In this thesis, we will study several problems about approximate mappings between C*-algebras.

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An (n, k, p, t) lotto design is a collection of k-subsets of a set X of n numbers wherein every p-subset of X must intersect at least one k-subset in t or more elements. L(n,k,p,t) is the minimum number of k-subsets which guarantees an intersection of at least t numbers between any p-subset of X and at least one of the k-subsets. To determine L(n,k,p,t) is the main goal of lotto design research. In previous work on lotto designs, other researchers used sequential algorithms to find bounds for L(n,k,p,t). We will determine the number of non-isomorphic optimal lotto designs on 5 or 6 blocks for n,k,p,t <= 20 and also improve lower bounds for L(n,k,p,t) >= 6 if possible by a more efficient implementation of a backtracking algorithm. / May 2005

Every Jordan isomorphism of triangular n¡Ñn matrices over F with characteristic 2 is either a isomorphism or a antiisomorphism while n is 2. But it is not true for n ¡Ù 3.

An (n, k, p, t) lotto design is a collection of k-subsets of a set X of n numbers wherein every p-subset of X must intersect at least one k-subset in t or more elements. L(n,k,p,t) is the minimum number of k-subsets which guarantees an intersection of at least t numbers between any p-subset of X and at least one of the k-subsets. To determine L(n,k,p,t) is the main goal of lotto design research. In previous work on lotto designs, other researchers used sequential algorithms to find bounds for L(n,k,p,t). We will determine the number of non-isomorphic optimal lotto designs on 5 or 6 blocks for n,k,p,t <= 20 and also improve lower bounds for L(n,k,p,t) >= 6 if possible by a more efficient implementation of a backtracking algorithm.

An (n, k, p, t) lotto design is a collection of k-subsets of a set X of n numbers wherein every p-subset of X must intersect at least one k-subset in t or more elements. L(n,k,p,t) is the minimum number of k-subsets which guarantees an intersection of at least t numbers between any p-subset of X and at least one of the k-subsets. To determine L(n,k,p,t) is the main goal of lotto design research. In previous work on lotto designs, other researchers used sequential algorithms to find bounds for L(n,k,p,t). We will determine the number of non-isomorphic optimal lotto designs on 5 or 6 blocks for n,k,p,t <= 20 and also improve lower bounds for L(n,k,p,t) >= 6 if possible by a more efficient implementation of a backtracking algorithm.

<p> In 1970, D. S. Ornstein introduced some new approximation concepts which enabled him to establish that the Shannon entropy of endomorphism was a complete invariant for a class of transformations known as Bernoulli shifts. This work of Ornstein contains powerful, deep and elegant techniques which have opened up a new period in the theory of measure preserving transformations, or as it is usually called, in ergodic theory.</p> <p> This thesis contains the study of two new classes of entropies, the γ-entropy and the δ-entropy, where each of these two classes have Shannon's entropy as a member. The algebraic and analytic properties of these entropies and their characterizations are discussed. Finally, the δ-entropy of endomorphism is defined and it has been used in solving the isomorphism problem for Bernoulli shifts. Thus, it is shown that the isomorphism problem for Bernoulli shifts holds not only for one entropy, but for an infinite class of entropies introduced in this thesis.</p> / Thesis / Doctor of Philosophy (PhD)

This thesis proposes an expressive yet efficient declarative framework for graph matching in constraint programming (CP), and focuses on efficient algorithms to solve the subgraph isomorphism problem. The framework is based on graph and map variables, and specific graph morphism constraints. This allows to model and solve various graph matching problems, avoiding the tedious development of dedicated and specific algorithms. A specialized filtering algorithm is proposed for the subgraph isomorphism problem, which uses the semantic of the problem as well as the global structure of the two input graphs. It is shown that it is the state-of-the-art filtering algorithm, compared to dedicated algorithms and other CP approaches. Various search techniques from CP are also extended to the subgraph isomorphism problem. An automatic detection and exploitation of symmetries for the subgraph isomorphism problem is proposed, together with a decomposition approach of the search. The significance of this thesis lies in the fact that, even though the framework is expressive, CP can be considered as the state-of-the-art for subgraph isomorphism, outperforming the dedicated known algorithms on current benchmarks.

The main theme of this thesis is to study the isometric algebra isomorphisms and the bipositive algebra isomorphisms between various Banach algebras associated with locally compact groups. Let $LUC(G)$ denote the $C^*$-algebra of left uniformly continuous functions with the uniform norm and let $C_0(G)^{\perp}$ denote the annihilator of $C_0(G)$ in $LUC(G)^*$. In Chapter 2 of this thesis, among other results, we show that if $G$ is a locally compact group and $H$ is a discrete group then whenever there exists a weak-star continuous isometric isomorphism between $C_0(G)^{\perp}$ and $C_0(H)^{\perp}$, $G$ is isomorphic to $H$ as a topological group. In particular, when $H$ is discrete $C_0(H)^{\perp}$ determines $H$ within the class of locally compact topological groups. In Chapter 3 of this thesis, we show that if $M(G,\omega_1)$ (the weighted measure algebra on $G$) is isometrically algebra isomorphic to $M(H,\omega_2)$, then the underlying weighted groups are isomorphic, i.e. there exists an isomorphism of topological groups $\phi:G\to H$ such that $\small{\displaystyle{\frac{\omega_1}{\omega_2\circ\phi}}}$ is multiplicative. Similarly, we show that any weighted locally compact group $(G,\omega)$ is completely determined by its Beurling group algebra $L^1(G,\omega)$, $LUC(G,\omega^{-1})^*$ and $L^1(G,\omega)^{**}$, when the two last algebras are equipped with an Arens product. Here, $LUC(G,\omega^{-1})$ is the weighted analogue of $LUC(G)$, for weighted locally compact groups. In Chapter 4 of this thesis, we show that the order structure combined with the algebra structure of each of the Banach algebras $L^1(G,\omega)$, $M(G,\omega)$, $LUC(G,\omega^{-1})^*$ and $L^1(G,\omega)^{**}$ completely determines the underlying topological group structure together with a constraint on the weight. In particular, we obtain new proofs for a previously known result of Kawada and results of Farhadi as special cases of our results. Finally, we provide an example of a bipositive algebra isomorphism between Beurling measure algebras that is not an isometry. We conclude this thesis with a selective list of open problems. / February 2016

<p> The major results of this work include an isomorphism theorem for B-complete barrelled spaces with similar bases and a theorem which shows that the cone associated with a separating biorthogonal system in a perfect C.N.S. has a basis. We also obtain some applications of the former result in the case of dual generalized bases and some results concerning Schauder bases in countably barrelled spaces.</p> / Thesis / Doctor of Philosophy (PhD)