Density functional theory has been used in combination with the correlation consistent and polarization consistent basis sets to investigate the structures and energetics for a series of first-row closed shell and several second-row molecules of potential importance in atmospheric chemistry. The impact of basis set choice upon molecular description has been examined, and irregular convergence of molecular properties with respect to increasing basis set size for several functionals and molecules has been observed. The possible reasons and solutions for this unexpected behavior including the effect of contraction and uncontraction, of the basis set diffuse sp basis functions, basis set superposition error (BSSE) and core-valence sets also have been examined.

Employing correlation consistent basis sets coupled with electronic structure methods has enabled accurate predictions of chemical properties for second- and third-row main group and transition metal molecular species. For third-row (Ga-Kr) molecules, the performance of the correlation consistent basis sets (cc-pVnZ, n=D, T, Q, 5) for computing energetic (e.g., atomization energies, ionization energies, electron and proton affinities) and structural properties using the ab initio coupled cluster method including single, double, and quasiperturbative triple excitations [CCSD(T)] and the B3LYP density functional method was examined. The impact of relativistic corrections on these molecular properties was determined utilizing the Douglas-Kroll (cc-pVnZ-DK) and pseudopotential (cc-pVnZ-PP) forms of the correlation consistent basis sets. This work was extended to the characterization of molecular properties of novel chemically bonded krypton species, including HKrCl, FKrCF3, FKrSiF3, FKrGeF3, FKrCCF, and FKrCCKrF, and provided the first evidence of krypton bonding to germanium and the first di-krypton system. For second-row (Al-Ar) species, the construction of the core-valence correlation consistent basis sets, cc-pCVnZ was reexamined, and a revised series, cc-pCV(n+d)Z, was developed as a complement to the augmented tight-d valence series, cc-pV(n+d)Z. Benchmark calculations were performed to show the utility of these new sets for second-row species. Finally, the correlation consistent basis sets were used to study the structural and spectroscopic properties of Au(CO)Cl, providing conclusive evidence that luminescence in the solid-state can be attributed to oligomeric species rather than to the monomer.

The concept of (b, s)-continuous functions in topological spaces is introduced and studied. Some of their characteristic properties are considered. Also we investigate the relationships between these classes of functions and other classes of functions.

As more of the modern world is connected to the Internet, threats can reach further than ever before. Attacks happen all the time and many have serious consequences that disrupts the daily processes of people and companies, possibly causing lasting damage. To fight back, defensive tools are used to find and counter attacks. One of these tools is Snort. Snort finds malicious data packets and warns the user and counters the found attack. Snort relies on a list of signatures of different attacks, called a rule set, to know what is malicious. Many rule sets are available as paid subscriptions, but there are free alternatives. But how well can Snort defend a network using these free rule sets? By designing a network for experimentation and populating it with realistic background traffic, a group of rule sets are evaluated using a set of common attacks and tools. The performance hit when defending in a high speed, high bandwidth environment is evaluated as well. The results favour the Emerging Threats rule set. As for performance, Snort could not handle the most extreme amounts of traffic, with the rate of dropped packets making security dubious, but that occurred at the absolute peak of what consumer hardware can provide.

Union-free families of subsets of [n] = {1, . . ., n} have been studied in Frankl and Füredi (Eur J Combin 5:127-131, 1984). In this paper, we provide a complete characterization of maximal symmetric difference-free families of subsets of [n].

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1 Title: Collections of compact sets in descriptive set theory Author: Václav Vlasák Department: Department of Mathematical Analysis Supervisor: Doc. RNDr. Miroslav Zelený, Ph.D. Author's e-mail address: vlasakmm@volny.cz Abstract: This work consists of three articles. In Chapter 2, we dissert on the connections between complexity of a function f from a Polish space X to a Polish space Y and complexity of the set C(f) = {K ∈ K(X); f K is continuous}, where K(X) denotes the space of all compact subsets of X equipped with the Vietoris topology. We prove that if C(f) is analytic, then f is Borel; and assuming ∆1 2-Determinacy we show that f is Borel if and only if C(f) is coanalytic. Similar results for projective classes are also presented. In Chapter 3, we continue in our investigation of collection C(f) and also study its restriction on convergent sequences (C(f)). We prove that C(f) is Borel if and only if f is Borel. Similar results for projective classes are also presented. The Chapter 4 disserts on HN -sets, which form an important subclass of the class of sets of uniqueness for trigonometric series. We investigate the size of these classes which is reflected by the family of measures called polar which annihilate all the sets belonging to the given class. The main aim of this chapter is to answer in...

Let K be a compact subset of Cn . The polynomially convex hull of K is defined as The compact set K is said to be polynomially convex if = K. A closed subset is said to be locally polynomially convex at if there exists a closed ball centred at z such that is polynomially convex. The aim of this thesis is to derive easily checkable conditions to detect polynomial convexity in certain classes of sets in This thesis begins with the basic question: Let S1 and S2 be two smooth, totally real surfaces in C2 that contain the origin. If the union of their tangent planes is locally polynomially convex at the origin, then is locally polynomially convex at the origin? If then it is a folk result that the answer is, “Yes.” We discuss an obstruction to the presumed proof, and use a different approach to provide a proof. When dimR it turns out that the positioning of the complexification of controls the outcome in many situations. In general, however, local polynomial convexity of also depends on the degeneracy of the contact of T0Sj with We establish a result showing this. Next, we consider a generalization of Weinstock’s theorem for more than two totally real planes in C2 . Using a characterization, recently found by Florentino, for simultaneous triangularizability over R of real matrices, we present a sufficient condition for local polynomial convexity at of union of finitely many totally real planes is C2 . The next result is motivated by an approximation theorem of Axler and Shields, which says that the uniform algebra on the closed unit disc generated by z and h — where h is a nowhereholomorphic harmonic function on D that is continuous up to ∂D — equals . The abstract tools used by Axler and Shields make harmonicity of h an essential condition for their result. We use the concepts of plurisubharmonicity and polynomial convexity to show that, in fact, the same conclusion is reached if h is replaced by h+ R, where R is a nonharmonic perturbation whose Laplacian is “small” in a certain sense. Ideas developed for the latter result, especially the role of plurisubharmonicity, lead us to our final result: a characterization for compact patches of smooth, totallyreal graphs in to be polynomially convex.

Master of Science / Department of Mathematics / Sarah Reznikoff / A (1, ≤ 2)-identifying code is a subset of the vertex set C of a graph such that each pair of vertices intersects C in a distinct way. This has useful applications in locating errors in multiprocessor networks and threat monitoring. At the time of writing, there is no simply-stated rule that will indicate if a graph is (1, ≤ 2)-identifiable. As such, we discuss properties that must be satisfied by a valid (1, ≤ 2)-identifying code, characteristics of a graph which preclude the existence of a (1, ≤ 2)-identifying code, and relationships between the maximum degree and order of (1, ≤ 2)-identifiable graphs. Additionally, we show that (1, ≤ 2)-identifiable graphs have no forbidden induced subgraphs and provide a list of (1, ≤ 2)-identifiable graphs with minimum (1, ≤ 2)-identifying codes indicated.

The objective of this thesis is to develop a strategy that allows robots to safely execute open-loop motion patterns for pre-computed time durations when facing interruptions in communication. By computing the time horizon in which collisions with other robots are impossible, this method allows the robots to move safely despite having no updated information about the environment. As the complexity of multi-robot systems increase, communication failures in the form of packet losses, saturated network channels and hardware failures are inevitable. This thesis is motivated by the need to increase the robustness of operation in the face of such failures. The advantage of this strategy is that it prevents the jerky and unpredictable motion behaviour which often plague robotic systems experiencing communication issues. To compute the safe time horizon, the first step involves constructing reachable sets around the robots to determine the set of all positions that can be reached by the robot in a given amount of time. In order to avoid complications arising from the non-convexity of these reachable sets, analytical expressions for minimum area ellipses enclosing the reachable sets are obtained. By using a fast gradient descent based technique, intersections are computed between a robot’s trajectory and the reachable sets of other robots. This information is then used to compute the safe time horizon for each robot in real time. To this end, provable safety guarantees are formulated to ensure collision avoidance. This strategy has been verified in simulation as well as on a team of two-wheeled differential drive robots on a multi-robot testbed.