Concerning eight mutually orthogonal latin squares

In this article, we provide a direct construction for 8 mutually orthogonal latin squares (MOLS)(48). Using this design together with one of Wilson's recursive constructions produces 8 new MOLS(v) for 88 other values of v. We also mention a few other new sets of 8 and 12 MOLS obtained recursively. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 255–261, 2007     

Group divisible designs in MOLS of order ten

The maximum number of mutually orthogonal latin squares (MOLS) of order 10 is known to be between 2 and 6. A hypothetical set of four MOLS must contain at least one of the types of group divisible designs (GDDs) classified here. The proof is based on a dimension argument modified from work by Dougherty. The argument has recently led to the discovery of a counterexample to Moorhouse’s conjecture on the rank of nets, found by Howard and Myrvold. Although it is known that even three MOLS can admit no nontrivial symmetry group, we are hopeful this classification via GDDs and dimension can offer some structure to aid the eventual goal of exhausting the search for four MOLS of order 10.     

Remarks on Sade's disproof of the Euler conjecture with an application to latin squares orthogonal to their transpose

Sade's singular direct product method for constructing pairs of mutually orthogonal latin squares (MOLS), published in 1960, gives counter-examples for all n > 482 to the Euler conjecture that pairs of MOLS of order n do not exist whenever n ≡ 2 (mod 4).     

On convex modified output least-squares for elliptic inverse problems: stability,regularization, applications,and numerics

Abstract

Inverse problems of identifying parameters in partial differential equations constitute an important class of problems with diverse real-world applications. These identification problems are commonly explored in an optimization framework and there are many optimization formulations having their own advantages and disadvantages. Although a non-convex output least-squares (OLS) objective is commonly used, a convex-modified output least-squares (MOLS) has shown encouraging results in recent years. In this work, we focus on various aspects of the MOLS approach. We devise a rigorous (quadratic and non-quadratic) regularization framework for the identification of smooth as well as discontinuous coefficients. This framework subsumes the total variation regularization that has attracted a great deal of attention in identifying sharply varying coefficients and also in image processing. We give new existence results for the regularized optimization problems for OLS and MOLS. Restricting to the Tikhonov (quadratic) regularization, we carry out a detailed study of various stability aspects of the inverse problem under data perturbation and give new stability estimates for general inverse problems using OLS and MOLS formulations. We give a discretization scheme for the continuous inverse problem and prove the convergence of the discrete inverse problem to the continuous one. We collect discrete formulas for OLS and MOLS and compute their gradients and Hessians. We present applications of our theoretical results. To show the feasibility of the MOLS framework, we also provide computational results for the inverse problem of identifying parameters in three different classes of partial differential equations .     

Local permutation polynomials and the action of e-Klenian groups

Permutation polynomials of finite fields have many applications in Coding Theory, Cryptography and Combinatorics. In the first part of this paper we present a new family of local permutation polynomials based on a class of symmetric subgroups without fixed points, the so called e-Klenian groups. In the second part we use the fact that bivariate local permutation polynomials define Latin Squares, to discuss several constructions of Mutually Orthogonal Latin Squares (MOLS) and, in particular, we provide a new construction of MOLS on size a prime power.     

Concerning seven and eight mutually orthogonal Latin squares

In this paper, three new direct Mutually Orthogonal Latin Squares (MOLS) constructions are presented for 7 MOLS(24), 7 MOLS(75) and 8 MOLS(36); then using recursive methods, several new constructions for 7 and 8 MOLS are obtained. These reduce the largest value for which 7 MOLS are unknown from 780 to 570, and the largest odd value for which 8 MOLS are unknown from 1935 to 419. © 2003 Wiley Periodicals, Inc.     

Algebraic representation of affine MDS-codes and mutually orthogonal Latin squares

An algebraic representation of affine MDS-codes and of mutually orthogonal Latin squares (MOLS) is given by introducing the term of a partial ternary. The extension respectively lengthening of partial ternaries, MDS-codes and MOLS is discussed.     

Searching for Mutually Orthogonal Latin Squares via integer and constraint programming

This paper applies algorithms integrating Integer Programming (IP) and Constraint Programming (CP) to the Mutually Orthogonal Latin Squares (MOLS) problem. We investigate the behaviour of these algorithms against traditional IP and CP schemes. Computational results are obtained with respect to various aspects of the algorithms, using instances of the 2 MOLS and 3 MOLS problems. The benefits of integrating IP with CP on this feasibility problem are clearly exhibited, especially in large problem instances.     

Existence of Five MOLS of Orders 18 and 60

In this article, we provide direct constructions for five mutually orthogonal Latin squares (MOLS) of orders and 60. For , these come from a new (60, 6, 1) difference matrix. For , the required construction is obtained by combining two different methods that were used in the constructions of four MOLS(14) and eight MOLS(36).     

Mols with holes

We consider sets of MOLS (mutually orthogonal Latin squares) having holes, corresponding to missing sub-MOLS, which are disjoint and spanning. We give several constructions for sets of MOLS with holes.     

Prokhorov Blocks and Strong Law of Large Numbers Under Rearrangements

We find conditions on a sequence of random variables to satisfy the strong law of large numbers (SLLN) under a rearrangement. It turns out that these conditions are necessary and sufficient for the permutational SLLN (PSLLN). By PSLLN we mean that the SLLN holds under almost all simple permutations within blocks the lengths of which grow exponentially (Prokhorov blocks). In the case of orthogonal random variables it is shown that Kolmogorov's condition, that is known not to be sufficient for SLLN, is actually sufficient for PSLLN. It is also shown that PSLLN holds for sequences that are strictly stationary with finite first moments. In the case of weakly stationary sequences a Gaposhkin result implies that SLLN and PSLLN are equivalent. Finally we consider the case of general norming and generalization of the Nikishin theorem. The methods of proof uses on the one hand the idea of Prokhorov blocks and Garsia's construction of product measure on the space of simple permutations, and on the other hand, a maximal inequality for permutations.     

Critical Relations of Crowns in Critical Times of Coronavirus Depression

The critical relations are the building blocks of the relational clone of a relational structure with respect to the relational operations intersection and direct product. In this paper we describe the critical relations of crowns. As a consequence, we obtain that the subpower membership problem for any crown is polynomial-time solvable.

     

On the existence of MOLS with equal-sized holes

We consider a pair of MOLS (mutually orthogonal Latin squares) having holes, corresponding to missing sub-MOLS, which are disjoint and spanning It is shown that a pair of MOLS withn holes of sizeh exist forh 2 if and only ifn 4 For SOLS (self-orthogonal Latin squares) with holes, we have the same result, with two possible exceptions SOLS with 7 or 13 holes of size 6     

Some general constructions of geodetic blocks

A general construction procedure for geodetic blocks, starting from an arbitrary geodetic block is given, which unifies and generalizes many of the known general procedures. The construction of geodetic blocks homeomorphic to a given one is also analysed and the problem of Bosák on the existence of such a graph for the Hoffman-Singleton graph is settled in the affirmative. A simple characterization theorem for geodetic blocks is given and the existence of geodetic blocks with given girth and diameter is investigated.     

Divisible difference families from Galois rings GR(4,n) and Hadamard matrices

We give a new construction of difference families generalizing Szekeres’s difference families Szekeres (Enseignment Math 15:269–278, 1969). As an immediate consequence, we obtain some new examples of difference families with several blocks in multiplicative subgroups of finite fields. We also prove that there exists an infinite family of divisible difference families with two blocks in a unit subgroup of the Galois ring \(GR(4,n)\) . Furthermore, we obtain a new construction method of symmetric Hadamard matrices by using divisible difference families and a new array.     

Locally-finite connected-homogeneous digraphs

A digraph is connected-homogeneous if any isomorphism between finite connected induced subdigraphs extends to an automorphism of the digraph. We consider locally-finite connected-homogeneous digraphs with more than one end. In the case that the digraph embeds a triangle we give a complete classification, obtaining a family of tree-like graphs constructed by gluing together directed triangles. In the triangle-free case we show that these digraphs are highly arc-transitive. We give a classification in the two-ended case, showing that all examples arise from a simple construction given by gluing along a directed line copies of some fixed finite directed complete bipartite graph. When the digraph has infinitely many ends we show that the descendants of a vertex form a tree, and the reachability graph (which is one of the basic building blocks of the digraph) is one of: an even cycle, a complete bipartite graph, the complement of a perfect matching, or an infinite semiregular tree. We give examples showing that each of these possibilities is realised as the reachability graph of some connected-homogeneous digraph, and in the process we obtain a new family of highly arc-transitive digraphs without property Z.     

Truncated Transversal Designs: A New Lower Bound on the Number of Idempotent MOLS of Side n

A truncated transversal design TTD of type g^km¹ is a {k, k+1}-GDD of type g^km¹ in which each point on the group of size m lies only in blocks of size k+1. Thus a TTD of type g^km¹ is equivalent to a transversal design TD (k, g) having m disjoint parallel classes of blocks. We employ a new construction developed by the author (1993, J. Combin. Des.1, 15–26) to show that if g₁<g₂ and if there exists a TD (k, g₁) and a TD (k+1, g₂), then there exists a TTD of type (g₁g₂)^km¹ for any 0m(g₂ div g₁) g²₁. As a corollary, we obtain a new lower bound on the number of mutually orthogonal idempotent latin squares of side g: if g₁<g₂ and there exist r MOLS of side g₁ and r+1 MOLS of side g₂ , then N(1 ^g₁g₂)r.     

Constructing highly arc transitive digraphs using a direct fibre product

We introduce a construction of highly arc transitive digraphs using a direct fibre product. This product generalizes some known classes of highly arc transitive digraphs but also allows us to construct new ones. We use the product to obtain counterexamples to a conjecture advanced by Cameron, Praeger and Wormald on the structure of certain highly arc transitive digraphs.     

关于积图点可区别边染色的若干结论

如果图G的一个正常边染色满足任意两个不同点的关联边色集不同, 则称为点可区别边染色(VDEC), 其所用最少颜色数称为点可区别边色数. 利用构造法给出了积图点可区别边染色的一个结论, 得到了关于积图点可区别边色数的若干结果, 并且给出25个具体积图的点可区别边色数, 验证了它们满足点可区别边染色猜想(VDECC).     

An Iterative Construction Heuristic for the Ore Selection Problem

The ore selection problem involves choosing a processing option for a number of mining blocks that maximises the expected payoff for a given level of financial risk. An innovative neighbourhood search heuristic is proposed for the ore selection problem. This iterative construction heuristic employs a stochastic demolition and reconstruction strategy. Computational experiments with this heuristic for two ore selection problem instances, one involving 2,500 blocks and the other involving 78,000 blocks, are given. These problem instances are made publicly available for use by future workers. Our computational experiments indicate that the proposed heuristic produces better quality solutions faster than a relay hybrid (constructive-simulated annealing) heuristic.