...{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
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\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
306 bytes (52 words) - 21:02, 15 May 2017