On the density of 2‐colorable 3‐graphs in which any four points span at most two edges

Let ex₂(n, K) be the maximum number of edges in a 2‐colorable K‐free 3‐graph (where K={123, 124, 134} ). The 2‐chromatic Turán density of K is $\pi_{2}({K}_{4}^-) =lim_{{n}\to \infty} {ex}_{2}({n}, {K}_{4}^-)/\left(_{3}^{n}\right)Let ex₂(n, K) be the maximum number of edges in a 2‐colorable K‐free 3‐graph (where K={123, 124, 134} ). The 2‐chromatic Turán density of K is $\pi_{2}({K}_{4}^-) =lim_{{n}\to \infty} {ex}_{2}({n}, {K}_{4}^-)/\left(_{3}^{n}\right)$. We improve the previously best known lower and upper bounds of 0.25682 and 3/10?ε, respectively, by showing that This implies the following new upper bound for the Turán density of K In order to establish these results we use a combination of the properties of computer‐generated extremal 3‐graphs for small n and an argument based on “super‐saturation”. Our computer results determine the exact values of ex(n, K) for n≤19 and ex₂(n, K) for n≤17, as well as the sets of extremal 3‐graphs for those n. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 105–114, 2010     

Global attractors for 2‐D wave equations with displacement‐dependent damping

In this paper the long‐time behaviour of the solutions of 2‐D wave equation with a damping coefficient depending on the displacement is studied. It is shown that the semigroup generated by this equation possesses a global attractor in H(Ω) × L₂(Ω) and H²(Ω)∩H(Ω) × H(Ω). Copyright © 2009 John Wiley & Sons, Ltd.     

On determinantal representation for the generalized inverse A and its applications

In this paper, first we establish a determinantal representation for the group inverse A_g of a square matrix A. Based on this, a determinantal representation for the generalized inverse A is presented. As an application, we give a determinantal formula for the unique solution of the general restricted linear system: Ax=b(x ∈ T, b ∈ AT and dim(AT)=dim(T)), which reduces to the common Cramer rule if A is non‐singular. These results extend our earlier work. Copyright © 2006 John Wiley & Sons, Ltd.     

Approximation and eigenvalue extrapolation of biharmonic eigenvalue problem by nonconforming finite element methods

In this paper, we analyze the biharmonic eigenvalue problem by two nonconforming finite elements, Q and E Q. We obtain full order convergence rate of the eigenvalue approximations for the biharmonic eigenvalue problem based on asymptotic error expansions for these two nonconforming finite elements. Using the technique of eigenvalue error expansion, the technique of integral identities, and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Finite element method by using quartic B‐splines

In this article, we discuss a kind of finite element method by using quartic B‐splines to solve Dirichlet problem for elliptic equations. Bivariate spline proper subspace of S(Δ) satisfying homogeneous boundary conditions on Type‐2 triangulations and quadratic B‐spline interpolating boundary functions are primarily constructed. Linear and nonlinear elliptic equations are solved by Galerkin quartic B‐spline finite element method. Numerical examples are provided to illustrate the proposed method is flexible. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 818–828, 2011     

Constructions for Circulant and Group‐Developed Generalized Weighing Matrices

An elementary construction yields a new class of circulant (so‐called “Butson‐type”) generalized weighing matrices, which have order and weight n², all of whose entries are nth roots of unity, for all positive integers , where . The idea is extended to a wider class of constructions giving various group‐developed generalized weighing matrices.     

On Ramsey‐type positional games

Let K denote the graph obtained from the complete graph K_s+t by deleting the edges of some K_t‐subgraph. We prove that for each fixed s and sufficiently large t, every graph with chromatic number s+t has a K minor. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 343–350, 2010     

Acyclic edge chromatic number of outerplanar graphs

A proper edge coloring of a graph G is called acyclic if there is no 2‐colored cycle in G. The acyclic edge chromatic number of G, denoted by χ(G), is the least number of colors in an acyclic edge coloring of G. In this paper, we determine completely the acyclic edge chromatic number of outerplanar graphs. The proof is constructive and supplies a polynomial time algorithm to acyclically color the edges of any outerplanar graph G using χ(G) colors. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 22–36, 2010     

Cycle decompositions of complete multigraphs

In this paper we establish necessary and sufficient conditions for decomposing the complete multigraph λK_n into cycles of length λ, and the λ‐fold complete symmetric digraph λK into directed cycles of length λ. As a corollary to these results we obtain necessary and sufficient conditions for decomposing λK_n (respectively, λK) into cycles (respectively, directed cycles) of prime length. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 85–93, 2010     

On an elementary operator with M‐hyponormal operator entries

A Hilbert space operator is M‐hyponormal if there exists a positive real number M such that for all . Let be M‐hyponormal and let denote either the generalized derivation or the elementary operator . We prove that if are M‐hyponormal, then satisfies the generalized Weyl's theorem and satisfies the generalized a‐Weyl's theorem for every f that is analytic on a neighborhood of .     

Some combinatorial principles for trees and applications to tree families in Banach spaces

Suppose that is a normalized family in a Banach space indexed by the dyadic tree S. Using Stern's combinatorial theorem we extend important results from sequences in Banach spaces to tree‐families. More precisely, assuming that for any infinite chain β of S the sequence is weakly null, we prove that there exists a subtree T of S such that for any infinite chain β of T the sequence is nearly (resp., convexly) unconditional. In the case where is a family of continuous functions, under some additional assumptions, we prove the existence of a subtree T of S such that for any infinite chain β of T, the sequence is unconditional. Finally, in the more general setting where for any chain β, is a Schauder basic sequence, we obtain a dichotomy result concerning the semi‐boundedly completeness of the sequences .     

On series of signed vectors and their rearrangements

Let x₁,…,x_m∈ \input amssym $ \Bbb R$ n be a sequence of vectors with ∥x_i∥₂ ≤ 1 for all i. It is proved that there are signs ε₁,…,ε_m = ±1 such that where C₁, C₂ are some numerical constants. It is also proved that there are signs ε,…,ε = ±1 and a permutation π of {1,…,m} such that where C^′ is some other numerical constant. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Minors in ‐ Chromatic Graphs,II

Let denote the graph obtained from the complete graph by deleting the edges of some ‐subgraph. The author proved earlier that for each fixed s and , every graph with chromatic number has a minor. This confirmed a partial case of the corresponding conjecture by Woodall and Seymour. In this paper, we show that the statement holds already for much smaller t, namely, for .     

Simple Signed Steiner Triple Systems

Let X be a v‐set, be a set of 3‐subsets (triples) of X, and be a partition of with . The pair is called a simple signed Steiner triple system, denoted by ST, if the number of occurrences of every 2‐subset of X in triples is one more than the number of occurrences in triples . In this paper, we prove that exists if and only if , , and , where and for , . © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 332–343, 2012     

Reversible and DRAD difference sets in

The existence of Hadamard difference sets has been a central question in design theory. Reversible difference sets have been studied extensively. Dillon gave a method for finding reversible difference sets in groups of the form (C)². DRAD difference sets are a newer concept. Davis and Polhill showed the existence of DRAD difference sets in the same groups as Dillon. This article determines the existence of reversible and DRAD difference sets in groups of the form (C)³. These are the only abelian 2‐groups outside of direct products of C₄ and (C)² known to contain reversible and DRAD difference sets. © 2011 Wiley Periodicals, Inc. J Combin Designs 20:58–67, 2012     

On induced Ramsey numbers for k‐uniform hypergraphs

Let denote the complete k‐uniform k‐partite hypergraph with classes of size t and the complete k‐uniform hypergraph of order s. One can show that the Ramsey number for and satisfies when t = s^o(1) as s →∞. The main part of this paper gives an analogous result for induced Ramsey numbers: Let be an arbitrary k‐partite k‐uniform hypergraph with classes of size t and an arbitrary k‐graph of order s. We use the probabilistic method to show that the induced Ramsey number (i.e. the smallest n for which there exists a hypergraph such that any red/blue coloring of yields either an induced red copy of or an induced blue copy of ) satisfies . © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 5–20, 2016     

4‐*GDD(6n)s and Related Optimal Quaternary Constant‐Weight Codes

Constant‐weight codes (CWCs) have played an important role in coding theory. To construct CWCs, a K‐GDD (where GDD is group divisible design) with the “star” property, denoted by K‐*GDD, was introduced, in which any two intersecting blocks intersect in at most two common groups. In this paper, we consider the existence of 4‐*GDDs. Previously, the necessary conditions for existence were shown to be sufficient for , and also sufficient for with prime powers and . We continue to investigate the existence of 4‐*GDD(6ⁿ)s and show that the necessary condition for the existence of a 4‐*GDD(6ⁿ), namely, , is also sufficient. The known results on the existence of optimal quaternary (n, 5, 4) CWCs are also extended.     

Decomposition of Complete Graphs into Isomorphic Complete Bipartite Graphs

A decomposition of a complete graph into disjoint copies of a complete bipartite graph is called a ‐design of order n. The existence problem of ‐designs has been completely solved for the graphs for , for , K_{2, 3} and K_{3, 3}. In this paper, I prove that for all , if there exists a ‐design of order N, then there exists a ‐design of order n for all (mod ) and . Giving necessary direct constructions, I provide an almost complete solution for the existence problem for complete bipartite graphs with fewer than 18 edges, leaving five orders in total unsolved.     

On the Choice Number of Packings

In this note, we show that for positive integers s and k, there is a function such that every t‐ packing with at least edges, , has choice number greater than s. Consequently, for integers s, k, t, and λ there is a such that every t‐ design with has choice number greater than s. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 504‐507, 2012     

A note on equivalence relations

We consider the equivalence relations on induced by the Banach subspaces . We show that if , then there is no Borel reduction from the equivalence relation , where X is a Banach space, to .