The purpose of this chapter is to develop a form of a "free" Boolean algebra with Σ as a base, by imposing the usual Boolean operations on the set Σ and thus generating new elements freely within explicitly prescribed restrictions.

The primary purpose of this paper is to state a set of postulates for Boolean algebra and show the characteristic theorems derivable from them, and to unify in one paper the more important methods of representing Boolean algebra and show their equivalence.

An orthomorphism, π, of a group, (G, +), is a permutation of G with the property that the map x → -x + π(x) is also a permutation. In this paper, we consider orthomorphisms of the additive group of binary n-tuples, Zn2. We use known orthomorphism preserving functions to prove a uniformity in the cycle types of orthomorphisms that extend certain partial orthomorphisms, and prove that extensions of particular sizes of partial orthomorphisms exist. Further, in studying the action of conjugating orthomorphisms by automorphisms, we find several symmetries within the orbits and stabilizers of this action, and other orthomorphism-preserving functions. In addition, we prove a lower bound on the number of orthomorphisms of Zn2 using the equivalence of orthomorphisms to transversals in Latin squares. Lastly, we present a Monte Carlo method for generating orthomorphisms and discuss the results of the implementation.

The purpose of this section is to define a boolean algebra and to determine some of the important properties of it. A boolean algebra is a set B with two binary operations, join and meet, denoted by + and juxtaposition respectively, and a unary operation, complement ation, denoted by ', which satisfy the following axioms: (1) for all a,b ∑ B (that is, for all a,b elements of B) a + b = b + a and a b = b a, (the commutative laws), (2) for all a,b,c ∑ B, a + b c =(a + b) (a + b) and a (b + c) = a b + a c, (the distributive laws), (3) there exists 0 ∑B such that for each a ∑B, a + 0 = a, and there exists 1 ∑B such that for each a ∑ B, a 1 = a, (4) for each a∑B, a + a' = 1 and a a' = 0. If a + e = a for all a in B then 0 = 0 + e = e + 0 = e, so that there is exactly one element in B which satisfies the first half of axiom 3, namely 0. Similarly there is exactly one element in B which satisfies the second half of axiom 3, namely 1. The O and 1 as defined above will be called the distinguished elements.

Philosophiae Doctor - PhD / Understanding the geometries of points in space has been attractive to mathematicians for ages. As a model, twelve years ago, Kurz and Meyer [32] considered point sets in the m-dimensional a ne space Fmq over a nite eld Fq with q = pr elements, p prime, where each squared Euclidean distance of two points is a square in Fq: The latter points are said to be at integral distance in Fmq , and the sets above are called integral point sets.

This thesis presents an interactive computerised teaching suite developed for the design of combinatorial and sequential logic circuits. This suite fills a perceived gap in the currently available computer-based teaching software for digital logic design. Several existing digital logic educational software are available, however these existing programs were found to be unsuitable for our use in providing alternative mode subject delivery. This prompted the development of a Microsoft Windows TM tutorial suite, called WinLogiLab. WinLogiLab comprises of a set of tutorials that uses student provided input data, to perform the initial design steps for digital Combinatorial and Sequential logic circuits. The combinatorial tutorials are designed to show the link between Boolean Algebra and Digital Logic circuits, and follows the initial design steps: from Boolean algebra, truth tables, to Exact and the Heuristic minimisation techniques, to finally produce the combinatorial circuit. Similarly, the sequential tutorials can design simple State Machine Counters, and can model more complex Finite State Automata.

A Boolean partition algebra is a pair $(B,F)$ where $B$ is a Boolean algebra and $F$ is a filter on the semilattice of partitions of $B$ where $\bigcup F=B\setminus\{0\}$. In this dissertation, we shall investigate the algebraic theory of Boolean partition algebras and their connection with uniform spaces. In particular, we shall show that the category of complete non-Archimedean uniform spaces is equivalent to a subcategory of the category of Boolean partition algebras, and notions such as supercompleteness of non-Archimedean uniform spaces can be formulated in terms of Boolean partition algebras.

Orientador: Reginaldo Palazzo Júnior / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-19T17:49:58Z (GMT). No. of bitstreams: 1 Oliveira_AndersonJosede_M.pdf: 1619063 bytes, checksum: 79a49301084eecde745f0e73cddfc1fa (MD5) Previous issue date: 2012 / Resumo: Uma área de pesquisa em franca expansão é a modelagem matemática do código genético, por meio da qual pode-se identificar as características e propriedades do mesmo. Neste trabalho apresentamos alguns modelos matemáticos aplicados à biologia, especificamente relacionado ao código genético. Os objetivos deste trabalho são: a) caracterização da hidropaticidade dos aminoácidos através da construção de reticulados booleanos e diagramas de Hasse associados a cada rotulamento do código genético, b) proposta de um algoritmo soma com transporte para efetuar a soma entre códons, ferramenta importante em análises mutacionais, c) representação polinomial dos códons do código genético, d) comparação dos resultados dos rotulamentos A, B e C em cada uma das modelagens construídas, e) análise do comportamento dos aminoácidos em cada um dos rotulamentos do código genético. Os resultados encontrados permitem a utilização de tais ferramentas em diversas áreas do conhecimento como bioinformática, biomatemática, engenharia genética, etc, devido a interdisciplinaridade do trabalho, onde elementos de biologia, matemática e engenharia foram utilizados / Abstract: A research area in frank expansion is the mathematical modeling of the genetic code, through can identify the characteristics and properties of them. In this paper we present some mathematical models applied to biology, specifically related to the genetic code. The aims of this work are: a) a characterization of the hydropathy of the amino acids through the construction of boolean lattices and Hasse diagrams associated with each labeling of the genetic code, b) the proposal of a sum algorithm of transportation to make the sum of codons, important tool in mutational analysis, c) a polynomial representation of the codons of the genetic code, d) a comparing of the results of the A, B and C labels in each of the built modeling, e) an analysis of the behavior of the amino acids in each of the labels of the genetic code. The results allow the use of such tools in a lot of areas like bio informatics, biomathematics, genetic engineering, etc., due to the interdisciplinary of the paper, where elements of biology, mathematics and engineering were used / Mestrado / Telecomunicações e Telemática / Mestre em Engenharia Elétrica

Spectral techniques in digital logic design have been known for more than thirty years. They have been used for Boolean function classification, disjoint decomposition, parallel and serial linear decomposition, spectral translation synthesis (extraction of linear pre- and post-filters), multiplexer synthesis, prime implicant extraction by spectral summation, threshold logic synthesis, estimation of logic complexity, testing, and state assignment. This dissertation resolves many important issues concerning the efficient application of spectral methods used in the computer-aided design of digital circuits. The main obstacles in these applications were, up to now, memory requirements for computer systems and lack of the possibility of calculating spectra directly from Boolean equations. By using the algorithms presented here these obstacles have been overcome. Moreover, the methods presented in this dissertation can be regarded as representatives of a whole family of methods and the approach presented can be easily adapted to other orthogonal transforms used in digital logic design. Algorithms are shown for Adding, Arithmetic, and Reed-Muller transforms. However, the main focus of this dissertation is on the efficient computer calculation of Rademacher-Walsh spectra of Boolean functions, since this particular ordering of Walsh transforms is most frequently used in digital logic design. A theory has been developed to calculate the Rademacher-Walsh transform from a cube array specification of incompletely specified Boolean functions. The importance of representing Boolean functions as arrays of disjoint ON- and DC- cubes has been pointed out, and an efficient new algorithm to generate disjoint cubes from non-disjoint ones has been designed. The transform algorithm makes use of the properties of an array of disjoint cubes and allows the determination of the spectral coefficients in an independent way. By such an approach each spectral coefficient can be calculated separately or all the coefficients can be calculated in parallel. These advantages are absent in the existing methods. The possibility of calculating only some coefficients is very important since there are many spectral methods in digital logic design for which the values of only a few selected coefficients are needed. Most of the current methods used in the spectral domain deal only with completely specified Boolean functions. On the other hand, all of the algorithms introduced here are valid, not only for completely specified Boolean functions, but for functions with don't cares. Don't care minterms are simply represented in the form of disjoint cubes. The links between spectral and classical methods used for designing digital circuits are described. The real meaning of spectral coefficients from Walsh and other orthogonal spectra in classical logic terms is shown. The relations presented here can be used for the calculation of different transforms. The methods are based on direct manipulations on Karnaugh maps. The conversion start with Karnaugh maps and generate the spectral coefficients. The spectral representation of multiple-valued input binary functions is proposed here for the first time. Such a representation is composed of a vector of Walsh transforms each vector is defined for one pair of the input variables of the function. The new representation has the advantage of being real-valued, thus having an easy interpretation. Since two types of codings of values of binary functions are used, two different spectra are introduced. The meaning of each spectral coefficient in classical logic terms is discussed. The mathematical relationships between the number of true, false, and don't care minterms and spectral coefficients are stated. These relationships can be used to calculate the spectral coefficients directly from the graphical representations of binary functions. Similarly to the spectral methods in classical logic design, the new spectral representation of binary functions can find applications in many problems of analysis, synthesis, and testing of circuits described by such functions. A new algorithm is shown that converts the disjoint cube representation of Boolean functions into fixed-polarity Generalized Reed-Muller Expansions (GRME). Since the known fast algorithm that generates the GRME, based on the factorization of the Reed-Muller transform matrix, always starts from the truth table (minterms) of a Boolean function, then the described method has advantages due to a smaller required computer memory. Moreover, for Boolean functions, described by only a few disjoint cubes, the method is much more efficient than the fast algorithm. By investigating a family of elementary second order matrices, new transforms of real vectors are introduced. When used for Boolean function transformations, these transforms are one-to-one mappings in a binary or ternary vector space. The concept of different polarities of the Arithmetic and Adding transforms has been introduced. New operations on matrices: horizontal, vertical, and vertical-horizontal joints (concatenations) are introduced. All previously known transforms, and those introduced in this dissertation can be characterized by two features: "ordering" and "polarity". When a transform exists for all possible polarities then it is said to be "generalized". For all of the transforms discussed, procedures are given for generalizing and defining for different orderings. The meaning of each spectral coefficient for a given transform is also presented in terms of standard logic gates. There exist six commonly used orderings of Walsh transforms: Hadamard, Rademacher, Kaczmarz, Paley, Cal-Sal, and X. By investigating the ways in which these known orderings are generated the author noticed that the same operations can be used to create some new orderings. The generation of two new Walsh transforms in Gray code orderings, from the straight binary code is shown. A recursive algorithm for the Gray code ordered Walsh transform is based on the new operator introduced in this presentation under the name of the "bi-symmetrical pseudo Kronecker product". The recursive algorithm is the basis for the flow diagram of a constant geometry fast Walsh transform in Gray code ordering. The algorithm is fast (N 10g2N additions/subtractions), computer efficient, and is implemented

Truth tables and circuit diagrams fall short in many ways including their abilities to evaluate and manipulate combinational logic. By using algebraic methods to represent logic expressions, we can apply properties and identities to improve performance.