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Cette thèse porte sur certaines extensions des automates pondérés, et étudie les séries qu’ils réalisent en fonction de la nature des poids.Ces extensions se distinguent par les mouvements supplémentaires autorisés à la tête de lecture de l’automate : retour au début du mot pour les automates circulaires, changement de sens de lecture pour les automates boustrophédons.Dans le cas général, les automates pondérés circulaires sont plus puissants que les automates unidirectionnels classiques, et moins puissants que les boustrophédons.On introduit de plus les expressions de Hadamard, qui sont une extension des expressions rationnelles et qui permettent de dénoter le comportement des automates circulaires. Les aspects algorithmiques de cette conversion sont étudiés dans le cas où les poids appartiennent à un semi-anneau rationnellement additif.On montre que lorsque les poids sont des nombres rationnels, réels ou complexes, les automates circulaires sont aussi expressifs que les boustrophédons.Enfin, si les poids forment un bi-monoïde localement fini, les automates boustrophédons ne sont pas plus expressifs que les automates pondérés classsiques. / This thesis deals with some extensions of weighted automata,and studies the series they can realisedepending on the nature of their weigths.These extensions are characterised by howthe input head of the automaton is allowed to move:rotating automata can go back at the beginning of the word,and two-way automata can change the reading direction.In the general setting, weigthed rotating automata are morepowerful than classical one-way automata, and less powerfulthan two-way ones.Moreover, we introduce Hadamard expressions,which are an extension of rational expressions and can denotethe behaviour of rotating automata.The algorithms for this conversion are studied when the weights belong toa rationally additive semiring.Then, rotating automata are shown as expressive as two-way automatain the case of rational, real or complex numbers.It is also proved that two-way and one-way automataare equivalent when weighted on a locally finite bimonoid.

在本篇論文中我們使用矩陣來建構可分組設計(GDD), 我們列出了兩種型 式的建構, 第一種 -- 起因於 W.H. Haemers -- A .crtimes. J + I .crtimes. D, 利用此種建構我們將所有符合 r - .lambda.1 = 1 的 (m,n,k,.lambda.1,.lambda.2) GDD 分成三類: (i) A=0 或 J-I, (ii) A 為 .mu. - .lambda. = 1 強則圖的鄰接矩陣, (iii) J-2A 為斜對稱 矩陣的核心。第二種型式為 A .crtimes. D + .Abar .crtimes. .Dbar ,此種方法可以建構出 b=4(r-.lambda.2) 的正規和半正規 GDD 。另外在 論文中, 我們研究在這些建構中出現的相關題目。 / In this thesis we use matrices to construct group divisible designs (GDDs). We list two type of constructions, the first type is -- due to W.H. Heamers -- A .crtimes. J + I .crtimes. D and use this construction we classify all the (m,n,k,. lambda.1, .lambda.2) GDD with r - .lambda.1 = 1 in three classes according to (i) A = 0 or J-I, (ii) A is the adjacency matrix of a strongly regular graph with .mu. - .lambda. = 1, (iii) J - 2A is the core of a skew-symmetric Hadamard matrix. The second type is A .crtimes. D + .Abar .crtimes. .Dbar , this type can construct many regular and semi-regular GDDs with b=4(r-.lambda.2). In the thesis we investigate related topics that occur in these constructions.

Boolean functions are the main building blocks of most cipher systems. Various aspects of their cryptological characteristics are examined and investigated by many researchers from different fields. This thesis has no claim to obtain original results but consists in an attempt at giving a unified survey of the main results of the subject. In this thesis, the theory of boolean functions is presented in details, emphasizing some important cryptological properties such as balance, nonlinearity, strict avalanche criterion and propagation criterion. After presenting many results about these criteria with detailed proofs, two upper bounds and two lower bounds on the nonlinearity of a boolean function due to Zhang and Zheng are proved. Because of their importance in the theory of boolean functions, construction of Sylvester-Hadamard matrices are shown and most of their properties used in cryptography are proved. The Walsh transform is investigated in detail by proving many properties. By using a property of Sylvester-Hadamard matrices, the fast Walsh transform is presented and its application in finding the nonlinearity of a boolean function is demonstrated. One of the most important classes of boolean functions, so called bent functions, are presented with many properties and by giving several examples, from the paper of Rothaus. By using bent functions, relations between balance, nonlinearity and propagation criterion are presented and it is shown that not all these criteria can be simultaneously satisfied completely. For this reason, several constructions of functions optimizing these criteria which are due to Seberry, Zhang and Zheng are presented.

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Empirical likelihood (EL) is a recently developed nonparametric method of statistical inference. It has been shown by Owen (1988,1990) and many others that empirical likelihood ratio (ELR) method can be used to produce nice confidence intervals or regions. Owen (1988) shows that -2logELR converges to a chi-square distribution with one degree of freedom subject to a linear statistical functional in terms of distribution functions. However, a generalization of Owen's result to the right censored data setting is difficult since no explicit maximization can be obtained under constraint in terms of distribution functions. Pan and Zhou (2002), instead, study the EL with right censored data using a linear statistical functional constraint in terms of cumulative hazard functions. In this dissertation, we extend Owen's (1988) and Pan and Zhou's (2002) results subject to non-linear but Hadamard differentiable statistical functional constraints. In this purpose, a study of differentiable functional with respect to hazard functions is done. We also generalize our results to two sample problems. Stochastic process and martingale theories will be applied to prove the theorems. The confidence intervals based on EL method are compared with other available methods. Real data analysis and simulations are used to illustrate our proposed theorem with an application to the Gini's absolute mean difference.

This thesis is comprised of three main parts: The Hermite-Hadamard inequality on discrete time scales, the fractional Hermite-Hadamard inequality, and Karush-Kuhn- Tucker conditions on higher dimensional discrete domains. In the first part of the thesis, Chapters 2 & 3, we define a convex function on a special time scale T where all the time points are not uniformly distributed on a time line. With the use of the substitution rules of integration we prove the Hermite-Hadamard inequality for convex functions defined on T. In the fourth chapter, we introduce fractional order Hermite-Hadamard inequality and characterize convexity in terms of this inequality. In the fifth chapter, we discuss convexity on n{dimensional discrete time scales T = T1 × T2 × ... × Tn where Ti ⊂ R , i = 1; 2,…,n are discrete time scales which are not necessarily periodic. We introduce the discrete analogues of the fundamental concepts of real convex optimization such as convexity of a function, subgradients, and the Karush-Kuhn-Tucker conditions. We close this thesis by two remarks for the future direction of the research in this area.

In this thesis we introduce selected classes of matrices, whose entries are only numbers −1, 1, 0. We combine existing results from various fields of Mathematics and enrich them with specific examples and explanations, with the aim of making the understanding of the text easier. Thanks to that, the reader can comprehend the theory and look under the hood of non-trivial applications. We will start with introducing adjacency matrices and covering of complete graphs with complete bipartite graphs. Then we follow with Hadamard matrices and will show the conditions for their constructions. Incidence matrices of the set systems will help us solve the combinatorial problem of the Odd-town clubs. Finally, we will prove the Cayley formula about the spanning trees of the complete graph, using incidence matrices.

In this thesis we study the application of trace maps over Galois fields and Galois rings in the construction of non-binary linear and non-linear codes and mutually unbiased bases. Properties of the trace map over the Galois fields and Galois rings has been used very successfully in the construction of cocyclic Hadamard, complex Hadamard and Butson Hadamard matrices and consequently to construct linear codes over integers modulo prime and prime powers. These results provide motivation to extend this work to construct codes over integers modulo . The prime factorization of integers paved the way to focus our attention on the direct product of Galois rings and Galois fields of the same degree. We define a new map over the direct product of Galois rings and Galois fields by using the usual trace maps. We study the fundamental properties of the this map and notice that these are very similar to that of the trace map over Galois rings and Galois fields. As such this map called the trace-like map and is used to construct cocyclic Butson Hadamard matrices and consequently to construct linear codes over integers modulo . We notice that the codes construct in this way over the integers modulo 6 is simplex code of type . A further generalization of the trace-like map called the weighted-trace map is defined over the direct product of Galois rings and Galois fields of different degrees. We use the weighted-trace map to construct some non-linear codes and mutually unbiased bases of odd integer dimensions. Further more we study the distribution of over the Galois fields of degree 2 and use it to construct 2-dimensional, two-weight, self-orthogonal codes and constant weight codes over integers modulo prime.

Neurological and neuroendocrine disorders result from signaling dysregulation at the molecular, cellular, and multi-cellular levels. This dissertation presents the development of separation methods, using capillary zone electrophoresis (CZE) and capillary liquid chromatography (CLC), for detecting and quantifying small molecules, peptides, and proteins involved in cellular signaling. CZE is a rapid separation technique, making it ideal for monitoring cellular dynamics with high temporal resolution. An ultraviolet - light emitting diode was used for photolytic optical gating of caged fluorophore-labeled biogenic amines, common functional groups in neurotransmitters. Additionally, a novel caged fluorophore with faster reaction kinetics than commercially available dyes was used to label reduced thiols and primary amines in the presence of o-phthalaldehyde. Together this light source and novel caged dye illustrate the utility of these methods for monitoring chemical dynamics during continuous sampling. Many cellular second messengers, including inositol phosphates, are known to exist within the cell, but their dynamics and intermolecular interactions are poorly understood since they lack chromophores or electroactive functional groups making direct detection difficult. Utilizing CZE with capacitive coupled contactless conductivity detection (C4D), biological phosphates were separated and detected based on their high anionic charge, suggesting the utility of C4D in label-free detection of biological molecules. The techniques described above require higher sensitivity to monitor physiologically relevant analyte concentrations; therefore, Hadamard transform capillary electrophoresis (HTCE) was used as a multiplexing method in which multiple separations were performed simultaneously. HTCE resulted in increased sensitivity by decreasing the random background noise. Peptides and proteins propagate signals within or between cells; yet, they are difficult to separate and detect by CZE since their highly charged surfaces result in non-specific adsorption to the capillary wall. To minimize these interactions, stable hybrid phospholipid bilayers were prepared as capillary coatings for CZE separations of cationic proteins. Additionally, stabilized phospholipid bilayer coatings were formed on silica particles through redox polymerization of synthetic, polymerizable lipids. These bilayers were stable after exposure to surfactant, organic solvents, and after storage for one month, suggesting their value as lipid chromatography stationary phases for future incorporation of transmembrane proteins to analyze binding interactions with small molecules.