Mathematics Commons^™

Efficient Conformal Binary Classification Under Nearest Neighbor, Maxwell Lovig

Undergraduate Mathematics Day: Past Content

There are many types of statistical inferences that can be used today: Frequentist, Bayesian, Fiducial, and others. However, Vovk introduced a new version of statistical inference known as Conformal Predictions. Conformal Predictions were designed to reduce the assumptions of standard prediction methods. Instead of assuming all observations are drawn independently and identically distributed, we instead assume exchangeability. Meaning, all N! possible orderings of our N observations are equally likely. This is more applicable to fields such as machine learning where assumptions may not be easily satisfied. In the case of binary classification, Vovk provided the nearest neighbors (NN) measure which …

The Parker Problem In Hall Magnetohydrodynamics Analytical And Numerical Solutions, Chad Malott

Electronic Theses and Dissertations, 2020-2023

In this thesis we follow on the mathematical aspects of the previous work of Shivamoggi (2009) on the Parker problem in Hall magnetohydrodynamics (MHD). We will present an analysis involving detailed analytical and numerical solutions to the Parker problem in Hall MHD. We give an analytical formulation for the Parker problem in Hall MHD, involving an initial value problem (IVP) associated with a first order Riccati equation (RE). We present Mathematica software exact solutions directly with special functions and more straightforward solutions that use the change of variables and power series methods without special functions. We give an asymptotic formulation …

Differentially Private Fractional Frequency Moments Estimation With Polylogarithmic Space, Lun Wang, Iosif Pinelis, Dawn Song

Michigan Tech Publications, Part 1

We prove that Fp sketch, a well-celebrated streaming algorithm for frequency moments estimation, is differentially private as is when p ∈ (0, 1]. Fp sketch uses only polylogarithmic space, exponentially better than existing DP baselines and only worse than the optimal non-private baseline by a logarithmic factor. The evaluation shows that Fp sketch can achieve reasonable accuracy with differential privacy guarantee. The evaluation code is included in the supplementary material.

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.

Spin(7) Orientifolds And 2d N = (0, 1) Triality, Sebastían Franco, Alessandro Mininno, Ángel M. Uranga, Xingyang Yu

Publications and Research

We present a new, geometric perspective on the recently proposed triality of 2d N = (0, 1) gauge theories, based on its engineering in terms of D1-branes probing Spin(7) orientifolds. In this context, triality translates into the fact that multiple gauge theories correspond to the same underlying orientifold. We show how Spin(7) orientifolds based on a particular involution, which we call the universal involution, give rise to precisely the original version of N = (0, 1) triality. Interestingly, our work also shows that the space of possibilities is significantly richer. Indeed, general Spin(7) orientifolds extend triality to theories that can …

Bft2: A General Class Of 2d N = (0, 2) Theories, 3-Manifolds And Toric Geometry, Sebastían Franco, Xingyang Yu

Publications and Research

We introduce and initiate the study of a general class of 2d N = (0, 2) quiver gauge theories, defined in terms of certain 2-dimensional CW complexes on oriented 3- manifolds. We refer to this class of theories as BFT₂’s. They are natural generalizations of Brane Brick Models, which capture the gauge theories on D1-branes probing toric Calabi-Yau 4-folds. The dynamics and triality of the gauge theories translate into simple transformations of the underlying CW complexes. We introduce various combinatorial tools for analyzing these theories and investigate their connections to toric Calabi-Yau manifolds, which arise as their …

Fano 3-Folds, Reflexive Polytopes And Brane Brick Models, Sebastían Franco, Rak-Kyeong Seong

Publications and Research

Reflexive polytopes in n dimensions have attracted much attention both in mathematics and theoretical physics due to their connection to Fano n-folds and mirror symmetry. This work focuses on the 18 regular reflexive polytopes corresponding to smooth Fano 3-folds. For the first time, we show that all 18 regular reflexive polytopes have corresponding 2d (0, 2) gauge theories realized by brane brick models. These 2d gauge theories can be considered as the worldvolume theories of D1-branes probing the toric Calabi-Yau 4-singularities whose toric diagrams are given by the associated regular reflexive polytopes. The generators of the mesonic moduli …

Interpolating The Riemann Zeta Function In The P-Adics, Rebecca Mamlet

Scripps Senior Theses

In this thesis, we develop the Kubota-Leopoldt Riemann zeta function in the p-adic integers. We follow Neil Koblitz's interpolation of Riemann zeta, using Bernoulli measures and p-adic integrals. The underlying goal is to better understand p-adic expansions and computations. We finish by connecting the Riemann zeta function to L-functions and their p-adic interpolations.

Efficient Numerical Optimization For Parallel Dynamic Optimal Power Flow Simulation Using Network Geometry, Rylee Sundermann

Electronic Theses and Dissertations

In this work, we present a parallel method for accelerating the multi-period dynamic optimal power flow (DOPF). Our approach involves a distributed-memory parallelization of DOPF time-steps, use of a newly developed parallel primal-dual interior point method, and an iterative Krylov subspace linear solver with a block-Jacobi preconditioning scheme. The parallel primal-dual interior point method has been implemented and distributed in the open-source PETSc library and is currently available. We present the formulation of the DOPF problem, the developed primal dual interior point method solver, the parallel implementation, and results on various multi-core machines. We demonstrate the effectiveness our proposed block-Jacobi …

Free Energies For Nonlinear Materials With Memory, John Murrough Golden

Articles

An exploration of representations of free energies and associated rates of dissipation for a broad class of nonlinear viscoelastic materials is presented in this work. Also included are expressions for the stress functions and work functions derivable from such free energies. For simplicity, only the scalar case is considered. Certain standard formulae are generalized to include higher power terms.

It is shown that the correct initial procedure in this context is to specify the rate of dissipation as a positive semi-definite functional and then to determine the free energy from this, rather than the other way around, which would be …

Mathematical Analysis Of The Van Der Waals Equation, Emil Prodanov

Articles

The parametric cubic van der Waals polynomial $p V^3 - (R T + b p) V^2 + a V - a b$ is analysed mathematically and some new generic features (theoretically, for any substance) are revealed: the temperature range for applicability of the van der Waals equation, $T > a/(4Rb)$, and the isolation intervals, at any given temperature between $a/(4Rb)$ and the critical temperature $8a/(27Rb)$, of the three volumes on the isobar--isotherm: $3b/2 < V_A \le 3b$, $ 2b < V_B < 4b/(3 - \sqrt{5})$, and $3b < V_C < b + RT/p$. The unstable states of the van der Waals model have also been generically localized: they lie in an interval within the isolation interval of $V_B$. In the case of unique intersection point of an isotherm with an isobar, the isolation interval of this unique volume is also determined. A discussion on finding the volumes $V_{A, B, C}$, on the premise of Maxwell's hypothesis, is also presented.

Global In Time Self-Interacting Dirac Fields In The De Sitter Space, Karen Yagdjian

School of Mathematical & Statistical Sciences Faculty Publications

In this paper the semilinear equation of the spin-12 fields in the de Sitter space is investigated. We prove the existence of the global in time small data solution in the expanding de Sitter universe. Then, under the Lochak–Majorana condition, we prove the existence of the global in time solution with large data. The sufficient conditions for the solutions to blow up in finite time are given for large data in the expanding and contracting de Sitter spacetimes. The influence of the Hubble constant on the lifespan is estimated.

P-Adic Analysis: A Quick Introduction, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

These notes aim to provide a fast introduction to p-adic analysis assuming basic knowledge in algebra and analysis. The text corresponds to the lecture notes for a Mini-Course in the L. Santal´o Research Summer School 2019. Palacio de la Magdalena, Santander, Spain, June 24-28, 2019.

Decomposing Manifolds In Low-Dimensions: From Heegaard Splittings To Trisections, Suixin "Cindy" Zhang

Honors Theses

The decomposition of a topological space into smaller and simpler pieces is useful for understanding the space. In 1898, Poul Heegaard introduced the concept of a Heegaard splitting, which is a bisection of a 3-manifold. Heegaard diagrams, which describe Heegaard splittings combinatorially, have been recognized as a powerful tool for classifying 3-manifolds and producing important invariants of 3-manifolds. Handle decomposition, invented by Stephen Smale in 1962, describes how an n-manifold can be constructed by successively adding handles. In 2012, Gay and Kirby introduced trisections of 4-manifold, which are a four-dimensional analogues of Heegaard splittings in dimension three. Trisection diagrams give …

On Rotationally Driven Nonlinear Inclined Polymeric Jet With Gravity Effect, Daniel N. Riahi, Saulo Orizaga

School of Mathematical & Statistical Sciences Faculty Publications

We consider rotationally driven nonlinear polymeric fiber jet, whose centerline is in an inclined plane, in the presence of gravity force. An empirical viscosity model is used for the polymeric fluid flow to investigate properties of the rotating inclined polymeric fiber jet. The aim of the study is to understand properties of such inclined fiber jet shape, which can be due to orientation change of an orifice in a rotating spinneret that generates such jet. Perturbation, asymptotic, scaling and numerical techniques are used to determine the nonlinear steady solutions for the jet quantities for different values of the parameters due …

Equitable Coloring Of Complete Tripartitle Graphs, Maxwell Vlam, Bailey Orehosky, Dominic Ditizio

Capstone Showcase

In this paper, we prove the Equitable Coloring Conjecture for variations of complete tripartite graphs with graphs K_n,n,n, K_n,n,2n, K_n,n,n+2, and K_n,n+2,n+4.

Rankings Of Mma Fighters, Michael Schaefer

All Graduate Theses, Dissertations, and Other Capstone Projects

Ranking is an essential process that allows sporting authorities to determine the relative performance of athletes. While ranking is straightforward in some sports, it is more complicated in MMA (mixed martial arts), where competition is often fragmented. This paper describes the mathematics behind four existing ranking algorithms: Elo’s System, Massey’s Method, Colley’s Method, and Google’s PageRank, and shows how to adapt them to rank MMA fighters in the UFC (Ultimate Fighting Championship). We also provide a performance analysis for each ranking method.

Math Department 2021-2022 Newsletter, Mathematics Department

Mathematics Newsletter

Welcome to our annual Mathematics Department Newsletter. As we are slowly coming out of the pandemic, we were able to resume many of our normal events.

Evolutionary Kuramoto Dynamics, Elizabeth A. Tripp, Feng Fu, Scott D. Pauls

Mathematics Faculty Publications

Biological systems have a variety of time-keeping mechanisms ranging from molecular clocks within cells to a complex interconnected unit across an entire organism. The suprachiasmatic nucleus, comprising interconnected oscillatory neurons, serves as a master-clock in mammals. The ubiquity of such systems indicates an evolutionary benefit that outweighs the cost of establishing and maintaining them, but little is known about the process of evolutionary development. To begin to address this shortfall, we introduce and analyse a new evolutionary game theoretic framework modelling the behaviour and evolution of systems of coupled oscillators. Each oscillator is characterized by a pair of dynamic behavioural …

A Distinguished Subgroup Of Compact Abelian Groups, Dikran Dikranjan, Wayne Lewis, Peter Loth, Adolf Mader

Mathematics Faculty Publications

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