Cette thèse est consacrée à une classe de groupes, appelés groupes de Kac-Moody, qui généralise de façon naturelle les groupes de Lie semi-simples, ou plus précisément, les groupes algébriques réductifs, dans un contexte infini-dimensionnel. On s'intéresse plus particulièrement au problème d'isomorphismes pour ces groupes, en vue d'obtenir un analogue infini-dimensionnel de la célèbre théorie des homomorphismes 'abstraits' de groupes algébriques simples, due à Armand Borel et Jacques Tits. Le problème d'isomorphismes qu'on étudie s'avère être un cas particulier d'un problème plus général, qui consiste à caractériser les homomorphismes de groupes algébriques vers les groupes de Kac-Moody, dont l'image est bornée. Ce problème peut à son tour s'énoncer comme un problème de rigidité pour les actions de groupes algébriques sur les immeubles, via l'action naturelle d'un groupe de Kac-Moody sur une paire d'immeubles jumelés. Les résultats partiels, relatifs à ce problème de rigidité, que nous obtenons, nous permettent d'apporter une solution complète au problème d'isomorphismes pour les groupes de Kac-Moody déployés. En particulier, on obtient un résultat de dévissage pour les automorphismes de ces objets. Celui-ci fournit à son tour une description complète de la structure du groupe d'automorphismes d'un groupe de Kac-Moody déployé sur un corps de caractéristique~$0$. Nos arguments permettent également de traiter de façon analogue certaines formes anisotropes de groupes de Kac-Moody complexes, appelées formes unitaires. On montre en particulier que la topologie Hausdorff naturelle que portent ces formes est un invariant de leur structure de groupe abstrait. Ceci généralise un résultat bien connu de H. Freudenthal pour les groupes de Lie compacts. Enfin, l'on s'intéresse aux homomorphismes de groupes de Kac-Moody à image fini-dimensionnelle, et l'on démontre la non-existence de tels homomorphismes à noyau central, lorsque le domaine est un groupe de Kac-Moody de type indéfini sur un corps infini. Ceci réduit un problème ouvert, dit problème de linéarité pour les groupes de Kac-Moody, au cas de corps de base finis.

One of the methods used for the construction of the classical Alexander horned sphere leads naturally to generalization to horned spheres of higher order. Let M2, denote the Alexander horned sphere. This is a 2-horned sphere of order 2. Denote by M 3 and M4, two 2-horned spheres of orders 3 and 4, respectively, constructed by such a generalization. The fundamental groups of the complements of M2, M3, and M4 are derived, and representations of these groups onto the Alternating Group, A5, are found. The form of the presentations of these fundamental groups leads to a more general class of groups, denoted by Gk, k ≥ 2. A set of homomorphisms ϴkl : Gk, k ≥ l ≥ 2 is found, which has a clear geometric meaning as applied to the groups G2, G3, and G4. Two theorems relating to direct systems of non-abelian groups are proved and applied to the groups Gk. The implication of these theorems is that the groups Gk, k≥2 are all free groups of countably infinite rank and that the embeddings of M2, M3, and M4 in E3 cannot be distinguished by means of fundamental groups. *33 pages)

This thesis investigates three areas of the theory of graph homomorphisms: cores of graphs, the homomorphism order, and quantum homomorphisms. A core of a graph X is a vertex minimal subgraph to which X admits a homomorphism. Hahn and Tardif have shown that, for vertex transitive graphs, the size of the core must divide the size of the graph. This motivates the following question: when can the vertex set of a vertex transitive graph be partitioned into sets which each induce a copy of its core? We show that normal Cayley graphs and vertex transitive graphs with cores half their size always admit such partitions. We also show that the vertex sets of vertex transitive graphs with cores less than half their size do not, in general, have such partitions. Next we examine the restriction of the homomorphism order of graphs to line graphs. Our main focus is in comparing this restriction to the whole order. The primary tool we use in our investigation is that, as a consequence of Vizing's theorem, this partial order can be partitioned into intervals which can then be studied independently. We denote the line graph of X by L(X). We show that for all n ≥ 2, for any line graph Y strictly greater than the complete graph Kₙ, there exists a line graph X sitting strictly between Kₙ and Y. In contrast, we prove that there does not exist any connected line graph which sits strictly between L(Kₙ) and Kₙ, for n odd. We refer to this property as being ``n-maximal", and we show that any such line graph must be a core and the line graph of a regular graph of degree n. Finally, we introduce quantum homomorphisms as a generalization of, and framework for, quantum colorings. Using quantum homomorphisms, we are able to define several other quantum parameters in addition to the previously defined quantum chromatic number. We also define two other parameters, projective rank and projective packing number, which satisfy a reciprocal relationship similar to that of fractional chromatic number and independence number, and are closely related to quantum homomorphisms. Using the projective packing number, we show that there exists a quantum homomorphism from X to Y if and only if the quantum independence number of a certain product graph achieves |V(X)|. This parallels a well known classical result, and allows us to construct examples of graphs whose independence and quantum independence numbers differ. Most importantly, we show that if there exists a quantum homomorphism from a graph X to a graph Y, then ϑ̄(X) ≤ ϑ̄(Y), where ϑ̄ denotes the Lovász theta function of the complement. We prove similar monotonicity results for projective rank and the projective packing number of the complement, as well as for two variants of ϑ̄. These immediately imply that all of these parameters lie between the quantum clique and quantum chromatic numbers, in particular yielding a quantum analog of the well known ``sandwich theorem". We also briefly investigate the quantum homomorphism order of graphs.

Homomorphic cryptography provides a third party with the ability to perform simple computations on encrypted data without revealing any information about the data itself. Typically, a third party can calculate one of the encrypted sum or the encrypted product of two encrypted messages. This is possible due to the fact that the encryption function is a group homomorphism, and thus preserves group operations. This makes homomorphic cryptosystems useful in a wide variety of privacy preserving protocols. A comprehensive survey of known homomorphic cryptosystems is provided, including formal definitions, security assumptions, and outlines of security proofs for each cryptosystem presented. Threshold variants of several homomorphic cryptosystems are also considered, with the first construction of a threshold Boneh-Goh-Nissim cryptosystem given, along with a complete proof of security under the threshold semantic security game of Fouque, Poupard, and Stern. This approach is based on Shoup's approach to threshold RSA signatures, which has been previously applied to the Paillier and Damg\aa rd-Jurik cryptosystems. The question of whether or not this approach is suitable for other homomorphic cryptosystems is investigated, with results suggesting that a different approach is required when decryption requires a reduction modulo a secret value. The wide variety of protocols utilizing homomorphic cryptography makes it difficult to provide a comprehensive survey, and while an overview of applications is given, it is limited in scope and intended to provide an introduction to the various ways in which homomorphic cryptography is used beyond simple addition or multiplication of encrypted messages. In the case of strong conditional oblivious tranfser, a new protocol implementing the greater than predicate is presented, utilizing some special properties of the Boneh-Goh-Nissim cryptosystem to achieve security against a malicious receiver.

Homomorphic cryptography provides a third party with the ability to perform simple computations on encrypted data without revealing any information about the data itself. Typically, a third party can calculate one of the encrypted sum or the encrypted product of two encrypted messages. This is possible due to the fact that the encryption function is a group homomorphism, and thus preserves group operations. This makes homomorphic cryptosystems useful in a wide variety of privacy preserving protocols. A comprehensive survey of known homomorphic cryptosystems is provided, including formal definitions, security assumptions, and outlines of security proofs for each cryptosystem presented. Threshold variants of several homomorphic cryptosystems are also considered, with the first construction of a threshold Boneh-Goh-Nissim cryptosystem given, along with a complete proof of security under the threshold semantic security game of Fouque, Poupard, and Stern. This approach is based on Shoup's approach to threshold RSA signatures, which has been previously applied to the Paillier and Damg\aa rd-Jurik cryptosystems. The question of whether or not this approach is suitable for other homomorphic cryptosystems is investigated, with results suggesting that a different approach is required when decryption requires a reduction modulo a secret value. The wide variety of protocols utilizing homomorphic cryptography makes it difficult to provide a comprehensive survey, and while an overview of applications is given, it is limited in scope and intended to provide an introduction to the various ways in which homomorphic cryptography is used beyond simple addition or multiplication of encrypted messages. In the case of strong conditional oblivious tranfser, a new protocol implementing the greater than predicate is presented, utilizing some special properties of the Boneh-Goh-Nissim cryptosystem to achieve security against a malicious receiver.

In this thesis, we study the question when a local automorphism of continuous functions, or in general, of an operator algebra, is an automorphism. We also study the question how to write an n-disjointness preserving operator as a finite sum of orthomorphisms locally.

This thesis investigates three areas of the theory of graph homomorphisms: cores of graphs, the homomorphism order, and quantum homomorphisms. A core of a graph X is a vertex minimal subgraph to which X admits a homomorphism. Hahn and Tardif have shown that, for vertex transitive graphs, the size of the core must divide the size of the graph. This motivates the following question: when can the vertex set of a vertex transitive graph be partitioned into sets which each induce a copy of its core? We show that normal Cayley graphs and vertex transitive graphs with cores half their size always admit such partitions. We also show that the vertex sets of vertex transitive graphs with cores less than half their size do not, in general, have such partitions. Next we examine the restriction of the homomorphism order of graphs to line graphs. Our main focus is in comparing this restriction to the whole order. The primary tool we use in our investigation is that, as a consequence of Vizing's theorem, this partial order can be partitioned into intervals which can then be studied independently. We denote the line graph of X by L(X). We show that for all n ≥ 2, for any line graph Y strictly greater than the complete graph Kₙ, there exists a line graph X sitting strictly between Kₙ and Y. In contrast, we prove that there does not exist any connected line graph which sits strictly between L(Kₙ) and Kₙ, for n odd. We refer to this property as being ``n-maximal", and we show that any such line graph must be a core and the line graph of a regular graph of degree n. Finally, we introduce quantum homomorphisms as a generalization of, and framework for, quantum colorings. Using quantum homomorphisms, we are able to define several other quantum parameters in addition to the previously defined quantum chromatic number. We also define two other parameters, projective rank and projective packing number, which satisfy a reciprocal relationship similar to that of fractional chromatic number and independence number, and are closely related to quantum homomorphisms. Using the projective packing number, we show that there exists a quantum homomorphism from X to Y if and only if the quantum independence number of a certain product graph achieves |V(X)|. This parallels a well known classical result, and allows us to construct examples of graphs whose independence and quantum independence numbers differ. Most importantly, we show that if there exists a quantum homomorphism from a graph X to a graph Y, then ϑ̄(X) ≤ ϑ̄(Y), where ϑ̄ denotes the Lovász theta function of the complement. We prove similar monotonicity results for projective rank and the projective packing number of the complement, as well as for two variants of ϑ̄. These immediately imply that all of these parameters lie between the quantum clique and quantum chromatic numbers, in particular yielding a quantum analog of the well known ``sandwich theorem". We also briefly investigate the quantum homomorphism order of graphs.

For many years, accounting discourse has been concerned with the measurement of attributes of accounting events. The concept of measurement forms one of the central pillars on which the preparation of financial statements is based. For example, financial statements can only be prepared if economic events meet the definition of an element of the financial statements and have a cost or value that can be measured with reliability. This has created the belief that accounting practices are practices of measurement. The principles of measurement require that every process of measurement be based on an appropriate theory of measurement. Given the belief in the accounting discipline that accounting practices are synonymous with practices of measurement, it is to be assumed that acceptable theories of measurement exist in the discipline. However, over the decades, researchers in this field have been unsuccessful in creating an acceptable theory of accounting measurement. This suggests that there is a gap between the accounting concept of measurement and the scientific principles of measurement. The main goals of this thesis are to determine whether the concept of measurement in accounting is in harmony with the principles of the representational theory of measurement and to develop a conceptual model of the accounting measurement problem. Representational theory establishes measurement in the social sciences. Accounting is regarded as a social science. This study applies the principles of the representational theory of measurement to determine the nature of the gap between the accounting concept of measurement and the principles of this theory. The very fact of referring to accounting as a measurement discipline implies the presence of an established and well-known theory of measurement in that discipline: a discipline cannot be referred to as a measurement discipline without the existence of such a theory. Therefore, the lack of success of researchers in creating a theory of accounting measurement suggests that accounting is not a measurement discipline and that accountants are currently unaware of this. To test this hypothesis: 1. A critical literature analysis was conducted, investigating whether the accounting concept of measurement is in harmony with the principles of the representational theory of measurement. The results of this analysis indicated that the accounting concept of measurement is not in harmony with the principles of the representational theory of measurement. 2. A questionnaire was sent to accountants, testing whether they were familiar with the principles of the representational theory of measurement and their application to the accounting discipline. The results of the survey indicated that accountants are not familiar with the principles of the representational theory of measurement or their application to the accounting discipline. The main contributions of this study lie in the fact that it reveals that current accounting practices are not in harmony with the principles of representational measurement. Accountants should be educated in the purpose of the principles of measurement in the social sciences in order to ensure their proper application during accounting measurement. / Thesis (DCom)--University of Pretoria, 2009. / Financial Management / unrestricted

Let K be a class of graphs. A pair (F,U) is a finite duality in K if U∈K, F is a finite set of graphs, and for any graph G in K we have G≤U if and only if F≤≰G for all F∈F, where "≤" is the homomorphism order. We also say U is a dual graph in K. We prove that the class of planar graphs has no finite dualities except for two trivial cases. We also prove that the class of toroidal graphs has no 5-colorable dual graphs except for two trivial cases. In a sharp contrast, for a higher genus orientable surface S we show that Thomassen's result (Thomassen, 1997 [17]) implies that the class, G(S), of all graphs embeddable in S has a number of finite dualities. Equivalently, our first result shows that for every planar core graph H except K1 and K4, there are infinitely many minimal planar obstructions for H-coloring (Hell and Nešetřil, 1990 [4]), whereas our later result gives a converse of Thomassen's theorem (Thomassen, 1997 [17]) for 5-colorable graphs on the torus.

We consider constrained variants of graph homomorphisms such as embeddings, monomorphisms, full homomorphisms, surjective homomorpshims, and locally constrained homomorphisms. We also introduce a new variation on this theme which derives from relations between graphs and is related to multihomomorphisms. This gives a generalization of surjective homomorphisms and naturally leads to notions of R-retractions, R-cores, and R-cocores of graphs. Both \\mbox{R-cores} and R-cocores of graphs are unique up to isomorphism and can be computed in polynomial time. The theory of the graph homomorphism order is well developed, and from it we consider analogous notions defined for orders induced by constrained homomorphisms. We identify corresponding cores, prove or disprove universality, characterize gaps and dualities. We give a new and significantly easier proof of the universality of the homomorphism order by showing that even the class of oriented cycles is universal. We provide a systematic approach to simplify the proofs of several earlier results in this area. We explore in greater detail locally injective homomorphisms on connected graphs, characterize gaps and show universality. We also prove that for every $d\\geq 3$ the homomorphism order on the class of line graphs of graphs with maximum degree $d$ is universal.