PHD, Kent State University, 2021, College of Arts and Sciences / Department of Mathematical Sciences

Linear preserver problems concern the characterization of linear operators on matrix spaces that leave invariant certain functions, subsets, relations, etc. We present several linear preserver problems whose solutions may be considered nonstandard since they differ significantly from classical results. In addition, we also discuss several related linear preserver problems with standard solutions to highlight the phenomena observed.

Committee: Mikhail Chebotar (Advisor); Joanne Caniglia (Committee Member); Feodor Dragan (Committee Member); Mark L. Lewis (Committee Member); Dmitry Ryabogin (Committee Member) Subjects: Mathematics

Master of Arts (MA), Bowling Green State University, 2022, Mathematics/Mathematics (Pure)

In this thesis, we study the free Lie algebra on two generators and a deformation of the free Lie bracket. Our goal is a hands-on derivation of relations which this deformed Lie bracket satisfies. The technical achievement that makes this possible is the identification of a basis for where the relations occur. Using that basis, we verify and extend the calculations found in Schneps (2006). An interesting connection to the Euler polynomials is also discussed.

Committee: Benjamin Ward Ph.D. (Committee Chair); Mihai Staic Ph.D. (Committee Member) Subjects: Mathematics

PHD, Kent State University, 2020, College of Arts and Sciences / Department of Mathematical Sciences

Maps characterized by their action on some equal products have often been studied in Functional Analysis and Linear Algebra. We will describe maps that act like derivations, homomorphisms, and Jordan homomorphisms on certain equal products over a variety of algebras. In particular, we will describe bijective additive maps preserving equal products on division algebras, thus solving a problem posed by Chebotar, Ke, Lee, and Shiao in 2005.

Committee: Mikhail Chebotar (Advisor) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 2018, Chemical Engineering

Similarity is the most pervasive concept in chemoinformatics and it provides direction for many of the problems which arise in that field. Similarity functions are mathematical tools for quantifying the similarity of one molecule with respect to another molecule. In this work, we developed a method for the quantification of the similarity of one molecule with respect to a set of molecules. This method requires a similarity function which is symmetric and positive definite. If the similarity function meets two additional mild requirements, namely if it is bound between zero and unity and is unity when evaluated on two identical molecules, then we say that the similarity function is extendable. In this case, the similarity of a molecule with respect to a set containing one molecule reduces to the original similarity function evaluated on those two molecules. We additionally stated and proved several properties of the extension of similarity functions. We then applied the extension of similarity functions to two problems in chemoinformatics. First, we used the extension of similarity functions as the basis for machine learning models for the prediction of various molecular endpoints. These machine learning models were compared to the kNN machine learning model. For each endpoint predicted, the model based on the extension of similarity functions was shown either comparable to or to be exceeding the kNN model. Second, we used the extension of similarity functions as the basis for defining the domain of applicability of a machine learning model. We applied this definition to a kNN model and showed that using the extension of similarity functions can be used to order predictions for the rational selection of molecules for further testing. We showed how doing so can increase the overall usefulness of a machine learning model. Finally, we stated several mathematical questions related to the extension of similarity functions which, if answered, (open full item for complete abstract)

Committee: James Rathman Dr (Advisor); Isamu Kusaka Dr (Committee Member); Aravind Asthagiri Dr (Committee Member) Subjects: Information Science; Mathematics; Molecular Chemistry; Pharmacy Sciences; Statistics; Theoretical Mathematics

Master of Arts (MA), Bowling Green State University, 2017, Mathematics

In this paper, we find all the quasiconformal maps on a particular non-rigid 2-step Carnot group. In particular, all quasiconformal maps on this Carnot group preserve the vertical direction. Given that a Carnot group is a Lie algebra with a group structure, we employ concepts from linear algebra and abstract algebra to gain information about the group. Utilizing the theory of Pansu differentiability along with the biLipschitz nature of quasisymmetric maps, we use an analytical approach to help determine the form of any quasiconformal map on the Carnot group. The main result has consequences for the rigidity of quasiisometries of negatively curved solvable Lie groups.

Committee: Xiangdong Xie (Advisor); Kit Chan (Committee Member) Subjects: Mathematics

PHD, Kent State University, 2017, College of Education, Health and Human Services / School of Teaching, Learning and Curriculum Studies

The purpose of this qualitative practitioner research study was to describe middle school algebra students' experiences of learning linear functions through kinesthetic movement. Participants were comprised of 8th grade algebra students. Practitioner research was used because I wanted to improve my teaching so students will have more success in learning mathematics. Since this research focused on the mental constructions made by students as they attempted to make sense of mathematics kinesthetically, it is grounded in the philosophical tenants of constructivism (Piaget & Vygotsky), math representation theory, and kinesthetic movement. This study utilized multiple data sources which included pre-and post-teacher-made assessments with state standardized problems, audio and video transcriptions of class, small group activities, individual discussions, learning style inventory, and attitude survey on kinesthetic learning. Data was collected and analyzed through triangulation. The results of this study have important curricular implications for math educators to understand how students can learn through kinesthetic movements. Educators can support their students learning by incorporating movement into their classrooms. Recommendations for future research based on unanticipated findings are suggested.

Committee: Caniglia Joanne Dr. (Advisor); Turner Steven Dr. (Committee Member); Martens Marianne Dr. (Committee Member); Gershon Walter Dr. (Committee Member) Subjects: Education; Mathematics Education; Middle School Education

BA, Oberlin College, 2015, Mathematics

Three formulations of the rank of a matrix that are equivalent in classical linear algebra give rise to distinct notions of rank over the tropical semiring. This paper explores these three concepts of tropical rank and their relationships with one another, working up to a proof of the inequality that relates the three.

Committee: Susan Colley (Advisor) Subjects: Mathematics

Master of Science, The Ohio State University, 2015, Computer Science and Engineering

Many scientific simulation codes represent physical systems by storing data at a set of discrete points. For example, a weather modeling software application may store data representing temperature, wind velocity, air pressure, humidity, etc. at a set of points in the atmosphere over a portion of the Earth's surface. Similarly, Computational Fluid Dynamics (CFD) software stores data representing fluid velocity, pressure, etc. at discrete points in the domain of the particular problem being solved. In many simulations, this set of discrete points is a structured grid; i.e., a finite n-dimensional regular lattice. One of the most important steps in many scientific simulations is the solution of a system of linear equations, where the unknowns of the system correspond to data elements at each of the discrete points used to model the system. If the simulation is based on a structured grid, this linear system will often have a special structure itself, and this structure may lead to more efficient techniques for solving the system than can be used for general sparse linear systems. Scientific simulations most often use an iterative technique such as the Conjugate Gradient Method or GMRES for solving linear systems.. It is well known that these iterative techniques converge much more quickly if a preconditioner is used. The ILU, or Incomplete LU Factorization, preconditioner is a good choice but it is not parallelizable is a straightforward way. This thesis examines techniques for parallelizing the application of the ILU preconditioner to linear systems arising from scientific simulations on structured grids. Various techniques are tested and timing results are recorded for different types and sizes of linear systems on structured grids.

Committee: P Sadayappan (Advisor); Atanas Rountev (Committee Member) Subjects: Computer Science

Master of Science, University of Akron, 2015, Mathematics

We consider the additive abelian group of all 3-by-3 matrices where the entries are from the ring of integers modulo 9. Ideally, we would like to construct and identify all those subgroups that are doubly-invariant under two endomorphisms of this group. However, we shall see that is a computationally large problem. Hence, in this thesis we consider a particular subgroup of the group and find all those doubly-invariant subgroups under two endomorphisms. In order to construct and identify the doubly invariant subgroups, we implement a methodical approach based on concepts from linear algebra and group theory. This problem is associated with classification problems for certain subgroups of wreath product finite groups of prime-power order and classification of their orbits [1].

Committee: Jeffrey M. Riedl Dr. (Advisor); James P. Cossey Dr. (Committee Member); Hung Nguyen Dr. (Committee Member) Subjects: Mathematics

Master of Mathematical Sciences, The Ohio State University, 2014, Mathematics

This paper is an educational approach to group characters through examples which introduces the beginner algebraist to representations and characters of finite groups. My hope is that this exploration might help the advanced undergraduate student discover some of the foundational tools of Character Theory. The prerequisite material for this paper includes some elementary Abstract and Linear Algebra. The basic groups used in the examples are intended to excited a student into exploration of groups they understand from their undergraduate studies. Throughout the section of examples there are exercises used to check understanding and give the reader opportunity to explore further. After taking a course in Abstract Algebra one might find that groups are not concrete objects. Groups model actions, rotations, reflections, movements, and permutations. Group representations turn these abstract sets of objects into sets of n X n matrices with real or complex entries, which can be easily handled by a computer for any number of calculations.

Committee: James Cogdell Dr. (Advisor); Warren Sinnott Dr. (Committee Member) Subjects: Mathematics

Master of Science, The Ohio State University, 2013, Aero/Astro Engineering

This thesis addresses the importance and issues of the robust control design of linear time-invariant (LTI) systems with real-time parameter uncertainties. It is known that most of the existing robust control techniques are fairly conservative when dealing with real time parameter uncertainty. Also, majority of these existing techniques use control gains that are essentially functions of the perturbation information. The robust control design algorithm proposed in this thesis differs from these traditional techniques by focusing on the control design in achieving a specific structure of the closed loop system matrix that guarantees a maximum stability robustness index as possible without the using any of the perturbation information. The determination of this specific desired structure of closed loop system matrix forms the focal point of this algoithm and is inspired by already existing principles in the field of ecology. Using this ecological backdrop, the desired closed loop matrix is determined to contain self regulated species with predator-prey interactions among these species. In matrix nomenclature, such a set of matrices are labelled as Target Pseudo-Symmetric (TPS) matrices and hence form the class of desirable closed-loop system matrices. Based on these TPS matrices, which capture the maximum robustness index for any LTI system, a robust control design is carried out such that the final closed loop system possesses a robustness index as close to this maximum as possible. The robust control design algorithm presented is based on minimizing the norm of an implicit error and is supported with several illustrative examples. This eco-inspired robust control algorithm exemplifies the strong correlation that exists between natural systems and engineering systems. Hence, the main goal of this thesis is to aid in the revival of research in the field of robust control using insights from ecological principles.

Committee: Rama Yedavalli Dr. (Advisor); Chia-Hsiang Menq Dr. (Committee Member) Subjects: Aerospace Engineering; Applied Mathematics; Automotive Engineering; Ecology; Engineering; Mathematics Education; Mechanical Engineering

PhD, University of Cincinnati, 2011, Arts and Sciences: Mathematical Sciences

Grobner bases are the single most important tool in applicable algebraic geometry. They are used to compute standard representatives in the residue classes of a polynomial ring modulo an ideal and they can be used as a step towards solving a system of polynomial equations. Applications in science and technology are abundant, particularly in cryptography and coding theory. Computation of Grobner bases is challenging. It has computational complexity double exponential in the worst-case and exponential in average. Although this makes Grobner bases intractable in many cases, a great deal of effort has been devoted to improve algorithms to compute faster larger Grobner bases. The concept of mutant polynomials, introduced by Ding in 2006, quantifies the deviation from the average case, leading to faster algorithms that profit on this degeneration. In this dissertation we introduce three algorithms aiming at improving Grobner bases computation that are inspired by mutant polynomials. The mutantF4 algorithm modifies Faugere's F4 by exploiting the presence of mutant polynomials. The MXL3 algorithm introduces a termination condition based on the absence of mutant polynomials. And the MGB algorithm combines MXL3 with the idea of preempted reduction to avoid storing large sets of polynomials. Each new idea achieved gains in speed and/or memory usage. The MGB algorithm is particularly successful, which was demonstrated by being the first algorithm to be able to compute a Grobner basis for a random system of 32 quadratic polynomials in 32 variables over GF(2). We also propose LASyz, a method to avoid redundant computation in Grobner bases computation, that is compatible with mutant algorithms. LASyz is a simple yet effective method to significantly reduce unproductive reductions to zero. It uses linear algebra to maintain a set of generators for the module of syzygies. LASyz is compatible with mutant Grobner basis algorithms thanks to its simplicity and the use of linear algebra (open full item for complete abstract)

Committee: Jintai Ding PhD (Committee Chair); Chris Christensen PhD (Committee Member); Dieter Schmidt PhD (Committee Member); Ning Zhong PhD (Committee Member) Subjects: Mathematics