On the determination and construction of optimal row-column designs having unequal row and column sizes

In this paper we consider experimental situations requiring usage of a row-column design where v treatments are to be applied to experimental units arranged in b ₁ rows and b ₂ columns where row i has size k _1i , i=1,..., b ₁ and column j has size k _2j , j=1,..., b ₂. Conditions analogous to those given in Kunert (1983, Ann. Statis., 11, 247–257) and Cheng (1978, Ann. Statist., 6, 1262–1272) are given which can often be used to establish the optimality of a given row-column design from the optimality of an associated block design. In addition, sufficient conditions are derived which guarantee the existence of an optimal row-column design which can be constructed by appropriately arranging treatments within blocks of an optimal block design.Visiting from the Indian Statistical Institute.     

QR factorization for row or column symmetric matrix

The problem of fast computing the QR factorization of row or column symmetric matrix is considered. We address two new algorithms based on a correspondence of Q and R matrices between the row or column symmetric matrix and its mother matrix. Theoretical analysis and numerical evidence show that, for a class of row or column symmetric matrices, the QR factorization using the mother matrix rather than the row or column symmetric matrix per se can save dramatically the CPU time and memory without loss of any numerical precision.     

Connectedness of Kronecker sum and partial Kronecker row sum of designs

In this paper we discuss connectedness of a design which is a Kronecker sum or a partial Kronecker row sum of any two equi-replicate and equi-block size designs.     

Efficiency of connected binary block designs when a single observation is unavailable

In this paper the problem of finding the design efficiency is considered when a single observation is unavailable in a connected binary block design. The explicit expression of efficiency is found for the resulting design when the original design is a balanced incomplete block design or a group divisible, singular or semiregular or regular with ₁>0, design. The efficiency does not depend on the position of the unavailable observation. For a regular group divisible design with ₁>0, the efficiency depends on the position of the unavailable observation. The bounds, both lower and upper, on the efficiency are given in this situation. The efficiencies of designs resulting from a balanced incomplete block design and a group divisible design are in fact high when a single observation is unavailable.The work of the first author is sponsored by the Air Force Office of Scientific Research under Grant AFOSR-90-0092.On leave from Indian Statistical Institute, Calcutta, India. The work of the third author was supported by a grant from the CMDS, Indian Institute of Management, Calcutta.     

Some E and MV-optimal designs for the two-way elimination of heterogeneity

Summary It is well known that in experimental settings wherev treatments are being tested inb blocks of sizek, a group divisible design having parametersλ ₂=λ₂+1 and whose correspondingC-matrix has maximal trace is both E and MV-optimal among all possible competing designs. In this paper, we show that under certain conditions, the E and MV-optimal group divisible block designs mentioned in the previous sentence can be used to construct E and MV-optimal row-column designs to handle experimental situations in which heterogeneity is to be eliminated in two directions and wherev treatments are being tested inb columns andk rows. Examples are given to illustrate how the results obtained can be applied. Research sponsored in part by National Science Foundation Grant No. DMS-8401943.     

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.     

On the connectedness of partially balanced block designs

The necessary and sufficient conditions for m-associate partially balanced block (PBB) designs to be connected are given. This generalizes the criterion for m-associate partially balanced incomplete block (PBIB) designs, which has originally been established by Ogawa, Ikeda and Kageyama (1984, Proceedings of the Seminar on Combinatorics and Applications, 248–255, Statistical Publishing Society, Calcutta).This work was partially supported by the Polish Academy of Sciences Grant No. MR I.1-2/2.     

Tabu search for the single row facility layout problem using exhaustive 2-opt and insertion neighborhoods

The single row facility layout problem (SRFLP) is the problem of arranging facilities with given lengths on a line, while minimizing the weighted sum of the distances between all pairs of facilities. The problem is NP-hard. In this paper, we present two tabu search implementations, one involving an exhaustive search of the 2-opt neighborhood and the other involving an exhaustive search of the insertion neighborhood. We also present techniques to significantly speed up the search of the two neighborhoods. Our computational experiments show that the speed up techniques are effective, and our tabu search implementations are competitive. Our tabu search implementations improved previously known best solutions for 23 out of the 43 large sized SRFLP benchmark instances.     

Construction of optimal supersaturated designs by the packing method

A supersaturated design is essentially a factorial design with the equal occurrence of levels property and no fully aliased factors in which the number of main effects is greater than the number of runs. It has received much recent interest because of its potential in factor screening experiments. A packing design is an important object in combinatorial design theory. In this paper, a strong link between the two apparently unrelated kinds of designs is shown. Several criteria for comparing supersaturated designs are proposed, their properties and connections with other existing criteria are discussed. A combinatorial approach, called the packing method, for constructing optimal supersaturated designs is presented, and properties of the resulting designs are also investigated. Comparisons between the new designs and other existing designs are given, which show that our construction method and the newly constructed designs have good properties.     

Pseudo generalized Youden designs

Youden square designs, or Youden rectangles, are classical objects in design theory. Extensions of these were introduced in 1958 by Kiefer and in 1981 by Cheng, in the form of generalized Youden designs (GYDs) and pseudo Youden designs (PYDs), respectively. In this paper, we introduce a common generalization of both these objects, which we call a pseudo generalized Youden design (PGYD). PGYDs share the statistically desirable optimality properties of GYDs and PYDs, and we show that they exist in situations where neither GYDs nor PYDs do. We determine some numerical necessary conditions for the existence of PGYDs, classify their existence for small parameter sets, and provide constructions for families of PGYDs using patchwork methods based on affine planes.     

The connectivity of the block-intersection graphs of designs

It is shown that the vertex connectivity of the block-intersection graph of a balanced incomplete block design,BIBD (v, k, 1), is equal to its minimum degree. A similar statement is proved for the edge connectivity of the block-intersection graph of a pairwise balanced design,PBD (v, K, 1). A partial result on the vertex connectivity of these graphs is also given. Minimal vertex and edge cuts for the corresponding graphs are characterized.Research supported in part by a B.C. Science Council G.R.E.A.T. Scholarship.Research supported in part by an NSERC Postdoctoral Fellowship.     

Semiconvergence analysis of the randomized row iterative method and its extended variants

The row iterative method is popular in solving the large‐scale ill‐posed problems due to its simplicity and efficiency. In this work we consider the randomized row iterative (RRI) method to tackle this issue. First, we present the semiconvergence analysis of RRI method for the overdetermined and inconsistent system, and derive upper bounds for the noise error propagation in the iteration vectors. To achieve a least squares solution, we then propose an extended version of the RRI (ERRI) method, which in fact can converge in expectation to the solution of the overdetermined or underdetermined, consistent or inconsistent systems. Finally, some numerical examples are given to demonstrate the convergence behaviors of the RRI and ERRI methods for these types of linear system.     

Nearly magic rectangles

Magic squares have been extremely useful and popular in combinatorics and statistics. One generalization of magic squares is magic rectangles which are useful for designing experiments in statistics. A necessary and sufficient condition for the existence of magic rectangles restricts the number of rows and columns to be either both odd or both even. In this paper, we generalize magic rectangles to even by odd nearly magic rectangles. We also prove necessary and sufficient conditions for the existence of a nearly magic rectangle, and construct one for each parameter set for which they exist.     

Old and new designs via difference multisets and strong difference families

Let X =((x_1,1,x_1,2,…,x_1,k),(x_2,1,x_2,2,…,x_2,k),…,(x_t,1,x_t,2,…,x_t,k)) be a family of t multisets of size k defined on an additive group G. We say that X is a t-(G,k,μ) strong difference family (SDF) if the list of differences (x_h,i-x_h,j∣h=1,…,t;i≠ j) covers all of G exactly μ times. If a SDF consists of a single multiset X, we simply say that X is a (G,k,μ) difference multiset. After giving some constructions for SDF's, we show that they allow us to obtain a very useful method for constructing regular group divisible designs and regular (or 1-rotational) balanced incomplete block designs. In particular cases this construction method has been implicitly used by many authors, but strangely, a systematic treatment seems to be lacking. Among the main consequences of our research, we find new series of regular BIBD's and new series of 1-rotational (in many cases resovable) BIBD's.     

New designs from circular nearrings

Nearrings are generalized rings in which addition is not in general abelian and only one distributive law holds. Some interesting combinatorial structures, as tactical configurations and balanced incomplete block designs (BIBDs) naturally arise when considering the class of planar and circular nearrings. We define the concept of disk and prove that in the case of field-generated planar circular nearrings it yields a BIBD. Such designs can be used in the construction of some classes of codes for which we are able to calculate the parameters.     

Row and column finite matrices

Consider the ring of all column finite matrices over a ring . We prove that each such matrix is conjugate to a row and column finite matrix if and only if is right Noetherian and is countable. We then demonstrate that one can perform this conjugation on countably many matrices simultaneously. Some applications and limitations are given.

     

Self-orthogonal codes constructed from orbit matrices of 2-designs and quotient matrices of divisible designs

In 2003 Harada and Tonchev showed a construction of self-orthogonal codes from orbit matrices of block designs with fixed-point-free automorphisms. We describe a construction of self-orthogonal codes from orbit matrices of 2-designs admitting certain automorphisms with fixed points (and blocks). Further, we present a construction of self-orthogonal codes from quotient matrices of divisible designs and divisible design graphs.     

Existential closure of block intersection graphs of infinite designs having infinite block size

A graph G is n ‐existentially closed ( n ‐e.c.) if for each pair ( A, B ) of disjoint subsets of V(G) with | A | + | B |≤ n there exists a vertex in V ( G )\( A ∪ B ) which is adjacent to each vertex in A and to no vertex in B . In this paper we study the n ‐existential closure property of block intersection graphs of infinite designs with infinite block size. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:317‐327, 2011