...$n$-dimensional unit cube $E = \{ (x_1,\ldots,x_n) : 0 \le x_i < 1\,\ i=1,\ldots n\}$. Here $\{x\}$ denotes the [[fractional part of a number]] $x$.
The sequence $\{P_j\}$, $j=1,2,\ldots$ is said to be uniformly distributed in $E$ if the equality
2 KB (349 words) - 12:10, 13 March 2018
...A \times \cdots \times A$ of a given set $A$, i.e. a set of tuples $(a_1,\ldots,a_n)$ of $n$ elements of $A$.
...the relation $R$. The notation $R(a_1,\ldots,a_n)$ signifies that $(a_1,\ldots,a_n) \in R$.
2 KB (284 words) - 19:27, 13 April 2024
...or there exists an $i$ such that $1 \le i \le \min(m,n)$ with $a_1 = b_1, \ldots, a_{i-1} = b_{i-1}$ and $a_i < b_i$. This word order is sometimes referred
...if it is strictly less than any of the circular shifts $a_i\ldots a_n a_1\ldots a_{i-1}$ ($2 \le i \le n$). One important characterization of these words y
6 KB (946 words) - 13:41, 20 March 2023
Let $\{1,2,\ldots,n\} = \cup_{i=1}^k \alpha_i$ be a partition of $\{1,2,\ldots,n\}$ into $k$ disjoint subsets. Then the corresponding Young subgroup of $S
...+ 1,\ldots, \lambda_i\}$, where $\lambda_0 = 0$ and $\lambda = (\lambda_1,\ldots,\lambda_k)$ is a [[partition]] of the natural number $n$, i.e. $\lambda_1 \
1 KB (163 words) - 17:28, 22 September 2017
...yer $i$, such that $\sum_{i=1}^n x_i = v(J)$ and $x_i\geq v(\{i\})$, $i=1,\ldots,n$.
764 bytes (121 words) - 18:18, 9 January 2016
...time processing the information $\xi = ( \ldots, \xi_{-1}, \xi_0, \xi_1, \ldots )$ generated by a stationary stochastic process
\xi_k\,,\ \ \ k = \ldots, -1,0,1, \ldots
1 KB (156 words) - 18:15, 19 October 2014
\Pf X = \sum_s \epsilon(s)x_{i_1j_1}\ldots x_{i_nj_n},
...j_\alpha\}$, where one may suppose that $i_\alpha<j_\alpha$, $\alpha=1,\ldots,n$, and where $\epsilon(s)$ is the sign of the permutation
1 KB (251 words) - 19:57, 30 November 2014
Consider a rectangle $R:= [a_1, b_1]\times \ldots \times [a_n, b_n]\subset \mathbb R^n$ and a function $f:R\to \mathbb R$. We
...k} (f, x) := f (x_1, \ldots, x_k+ h_k, \ldots, x_n) - f(x_1, \ldots, x_k, \ldots x_n)
4 KB (644 words) - 10:52, 10 December 2012
...tisfying the condition: For any finite $k\leq n$ the functions $\phi_1(x),\ldots,\phi_k(x)$ 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
H(z; \xi_1,\ldots,\xi_s) = \prod_{i=1}^s \frac{z \xi_i}{1 - \exp(-z\xi_i)} \ .
...ing the coefficient of $z^m$ in the power series expansion of $H(z; \xi_1,\ldots,\xi_m)$ where the $\sigma_i$ are the [[elementary symmetric function]]s of
1 KB (169 words) - 13:51, 4 November 2023
...n parameters, by a given vector $c = (c_1, \ldots, c_k)^T$ such that $c_1+\ldots + c_k = 0$. For example, the difference $\theta_1 - \theta_2 = (\theta_1, \
1 KB (152 words) - 08:11, 13 February 2024
...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
...of rectilinear segments $A_0A_1,\ldots,A_{n-1}A_n$, the ends $A_i$, $i=0,\ldots,n$, of which lie on a given planar or spatial curve, the points $A_i$ being
465 bytes (81 words) - 14:44, 29 April 2014
...ional volume of the $(n-1)$-dimensional simplex $OA_1\ldots A_{i-1}A_{i+1}\ldots A_n$. Then $s^2=\sum_{i=1}^ns_i^2$.
1 KB (160 words) - 14:17, 14 August 2014
...ion $F$, for an appropriate choice of constants $A_n$ and $B_n>0$, $n=1,2,\ldots$
1 KB (181 words) - 20:38, 8 November 2017
...ed from $g$ by the least-number operator if, for any natural numbers $k_1,\ldots,k_n,k$,
$$f(k_1,\ldots,k_n)=k$$
2 KB (391 words) - 07:06, 12 August 2014
(AB)_{ik} = \sum_{j=1}^n a_{ij} b_{jk}, \quad i=1,\ldots,m;\ \ j=1,\ldots,p.
(A \circ B)_{ij} = a_{ij} b_{ij}, \quad i=1,\ldots,m;\ \ j=1,\ldots,n.
2 KB (289 words) - 19:22, 11 November 2023
...near [[Regression|regression]] equation between $X_n$ and $X_{n-k}$, $k=1,\ldots,m$, i.e.
where $\beta_1, \ldots, \beta_m$ are constants and the random variables $\epsilon_n$ are identical
1 KB (180 words) - 01:22, 15 February 2024
...n a finite set of size $n$, given by $P = (p_1,\ldots,p_n)$ and $Q = (q_1,\ldots,q_n)$, the Jeffreys distance is
612 bytes (90 words) - 16:50, 12 June 2016
...he following: If $a_j^{(k)}$ is the $j$-th column of $A_k$, then for $k=1,\ldots,n$,
...-1)} - \frac{ a_ka_j^{(k-1)} }{ 1+a_ka_k^{(k-1)} } a_k^{(k-1)} \ ,\ \ j=1,\ldots,n\,.
2 KB (294 words) - 22:01, 19 November 2016
\begin{equation} \tag{a1} x ^ { n } + a _ { 1 } x ^ { n - 1 } + \ldots + a _ { n - 1 } x + a _ { n } = 0 \end{equation}
...is the $i$th [[elementary symmetric function]] of the roots $\alpha_{1} , \ldots , \alpha _ { n }$. See also [[Viète theorem|Viète theorem]].
3 KB (498 words) - 13:58, 20 March 2023
The Bell numbers $B_0,B_1,\ldots$ are given by
They are equal to $1,1,2,5,15,52,203,877,4140,\ldots$ ({{OEIS|A000110}}).
560 bytes (91 words) - 07:26, 7 November 2023
...x_1, \ldots, x_n$ are the polynomials $\sigma_k(x_1,\ldots,x_n)$ for $k=0,\ldots,n$ where the $k$-th polynomial is obtained by summing all distinct [[monomi
1,001 bytes (159 words) - 20:35, 13 September 2016
S _ {k} = \{ {( x _ {1}, \dots, x _ {k} ) } : {x _ {1} > 0, \ldots, x _ {k} > 0 , x _ {1} + \dots + x _ {k} = 1 } \}
...{ i=1 } ^ { k } x _ {i} ^ {\nu _ {i} - 1 } & \textrm{ if } ( x _ {1}, \ldots, x _ {k} ) \in S _ {k} , \\
2 KB (306 words) - 08:57, 8 April 2023
...$ of the interpolation polynomial $L_n(x)$ with respect to the nodes $x_0,\ldots,x_n$, based on the successive application of the formula
...x_0} \left\vert{ \begin{array}{cc} L_{(0,\ldots,k-1)} & x_0 - x \\ L_{(1,\ldots,k)} & x_k - x \end{array} }\right\vert
2 KB (335 words) - 14:03, 30 April 2023
...r every $i$ one can find both formulas $p$ and $\neg p$ among the $C_{i1},\ldots,C_{im_i}$, for some atomic formula $p$. Given any propositional formula $A$
938 bytes (159 words) - 09:28, 29 November 2014
...ponents $\lambda_i$, and especially the case $\lambda_i = i-1$ for $i = 1,\ldots,n$, when the alternant is a [[Vandermonde determinant]].
595 bytes (99 words) - 16:29, 16 January 2016
Consider a rectangle $R:= [a_1, b_1]\times \ldots \times [a_n, b_n]\subset \mathbb R^n$ and denote by
* $\Gamma$ the class of continuous $\gamma= (\gamma_1, \ldots, \gamma_n):[0,1]\to R$ such that each component $\gamma_j$ is nondecreasing
2 KB (323 words) - 09:38, 16 August 2013
f(x_1,\ldots,x_n) = f(\pi(x_1),\ldots,\pi(x_n)).
The symmetric polynomials form the algebra $S(x_1,\ldots,x_n)$ over $K$.
5 KB (801 words) - 20:34, 13 September 2016
If the direction field $l$ on $S$ has the form $l=(l_1,\ldots,l_n)$, where $l_i$ are functions of the points $P\in S$ such that $\sum_{i=
...ac{df}{dl}=\sum_{i=1}^nl_i(P)\frac{\partial f}{\partial x_i},\quad P=(x_1,\ldots,x_n),$$
1 KB (187 words) - 17:57, 30 July 2014
...ssions $A_1,\ldots,A_n$ by virtue of a derivation rule of $S_1$, and $A_1,\ldots,A_n$ are deducible in $S_2$, then $B$ is also deducible in $S_2$; and 4) th
613 bytes (102 words) - 20:17, 13 May 2017
...a sum-free subset of order $> \frac27 n$. The set of odd numbers in $\{1,\ldots,n\}$ is a sum-free set of order $\frac12 n$.
...rdős conjecture]], proved by Ben Green in 2003, asserts that the set $\{1,\ldots,n\}$ contains $O\left({2^{n/2}}\right)$ sum-free sets.
660 bytes (110 words) - 16:47, 23 November 2023
...type $(p,q)$ with the aid of $n^{p+q}$ components $T^{i_1\ldots i_p}_{j_1\ldots j_p}$ which transform as a result of a change of basis $e'_i = A_i^s e_s$ a
...A'^{i_p}_{s_p} A^{t_1}_{j_1} \cdots A^{t_q}_{j_q} T^{i_1\ldots i_p}_{j_1\ldots j_p}
2 KB (363 words) - 16:21, 18 November 2023
...antile $K_p$ corresponding to the value of $p$ equal to $j/100$, for $j=0,\ldots,99$. For a continuous strictly-monotone distribution function $F(x)$, the $
$j=0,\ldots,99$. In mathematical statistics, the set of percentiles gives a good pictur
800 bytes (116 words) - 09:00, 12 April 2014
...Each element of a vector ring is a vector $\mathbf{a} = (\ldots,a_\lambda,\ldots)$with coordinates in $R_\lambda$, and $\mathbf{a} \ge 0$ if and only if eac
755 bytes (120 words) - 21:01, 22 December 2014
...elements in which the element $i$ cannot occupy the $i$-th position, $i=1,\ldots,n$: a permutation with no fixed points. The problem of calculating the numb
...$i$-th element occupies different positions in each of them for all $i=1,\ldots,n$). The number $U_n$ is given by the formula:
2 KB (279 words) - 08:14, 3 December 2016
''over a ring $A$ in commuting variables $T_1,\ldots,T_N$''
where $F_k$ is a form of degree $k$ in $T_1,\ldots,T_N$ with coefficients in $A$. The minimal value of $k$ for which $F_k \ne
6 KB (1,093 words) - 08:26, 16 March 2023
A set of $n$ linearly-independent vectors $\mathbf{e}_i$ ($i=1,\ldots,n$) of $n$-dimensional affine space $A^n$, and a point $O$. The point $O$ i
...linearly independent in the affine sense, i.e. the vectors $p_0p_i$, $i=1,\ldots,n$, are linearly independent in the corresponding vector space. Independenc
1 KB (211 words) - 19:23, 21 January 2016
...e greatest common divisor of $a_1,\ldots,a_n$ is usually denoted by $(a_1,\ldots,a_n)$.
1) The greatest common divisor of $a_1,\ldots,a_n$ is divisible by any common divisor of these numbers.
4 KB (673 words) - 17:01, 26 October 2014
...t for any element $x \in X$ the sequence of values $y_n = f_n(x)$, $n=1,2,\ldots$ converges in the space $Y$. The function $f : x \mapsto \lim_n y_n$ is th
790 bytes (125 words) - 17:36, 31 December 2016
k[\Delta] = k[x_1,\ldots,x_n]/I_\Delta
...variables $\{x_1,\ldots,x_n\}$, and $I_\Delta$ is the [[ideal]] in $k[x_1,\ldots,x_n]$ generated by the non-faces of $\Delta$, i.e.,
3 KB (544 words) - 20:16, 8 November 2023
...is realizable (is a [[tautology]]) if and only if, for each $i$, $C_{i1},\ldots,C_{im_i}$ do not contain both the formulas $p$ and $\neg p$, where $p$ is a
...s a disjunctive form: for a given set of $m$ propositional variables $p_1,\ldots,p_m$, the normal form is that in which each term $\wedge C_{ij}$ contains e
2 KB (307 words) - 16:22, 18 November 2023
...\ 0 & d_2(X) & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & d_m(X) \end{array} \right)
where $d_i(X)$, $i=1,\ldots,m$, are square matrices, the remaining places being filled by zeros, and ea
2 KB (303 words) - 16:55, 10 April 2016
...en covering $\{\Gamma_0,\Gamma_1,\ldots\}$ of $X$ and a sequence $n_0,n_1,\ldots$ of positive integers greater than $N$ such that $\rho(f(x),f_{n_k}(x))<\ep
1,012 bytes (146 words) - 19:31, 9 November 2014
For a holomorphic function $f(z)$ of several complex variables $z=(z_1,\ldots,z_n)$, $n>1$, the Cauchy inequalities are
...partial z_1^{k_1}\cdots\partial z_n^{k_n}}\leq k_1!\cdots k_n!\frac{M(r_1,\ldots,r_n)}{r_1^{k_1}\cdots r_n^{k_n}}
4 KB (654 words) - 11:56, 10 June 2016
...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$, then
651 bytes (126 words) - 12:13, 9 November 2014
...h, for any formula $A(x)$, from a deduction of $A(\bar0),\ldots,A(\bar n),\ldots,$ it follows that one can infer the formula $\forall xA(x)$, where $\bar n$
961 bytes (151 words) - 21:23, 11 January 2017
b_1 = c_1 \epsilon_1,\ \ldots,\ b_k = c_k \epsilon_k
where $\epsilon_1,\ldots,\epsilon_k$ are invertible elements.
1 KB (153 words) - 16:17, 21 December 2014
$$x'=f(t,x),\quad x=(x_1,\ldots,x_n)\in\mathbf R^n,$$
$$x=\phi(t,C_1,\ldots,C_n),\quad (C_1,\ldots,C_n)\in C\subset\mathbf R^n,\label{2}\tag{2}$$
4 KB (713 words) - 15:28, 14 February 2020
...not expressible as a linear non-negative integer combination of the $a_1,\ldots,a_n$.
1 KB (154 words) - 21:17, 8 April 2018
...{i,i} = a_i$ ($i=1,\ldots,n$), $a_{i,i+1}=b_i$, and $a_{i+1,i}=c_i$ ($i=1,\ldots,n-1$), then a Jacobi matrix has the form
...i$ and $c_i$ ($i=1,\ldots,n-1$) are non-negative. If $b_ic_i>0$ for $i=1,\ldots,n-1$, then the roots of the [[Characteristic polynomial|characteristic poly
2 KB (372 words) - 01:37, 7 May 2022
...of the first $i$ elements is greater than or equal to zero for every $i=1,\ldots,2n$. The number of such sequences is given by the $n$-th Catalan number
...of the $i$-th step the position of the particle is $x = \eta_i$ for $i=1,\ldots,2n$.
3 KB (458 words) - 17:54, 26 December 2017
...is a [[Field|field]]. If $f$ vanishes on the common zeros of $f _ { 1 } , \ldots , f _ { m }$, then there are polynomials $a _ { 0 } , a _ { 1 } , \dots , a
\begin{equation*} a _ { 0 } ( 1 - x _ { 0 } f ) + a _ { 1 } f _ { 1 } + \ldots + a _ { m } f _ { m } = 1. \end{equation*}
2 KB (228 words) - 17:00, 1 July 2020
...centrated on a set of points of the form $\pm nh$, where $h>0$ and $n=1,2,\ldots$. It is a special case of a [[Lattice distribution|lattice distribution]].
207 bytes (33 words) - 15:09, 17 July 2014
...gebra]]. Let $V$ be a [[vector space]] over a field $K$; the vectors $a_1,\ldots,a_n$ are said to be linearly independent if
...$a_1,\ldots,a_n$ are said to be ''linearly dependent''. The vectors $a_1,\ldots,a_n$ are linearly dependent if and only if at least one of them is a linear
2 KB (325 words) - 19:52, 20 November 2014
* An inversion of a permutation $\pi$ on the ordered set $\{1,2,\ldots,n\}$ is a pair $i < j$ such that $\pi(i) > \pi(j)$
287 bytes (45 words) - 07:22, 2 December 2016
...}\langle Z \rangle$ be the [[free associative algebra]] on $Z = \{Z_1,Z_2,\ldots\}$ over the integers. Give $\mathbf{Z}\langle Z \rangle$ a [[Hopf algebra]]
\epsilon(Z_n) = 0,\ \ n=1,2,\ldots
4 KB (709 words) - 13:46, 20 March 2023
...dots,b_n)\omega$ whenever $a_i \,\pi\, b_i$, where $a_i, b_i \in A$, $i=1,\ldots,n$, and $\omega$ is an $n$-ary operation. Congruences in algebraic systems
2 KB (277 words) - 22:08, 12 November 2016
...unit $s$-dimensional cube $K_s = \{\mathbf{x} : 0 \le x_\nu < 1\,,\ \nu=1,\ldots,s \}$''
...f the type $V = \{\mathbf{x} : \alpha_\nu \le x_\nu < \beta_\nu\,,\ \nu=1,\ldots,s \}$, then, in formula (1), $\phi(\alpha;\omega)$ is usually replaced by $
5 KB (718 words) - 15:35, 8 October 2023
The infinite [[word]] over the alphabet $B = \{0,1,\ldots,b-1\}$ for given $b \ge 2$, obtained by concatenating the representations o
0\,1\,10\,11\,100\,101 \ldots \ .
997 bytes (135 words) - 19:33, 6 December 2023
...ots,X_n$ and $Y_1,\ldots,Y_n$ from $X$ and any sequence of exponents $i_1,\ldots,i_n \in \mathbf{N}$, the coefficients in $f$ of $X_1^{i_1} \cdots X_n^{i_n}
...ll totally ordered sets of indeterminates $X_1 < \ldots < X_n$ and $Y_1 < \ldots < Y_n$. For example,
4 KB (708 words) - 13:51, 20 March 2023
Consider a rectangle $R:= [a_1, b_1]\times \ldots \times [a_n, b_n]\subset \mathbb R^n$ and let
...sion $\Sigma^m$ of the rectangle $R$ into $2^m$ closed rectangles $R^m_1, \ldots, R^m_{2^m}$ having equal side lengths.
2 KB (261 words) - 12:31, 27 September 2012
...the points $x=(x_1,x_2,\ldots)$ for which $0\leq x_n\leq(1/2)^n$, $n=1,2,\ldots$. The Hilbert cube is a [[Compactum|compactum]] and is topologically equiva
1,008 bytes (144 words) - 19:09, 12 April 2014
...fields coincides with the class of functions in $n$ variables $\lambda_1,\ldots,\lambda_n$ differing only by a non-negative constant multiplier from $n$-di
1,020 bytes (144 words) - 16:56, 30 July 2014
\ldots&&\\
...ditional restrictions [[#References|[3]]] are imposed on the numbers $N_n,\ldots,N_1$, and $s$ is a sufficiently-large number which depends only on the natu
2 KB (254 words) - 19:09, 10 October 2017
...in the formalized language: "If $t_1,\ldots,t_n$ are terms, then $P(t_1,\ldots,t_n)$ is a formula". Thus, predicate symbols are syntactically used to form
1 KB (191 words) - 13:16, 14 February 2020
Consider a rectangle $R:= [a_1, b_1]\times \ldots \times [a_n, b_n]\subset \mathbb R^n$ and a function $f:R\to \mathbb R$. W
...k} (f, x) := f (x_1, \ldots, x_k+ h_k, \ldots, x_n) - f(x_1, \ldots, x_k, \ldots x_n)
4 KB (637 words) - 12:29, 10 February 2020
...ts $a,b\in G$ there is an integer $n=n(a,b)$ such that $[[\ldots[[a,b],b],\ldots],b]=1$, where $b$ is separated $n$ times and $[a,b]$ is the [[Commutator|co
1 KB (200 words) - 11:46, 26 April 2014
(z, x_1, \ldots, x_n, p_1, \ldots, p_n) \mapsto
(z', x_1', \ldots, x_n', p'_1, \ldots, p'_n)
3 KB (384 words) - 07:34, 13 February 2024
...in cyclic order. If the alphabet is identified with $\mathbf{Z}_m = \{0, \ldots, m-1 \}$ then the Lee distance between $x, y \in \mathbf{Z}_m^n$ is
939 bytes (133 words) - 19:39, 17 November 2023
...\left[ 1 - \frac{x^2}{2^2-\nu^2} + \frac{x^4}{(2^2-\nu^2)(4^2 - \nu^2)} + \ldots \right]
...nu\pi}{\pi}\left[ \frac{x}{1-\nu^2} - \frac{x^3}{(1-\nu^2) (3^2-\nu^2)} + \ldots \right]
2 KB (367 words) - 09:12, 22 February 2014
...incident with $a$. If $A = \{a_1,\ldots,a_n\}$ and $\mathfrak{B} = \{B_1,\ldots,B_m\}$ are finite sets, then the incidence matrix $\Sigma$, where $\Sigma_{
...bership. As before, if $V = \{v_1,\ldots,v_n\}$ and $\mathcal{E} = \{E_1,\ldots,E_m\}$ are finite sets, then the incidence matrix $\Sigma$, where $\Sigma_{
3 KB (475 words) - 20:29, 20 November 2023
A [[Continuum|continuum]] $X$ such that for each sequence $x_1,\ldots,x_n,\ldots,$ of points of $X$ converging towards a point $x$ and each subcontinuum $K\
1 KB (188 words) - 17:01, 13 June 2020
...dots, S_n \}$ of the partial sums $S_0 = 0$, $S_1=X_1$, $S_2=X_1 + X_2$, $\ldots$, $S_n = X_1+\cdots+X_n$, of a sequence $X_i$ of independent identically-di
where $\phi_n(t)$ is the [[characteristic function]] of $\max\{0, S_1, \ldots, S_n \}$ and $\psi_k(t)$ is the characteristic function of $\max\{0, S_k \}
6 KB (960 words) - 07:40, 18 November 2023
...umber $A$ in the strict sense if the $a_n$ are real and if for all $n=0,1,\ldots,$
...eal values, and if the numbers $a_n$ are real, then the signs of $a_1,a_2,\ldots,$ alternate and the series is enveloping in the strict sense. This series i
2 KB (290 words) - 15:50, 14 February 2020
by a binomial $x-a$, where all the coefficients $a,a_0,\ldots,a_n$ lie in a certain field, e.g. in the field of complex numbers. Any poly
b_{n-1} = a_n,\ \ b_{n-2} = a_{n-1} + a b_{n-1},\ \ldots\,,\ b_0 = a_1 + a b_1 \ ; \\
3 KB (445 words) - 18:09, 1 October 2017
...left[ 1 - \frac{x^2}{2^2-\nu^2} + \frac{x^4}{(2^2-\nu^2)(4^2 - \nu^2)} + \ldots \right]
...nu\pi}{\pi}\left[ \frac{x}{1-\nu^2} - \frac{x^3}{(1-\nu^2) (3^2-\nu^2)} + \ldots \right]
2 KB (361 words) - 09:02, 22 February 2014
...steps of multiples of $ \dfrac{1}{n} $ at the points defined by $ X_{(1)},\ldots,X_{(n)} $:
...nce, the empirical distribution corresponding to a random sample $ (X_{1},\ldots,X_{n}) $ is given by the family $ ({F_{n}}(x))_{x \in \mathbf{R}} $ of rand
5 KB (778 words) - 04:08, 22 June 2017
$$a,a+d,a+2d,\ldots,$$
...xample of an arithmetic progression is the series of natural numbers $1,2,\ldots$. The number of terms of an arithmetic progression can be bounded or unboun
992 bytes (162 words) - 19:19, 7 February 2013
...e vertex set $V = A \cup B$ where $A = \{a_1,\ldots,a_n\}$ and $B = \{b_1,\ldots,b_n\}$ .
1 KB (173 words) - 07:29, 14 November 2023
...e exist a [[greatest common divisor]] (a greatest common divisor of $(a_1,\ldots,a_n)$ has the form $\sum b_i a_i$, $b_i \in A$, a so-called Bezout identity
1 KB (190 words) - 19:52, 2 November 2014
...|[2]]]). (It can also be solved by other methods.) Let the values $f(P_1),\ldots,f(P_N)$ of a function be known. It is required to construct a formula for t
$$f(P)\approx g(f(P_1),\ldots,f(P_N),P);$$
4 KB (621 words) - 11:59, 12 August 2014
...value]] $\zeta$. Further, a general circulant with first row $(a_0, a_1, \ldots, a_{n-1})$ is equal to the polynomial $a(C) = a_0 I + a_1 C + \cdots + a_{n
1 KB (160 words) - 19:40, 17 November 2023
...\subseteq Q$ for some natural number $n$). The set of prime ideals $\{P_1,\ldots,P_n\}$ is uniquely determined by the ideal $I$ (the first uniqueness theore
2 KB (386 words) - 20:06, 5 October 2017
...with the exception of the points with abscissas $x=\pi n$, $n=0,\pm1,\pm2,\ldots$. The cotangent is an unbounded odd periodic function (with period $\pi$).
...a meromorphic function with poles at the points $z=\pi n$, $n=0,\pm1,\pm2,\ldots$.
1 KB (179 words) - 14:16, 14 February 2020
...ynomials in a finite set of variables $x_1,\ldots,x_n$ over $R$ is $R[x_1,\ldots,x_n]$. It is possible to speak of a ring of polynomials in an infinite set
...pace to canonical form (see [[Jordan matrix]]). For $n>1$ the ring $k[x_1,\ldots,x_n]$ is not a principal ideal ring.
6 KB (1,011 words) - 06:01, 30 September 2023
...f presentation of matrices is called general. By fixing the matrices $U_1,\ldots,U_q$ and leaving the matrix $U$ variable one obtains the partial problem of
...m (cf. [[#References|[4]]]). By appropriately fixing $U$ and varying $U_1,\ldots,U_q$ the unsolvability of the general formulation has been proved for $n=3$
3 KB (462 words) - 20:23, 31 July 2014
...$A(x)$, each formula of the infinite sequence $A(\bar0),\ldots,A(\bar n),\ldots,$ and the formula $\neg\forall x\ A(x)$ are provable, where $\bar 0$ is a c
1 KB (194 words) - 16:32, 30 December 2018
Numbers $\lambda_n$, $n=0,1,\ldots,$ such that the series $\sum_{n=0}^\infty\lambda_nu_n(x)$ converges almost-
...series of a function from $L_1$, the numbers $\lambda_n=1/\ln n$, $n=2,3,\ldots,$ are convergence multipliers ($\lambda_0$ and $\lambda_1$ can be chosen ar
896 bytes (152 words) - 16:01, 17 July 2014
...responding minors of the same order of $B$. More precisely, if $\alpha=(1,\ldots,m)$ and $\beta$
...[Multiindex|multi-index]] $(\beta_1,\ldots,\beta_m)$ with $1\leq \beta_1<\ldots<\beta_m\leq n$ of
4 KB (615 words) - 16:20, 24 November 2012
...and $(y_1,\ldots,y_k)$ there is $g\in G$ such that $y_i = (x_i,g)$, $i=1,\ldots,k$. An action is ''primitive'' if there is no non-trivial partition of $X$
2 KB (330 words) - 16:58, 25 November 2023
...ldots, a_s, b_1, \ldots, b_t$ such that both $a_1,\ldots, a_s$ and $b_1, \ldots, b_t$ appear in their original order. E.g.,
2 KB (377 words) - 13:44, 20 March 2023
...ects with weights $a_1 , \dots , a _ { n }$ and respective values $c_ 1 , \ldots , c _ { n }$, the problem is to pack as much value in the knapsack as possi
...follows. Given positive integers $c$, $a_1 , \dots , a _ { n }$, $c_ 1 , \ldots , c _ { n }$, the problem is to maximize $\sum c_ { i } x _ { i }$ subject
2 KB (354 words) - 15:30, 1 July 2020
...e [[Rational number|rational]] or [[algebraic number]]s and $\log\alpha_1,\ldots,\log\alpha_n$, with fixed branches of the logarithms, are linearly independ
...= \max \{|p_i|,|q_i|\}$ and $c_1>0$ depends only on the numbers $\alpha_1,\ldots,\alpha_n$. The methods by means of which non-trivial lower bounds for $|L|$
5 KB (776 words) - 08:31, 23 November 2023
...that $Q(x_1,\ldots,x_n) \ge 0$ for all $x_1,\ldots,x_n$ for which $Q(x_1,\ldots,x_n)$ is defined. Then $Q$ is a sum of squares of rational functions with c
2 KB (316 words) - 20:06, 21 September 2017
...s , i _ { k } } ^ { 1 , \ldots , k } | _ { q }$, with $1 \leq i _ { 1 } < \ldots < i _ { k } \leq n$ and with $T \in \operatorname { Mat } ( n ) \otimes \ma
...t a localization by allowing the $q$-minor $| T _ { 1, \dots, k } ^ { 1 , \ldots , k } | _ { q }$ to be invertible. The generators $z _ { s t }$ of the quan
6 KB (822 words) - 16:40, 2 February 2024
Let $S=\{a_1,\ldots,a_k\}$ be a finite set of positive integers with greatest common divisor $1
314 bytes (50 words) - 15:24, 10 August 2014
The undirected [[graph]] on a vertex set $\{v_1,\ldots,v_n\}$ in which any two distinct vertices are joined by an edge: denoted $K
328 bytes (54 words) - 16:45, 16 March 2023
...on of an [[Abelian function]]. A [[meromorphic function]] $f(z)$, $z=(z_1,\ldots,z_n)$, in the complex space $\mathbf C^n$, $n>1$, is called a quasi-Abelian
598 bytes (88 words) - 19:20, 6 December 2023
A system of [[cycle]]s $z_1,\ldots,z_n$ with the following properties: None of their non-trivial linear combin
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z^{p^n} = 1,\ \ n=1,2,\ldots
...nion of an ascending chain of cyclic groups $C_n$ of orders $p^n$, $n=1,2,\ldots$; more precisely, it is the [[inductive limit]]
3 KB (414 words) - 20:17, 16 October 2017
...points $x=\pm a,\pm3a,\pm5a,\ldots,$ with asymptotes $x=\pm2a,\pm4a,\pm6a,\ldots$. The points of intersection with the straight line $y=2a/\pi$ are points o
1 KB (230 words) - 05:54, 15 April 2023
variables $ x _ {1}, \ldots, x _ {n} $,
and any reals $ \alpha _ {1}, \ldots ,\alpha _ {n} $,
2 KB (294 words) - 20:27, 11 November 2023
...dots,x_N$ turn out to be real only for $N=1,\ldots,7$ and $N=9$. For $N=1,\ldots,7$ the nodes were calculated by P.L. Chebyshev. For $N\geq10$ among the nod
3 KB (420 words) - 16:58, 14 February 2020
...investigated. Let $G$ have vertices $\{v_1,\ldots,v_n\}$ and edges $e_1,\ldots,e_p$, and let $M$ be the $n \times p$ [[incidence matrix]] of $G$, so that
1 KB (180 words) - 17:39, 11 November 2023
A function $f(z)=u+iv$ of one or more complex variables $z=(z_1,\ldots,z_n)$ that is the complex conjugate of a holomorphic function $\overline{f(
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...er $F$ whose [[characteristic polynomial]] is irreducible in $F[X]$, $j=1,\ldots,k$ (cf. [[Irreducible polynomial]]). For a matrix $A$ over a field $F$, the
1 KB (227 words) - 18:07, 12 November 2017
''of a vector field $ \mathbf{a} $ at a point $ x = (x^{1},\ldots,x^{n}) $''
...$ \nabla \stackrel{\text{df}}{=} \left( \dfrac{\partial}{\partial x^{1}},\ldots,\dfrac{\partial}{\partial x^{n}} \right) $ and the vector $ \mathbf{a}(x) $
6 KB (941 words) - 05:04, 11 December 2016
...ts 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
...disjoint edges: equivalently, the [[complete multipartite graph]] $K_{2,2,\ldots,2}$.
401 bytes (51 words) - 17:37, 27 June 2023
...ensity $p(x)$ of the probability law of independent random variables $X_1,\ldots,X_n$ is symmetric about zero, that is, $p(x)=p(-x)$ for any $x$ from the do
Most often the hypothesis $H_1$ that all the random variables $X_1,\ldots,X_n$ have probability density $p(x-\Delta)$, $\Delta\neq0$, is considered a
1 KB (238 words) - 16:19, 19 April 2014
...obenius endomorphism $\phi$ takes the point $(x_1,\ldots,x_n)$ to $(x_1^q,\ldots,x_n^q)$.
1 KB (233 words) - 17:09, 10 January 2016
M(x_1,\ldots,x_n) = \frac{1}{k!} \sum_{ \emptyset \neq I \subseteq \{1,\ldots,n\} } (-1)^{n-|I|} F\left({\sum_{i \in I} x_i}\right) \ .
1 KB (207 words) - 15:05, 19 November 2023
...space $ \mathbf{P}_{\mathbb{k}}^{n} $. In homogeneous coordinates $ x_{0},\ldots,x_{n} $ on $ \mathbf{P}_{\mathbb{k}}^{n} $, a projective scheme is given by
{f_{1}}(x_{0},\ldots,x_{n}) = 0, \quad \ldots \quad {f_{r}}(x_{0},\ldots,x_{n}) = 0.
3 KB (412 words) - 13:07, 17 April 2023
$$[[\ldots[[x,y]y],\ldots]y]=1,$$
2 KB (245 words) - 10:51, 15 April 2014
...1, \ldots, x_{n-1})) = \frac{(-\nabla f, 1)}{\sqrt{1+|\nabla f|^2}} (x_1, \ldots, x_{n-1})\, ,
- \ldots - \frac{\partial f}{\partial x_{n-1}} (x') v_{n-1} (x', f (x'))\right)\, dx
4 KB (589 words) - 15:27, 4 February 2014
...<_0b_n)$ be the number of linear extensions in which $a_i<_0b_i$ for $i=1,\ldots,n$.
2 KB (323 words) - 17:05, 9 January 2016
...n function; $\alpha$ is a [[Multi-index notation|multi-index]] $(\alpha_1,\ldots,\alpha_n)$; $Du$ is a vector with components
...he components of a vector $q$; $\xi$ is an $n$-dimensional vector $(\xi_1,\ldots,\xi_n)$; and $\xi^\alpha = \xi_1^{\alpha_1} \cdots \xi_n^{\alpha_n} $. If s
2 KB (296 words) - 22:17, 2 May 2012
...f arbitrarily high degree, for example, any polynomial of the form $f(x_1,\ldots,x_{n-1})+x_n$ is absolutely irreducible.
The polynomial ring $k[x_1,\ldots,x_n]$ is factorial (cf. [[Factorial ring]]): Any polynomial splits into a p
3 KB (478 words) - 15:26, 30 December 2018
...^1}{\partial x_1} (y) & \frac{\partial \Phi^1}{\partial x_2} (y)&\qquad \ldots \qquad & \frac{\partial \Phi^1}{\partial x_n} (y)\\
...^2}{\partial x_1} (y) & \frac{\partial \Phi^2}{\partial x_2} (y)&\qquad \ldots \qquad & \frac{\partial \Phi^2}{\partial x_n} (y)\\
3 KB (384 words) - 14:28, 14 December 2012
...oduct is denoted by $\prod_{i\in I} X_i$, for a finite index set $I = \{1,\ldots,n\}$ one also uses the notations $\prod_{i=1}^n X_i$ and $X_1 \times \cdots
...arrow X$ and a $k$-ary operation $\omega$ the action of the function $f_1,\ldots,f_k\,\omega$ on one element $i\in I$ is defined by
4 KB (769 words) - 20:29, 7 February 2017
\text{diag}[A_1,A_2,\ldots,A_t,0,0,\ldots]
with $\alpha_i$ real numbers, $i = 1,\ldots,t$. The [[Jordan normal form]] $J$ of a complex skew-symmetric matrix posse
2 KB (339 words) - 06:23, 12 September 2016
...ace spanned by $1$ and random variables of the form $\tilde { h } _ { 1 } \ldots \tilde { h } _ { k }$ with $k \leq n$ and $h _ { j } \in H$ for $1 \leq j \
...ction of $L ^ { 2 } ( \mu )$ onto the space $G_n$. For $h _ { 1 } \otimes \ldots \otimes h _ { n } \in H ^ { \otimes n }$, define
9 KB (1,318 words) - 17:43, 1 July 2020
...[[reduced system of residues]] modulo $n$: the natural numbers $k \in \{1,\ldots,n\}$ that are relatively prime to $n$.
..._1,\ldots,a_k)$, $a_i \in \{1,\ldots,n\}$, such that $\mathrm{hcf}\{n,a_1,\ldots,a_k\} = 1$. Clearly, $J_1 = \phi$. The $J_k$ are [[multiplicative arithmet
3 KB (519 words) - 10:04, 14 December 2014
...to tell, with the aid of algorithms, whether $P_0$ is deducible from $P_1,\ldots,P_l$ by one application of the rule.
3 KB (431 words) - 19:19, 17 October 2014
...which a distribution function $F$ assumes values equal to $j/10$ for $j=1,\ldots,9$. If deciles exist, they give a fair idea about the shape of the distribu
395 bytes (69 words) - 09:20, 12 April 2014
with gaps (lacunas), so that the exponents $\lambda_1,\lambda_2,\ldots,$ do not run through all the natural numbers. Depending on the properties o
$$\lambda_{k+1}-\lambda_k>\theta\lambda_k,\quad k=0,1,\ldots,\quad \theta>0,$$
1 KB (203 words) - 06:24, 22 April 2012
subject to $g_i(x)\leq0$, $i=1,\ldots,m$.
421 bytes (64 words) - 17:27, 14 October 2014
\mbox{subject to} & g_i(x)\leq 0, & i=1,\ldots,p,\\
& h_j(x)=0, & j=1,\ldots,q,
5 KB (749 words) - 13:25, 3 May 2016
...ial f^1}{\partial x_1} (y) & \frac{\partial f^1}{\partial x_2} (y)&\qquad \ldots \qquad & \frac{\partial f^1}{\partial x_n} (y)\\
...al f^2}{\partial x_1} (y) & \frac{\partial f^2}{\partial x_2} (y)&\qquad \ldots \qquad & \frac{\partial f^2}{\partial x_n} (y)\\
4 KB (631 words) - 08:46, 18 November 2012
$$f(x_1,\ldots,x_n)=0,\label{*}\tag{*}$$
497 bytes (78 words) - 15:05, 14 February 2020
subject to $g_i(x)\leq0$, $i=1,\ldots,m$.
444 bytes (66 words) - 17:58, 14 October 2014
...P(y_1,\ldots,y_n)$, where $P$ is an $n$-place predicate variable and $y_1,\ldots,y_n$ are object variables. The formulas of predicate calculus are defined a
2 KB (252 words) - 16:42, 30 December 2018
...$d$ for which there exists a family of linear extensions $L_1=(X,{\le_1}),\ldots,L_d=(X,{\le_d})$ such that
x \le y \Leftrightarrow x \le_1 y,\ldots,x \le_d y \ .
2 KB (402 words) - 15:27, 10 January 2016
...} & { } & { \ddots } & { } & { \vdots } \\ { 0 } & { \ldots } & { 0 } & { p _ { k - 1 } } & { 0 } \end{array} \right), \end
...- 1 }$. A left eigenvector corresponding to $r$ has the form $[ y _ { 1 } \ldots y _ { k } ]$, where for $1 \leq j \leq k$,
6 KB (894 words) - 20:14, 4 February 2024
...i.e. the ring of polynomials with non-commuting indeterminates $\{a_1,a_2,\ldots\}$) following the rule:
...\right\rbrace$ is a linear basis for the free Lie algebra over $\{a_1,a_2,\ldots\}$.
3 KB (577 words) - 13:35, 20 March 2023
...y be encoded by the ''Parikh vector'' of multiplicities $(\epsilon_X(x_1),\ldots,\epsilon_X(x_n))$.
2 KB (265 words) - 19:01, 9 November 2023
...s]], introduced by N.H. Abel in {{Cite|Ab}}. If $a_1, \ldots, a_N$, $b_1, \ldots, b_N$, are given complex numbers and we set
...1)}{2\cdot 1} x^2 + \frac{m\cdot (m-1)\cdot (m-2)}{3\cdot 2\cdot 1} x^3 + \ldots$ u.s.w.", ''J. Reine Angew. Math.'' , '''1''' (1826) pp. 311–339
2 KB (382 words) - 13:06, 10 December 2013
...uple $\delta = (\delta_1,\ldots,\delta_s)$ consisting of the symbols $0,1,\ldots,g-1$ appears with asymptotic frequency $1/g^s$ in the sequence
\alpha_1,\ldots,\alpha_n,\ldots
6 KB (960 words) - 08:11, 4 November 2023
...ing an infinite matrix. Employing an infinite matrix $[a_{nk}]$, $n,k=1,2,\ldots,$ a given sequence $(s_n)$ is transformed into the sequence $\sigma_n$:
If the series on the right-hand side converges for all $n=1,2,\ldots,$ and if the sequence $\sigma_n$ has a limit $s$ for $n \rightarrow \infty$
3 KB (519 words) - 22:33, 8 May 2012
...ernoulli trials]]). Consider independent random variables $X_1,\ldots,X_n,\ldots$ which are identically distributed and assume one of two values 0 and 1 wit
2 KB (236 words) - 10:23, 8 February 2017
...$) extends to an isomorphism of $S(M)$ onto the polynomial algebra $A[X_1,\ldots,X_n]$ (see [[Ring of polynomials]]).
3 KB (487 words) - 18:21, 11 April 2017
If $f, g_1, \ldots, g_m: \mathbb R^n \supset \Omega \to \mathbb R$
...functions, a conditional extremal point of $f$ under the costraints $g_1, \ldots, g_m$ is an element $x^\star$ with the property that $f (x^\star)$ is the m
6 KB (985 words) - 17:11, 22 June 2014
...$a_i R b_i$, $i = 1,\ldots,n$, imply $(a_1,a_2,\ldots,a_n \omega) R (b_1,\ldots,b_n \omega)$. Thus, a tolerance is a natural generalization of the notion o
3 KB (402 words) - 21:38, 12 October 2014
...dots,b_n$ such that $a\in A$ is equivalent to the truth of $\phi(b_1\in B,\ldots,b_n\in B)$.
3 KB (414 words) - 21:57, 16 January 2016
...ary length. The result was conjectured by A. Baudet. Let $A \subset \{1,2,\ldots\}$ be a set of natural numbers and let $\bar d(A)$ be its upper [[asymptoti
Let $B(\ell,n)$ be a maximally large subset of $\{1,2,\ldots,n\}$ which contains no $\ell$ elements in arithmetic progression. Let $b(\e
3 KB (415 words) - 19:05, 20 November 2023
f (z)\, dz_1\wedge dz_2\wedge \ldots \wedge dz_n\, .
...[a_{i-1}, b_{i-1}]\times \partial \Delta \times [a_{i+1}, b_{i+1}] \times \ldots \times [a_n b_n]\, ,
2 KB (358 words) - 17:23, 12 January 2014
and for a finite series, indexed by $ \{ 0,\ldots, n \} $,
B _ {i} = A _ {i+1} ,\ i = 0, \ldots, n- 1.
2 KB (295 words) - 11:42, 12 January 2021
..., it is necessary that for any neighbourhood $O_K$, the family $\{O_k,U_1,\ldots,U_k\}$ is a uniform covering of the space $X$. A bordering of a space $X$ i
2 KB (294 words) - 16:43, 24 September 2017
A [[metric]] on a countably infinite set $X = \{x_1,x_2,\ldots\}$. For $i \ne j$ define $d(x_i,x_j) = 1 + 1/(i+j)$, and $d(x_i,x_i) = 0$.
500 bytes (74 words) - 19:07, 14 November 2023
...s \wedge d \zeta _ { n }$, $\langle a , b \rangle = a _ { 1 } b _ { 1 } + \ldots + a _ { n } b _ { n }$,
...pha | = 0 } ^ { m } \frac { ( | \alpha | + n - 1 ) ! } { \alpha _ { 1 } ! \ldots \alpha _ { n } ! } \left( \frac { \rho ^ { \prime } ( \zeta ) } { \langle \
7 KB (996 words) - 09:31, 3 February 2024
...[[Linear independence|Linear independence]]). I.e., if $\sigma _ { 1 } , \ldots , \sigma _ { t }$ are distinct homomorphisms $E \rightarrow F$, then for al
\begin{equation*} a _ { 1 } \sigma _ { 1 } ( u ) + \ldots + a _ { t } \sigma _ { t } ( u ) \neq 0. \end{equation*}
2 KB (275 words) - 16:55, 1 July 2020
...ory|probability theory]] it means the following theorem: Let $A _ { 1 } , \ldots , A _ { n }$ be events in a [[Probability space|probability space]] and
...( A _ { 1 } \bigcup \ldots \bigcup A _ { n } ) = S _ { 1 } - S _ { 2 } + \ldots + ( - 1 ) ^ { n - 1 } S _ { n }.
11 KB (1,700 words) - 06:06, 15 February 2024
...collection of values of the variables $(x_1,\ldots,x_n)$ for which $f(x_1,\ldots,x_n) = 0$.
3 KB (471 words) - 21:06, 3 January 2015
$$\phi(0),\phi(1),\ldots,\phi(n),\ldots,$$
2 KB (271 words) - 16:50, 2 November 2014
...$K_{a_1,\ldots,a_k}$. The complement of a complete multipartite $K_{a_1,\ldots,a_k}$ is a disjoint union of [[complete graph]]s $K_{a_1} \sqcup \cdots \sq
3 KB (475 words) - 13:16, 25 November 2023
...rs, is isomorphic to the direct product of distributive quasi-groups $Q_1,\ldots,Q_k$, where $Q_i$ has order $p_i^{\alpha_i}$ and is medial (i.e. satisfies
3 KB (425 words) - 22:16, 7 January 2017
$[\mathbf{v,w}]=[v_1,v_2,\ldots,v_I,w_1,\ldots,w_J]$ in
$\int^{\infty}_{-\infty}\ldots\int^{\infty}_{-\infty}
10 KB (1,679 words) - 18:34, 29 February 2024
If the functions $u_n$, $n=1,2,\ldots,$ are continuous and non-negative on a segment $[a,b]$ and if the sum of th
416 bytes (71 words) - 13:25, 9 August 2014
$[\mathbf{v,w}]=[v_1,v_2,\ldots,v_I,w_1,\ldots,w_J]$ in
$\int^{\infty}_{-\infty}\ldots\int^{\infty}_{-\infty}
10 KB (1,649 words) - 18:34, 29 February 2024
...} } ) \in \Omega$ for $t \in \mathbf{R}$. In particular, for $p _ { 1 } = \ldots = p _ { n } = 1$ this circular domain is a Cartan domain. Moreover, assume
...s \wedge d \zeta _ { n }$, $\langle a , b \rangle = a _ { 1 } b _ { 1 } + \ldots + a _ { n } b _ { n }$. Then $\operatorname { grad } \psi = ( \partial \psi
6 KB (978 words) - 17:44, 1 July 2020
...ardinal number]]s of non-empty finite sets. The set $\mathbf{N} = \{1,2,3,\ldots\}$ of all natural numbers, together with the operations of [[addition]] $({
...number with the set of its predecessors: $0 = \emptyset$, $n+1 = Sn = \{0,\ldots,n\}$. Here $S$ denotes "successor" . In this definition $0$ is taken to be
2 KB (319 words) - 19:08, 5 February 2016
...but can be displaced along given hypersurfaces $\psi_\mu = 0$, $\mu = 1, \ldots, p$.
...=1,\ldots,p$, as well as multipliers $\lambda_0$ and $\lambda_i(t)$, $i=1,\ldots,m$, such that, in addition to the boundary conditions (7), the following re
9 KB (1,496 words) - 08:59, 13 February 2024
\sum_{\substack{p_{1},\ldots,p_{m} \in \mathbf{N}_{0} \\ q_{1},\ldots,q_{m} \in \mathbf{N}_{0} \\ p_{1} + q_{1} > 0 \\ \vdots \\ p_{m} + q_{m} >
...rbrace{u,[u, \ldots [u}_{p_{m}},[\underbrace{v,[v, \ldots [v,v}_{q_{m}}]] \ldots ]]
6 KB (1,020 words) - 17:41, 4 May 2017
A composition series is a finite subset $\{a_0,\ldots,a_n\}$ of a partially ordered set with least element $0$ and greatest eleme
...nly if they are equally long and if there is a permutation $\sigma$ of $1,\ldots,m$ such that
3 KB (566 words) - 14:25, 3 September 2017
...coordinates $\rho=R$, $\phi=\pi2k/m$, $k=0,\ldots,m-1$. The vertices $B_1,\ldots,B_m$ of the curve have the coordinates $\rho=R+2r$, $\phi=\pi(2k+1)/m$. Whe
3 KB (486 words) - 21:16, 12 December 2017
...(G)$ are all normal subgroups. The series $Z_0(G)\subseteq Z_1(G)\subseteq\ldots$ is the ascending central series of $G$. Let $Z_\infty(G)=\bigcup_i Z_i(G)$
Let $G=\gamma_1(G)\supseteq\gamma_2(G)\supseteq\ldots$ be the lower central series of $G$, i.e. $\gamma_n(G)=[G,\gamma_{n-1}(G)]$
3 KB (508 words) - 13:05, 12 December 2013
...te system of residues]] modulo $m$. The smallest non-negative residues $0,\ldots,m-1$ or the absolutely smallest residues are the complete systems of residu
2 KB (318 words) - 13:55, 11 August 2014
...+\Delta x=(x_1+\Delta x_1,\dots,x_n+\Delta x_n)\in G$ a point $\xi=(\xi_1,\ldots,\xi_n)$ lying on the segment joining $x$ and $x+\Delta x$ and such that
..._i}\Delta x_i,\quad \xi_i=x_i+\theta\Delta x_i,\quad 0<\theta<1,\quad i=1,\ldots,n.
2 KB (312 words) - 08:41, 28 April 2016
$$\left\lbrace\sqrt\frac2\pi\cos nx\right\rbrace,\quad n=0,1,\ldots,$$
500 bytes (88 words) - 18:23, 14 August 2014
...|signature]] $\Sigma=(S,F)$ and a set $X$ of variables. These values $a_1,\ldots,a_n$ are elements of a [[Sigma-algebra (Computer Science)|$\Sigma$-algebra]
3 KB (467 words) - 12:21, 3 March 2013
The problem of determining in a domain $D$ of the variable $x = (x_1,\ldots,x_n)$ the solution $u(x)$ of a linear differential equation
471 bytes (78 words) - 21:19, 2 May 2012
...al in $\mathcal{H}$ if $x _ { 1 } , \ldots , x _ { n }$ and $\alpha_{1} , \ldots , \alpha _ { n }$ are chosen such as to minimize $\| r\|$. If $\mathcal{H}$
...f minimizing $\| r\|$ can also be considered for fixed nodes $x _ { 1 } , \ldots , x _ { n }$. These formulas are characterized by integrating the unique el
5 KB (662 words) - 20:04, 23 January 2021
...ange interpolation formula]]). For given interpolation nodes $x _ { 1 } < \ldots < x _ { m }$, the method consists in considering the interpolation process
For $x _ { 1 } , \ldots , x _ { m }$ being the zeros of the Jacobi polynomial (cf. [[Jacobi polynom
6 KB (756 words) - 20:24, 5 December 2023
The sequence of rational numbers $B_0,B_1,\ldots,$ discovered by Jacob Bernoulli [[#References|[1]]] in connection with the
$$n=0,1,\ldots,\quad m=1,2,\ldots.$$
4 KB (684 words) - 18:44, 5 October 2023
...R$ and the collection $\Pi$ of ordered $(N+1)$-ples of points $a_1<a_2 < \ldots < a_{N+1}\in I$,
...(f) := \sup \left\{ \sum_{i=1}^N \max \{-(f(a_{i+1})-f(a_i)), 0\} : (a_1, \ldots, a_{N+1})\in\Pi\right\}\, .
2 KB (271 words) - 08:48, 16 September 2012
...R$ and the collection $\Pi$ of ordered $(N+1)$-ples of points $a_1<a_2 < \ldots < a_{N+1}\in I$,
...(f) := \sup \left\{ \sum_{i=1}^N \max \{(f(a_{i+1})-f(a_i)), 0\} : (a_1, \ldots, a_{N+1})\in\Pi\right\}\, .
2 KB (269 words) - 08:46, 16 September 2012
A sequence $\{x_n\}$ such that for each $n=1,2,\ldots,$ one has $x_n>x_{n+1}$. Sometimes such a sequence is called strictly decre
564 bytes (90 words) - 21:39, 25 October 2014
ii) for any $\epsilon > 0$ one has $H(a_0,\ldots,a_n) = O\left({ n^{\epsilon n} }\right)$.
Then $f$ is called an $E$-function. Here, the notation $H(x_0,\ldots,x_n)$ stands for the so-called projective height (cf [[Height, in Diophanti
5 KB (740 words) - 18:03, 26 March 2021
...t one to solve systems of Diophantine equations, since the system $P_1=0, \ldots, P_k=0$ is equivalent to the equation
...d only if the exponential-Diophantine equation $K=L$ is solvable for $z_1,\ldots,z_n$. After this, for Davis' hypothesis it remained to be proved that a met
6 KB (845 words) - 19:39, 31 December 2014
...Suppose that $A_1,\ldots,A_k$ are independent sets. If $\mathcal{C}=\{C_1,\ldots,C_m\}$ is a partition of the set into chains, then each $C_i$ meets $\cup_{
...ion $\mathcal{P}$ of the partially ordered set into independent sets $I_1,\ldots,I_m$ to be $\sum_{j\le m}\min(|I_j|,k)$.
3 KB (524 words) - 21:52, 1 November 2017
where for $n = 0 , \ldots , N$ and some $r > 0$, the functions
...\lambda ) = \sum _ { n = 0 } ^ { N } ( \lambda + n ) ( \lambda + n - 1 ) \ldots ( \lambda + 1 ) a ^ { n _0} = \end{equation}
13 KB (1,803 words) - 07:40, 4 February 2024
...ile the point $y=(y_1,\ldots,y_m)$, representing a set of parameters $y_1,\ldots,y_m$, varies within some domain $G$ of the space $\mathbf R^m$.
...nuity and differentiability of $J(y)$ with respect to the parameters $y_1,\ldots,y_m$. If $J(y)$ is interpreted as a [[Lebesgue integral|Lebesgue integral]]
4 KB (785 words) - 17:33, 14 February 2020
...ual to the set of all finite linear combinations of elements $\{m_i : i=1,\ldots,n \}$ of $M$,
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...k } \end{array} \right) : = \{ X \subseteq [ n ] : | X | = k \} , k = 0 , \ldots , n. \end{equation*}
...begin{array} { c } { n } \\ { \lceil n / 2 \rceil } \end{array} \right) > \ldots > \left( \begin{array} { l } { n } \\ { n } \end{array} \right), \end{equat
4 KB (654 words) - 20:17, 25 January 2024
The statement that the number of [[sum-free set]]s contained in $\{1,\ldots,N\}$ is $O\left({2^{N/2}}\right)$. The conjecture was stated by Peter Came
558 bytes (83 words) - 14:02, 30 December 2015
...ngular algebraic variety of dimension $n$, then in local coordinates $x_1,\ldots,x_n$ a form $\omega$ can be written as
\omega = f(x_1,\ldots,x_n) \, dx_1 \wedge \cdots \wedge dx_n \ .
2 KB (317 words) - 16:19, 9 April 2023
Convergence of a sequence of [[random variable]]s $X_1,X_2,\ldots$ defined on a [[probability space]] $(\Omega,\mathcal{F},\mathbb{P})$, to a
691 bytes (91 words) - 19:30, 1 September 2017
...kes the numbers mutually prime with $m$ in the complete residue system $0,\ldots,m-1$ as reduced residue system.
521 bytes (82 words) - 12:45, 23 November 2014
..._ { n - 1 } ( x , y ) + y U _ { n - 2 } ( x , y ), }\\{ \quad n = 2 , 3 , \ldots ,}\end{cases} \right. \end{equation}
\begin{equation*} n = 0,1 , \ldots \end{equation*}
14 KB (2,016 words) - 07:46, 27 January 2024
...the set $Q(\Sigma,P,X)$ of <i>atomic formulas</i> consists of all $p(t_1,\ldots,t_n)$ for predicates $p\in P$ with type$(p)= s_1\times\cdots\times s_{ar(p
...dots,x_n)$ and $\exists x_1\in X_{s_1},\ldots,x_n\in X_{s_n}\colon p(x_1,\ldots,x_n)$ for predicates $p\in P$ with type$(p)= s_1\times\cdots\times s_{ar(p
9 KB (1,594 words) - 08:40, 20 May 2013
...able curve $ \gamma = \{ x = x(s) \mid 0 \leq s \leq S \} $, $ x = (x_{1},\ldots,x_{n}) $, where $ s $ is the arc-length; let $ F = F(x(s)) $ be a function
...the point $ x(s_{i - 1}) $ to the point $ x(s_{i}) $, where $ i \in \{ 1,\ldots,m \} $, and
9 KB (1,447 words) - 06:40, 1 December 2016
$$(M_1,F_1),\ldots,(M_i,F_i),\ldots,(M_n,F_n),\ldots,$$
...rarchy for $M$ is a finite partial hierarchy, $(M_1,F_1),\ldots,(M_i,F_i),\ldots,(M_n,F_n)$, and $n$ is called the length of the hierarchy.
7 KB (1,067 words) - 18:00, 4 August 2014
The distribution among the numbers $1,\ldots,m-1$ of those values of $x$ for which the congruence
...ttle is known about how these values are distributed among the numbers $1,\ldots,p-1$.
4 KB (628 words) - 19:38, 19 December 2014
\sigma(x_1 \ldots x_m) = [\ldots[x_1,x_2],\ldots, x_m]
3 KB (564 words) - 19:53, 15 March 2023
A sequence of real numbers $x_n$ such that for all $n=1,2,\ldots,$ the inequality $x_n<x_{n+1}$ is satisfied. Such a sequence is sometimes c
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...ma 1 } , \ldots , X _ { \sigma ( k + 1 ) } ) , X _ { \sigma ( k + 2 ) } , \ldots ) \end{equation*}
...1 } , \ldots , X _ { \sigma k } ) ) ( \omega ( X _ { \sigma ( k + 1 ) } , \ldots , X _ { \sigma ( k + 1 ) } ) ) + \end{equation*}
14 KB (2,077 words) - 12:23, 12 December 2020
where $C=0.261497\ldots$.
659 bytes (104 words) - 15:21, 10 April 2023
...} \operatorname { dim } \operatorname { lin } \{ x , T x , T ^ { 2 } x , \ldots \} = N \end{equation*}
...able function $f$ in an interval containing different points $t _ { 1 } , \ldots , t _ { n }$,
10 KB (1,502 words) - 22:40, 12 December 2020
...f the property: "to be a word of length $n$ over a given alphabet $\{a_1,\ldots,a_k\}$".
In the article above, the natural numbers are $1,2,\ldots$ (i.e. excluding $0$).
5 KB (788 words) - 19:52, 15 January 2021
...f there are an $L$-formula $\phi(x_1,\ldots,x_n,y_1,\ldots,y_k)$ and $b_1,\ldots,b_k \in M$ such that
A = \{ (a_1,\ldots,a_n) \in M^n : \phi(a_1,\ldots,a_n,b_1,\ldots,b_k)\ \text{is true in}\ M \} \ .
5 KB (799 words) - 09:27, 26 November 2016
* $f(t_1,\ldots,t_n)$ for $f\in F$ with type$(f)= s_1\times\cdots\times s_{ar(f)} \longrig
...ongrightarrow s$ and terms $t_i\in T_{s_i}(\Sigma,X)$, it holds $V(f(t_1,\ldots,t_n)) := V(t_1)\cup\cdots\cup V(t_n)$
5 KB (955 words) - 16:09, 28 January 2013
...axes is given through the origin having a basic system of directions $v_1,\ldots,v_n$, then the $i$-th coordinate of the position vector with respect to thi
595 bytes (104 words) - 21:12, 14 April 2014
...th corrdinate (in the standared basis) of $w_j$. Moreover, $|\omega (v_1, \ldots, v_n)|$ is the [[Lebesgue measure]] (i.e. the $n$-dimensional volume) of th
Consider a local coordinate patch $U$ on $\Sigma$. Let $x_1, \ldots, x_n$ be the corresponding coordinates and denote by $g_{ij} (x)$ the inner
7 KB (1,061 words) - 06:58, 29 June 2014
$$(-\infty,a_1),(a_1,a_2),\ldots,(a_n,\infty),$$
599 bytes (102 words) - 17:12, 30 December 2018
...licity. The helicity $\lambda$ takes values $\lambda = 0, 1/2, 1, 3/2, 2, \ldots$. The dimension of the representation space equals one in this case.
2 KB (347 words) - 19:55, 12 January 2017
...the type $\{f : f(x_i) \in v_{f_0(x_i)} \,,\ i=1,\ldots n \}$, where $x_1,\ldots,x_n$ is a finite set of points in $X$ and $v_{f_0(x_i)} \in V_{f_0(x_i)}$
2 KB (386 words) - 17:27, 18 November 2023
...$, is defined by a density function $ p(x |\theta) $, where $ x = (x_{1},\ldots,x_{n})^{\intercal} $ and $ \theta \in \Theta \subseteq \mathbb{R} $. Suppos
...riables subject to the same normal law $ N(\theta,1) $, then $ T = (X_{1},\ldots,X_{n}) / n $ is an efficient estimator of the unknown mean $ \theta $.
5 KB (824 words) - 10:31, 16 July 2021
$$k=0,\ldots,n-1,\quad\phi=\arg\left(-\frac ba\right).$$
524 bytes (94 words) - 17:30, 23 December 2014
...l probability mass of $ \dfrac{1}{n} $ on $ X_{i} $ for each $ i \in \{ 1,\ldots,n \} $. For example, with a sample size of $ n = 5 $ and distinct observati
...tatistical estimation]]), and let $ T_{n} \stackrel{\text{df}}{=} T(X_{1},\ldots,X_{n}) $. The object of interest is then the cdf $ G_{n} $ of the random va
9 KB (1,419 words) - 05:52, 15 June 2017
...r the set $\Sigma$ of points $x$ which satisfy the constraints $g_1 (x) = \ldots = g_m (x) = 0$. The method of Lagrange multipliers is particularly useful b
Assume $f, g_1, \ldots, g_m$ are $C^1$ function of $n$ variables and let $x^\star$ be a local maxi
11 KB (1,859 words) - 23:04, 27 June 2014
With the use of the sequence of partial sums $S_n(f; x)$, $n=1, 2, \ldots$, the notion of convergence of the series $S[f]$ is introduced and its sum
S_n(f; x) = \frac1\pi \int_{-\pi}^\pi f(x+t) D_n() \, dt, \qquad n=0,1,\ldots,
6 KB (1,047 words) - 11:37, 13 February 2024
One of the most general methods in additive number theory. Let $X_1,\ldots,X_k$ be arbitrary sets of natural numbers, let $N$ be a natural number and
...ural number is the sum of $k$ terms taken respectively from the sets $X_1,\ldots,X_k$. Now let $s$ be a complex number and
5 KB (868 words) - 21:24, 18 November 2016
...ten as non-periodic decimal fractions, e.g. $\sqrt2=1.41\ldots$, $\pi=3.14\ldots$. Irrational numbers determine cuts (cf. [[Dedekind cut|Dedekind cut]]) in
2 KB (331 words) - 10:10, 13 April 2014
...or goodness of fit. The essence of such a test is the following. Let $X_1,\ldots,X_n$ be independent identically-distributed random variables whose distribu
643 bytes (97 words) - 19:24, 16 April 2014
...{ 1 } \wedge \ldots \wedge d w _ { k - 1 } \wedge d w _ { k + 1 } \wedge \ldots \wedge d w _ { n }$, and $Z _ { f }$ is the set of zeros of the mapping (a1
where the degree of $p_j$ is less than $ k_{j }$ for $j = 1 , \ldots , n$.
7 KB (1,090 words) - 10:25, 16 March 2024
...c { ( - 1 ) ^ { k } } { ( 2 k + 1 ) ^ { 2 } } \approx0.915965594177219015 \ldots. \end{equation}
...bf{Z} _ { 0 }^- , \quad \mathbf{Z} _ { 0 } ^ { - } : = \{ 0 , - 1 , - 2 , \ldots \}, \end{equation*}
4 KB (577 words) - 14:03, 11 November 2023
...of words of length $n$ over $A$. Let $u=(u_1,\ldots,u_n)$ and $v = (v_1,\ldots,v_n)$ be words in $C$. The Hamming distance $d(u,v)$ is defined as the num
2 KB (333 words) - 16:49, 23 November 2023
...t $\nu(Q)$ be a real-valued function defined on all subsets $Q$ of $\{1,2,\ldots,n\}$ and consider the problem $\max \nu(Q)$.
...t$, consider $\max \nu(Q^{t-1} \cup \{j\})$ and choose any $j_t \in \{1,2,\ldots,n\}$ that realizes this maximum; if $\nu(Q^{t-1} \cup \{j_t\}) \le \nu(Q^{t
2 KB (387 words) - 15:16, 20 November 2014
...x _ { 1 } ) , \ldots , F _ { n } ( t , x _ { 1 } , \ldots , x _ { n } ) , \ldots )$ be a sequence of probability distribution functions $F _ { n } ( t )$ de
...x _ { 1 } ) , \ldots , F _ { n } ( 0 , x _ { 1 } , \ldots , x _ { n } ) , \ldots ). \end{equation*}
10 KB (1,427 words) - 07:38, 7 February 2024
x = a \frac{ 1}{ 1+ \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \ldots}}} },
where $1,1,2,3,5,8,13,21,\ldots,$ are the [[Fibonacci numbers|Fibonacci numbers]].
2 KB (325 words) - 21:24, 3 April 2015
...rn $A_1,\ldots,A_m$ have polar coordinates $\rho=R$, $\phi=2k\pi/m$, $k=0,\ldots,m-1$. If $m$ is irrational, the number of branches is infinite, and the poi
2 KB (406 words) - 18:25, 20 February 2024
...els: closed points, generic points of curves, generic points of surfaces,$\ldots$, the generic point of the $n$-dimensional [[affine space]].
3 KB (403 words) - 08:08, 26 November 2023
''$\sigma$ of a set $X = \{1,2,\ldots,n\}$ into itself''
782 bytes (121 words) - 17:57, 22 November 2016
...in some coordinate system $(x_1,\ldots,x_n)$, if $x_k=\phi_k(y)$, $y=(y_1,\ldots,y_n)$, is some transformation of coordinates inducing a corresponding mappi
3 KB (392 words) - 21:35, 24 July 2012
\sum_{n_1, \ldots n_k} a_{n_1 \ldots n_k}\, .
\sum_{n_1=0}^\infty \ldots \sum_{n_k=0}^\infty a_{n_1\ldots n_k}
5 KB (821 words) - 09:35, 16 August 2013
...{ k }$ into all minimal idempotent elements. Thus, $A = A _ { 1 } \oplus \ldots \oplus A _ { k }$, where $A _ { i } = A \cdot e _ { i } = \mathbf{R} \cdot
...[ x _ { 1 } , \dots , x _ { n } ]$, ${\cal A} \subset \langle x ^ { 1 } , \ldots , x _ { n } \rangle ^ { 2 }$, then the width of $A = \mathbf{R} [ x _ { 1 }
7 KB (1,051 words) - 06:27, 15 February 2024
* $f^A(t_1^A,\ldots,t_n^A)\in A^T$ for all $f\in F$ with arity ar$(f)=n$ and with type$(f)= s_
* $G_{i+1}:=G_i\cup \{f^A(a_1,\ldots,a_n)\in A^T\mid f\in F, \mathrm{ar}(f)=n, \mathrm{type}(f)= s_1\times\cdot
6 KB (940 words) - 19:31, 20 February 2013
$$G=\{1=a^0=a^n,a,\ldots,a^{n-1}\}.$$
666 bytes (123 words) - 06:43, 21 March 2024
a) superposition (composition) of functions: If $f,\alpha_1,\ldots,a_m \in F$, then one says that the function
\phi(x_1,\ldots,x_n) = f(\alpha_1(x_1,\ldots,x_n),\ldots,\alpha_m(x_1,\ldots,x_n))
18 KB (2,635 words) - 19:56, 5 September 2017
...infinitely many double points, for which $\rho=ak^2\pi^2-l$, where $k=1,2,\ldots$. The Galilean spiral is a so-called algebraic spiral (cf. [[Spirals|Spiral
750 bytes (121 words) - 19:26, 16 March 2023
A [[Young diagram]] of order $m$ in whose cells the different numbers $1,\ldots,m$ have been inserted in some order, ''e.g.''
...tition]] of $m$ ($\kappa=(\kappa_1,\ldots,\kappa_t)$, $\kappa_i \in \{0,1,\ldots\}$, $\kappa_1+\cdots+\kappa_t=m$) as well as its corresponding [[Young diag
5 KB (725 words) - 03:36, 15 February 2024
where $q_1,\ldots,q_s\in K$ and $s$ is as small as possible, then $q_1,\ldots,q_s\in A$.
2 KB (407 words) - 19:17, 19 April 2012
...s in the unit poly-disc $U _ { 1 } = \{ z : | z _ { j } | < 1 ,\; j = 1 , \ldots , n \}$ and the estimate
..., \dots , z _ { n } )$, $z ^ { \alpha } = z _ { 1 } ^ { \alpha _ { 1 } } \ldots z _ { n } ^ { \alpha _ { n } }$.
7 KB (1,065 words) - 09:52, 11 November 2023
Given a [[Polynomial|polynomial]] $P ( x _ { 1 } , \ldots , x _ { n } )$ with complex coefficients, the logarithmic Mahler measure $m
...g } | P ( e ^ { i t_{1} } , \ldots , e ^ { i t _ { n } } ) | d t _ { 1 } \ldots d t _ { n }. \end{equation*}
7 KB (1,101 words) - 14:50, 27 January 2024
...gular analytic, or holomorphic, function of $n$ complex variables $z=(z_1,\ldots,z_n)$, $n\geq1$, defined on an (open) domain $D$ of the complex space $\mat
...athbb{C}^m$, that is, $f=(f_1,\ldots, f_m)$, $m\geq 1$, where $f_j$, $j=1,\ldots,m$, are holomorphic functions on $D$, $f(z)\neq\textrm{const}$ and $\lVert
4 KB (614 words) - 06:23, 12 October 2023
...In particular, $A$ is nilpotent if and only if $\tr A^m = 0$ for all $m=1,\ldots,n$.
2 KB (389 words) - 19:21, 11 November 2023
...are characterised by the property that in any odd length closed path $x_1,\ldots,x_{2n+1}$ with $n \ge 2$ (so all $x_i,x_{i+1}$ are adjacent) there exists a
741 bytes (122 words) - 18:48, 14 November 2023
...dots , x _ { n } )$, $\mathbf{k} \cdot \mathbf{x} = k _ { 1 } x _ { 1 } + \ldots + k _ { n } x _ { n }$. Unlike in the one-dimensional case, there is no nat
3 KB (468 words) - 13:55, 21 January 2024
...rary field (or over the ring of integers), regarded as schemes; let $u_0, \ldots, u_n$ be projective coordinates in $\PP^n$, and let $v_{i_0 \cdots i_n}$, $
where $i_0 + j_0 = k_0 + r_0, \ldots, i_n + j_n = k_n + r_n$. For instance, $v_2(\PP^1)$ is the curve represente
2 KB (408 words) - 23:35, 22 October 2018
...by degree $n$ and rank $\nu$ — the maximal degree of the polynomials $f_0,\ldots,f_n$. For example, for curves of order three, $n=1$, $\nu=2$; for the [[Arc
3 KB (434 words) - 13:49, 9 April 2023
...cients in $R$. The pole assignment problem asks the following. Given $r_1,\ldots,r_n$, does there exist an $(m \times n)$-matrix $F$, called a feedback matr
...$(A,B)$ is a ''coefficient assignable'' pair of matrices if for all $a_1,\ldots,a_n \in R$ there is an $(m\times n)$-matrix $F$ such that $A+BF$ has charac
5 KB (799 words) - 20:38, 12 November 2017
...to the set of all finite [[linear combination]]s of elements $\{m_i : i=1,\ldots,n \}$ of $A$. If the linear span of $A$ is $M$, then $A$ is a ''[[spanning
855 bytes (145 words) - 08:49, 26 November 2023
...er (cf. [[Division]]). A number $n$ divisible by each of the numbers $a,b,\ldots,k$ is called a ''common multiple'' of these numbers. Among all common multi
951 bytes (151 words) - 20:23, 2 November 2016
...of the $n$-ary functional constant $g$ if and only if the equation $g(t_1,\ldots,t_n)=t_{n+1}$ is a member of the maximal set. This interpretation is a mode
6 KB (950 words) - 12:20, 30 December 2015
ii) there exists a $c>0$ such that $H(a_0,a_1,\ldots,a_n) = O(c^n)$. Then $f$ is called a $G$-function. Here, the notation $H$ s
H(x_0,\ldots,x_n) = \prod_\nu \max(|x_0|_\nu,\ldots,|x_n|_\nu)
5 KB (818 words) - 16:38, 28 October 2017
...)}_i = T_{s_i}(\Sigma)$, it holds $f^{T(\Sigma)}(t_1,\ldots,t_n):= f(t_1,\ldots,t_n)$
3 KB (426 words) - 17:09, 8 February 2013
...esian orthogonal coordinate system|Cartesian]]) coordinate system $x=(x_1,\ldots,x_n)$ and $y=(y_1,\dots,y_n)$ is given by the formula \begin{equation} (x,y
872 bytes (132 words) - 14:38, 1 November 2023
If $A = \{a_1,\ldots,a_n\}$ and $\mathfrak{B} = \{B_1,\ldots,B_m\}$ are finite sets, then the properties of the incidence system $S$ can
3 KB (488 words) - 19:37, 7 November 2023
For analytic functions of several complex variables $z = (z_1, \ldots, z_n)$, with $z_k = x_k + iy_k$ , the Cauchy–Riemann equations is given
& u (x_1, \ldots, x_n, y_1, \ldots , y_n) = {\rm Re}\, f (x_1+iy_1, \ldots, x_n + iy_n)\\
7 KB (1,063 words) - 12:51, 22 March 2023