...e axiom of extensionality. Let $\text{ZF}^-$ be the system obtained from $\text{ZF}$ by removing the axiom of extensionality and by replacing formulas of t
...that there exists an [[Interpretation|interpretation]] of $\text{ZF}$ in $\text{ZF}^-$. A similar assertion is valid for the theory of types.
2 KB (361 words) - 12:02, 9 April 2014
The set $\text{ln}(\sigma,T)$ of those simplices from the [[closed star]] $\text{St}(\sigma,T)$ (the union of the simplices in $T$ containing $\sigma$) whic
333 bytes (52 words) - 20:16, 13 May 2017
\frac{\partial \rho}{\partial t } + \text{div}(\rho \mathbf{v}) \equiv \frac{\partial \rho}{\partial t} + \frac{\part
...velocity on the coordinate axes. If the fluid is incompressible $ (\rho = \text{const}) $, then the continuity equation takes the form
1 KB (175 words) - 12:28, 30 December 2018
...s}\frac\pi2&\text{if }\beta<\alpha,\\\frac\pi4&\text{if }\beta=\alpha,\\0&\text{if }\beta>\alpha,\end{cases}$$
799 bytes (121 words) - 14:14, 14 February 2020
...\frac{x^2}{(1+x^2)^n} = \begin{cases} 0 & \text{if}\ x = 0 \ , \\ 1+x^2 & \text{otherwise} \ . \end{cases}
...\sin nx}{n} = \begin{cases} 0 & \text{if}\ x = 0 \ , \\ \frac{\pi-x}{2} & \text{if} \ 0 < x < \pi \ . \end{cases}
938 bytes (131 words) - 17:19, 20 November 2016
...d on the set of truth values (cf. [[Truth value|Truth value]]) $\{\text T,\text F\}$ and taking values in this set. With every [[Logical operation|logical
...ace function defined on a set $M$ and taking values in the set $\{\text T,\text F\}$. Such functions are used in mathematical logic as an analogue of the c
806 bytes (143 words) - 16:55, 2 November 2014
\ \ \,\,1\quad \text{if } x>0,\\
\ \ \,\,0\quad \text{if } x=0,\\
630 bytes (102 words) - 15:41, 23 November 2012
...where $n$ is a natural number, are also periods of $f$. For a function $f=\text{const.}$ on an axis or on a plane, any number $T\ne0$ is a period; for the
...= \begin{cases} 1 &\text{if}\ x\ \text{is rational} \\ 0 & \text{if}\ x\ \text{is irrational} \end{cases} \ ,
1 KB (227 words) - 21:30, 18 November 2017
I_{ij} = \delta_{ij} = \begin{cases} 1 & \text{if}\ i =j \\ 0 & \text{otherwise} \end{cases}
519 bytes (74 words) - 11:25, 2 April 2018
...ham cohomology groups of $X$. If $X$ is a smooth complete variety and if $\text{char } k=0$, then de Rham cohomology is a special case of
$$H_{\text{dR}}^p (X/k) \cong H^p (X^\text{an}, \mathbf{C}), \qquad p \ge 0,$$
2 KB (380 words) - 01:16, 23 July 2018
$$||B\& C||=T \iff ||B||=T \text{ and } ||C||=T,$$
$$||B \vee C||=T \iff ||B||=T \text{ or } ||C||=T,$$
3 KB (439 words) - 22:34, 16 June 2014
...operators on $P$ over $F$ (the [[regular representation]]), and the ring $\text{Hom}_F(P,P)$ of all linear operators on $P$ over $F$ is generated by $P$ an
...mension. Moreover, the dimension of $P$ over $E$ equals the dimension of $\text{Hom}_E(P,P)$ over $P$.
3 KB (470 words) - 19:10, 9 November 2016
The [[transformation]] on $X$, often denoted $1_X$ or $\text{id}_X$ defined by $x \mapsto x$ for all $x \in X$.
129 bytes (26 words) - 20:14, 24 September 2016
A [[C*-algebra|$ C^{*} $-algebra]] $ \text{Fun}_{q} \left( \Bbb{S}^{2}_{c} \right) $ generated by two elements $ A $ a
...to \text{Fun}_{q} \left( \Bbb{S}^{2}_{c} \right) \otimes {\text{Fun}_{q}}(\text{SU}(2)),
5 KB (814 words) - 07:12, 24 January 2024
...a finite [[Non-cooperative game|non-cooperative game]] $ \Gamma \stackrel{\text{df}}{=} \langle J,(S_{i})_{i \in J},(H_{i})_{i \in J} \rangle $, where
* $ H_{i}: S \stackrel{\text{df}}{=} \prod_{i \in J} S_{i} \to \mathbb{R} $ is a pay-off function for pl
2 KB (296 words) - 06:03, 8 April 2023
...uction in its full extent, that is responsible for the incompleteness of $\text{FA}$ (see [[Gödel incompleteness theorem|Gödel incompleteness theorem]]).
2 KB (375 words) - 17:00, 30 December 2018
...in any compact subgroup of $G$. For example, $K = \text{SO}(n)$ for $G = \text{GL}(n,\mathbf{R})$; $K=\{e\}$ for a solvable simply-connected Lie group $G$
...group $G$ maximal compact subgroups need not exist (for example, if $G = \text{GL}(V)$, where $V$ is an infinite-dimensional Hilbert space), and if they d
1 KB (233 words) - 17:30, 16 July 2024
...rac{a}{2}}\right) = \begin{cases}0,&a\ \text{even},\\(-1)^{(a^2-1)/8},&a\ \text{odd}.\end{cases}
857 bytes (137 words) - 16:43, 23 November 2023
...=\begin{cases}x-\lfloor x\rfloor&\text{for }x\geq0,\\x-\lfloor x\rfloor-1&\text{for }x<0.\end{cases}$$
1 KB (158 words) - 12:37, 27 August 2014
...has the same cardinality as }A\}$ (the Chang quantifier), $\{B\subset A:B\text{ is uncountable}\}$, etc.
...la $Qxy(\Phi(x),\Psi(y))$, which is interpreted by $\{(B,C):B\text{ and }C\text{ have the same cardinality}\}$). Even more general is the Lindström quanti
3 KB (468 words) - 11:33, 30 December 2018