Mathematics Commons^™

Logic, Co-Ordination And The Envelope Of Our Beliefs, Rohit J. Parikh

Publications and Research

Each of us has a story which we can think of as a set of beliefs, hopefully consistent. We make our decisions in view of our beliefs which may be probabilistic, in the general case, but simple yes or no as in this paper. Our beliefs are our envelope just as the shell of a tortoise is its envelope.

Decision theory - or single agent game theory tells us when to make the best choice in a game of us against nature. But nature has no desire to further or frustrate our efforts. Nature is mysterious but not malign.

Things …

Conflict Dynamics In Scale-Free Networks With Degree Correlations And Hierarchical Structure, Eduardo Jacobo-Villegas, Bibiana Obregón-Quintana, Lev Guzmán-Vargas, Larry S. Liebovitch

Publications and Research

We present a study of the dynamic interactions between actors located on complex networks with scale-free and hierarchical scale-free topologies with assortative mixing, that is, correlations between the degree distributions of the actors. The actor’s state evolves according to a model that considers its previous state, the inertia to change, and the influence of its neighborhood. We show that the time evolution of the system depends on the percentage of cooperative or competitive
interactions. For scale-free networks, we find that the dispersion between actors is higher when all interactions are either cooperative or competitive, while a balanced presence of interactions …

Asymptotic Classes, Pseudofinite Cardinality And Dimension, Alexander Van Abel

Dissertations, Theses, and Capstone Projects

We explore the consequences of various model-theoretic tameness conditions upon the behavior of pseudofinite cardinality and dimension. We show that for pseudofinite theories which are either Morley Rank 1 or uncountably categorical, pseudofinite cardinality in ultraproducts satisfying such theories is highly well-behaved. On the other hand, it has been shown that pseudofinite dimension is not necessarily well-behaved in all ultraproducts of theories which are simple or supersimple; we extend such an observation by constructing simple and supersimple theories in which pseudofinite dimension is necessarily ill-behaved in all such ultraproducts. Additionally, we have novel results connecting various forms of asymptotic classes …

Generalization Of Bi-Canonical Degrees, Joseph Brennan, Laura Ghezzi, Jooyoun Hong, Wolmer Vasconcelos

Publications and Research

We discuss invariants of Cohen-Macaulay local rings that admit a canonical module ω. Attached to each such ring R, when ω is an ideal, there are integers–the type of R, the reduction number of ω–that provide valuable metrics to express the deviation of R from being a Gorenstein ring. In (Ghezzi et al. in JMS 589:506–528, 2017) and (Ghezzi et al. in JMS 571:55–74, 2021) we enlarged this list with the canonical degree and the bi-canonical degree. In this work we extend the bi-canonical degree to rings where ω is not necessarily an ideal. We also discuss generalizations to rings …

The Local Lifting Problem For Curves With Quaternion Actions, George Mitchell

Dissertations, Theses, and Capstone Projects

The lifting problem asks whether one can lift Galois covers of curves defined over positive characteristic to Galois covers of curves over characteristic zero. The lifting problem has an equivalent local variant, which asks if a Galois extension of complete discrete valuation rings over positive characteristic, with algebraically closed residue field, can be lifted to characteristic zero. In this dissertation, we content ourselves with the study of the local lifting problem when the prime is 2, and the Galois group of the extension is the group of quaternions. In this case, it is known that certain quaternion extensions cannot be …

The Zariski-Riemann Space As A Universal Model For The Birational Geometry Of A Function Field, Giovan Battista Pignatti Morano Di Custoza

Dissertations, Theses, and Capstone Projects

Given a function field $K$ over an algebraically closed field $k$, we propose to use the Zariski-Riemann space $\ZR (K/k)$ of valuation rings as a universal model that governs the birational geometry of the field extension $K/k$. More specifically, we find an exact correspondence between ad-hoc collections of open subsets of $\ZR (K/k)$ ordered by quasi-refinements and the category of normal models of $K/k$ with morphisms the birational maps. We then introduce suitable Grothendieck topologies and we develop a sheaf theory on $\ZR (K/k)$ which induces, locally at once, the sheaf theory of each normal model. Conversely, given a sheaf …

Thickened Surfaces, Checkerboard Surfaces, And Quantum Link Invariants, Joseph W. Boninger

Dissertations, Theses, and Capstone Projects

This dissertation has two parts, each motivated by an open problem related to the Jones polynomial. The first part addresses the Volume Conjecture of Kashaev, Murakami, and Murakami. We define a polynomial invariant, J^T_n, of links in the thickened torus, which we call the nth toroidal colored Jones polynomial, and we show J^T_n satisfies many properties of the original colored Jones polynomial. Most significantly, J^T_n exhibits volume conjecture behavior. We prove a volume conjecture for the 2-by-2 square weave, and provide computational evidence for other links. We also give two equivalent constructions …

Prime Factors: America’S Prioritization Of Literacy Over Numeracy And Its Relationship To Systemic Inequity, Troy Smith

Dissertations, Theses, and Capstone Projects

For much of American history, literacy has been prioritized in K-12 education and society, at large, at the expense of numeracy. This lack of numerical emphasis has established innumeracy as an American cultural norm that has resulted in America not producing a sufficient number of numerate citizens, and ranking poorly on mathematical performance in international comparisons. This paper investigates the decisions and circumstances that led to this under prioritization, along with the public and cultural impact of said actions. Toward this end, literature regarding contemporary and historical influences on American mathematics education (e.g., civic, policy, and parental) was reviewed. The …

Mathematics For Liberal Arts (Ma301), Lecture Notes, Lyubomir I. Boyadzhiev

Open Educational Resources

These lecture notes are designed to provide the students majoring in the liberal arts with an understanding and appreciation of mathematics as a lively, interesting, and surprisingly rich human activity with many fascinating applications. The text, emphasizing strongly intuitive thinking and visualization, is a collection of topics chosen to show the open-minded readers that:

• The connection between the mathematics presented in the course and down-to-earth, concrete real-life problems is transparent and immediate.
• Modern mathematical discoveries do not have to be the exclusive province of professional mathematicians.
• There is an important aesthetic component in mathematics, and just as in art and …

Trapped Surfaces, Topology Of Black Holes, And The Positive Mass Theorem, Lan-Hsuan Huang, Dan A. Lee

Publications and Research

No abstract provided.

Combinatorial Optimization With Photonics-Inspired Clock Models, Mostafa Honari-Latifpour, Matthew S. Mills, Mohammad-Ali Miri

Publications and Research

NP-hard combinatorial optimization problems are in general hard problems that their computational complexity grows faster than polynomial scaling with the size of the problem. Thus, over the years there has been a great interest in developing unconventional methods and algorithms for solving such problems. Here, inspired by the nonlinear optical process of q-photon down-conversion, in which a photon is converted into q degenerate lower energy photons, we introduce a nonlinear dynamical model that builds on coupled single-variable phase oscillators and allows for efficiently approximating the ground state of the classical q-state planar Potts Hamiltonian. This reduces the exhaustive search in …

Amm Problem #12279, Brad Isaacson

Publications and Research

No abstract provided.

Extractable Entanglement From A Euclidean Hourglass, Takanori Anegawa, Norihiro Iizuka, Daniel Kabat

Publications and Research

We previously proposed that entanglement across a planar surface can be obtained from the partition function on a Euclidean hourglass geometry. Here we extend the prescription to spherical entangling surfaces in conformal field theory. We use the prescription to evaluate log terms in the entropy of a conformal field theory in two dimensions, a conformally coupled scalar in four dimensions, and a Maxwell field in four dimensions. For Maxwell we reproduce the extractable entropy obtained by Soni and Trivedi. We take this as evidence that the hourglass prescription provides a Euclidean technique for evaluating extractable entropy in quantum field theory.