the magnetic field $ \vec{B} $
\frac{1}{\omega _ {B} B }
4 KB (588 words) - 19:36, 5 June 2020
$#C+1 = 10 : ~/encyclopedia/old_files/data/B110/B.1100830 Brafman polynomials
B _ {n} ^ {p} ( a _ {1} \dots a _ {r} ;b _ {1} \dots b _ {s} ;x ) =
2 KB (357 words) - 06:29, 30 May 2020
$$(ab)c+(bc)a+(ca)b=0$$
(the Jacobi identity), where $a,b,c$ are any elements of $A$. The first of these conditions implies that $A$
470 bytes (73 words) - 16:01, 22 July 2014
$\def\b{\beta}$
\a\pm\b&=(\a_1\pm\b_1,\dots,\a_n\pm\b_n)\in\Z^n,
3 KB (512 words) - 06:16, 13 June 2022
...|#1\right|}(\vect a,\vect b)$ of two non-zero vectors $\vect a$ and $\vect b$''
\[(\vect a,\vect b)=\modulus{\vect a}\modulus{\vect b}\cos\phi.\]
2 KB (338 words) - 04:43, 22 March 2013
\int\limits _ { a } ^ { b } f ( x) dx \cong \
\frac{b - a }{6}
3 KB (376 words) - 08:14, 6 June 2020
$$A(s)\frac{du}{dn}+B(s)\frac{du}{ds}+c(s)u=f(s),$$
...problem in case $A(s)=1$, $c(s)=0$ and the contour $L$ and the functions $B(s)$ and $f(s)$ are analytic.
734 bytes (129 words) - 19:41, 14 August 2014
$#C+1 = 32 : ~/encyclopedia/old_files/data/B015/B.0105380 Bayes formula
Then the a posteriori probability $ {\mathsf P} (A _ {i} \mid B) $
3 KB (402 words) - 10:33, 29 May 2020
A mapping $f$ from a set $A$ to a set $B$ is
an (ordered) triple $ f = (A,B,G_f) $ where $ G_f \subset A \times B $
4 KB (657 words) - 11:54, 20 March 2016
...sociated to $x$ are determined in the blocks of $C _ { G } ( x )$ sent to $B$ by the Brauer correspondence (cf. also [[Brauer first main theorem|Brauer
3 KB (540 words) - 17:43, 1 July 2020
...ecursive function]] $\phi$ such that $x \in A$ if and only if $\phi(x) \in B$. This defines a [[pre-order]] on sets of natural numbers, and the equiva
...here is a computable permutation $\pi$ of $\mathbf{N}$ that maps $A$ onto $B$. The classes of $1$-complete, $\mathrm{m}$-complete and [[creative set]]s
2 KB (238 words) - 12:03, 23 November 2023
...it must also pass through a point between $A$ and $C$ or a point between $B$ and $C$.
2 KB (269 words) - 15:09, 9 April 2014
...nuous function on this segment, then the series converges uniformly on $[a,b]$. Dini's theorem can be generalized to the case when an arbitrary [[Compac
416 bytes (71 words) - 13:25, 9 August 2014
....R. Butler. Namely, $\text{Hom}_A( T , - )$ and its adjoint $- \otimes _ { B } T$ give an equivalence between the subcategories
...l{Y} ( T _ { A } ) = \left\{ N _ { B } : \operatorname { Tor } _ { 1 } ^ { B } ( N , T ) = 0 \right\}, \end{equation*}
3 KB (507 words) - 16:45, 1 July 2020
let $ B = X / G $
be the orbit space, and let $ p : X \rightarrow B $
6 KB (989 words) - 19:40, 5 June 2020
...that of the form $x\to a^x$ a power function); $a$ is called the base and $b$ is called the exponent.
330 bytes (54 words) - 22:02, 11 April 2014
B = \lim_{n\to\infty} \left[n\left(\frac{a_n}{a_{n+1}}-1\right)-1\right]\ln n
exists. If $B>1$ then the series converges and if $B<1$, then the series diverges. If the limit is $1$, then the convergence can
843 bytes (133 words) - 10:25, 16 March 2023
L _ {n} ^ \Phi ( t) = \int\limits _ { a } ^ { b }
...1 } ^ { n } \phi _ {k} ( x) \phi _ {k} ( t) \right | d x ,\ t \in [ a , b ] ,
1 KB (217 words) - 19:29, 28 February 2021
Let $ \mathfrak B $
with respect to $ \mathfrak B $
11 KB (1,715 words) - 22:10, 5 June 2020
H_{\mathfrak{L}} (A,B) = H_{\mathfrak{K}}(A,B) \cap \mathrm{Mor}(\mathfrak{L})
...ated by the hom-functors (morphism functors, $A \mapsto H_{\mathfrak{D}}(A,B)$ is isomorphic to $\mathfrak{D}$ (cf. also [[Functor]]). This result enabl
2 KB (351 words) - 19:56, 5 January 2018
B ( t _ {1} , t _ {2} ) = \
B ^ {\prime\prime} ( t _ {1} , t _ {2} ) =
2 KB (301 words) - 08:23, 6 June 2020
and $ ( b , \kappa ) $
Let $ B $
2 KB (376 words) - 08:01, 6 June 2020
For any function $ f \in L _ {2} [ a , b ] $,
on $ [ a , b ] $
2 KB (317 words) - 12:37, 6 January 2024
...centralizer $K$ in $G$; indeed $K$ is isomorphic to the quotient group $G/B$.
1 KB (238 words) - 15:02, 19 November 2023
...e equal parts of a unit. It is denoted by the symbol $a/b$, where $a$ and $b\ne 0$ are integers (cf.
...fraction may also be considered as the ratio produced by dividing $a$ by $b$.
4 KB (649 words) - 17:15, 9 December 2013
...}[x]$. Then the prime ideal factorization of the ideal $\mathfrak{p}B$ in $B$ is
\mathfrak{p}B = \mathfrak{P}_1^{e_1} \cdots \mathfrak{P}_r^{e_r}
2 KB (331 words) - 11:31, 17 September 2017
...such that $A \cup B = P$, $A \cap B$ is empty, $A \subseteq B^\nabla$ and $B \subseteq A^\Delta$, where
B^\nabla = \{ x \in P : x \le b \ \text{for all}\ b \in B \} \ ,
3 KB (509 words) - 06:25, 28 March 2023
L(a,s) + L(b,s) = \sum_{n=1}^\infty (a+b)(n) n^{-s}
L(a,s) \cdot L(b,s) = \sum_{n=1}^\infty (a*b)(n) n^{-s}
2 KB (358 words) - 17:25, 11 November 2023
be a probability space, $ \mathfrak B $
A function $ Q ( \omega , B ) $
4 KB (599 words) - 17:46, 4 June 2020
...he [[Fibonacci word]] on $A = \{a,b\}$ is invariant under $a \mapsto ab$, $b \mapsto a$.
1 KB (168 words) - 09:30, 12 October 2023
a+(b+c) = (a+b) + c\ \ \text{and}\ \ a(bc) = (ab)c\ .
a \star (b \star c) = (a \star b) \star c
1 KB (190 words) - 20:52, 7 January 2016
...are said to be similar if there exists a [[bijection]] $f : A \rightarrow B$ such that for any $x,y \in A$ it follows from $x\, R\, y$ that $f(x)\, S\,
493 bytes (85 words) - 18:37, 6 October 2016
(here $ \partial B $
is the boundary of $ B $
1 KB (212 words) - 22:17, 5 June 2020
b _ {ij} = \
b _ {ijk} = a _ {ijk} - {
3 KB (382 words) - 02:34, 14 September 2022
''of two fractions $a/b$ and $c/d$ with positive denominators''
...nt of two fractions is positioned between them, i.e. if $(a/b)\leq(c/d)$, $b,d>0$, then
1 KB (218 words) - 13:00, 1 September 2014
Suppose now that the equation $A + B + C = 0$ holds for coprime integers $A,B,C$. The conjecture asserts that for every $\epsilon > 0$ there exists $\k
|A|, |B|, |C| < \kappa(\epsilon) r(ABC)^{1+\epsilon} \ .
2 KB (362 words) - 19:28, 14 November 2023
and a pure crystal $ B $(
and $ B $
3 KB (501 words) - 08:12, 6 June 2020
...sentence with parameters from $A$ is true in $(A,|A|)$ if it is true in $(B,|A|)$. An existential sentence with parameters from $A$ is a [[Closed formu
...d $\alpha$ is a cardinal number greater than the cardinality of $B$, then $B$ admits an embedding, fixing the elements of $A$, in every $\alpha$-saturat
3 KB (454 words) - 20:57, 22 December 2018
module $ B $
A \otimes _ {R} C \rightarrow B \otimes _ {R} C
4 KB (701 words) - 08:04, 21 January 2024
\begin{equation*} \vec { F } = q ( \vec { E } + \vec { v } \times \vec { B } ), \end{equation*}
...$\overset{\rightharpoonup }{ E }$, the vector $\overset{\rightharpoonup} { B }$ satisfies the [[Maxwell equations|Maxwell equations]].
2 KB (288 words) - 17:01, 1 July 2020
\int\limits_a^bf(x)\,dx = F(b)-F(a).
If $f$ is [[ Lebesgue integral | Lebesgue integrable]] over $[a,b]$ and $F$ is defined by
1 KB (209 words) - 20:49, 8 December 2013
that is square-integrable on $ [ a , b ] \times [ a , b ] $
for almost-all $ ( x , y ) \in [ a , b ] \times [ a , b ] $.
4 KB (551 words) - 16:33, 6 January 2024
\int\limits _ { a } ^ { b } f ( x) dx \cong \
( b - a)
4 KB (610 words) - 08:02, 6 June 2020
...can be uniquely extended to a continuous homomorphism $\hat A \rightarrow B$.
1 KB (174 words) - 15:55, 21 December 2014
$$a\ddot q+b\dot q+cq=H\sin(pt+\delta),$$
where $q$ is a generalized coordinate, $a,b,c$ are constant parameters characterizing the system, and $H$, $p$, $\delta
2 KB (305 words) - 07:04, 27 May 2023
\wedge \left ( \lor _ {i \in I } b _ {i} \right ) = \
\lor _ {i \in I } ( a _ {i} \wedge b _ {i} )
2 KB (231 words) - 17:45, 4 June 2020
...tness, or more generally compactness in an interval of cardinals $ [ a , b ] $,
or $ [ a , b ] $-compactness, expressible in three equivalent forms: 1) each set $ M \
4 KB (577 words) - 08:32, 16 June 2022
...A$ contains a subset $B$ of cardinality $\le \mathfrak{t}$ with $x \in\bar B$ . The tightness $t(X)$ is the least upper bound of the local tightness.
566 bytes (91 words) - 20:48, 5 December 2023
...into the space $ C([a,b]) $ of all continuous functions on $ a \leq t \leq b $, because the operator $ \dfrac{\mathrm{d}}{\mathrm{d}{t}} $ takes the bou
...$ commutes with a bounded operator $ B $ if and only if $ B A \subseteq A B $.
4 KB (675 words) - 04:55, 1 March 2017
...from $\Gamma,A\vdash C$ and $\Gamma,B\vdash C$ follows that $\Gamma,A\lor B\vdash C$. In a number of cases, deducible rules have the following structur
2 KB (266 words) - 19:22, 17 October 2014
A) $T(a,0,c)=T(0,b,c)=c$ for all $a,b,c\in R$;
B) $T(a,1,0)=T(1,a,0)=a$ for all $a\in R$;
4 KB (678 words) - 08:05, 10 December 2023
a _ {11} x _ {1} + \dots + a _ {1n} x _ {n} = b _ {1} ,
a _ {n1} x _ {1} + \dots + a _ {nn} x _ {n} = b _ {n}
1 KB (214 words) - 17:31, 5 June 2020
...trum associated to a certain structure series (cf. [[B-Phi-structure| $ ( B, \phi ) $-
Let $ ( B _ {n} , \phi _ {n} , g _ {n} ) $
3 KB (438 words) - 08:25, 6 June 2020
\phi ( x) - \lambda \int\limits _ { a } ^ { b } K ( x , s ) \phi ( s) d s = f ( x) .
\{ a \leq x \leq b , a \leq s \leq b \}
4 KB (616 words) - 18:31, 7 June 2020