Multi‐layered Youden Rectangles

Multi‐layered Youden rectangles are introduced. These new designs include double and triple Youden rectangles as subdesigns. Though many double Youden rectangles are known, the triples, introduced in 1994, have to now yielded few nontrivial examples. Two infinite series of multi‐layered Youden rectangles are constructed, and so also many new triple Youden rectangles.     

Pseudo quasi‐3 designs and their applications to coding theory

We define a pseudo quasi‐3 design as a symmetric design with the property that the derived and residual designs with respect to at least one block are quasi‐symmetric. Quasi‐symmetric designs can be used to construct optimal self complementary codes. In this article we give a construction of an infinite family of pseudo quasi‐3 designs whose residual designs allow us to construct a family of codes with a new parameter set that meet the Grey Rankin bound. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 411–418, 2009     

Complete Sets of Mutually Orthogonal Hypercubes and Their Connections to Affine Resolvable Designs

. Recently, Laywine and Mullen proved several generalizations of Bose's equivalence between the existence of complete sets of mutually orthogonal Latin squares of order n and the existence of affine planes of order n. Laywine further investigated the relationship between sets of orthogonal frequency squares and affine resolvable balanced incomplete block designs. In this paper we generalize several of Laywine's results that were derived for frequency squares. We provide sufficient conditions for construction of an affine resolvable design from a complete set of mutually orthogonal Youden frequency hypercubes; we also show that, starting with a complete set of mutually equiorthogonal frequency hypercubes, an analogous construction can always be done. In addition, we give conditions under which an affine resolvable design can be converted to a complete set of mutually orthogonal Youden frequency hypercubes or a complete set of mutually equiorthogonal frequency hypercubes.     

Quadrature Methods for Bayesian Optimal Design of Experiments With Nonnormal Prior Distributions

Many optimal experimental designs depend on one or more unknown model parameters. In such cases, it is common to use Bayesian optimal design procedures to seek designs that perform well over an entire prior distribution of the unknown model parameter(s). Generally, Bayesian optimal design procedures are viewed as computationally intensive. This is because they require numerical integration techniques to approximate the Bayesian optimality criterion at hand. The most common numerical integration technique involves pseudo Monte Carlo draws from the prior distribution(s). For a good approximation of the Bayesian optimality criterion, a large number of pseudo Monte Carlo draws is required. This results in long computation times. As an alternative to the pseudo Monte Carlo approach, we propose using computationally efficient Gaussian quadrature techniques. Since, for normal prior distributions, suitable quadrature techniques have already been used in the context of optimal experimental design, we focus on quadrature techniques for nonnormal prior distributions. Such prior distributions are appropriate for variance components, correlation coefficients, and any other parameters that are strictly positive or have upper and lower bounds. In this article, we demonstrate the added value of the quadrature techniques we advocate by means of the Bayesian D-optimality criterion in the context of split-plot experiments, but we want to stress that the techniques can be applied to other optimality criteria and other types of experimental designs as well. Supplementary materials for this article are available online.     

Some results on resolvable incomplete block designs

This paper is concerned with the uniformity of a certain kind of resolvable incomplete block (RIB for simplicity) design which is called the PRIB design here. A sufficient and necessary condition is obtained, under which a PRIB design is the most uniform in the sense of a discrete discrepancy measure, and the uniform PRIB design is shown to be connected. A construction method for such designs via a kind of U-type designs is proposed, and an existence result of these designs is given. This method sets up an important bridge between PRIB designs and U-type designs.     

混水平均匀设计的构造

我们用离散偏差来度量部分因子设计的均匀性,本文的目的在于寻找一些构造混水平均匀设计的方法,这些方法比文献中已有的方法更简单且计算成本更低.我们得到了离散偏差的一个下界,如果一个U 型设计的离散偏差值达到这个下界,那么该设计是—个均匀设计.我们建立了均匀设计与组合设计理论中一致可分解设计之间的联系.通过一致可分解设计,我们提出了一些构造均匀设计的新方法,同时也给出了许多均匀设计存在的无穷类.     

Superpure digraph designs

A digraph design is a decomposition of a complete (symmetric) digraph into copies of pre‐specified digraphs. Well‐known examples for digraph designs are Mendelsohn designs, directed designs or orthogonal directed covers. A digraph design is superpure if any two of the subdigraphs in the decomposition have no more than two vertices in common. We give an asymptotic existence theorem for superpure digraph designs, which is a variation of an earlier result of Lamken and Wilson J Combin Theory Ser A 89: 149–200, 2000. As an immediate consequence, we obtain new results for supersimple designs and pure perfect Mendelsohn designs. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 239–255, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10013     

Balanced generalized handcuffed designs

The definition of balanced generalized handcuffed designs (BHD) is of course more specific than that of the generalized handcuffed designs that we introduced in 1987.

In the first part of this paper, we present a particular property of a BHD, which is not necessarily that of a generalized handcuffed design.

Then, we provide the reader with a general procedure that enables one to obtain such designs, and is called a ‘difference method’. We also show how this difference method can be made more useful in the case where the set V on which a BHD is constructed is the residue classes of integers mod V.

The third part of this paper deals with the problem of the existence of a BHD; and a solution is given for a particular case. We assume that the method applied for solving this problem will allow for the constructing of many more theorems analogous to Theorem 3.     

Generalized covering designs and clique coverings

Inspired by the “generalized t‐designs” defined by Cameron [P. J. Cameron, Discrete Math 309 (2009), 4835–4842], we define a new class of combinatorial designs which simultaneously provide a generalization of both covering designs and covering arrays. We then obtain a number of bounds on the minimum sizes of these designs, and describe some methods of constructing them, which in some cases we prove are optimal. Many of our results are obtained from an interpretation of these designs in terms of clique coverings of graphs. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:378‐406, 2011     

General Constructions for Double Group Divisible Designs and Double Frames

We generalize the concept of an incomplete double group divisible design and describe some recursive constructions for such a generalized new design. As a consequence, we obtain a general recursive construction for group divisible designs, which unifies many important recursive constructions for various types of combinatorial designs. We also introduce the concept of a double frame. After providing a preliminary result on the number of partial resolution classes, we describe a general construction for double frames. This construction method can unify many known recursive constructions for frames.     

Unitals and Unitary Polarities in Symmetric Designs

We extend the notion of unital as well as unitary polarity from finite projective planes to arbitrary symmetric designs. The existence of unitals in several families of symmetric designs has been proved. It is shown that if a unital in a point-hyperplane design PG _d-1(d,q) exists, then d = 2 or 3; in particular, unitals and ovoids are equivalent in case d = 3. Moreover, unitals have been found in two designs having the same parameters as the PG ₄(5,2), although the latter does not have a unital. It had been not known whether or not a nonclassical design exists, which has a unitary polarity. Fortunately, we have discovered a unitary polarity in a symmetric 2-(45,12,3) design. To a certain extent this example seems to be exceptional for designs with these parameters.     

A construction of balanced arrays of strengtht and some related incomplete block designs

Summary Saha [6] has shown the equivalence between a ‘tactical system’ (or at-design) and a 2-symbol balanced array (BA) of strengtht. The implicit method of construction of BA in that paper has been generalized herein to that of ans-symbol BA of strengtht. Some BIB and PBIB designs are also constructed from these arrays. Majindar [2], Vanstone [8] and Saha [6] have all shown that the existence of a symmetrical BIBD forv treatments implies the existence of six more BIBD's forv treatments in (v/2) blocks. An analogue of this result has been obtained for a large class of PBIB designs in this paper.     

Noncooperative Games with Vector Payoffs Under Relative Pseudomonotonicity

We consider the Nash equilibrium problem with vector payoffs in a topological vector space. By employing the recent concept of relative (pseudo) monotonicity, we establish several existence results for vector Nash equilibria and vector equilibria. The results strengthen in a major way existence results for vector equilibrium problems which were based on the usual (generalized) monotonicity concepts.     

Complementary design theory for sliced equidistance designs

The complementary design theory is powerful for searching for an optimal design when its complementary design is smaller. This paper introduces a new class of sliced equidistance designs and develops the corresponding complementary design theory under the generalized minimum aberration criterion. Two rules are established to search for a generalized minimum aberration design through its complementary design in a sliced equidistance design. As a result, the developed theory covers the related results for the whole designs being saturated designs as special cases. Some examples are presented to illustrate its usefulness.     

λKv的最大K2,3填充设计和最小K2,3覆盖设计

对于一个有限简单图G,λKv的G-设计(G-填充,G-覆盖),记为(v,G,λ)-GD((v,G,λ)-PD,(v,G,λ)-CD),是一个(X,B),其中X是Kb的顶点集,B是Kv的子图族,每个子图(称为区组)均同构于G,且Kv中任一边都恰好(最多,至少)出现在B的λ个区组中.一个填充(覆盖)设计称为是最大(最小)的,如果没有其它的这种填充(覆盖)设计具有更多(更少)的区组.本文对于λ>1确定了(v,K2,3,λ)-GD的存在谱,并对任意λ构造了λKv的最大K2,3-填充设计和最小K2,3-覆盖设计.     

Common intersection designs

Motivated by an application to sensor networks, Lee and Stinson [ 6 ] defined a new type of set system termed a “common intersection design.” Briefly, a µ‐common intersection design is a 1‐design in which no pair of points occurs in more than one block, and in which any two disjoint blocks intersect at least µ blocks in common. In general, we want to maximize µ as a function of the other parmameters of the design. In this paper, we analyze combinatorial properties of common intersection designs. We determine necessary conditions for “optimal” common intersection designs and provide several existence results. Connections with other types of designs are pointed out. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 251–269, 2006     

具有特定纯净效应折中设计的若干理论及构造

许多因析试验中,试验者只关心指定的一部分因子效应的估计效果.针对此类问题,Addelman(1962)首次提出了折中设计的方法,并定义纯净折中设计以保证指定的因子效应被有效地估计出来,但此类纯净折中设计的分辨度限定为Ⅳ.本文研究了四类全新的折中设计,指定因子效应的集合分别记为{G1,G1×G1}、{G1,G1×G1,G...     

Editorial

Complete randomization for many industrial and agricultural experiments is frequently impractical due to constraints in time, cost, or existence of one or more hard-to-change factors. In these situations, restrictions on randomization may lead to split-plot designs, allowing certain factor levels to be randomly applied to the whole plot units and remaining factor levels randomly applied to the subplot units. Separate random errors in whole and subplot units from the two randomizations results in a compound symmetric error structure, which affects estimation, inference, and choice of design. In this article, we consider the prediction properties of split-plot designs, expanding the comparison between designs beyond parameter estimation properties and present three-dimensional variance dispersion graphs. These graphs can be used for evaluating the prediction capability of split-plot designs as well as for developing design strategies. We demonstrate, through several examples, that three-dimensional variance dispersion plots offer a more comprehensive study of competing designs than what is offered when comparing designs with single number optimality criteria (such as D-, G-, and V-criteria).     

区组大小为 4 的二重超单可分解的可分组设计

自 1992 年 Gronau 和 Mullin 提出超单设计的概念以来, 很多研究者参与了超单设计的研究. 超单设计在编码等方面也有广泛的应用. 超单可分组设计是超单设计的重要组成部分. 本文我们主要研究区组大小为4 的二重超单可分解的可分组设计, 并基本解决了此类设计的存在性问题.     

Pseudo almost periodicity of fractional integro-differential equations with impulsive effects in Banach spaces

In this paper, for the impulsive fractional integro-differential equations involving Caputo fractional derivative in Banach space, we investigate the existence and uniqueness of a pseudo almost periodic PC-mild solution. The working tools are based on the fixed point theorems, the fractional powers of operators and fractional calculus. Some known results are improved and generalized. Finally, existence and uniqueness of a pseudo almost periodic PC-mild solution of a two-dimensional impulsive fractional predator-prey system with diffusion are investigated.