Browsing by Author "Elgart, Alexander"

Now showing 1 - 14 of 14
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    The Born-Oppenheimer Approximation for Triatomic Molecules with Large Angular Momentum in Two Dimensions
    Bowman, Adam Shoresworth (Virginia Tech, 2010-12-08)
    We study the Born-Oppenheimer approximation for a symmetric linear triatomic molecule in two space dimensions. We compute energy levels up to errors of order ε⁵, uniformly for three bounded vibrational quantum numbers n₁, n₂, and n₃; and nuclear angular momentum quantum numbers â ≤ kε-3/4 for k > 0. Here the small parameter ε is the fourth root of the ratio of the electron mass to an average nuclear mass.
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    Bounds for Bilinear Analogues of the Spherical Averaging Operator
    Sovine, Sean Russell (Virginia Tech, 2022-05-12)
    This thesis contains work from the author's papers Palsson and Sovine (2020); Iosevich, Palsson, and Sovine (2022); and Palsson and Sovine (2022) with coauthors Eyvindur Palsson and Alex Iosevich. These works establish new $L^p$-improving, quasi-Banach, and sparse bounds for several bilinear and multilinear operators that generalize the linear spherical average to the multilinear setting, and maximal variants of these operators, with an emphasis on the triangle averaging operator and the bilinear spherical averaging operator.
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    Conical Intersections and Avoided Crossings of Electronic Energy Levels
    Gamble, Stephanie Nicole (Virginia Tech, 2021-01-14)
    We study the unique phenomena which occur in certain systems characterized by the crossing or avoided crossing of two electronic eigenvalues. First, an example problem will be investigated for a given Hamiltonian resulting in a codimension 1 crossing by implementing results by Hagedorn from 1994. Then we perturb the Hamiltonian to study the system for the corresponding avoided crossing by implementing results by Hagedorn and Joye from 1998. The results from these demonstrate the behavior which occurs at a codimension 1 crossing and avoided crossing and illustrates the differences. These solutions may also be used in further studies with Herman-Kluk propagation and more. Secondly, we study codimension 2 crossings by considering a more general type of wave packet. We focus on the case of Schrödinger equation but our methods are general enough to be adapted to other systems with the geometric conditions therein. The motivation comes from the construction of surface hopping algorithms giving an approximation of the solution of a system of Schrödinger equations coupled by a potential admitting a conical intersection, in the spirit of Herman-Kluk approximation (in close relation with frozen/thawed approximations). Our main Theorem gives explicit transition formulas for the profiles when passing through a conical crossing point, including precise computation of the transformation of the phase and its proof is based on a normal form approach.
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    Dynamic Simulation of Power Systems using Three Phase Integrated Transmission and Distribution System Models: Case Study Comparisons with Traditional Analysis Methods
    Jain, Himanshu (Virginia Tech, 2017-01-10)
    Solar PV-based distributed generation has increased significantly over the last few years, and the rapid growth is expected to continue in the foreseeable future. As the penetration levels of distributed generation increase, power systems will become increasingly decentralized with bi-directional flow of electricity between the transmission and distribution networks. To manage such decentralized power systems, planners and operators need models that accurately reflect the structure of, and interactions between the transmission and distribution networks. Moreover, algorithms that can simulate the steady state and dynamics of power systems using these models are also needed. In this context, integrated transmission and distribution system modeling and simulation has become an important research area in recent years, and the primary focus so far has been on studying the steady state response of power systems using integrated transmission and distribution system models. The primary objective of this dissertation is to develop an analysis approach and a program that can simulate the dynamics of three phase, integrated transmission and distribution system models, and use the program to demonstrate the advantages of evaluating the impact of solar PV-based distributed generation on power systems dynamics using such models. To realize this objective, a new dynamic simulation analysis approach is presented, the implementation of the approach in a program is discussed, and verification studies are presented to demonstrate the accuracy of the program. A new dynamic model for small solar PV-based distributed generation is also investigated. This model can interface with unbalanced networks and change its real power output according to the incident solar irradiation. Finally, application of the dynamic simulation program for evaluating the impact of solar PV units using an integrated transmission and distribution system model is discussed. The dissertation presents a new approach for studying the impact of solar PV-based distributed generation on power systems dynamics, and demonstrates that the solar PV impact studies performed using the program and integrated transmission and distribution system models provide insights about the dynamic response of power systems that cannot be obtained using traditional dynamic simulation approaches that rely on transmission only models.
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    Eigenvalue Statistics for Random Block Operators
    Schmidt, Daniel F. (Virginia Tech, 2015-04-28)
    The Schrodinger Hamiltonian for a single electron in a crystalline solid with independent, identically distributed (i.i.d.) single-site potentials has been well studied. It has the form of a diagonal potential energy operator, which contains the random variables, plus a kinetic energy operator, which is deterministic. In the less-understood cases of multiple interacting charge carriers, or of correlated random variables, the Hamiltonian can take the form of a random block-diagonal operator, plus the usual kinetic energy term. Thus, it is of interest to understand the eigenvalue statistics for such operators. In this work, we establish a criterion under which certain random block operators will be guaranteed to satisfy Wegner, Minami, and higher-order estimates. This criterion is phrased in terms of properties of individual blocks of the Hamiltonian. We will then verify the input conditions of this criterion for a certain quasiparticle model with i.i.d. single-site potentials. Next, we will present a progress report on a project to verify the same input conditions for a class of one-dimensional, single-particle alloy-type models. These two results should be sufficient to demonstrate the utility of the criterion as a method of proving Wegner and Minami estimates for random block operators.
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    Impact of Discretization Techniques on Nonlinear Model Reduction and Analysis of the Structure of the POD Basis
    Unger, Benjamin (Virginia Tech, 2013-11-19)
    In this thesis a numerical study of the one dimensional viscous Burgers equation is conducted. The discretization techniques Finite Differences, Finite Element Method and Group Finite Elements are applied and their impact on model reduction techniques, namely Proper Orthogonal Decomposition (POD), Group POD and the Discrete Empirical Interpolation Method (DEIM), is studied. This study is facilitated by examination of several common ODE solvers. Embedded in this process, some results on the structure of the POD basis and an alternative algorithm to compute the POD subspace are presented. Various numerical studies are conducted to compare the different methods and the to study the interaction of the spatial discretization on the ROM through the basis functions. Moreover, the results are used to investigate the impact of Reduced Order Models (ROM) on Optimal Control Problems. To this end, the ROM is embedded in a Trust Region Framework and the convergence results of Arian et al. (2000) is extended to POD-DEIM. Based on the convergence theorem and the results of the numerical studies, the emphasis is on implementation strategies for numerical speedup.
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    The Mattila-Sjölin Problem for Triangles
    Romero Acosta, Juan Francisco (Virginia Tech, 2023-05-08)
    This dissertation contains work from the author's papers [35] and [36] with coauthor Eyvindur Palsson. The classic Mattila-Sjolin theorem shows that if a compact subset of $mathbb{R}^d$ has Hausdorff dimension at least $frac{(d+1)}{2}$ then its set of distances has nonempty interior. In this dissertation, we present a similar result, namely that if a compact subset $E$ of $mathbb{R}^d$, with $d geq 3$, has a large enough Hausdorff dimension then the set of congruence classes of triangles formed by triples of points of $E$ has nonempty interior. These types of results on point configurations with nonempty interior can be categorized as extensions and refinements of the statement in the well known Falconer distance problem which establishes a positive Lebesgue measure for the distance set instead of it having nonempty interior
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    Noncommutative Kernels
    Marx, Gregory (Virginia Tech, 2017-07-17)
    Positive kernels and their associated reproducing kernel Hilbert spaces have played a key role in the development of complex analysis and Hilbert-space operator theory, and they have recently been extended to the setting of free noncommutative function theory. In this paper, we develop the subject further in a number of directions. We give a characterization of completely positive noncommutative kernels in the setting of Hilbert C*-modules and Hilbert W*-modules. We prove an Arveson-type extension theorem for completely positive noncommutative kernels, and we show that a uniformly bounded noncommutative kernel can be decomposed into a linear combination of completely positive noncommutative kernels.
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    A note on the switching adiabatic theorem
    Elgart, Alexander; Hagedorn, George A. (AIP Publishing, 2012-10)
    We derive a nearly optimal upper bound on the running time in the adiabatic theorem for a switching family of Hamiltonians. We assume the switching Hamiltonian is in the Gevrey class G(alpha) as a function of time, and we show that the error in adiabatic approximation remains small for running times of order g(-2) vertical bar ln g vertical bar(6 alpha). Here g denotes the minimal spectral gap between the eigenvalue(s) of interest and the rest of the spectrum of the instantaneous Hamiltonian. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4748968]
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    On the efficiency of Hamiltonian-based quantum computation for low-rank matrices
    Cao, Zhenwei; Elgart, Alexander (AIP Publishing, 2012-03)
    We present an extension of adiabatic quantum computing (AQC) algorithm for the unstructured search to the case when the number of marked items is unknown. The algorithm maintains the optimal Grover speedup and includes a small counting subroutine. Our other results include a lower bound on the amount of time needed to perform a general Hamiltonian-based quantum search, a lower bound on the evolution time needed to perform a search that is valid in the presence of control error and a generic upper bound on the minimum eigenvalue gap for evolutions. In particular, we demonstrate that quantum speedup for the unstructured search using AQC type algorithms may only be achieved under very rigid control precision requirements. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3690045]
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    Quantum evolution: The case of weak localization for a 3D alloy-type Anderson model and application to Hamiltonian based quantum computation
    Cao, Zhenwei (Virginia Tech, 2012-12-11)
    Over the years, people have found Quantum Mechanics to be extremely useful in explaining various physical phenomena from a microscopic point of view. Anderson localization, named after physicist P. W. Anderson, states that disorder in a crystal can cause non-spreading of wave packets, which is one possible mechanism (at single electron level) to explain metalinsulator transitions. The theory of quantum computation promises to bring greater computational power over classical computers by making use of some special features of Quantum Mechanics. The first part of this dissertation considers a 3D alloy-type model, where the Hamiltonian is the sum of the finite difference Laplacian corresponding to free motion of an electron and a random potential generated by a sign-indefinite single-site potential. The result shows that localization occurs in the weak disorder regime, i.e., when the coupling parameter λ is very small, for energies E ≤ −Cλ² . The second part of this dissertation considers adiabatic quantum computing (AQC) algorithms for the unstructured search problem to the case when the number of marked items is unknown. In an ideal situation, an explicit quantum algorithm together with a counting subroutine are given that achieve the optimal Grover speedup over classical algorithms, i.e., roughly speaking, reduce O(2n ) to O(2n/2 ), where n is the size of the problem. However, if one considers more realistic settings, the result shows this quantum speedup is achievable only under a very rigid control precision requirement (e.g., exponentially small control error).
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    The Role of Students' Gestures in Offloading Cognitive Demands on Working Memory in Proving Activities
    Kokushkin, Vladislav (Virginia Tech, 2023-02-03)
    This study examines how undergraduate students use hand gestures to offload cognitive demands on their working memory (WM) when they are engaged in three major proving activities: reading, presenting, and constructing proofs of mathematical conjectures. Existing research literature on the role of gesturing in cognitive offloading has been limited to the context of elementary mathematics but has shown promise for extension to the college level. My framework weaves together theoretical constructs from mathematics education and cognitive psychology: gestures, WM, and mathematical proofs. Piagetian and embodied perspectives allow for the integration of these constructs through positioning bodily activity at the core of human cognition. This framework is operationalized through the methodology for measuring cognitive demands of proofs, which is used to identify the set of mental schemes that are activated simultaneously, as well as the places of potential cognitive overload. The data examined in this dissertation includes individual clinical interviews with six undergraduate students enrolled in different sections of the Introduction to Proofs course in Fall 2021 and Spring 2022. Each student participated in seven interviews: two WM assessments, three proofs-based interviews, a stimulated recall interview (SRI), and post-interview assessments. In total, 42 interviews were conducted. The participants' hand gesturing and mathematical reasoning were qualitatively analyzed. Ultimately, students' reflections during SRIs helped me triangulate the initial data findings. The findings suggest that, in absence of other forms of offloading, hand gesturing may become a convenient, powerful, although not an exclusive offloading mechanism: several participants employed alternative mental strategies in overcoming the cognitive overload they experienced. To better understand what constitutes the essence of cognitive offloading via hand gesturing, I propose a typology of offloading gestures. This typology differs from the existing classification schemes by capturing the cognitive nuances of hand gestures rather than reflecting their mechanical characteristics or the underlying mathematical content. Employing the emerged typology, I then show that cognitive offloading takes different forms when students read or construct proofs, and when they present proofs to the interviewer. Finally, I report on some WM-related issues in presenting and constructing proofs that can be attributed to the potential side effects of mathematical chunking. The dissertation concludes with a discussion of the limitations and practical implications of this project, as well as foreshadowing the avenues for future research.
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    Spectra of Periodic Schrödinger Operators on the Octagonal Lattice
    Storms, Rebecah Helen (Virginia Tech, 2020-06-25)
    We consider the spectrum of the Schrödinger operator on an octagonal lattice using the Floquet-Bloch transform of the Laplacian. We will first consider the spectrum of the Laplacian in detail and prove various properties thereof, including spectral-band limits and locations of singularities. In addition, we will prove that Schrödinger operators with 1-1 periodic potentials can open at most two gaps in the spectrum precisely at energies $pm1$, and that a third gap can open at 0 for 2-2 periodic potentials. We describe in detail the structure of these operators for higher periods, and motivate our expectations of their spectra.
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    Toward a Rigorous Justification of the Three-Body Impact Parameter Approximation
    Bowman, Adam (Virginia Tech, 2014-03-06)
    The impact parameter (IP) approximation is a semiclassical model in quantum scattering theory wherein N large masses interact with one small mass. We study this model in one spatial dimension using the tools of time-dependent scattering theory, considering a system of two large-mass particles and one small-mass particle. We demonstrate that the model's predictive power becomes arbitrarily good as the masses of the two heavy particles are made larger by studying the S-matrix for a particular scattering channel. We also show that the IP wave functions can be made arbitrarily close to the full three-body solution, uniformly in time, provided one of the large masses is fixed in place, and that such a result probably will not hold if we allow all the masses to move.