<p> Motivated by the problem of optimizing sensor network covers, we generalize the persistent homology of simplicial complexes over a single radial parameter to the context of multiple radial parameters. The persistent homology of so-called multiradial (multi)filtrations is identified as a special case of multidimensional persistence. Specifically, we exhibit that the persistent homology of (multi)filtrations corresponds to both generalized persistence modules of the form <b>[special characters omitted]</b> and (multi)graded modules over a polynomial ring. The stability of persistence barcodes/diagrams of multiradial filtrations is derived, along with explicit bounds associated to perturbations in both radii and vertex position. A strengthening of the Vietoris-Rips lemma of [DSG07, p. 346] to the setting of multiple radial parameters is obtained. We also use the categorical framework of [BDSS15] to show the persistent homology modules of multiradial (multi)filtrations are stable.</p>

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There are two basic approaches to pattern recognition: decision-theoretic and syntactic. However, in actual applications, a combination of both may be needed. One such hybrid technique consists of syntactic method coupled with stochasticity in its grammar. Randomness in the syntactic case is caused due to noise and insufficient information about characteristics of pattern classes. To absorb the effect of this randomness, the grammar must be generalized to include the probabilities of production rules. In this paper, a preliminary discussion of issues involved with hybrid techniques, in general, and stochastic grammars, in particular, is provided. An efficient algorithm for an automatic learning of production probabilities is devised. Concepts are illustrated via examples.

Motivated by potential applications to theoretical computer science, in particular those areas where the Curry-Howard correspondence plays an important role, as well as by the ongoing search in pure mathematics for feasible approaches to higher category theory, we undertake a detailed study of a new mathematical abstraction, the generalized category. It is a partially defined monoid equipped with endomorphism maps defining sources and targets on arbitrary elements, possibly allowing a proximal behavior with respect to composition. We first present a formal introduction to the theory of generalized categories. We describe functors, equivalences, natural transformations, adjoints, and limits in the generalized setting. Next we indicate how the theory of monads extends to generalized categories, and discuss applications to computer science. In particular we discuss implications for the functional programming paradigm, and discuss how to extend categorical semantics to the generalized setting. Next, we present a variant of the calculus of deductive systems developed in the work of Lambek, and give a generalization of the Curry-Howard-Lambek theorem giving an equivalence between the category of typed lambda-calculi and the category of cartesian closed categories and exponential-preserving morphisms that leverages the theory of generalized categories. Next, we develop elementary topos theory in the generalized setting of ideal toposes, by building upon the formalism we have previously developed for the extension of the Curry-Howard-Lambek theorem. In particular, we prove that ideal toposes possess the same Heyting algebra structure and squares of adjoints that ordinary toposes do. Finally, we develop generalized sheaves, and show that such categories form ideal toposes. We extend Lawvere and Tierney's theorem relating j-sheaves and sheaves in the sense of Grothendieck to the generalized setting.

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<p> George Boole's transformation f(x)=x-1/x is an important example of a Lebesgue measure-preserving transformation of the real line. Generalized Boole transformations with finitely many singularities have been widely studied, and they are known to be measure-preserving, ergodic, conservative, pointwise ergodic, exact, and quasi-finite. We extend this work by considering a certain family of generalized Boole transformations that have infinitely many singularities. We assume that the closure of the set of singularities has Lebesgue measure zero. Transformations in this family are also known to be Lebesgue measure-preserving, and we prove that they are ergodic, conservative, pointwise dual ergodic, exact, and quasi-finite. We find the wandering rates and return sequences of these transformations, and under some further assumptions, we obtain a formula for their entropy. We also investigate the c-isomorphism of these transformations.</p>

<p> This dissertation contains a variety of results in quantitative combinatorial geometry, as well as applications to optimization and number theory. </p><p> We use Ehrhart theory, the study of the number of lattice points in polytopes, to prove a Rainbow Ramsey analogue of Richard Rado's 1933 result in Ramsey Theory. There were some conjectures as to what this analogue would be; they are disproven by our result. We also prove a few corollaries. </p><p> A portion of this dissertation contains new quantitative variants of classical convexity theorems: whereas the classical theorems have hypotheses and conclusions requiring certain sets to be nonempty, their quantitative variants require that the sets have a certain "size" (volume, diameter, etc). The three classical theorems we quantize are Carath&eacute;odory's Theorem, Helly's Theorem, and Tverberg's Theorem. The Helly portion contains new non-quantitative results as well. The Tverberg portion proves that any Helly-type theorem implies a corresponding Tverberg-type theorem. </p><p> A <i>maximal 1-lattice polytope</i> is a lattice polytope containing exactly one interior lattice point whose facets each contain at least one interior lattice point. In this dissertation, we bound the size of all maximal 1-lattice pyramids and prisms. These bounds may be used with a brute-force algorithm to quickly enumerate all maximal 1-lattice pyramids and prisms up to equivalence. </p><p> <i>S</i>-optimization, a generalization of continuous, integer, and mixed-integer optimization in which variables take values from a set <i> S</i>, is introduced and studied in the last part of this dissertation. We generalize two traditional optimization algorithms to <i>S</i>-optimization: the chance-constrained algorithm of Calafiore and Campi, and the sampling algorithm of Clarkson. Interestingly, the complexities of the generalized algorithms are dependent upon the same Helly numbers studied elsewhere in this dissertation.</p>