Doctor of Philosophy, The Ohio State University, 2021, Mathematics

We show that for a large class of algebraic model categories, the compact algebraic model categories, the projective model structure on the functor category of any diagram exists and is an algebraic model category. For a large class of these compact algebraic model categories, the projective algebraic model structures themselves will be compact. This generalizes a result for cofibrantly generated algebraic model categories. To prove our result, we fix an issue with and generalize Garner's construction of free algebraic weak factorization systems and more fully develop the theory of algebraic model categories. We then present an easy proof that the h-model structure on k-spaces is a compact algebraic model structure. This gives a method for computing homotopy colimits of any shape of diagram in the h-model structure. We also define quasiaccessible categories, which both generalize locally presentable categories and include the categories of topological spaces and k-spaces. We define quasiaccessible model structures on quasiaccessible categories, prove they have associated algebraic model structures, and show how the Bousfield-Friedlander theorem can be applied to produce a Bousfield localization of a quasiaccessible category that is itself an algebraic model category. We then prove that the h-model structure on topological spaces is a quasiaccessible model structure. We conclude with a characterization of certain accessible model categories inspired by Smith's theorem for combinatorial model categories. The results of this thesis provide general methods for dealing with large classes of noncofibrantly generated model structures on reasonably well-behaved categories.

Committee: Sanjeevi Krishnan Dr (Advisor); John Harper Dr (Committee Member); Crichton Ogle Dr (Committee Member) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 1982, Graduate School

Committee: Not Provided (Other) Subjects: Mathematics

Master of Arts, The Ohio State University, 1940, Graduate School

Committee: Not Provided (Other) Subjects:

Master of Science, The Ohio State University, 1970, Graduate School

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Master of Arts, The Ohio State University, 1929, Graduate School

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Master of Science, The Ohio State University, 1971, Graduate School

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Master of Science, The Ohio State University, 1966, Graduate School

Committee: Not Provided (Other) Subjects:

Master of Science, The Ohio State University, 2007, Graduate School

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Master of Arts, The Ohio State University, 1960, Graduate School

Committee: Not Provided (Other) Subjects:

Doctor of Philosophy, The Ohio State University, 2024, Mathematics

This thesis explores deloopings of monoids and a natural generalization of group extensions for monoids called Schreier extensions. We provide explicit computations of finitely generated presentations for several classes of Schreier extensions. We also examine how Schreier extensions relate to a certain twofold delooping K(M,2) of a commutative monoid M. We prove two classification theorems that identify central Schreier extensions with certain simplicial maps (and isomorphism classes of such extensions with certain homotopy classes) and relate those maps to K(M,2). More specifically, given a commutative monoid M, the set of simplicial maps from the nerve of a monoid H to K(M,2) corresponds to a subset of the central Schreier extensions, and when M is cancellative, there is a natural extension of K(M,2) that classifies all central Schreier extensions. Next, we define a generalization of simplicial (sheaf) cohomology, prove ordinal subdivision invariance for H1, and show the failure of ordinal subdivision invariance for H2. Finally, we tackle the higher categorical structure of deloopings of monoids. We prove K(M,2) is not quite a weak complicial set with a natural thin structure, but when M is cancellative the natural extension of K(M,2) is weak complicial. We then discuss an interpretation of iterated deloopings of a monoid as a rewriting system.

Committee: Sanjeevi Krishnan (Advisor) Subjects: Mathematics

Doctor of Philosophy (PhD), Ohio University, 2024, Mathematics (Arts and Sciences)

A thorough analysis of the semigroup properties of the magma monoid (M(S ), ◁) is undertaken. Properties considered include Green's relations, idempotent elements, and regular elements. Noticing both that the so-called one-value and two-value graph magmas are subsemigroups of (M(S ), ◁) and that both of these families are induced by simple directed graphs gives us opportunities to lift the ◁ operation to operations ◁1 and ◁2 on certain families of graphs. The study of these lifts and the corresponding analyses of their semigroup properties are also pursued. In particular, we focus on the analysis of to what extent the kernel-cokernel and cokernel-kernel decompositions in (M(S ), ◁) lift to these graph semigroups. We also consider the notion of collaborations of binary operations that generalizes the graph-induced constructions above and focuses on the collaborative span of various families of operations.

Committee: Sergio R. Lopez-Permouth (Advisor); Sergio Ulloa (Committee Member); Adam Fuller (Committee Member); Alexei Davydov (Committee Member) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 2023, Chemistry

Algebraic diagrammatic construction (ADC) theory has been found to be a relatively low-cost method with attractive features for the study of photochemical problems. However, because the original formulation of ADC is formulated with a single Slater determinant approximating the ground-state wavefunction, it is insufficient in reproducing accurate photochemical results for systems that exhibit strong correlation effects. In this thesis, I present new approximations in multireference algebraic diagrammatic construction (MR-ADC) theory for simulating electronic excitations of strongly-correlated molecular systems. First, I present the theory and implementation of new strict and extended second-order MR-ADC methods (MR-ADC(2) and MR-ADC(2)-X, respectively) and benchmark these methods for low-lying excited states in several small molecules, including the carbon dimer, ethylene, and butadiene. Next, I present an implementation of MR-ADC methods that incorporates the core-valence separation (CVS) approximation, providing efficient access to simulating core-excited states. The potential of CVS-MR-ADC for systems with multiconfigurational electronic structure is examined by calculating the K-edge XAS spectrum of the ozone molecule and the dissociation curve of core-excited molecular nitrogen. I conclude with an efficient implementation of CVS-MR-ADC, reformulated in terms of spin-free quantities, and present preliminary data on core-excitations in large chemical systems that were not computationally feasible using spin-orbital quantities.

Committee: Alexander Sokolov (Advisor); Bern Kohler (Committee Member); John Herbert (Committee Member) Subjects: Physical Chemistry

Doctor of Philosophy, The Ohio State University, 2023, Mathematics

Working in the context of operadic algebras in modules over the sphere spectrum, we study completions with respect to invariants centered away from the base point—that is, centered at a fixed operadic algebra $Y$. We show that for retractive objects admitting $0$-connected structural maps $Y\to X$, the Bousfield-Kan completion map $X\to X^{\sma}_{\Omega_Y^k\Sigma_Y^k}$ is an equivalence for $1\le k\le\infty$. This generalizes completion results of Blomquist and Ching-Harper when $Y=\ast$. The manner of our attack will require us to pick up and develop Hovey's stabilization machinery and carefully study the homotopy theory and stabilization of categories of retractive objects.

Committee: John Harper (Advisor); Vidhyanath Rao (Committee Member); Niles Johnson (Committee Member) Subjects: Mathematics

Doctor of Philosophy (PhD), Ohio University, 2023, Electrical Engineering & Computer Science (Engineering and Technology)

This dissertation presents Zar: a formally verified compilation pipeline from discrete probabilistic programs in the conditional probabilistic guarded command language (cpGCL) to proved-correct executable samplers in the random bit model. Zar exploits the key idea that discrete probability distributions can be reduced to unbiased coin-flipping schemes. The compiler pipeline first translates cpGCL programs into choice-fix trees, an intermediate representation suitable for reduction of biased probabilistic choices. Choice-fix trees are then translated to coinductive interaction trees for execution within the random bit model. The correctness of the composed translations establishes the sampling equidistribution theorem: compiled samplers are correct with respect to the conditional weakest pre-expectation (cwp) semantics of their cpGCL source programs. Zar is implemented and fully verified in the Coq proof assistant. We extract verified samplers to OCaml and Python and empirically validate them on a number of illustrative examples. We additionally present AlgCo (Algebraic Coinductives), a practical framework for inductive reasoning over coinductive types such as conats, streams, and infinitary trees with finite branching factor, developed during the course of this work to enable convenient formal reasoning for coinductive samplers generated by Zar. The key idea is to exploit the notion of algebraic CPO from domain theory to define continuous operations over coinductive types via primitive recursion on "dense" collections of their elements, enabling a convenient strategy for reasoning about algebraic coinductives by straightforward proofs by induction. We implement the AlgCo library in Coq and demonstrate its utility by verifying a stream variant of the sieve of Eratosthenes, a regular expression library based on coinductive tries, and weakest pre-expectation semantics for potentially nonterminating sampling processes over discrete probability distributions in the r (open full item for complete abstract)

Committee: David Juedes (Advisor); James Stewart (Committee Member); Vladimir Uspenskiy (Committee Member); Jundong Liu (Committee Member); Anindya Banerjee (Committee Member); David Chelberg (Committee Member) Subjects: Computer Science

Master of Arts (MA), Bowling Green State University, 1953, Mathematics

Committee: Frank C. Ogg (Advisor) Subjects: Mathematics

Master of Arts (MA), Bowling Green State University, 1953, Mathematics

Committee: Frank C. Ogg (Advisor) Subjects: Mathematics

Master of Science, The Ohio State University, 2022, Mechanical Engineering

The development of next generation electrode materials provides the opportunity to significantly increase the energy density of lithium-ion batteries. These materials form alloy compounds with lithium and have specific capacities that are much higher than of graphite. Of these materials, silicon, which has a theoretical capacity of ~4200 mAh/g, is the closest to commercialization. However, silicon experiences large volume changes during the lithiation and delithiation processes. This ultimately leads to stress generation within the particle causing fracture, loss of active material, and rapid loss of cell capacity. The differences in behavior for silicon-based anodes compared to traditional graphite anodes highlights the need for the development of physics-based models to capture the effects of volume change and stress generation on the solid-state diffusion process. One such model proposed by Christensen and Newman describes the effects of diffusion, volume change, and stress generation in a spherical electrode particle. This mathematical model takes the form of an index-2 set of differential and algebraic equations which require implicit numerical methods to obtain a solution. Due to the mathematical complexity and high computational time and memory requirements, this model is not suited for controls or estimation-based applications. This thesis presents a mathematical reformulation of the Christensen-Newman equations using index-reduction to obtain a semi-explicit index-1 version of the model. Index reduction allows for the differential and algebraic equations to be decoupled enabling the use of explicit time marching methods. This reformulation will enable the integration of this model into larger cell level frameworks as well as estimation and controls-based applications. The reduced index model is verified against a fully implicit benchmark solution for a graphite anode. A local sensitivity analysis is performed to ascertain the effects of the mechanical (open full item for complete abstract)

Committee: Marcello Canova (Advisor); Jung Hyun Kim (Committee Member) Subjects: Mechanical Engineering

Doctor of Philosophy, The Ohio State University, 2021, Teaching and Learning

Understanding and reasoning with functions permeates mathematics education standards at all grade levels. Research shows students take multiple approaches to building functions. Deep content knowledge of functions may affect how pre-service teachers link student reasoning with mathematical ideas. To extend and connect the research literature on pre-service teachers' understanding of functions, this study elaborates on how pre-service teachers reason with different families of functions. This dissertation study investigates pre-service mathematics teachers' reasoning with linear, quadratic, and exponential functions in a variety of representations. This study was conceptualized within an algebraic and functional thinking framework. The research questions guiding this study are the following: (1) In what ways do pre-service teachers demonstrate their reasoning with functions? (2) How do various representations of function tasks influence pre-service teachers' reasoning with functions? The data analyzed in this study were from six semi-structured task-based clinical interviews conducted with three pre-service mathematics teachers. The results show pre-service mathematics teachers used multiple approaches to create rules for linear, quadratic, and exponential functions. The pre-service teachers demonstrated their reasoning with functions through five major strategies: covariation, correspondence, decomposition, composition, and graphical structure. The results also show that the representation of a function can influence pre-service teachers' approach. Visualization and composition played important roles in growth pattern tasks. Graphical structure could lead to function identification when prototypical features of the function were salient in the graph. Covariation, correspondence, and decomposition were often used with tables. Tables were also frequently created by pre-service teachers while engaging with tasks displaying functions in other representations.

Committee: Patricia Brosnan (Advisor); Michael Battista (Committee Member); Mandy Smith (Committee Member) Subjects: Mathematics Education

Doctor of Philosophy, The Ohio State University, 2021, Mathematics

This thesis contains two different projects on toric geometry. The first project describes the operational K-theory introduced by D. Anderson and S. Payne in [3] for toric varieties, via the introduction of a ring of Grothendieck weights. We prove several properties of Grothendieck weights, which combinatorially characterize them in low dimensions. The second project introduces generalized Rogers-Szego polynomials, which depend on the data of a smooth lattice polytope P. For P an interval these specialize to the polynomials studied in [39]. We prove a q-series identity for these functions involving certain q-hypergeometric functions introduced in [25] and separately in [32]. The identity is a q-deformation of the well-known identity of Brion [9] in Ehrhart theory, and is proved via equivariant K-theory on quasimap spaces. We finish by proving some combinatorial properties of generalized Rogers-Szego polynomials.

Committee: David Anderson (Advisor); Hsian-Hua Tseng (Committee Member); Maria Angelica Cueto (Committee Member) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 2021, Mathematics

We work in the setting of algebras over a reduced operad in spectra. Our first main result establishes a spectral algebra analog of the Bousfield-Kan “fibration lemma” under appropriate conditions. In particular, we show that completion with respect to topological Quillen homology (or TQ-completion, for short) preserves homotopy fibration sequences provided that the base and total O-algebras are connected. Our argument essentially boils down to proving that the natural map from the homotopy fiber to its TQ-completion tower induces a pro-isomorphism on homotopy groups. More generally, we also show that similar results remain true if we replace “homotopy fibration sequence” with “homotopy pullback square.” Our second main result concerns the convergence of the Taylor tower of the identity functor in O-algebras. Specifically, we show that if A is a (−1)-connected O-algebra with 0-connected TQ-homology spectrum TQ(A), then there is a natural weak equivalence between the limit of the Taylor tower of the identity functor evaluated on A and the TQ-completion of A. Since, in this context, the identity functor is only known to be 0-analytic, this result extends knowledge of the Taylor tower of the identity beyond its “radius of convergence.”

Committee: John Harper (Advisor); Niles Johnson (Committee Member); Crichton Ogle (Committee Member) Subjects: Mathematics