On the Enumeration of ‐Optimal and Minimax‐Optimal k‐Circulant Supersaturated Designs

In recent years, several methods have been proposed for constructing ‐optimal and minimax‐optimal supersaturated designs (SSDs). However, until now the enumeration problem of such designs has not been yet considered. In this paper, ‐optimal and minimax‐optimal k‐circulant SSDs with 6, 10, 14, 18, 22, and 26 runs, factors and are enumerated in a computer search. We have also enumerated all ‐optimal and minimax‐optimal k‐circulant SSDs with (mod 4) and . The computer search utilizes the fact that theses designs are equivalent to certain 1‐rotational resolvable balanced incomplete block designs. Combinatorial properties of these resolvable designs are used to restrict the search space.     

Comparing methods for design follow‐up: revisiting a metal‐cutting case study

Adding another fraction to an initial fractional factorial design is often required to resolve ambiguities with respect to aliasing of factorial effects from the initial experiment and/or to improve estimation precision. Multiple techniques for design follow‐up exist; the choice of which is often made on the basis of the initial design and its analysis, resources available, experimental objectives, and so on. In this paper, we compare four design follow‐up strategies: foldover, semifoldover, D‐optimal, and Bayesian (MD‐optimal) in the context of a metal‐cutting case study previously utilized to compare fractional factorials of different run sizes. Follow‐up designs are compared for each of a , , and Plackett–Burman initial experiments. Our empirical results suggest that a single follow‐up strategy does not outperform all others in every situation. This case study serves to illustrate design augmentation possibilities for practitioners and provides some basis for the selection of a follow‐up experiment. Copyright © 2013 John Wiley & Sons, Ltd.     

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 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     

Finiteness Results in Triangle‐Free Quasi‐Symmetric Designs

Triangle‐free quasi‐symmetric 2‐ designs with intersection numbers ; and are investigated. Possibility of triangle‐free quasi‐symmetric designs with or is ruled out. It is also shown that, for a fixed x and a fixed ratio , there are only finitely many triangle‐free quasi‐symmetric designs. © 2012 Wiley Periodicals, Inc. J Combin Designs 00: 1‐6, 2012     

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     

Degree‐ and Orbit‐Balanced Γ‐Designs When Γ Has Five Vertices

A Γ‐design of the complete graph is a set of subgraphs isomorphic to Γ (blocks) whose edge‐sets partition the edge‐set of . is balanced if the number of blocks containing x is the same number of blocks containing y for any two vertices x and y. is orbit‐balanced, or strongly balanced, if the number of blocks containing x as a vertex of a vertex‐orbit A of Γ is the same number of blocks containing y as a vertex of A, for any two vertices x and y and for every vertex‐orbit A of Γ. We say that is degree‐balanced if the number of blocks containing x as a vertex of degree d in Γ is the same number of blocks containing y as a vertex of degree d in Γ, for any two vertices x and y and for every degree d in Γ. An orbit‐balanced Γ‐design is also degree‐balanced; a degree‐balanced Γ‐design is also balanced. The converse is not always true. We study the spectrum for orbit‐balanced, degree‐balanced, and balanced Γ‐designs of when Γ is a graph with five vertices, none of which is isolated. We also study the existence of balanced (respectively, degree‐balanced) Γ‐designs of which are not degree‐balanced (respectively, not orbit‐balanced).     

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     

Group Divisible Designs with Block Size Four and Group Type for

Nonuniform group divisible designs (GDDs) have been studied by numerous researchers for the past two decades due to their essential role in the constructions for other types of designs. In this paper, we investigate the existence problem of ‐GDDs of type for . First, we determine completely the spectrum of ‐GDDs of types and . Furthermore, for general cases, we show that for each and , a ‐GDD of type exists if and only if , and , except possibly for , and .     

The Existence and Construction of (K5∖e)‐Designs of Orders 27, 135, 162, and 216

The problem of the existence of a decomposition of the complete graph into disjoint copies of has been solved for all admissible orders n, except for 27, 36, 54, 64, 72, 81, 90, 135, 144, 162, 216, and 234. In this paper, I eliminate 4 of these 12 unresolved orders. Let Γ be a ‐design. I show that divides 2^k3 for some and that . I construct ‐designs by prescribing as an automorphism group, and show that up to isomorphism there are exactly 24 ‐designs with as an automorphism group. Moreover, I show that the full automorphism group of each of these designs is indeed . Finally, the existence of ‐designs of orders 135, 162, and 216 follows immediately by the recursive constructions given by G. Ge and A. C. H. Ling, SIAM J Discrete Math 21(4) (2007), 851–864.     

Generating Nonisomorphic Quadrangular Embeddings of a Complete Graph

We construct (resp. ) index one current graphs with current group such that the current graphs have different underlying graphs and generate nonisomorphic orientable (resp. nonorientable) quadrangular embeddings of the complete graph , (resp. ).     

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     

Flag‐Transitive 2‐ Symmetric Designs with Sporadic Socle

Let be a nontrivial 2‐ symmetric design admitting a flag‐transitive, point‐primitive automorphism group G of almost simple type with sporadic socle. We prove that there are up to isomorphism six designs, and must be one of the following: a 2‐(144, 66, 30) design with or , a 2‐(176, 50, 14) design with , a 2‐(176, 126, 90) design with or , or a 2‐(14,080, 12,636, 11,340) design with .     

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.     

Harmonic Distributions for Equitable Partitions of a Hypercube

We provide general criteria for orthogonal arrays and t‐designs on equitable partitions of a hypercube by exploring harmonic distributions. Generalized harmonic weight enumerators for real‐valued functions of are introduced and applied to eigenfunctions of the adjacency matrix of . Using this, expressions for harmonic distributions are established for every cell of an equitable partition π of . Moreover, for any given cell in the partition π, the strength of the cell as an orthogonal array is explicitly expressed, and also a characterization of a t‐design of that cell is established. We also compute strengths of cells and find t‐designs from cells based on constructions of Krotov, Borges, Rifa, and Zinoviev.     

Determinants of classical SG‐pseudodifferential operators

We introduce a generalized trace functional TR in the spirit of Kontsevich and Vishik's canonical trace for classical SG‐pseudodifferential operators on and suitable manifolds, using a finite‐part integral regularization technique. This allows us to define a zeta‐regularized determinant for parameter‐elliptic operators , , . For , the asymptotics of as and of as are derived. For suitable pairs we show that coincides with the so‐called relative determinant .     

The Large‐Time Solution of Burgers' Equation with Time‐Dependent Coefficients. I. The Coefficients Are Exponential Functions

In this paper, we consider an initial‐value problem for Burgers' equation with variable coefficients where x and t represent dimensionless distance and time, respectively, and , are given functions of t. In particular, we consider the case when the initial data have algebraic decay as , with as and as . The constant states and are problem parameters. Two specific initial‐value problems are considered. In initial‐value problem 1 we consider the case when and , while in initial‐value problem 2 we consider the case when and . The method of matched asymptotic coordinate expansions is used to obtain the large‐t asymptotic structure of the solution to both initial‐value problems over all parameter values.     

Cycles Avoiding a Color in Colorful Graphs

The Ramsey numbers of cycles imply that every 2‐edge‐colored complete graph on n vertices contains monochromatic cycles of all lengths between 4 and at least . We generalize this result to colors by showing that every k‐edge‐colored complete graph on vertices contains ‐edge‐colored cycles of all lengths between 3 and at least .     

Weak Musielak–Orlicz Hardy spaces and applications

Let satisfy that , for any given , is an Orlicz function and is a Muckenhoupt weight uniformly in . In this article, the authors introduce the weak Musielak–Orlicz Hardy space via the grand maximal function and then obtain its vertical or its non–tangential maximal function characterizations. The authors also establish other real‐variable characterizations of , respectively, in terms of the atom, the molecule, the Lusin area function, the Littlewood–Paley g‐function or ‐function. All these characterizations for weighted weak Hardy spaces (namely, and with and ) are new and part of these characterizations even for weak Hardy spaces (namely, and with ) are also new. As an application, the boundedness of Calderón–Zygmund operators from to in the critical case is presented.     

k‐Kings in k‐Quasitransitive Digraphs

Let D be a digraph with vertex set and arc set . A vertex x is a k‐king of D, if for every , there is an ‐path of length at most k. A subset N of is k‐independent if for every pair of vertices , we have and ; it is l‐absorbent if for every there exists such that . A ‐kernel of D is a k‐independent and l‐absorbent subset of . A k‐kernel is a ‐kernel. A digraph D is k‐quasitransitive, if for any path of length k, x₀ and are adjacent. In this article, we will prove that a k‐quasitransitive digraph with has a k‐king if and only if it has a unique initial strong component and the unique initial strong component is not isomorphic to an extended ‐cycle where each has at least two vertices. Using this fact, we show that every strong k‐quasitransitive digraph has a ‐kernel.