Recent work has established that a group theoretical viewpoint of completely integrable dynamical systems with N degrees of freedom yields an algorithm that provides new information concerning the symmetry transformation group structure of this class of dynamical systems. The work presented here rests heavily on the results presented in reference and it is recommended that the reader consult this reference for a more rigorous discussion of the results given in this thesis.

In this dissertation we develop aspects of ergodic theory for C*-dynamical systems for which the C*-algebras are allowed to be noncommutative. We define four ergodic properties, with analogues in classic ergodic theory, and study C*-dynamical systems possessing these properties. Our analysis will show that, as in the classical case, only certain combinations of these properties are permissable on C*-dynamical systems. In the second half of this work, we construct concrete noncommutative C*-dynamical systems having various permissable combinations of the ergodic properties. This shows that, as in classical ergodic theory, these ergodic properties continue to be meaningful in the noncommutative case, and can be useful to classify and analyse C*-dynamical systems. / Dissertation (MSc)--University of Pretoria, 2013. / gm2014 / Mathematics and Applied Mathematics / unrestricted

We study the Cohen-Macaulay property of Rees algebras of modules of Kähler differentials. When the module of differentials has projective dimension one, it is known that condition $F_1$ is sufficient for the Rees algebra to be Cohen-Macaulay. The converse was proved if the module of differentials is already $F_0$. We weaken the condition $F_0$ globally by assuming some homogeneity condition.<br> <br> We are also interested in the defining ideal of the Rees algebra of a Jacobian module. If the Jacobian module is an ideal, we prove a formula for computing the defining ideal. Using the formula, we give an explicit description of the defining ideal in the monomial case. From there, we characterize the Cohen-Macaulay property of the Rees algebra.<br> <br> In the last chapter, we study Gorenstein linkage mostly in the graded case. In particular, we give an explicit example of a class of monomial ideals that are in the homogeneous Gorenstein linkage class of a complete intersection. To do so, we prove a Gorenstein double linkage construction that is analogous to Gorenstein biliaison.

The differential geometry software package in Maple has the necessary tools and commands to automate the classification process for complex simple Lie algebras. The purpose of this thesis is to write the programs to complete the classification for real simple Lie algebras. This classification is difficult because the Cartan subalgebras are not all conjugate as they are in the complex case. For the process of the real classification, one must first identify a maximally noncompact Cartan subalgebra. The process of the Cayley transform is used to find this specific Cartan subalgebra. This Cartan subalgebra is used to find the simple roots for the given real simple Lie algebra. With this information, we can then create a Satake diagram. Then we match our given algebra's Satake diagram to a Satake diagram of a known algebra. The programs explained in this thesis complete this process of classification.

This thesis was concerned with classifying the real indecomposable solvable Lie algebras with codimension one nilradicals of dimensions two through seven. This thesis was organized into three chapters. In the first, we described the necessary concepts and definitions about Lie algebras as well as a few helpful theorems that are necessary to understand the project. We also reviewed many concepts from linear algebra that are essential to the research. The second chapter was occupied with a description of how we went about classifying the Lie algebras. In particular, it outlined the basic premise of the classification: that we can use the automorphisms of the nilradical of the Lie algebra to find a basis with the simplest structure equations possible. In addition, it outlined a few other methods that also helped find this basis. Finally, this chapter included a discussion of the canonical forms of certain types of matrices that arose in the project. The third chapter presented a sample of the classification of the seven-dimensional Lie algebras. In it, we proceeded step-by-step through the classification of the Lie algebras whose nilradical was one of four specifically chosen because they were representative of the different types that arose during the project. In the appendices, we presented our results in a list of the multiplication tables of the isomorphism classes found.

No description available.

Octonion algebras are certain algebras with a multiplicative quadratic form. In 2019, Alsaody and Gille showed that, for octonion algebras over unital commutative rings, there is an equivalence between isotopes and isometric quadratic forms. The contravariant equivalence from unital commutative rings to affine schemes, sending a ring to its spectrum, leads us to a question: can the equivalence of isometry and isotopy be generalized to octonion algebras over a (not necessarily affine) scheme? We present the basic definitions and properties of octonion algebras, both over rings and over schemes. Then we show that an isotope of an octonion algebra C over a scheme is isomorphic to a twist by an Aut(C)–torsor. We conclude the thesis by giving an affirmative answer to our question.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第23564号 / 理博第4758号 / 新制||理||1682(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 泉 正己, 教授 COLLINS Benoit Vincent Pierre, 教授 加藤 毅 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

<p> This thesis deals with metrizations of sets of conditional entropies and sets of sub-σ-algebras. G Co Rajski's Theorem ([9]) on the metric space of discrete probability distributions can be deduced as a particular case of a theorem on the metric space of sub-σ-algebras given in Chapter III, the proof of which is comparatively very concise. The completeness of this metric space and some other properties are also proved. </p> / Thesis / Master of Science (MSc)

In the first part of the dissertation, we construct two-dimensional TQFTs which categorify the evaluations of circles in Kuperberg’s 𝐵₂ spider. We give a purely combinatorial evaluation formula for these TQFTs and show that it is compatible with the trace map on the corresponding commutative Frobenius algebras. Furthermore, we develop a theory of Θ-foams and their combinatorial evaluations to lift the ungraded evaluation of the Θ-web, thus paving a way for categorifying 𝐵₂ webs to 𝐵₂ foams. In the second part of the dissertation, we study the calculus of unoriented 𝔰𝔩₃ webs and foams. We focus on webs with a small number of boundary points. We obtain reducible collections and consider bilinear forms on these collections given by pairings of webs. We give web categories stable under the action of certain endofunctors and derive relations between compositions of these endofunctors.