...$n$-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