No description available.

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

M.Sc.

This thesis is concerned With showing the relations among Boolean lattices, Boolean rings and Boolean spaces, It establishes that the categories of Boolean lattices and proper Boolean lattice homomorphisms, Boolean spaces and proper continuous maps, Boolean rings and proper ring homomorphisms are equivalent to each other, In the final chapter the notion of a Boolean semi-group is used to obtain an alternate characterization of a Boolean lattice. / Thesis / Master of Science (MS)

Yes / Coherent spaces spanned by a nite number of coherent states, are introduced. Their coherence properties are studied, using the Dirac contour representation. It is shown that the corresponding projectors resolve the identity, and that they transform into projectors of the same type, under displacement transformations, and also under time evolution. The set of these spaces, with the logical OR and AND operations is a distributive lattice, and with the logical XOR and AND operations is a Boolean ring (Stone's formalism). Applications of this Boolean ring into classical CNOT gates with n-ary variables, and also quantum CNOT gates with coherent states, are discussed.

This dissertation attempts to answer some of the vital questions involved in the genetic regulatory networks: inference, optimization and robustness of the mathe- matical models. Network inference constitutes one of the central goals of genomic signal processing. When inferring rule-based Boolean models of genetic regulations, the same values of predictor genes can correspond to di®erent values of the target gene because of inconsistencies in the data set. To resolve this issue, a consistency-based inference method is developed to model a probabilistic genetic regulatory network, which consists of a family of Boolean networks, each governed by a set of regulatory functions. The existence of alternative function outputs can be interpreted as the result of random switches between the constituent networks. This model focuses on the global behavior of genetic networks and re°ects the biological determinism and stochasticity. When inferring a network from microarray data, it is often the case that the sample size is not su±ciently large to infer the network fully, such that it is neces- sary to perform model selection through an optimization procedure. To this end, the network connectivity and the physical realization of the regulatory rules should be taken into consideration. Two algorithms are developed for the purpose. One algo- rithm ¯nds the minimal realization of the network constrained by the connectivity, and the other algorithm is mathematically proven to provide the minimally connected network constrained by the minimal realization. Genetic regulatory networks are subject to modeling uncertainties and perturba- tions, which brings the issue of robustness. From the perspective of network stability, robustness is desirable; however, from the perspective of intervention to exert in- °uence on network behavior, it is undesirable. A theory is developed to study the impact of function perturbations in Boolean networks: It ¯nds the exact number of a®ected state transitions and attractors, and predicts the new state transitions and robust/fragile attractors given a speci¯c perturbation. Based on the theory, one algorithm is proposed to structurally alter the network to achieve a more favorable steady-state distribution, and the other is designed to identify function perturbations that have caused changes in the network behavior, respectively.

In this thesis we are concerned with arithmetic in a certain partially ordered, commutative semigroup D. The first chapter investigates the class of countable Boolean algebras from which this semigroup arises. The elements of D correspond to the isomorphism classes of the Boolean algebras under consideration. In Chapter 2 we begin the study of the semigroup structure of D. D is axiomatically described by three groups of axioms. It is proved that these axioms are categorical. The ordering of D is used to investigate the multiplication. The set of T of torsion elements of D (elements with only finite many distinct powers), form a subsemigroup whose structure is studied. There is a natural torsion free quotient D/T whose structure is also investigated. In Chapter 3, the axioms are used to characterize elements s of T in terms of the arithmetic in the subsemigroup generated by the elements that are smaller than s. The characterization is used to determine elements of T that cover a single element. In the last part of Chapter 3, we obtain some sufficient, purely combinatorial conditions for an element to have infinite order.

No description available.

The purpose of this thesis is to investigate certain properties of rings, ideals, and a special type of ring called a Boolean ring.

<p>This dissertation focuses on Boolean networks with a view to their applications in Systems Biology. We study two notions of stability, based on Hamming distance and on maintenance of a stable period length. Algorithms are given for the determination of Boolean networks from both complete and partial dynamics. The dynamics of ring networks are systematically studied. An algebraic structure is developed for derivation of adjacency matrices for the dynamics of Boolean networks from simple building blocks, both by edge-swapping and by gluing simple building blocks. Some results are implemented in Python and conclusions drawn for theta networks, a class of networks only slightly more complex than rings. A short section on applications to a known biological system closes the dissertation.</p> / Dissertation