Convergence in probability and almost sure with applications

A necessary and sufficient condition is given for the convergence in probability of a stochastic process {X_t}. Moreover, as a byproduct, an almost sure convergent stochastic process {Y_t} with the same limit as {X_t} is identified. In a number of cases {Y_t} reduces to {X_t} thereby proving a.s. convergence. In other cases it leads to a different sequence but, under further assumptions, it may be shown that {X_t} and {Y_t} are a.s. equivalent, implying that {X_t} is a.s. convergent. The method applies to a number of old and new cases of branching processes providing an unified approach. New results are derived for supercritical branching random walks and multitype branching processes in varying environment.     

Approximation operators and tauberian constants

The explicit expression of the smallest constantC satisfying $$\mathop {lim}\limits_{\lambda \to \infty } \left| {t_{n(\lambda )}^{(1)} - t_{m(\lambda )}^{(2)} } \right| \leqq C. \mathop {lim sup}\limits_{n \to \infty } \left| {d_n } \right|$$ for all sequences {s _n} satisfying lim sup _n→∞ |d _n| <∞, where {t _n ⁽¹⁾ }, {t _n ⁽²⁾ } are two generalised Hausdorff transforms of {s _n }, {d _n} is the generalised (C, α)-transform (0≦α≦1) of {λ _n a _n} andn(λ, m(λ) are suitably related, is obtained. These results are obtained by using new properties of positive approximation operators and generalised Bernstein approximation operators.     

Banach spaces in which a theorem of orlicz is not true

Let the Banach space X be such that for every numerical sequencet _n ↘0 there exists in X an unconditionally convergent series σx_n, the terms of which are subject to the condition ∥x_n∥=t_n (n=1,2,...). Then $$\mathop {sup}\limits_n \mathop {inf}\limits_{X_n } d(X_n ,l_\infty ^n )< \infty ,$$ where X_n ranges over all the n-dimensional subspaces of X.     

TheT-ideal generated by the standard identitys 3[x 1,x 2,x 3]

LetK = T_o(s₃), {c_n} its codimensions, {l_n} its colengths and {Χ_n} its sequence of co-characters. For 9≦n, c_n =2n - 1 or c_n =n(n + l)/2- 1, 3≦l_n ≦4 and χ_n =[n] + 2[n-1,1] + α[n-2,2] + β[2²,1^n?4] where α + β≦l.     

On the summability factors of infinite series

The author has established that if [λ_n] is a convex sequence such that the series Σn ^-1λ_n is convergent and the sequence {K _n} satisfies the condition |K _n|=O[log(n+1)]^k(C, 1),k?0, whereK _n denotes the (R, logn, 1) mean of the sequence {n log (n+1)a _n}, then the series Σlog(n+1)^1-kλ_n a _n is summable |R, logn, 1|. The result obtained for the particular casek=0 generalises a previous result of the author [1].     

A construction of Gray codes inducing complete graphs

A binary Gray code G(n) of length n, is a list of all 2ⁿn-bit codewords such that successive codewords differ in only one bit position. The sequence of bit positions where the single change occurs when going to the next codeword in G(n), denoted by S(n)?s₁,s₂,…,s_2ⁿ-1, is called the transition sequence of the Gray code G(n). The graph G_G(n) induced by a Gray code G(n) has vertex set {1,2,…,n} and edge set {{s_i,s_i+1}:1?i?2ⁿ-2}. If the first and the last codeword differ only in position s_2ⁿ, the code is cyclic and we extend the graph by two more edges {s_2ⁿ-1,s_2ⁿ} and {s_2ⁿ,s₁}. We solve a problem of Wilmer and Ernst [Graphs induced by Gray codes, Discrete Math. 257 (2002) 585-598] about a construction of an n-bit Gray code inducing the complete graph K_n. The technique used to solve this problem is based on a Gray code construction due to Bakos [A. Ádám, Truth Functions and the Problem of their Realization by Two-Terminal Graphs, Akadémiai Kiadó, Budapest, 1968], and which is presented in D.E. Knuth [The Art of Computer Programming, vol. 4, Addison-Wesley as part of “fascicle” 2, USA, 2005].     

Dimensional results for Cartesian products of homogeneous Moran sets

M(J, {m _s * n _s }, {c _s }) be the collection of Cartesian products of two homogenous Moran sets with the same ratios {c _s } where J = [0, 1]×[0, 1]. Then the maximal and minimal values of the Hausdorff dimensions for the elements in M are obtained without any restriction on {m _s n _s } or {c _s }.     

Normal family and the sequence of omitted functions

Let {fn} be a sequence of meromorphic functions on a plane domain D, whose zeros and poles have multiplicity at least 3. Let {hn} be a sequence of meromorphic functions on D, whose poles are multiple, such that {hn} converges locally uniformly in the spherical metric to a function h which is meromorphic and zero-free on D.If fn≠hn, then {fn} is normal on D.     

Quadratic Decomposition of a Laguerre-Hahn Polynomial Sequence II

In [4] the authors proved that a quasi-symmetric orthogonal polynomial sequence {R_n}_{n 3 0}{{\{R_{n}\}}_{n \ge 0}} is a Laguerre-Hahn sequence if and only if the component {P_n}_{n 3 0}{{\{P_{n}\}}_{n \ge 0}} in its quadratic decomposition is also a Laguerre-Hahn sequence. In this paper and under these conditions, we deduce the class s of the Laguerre-Hahn sequence {R_n}_{n 3 0}{{\{R_{n}\}}_{n \ge 0}}. More precisely, if s′ is the class of {P_n}_{n 3 0}{{\{P_{n}\}}_{n \ge 0}} then 2s′ ≤ s ≤ 2s′ + 3. On the other hand the polynomial coefficients of the Riccati equation satisfied by the Stieltjes function corresponding to {R_n}_{n 3 0}{{\{R_{n}\}}_{n \ge 0}} are given in terms of those of {P_n}_{n 3 0}{{\{P_{n}\}}_{n \ge 0}}. As an application, we determine all non-symmetric quasi-symmetric Laguerre-Hahn sequences of class one.     

Metrics and undirected cuts

Suppose thatG is an undirected graph whose edges have nonnegative integer-valued lengthsl(e), and that {s ₁,t ₁},?, {s _m ,t _m } are pairs of its vertices. Can one assign nonnegative weights to the cuts ofG such that, for each edgee, the total weight of cuts containinge does not exceedl(e) and, for eachi, the total weight of cuts ‘separating’s _i andt _i is equal to the distance (with respect tol) betweens _i andt _i ? Using linear programming duality, it follows from Papernov's multicommodity flow theorem that the answer is affirmative if the graph induced by the pairs {s ₁,t ₁},?, {s _m ,t _m } is one of the following: (i) the complete graph with four vertices, (ii) the circuit with five vertices, (iii) a union of two stars. We prove that if, in addition, each circuit inG has an even length (with respect tol) then there exists a suitable weighting of the cuts with the weights integer-valued; moreover, an algorithm of complexity O(n ³) (n is the number of vertices ofG) is developed for solving such a problem. Also a class of metrics decomposable into a nonnegative linear combination of cut-metrics is described, and it is shown that the separation problem for cut cones isNP-hard.     

Orthogonal Laurent polynomials and the strong Hamburger moment problem

This paper is concerned with the strong Hamburger moment problem (SHMP): For a given double sequence of real numbers C = {c_n}^∞_?∞, does there exist a real-valued, bounded, non-decreasing function ψ on (?∞, ∞) with infinitely many points of increase such that for every integer n, c_n = ∝^∞_?∞ (?t)ⁿdψ(t)? Necessary and sufficient conditions for the existence of such a function ψ are given in terms of the positivity of certain Hankel determinants associated with C. Our approach is made through the study of orthogonal (and quasi-orthogonal) Laurent polynomials (referred to here as L-polynomials) and closely related Gaussian-type quadrature formulas. In the proof of sufficiency an inner product for L-polynomials is defined in terms of the given double sequence C. Since orthogonal L-polynomials are believed to be of interest in themselves, some examples of specific systems are considered.     

Взаимоотношение меж ду сходимостью подпоследовательно стей частных сумм числово го ряда и его суммируе мостью методами (C, α) и Абеля

Numerical series \(\mathop \Sigma \limits_{n = 0}^\infty u_n\) with partial sumss _n are studied under the assumption that a subsequence \(\left\{ {S_{n_k } } \right\}_{k = 0}^\infty\) of the partial sums is convergent. Then a sequence {η _k } is chosen, by means of which a majorant of the termsu _n is constructed. Conditions on {n _k } and {η _k } are found which imply the (C, 1)-summability of the series∑ u _n (Theorem 1). In the meanwhile, it is proved that the (C, 1)-means in Theorem 1 cannot be replaced by (C, α)-means, if 0<α<1 (Theorem 2). On the other hand, if the assumption in Theorem 1 is not satisfied, then in certain cases the series∑ u _n preserves the property of (C, 1)-summability (Theorems 4 and 5), while in other cases it is not summable even by Abel means (Theorems 3 and 6).     

Universal meromorphic approximation on Vitushkin sets

The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ _n } ? ?, there exists a function ?, meromorphic on ?, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {n _k } of ? such that $ \left\{ {\varphi \left( {z + \lambda _{n_k } } \right)} \right\} The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ _n } ⊂ ℂ, there exists a function ϕ, meromorphic on ℂ, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {n _k } of ℕ such that converges to f(z) uniformly on K. A similar result is obtained for arbitrary domains G ≠ ℂ. Moreover, in case {λ _n }={n} the function ϕ is frequently universal in terms of Bayart/Grivaux [3]. Original Russian Text ? W.Luh, T.Meyrath, M.Niess, 2008, published in Izvestiya NAN Armenii. Matematika, 2008, No. 6, pp. 66–75.     

Regions of meromorphy and value distribution of geometrically converging rational functions

Let D be a region, {rn}_n∈N a sequence of rational functions of degree at most n and let each rn have at most m poles in D, for m∈N fixed. We prove that if {rn}_n∈N converges geometrically to a function f on some continuum S⊂D and if the number of zeros of rn in any compact subset of D is of growth o(n) as n→∞, then the sequence {rn}_n∈N converges m₁-almost uniformly to a meromorphic function in D. This result about meromorphic continuation is used to obtain Picard-type theorems for the value distribution of m₁-maximally convergent rational functions, especially in Padé approximation and Chebyshev rational approximation.     

Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences

In 2012,Zhi-Wei Sun posed many conjectures about the monotonicity of sequences of form {n√zn},where {zn} is a familiar number-theoretic or combinatorial sequence. We show that if the sequence {zn+1/zn}is increasing(resp.,decreasing),then the sequence {n√zn} is strictly increasing(resp.,decreasing) subject to a certain initial condition. We also give some sufficient conditions when {zn+1/zn} is increasing,which is equivalent to the log-convexity of {zn}. As consequences,a series of conjectures of Zhi-Wei Sun are verified in a unified approach.     

Sun's Log-concavity Conjecture on the Catalan–Larcombe–French Sequence

Let {P_n}n≥0 denote the Catalan-Larcombe-French sequence,which naturally came from the series expansion of the complete elliptic integral of the first kind.In this paper,we prove the strict log-concavity of the sequence {P_n~(1/n)}_n≥1,which was originally conjectured by Z.W.Sun.We also obtain the strict log-concavity of the sequence {V_n~(1/n)}_n≥1,where {V_n}_n≥0 is the Fennessey-LarcombeFrench sequence arising from the series expansion of the complete elliptic integral of the second kind.     

Functions generating normal Toeplitz matrices

To any complex function there corresponds a Fourier series, which is often associated with a sequence {T _n} of Toeplitz n × n matrices. Functions whose Fourier series generate sequences of normal Toeplitz matrices are classified, and a procedure for constructing Fourier series for which the sequence {T _n} contains an infinite subsequence of normal matrices is described.     

Some results related with statistical convergence and Berezin symbols

The concept of discrete statistical Abel convergence is introduced. In terms of Berezin symbols we present necessary and sufficient condition under which a series with bounded sequence {an}_n?0 of complex numbers is discrete statistically Abel convergent. By using concept of statistical convergence we also give slight strengthening of a result of Gokhberg and Krein on compact operators.     

Diophantine inequalities with mixed powers

Let {λ_i}_{i = 1}^s (s ≥ 2) be a finite sequence of non-zero real numbers, not all of the same sign and in which not all the ratios $\text{λ}{}_{\mathrm{i}}\text{λ}{}_{\mathrm{j}}$ are rational. A given sequence of positive integers {n_i}_{i = 1}^s is said to have property (P) (($\text{P}{}^1$) respectively) if for any {λ_i}_{i = 1}^s and any real number η, there exists a positive constant σ, depending on {λ_i}_{i = 1}^s and {n_i}_{i = 1}^s only, so that the inequality |η + Σ_{i = 1}^sλ_ix_iⁿⁱ| < (max x_i)^?σ has infinitely many solutions in positive integers (primes respectively) x₁, x₂,…, x_s. In this paper, we prove the following result: Given a sequence of positive integers {n_i}_{i = 1}^∞, a necessary and sufficient condition that, for any positive integer j, there exists an integer s, depending on {n_i}_{i = j}^∞ only, such that {n_i}_{i = j}^{j + s ? 1} has property (P) (or ($\text{P}{}^1$)), is that Σ_{i = 1}^∞n_i^?1 = ∞. These are parallel to some striking results of G. A. Fre?man, E. J. Scourfield and K. Thanigasalam.     

Mean-Field Conditions for Percolation on Finite Graphs

Let {G _n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G _n | = n. Denote by p ^t (v, v) the return probability after t steps of the non-backtracking random walk on G _n . We show that if p ^t (v, v) has quasi-random properties, then critical bond-percolation on G _n behaves as it would on a random graph. More precisely, if $\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}$ then the size of the largest component in p-bond-percolation with ${p =\frac{1+O(n^{-1/3})}{d-1}}Let {G _n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G _n | = n. Denote by p ^t (v, v) the return probability after t steps of the non-backtracking random walk on G _n . We show that if p ^t (v, v) has quasi-random properties, then critical bond-percolation on G _n behaves as it would on a random graph. More precisely, if

lim sup  n n1/3 ?t = 1n1/3 tpt(v,v) < ￥,\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}