This thesis work is about commutativity which is a very important topic in mathematics, physics, engineering and many other fields. Two processes are said to be commutative if the order of "operation" of these processes does not matter. A typical example of two processes in real life that are not commutative is the process of opening the door and the process of going through the door. In mathematics, it is well known that matrix multiplication is not always commutative. Commutating operators play an essential role in mathematics, physics engineering and many other fields. A typical example of the importance of commutativity comes from signal processing. Signals pass through filters (often called operators on a Hilbert space by mathematicians) and commutativity of two operators corresponds to having the same result even when filters are interchanged. Many important relations in mathematics, physics and engineering are represented by operators satisfying a number of commutation relations. In chapter two of this thesis we treat commutativity of monomials of operatos satisfying certain commutation relations in relation to one-dimensional dynamical systems. We derive explicit conditions for commutativity of the said monomials in relation to the existence of periodic points of certain one-dimensional dynamical systems. In chapter three, we treat the crossed product algebra for the algebra of piecewise constant functions on given set, describe the commutant of this algebra of functions which happens to be the maximal commutative subalgebra of the crossed product containing this algebra. In chapter four, we give a characterization of the commutant for the algebra of piecewise constant functions on the real line, by comparing commutants for a non decreasing sequence of algebras.

In this thesis, We investigate the problem of when a C*-algebra is commutative through continuous functional calculus, The principal results are that: (1) A C*-algebra A is commutative if and only if e^(ix)e^(iy)=e^(iy)e^(ix), for all self-adjoint elements x,y in A. (2) A C*-algebra A is commutative if and only if e^(x)e^(y)=e^(y)e^(x) for all positive elements x,y in A. We will give an extension of (2) as follows: Let f:[a,b]-->[c,d] be any continuous strictly monotonic function where a,b,c,d in R, a<b,c<d. Then a C*-algebra A is commutative if and only if f(x)f(y)=f(y)f(x), for all self-adjoint elements x,y in A with spec(x) in [a,b] and spec(y) in [a,b].

A partly (anti-)commutative quiver algebra is a quiver algebra bound by an (anti-)commutativity ideal, that is, a quadratic ideal generated by monomials and (anti-)commutativity relations. We give a combinatorial description of the ideals and the associated generator graphs, from which one can quickly determine if the ideal is admissible or not. We describe the center of a partly (anti-)commutative quiveralgebra and state necessary and sufficient conditions for the center to be finitely genteratedas a K-algebra.Examples are provided of partly (anti-)commutative quiver algebras that are Koszul algebras. Necessary and sufficient conditions for finite generation of the Hochschild cohomology ring modulo nilpotent elements for a partly (anti-)commutative Koszul quiver algebra are given.

Orientador: Prof. Dr. Alysson Fábio Ferrari / Tese (doutorado) - Universidade Federal do ABC, Programa de Pós-Graduação em Física, 2017. / As deformações do espaço-tempo têm sido bastante estudadas desde diferentes abordagens tais como a não comutatividade canônica e deformações via álgebras de Hopf, com a motivação de que estas deformações podem aparecer a escalas de altas energias, como por exemplo a escala de Planck. De igual forma, pode-se buscar estender deformações para a estrutura do superespaço e a supersimetria, e assim estudar o comportamento clássico e quântico, como a invariância supersimétrica e renormalizabilidade, em modelos definidos sobre estas estruturas. Dois tipos de deformações possíveis da supersimetria foram estudadas neste trabalho. O primeiro deles envolve a introdução de um produto não comutativo em (3+1) dimensões, que embora seja um produto não associativo e que quebra a álgebra da supersimetria, permite construir um modelo de Wess-Zumino com correções de derivadas de ordem superior do tipo Lee-wick, e que resultam ser invariante sob as transformações da SUSY usual. O segundo tipo de deformação estudado utiliza o conceito de álgebras de Hopf, através de um twist de Drinfel¿d. No caso do modelo de Wess-Zumino em (2 + 1) dimensões, veremos que apesar de que as estruturas sejam construídas de forma consistente e seja possível preservar a álgebra da SUSY usando geradores deformados, o modelo resulta não ser invariante sob esta última e não renormalizável. Também foi usado o formalismo de twist para um modelo de Chern-simons com SUSY N = 2 em (2 + 1) dimensões, que permite construir um modelo invariante de calibre, no entanto a invariância da SUSY não seja evidente. Neste modelo, embora em principio a álgebra da SUSY pode ser preservada pelo uso de geradores deformados, estes tornam-se bastante complicados, dificultando a prova da invariância supersimétrica. Pode-se concluir que existem diferentes formas de deformar as estruturas algébricas da supersimetria e que devido aos vínculos de cada modelo em específico torna-se difícil a construção de modelos que preservem algumas das propriedades importantes de modelos supersimétricos que se estudam, tais como a invariância e renormalização. / The space-time deformations have been well studied using different approaches, like as canonical commutativity and deformations via Hopf algebras, with the motivation of such deformations can appear in high scale energies, for example, planck scales. The same way, they can extend deformations to superspace and supersymmetry structures, and thus, study the quantum and classical behavior, like as the supersymmetry invariance and renormalizability, in models defined on these structures. Two classes of possible transformation of supersymmetry were studied in this work. The first one involves the introduction of one non commutative product in (3 + 1) dimensions, although it is not associative and breaks the supersymmetry algebra. It allows the construction of a Wess- Zumino model with higher order derivatives corrections like as Lee-Wick models, and it is invariant under usual SUSY transformations. The second deformation class studied utilizes the Hopf algebra concept, through Drinfel¿d twist. In the Wess-Zumino case in (2 + 1) dimensions, we can observe, although, the construction of the algebraic structure is consistent and it is possible preserve the SUSY algebra using deformed generators, the model is not invariant under this last and non renormalizable, also the twist formalism was used to Chern-Simons model N = 2 in (2 + 1) dimensions, it allows to construct an invariant gauge model, however the SUSY invariance is not evident. In this model, although the SUSY algebra can be preserved using the deformed generators, they become complicated, making it difficult to prove the supersymmetric invariance. It is possible to conclude that there are different ways to deform the algebraic structures of supersymmetry and because of the constraints of each specific model, it is difficult the construction of models which preserve some important properties of supersymmetry models studies, like as invariance and renormalizability.

Ett av grundskolans uppdrag är att elever ska utveckla kunskap om de fyra räknesättens olika egenskaper. En sådan egenskap är kommutativitet för räknesättet addition, vilket innebär att termers rumsliga placering inte har betydelse för summan. Inom kunskapsområdet aritmetik och algebra är kunskap om den kommutativa egenskapen av vikt. I denna studie intervjuades tio elever i årskurs 2. Intervjuerna var semistrukturerade, vilket bland annat innebar att en intervjuguide följdes. Syftet var att kvalitativt beskriva hur elever resonerar kring operationer med tydlig kommutativitet genom att också använda operationer utan kommutativa egenskaper. Analysen visar att eleverna kan fokusera på olika aspekter av kommutativitet, de kan fokusera på summan, på termerna eller på operationen. Studien visar också att det förekommer elever som övergeneraliserar den kommutativa egenskapen till att gälla vid uttryck med subtraktion. / According to the curriculum, students in elementary school should develop their understanding of the different properties of addition, subtraction, multiplication and division. One property is commutativity for addition. Which means that the terms’ spatial position does not change the sum. A solid understanding of the commutative property is of importance in arithmetic but also in algebra. This qualitative study is based on interviews with ten students in second grade. The purpose of the study is to investigate how students reason when they meet sequences with commutative and non-commutative expressions. The result is that students tend to describe commutativity by focusing on either the sum, the terms or the operation. Students in the study also overgeneralize the commutative property to expression with subtraction.

The subgroup commutativity degree sd(G) of a finite group G was introduced almost ten years ago and deals with the number of commuting subgroups in the subgroups lattice L(G) of G. The extremal case sd(G) = 1 detects a class of groups classified by Iwasawa in 1941 (in fact sd(G) represents a probabilistic measure which allows us to understand how far is G from the groups of Iwasawa). Among them we have sd(G) = 1 when L(G) is distributive, that is, when G is cyclic. The characterization of a cyclic group by the distributivity of its lattice of subgroups is due to a classical result of Ore in 1938. Therefore sd(G) is strongly related to structural properties of L(G). Here we introduce a new notion of probability gsd(G) in which two arbitrary sublattices S(G) and T(G) of L(G) are involved simultaneously. In case S(G) = T(G) = L(G), we find exactly sd(G). Upper and lower bounds in terms of gsd(G) and sd(G) are among our main contributions, when the condition S(G) = T(G) = L(G) is removed. Then we investigate the problem of counting the pairs of commuting subgroups via an appropriate graph. Looking at the literature, we noted that a similar problem motivated the permutability graph of non–normal subgroups ΓN (G) in 1995, that is, the graph where all proper non– normal subgroups of G form the vertex set of ΓN (G) and two vertices H and K are joined if HK = KH. The graph ΓN (G) has been recently generalized via the notion of permutability graph of subgroups Γ(G), extending the vertex set to all proper subgroups of G and keeping the same criterion to join two vertices. We use gsd(G), in order to introduce the non–permutability graph of subgroups ΓL(G) ; its vertices are now given by the set L(G) − CL(G)(L(G)), where CL(G)(L(G)) is the smallest sublattice of L(G) containing all permutable subgroups of G, and we join two vertices H, K of ΓL(G) if HK 6= KH. We finally study some classical invariants for ΓL(G) and find numerical relations between the number of edges of ΓL(G) and gsd(G).

Ett område inom matematik som elever arbetar mycket med i skolan är aritmetik, vilket innebär hur de fyra räknesätten fungerar och kan användas. Dessa räknesätt har olika egenskaper, en av dessa egenskaper, som innehas av addition och mutliplikation, är kommutativitet. Denna egenskap innebär att termers rumsliga placering inte har betydelse för summan. För att eleverna ska utveckla goda kunskaper inom aritmetik och algebra är det därför av vikt att de lär sig om den kommutativa egenskapen. För att kunna skapa goda förutsättningar för elever behöver vi som lärare veta mer om hur elever förstår kommutativitet. Syftet med studien är därför att utforska hur elever i lågstadieåldern resonerar om och använder kommutativitet. I den här studien har elever i förskoleklassen upp till årskurs 3 intervjuats. Intervjuerna var semistrukturerade och individuella. Resultatet visar att när elever resonerar om kommutativitet, har några fokus på summan och några har fokus på termerna. I studien har det även framkommit att flertalet av eleverna övergeneraliserar kommutativitet och tillämpar egenskapen vid subtraktion, vilket överensstämmer väl med vad man sett i tidigare forskning. Elever använder olika beskrivningar när de resonerar om kommutativitet, där framförallt fyra var tydligt framträdande i studien: det spelar ingen roll vilken plats talen står på, de har bytt plats, de har vänt på siffrorna och de har bytt håll. Slutsatsen i studien är att förståelsen för kommutativitet är viktig för att tillförskaffa sig effektiva och användbara strategier i aritmetik. / One area of mathematics that students learn in school is arithmetic, where the four operations are found. These operations have different properties. One of those properties, valid for addition and multiplication, is commutativity. For addition, commutativity means that the terms’ spatial position does not change the sum. For example, 5+2 is equal to 2+5. For students to develop their knowledge of arithmetic, it is important that they also learn about commutativity.  Therefore, the aim of the study is to explore how student in primary school discuss and use commutativity in addition. Interviews have been made with student in the preschool class to grade 3. The interviews were semi-structured and individual. It was found that the students reason about commutativity in different ways, some focusing on the sum and some focusing on the terms. The study also shows that most students overgeneralize commutativity and apply it in subtraction which is in argument with findings from previous research. Students used different explanations when they described commutativity: the numbers spatial position doesn’t matter, they have changed place, the numbers are turned around and they have changed direction. The conclusion of the study is that understanding commutativity is important in providing effective and useful strategies in arithmetic.

Aufgabengeeignete Rechenstrategien flexibel zu nutzen ist ein wichtiges Ziel mathematischer Bildung und Bestandteil der Bildungsstandards der Grundschulmathematik. Kinder sollen spontan entscheiden, ob sie arithmetische Aufgaben in üblicher Weise berechnen oder ob sie Zeit und Aufwand investieren, um nach Vereinfachungsstrategien zu suchen und diese anzuwenden. Der Schwerpunkt der aktuellen Arbeit ist, wie Schüler beim flexiblen Erkennen und Anwenden von Vereinfachungsstrategien unterstützt werden können. Kontextfaktoren werden untersucht, welche die spontane Nutzung von Vereinfachungsstrategien unterstützen und den Transfer zwischen ihnen beeinflussen. Kognitive Theorien über die Entwicklung von mathematischen Konzepten und Strategien wurden mit Erkenntnissen aus der Expertise Forschung verbunden, welche die Unterschiede in der Flexibilität zwischen Experten und Novizen offen legen. Im Rahmen der iterativen Entwicklung von mathematischen Konzepten könnte ein erfolgreiches Erkennen und Anwenden einer Vereinfachungsstrategie von Faktoren, die konzeptionelles und/oder prozedurales Wissen aktivieren, profitieren. Am Beispiel von Vereinfachungsstrategien, die auf dem Kommutativgesetz (a + b = b + a) basieren, werden drei Kontextfaktoren (Instruktion, Assoziation und Schätzen), die spontanen Strategiegebrauch unterstützen oder behindern, untersucht. Insgesamt zeigt die Dissertation, dass spontane Strategienutzung durch bestimmte Kontextfaktoren unterstützt und durch Andere behindert werden kann. Diese Kontextfaktoren können im Prinzip in der Schulumgebung gesteuert werden. / Flexible use of task-appropriate solving strategies is an important goal in mathematical education and educational standard of elementary school mathematics. Children need to decide spontaneously whether they calculate arithmetic problems the usual way or whether they invest time and effort to search for shortcut options and apply them. The focus of the current work lies on how students can be supported in spotting and applying shortcut strategies flexibly. Contextual factors are investigated that support the spontaneous usage of shortcuts and influences the transfer between them. Cognitive theories about how mathematical concepts and strategies develop were combined with findings from research on expertise, which disclose differences between the flexibility of experts and novices. In line with iterativ development of mathematical concepts successfully spotting and applying a shortcut might thus benefit from factors activating conceptual and/or procedural knowledge. Shortcuts based on commutativity (a + b = b + a) are used as a test case to investigat three contextual factors (instruction, association and estimation), which support or hinder spontaneous strategy use. Overall, the dissertation shows that spontaneous strategy use can be supported by some contextual factors and impeded by others. These contextual factors can, in principle, be controlled in school environment.

This dissertation will seek to explore if the properties of some of the key results on semigroups and their compacti cations under the operation of addition also apply under the operation of multiplication. Consider- able emphasis will be placed on the semigroup N of the set of natural numbers and its compacti cation N. Furthermore, the dissertation will discuss the smallest ideal of a semi- group and highlight some of its fundamental properties. / Mathematics / M. Sc. (Mathematics)

This thesis tackles the problem of automatically proving the validity of mathematical formulas generated by program verification tools. In particular, it focuses on Satisfiability Modulo Theories (SMT): a young research topic that has seen great advances during the last decade. The solvers of this family have various applications in hardware design, program verification, model checking, etc.SMT solvers offer a good compromise between expressiveness and efficiency. They rely on a tight cooperation between a SAT solver and a combination of decision procedures for specific theories, such as the free theory of equality with uninterpreted symbols, linear arithmetic over integers and rationals, or the theory of arrays.This thesis aims at improving the efficiency and the expressiveness of the Alt-Ergo SMT solver. For that, we designed a new decision procedure for the theory of linear integer arithmetic. This procedure is inspired by Fourier-Motzkin's method, but it uses a rational simplex to perform computations in practice. We have also designed a new combination framework, capable of reasoning in the union of the free theory of equality, the AC theory of associative and commutativesymbols, and an arbitrary signature-disjoint Shostak theory. This framework is a modular and non-intrusive extension of the ground AC completion procedure with the given Shostak theory. In addition, we have extended Alt-Ergo with existing decision procedures to integrate additional interesting theories, such as the theory of enumerated data types and the theory of arrays. Finally, we have explored preprocessing techniques for formulas simplification as well as the enhancement of Alt-Ergo's SAT solver.