Mathematics Commons^™

Some Proofs Regarding Minami Estimates And Local Eigenvalue Statistics For Some Random Schrödinger Operator Models, Samuel Herschenfeld

Theses and Dissertations--Mathematics

We provide three proofs on different, but related models in the field of random Schrödinger operators. All three results are motivated by the desire to extend results and techniques on eigenvalue statistics or Minami estimates (an essential ingredient Poisson eigenvalue statistics).

Chapters 2 and 4 are explorations of the only two known techniques for proving Minami estimates for continuum Minami estimates. In Chapter 2, we provide an alternative and simplified proof of Klopp that holds in d = 1. Chapter 4 is an application of the techniques of Dietlein and Elgart to prove a Minami estimate for finite rank lattice …

On Properties Of Positive Semigroups In Lattices And Totally Real Number Fields, Siki Wang

CMC Senior Theses

In this thesis, we give estimates on the successive minima of positive semigroups in lattices and ideals in totally real number fields. In Chapter 1 we give a brief overview of the thesis, while Chapters 2 – 4 provide expository material on some fundamental theorems about lattices, number fields and height functions, hence setting the necessary background for the original results presented in Chapter 5. The results in Chapter 5 can be summarized as follows. For a full-rank lattice L ⊂ Rd, we are concerned with the semigroup L+ ⊆ L, which denotes the set of all vectors with nonnegative …

Computation Of Effective Properties Of Smart Composite Materials With Generalized Periodicity Using The Two-Scales Asymptotic Homogenization Method, David Guinovart Sanjuan

Electronic Theses and Dissertations, 2020-2023

In this work, a general mathematical models for flexoelectric heterogeneous equilibrium boundary value problems are considered. A methodology to find the local problems and the effective properties of flexoelectric composites with generalized periodicity is presented, using a two-scales asymptotic homogenization method. The model of the homogenized boundary values problem is presented. A procedure to solve the local problems of stratified multilayered composites with wavy geometry with perfect contact at the interface is proposed. Further, a study of a multilayered piezoelectric composite with imperfect contact at the interface and the influence of the flexoelectric constituents in the behavior of heterogeneous structures …

Dimers, Orientifolds And Anomalies, Riccardo Argurio, Matteo Bertolini, Sebastían Franco, Eduardo Garcia-Valdecasas, Shani Meynet, Antoine Pasternak, Valdo Tatitscheff

Publications and Research

We study 4d N = 1 gauge theories engineered via D-branes at orientifolds of toric singularities, where gauge anomalies are cancelled without the introduction of non-compact flavor branes. Using dimer model techniques, we derive geometric criteria for establishing whether a given singularity can admit anomaly-free D-brane configurations purely based on its toric data and the type of orientifold projection. Our results therefore extend the dictionary between geometric properties of singularities and physical properties of the corresponding gauge theories.

Light-Ray Moments As Endpoint Contributions To Modular Hamiltonians, Daniel Kabat, Gilad Lifschytz, Phuc Nguyen, Debajyoti Sarkar

Publications and Research

We consider excited states in a CFT, obtained by applying a weak unitary perturbation to the vacuum. The perturbation is generated by the integral of a local operator J⁽ⁿ⁾ of modular weight n over a spacelike surface passing through x = 0. For |n| ≥ 2 the modular Hamiltonian associated with a division of space at x = 0 picks up an endpoint contribution, sensitive to the details of the perturbation (including the shape of the spacelike surface) at x = 0. The endpoint contribution is a sum of light-ray moments of the perturbing operator …

Data-Driven Reconstruction And Encoding Of Sparse Stimuli Across Convergent Sensory Layers From Downstream Neuronal Network Dynamics, Victor J. Barranca, Yolanda Hu , '22, Zoe Porterfield , '22, Samuel Rothstein , '21, Alex Xuan , '22

Mathematics & Statistics Faculty Works

Neuronal networks vary dramatically in size, connectivity structure, and functionality across downstream layers of the brain. This raises the question of whether information is lost as it is re-encoded along compressive and expansive pathways. In this work, we develop a potential data-driven mechanism for the preservation of information in the activity of neuronal networks across downstream layers, which uses the widespread linearity of individual neuronal responses to sufficiently strong ramped artificial inputs to fit a linear input-output mapping across the network. We analyze the dynamics of several families of two-layer neuronal network models, where the input components far outnumber the …

Commuting Perturbations Of Operator Equations, Xue Xu, Jiu Ding

Faculty Publications

Let X be a Banach space and let T: X →X be a bounded linear operator with closed range. We study a class of commuting perturbations of the corresponding operator equation, using the concept of the spectral radius of a bounded linear operator. Our results extend the classic perturbation theorem for invertible operators and its generalization for arbitrary operators under the commutability assumption.

Abelian P-Groups With Minimal Full Inertia, Brendan Goldsmith, Luigi Salce

Articles

The class of abelian p-groups with minimal full inertia, that is, satisfying the property that fully inert subgroups are commensurable with fully invariant subgroups is investigated, as well as the class of groups not satisfying this property; it is known that both the class of direct sums of cyclic groups and that of torsion-complete groups are of the first type. It is proved that groups with “small" endomorphism ring do not satisfy the property and concrete examples of them are provided via Corner’s realization theorems. Closure properties with respect to direct sums of the two classes of groups are also …

On The Bassian Property For Abelian Groups, Andrey R. Chekhlov, Peter V. Danchev, Brendan Goldsmith

Articles

We define an object (group, ring, module, algebra, etc.) to be Bassian if it is not possible to embed it in a proper homomorphic image of itself. Here we study this concept for Abelian groups and achieve a complete characterization of all such groups in terms of their associated torsion-free and p-primary ranks.

Five-Wave Resonances In Deep Water Gravity Waves: Integrability, Numerical Simulations And Experiments, Dan Lucas, Marc Perlin, Dian-Yong Liu, Shane Walsh, Rossen Ivanov, Miguel D. Bustamante

Articles

In this work we consider the problem of finding the simplest arrangement of resonant deep water gravity waves in one-dimensional propagation, from three perspectives: Theoretical, numerical and experimental. Theoretically this requires using a normal-form Hamiltonian that focuses on 5-wave resonances. The simplest arrangement is based on a triad of wave vectors K₁ + K₂ = K₃ (satisfying specific ratios) along with their negatives, corresponding to a scenario of encountering wave packets, amenable to experiments and numerical simulations. The normal-form equations for these encountering waves in resonance are shown to be non-integrable, but they admit an integrable reduction …

A Method For Locating The Real Roots Of The Symbolic Quintic Equation Using Quadratic Equations, Emil Prodanov

Articles

A method is proposed with which the locations of the roots of the monic symbolic quintic polynomial $x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0$ can be determined using the roots of two {\it resolvent} quadratic polynomials: $q_1(x) = x^2 + a_4 x + a_3$ and $q_2(x) = a_2 x^2 + a_1 x + a_0$, whose coefficients are exactly those of the quintic polynomial. The different cases depend on the coefficients of $q_1(x)$ and $q_2(x)$ and on some specific relationships between them. The method is illustrated with the full analysis of one of …

Bayesian Dynamic Network Actor Models With Application To South Korean Covid-19 Patient Movement Data, Antonio Arrizza, Alberto Caimo

Articles

Motivated by the ongoing COVID-19 pandemic, this article intro- duces Bayesian dynamic network actor models for the analysis of infected individuals’ movements in South Korea during the first three months of 2020. The relational event data modelling framework makes use of network statistics capturing the structure of movement events from and to several country’s municipalities. The fully probabilistic Bayesian approach allows to quantify the uncertainty associated to the relational tendencies explaining where and when movement events are established and where they are directed. The observed patient movements’ patterns at an early stage of the pandemic can provide interesting insights about …

Climate Change Impacts On Wind Energy Generation In Ireland, Eadaoin Doddy Clarke, Conor Sweeney, Frank Mcdermott, Seánie Griffin, Joao Monteiro Correia, Paul Nolan, Laura Cooke

Articles

An ensemble of high-resolution regional climate model simulation data is used to examine the impacts of climate change on offshore and onshore wind energy genera- tion in Ireland. Two Representative Concentration Pathway (RCP) scenarios (RCP 4.5 and 8.5) are analysed for the mid-term (2041–2060) and the long-term (2081–2100) future. Wind energy is projected to decrease (≤2%) overall in future climate scenarios. Changes are evident by mid-century and are more pronounced by late 21st century, particularly for RCP 8.5 offshore. Seasonally, wind energy is projected to decrease by less than 6% in summer and to increase slightly in winter (up to …

Floor Plan Assignment In Elementary Mathematics Education, Tunde Jakab

Open Educational Resources

In this upper elementary mathematics education assignment, the prospective teachers gain hands-on experience in measuring distances in feet and inches, calculating areas, and converting distance and area measurements. Moreover, they solve a real-life situation by choosing the most economical tiles for their kitchen. This last part (3) of the assignment develops critical thinking and expressing one's thought processes. Part 3 can be used as an in-class discussion, which further promotes reasoning skills.

Tiling Representations Of Zeckendorf Decompositions, John Lentfer

HMC Senior Theses

Zeckendorf’s theorem states that every positive integer can be decomposed uniquely into a sum of non-consecutive Fibonacci numbers (where f₁ = 1 and f₂ = 2). Previous work by Grabner and Tichy (1990) and Miller and Wang (2012) has found a generalization of Zeckendorf’s theorem to a larger class of recurrent sequences, called Positive Linear Recurrence Sequences (PLRS’s). We apply well-known tiling interpretations of recurrence sequences from Benjamin and Quinn (2003) to PLRS’s. We exploit that tiling interpretation to create a new tiling interpretation specific to PLRS’s that captures the behavior of the generalized Zeckendorf’s theorem.

On The Tropicalization Of Lines Onto Tropical Quadrics, Natasha Crepeau

HMC Senior Theses

Tropical geometry uses the minimum and addition operations to consider tropical versions of the curves, surfaces, and more generally the zero set of polynomials, called varieties, that are the objects of study in classical algebraic geometry. One known result in classical geometry is that smooth quadric surfaces in three-dimensional projective space, $\mathbb{P}^3$, are doubly ruled, and those rulings form a disjoint union of conics in $\mathbb{P}^5$. We wish to see if the same result holds for smooth tropical quadrics. We use the Fundamental Theorem of Tropical Algebraic Geometry to outline an approach to studying how lines lift onto a tropical …

Global Dynamics Of Generalized Second-Order Beverton–Holt Equations Of Linear And Quadratic Type, Elliott J. Bertrand, Mustafa R. S. Kulenović

Mathematics Faculty Publications

We investigate second-order generalized Beverton–Holt difference equations ... In particular, we will investigate the local and global dynamics in the event f is a certain type of linear or quadratic polynomial, and we explore the existence problem of period-two solutions.

Upward-Closed Hereditary Families In The Dominance Order, Michael D. Barrus, Jean A. Guillaume

Mathematics Faculty Publications

The majorization relation orders the degree sequences of simple graphs into posets called dominance orders. As shown by Ruch and Gutman (1979) and Merris (2002), the degree sequences of threshold and split graphs form upward-closed sets within the dominance orders they belong to, i.e., any degree sequence majorizing a split or threshold sequence must itself be split or threshold, respectively. Motivated by the fact that threshold graphs and split graphs have characterizations in terms of forbidden induced subgraphs, we define a class F of graphs to be dominance monotone if whenever no realization of e contains an element F as …

On The Distribution Of Central Values Of Hecke L-Functions, Daniel White

Bryn Mawr College Dissertations and Theses

Questions regarding the behavior of the Riemann zeta function on the critical line 1/2 + it can be naturally interpreted as questions regarding the family of L-functions over Q associated to the archimedian characters ψ (k) = k -it at the center point 1/2. There are many families of characters besides those strictly of archimedean-type, especially as one expands their scope to proper finite extensions of Q. Consideration of these Hecke characters leads immediately to analogous questions concerning their associated L-functions.

Using tools from p-adic analysis which are analogues of traditional archimedean techniques, we prove the q-aspect analogue …

On P-Adic Modules With Isomorphic Endomorphism Algebras, Brendan Goldsmith, Noel White

Articles

We investigate pairs of modules over the ring of p-adic integers having isomorphic endomorphism algebras. In many cases this forces the modules to be isomorphic but there are two exceptional situations where isomorphism does not follow.