Discrete Mathematics and Combinatorics Commons^™

Generating Polynomials Of Exponential Random Graphs, Mohabat Tarkeshian

Electronic Thesis and Dissertation Repository

The theory of random graphs describes the interplay between probability and graph theory: it is the study of the stochastic process by which graphs form and evolve. In 1959, Erdős and Rényi defined the foundational model of random graphs on n vertices, denoted G(n, p) ([ER84]). Subsequently, Frank and Strauss (1986) added a Markov twist to this story by describing a topological structure on random graphs that encodes dependencies between local pairs of vertices ([FS86]). The general model that describes this framework is called the exponential random graph model (ERGM).

In the past, determining when a probability distribution has strong …

Complex-Valued Approach To Kuramoto-Like Oscillators, Jacqueline Bao Ngoc Doan

Electronic Thesis and Dissertation Repository

The Kuramoto Model (KM) is a nonlinear model widely used to model synchrony in a network of oscillators – from the synchrony of the flashing fireflies to the hand clapping in an auditorium. Recently, a modification of the KM (complex-valued KM) was introduced with an analytical solution expressed in terms of a matrix exponential, and consequentially, its eigensystem. Remarkably, the analytical KM and the original KM bear significant similarities, even with phase lag introduced, despite being determined by distinct systems. We found that this approach gives a geometric perspective of synchronization phenomena in terms of complex eigenmodes, which in turn …

An Implementation Of Integrated Information Theory In Resting-State Fmri, Idan E. Nemirovsky

Electronic Thesis and Dissertation Repository

Integrated Information Theory (IIT) is a framework developed to explain consciousness, arguing that conscious systems consist of interacting elements that are integrated through their causal properties. In this study, we present the first application of IIT to functional magnetic resonance imaging (fMRI) data and investigate whether its principal metric, Phi, can meaningfully quantify resting-state cortical activity patterns. Data was acquired from 17 healthy subjects who underwent sedation with propofol, a short acting anesthetic. Using PyPhi, a software package developed for IIT, we thoroughly analyze how Phi varies across different networks and throughout sedation. Our findings indicate that variations in Phi …

Distribution Of The P-Torsion Of Jacobian Groups Of Regular Matroids, Sergio R. Zapata Ceballos

Electronic Thesis and Dissertation Repository

Given a regular matroid $M$ and a map $\lambda\colon E(M)\to \N$, we construct a regular matroid $M_\lambda$. Then we study the distribution of the $p$-torsion of the Jacobian groups of the family $\{M_\lambda\}_{\lambda\in\N^{E(M)}}$. We approach the problem by parameterizing the Jacobian groups of this family with non-trivial $p$-torsion by the $\F_p$-rational points of the configuration hypersurface associated to $M$. In this way, we reduce the problem to counting points over finite fields. As a result, we obtain a closed formula for the proportion of these groups with non-trivial $p$-torsion as well as some estimates. In addition, we show that the …

Of Matroid Polytopes, Chow Rings And Character Polynomials, Ahmed Ashraf

Electronic Thesis and Dissertation Repository

Matroids are combinatorial structures that capture various notions of independence. Recently there has been great interest in studying various matroid invariants. In this thesis, we study two such invariants: Volume of matroid base polytopes and the Tutte polynomial. We gave an approach to computing volume of matroid base polytopes using cyclic flats and apply it to the case of sparse paving matroids. For the Tutte polynomial, we recover (some of) its coefficients as degrees of certain forms in the Chow ring of underlying matroid. Lastly, we study the stability of characters of the symmetric group via character polynomials. We show …

Approximation Algorithms For Problems In Makespan Minimization On Unrelated Parallel Machines, Daniel R. Page

Electronic Thesis and Dissertation Repository

A fundamental problem in scheduling is makespan minimization on unrelated parallel machines (R||C_max). Let there be a set J of jobs and a set M of parallel machines, where every job J_j ∈ J has processing time or length p_i,j ∈ ℚ⁺ on machine M_i ∈ M. The goal in R||C_max is to schedule the jobs non-preemptively on the machines so as to minimize the length of the schedule, the makespan. A ρ-approximation algorithm produces in polynomial time a feasible solution such that its objective value is within a multiplicative factor ρ of …

Topological Recursion And Random Finite Noncommutative Geometries, Shahab Azarfar

Electronic Thesis and Dissertation Repository

In this thesis, we investigate a model for quantum gravity on finite noncommutative spaces using the topological recursion method originated from random matrix theory. More precisely, we consider a particular type of finite noncommutative geometries, in the sense of Connes, called spectral triples of type ${(1,0)} \,$, introduced by Barrett. A random spectral triple of type ${(1,0)}$ has a fixed fermion space, and the moduli space of its Dirac operator ${D=\{ H , \cdot \} \, ,}$ ${H \in {\mathcal{H}_N}}$, encoding all the possible geometries over the fermion space, is the space of Hermitian matrices ${\mathcal{H}_N}$. A distribution of the …

Advances In Graph-Cut Optimization: Multi-Surface Models, Label Costs, And Hierarchical Costs, Andrew T. Delong

Electronic Thesis and Dissertation Repository

Computer vision is full of problems that are elegantly expressed in terms of mathematical optimization, or energy minimization. This is particularly true of "low-level" inference problems such as cleaning up noisy signals, clustering and classifying data, or estimating 3D points from images. Energies let us state each problem as a clear, precise objective function. Minimizing the correct energy would, hypothetically, yield a good solution to the corresponding problem. Unfortunately, even for low-level problems we are confronted by energies that are computationally hard—often NP-hard—to minimize. As a consequence, a rather large portion of computer vision research is dedicated to proposing …

Arrangements Of Submanifolds And The Tangent Bundle Complement, Priyavrat Deshpande

Electronic Thesis and Dissertation Repository

Drawing parallels with the theory of hyperplane arrangements, we develop the theory of arrangements of submanifolds. Given a smooth, finite dimensional, real manifold $X$ we consider a finite collection $\A$ of locally flat codimension $1$ submanifolds that intersect like hyperplanes. To such an arrangement we associate two posets: the \emph{poset of faces} (or strata) $\FA$ and the \emph{poset of intersections} $L(\A)$. We also associate two topological spaces to $\A$. First, the complement of the union of submanifolds in $X$ which we call the \emph{set of chambers} and denote by $\Ch$. Second, the complement of union of tangent bundles of these …

Characteristic Polynomial Of Arrangements And Multiarrangements, Mehdi Garrousian

Electronic Thesis and Dissertation Repository

This thesis is on algebraic and algebraic geometry aspects of complex hyperplane arrangements and multiarrangements. We start by examining the basic properties of the logarithmic modules of all orders such as their freeness, the cdga structure, the local properties and close the first chapter with a multiarrangement version of a theorem due to M. Mustata and H. Schenck.

In the next chapter, we obtain long exact sequences of the logarithmic modules of an arrangement and its deletion-restriction under the tame conditions. We observe how the tame conditions transfer between an arrangement and its deletion-restriction.

In chapter 3, we use some …