Searching for incomplete self orthogonal latin squares : a targeted and parallel approach

The primary purpose of this dissertation is in the search for new methods in which to search for Incomplete Self Orthogonal Latin Squares. As such a full understanding of the structures involved must be examined, starting from basic Latin Squares. The structures will be explained and built upon in order to cover Mutually Orthogonal Latin Squares, Frame Latin Squares and Self Orthogonal Latin Squares. In addition the related structure Orthogonal Arrays, will be explained as they relate to Incomplete Self Orthogonal Latin Squares. This paper also dedicates time to explaining basic search methods and optimizations that can be done. The two search methods of focus are the backtracking algorithm and heuristic searches. In our 6nal method the two will work together to achieve an improved result. The methods currently being used to search in parallel are also provided, along with the necessary backup to there structure. The main research of this paper is focused on the search for Incomplete Self Orthogonal Squares. This is done by breaking down the problem into four separate areas of the square. By separating the blocks it enables us to work on a smaller problem while eliminating many incorrect solutions. The solution methodology is broken up into three steps and systematically solving the individual areas of the square. By taking advantage of the properties of squares to constrain our search as much as possible we succeeded in reducing the total search time significantly. Unfortunately, even with our improvement in the overall search time, no open incomplete self orthogonal latin square problems could be solved. Full results and comparisons to existing methods are provided

A random Hall-Paige conjecture

A complete mapping of a group $G$ is a bijection $\phi\colon G\to G$ such that $x\mapsto x\phi(x)$ is also bijective. Hall and Paige conjectured in 1955 that a finite group $G$ has a complete mapping whenever $\prod_{x\in G} x$ is the identity in the abelianization of $G$. This was confirmed in 2009 by Wilcox, Evans, and Bray with a proof using the classification of finite simple groups. In this paper, we give a combinatorial proof of a far-reaching generalisation of the Hall-Paige conjecture for large groups. We show that for random-like and equal-sized subsets $A,B,C$ of a group $G$, there exists a bijection $\phi\colon A\to B$ such that $x\mapsto x\phi(x)$ is a bijection from $A$ to $C$ whenever $\prod_{a\in A} a \prod_{b\in B} b=\prod_{c\in C} c$ in the abelianization of $G$. Using this result, we settle the following conjectures for sufficiently large groups. (1) We confirm in a strong form a conjecture of Snevily by characterising large subsquares of multiplication tables of finite groups that admit transversals. Previously, this characterisation was known only for abelian groups of odd order. (2) We characterise the abelian groups that can be partitioned into zero-sum sets of specific sizes, solving a problem of Tannenbaum, and confirming a conjecture of Cichacz. (3) We characterise harmonious groups, that is, groups with an ordering in which the product of each consecutive pair of elements is distinct, solving a problem of Evans. (4) We characterise the groups with which any path can be assigned a cordial labelling. In the case of abelian groups, this confirms a conjecture of Patrias and Pechenik

Synchronizing permutation groups and graph endomorphisms

The current thesis is focused on synchronizing permutation groups and on graph endo- morphisms. Applying the implicit classification of rank 3 groups, we provide a bound on synchronizing ranks of rank 3 groups, at first. Then, we determine the singular graph endomorphisms of the Hamming graph and related graphs, count Latin hypercuboids of class r, establish their relation to mixed MDS codes, investigate G-decompositions of (non)-synchronizing semigroups, and analyse the kernel graph construction used in the theorem of Cameron and Kazanidis which identifies non-synchronizing transformations with graph endomorphisms [20]. The contribution lies in the following points: 1. A bound on synchronizing ranks of groups of permutation rank 3 is given, and a complete list of small non-synchronizing groups of permutation rank 3 is provided (see Chapter 3). 2. The singular endomorphisms of the Hamming graph and some related graphs are characterised (see Chapter 5). 3. A theorem on the extension of partial Latin hypercuboids is given, Latin hyper- cuboids for small values are counted, and their correspondence to mixed MDS codes is unveiled (see Chapter 6). 4. The research on normalizing groups from [3] is extended to semigroups of the form , and decomposition properties of non-synchronizing semigroups are described which are then applied to semigroups induced by combinatorial tiling problems (see Chapter 7). 5. At last, it is shown that all rank 3 graphs admitting singular endomorphisms are hulls and it is conjectured that a hull on n vertices has minimal generating set of at most n generators (see Chapter 8)

Computer construction of experimental plans

Experimental plans identify the treatment allocated to each unit and they are necessary for the supervision of most comparative experiments. Few computer programs have been written for constructing experimental plans but many for analysing data arising from designed experiments. In this thesis the construction of experimental plans is reviewed so as to determine requirements for a computer program. One program, DSIGNX, is described. Four main steps in the construction are identified: declaration, formation of the unrandomized plan (the design), randomization and output. The formation of the design is given most attention. The designs considered are those found to be important in agricultural experimentation and a basic objective is set that the 'proposed' program should construct most designs presented in standard texts (e.g. Cochran and Cox (1957)) together with important designs which have been developed recently. Topics discussed include block designs, factorial designs, orthogonal Latin squares and designs for experiments with non-independent observations. Some topics are discussed in extra detail; these include forming standard designs and selecting defining contrasts in symmetric factorial experiments, general procedures for orthogonal Latin squares and constructing serially balanced designs. Emphasis is placed on design generators, especially the design key and generalized cyclic generators, because of their versatility. These generators are shown to provide solutions to most balanced and partially balanced incomplete block designs and to provide efficient block designs and row and column designs. They are seen to be of fundamental importance in constructing factorial designs. Other versatile generators are described but no attempt is made to include all construction techniques. Methods for deriving one design from another or for combining two or more designs are shown to extend the usefulness of the generators. Optimal design procedures and the evaluation of designs are briefly discussed. Methods of randomization are described including automatic procedures based on defined block structures and some forms of restricted randomization for the levels of specified factors. Many procedures presented in the thesis have been included in a computer program DSIGNX. The facilities provided by the program and the language are described and illustrated by practical examples. Finally, the structure of the program and its method of working are described and simplified versions of the principal algorithms presented

Medical image enhancement

Each image acquired from a medical imaging system is often part of a two-dimensional (2-D) image set whose total presents a three-dimensional (3-D) object for diagnosis. Unfortunately, sometimes these images are of poor quality. These distortions cause an inadequate object-of-interest presentation, which can result in inaccurate image analysis. Blurring is considered a serious problem. Therefore, “deblurring” an image to obtain better quality is an important issue in medical image processing. In our research, the image is initially decomposed. Contrast improvement is achieved by modifying the coefficients obtained from the decomposed image. Small coefficient values represent subtle details and are amplified to improve the visibility of the corresponding details. The stronger image density variations make a major contribution to the overall dynamic range, and have large coefficient values. These values can be reduced without much information loss

Proceedings of AUTOMATA 2010: 16th International workshop on cellular automata and discrete complex systems

International audienceThese local proceedings hold the papers of two catgeories: (a) Short, non-reviewed papers (b) Full paper

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Computer science: the hardware software and heart of IT

1st edition, 201