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Articles 1 - 30 of 298
Full-Text Articles in Physical Sciences and Mathematics
Intersection Cohomology Of Rank One Local Systems For Arrangement Schubert Varieties, Shuo Lin
Intersection Cohomology Of Rank One Local Systems For Arrangement Schubert Varieties, Shuo Lin
Doctoral Dissertations
In this thesis we study the intersection cohomology of arrangement Schubert varieties with coefficients in a rank one local system on a hyperplane arrangement complement. We prove that the intersection cohomology can be computed recursively in terms of certain polynomials, if a local system has only $\pm 1$ monodromies. In the case where the hyperplane arrangement is generic central or equivalently the associated matroid is uniform and the local system has only $\pm 1$ monodromies, we prove that the intersection cohomology is a combinatorial invariant. In particular when the hyperplane arrangement is associated to the uniform matroid of rank $n-1$ …
When To Hold And When To Fold: Studies On The Topology Of Origami And Linkages, Mary Elizabeth Lee
When To Hold And When To Fold: Studies On The Topology Of Origami And Linkages, Mary Elizabeth Lee
Doctoral Dissertations
Linkages and mechanisms are pervasive in physics and engineering as models for a
variety of structures and systems, from jamming to biomechanics. With the increase
in physical realizations of discrete shape-changing materials, such as metamaterials,
programmable materials, and self-actuating structures, an increased understanding
of mechanisms and how they can be designed is crucial. At a basic level, linkages
or mechanisms can be understood to be rigid bars connected at pivots around which
they can rotate freely. We will have a particular focus on origami-like materials, an
extension to linkages with the added constraint of faces. Self-actuated versions typ-
ically start …
Thermodynamic Laws Of Billiards-Like Microscopic Heat Conduction Models, Ling-Chen Bu
Thermodynamic Laws Of Billiards-Like Microscopic Heat Conduction Models, Ling-Chen Bu
Doctoral Dissertations
In this thesis, we study the mathematical model of one-dimensional microscopic heat conduction of gas particles, applying both both analytical and numerical approaches. The macroscopic law of heat conduction is the renowned Fourier’s law J = −k∇T, where J is the local heat flux density, T(x, t) is the temperature gradient, and k is the thermal conductivity coefficient that characterizes the material’s ability to conduct heat. Though Fouriers’s law has been discovered since 1822, the thorough understanding of its microscopic mechanisms remains challenging [3] (2000). We assume that the microscopic model of heat conduction is a hard ball system. The …
Positive Factorizations Via Planar Mapping Classes And Braids, Richard E. Buckman
Positive Factorizations Via Planar Mapping Classes And Braids, Richard E. Buckman
Doctoral Dissertations
In this thesis we seek to better understand the planar mapping class group in
order to find factorizations of boundary multitwists, primarily to generate and study
symplectic Lefschetz pencils by lifting these factorizations. Traditionally this method
is applied to a disk or sphere with marked points, utilizing factorizations in the stan-
dard and spherical braid groups, whereas in our work we allow for multiple boundary components. Dehn twists along these boundaries give rise to exceptional sections of Lefschetz fibrations over the 2–sphere, equivalently, to Lefschetz pencils with base points. These methods are able to derive an array of known examples …
Semi-Infinite Flags And Zastava Spaces, Andreas Hayash
Semi-Infinite Flags And Zastava Spaces, Andreas Hayash
Doctoral Dissertations
ABSTRACT SEMI-INFINITE FLAGS AND ZASTAVA SPACES SEPTEMBER 2023 ANDREAS HAYASH, B.A., HAMPSHIRE COLLEGE M.S., UNIVERSITY OF MASSACHUSETTS AMHERST Ph.D, UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Ivan Mirković We give an interpretation of Dennis Gaitsgory’s semi-infinite intersection cohomol- ogy sheaf associated to a semisimple simply-connected algebraic group in terms of finite-dimensional geometry. Specifically, we construct machinery to build factoriza- tion spaces over the Ran space from factorization spaces over the configuration space, and show that under this procedure the compactified Zastava space is sent to the support of the semi-infinite intersection cohomology sheaf in the Beilinson-Drinfeld Grassmannian. We also construct …
Facets Of The Union-Closed Polytope, Daniel Gallagher
Facets Of The Union-Closed Polytope, Daniel Gallagher
Doctoral Dissertations
In the haze of the 1970s, a conjecture was born to unknown parentage...the union-closed sets conjecture. Given a family of sets $\FF$, we say that $\FF$ is union-closed if for every two sets $S, T \in \FF$, we have $S \cup T \in \FF$. The union-closed sets conjecture states that there is an element in at least half of the sets of any (non-empty) union-closed family. In 2016, Pulaj, Raymond, and Theis reinterpreted the conjecture as an optimization problem that could be formulated as an integer program. This thesis is concerned with the study of the polytope formed by taking …
Supplementary Code For "Chain Trajectories, Domain Shapes And Terminal Boundaries In Block Copolymers", Benjamin R. Greenvall, Michael S. Dimitriyev, Gregory M. Grason
Supplementary Code For "Chain Trajectories, Domain Shapes And Terminal Boundaries In Block Copolymers", Benjamin R. Greenvall, Michael S. Dimitriyev, Gregory M. Grason
Data and Datasets
Supplementary code used for the publication "Chain trajectories, domain shapes and terminal boundaries in block copolymers" by Benjamin R. Greenvall, Michael S. Dimitriyev, and Gregory M. Grason. Includes code for extracting and analyzing polar order and chain trajectories from self-consistent field calculations of block copolymers.
Applications Of Statistical Physics To Ecology: Ising Models And Two-Cycle Coupled Oscillators, Vahini Reddy Nareddy
Applications Of Statistical Physics To Ecology: Ising Models And Two-Cycle Coupled Oscillators, Vahini Reddy Nareddy
Doctoral Dissertations
Many ecological systems exhibit noisy period-2 oscillations and, when they are spatially extended, they undergo phase transition from synchrony to incoherence in the Ising universality class. Period-2 cycles have two possible phases of oscillations and can be represented as two states in the bistable systems. Understanding the dynamics of ecological systems by representing their oscillations as bistable states and developing dynamical models using the tools from statistical physics to predict their future states is the focus of this thesis. As the ecological oscillators with two-cycle behavior undergo phase transitions in the Ising universality class, many features of synchrony and equilibrium …
A Representation For Cmc 1 Surfaces In H^3 Using Two Pairs Of Spinors, Tetsuya Nakamura
A Representation For Cmc 1 Surfaces In H^3 Using Two Pairs Of Spinors, Tetsuya Nakamura
Doctoral Dissertations
For Bryant's representation $\Phi\colon \widetilde{M} \rightarrow \SL_2(\C)$ of a constant mean curvature (CMC) $1$ surface $f\colon M\rightarrow \Hyp^3$ in the $3$-dimensional hyperbolic space $\Hyp^3$, we will give a formula expressed only by the global $\tbinom{P}{Q}$ and local $\tbinom{p}{q}$ spinors and their derivatives. We will see that this formula is derived from the Klein correspondence, understanding $\Phi$ as a null curve immersion into a $3$-dimensional quadric. We will show that, if $f$ is a CMC $1$ surface with smooth ends modeled on a compact Riemann surface, the linear change of $\tbinom{P}{Q}\oplus \tbinom{p}{-q}$ by some $\Sp(\C^4)$ matrices gives rise to a transformtion …
Combinatorial Algorithms For Graph Discovery And Experimental Design, Raghavendra K. Addanki
Combinatorial Algorithms For Graph Discovery And Experimental Design, Raghavendra K. Addanki
Doctoral Dissertations
In this thesis, we study the design and analysis of algorithms for discovering the structure and properties of an unknown graph, with applications in two different domains: causal inference and sublinear graph algorithms. In both these domains, graph discovery is possible using restricted forms of experiments, and our objective is to design low-cost experiments. First, we describe efficient experimental approaches to the causal discovery problem, which in its simplest form, asks us to identify the causal relations (edges of the unknown graph) between variables (vertices of the unknown graph) of a given system. For causal discovery, we study algorithms …
On The Iwasawa Invariants Of Nonordinary Modular Forms, Rylan J. Gajek-Leonard
On The Iwasawa Invariants Of Nonordinary Modular Forms, Rylan J. Gajek-Leonard
Doctoral Dissertations
We extend known results on the behavior of Iwasawa invariants attached to Mazur-Tate elements for p-nonordinary modular forms of weight k=2 to higher weight modular forms with a_p=0. This is done by using a decomposition of the p-adic L-function due to R. Pollack in order to construct explicit lifts of Mazur-Tate elements to the full Iwasawa algebra. We then study the behavior of Iwasawa invariants upon projection to finite layers, allowing us to express the invariants of Mazur-Tate elements in terms of those coming from plus/minus p-adic L-functions. Our results combine with work of Pollack and Weston to relate the …
General Covariance With Stacks And The Batalin-Vilkovisky Formalism, Filip Dul
General Covariance With Stacks And The Batalin-Vilkovisky Formalism, Filip Dul
Doctoral Dissertations
In this thesis we develop a formulation of general covariance, an essential property for many field theories on curved spacetimes, using the language of stacks and the Batalin-Vilkovisky formalism. We survey the theory of stacks, both from a global and formal perspective, and consider the key example in our work: the moduli stack of metrics modulo diffeomorphism. This is then coupled to the Batalin-Vilkovisky formalism–a formulation of field theory motivated by developments in derived geometry–to describe the associated equivariant observables of a theory and to recover and generalize results regarding current conservation.
An Optimal Transportation Theory For Interacting Paths, Rene Cabrera
An Optimal Transportation Theory For Interacting Paths, Rene Cabrera
Doctoral Dissertations
In this work we study a modification of the Monge-Kantorovich problem taking into account path dependence and interaction effects between particles. We prove existence of solutions under mild conditions on the data, and after imposing stronger conditions, we characterize the minimizers by relating them to an auxiliary Monge-Kantorovich problem of the more standard kind. With this notion of how particles interact and travel along paths, we produce a dual problem. The main novelty here is to incorporate an interaction effect to the optimal path transport problem. This covers for instance, N-body dynamics when the underlying measures are discrete. Lastly, …
Extensions And Bijections Of Skew-Shaped Tableaux And Factorizations Of Singer Cycles, Ga Yee Park
Extensions And Bijections Of Skew-Shaped Tableaux And Factorizations Of Singer Cycles, Ga Yee Park
Doctoral Dissertations
This dissertation is in the field of Algebraic and Enumerative Combinatorics. In the first part of the thesis, we study the generalization of Naruse hook-length formula to mobile posets. Families of posets like Young diagrams of straight shapes and d-complete posets have hook-length product formulas to count linear extensions, whereas families like Young diagrams of skew shapes have determinant or positive sum formulas like the Naruse hook-length formula (NHLF). In 2020, Garver et. al. gave determinant formulas to count linear extensions of a family of posets called mobile posets that refine d-complete posets and border strip skew shapes. We give …
Anticanonical Models Of Smoothings Of Cyclic Quotient Singularities, Arie A. Stern Gonzalez
Anticanonical Models Of Smoothings Of Cyclic Quotient Singularities, Arie A. Stern Gonzalez
Doctoral Dissertations
In this thesis we study anticanonical models of smoothings of cyclic quotient singularities. Given a surface cyclic quotient singularity $Q\in Y$, it is an open problem to determine all smoothings of $Y$ that admit an anticanonical model and to compute it. In \cite{HTU}, Hacking, Tevelev and Urz\'ua studied certain irreducible components of the versal deformation space of $Y$, and within these components, they found one parameter smoothings $\Y \to \A^1$ that admit an anticanonical model and proved that they have canonical singularities. Moreover, they compute explicitly the anticanonical models that have terminal singularities using Mori's division algorithm \cite{M02}. We study …
Moving Polygon Methods For Incompressible Fluid Dynamics, Chris Chartrand
Moving Polygon Methods For Incompressible Fluid Dynamics, Chris Chartrand
Doctoral Dissertations
Hybrid particle-mesh numerical approaches are proposed to solve incompressible fluid flows. The methods discussed in this work consist of a collection of particles each wrapped in their own polygon mesh cell, which then move through the domain as the flow evolves. Variables such as pressure, velocity, mass, and momentum are located either on the mesh or on the particles themselves, depending on the specific algorithm described, and each will be shown to have its own advantages and disadvantages. This work explores what is required to obtain local conservation of mass, momentum, and convergence for the velocity and pressure in a …
Windows In Algebraic Geometry And Applications To Moduli, Sebastian Torres
Windows In Algebraic Geometry And Applications To Moduli, Sebastian Torres
Doctoral Dissertations
We apply the theory of windows, as developed by Halpern-Leistner and by Ballard, Favero and Katzarkov, to study certain moduli spaces and their derived categories. Using quantization and other techniques we show that stable GIT quotients of $(\mathbb{P}^1)^n$ by $PGL_2$ over an algebraically closed field of characteristic zero satisfy a rare property called Bott vanishing, which states that $\Omega^j_Y \otimes L$ has no higher cohomology for every j and every ample line bundle L. Similar techniques are used to reprove the well known fact that toric varieties satisfy Bott vanishing. We also use windows to explore derived categories of moduli …
Equivariant Smoothings Of Cusp Singularities, Angelica Simonetti
Equivariant Smoothings Of Cusp Singularities, Angelica Simonetti
Doctoral Dissertations
Let $p \in X$ be the germ of a cusp singularity and let $\iota$ be an antisymplectic involution, that is an involution free on $X\setminus \{p\}$ and such that there exists a nowhere vanishing holomorphic 2-form $\Omega$ on $X\setminus \{p\}$ for which $\iota^*(\Omega)=-\Omega$. We prove that a sufficient condiition for such a singularity equipped with an antisymplectic involution to be equivariantly smoothable is the existence of a Looijenga (or anticanonical) pair $(Y,D)$ that admits an involution free on $Y\setminus D$ and that reverses the orientation of $D$.
Equivariant Surgeries And Irreducible Embeddings Of Surfaces In Dimension Four, Andrew J. Havens
Equivariant Surgeries And Irreducible Embeddings Of Surfaces In Dimension Four, Andrew J. Havens
Doctoral Dissertations
We construct families of smoothly irreducible embeddings of surfaces in the 4-sphere, corresponding to a range of normal Euler numbers. We also describe a procedure to produce equivariant symplectic sums of real symplectic 4-manifolds. For explicit real symplectic involutions on pairs of symplectic 4-manifolds the conditions for the existence of equivariant symplectic sums can be detected combinatorially. Such sums are sought for potential new constructions of families of irreducible knotted surfaces in fixed four manifolds.
Hyperbolicity And Certain Statistical Properties Of Chaotic Billiard Systems, Kien T. Nguyen
Hyperbolicity And Certain Statistical Properties Of Chaotic Billiard Systems, Kien T. Nguyen
Doctoral Dissertations
In this thesis, we address some questions about certain chaotic dynamical systems. In particular, the objects of our studies are chaotic billiards. A billiard is a dynamical system that describe the motions of point particles in a table where the particles collide elastically with the boundary and with each other. Among the dynamical systems, billiards have a very important position. They are models for many problems in acoustics, optics, classical and quantum mechanics, etc.. Despite of the rather simple description, billiards of different shapes of tables exhibit a wide range of dynamical properties from being complete integrable to chaotic. A …
On The Universal Ordinary Deformation Ring For Ordinary Modular Deformation Problems, Victoria L. Day
On The Universal Ordinary Deformation Ring For Ordinary Modular Deformation Problems, Victoria L. Day
Doctoral Dissertations
Let f be an ordinary newform of weight k at least 3 and level N. Let p be a prime of the number field generated by the Fourier coefficients of f. Assume that f is p-ordinary. We consider the residual mod p Galois representation coming from f and prove that for all but finitely many primes the associated universal ordinary deformation ring is isomorphic to a one variable power series ring.
A Cone Conjecture For Log Calabi-Yau Surfaces, Jennifer Li
A Cone Conjecture For Log Calabi-Yau Surfaces, Jennifer Li
Doctoral Dissertations
In 1993, Morrison conjectured that the automorphism group of a Calabi-Yau 3-fold acts on its nef cone with a rational polyhedral fundamental domain. In this thesis, we prove a version of this conjecture for log Calabi-Yau surfaces. In particular, for a generic log Calabi-Yau surface with singular boundary, the monodromy group acts on the nef effective cone with a rational polyhedral fundamental domain. In addition, the automorphism group of the unique surface with a split mixed Hodge structure in each deformation type acts on the nef effective cone with a rational polyhedral fundamental domain. We also prove that, given a …
Compact Representations Of Uncertainty In Clustering, Craig Stuart Greenberg
Compact Representations Of Uncertainty In Clustering, Craig Stuart Greenberg
Doctoral Dissertations
Flat clustering and hierarchical clustering are two fundamental tasks, often used to discover meaningful structures in data, such as subtypes of cancer, phylogenetic relationships, taxonomies of concepts, and cascades of particle decays in particle physics. When multiple clusterings of the data are possible, it is useful to represent uncertainty in clustering through various probabilistic quantities, such as the distribution over partitions or tree structures, and the marginal probabilities of subpartitions or subtrees. Many compact representations exist for structured prediction problems, enabling the efficient computation of probability distributions, e.g., a trellis structure and corresponding Forward-Backward algorithm for Markov models that model …
Evolutionary Dynamics Of Bertrand Duopoly, Julian Killingback, Timothy Killingback
Evolutionary Dynamics Of Bertrand Duopoly, Julian Killingback, Timothy Killingback
Computer Science Department Faculty Publication Series
Duopolies are one of the simplest economic situations where interactions between firms determine market behavior. The standard model of a price-setting duopoly is the Bertrand model, which has the unique solution that both firms set their prices equal to their costs-a paradoxical result where both firms obtain zero profit, which is generally not observed in real market duopolies. Here we propose a new game theory model for a price-setting duopoly, which we show resolves the paradoxical behavior of the Bertrand model and provides a consistent general model for duopolies.
Generalized Catalan Numbers From Hypergraphs, Paul E. Gunnells
Generalized Catalan Numbers From Hypergraphs, Paul E. Gunnells
Mathematics and Statistics Department Faculty Publication Series
The Catalan numbers Cn ∈ {1,1,2,5,14,42,…} form one of the most venerable sequences in combinatorics. They have many combinatorial interpretations, from counting bracketings of products in non-associative algebra to counting plane trees and noncrossing set partitions. They also arise in the GUE matrix model as the leading coefficient of certain polynomials, a connection closely related to the plane trees and noncrossing set partitions interpretations. In this paper we defi ne a generalization of the Catalan numbers. In fact we defi ne an infinite collection of generalizations Cn(m) , m >= 1, with m = 1 giving the …
Filaments, Fibers, And Foliations In Frustrated Soft Materials, Daria Atkinson
Filaments, Fibers, And Foliations In Frustrated Soft Materials, Daria Atkinson
Doctoral Dissertations
Assemblies of one-dimensional filaments appear in a wide range of physical systems: from biopolymer bundles, columnar liquid crystals, and superconductor vortex arrays; to familiar macroscopic materials, like ropes, cables, and textiles. Interactions between the constituent filaments in such systems are most sensitive to the distance of closest approach between the central curves which approximate their configuration, subjecting these distinct assemblies to common geometric constraints. Dual to strong dependence of inter-filament interactions on changes in the distance of closest approach is their relative insensitivity to reptations, translations along the filament backbone. In this dissertation, after briefly reviewing the mechanics and …
Compactifications Of Cluster Varieties Associated To Root Systems, Feifei Xie
Compactifications Of Cluster Varieties Associated To Root Systems, Feifei Xie
Doctoral Dissertations
In this thesis we identify certain cluster varieties with the complement of a union of closures of hypertori in a toric variety. We prove the existence of a compactification $Z$ of the Fock--Goncharov $\mathcal{X}$-cluster variety for a root system $\Phi$ satisfying some conditions, and study the geometric properties of $Z$. We give a relation of the cluster variety to the toric variety for the fan of Weyl chambers and use a modular interpretation of $X(A_n)$ to give another compactification of the $\mathcal{X}$-cluster variety for the root system $A_n$.
Elliptic Curves And Power Residues, Vy Thi Khanh Nguyen
Elliptic Curves And Power Residues, Vy Thi Khanh Nguyen
Doctoral Dissertations
Let E1 x E2 over Q be a fixed product of two elliptic curves over Q with complex multiplication. I compute the probability that the pth Fourier coefficient of E1 x E2, denoted as ap(E1) + ap(E2), is a square modulo p. The results are 1/4, 7/16, and 1/2 for different imaginary quadratic fields, given a technical independence of the twists. The similar prime densities for cubes and 4th power are 19/54, and 1/4, respectively. I also compute the probabilities without the technical …
Comparison Of Three Dimensional Selfdual Representations By Faltings-Serre Method, Lian Duan
Comparison Of Three Dimensional Selfdual Representations By Faltings-Serre Method, Lian Duan
Doctoral Dissertations
In this thesis, we prove that, a selfdual 3-dimensional Galois representation constructed by van Geemen and Top is isomorphic to a quadratic twist of the symmetric square of the Tate module of an elliptic curve. This is an application of our refinement of the Faltings-Serre method to 3-dimensional Galois representations with ground field not equal to Q. The proof makes use of the Faltings-Serre method, $\ell$-adic Lie algebra, and Burnside groups.
Bruhat-Tits Buildings And A Characteristic P Unimodular Symbol Algorithm, Matthew Bates
Bruhat-Tits Buildings And A Characteristic P Unimodular Symbol Algorithm, Matthew Bates
Doctoral Dissertations
Let k be the finite field with q elements, let F be the field of Laurent series in the variable 1/t with coefficients in k, and let A be the polynomial ring in the variable t with coefficients in k. Let SLn(F) be the ring of nxn-matrices with entries in F, and determinant 1. Given a polynomial g in A, let Gamma(g) subset SLn(F) be the full congruence subgroup of level g. In this thesis we examine the action of Gamma(g) on the Bruhat-Tits building Xn associated to SLn(F) for n equals 2 and n equals 3. Our first main …