Doctor of Philosophy, The Ohio State University, 2013, Computer Science and Engineering

Most of the traditional High End Computing (HEC) applications and current petascale applications are written using the Message Passing Interface (MPI) programming model. Consequently, MPI communication primitives (both point to point and collectives) are extensively used across various scientific and HEC applications. The large-scale HEC systems on which these applications run, by necessity, are designed with multiple layers of switches with different topologies like fat-trees (with different kinds of over-subscription), meshes, torus, etc. Hence, the performance of an MPI library, and in turn the applications, is heavily dependent upon how the MPI library has been designed and optimized to take the system architecture (processor, memory, network interface, and network topology) into account. In addition, parallel jobs are typically submitted to such systems through schedulers (such as PBS and SLURM). Currently, most schedulers do not have the intelligence to allocate compute nodes to MPI tasks based on the underlying topology of the system and the communication requirements of the applications. Thus, the performance and scalability of a parallel application can suffer (even using the best MPI library) if topology-aware scheduling is not employed. Moreover, the placement of logical MPI ranks on a supercomputing system can significantly affect overall application performance. A naive task assignment can result in poor locality of communication. Thus, it is important to design optimal mapping schemes with topology information to improve the overall application performance and scalability. It is also critical for users of High Performance Computing (HPC) installations to clearly understand the impact IB network topology can have on the performance of HPC applications. However, no currently existing tool allows users of such large scale clusters to analyze and to visualize the communication pattern of their MPI based HPC applications in a network topolo (open full item for complete abstract)

Committee: Dhabaleswar Kumar Panda Dr (Advisor); Ponnuswamy Sadayappan Dr (Committee Member); Radu Teodorescu Dr (Committee Member); Karen Tomko Dr (Committee Member) Subjects: Computer Engineering; Computer Science

Doctor of Philosophy (Ph.D.), Bowling Green State University, 2023, Mathematics

An interesting topic of study for a hypercyclic operator T on a topological vector space X has been whether X has an infinite-dimensional, closed subspaces consisting entirely, except for the zero vector, of hypercyclic vectors. These subspaces are called hypercyclic subspaces. The existence of a strictly weakly hypercyclic operator T, which is a weakly hypercyclic operator that is not norm hypercyclic on a Hilbert space H has been shown by Chan and Sanders. However, it is not known whether there exists a strictly weakly hypercyclic subspace of H. We first show that the left multiplication operator LT with the aforementioned strictly weakly hypercyclic operator T is a strictly WOT-hypercyclic operator on the operator algebra B(H). Then we obtain a sufficient condition for an operator T on a Hilbert space H to have a strictly weakly hypercyclic subspace. After that we construct an operator that satisfies these conditions and therefore prove the existence of a strictly weakly hypercyclic subspace.

Committee: Kit Chan Ph.D. (Committee Chair); Christopher Kluse Ph.D. (Other); Mihai Staic Ph.D. (Committee Member); Juan Bes Ph.D. (Committee Member) Subjects: Mathematics

Master of Science (M.S.), University of Dayton, 2021, Aerospace Engineering

With more efficient computational capabilities, the use of topology optimization (TO) is becoming more common for many different types of structural design problems. Rapid prototyping and testing is often used to further validate optimized designs, but depending on a design's complexity, the structural behavior of physical models can vary significantly compared to that of their computational counterparts. For graph-based topologies such differences are caused, in part, by a need to realize finite-thickness structures from the infinitely thin geometries described by graph theory. Other differences are caused by limitations on manufacturing processes such as the need to fabricate large models from smaller components. While additive manufacturing (AM) can be more conducive for fabrication of complex topologies, its limitations are generally less understood than those for traditional subtractive manufacturing processes. Understanding and incorporating limitations on AM into a TO process in the form of added constraints would allow the algorithm to produce not only optimal designs, but also those that are feasible for AM. In this work, two specific AM constraints are characterized for Lindenmayer system (L-system) graph-based topologies of a multi-material, diamond-shaped, morphing airfoil in supersonic flow. One constraint is related to the feasible generation of thick structural members from the infinitely thin beams of graph-based topologies. To characterize the effects of geometric overlap, structural behavior of finite element models made of lower-fidelity beam elements is compared to that of finite element models made of higher-fidelity volume elements. Results indicate that at intersections where 10% or more of a member's length is overlapped, there will be significant variations in stress and effective torsional stiffness when thin members are converted to thick members. The second AM constraint characterized in this work is related to partitioning of large mo (open full item for complete abstract)

Committee: Markus Rumpfkeil (Committee Chair); Richard Beblo (Committee Member); Alexander Pankonien (Committee Member); Raymond Kolonay (Committee Member) Subjects: Aerospace Engineering; Engineering; Mechanical Engineering

Master of Sciences, Case Western Reserve University, 2019, Physics

Space is locally flat and smooth. These geometric properties, however, give us no hints about its topology. Recent searches for universal topology seek to evaluate the Bayesian likelihood of both a set of parameters for an explicit topo- logical model and the Λ Cold Dark Matter (ΛCDM) cosmological parameters [3, 6]. The chief element of such a computation is the correlation matrix of topological eigenspectra. This thesis improves on such searches by providing a new parameteriza- tion of the three-torus; expanding the search space and introducing infinitely many more unique eigenspectra. For each quotient space of the three-torus we: provide a method for finding and explicitly calculate all eigenmode functions. Furthermore, we derive formulae for computing the angular mode-mode corre- lation matrix for full and cut sky CMB temperature data. Finally, we present code for computing these matrices for a given set of cosmological and topological parameters.

Committee: Glenn Starkman (Committee Chair); Craig Copi (Committee Member); Corbin Covault (Committee Member) Subjects: Astrophysics; Physics

Master of Science in Mathematics, Youngstown State University, 2023, Department of Mathematics and Statistics

This thesis investigates a topological quotient space (ℝ/ ~) where ℝ typically carries the usual topology and the equivalence relation ~ stipulates x ~ y ↔ t(x) = t(y) where t is Thomae's function from Thomae's 1875 work, Einleitung in die Theorie der bestimmten Integrale [1]. Various topological properties of (ℝ/ ~) were examined, including compactness, separation axioms, and countability axioms. Under a few popular real-line topologies, the quotient space was often no more separable than T0. The thesis devotes much of its attention toward a conjecture that postulates whether every real number can be the limit of a sequence of simplest form fractions with non-decreasing denominator. Despite many efforts and trials, a proof of the conjecture was not produced; instead, proofs of various propositions involving prime decomposition of integers were produced. Once this closer inspection of prime decomposition of integers was completed, a weaker form of the conjecture was proven and thereafter applied in a topological context. The thesis closes this topic by asserting observations about the space (ℝ/ ~) that would follow if the conjecture could be proven true.

Committee: Jamal Tartir PhD (Advisor); Eric Wingler PhD (Committee Member); Stephen Rodabaugh PhD (Committee Member) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 2018, Mathematics

We study completion with respect to the iterated suspension functor on 풪-algebras, where 풪 is a reduced operad in symmetric spectra. It turns out it is the unit of a derived adjunction comparing 풪-algebras with coalgebras over the associated iterated suspension-loop homotopical comonad via the iterated suspension functor. We prove that this derived adjunction becomes a derived equivalence when restricted to 0-connected 풪-algebras and r connected Σr Ωr -coalgebras. We also consider the dual picture, using iterated loops to build a cocompletion map from algebras over the iterated loop-suspension homotopical monad to 풪-algebras. This is the counit of a derived adjunction, which we prove is a derived equivalence when restricting to r-connected 풪-algebras and 0-connected Ωr Σr-algebras. The final chapter also goes through similar analysis in the context of spaces, for iterated suspension, stabilization, and integral chains, as well as the dual picture for iterated loops.

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

Doctor of Philosophy, The Ohio State University, 2013, Mathematics

Gromov's systolic estimate, first proved in [2], is considered one of the deepest results in systolic geometry. It states that, for an essential Riemannian n-manifold M, the length of the shortest noncontractible loop, or systole, of M, denoted Sysp1 (M) satisfies The formula can be viewed in the abstract of the actual dissertation on Ohio Link. where the constant Cn only depends on n and not on M. We will prove a discrete version of related theorems for triangulated surfaces. The argument involves creating a special Riemannian metric on a triangulated surface whose total volume is close to the number of facets in the triangulation. This metric then allows one to convert Riemannian geodesics to homotopic edge paths of controlled length. The proof of the analogous inequality in the case of a triangulated triangles then follows easily from these facts. We then apply our discrete version to facts about triangulations of orientable surfaces. Given a triangulated, orientable, closed surface with x 2-simplices, we can ask how many 3-simplices are required to “fill” the triangulation: that is, produce a triangulated 3-manifold whose boundary triangulation is the triangulated surface with which we started. Our method produces such a 3-manifold with no more than O(x log2 x) simplices. We will also prove that a discrete version of this inequality implies the Riemannian ii version. The proof of this fact involves creating a triangulation of a Riemannian manifold that is in some sense aware of the geometry of the manifold. We embed M in Rm using the Nash Embedding Theorem [5] and use an argument of Whitney's [6] to produce a triangulation whose simplices are large in volume relative to their edges in the metric of Rm. By working on a small enough scale, one obtains information about the geometry of the simplices embedded in M in the induced path metric. Since M is isometrically embedded, this gives the result. Again, once this obtained, proving t (open full item for complete abstract)

Committee: Jean-Francios Lafont (Advisor); Michael Davis (Committee Member); Matthew Kahle (Committee Member) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 2012, Computer Science and Engineering

Scalar functions are virtually ubiquitous in scientific research. A vast amount of research has been conducted in visualization and exploration of low-dimensional data during the last few decades, but adapting these techniques to high-dimensional, topologically-complex data remains challenging. Traditional metric-preserving dimensionality reduction techniques suffer when the intrinsic dimension of data is high, as the metric cannot generally survive projection into low dimensions. The metric distortion can be arbitrarily large, and preservation of topological structure is not guaranteed, resulting in a misleading view of the data. When preservation of geometry is not possible, topological analysis provides a promising alternative. As an example, simplicial homology characterizes the structure of a topological space (i.e. a simplicial complex) via its intrinsic topological features of various dimensions. Unfortunately, this information can be abstract and difficult to comprehend. The ranks of these homology groups (the Betti numbers) offer a simpler, albeit coarse, interpretation as the number of voids of each dimension. In high dimensions, these approaches suffer from exponential time complexity, which can render them impractical for use with real data. In light of these difficulties, we turn to an alternative type of topological characterization. We investigate the Reeb graph as a visualization and analysis tool for such complex data. The Reeb graph captures the topology of the set of level sets of a scalar function, providing a simple, intuitive, and informative topological representation. We present the first sub-quadratic expected time algorithm for computing the Reeb graph of an arbitrary simplicial complex, opening up the possibility of using the Reeb graph as a tool for understanding high-dimensional data. While the Reeb graph effectively captures some topological structure, it is still somewhat terse. The Morse-Smale complex summarizes a scalar function by b (open full item for complete abstract)

Committee: Yusu Wang PhD (Advisor); Tamal Dey PhD (Committee Member); Rephael Wenger PhD (Committee Member) Subjects: Bioinformatics; Computer Science

Doctor of Philosophy (Ph.D.), Bowling Green State University, 2009, Mathematics

The following dissertation investigates algebraic frames. Formally speaking, a frame is a complete lattice which satisfies a strengthened distributive law where finite infima distribute over arbitrary suprema. In particular, we are interested in focussing on a certain space associated with an algebraic frame: the space of minimal prime elements. In the first half of the dissertation we will investigate different interesting properties of these topological spaces in terms of the algebraic properties of the frame. In one of our main results we state internal conditions of an algebraic frame which will ensure its minimal prime element space is compact.In Chapter 5 we will describe the radical of an algebraic frame. This is a generalization in context to the frame of radical ideals of a commutative ring with identity. We will demonstrate that the radical of an algebraic frame is an algebraic frame. The last part of the dissertation focuses on extensions of algebraic frames. We will generalize the notions of rigid extension, r-extension and r*-extension which are known in the theory of lattice-ordered groups. Our main result will characterize rigid extensions in several ways. We will answer the following question: “Which type of extensions between two algebraic frames will ensure a homeomorphism between their corresponding minimal prime element spaces?” This question had been looked at and answered for lattice-ordered groups by Conrad and Martinez in [4] and later by McGovern in [17]. We will also provide an important example from the theory of rings of continuous functions. In this example, we will construct an extension of algebraic frames which will demonstrate that an r*-extension and an r-extension are two different concepts. In the end we will provide several open questions which may lead to future study.

Committee: Warren Wm. McGovern PhD (Advisor); Rieuwert J. Blok PhD (Committee Member); Kit C. Chan PhD (Committee Member); Ron Lancaster PhD (Committee Member) Subjects: Mathematics

Bachelor of Science (BS), Ohio University, 2013, Mathematics

I present a fairly comprehensive study of simplicial homology theory. Starting with a brief overview of group theory to aid novices in understanding some structure, I present simplices, orientation of simplices, Chain, Cycle, Boundary, and Homology groups. After doing some example computations, I compute the Homology Groups of closed surfaces, as well as demonstrating a method for finding a space whose first homology group is isomorphic to any finitely-generated Abelian group that we desire.

Committee: Todd Eisworth PhD (Advisor) Subjects: Mathematics

Master of Arts, The Ohio State University, 1950, Graduate School

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Master of Science, The Ohio State University, 1962, Graduate School

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Master of Science, The Ohio State University, 1954, Graduate School

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Master of Science, The Ohio State University, 1964, Graduate School

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

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Master of Science, The Ohio State University, 1954, Graduate School

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

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Master of Arts, The Ohio State University, 1951, Graduate School

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

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Master of Science, The Ohio State University, 1964, Graduate School

Committee: Not Provided (Other) Subjects: