Sperner''s theorem with constraints

A set X of subsets of an n-element set S is called an anti-chain if no two elements of X are related by set-wise inclusion. Sperner showed [8] that max |X|=(ⁿ_[n/2]), where |X| denotes the number of elements in X and the maximum is taken over all anti-chains of subsets of S.

Let non-negative integers i_o<n and m_io≠0, m_io+1,…m_n be given. In this paper we give an algorithm for calculating max |X| where the maximum is taken only over anti-chains containing exactly m_i i-element subsets of S for i_o i n.     

MEROMORPHIC FUNCTIONS SHARING TWO FINITE SETS

Let S1 ＝ {∞} and S2 ＝ {w: Ps(w)＝ 0}, Ps(w) being a uniqueness polynomial under some restricted conditions. Then, for any given nonconstant meromorphic function f, there exist at most finitely many nonconstant meromorphic functions g such that f-1(Si) ＝ g-1(Si)(i ＝ 1,2), where f-1(Si) and g-1(Si) denote the pull-backs of Si considered as a divisor, namely, the inverse images of Si counted with multiplicities, by f and g respectively.     

Flat portions on the boundary of the numerical ranges of certain Toeplitz matrices

We give criterions for a flat portion to exist on the boundary of the numerical range of a matrix. A special type of Teoplitz matrices with flat portions on the boundary of its numerical range are constructed. We show that there exist 2 × 2 nilpotent matrices A₁,A₂, an n  × n nilpotent Toeplitz matrix N_n, and an n  × n cyclic permutation matrix S_n(s) such that the numbers of flat portions on the boundaries of W(A₁⊕N_n) and W(A₂⊕S_n(s)) are, respectively, 2(n - 2) and 2n.     

Cycle decompositions of crowns

For an integer n3, the crown S_n⁰ is defined to be the graph with vertex set {x₀,x₁,…,x_n−1,y₀,y₁,…,y_n−1} and edge set {x_iy_j: 0i,jn−1, i≠j}. In this paper we give some sufficient conditions for the edge decomposition of the crown into isomorphic cycles.     

On some Rado numbers for generalized arithmetic progressions

The 2-color Rado number for the equation x₁+x₂−2x₃=c, which for each constant we denote by S₁(c), is the least integer, if it exists, such that every 2-coloring, Δ : [1,S₁(c)]→{0,1}, of the natural numbers admits a monochromatic solution to x₁+x₂−2x₃=c, and otherwise S₁(c)=∞. We determine the 2-color Rado number for the equation x₁+x₂−2x₃=c, when additional inequality restraints on the variables are added. In particular, the case where we require x₂<x₃<x₁, is a generalization of the 3-term arithmetic progression; and the work done here improves previously established upper bounds to an exact value.     

Some results on integral sum graphs

Let Z denote the set of all integers. The integral sum graph of a finite subset S of Z is the graph (S,E) with vertex set S and edge set E such that for u,vS, uvE if and only if u+vS. A graph G is called an integral sum graph if it is isomorphic to the integral sum graph of some finite subset S of Z. The integral sum number of a given graph G, denoted by ζ(G), is the smallest number of isolated vertices which when added to G result in an integral sum graph. Let x denote the least integer not less than the real x. In this paper, we (i) determine the value of ζ(K_n−E(K_r)) for r2n/3−1, (ii) obtain a lower bound for ζ(K_n−E(K_r)) when 2r<2n/3−1 and n5, showing by construction that the bound is sharp when r=2, and (iii) determine the value of ζ(K_r,r) for r2. These results provide partial solutions to two problems posed by Harary (Discrete Math. 124 (1994) 101–108). Finally, we furnish a counterexample to a result on the sum number of K_r,s given by Hartsfiedl and Smyth (Graphs and Matrices, R. Rees (Ed.), Marcel, Dekker, New York, 1992, pp. 205–211).     

A chromatic partition polynomial

A polynomial in two variables is defined by C_n(x,t)=Σ_πΠnx(Gπ,x)t^|π|, where Π_n is the lattice of partitions of the set {1, 2, …, n}, G_π is a certain interval graph defined in terms of the partition gp, χ(G_π, x) is the chromatic polynomial of G_π and |π| is the number of blocks in π. It is shown that , where S(n, i) is the Stirling number of the second kind and (x)_i = x(x − 1) ··· (x − i + 1). As a special case, C_n(−1, −t) = A_n(t), where A_n(t) is the nth Eulerian polynomial. Moreover, A_n(t)=Σ_πΠna_πt^|π| where a_π is the number of acyclic orientations of G_π.     

The existence of Schröder designs with equal-sized holes

A holey Schröder design of type h₁^n₁h₂^n₂ … h_k^n_k (HSD(h₁^n₁h₂^n₂ … h_k^n_k)) is equivalent to a frame idempotent Schröder quasigroup (FISQ(h₁^n₁h₂^n₂ … h_k^n_k)) of order n with n_i missing subquasigroups (holes) of order h_i, (1 i k), which are disjoint and spanning, that is, Σ_{1 i k} n_ih_i = n. In this paper, it is shown that an HSD(hⁿ) exists if and only if h²n(n − 1) 0 (mod 4) with expceptions (h, n) ε {{(1,5),(1,9),(2,4)}} and the possible exception of (h, n) = (6,4).     

Individual behaviors of oriented walks

Given an infinite sequence t=(_k)_k of −1 and +1, we consider the oriented walk defined by S_n(t)=∑_k=1ⁿ₁₂…_k. The set of t's whose behaviors satisfy S_n(t)bn^τ is considered ( and 0<τ1 being fixed) and its Hausdorff dimension is calculated. A two-dimensional model is also studied. A three-dimensional model is described, but the problem remains open.     

An extension theorem for t-designs

In 1994, van Trung (Discrete Math. 128 (1994) 337–348) [9] proved that if, for some positive integers d and h, there exists an S_λ(t,k,v) such that

then there exists an S_λ(v−t+1)(t,k,v+1) having v+1 pairwise disjoint subdesigns S_λ(t,k,v). Moreover, if B_i and B_j are any two blocks belonging to two distinct such subdesigns, then d|B_i∩B_j|<k−h. In 1999, Baudelet and Sebille (J. Combin. Des. 7 (1999) 107–112) proved that if, for some positive integers, there exists an S_λ(t,k,v) such that

where m=min{s,v−k} and n=min{i,t}, then there exists an

having pairwise disjoint subdesigns S_λ(t,k,v). The purpose of this paper is to generalize these two constructions in order to produce a new recursive construction of t-designs and a new extension theorem of t-designs.     

About “On certain incomplete statistics” by Lima et al.

Lima et al. recently claim that (Chaos, Solitons & Fractals 19 (2004) 1005) the entropy for the incomplete statistics based on the normalization ∑_ip_i^q=1 should be S=−∑_ip_i^2q−1ln_qp_i instead of S=−∑_ip_i^qln_qp_i initially proposed by Wang. We indicate here that this conclusion is a result of erroneous use of temperature definition for the incomplete statistics.     

Equality of higher numerical ranges of matrices and a conjecture of Kippenhahn on Hermitian pencils

Let M_n be the algebra of all n × n complex matrices. For 1 k n, the kth numerical range of A M_n is defined by W_k(A) = (1/k)∑_j^k=1x_j^*Ax_j : x₁, …, x_k is an orthonormal set in ⁿ]. It is known that tr A/n = W_n(A) W_n−1(A) W₁(A). We study the condition on A under which W_m(A) = W_k(A) for some given 1 m < k n. It turns out that this study is closely related to a conjecture of Kippenhahn on Hermitian pencils. A new class of counterexamples to the conjecture is constructed, based on the theory of the numerical range.     

On certain incomplete statistics

The foundations of the incomplete statistics recently proposed by Wang is rediscussed in the context of the canonical statistical ensemble. It is found that the incomplete normalization condition, ∑p^q_i=1 (i=1,…,w), where p_i is the probability of a given microstate, is not compatible with the entropic non-extensive formula proposed by Tsallis. It is proved that the entropic function proposed by Wang must be written as S_q=−k_B∑p_i^2q−1ln_qp_i, whereas the form proposed by Tsallis namely, S_q=−k_B∑p_i^qln_qp_i, is directly associated with the standard normalization condition (∑_ip_i=1).     

Complexity of projected images of convex subdivisions

Let S be a subdivision of ^d into n convex regions. We consider the combinatorial complexity of the image of the (k - 1)-skeleton of S orthogonally projected into a k-dimensional subspace. We give an upper bound of the complexity of the projected image by reducing it to the complexity of an arrangement of polytopes. If k = d − 1, we construct a subdivision whose projected image has Ω(n^(3d−2)/2) complexity, which is tight when d 4. We also investigate the number of topological changes of the projected image when a three-dimensional subdivision is rotated about a line parallel to the projection plane.     

Weight distribution of the bases of a binary matroid

Let M be a weighted binary matroid and w₁ < … < w_m be the increasing sequence of all possible distinct weights of bases of M. We give a sufficient condition for the property that w₁, …, w_m is an arithmetical progression of common difference d. We also give conditions which guarantee that w_i+1 − w_i ≤ d, 1 ≤ i ≤ m −1. Dual forms for these results are given also.     

Extremal properties of strong quadrature weights and maximal mass results for truncated strong moment problems

A bisequence of complex numbers {μ_n}_−∞^∞ determines a strong moment functional satisfying L[xⁿ] = μ_n. If is positive-definite on a bounded interval (a,b) R{0}, then has an integral representation , n=0, ±1, ±2,…, and quadrature rules {w_ni,x_ni} exist such that μ_k = ∑_i=i^n_ns_ni^kw_ni. This paper is concerned with establishing certain extremal properties of the weights w_ni and using these properties to obtain maximal mass results satisfied by distributions ψ(x) representing when only a finite bisequence of moments {μ_k}_k=−nⁿ⁻¹ is given.     

Strong approximation for cross-covariances of linear variables with long-range dependence

Suppose {_k, −∞ < k < ∞} is an independent, not necessarily identically distributed sequence of random variables, and {c_j}^∞_j=0, {d_j}^∞_j=0 are sequences of real numbers such that Σ_jc²_j < ∞, Σ_jd²_j < ∞. Then, under appropriate moment conditions on {_k, −∞ < k < ∞}, y_k Σ^∞_j=0c_jk-j, z_k Σ^∞_j=0d_jk-j exist almost surely and in ⁴ and the question of Gaussian approximation to S_[t]Σ^[t]_k=1 (y_k z_k − E{y_k z_k}) becomes of interest. Prior to this work several related central limit theorems and a weak invariance principle were proven under stationary assumptions. In this note, we demonstrate that an almost sure invariance principle for S_[t], with error bound sharp enough to imply a weak invariance principle, a functional law of the iterated logarithm, and even upper and lower class results, also exists. Moreover, we remove virtually all constraints on _k for “time” k ≤ 0, weaken the stationarity assumptions on {_k, −∞ < k < ∞}, and improve the summability conditions on {c_j}^∞_j=0, {d_j}^∞_j=0 as compared to the existing weak invariance principle. Applications relevant to this work include normal approximation and almost sure fluctuation results in sample covariances (let d_j = c_j-m for j ≥ m and otherwise 0), quadratic forms, Whittle's and Hosoya's estimates, adaptive filtering and stochastic approximation.     

A faster 2-approximation algorithm for the minmax p-traveling salesmen problem on a tree

Given an edge-weighted tree T and an integer p1, the minmax p-traveling salesmen problem on a tree T asks to find p tours such that the union of the p tours covers all the vertices. The objective is to minimize the maximum of length of the p tours. It is known that the problem is NP-hard and has a (2−2/(p+1))-approximation algorithm which runs in O(p^p−1n^p−1) time for a tree with n vertices. In this paper, we consider an extension of the problem in which the set of vertices to be covered now can be chosen as a subset S of vertices and weights to process vertices in S are also introduced in the tour length. For the problem, we give an approximation algorithm that has the same performance guarantee, but runs in O((p−1)!·n) time.     

Boundedness and asymptotic behavior of the solutions of a fuzzy difference equation

In this paper we study the existence, the uniqueness, the boundedness and the asymptotic behavior of the positive solutions of the fuzzy difference equation x_n+1=∑_i=0^kA_i/x_n−i^p_i, where k{1,2,…,}, A_i, i{0,1,…,k}, are positive fuzzy numbers, p_i, i{0,1,…,k}, are positive constants and x_i, i{−k,−k+1,…,0}, are positive fuzzy numbers.     

Convexoid and generalized derivations

Let E,F be two Banach spaces and let S be a symmetric norm ideal of L(E,F). For AL(F) and BL(E) the generalized derivation δ_S,A,B is the operator on S that sends X to AX−XB. A bounded linear operator is said to be convexoid if its (algebraic) numerical range coincides with the convex hull of its spectrum. We show that δ_S,A,B is convexoid if and only if A and B are convexoid.