Aizenman's Theorem for Orthogonal Polynomials on the Unit Circle

For suitable classes of random Verblunsky coefficients, including independent, identically distributed, rotationally invariant ones, we prove that if $$ \bbE \biggl( \int\f{d\theta}{2\pi} \biggl|\biggl( \f{\calC + e^{i\theta}}{\calC-e^{i\theta}} \biggr)_{k\ell}\biggr|^p \biggr) \leq C_1 e^{-\kappa_1 \abs{k-\ell}} $$ for some $\kappa_1 < 0$ and $p < 1$, then for suitable $C_2$ and $\kappa_2 >0$, $$ \bbE \Bigl( \sup_n \abs{(\calC^n)_{k\ell}}\Bigr) \leq C_2 e^{-\kappa_2 \abs{k-\ell}}. $$ Here $\calC$ is the CMV matrix.     

Minimum degree of bipartite graphs and the existence ofk-factors

LetG be a bipartite graph with bipartition (X, Y) andk a positive integer. If (i) $$\left| X \right| = \left| Y \right|,$$ (ii) $$\delta (G) \geqslant \left\lceil {\frac{{\left| X \right|}}{2}} \right\rceil \geqslant k,$$ \(\left| X \right| \geqslant 4k - 4\sqrt k + 1\) when |X| is odd and |X| ≥ 4k ? 2 when |X| is even, thenG has ak-factor.     

Martingale inequalities for spline sequences

Positivity - We show that D.&nbsp;Lépingle’s $$L_1(\ell _2)$$-inequality $$\begin{aligned} \left\| \left( \sum _n {\mathbb {E}}[f_n | {\mathscr {F}}_{n-1}]^2 \right) ^{1/2}\right\|...     

The binding number of a graph and its pancyclism

The binding number is an important parameter of a graph.The binding number of a graph G,bind (G),is the largest real number c such that|N(X)|=min(c|X|,|V(G)|) for every X(?)V(G).In this paper,we prove the Woodall's conjecture:In bind (G)≥3/2,then G is pancyclic;andobtain some interesting results.     

Freiman Cover Ideals of Unmixed Bipartite Graphs

Mathematical Notes - An equigenerated monomial ideal $$I$$ in the polynomial ring $$R=k[z_1,\ldots, z_n]$$ is a Freiman ideal if $$\mu(I^2)=\ell(I)\mu(I)-{\ell(I)\choose 2}$$ , where $$\ell(I)$$ is...     

图中相互独立的4圈和含4个点的路

设k是一个正整数，G是一个顶点数为|G|=4k的图. 假设σ\-2(G)≥4k-1, 则G有一个支撑子图含k-1个4圈和一条顶点数为4的路，使得所有这些圈和路都是相互独立的. 设G=(V\-1,V \-2;E)是一个二分图使得|V\-1|=|V\-2|=2k. 如果对G中每一对满足x∈V\-1和y∈V\-2的不 相邻的顶点x和y 都有d(x)+d(y)≥2k+1, 则G包含k-1个相互独立的4圈和一条顶点数为4的路，使得所有这些圈和路都是相互独立的，并且此度条件是最好的.     

THE ASYMPTOTIC BEHAVIOUR OF ANALYTIC FUNCTIONS

AIn this paper, the author obtains the following results:(1) If Taylor coeffiients of a function satisfy the conditions:(i),(ii),(iii)A_k=O(1/k) the for any h>0 the function φ(z)=exp{w(z)} satisfies the asymptotic equality the case h>1/2 was proved by Milin.(2) If f(z)=z α_2z~2 …∈S~* and,then for λ>1/2     

Inequalities for Taylor series involving the divisor function

Czechoslovak Mathematical Journal - Let $$T(q) = \sum\limits_{k = 1}^\infty {d(k){q^k},\,\,\,\,\left| q \right| &lt; 1,} $$ where d(k) denotes the number of positive divisors of the natural...     

Asymptotic behavior of sums of the type\sum\limits_{\kappa = m}^{n - 1} {\exp } (i\omega \sqrt {n\kappa } ) as n,m → ∞

The asymptotic behavior asn, m → ∞ of the sum $$\sum\limits_{\kappa ,\ell = m}^{n - 1} {\exp \left[ {i\omega \sqrt n \left( {\sqrt \kappa + \sqrt \ell } \right)} \right]} \Phi \left( {1 - \frac{{\left| {\sqrt \kappa - \sqrt \ell } \right|}}{\Delta }} \right)$$ is studied where π(t)=0 for t?0 and φ(t)=t for t > 0.     

Split Casimir operator and solutions of the Yang–Baxter equation for the $$osp(M|N)$$ and $$s\ell(M|N)$$ Lie superalgebras,higher Casimir operators,and the Vogel parameters

Theoretical and Mathematical Physics - We find the characteristic identities for the split Casimir operator in the defining and adjoint representations of the $$osp(M|N)$$ and $$s\ell(M|N)$$ Lie...     

A Generalized Measure Concentrated on a Closed Set of Measure Zero

When E is a closed set of measure zero in the dyadic group and the Walsh series satisfies

Inequalities for Alternating Trigonometric Sums

In 1970, J.B. Kelly proved that $$\begin{array}{ll}0 \leq \sum\limits_{k=1}^n (-1)^{k+1} (n-k+1)|\sin(kx)| \quad{(n \in \mathbf{N}; \, x \in \mathbf{R})}.\end{array}$$ We generalize and complement this inequality. Moreover, we present sharp upper and lower bounds for the related sums $$\begin{array}{ll} & \sum\limits_{k=1}^{n} (-1)^{k+1}(n-k+1) | \cos(kx) | \quad {\rm and}\\ & \quad{\sum\limits_{k=1}^{n} (-1)^{k+1}(n-k+1)\bigl( | \sin(kx) | + | \cos(kx)| \bigr)}.\end{array}$$      

一类振荡积分算子在Wiener 共合空间上的有界性

假设a,b0并且K_(a,b)(x)=(e~(i|x|~(-b)))/(|x|~(n+a))定义强奇异卷积算子T如下:Tf(x)=(K_(a,b)*f)(x),本文主要考虑了如上定义的算子T在Wiener共合空间W(FL~p,L~q)(R~n)上的有界性.另一方面,设α,β0并且γ(t)=|t|~k或γ(t)=sgn(t)|t|~k.利用振荡积分估计,本文还研究了算子T_(α,β)f(x,y)=p.v∫_(-1)~1f(x-t,y-γ(t))(e~(2πi|t|~(-β)))/(t|t|~α)dt及其推广形式∧_(α,β)f(x,y,z)=∫_(Q~2)f(x-t,y-s,z-t~ks~j)e~(-2πit)~(-β_1_s-β_2)t~(-α_1-1)s~(-α_2-1)dtds在Wiener共合空间W(FL~p,L~q)上的映射性质.本文的结论足以表明,Wiener共合空间是Lebesgue空间的一个很好的替代.     

Solvability of the Dirichlet problem for degenerate quasilinear elliptic equations

In a bounded domain of the n -dimensional (n?2) space one considers a class of degenerate quasilinear elliptic equations, whose model is the equation $$\sum\limits_{i = 1}^n {\frac{{\partial F}}{{\partial x_i }}} (a^{\ell _i } (u)\left| {u_{x_i } } \right|^{m_i - 2} u_{x_i } ) = f(x),$$ where x =(x₁,..., x_r), l_i?0, m_i>1, the function f is summable with some power, the nonnegative continuous function a(u) vanishes at a finite number of points and satisfies \(\frac{{lim}}{{\left| u \right| \to \infty }}a(u) > 0\) . One proves the existence of bounded generalized solutions with a finite integral $$\int\limits_\Omega {\sum\limits_{i = 1}^n {a^{\ell _i } (u)\left| {u_{x_i } } \right|^{m_i } dx} }$$ of the Dirichlet problem with zero boundary conditions.     

Markov-Bernstein-type inequalities for classes of polynomials with restricted zeros

We prove that an absolute constantc>0 exists such that

A Change-of-Variable Formula with Local Time on Curves

Let be a continuous semimartingale and let be a continuous function of bounded variation. Setting and suppose that a continuous function is given such that F is C^1,2 on and F is on . Then the following change-of-variable formula holds: where is the local time of X at the curve b given by and refers to the integration with respect to . A version of the same formula derived for an Itô diffusion X under weaker conditions on F has found applications in free-boundary problems of optimal stopping.     

On the properties of the strong Cesàro summability and strong convergence of series

Говорят, что ряд \(\mathop \sum \limits_{k = 0}^\infty a_k \) сумм ируется к s в смысле (С, gа), gа >?1, если $$\sigma _n^{(k)} - s = o(1),n \to \infty ,$$ в смысле [C,α] _λ , α<0, λ>0, если $$\frac{1}{{n + 1}}\mathop \sum \limits_{k = 0}^n \left| {\sigma _k^{(\alpha - 1)} - s} \right|^\lambda = o(1),n \to \infty ,$$ и в смысле [C,0] _λ , λ>0, если $$\frac{1}{{n + 1}}\mathop \sum \limits_{k = 0}^n \left| {(k + 1)(s_k - 1) - k(s_{k - 1} - 1)} \right|^\lambda = o(1),n \to \infty ,$$ где σ _n ^(α) обозначаетn-ое ч езаровское среднее р яда. Суммируемость [C,α] _λ , α>?1, λ ≧1 о значает, что $$\mathop \sum \limits_{k = 0}^\infty k^{\lambda - 1} \left| {\sigma _k^{(\alpha )} - \sigma _{k - 1}^{(\alpha )} } \right|^\lambda< \infty .$$ В данной статье содер жится продолжение ис следований свойств [C,α] _λ -суммиру емо сти, которые начали Винн, Х ислоп, Флетт, Танович-М иллер и автор, в частности свя зей между указанными методами суммирования. Наконец, даны некотор ые простые приложени я к вопросам суммируемости ортог ональных рядов.     

On spectra ofp-hyponormal operators

The purpose of this paper is to show the following: Let 0<p<1/2. IfT=U|T| is a p-hyponormal operator with a unitaryU on a Hilbert space, then $$\sigma (T) = \mathop \cup \limits_{0 \leqslant k \leqslant 1} \sigma (T_{\left[ k \right]} ),$$ where $$T_{\left[ k \right]} = U[(1 - k)S_U^ - (\left| T \right|^{2p} ) + kS_U^ + (\left| T \right|^{2p} ]^{\tfrac{1}{{2p}}} $$ andS _U ^± (T) denote the polar symbols ofT.     

On integral graphs which belong to the class\overline {\alpha K_a \cup \beta K_b }

LetG be a simple graph and let $\bar G$ denotes its complement. We say thatG is integral if its spectrum consists entirely of integers. If $\overline {\alpha K_a \cup \beta K_b } $ is integral we show that it belongs to the class of integral graphs $$\overline {[\frac{{kt}}{\tau }x_o + \frac{{mt}}{\tau }z]K(t + \ell n)k + \ell m \cup [\frac{{kt}}{\tau }y_o + \frac{{(t + \ell n)k + \ell m}}{\tau }z]nK\ell m,} $$ where (i) t, k, l, m, n ∈ ? such that (m, n) =1, (n, t) =1 and (l, t)=1; (ii) τ=((t+ln)k+lm, mt) such that τ| kt; (iii) (x₀, y₀) is aparticular solution of the linear Diophantine equation ((t+ln)k+lm)x-(mt)y=τ and (iv) z≥z₀ where z₀ is the least integer such that $(\frac{{kt}}{\tau }x_0 + \frac{{mt}}{\tau }z_0 ) \geqslant 1$ and $(\frac{{kt}}{\tau }y_0 + \frac{{(t + \ell n)k + \ell m}}{\tau }z_0 ) \geqslant 1$ .     

An exponential Diophantine equation related to the sum of powers of two consecutive terms of a Lucas sequence and x-coordinates of Pell equations

Periodica Mathematica Hungarica - For the Fibonacci sequence the identity $$F_n^2+F_{n+1}^2 = F_{2n+1}$$ holds for all $$n \ge 0$$ . Let $${\mathcal {X}}:= (X_\ell )_{\ell \ge 1}$$ be the sequence...