...$n$-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