Some extensions of a property of linear representation functions

Let A={a₁,a₂,…}(a₁<a₂<) be an infinite sequence of nonnegative integers. Let k≥2 be a fixed integer and for , let R_k(A,n) be the number of solutions of a_i1++a_ik=n,a_i1,…,a_ikA, and let and denote the number of solutions with the additional restrictions a_i1<<a_ik, and a_i1≤≤a_ik respectively. Recently, Horváth proved that if d>0 is an integer, then there does not exist n₀ such that for n>n₀. In this paper, we obtain the analogous results for R_k(A,n), and .     

An Inequality on the Size of a Set in a Cartesian Product

Let Ω be a finite subset of the Cartesian productW₁  ×   × W_nof n sets. ForA    {1, 2, , n }, denote by Ω_Athe projection ofΩ onto the Cartesian product of W_i, i   A. Generalizing an inequality given in an article by Shen, we prove that | Ω |² ≤  |Ω_A1 || Ω_Ak| provided that { A₁, , A_k} is a double cover of {1, 2, , n }. This inequality is applied to give some bounds on the numbers of special subgraphs of a graph.     

Constructing set systems with prescribed intersection sizes

Let f be an n-variable polynomial with positive integer coefficients, and let be a set system on the n-element universe. We define a set system and prove that f(H_i1∩H_i2∩∩H_ik)=|G_i1∩G_i2∩∩G_ik|, for any 1km, where f(H_i1∩H_i2∩∩H_ik) denotes the value of f on the characteristic vector of H_i1∩H_i2∩∩H_ik. The construction of is a straightforward polynomial-time algorithm from the set system and the polynomial f. In this paper we use this algorithm for constructing set systems with prescribed intersection sizes modulo an integer. As a by-product of our method, some upper bounds on the number of sets in set systems with prescribed intersection sizes are extended.     

Primitive normal polynomials with a prescribed coefficient

In this paper, we established the existence of a primitive normal polynomial over any finite field with any specified coefficient arbitrarily prescribed. Let n15 be a positive integer and q a prime power. We prove that for any aF_q and any 1m<n, there exists a primitive normal polynomial f(x)=xⁿ−σ₁xⁿ⁻¹++(−1)ⁿ⁻¹σ_n−1x+(−1)ⁿσ_n such that σ_m=a, with the only exceptions σ₁≠0. The theory can be extended to polynomials of smaller degree too.     

Roots under convolution of sequences

The convolution a * b of the sequences a = a₀, a₁, a₂, and b is the sequence with elements ∑₀ⁿ a_kb_{n − k}. One sets 1, 1, 1, equal to σ. Given that a * a with a ≥ 0 is close to σ * σ, how close is a to σ? More generally, one asks how close a is to σ if the p-th convolution power, a*P with a ≥ 0, is close to σ*P. Power series and complex analysis form a natural tool to estimate the ‘summed deviation’ ρ = σ * (a — σ) in terms of b = a * a — σ * σ or b = a*P − σ*P. Optimal estimates are found under the condition ∑_k=0ⁿ b_k² = %plane1D;512;(n^{2β + 1}) whenever −½ < β < p − 1. It is not known what the optimal estimates are for the special case b_n = %plane1D;512;(n^β).     

Linked graphs with restricted lengths

A graph G is k-linked if G has at least 2k vertices, and for every sequence x₁,x₂,…,x_k,y₁,y₂,…,y_k of distinct vertices, G contains k vertex-disjoint paths P₁,P₂,…,P_k such that P_i joins x_i and y_i for i=1,2,…,k. Moreover, the above defined k-linked graph G is modulo (m₁,m₂,…,m_k)-linked if, in addition, for any k-tuple (d₁,d₂,…,d_k) of natural numbers, the paths P₁,P₂,…,P_k can be chosen such that P_i has length d_i modulo m_i for i=1,2,…,k. Thomassen showed that, for each k-tuple (m₁,m₂,…,m_k) of odd positive integers, there exists a natural number f(m₁,m₂,…,m_k) such that every f(m₁,m₂,…,m_k)-connected graph is modulo (m₁,m₂,…,m_k)-linked. For m₁=m₂=…=m_k=2, he showed in another article that there exists a natural number g(2,k) such that every g(2,k)-connected graph G is modulo (2,2,…,2)-linked or there is XV(G) such that |X|4k−3 and G−X is a bipartite graph, where (2,2,…,2) is a k-tuple.We showed that f(m₁,m₂,…,m_k)max{14(m₁+m₂++m_k)−4k,6(m₁+m₂++m_k)−4k+36} for every k-tuple of odd positive integers. We then extend the result to allow some m_i be even integers. Let (m₁,m₂,…,m_k) be a k-tuple of natural numbers and ℓk such that m_i is odd for each i with ℓ+1ik. If G is 45(m₁+m₂++m_k)-connected, then either G has a vertex set X of order at most 2k+2ℓ−3+δ(m₁,…,m_ℓ) such that G−X is bipartite or G is modulo (2m₁,…,2m_ℓ,m_ℓ+1,…,m_k)-linked, where

Our results generalize several known results on parity-linked graphs.     

An iterative approximation procedure for the distribution of the maximum of a random walk

Let I(F) be the distribution function (d.f.) of the maximum of a random walk whose i.i.d. increments have the common d.f. F and a negative mean. We derive a recursive sequence of embedded random walks whose underlying d.f.'s F_k converge to the d.f. of the first ladder variable and satisfy FF₁F₂ on [0,∞) and I(F)=I(F₁)=I(F₂)=. Using these random walks we obtain improved upper bounds for the difference of I(F) and the d.f. of the maximum of the random walk after finitely many steps.     

Stable extendibility of vector bundles over lens spaces mod 3 and the stable splitting problem

Let Lⁿ(3) denote the (2n+1)-dimensional standard lens space mod 3. In this paper, we study the conditions for a given real vector bundle over Lⁿ(3) to be stably extendible to L^m(3) for every mn, and establish the formula on the power ζ^k=ζζ (k-fold) of a real vector bundle ζ over Lⁿ(3). Moreover, we answer the stable splitting problem for real vector bundles over Lⁿ(3) by means of arithmetic conditions.     

On sharp Burkholder–Rosenthal-type inequalities for infinite-degree U-statistics

In this paper, we present a method that allows one to obtain a number of sharp inequalities for expectations of functions of infinite-degree U-statistics. Using the approach, we prove, in particular, the following result: Let D be the class of functions f :R₊→R₊ such that the function f(x+z)−f(x) is concave in xR₊ for all zR₊. Then the following estimate holds: for all fD and all U-statistics ∑_1i1<<ilnY_i1,…,il(X_i1,…,X_il) with nonnegative kernels Y_i1,…,il :R^l→R₊, 1i_kn; i_r≠i_s, r≠s; k,r,s=1,…,l; l=0,…,m, in independent r.v.'s X₁,…,X_n. Similar inequality holds for sums of decoupled U-statistics. The class D is quite wide and includes all nonnegative twice differentiable functions f such that the function f″(x) is nonincreasing in x>0, and, in particular, the power functions f(x)=x^t, 1<t2; the power functions multiplied by logarithm f(x)= (x+x₀)^t ln(x+x₀), 1<t<2, x₀max(e^{(3t2−6t+2)/(t(t−1)(2−t))},1); and the entropy-type functions f(x)=(x+x₀)ln(x+x₀), x₀1. As an application of the results, we determine the best constants in Burkholder–Rosenthal-type inequalities for sums of U-statistics and prove new decoupling inequalities for those objects. The results obtained in the paper are, to our knowledge, the first known results on the best constants in sharp moment estimates for U-statistics of a general type.     

Existence results for -point boundary value problem of second order ordinary differential equations

This study concerns the existence of positive solutions to the boundary value problemwhere ξ_i(0,1) with 0<ξ₁<ξ₂<<ξ_n-2<1, a_i, b_i[0,∞) with and . By applying the Krasnoselskii's fixed-point theorem in Banach spaces, some sufficient conditions guaranteeing the existence of at least one positive solution or at least two positive solutions are established for the above general n-point boundary value problem.