Mathematics Commons^™

The Schwartz Space: Tools For Quantum Mechanics And Infinite Dimensional Analysis, Jeremy Becnel, Ambar Sengupta

Faculty Publications

An account of the Schwartz space of rapidly decreasing functions as a topological vector space with additional special structures is presented in a manner that provides all the essential background ideas for some areas of quantum mechanics along with infinite-dimensional distribution theory.

On A Multi-Dimensional Singular Stochastic Control Problem: The Parabolic Case, Nhat Do Minh Nguyen

Wayne State University Dissertations

This dissertation considers a stochastic dynamic system which is governed by a multidimensional diffusion process with time dependent coefficients. The control acts additively on the state of the system. The objective is to minimize the expected cumulative cost associated with the position of the system and the amount of control exerted. It is proved that Hamilton-Jacobi-Bellman’s equation of the problem has a solution, which corresponds to the optimal cost of the problem. We also investigate the smoothness of the free boundary arising from the problem.

In the second part of the dissertation, we study the backward parabolic problem for a …

New Characterizations Of Sobolev Spaces On Heisenberg And Carnot Groups And High Order Sobolev Spaces On Eucliean Spaces, Xiaoyue Cui

Wayne State University Dissertations

This dissertation focuses on new characterizations of Sobolev spaces .

It encompasses an in-depth study of Sobolev spaces on Heisenberg groups, as well as Carnot groups, second order and high order Sobolev spaces on Euclidean spaces.

The Topology Of Tile Invariants, Michael P. Hitchman

Michael P. Hitchman

In this note, we use techniques in the topology of 2-complexes to recast some tools that have arisen in the study of planar tiling questions. With spherical pictures, we show that the tile counting group associated to a set T of tiles and set of regions tileable by T is isomorphic to a quotient of the second homology group of a 2-complex built from T. In this topological setting, we derive some well-known tile invariants, one of which we apply to the solution of a tiling question involving modified rectangles.

Geometry: Drawing From The Islamic Tradition, Carol Bier

Carol Bier

Getting students involved in careful observation and analysis and encouraging their exploration of cultural forms of expression is an excellent means of introducing mathematical ideas. Geometric patterns abound in Islamic art and architecture. Exhibiting great ingenuity over the centuries, Muslim artists and craftsmen created beautiful patterns to adorn architectural monuments and exquisite objects. The Alhambra in Spain and the Taj Mahal in India offer the most famous examples of extraordinary patterns using brick and glazed tile, or carved and inlaid marble. Other examples of patterns are made using metal, wood, and fiber. Students may gain conceptual and theoretical understanding of …

Inżynieria Chemiczna Ćw., Wojciech M. Budzianowski

Wojciech Budzianowski

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Tematyka Prac Doktorskich, Wojciech M. Budzianowski

Wojciech Budzianowski

No abstract provided.

Primary Spaces, Mackey’S Obstruction, And The Generalized Barycentric Decomposition, Patrick Iglesias-Zemmour, François Ziegler

François Ziegler

We call a hamiltonian N-space primary if its moment map is onto a single coadjoint orbit. The question has long been open whether such spaces always split as (homogeneous) × (trivial), as an analogy with representation theory might suggest. For instance, Souriau’s barycentric decomposition theorem asserts just this when N is a Heisenberg group. For general N, we give explicit examples which do not split, and show instead that primary spaces are always flat bundles over the coadjoint orbit. This provides the missing piece for a full “Mackey theory” of hamiltonian G-spaces, where G is an overgroup in which N …

Transitions From Order To Disorder In Multi-Dark And Multi-Dark-Bright Soliton Atomic Clouds, Wenlong Wang, Panos Kevrekidis

Panos Kevrekidis

We have performed a systematic study quantifying the variation of solitary wave behavior from that of an ordered cloud resembling a “crystalline” configuration to that of a disordered state that can be characterized as a soliton “gas.” As our illustrative examples, we use both one-component, as well as two-component, one-dimensional atomic gases very close to zero temperature, where in the presence of repulsive interatomic interactions and of a parabolic trap, a cloud of dark (dark-bright) solitons can form in the one- (two-) component system. We corroborate our findings through three distinct types of approaches, namely a Gross-Pitaevskii type of partial …

Pathwise Sensitivity Analysis In Transient Regimes, Georgios Arampatzis, Markos Katsoulakis, Yannis Pantazis

Markos Katsoulakis

The instantaneous relative entropy (IRE) and the corresponding instantaneous Fisher information matrix (IFIM) for transient stochastic processes are presented in this paper. These novel tools for sensitivity analysis of stochastic models serve as an extension of the well known relative entropy rate (RER) and the corresponding Fisher information matrix (FIM) that apply to stationary processes. Three cases are studied here, discrete-time Markov chains, continuous-time Markov chains and stochastic differential equations. A biological reaction network is presented as a demonstration numerical example.

Birational Geometry Of Cluster Algebras, Mark Gross, Paul Hacking, Sean Keel

Paul Hacking

We give a geometric interpretation of cluster varieties in terms of blowups of toric varieties. This enables us to provide, among other results, an elementary geometric proof of the Laurent phenomenon for cluster algebras (of geometric type), extend Speyer's example [Spe13] of upper cluster algebras which are not finitely generated, and show that the Fock-Goncharov dual basis conjecture is usually false.

Generating Functions, Polynomials And Vortices With Alternating Signs In Bose-Einstein Condensates, Anna M. Barry, Farshid Hajir, P. G. Kevrekidis

Farshid Hajir

In this work, we construct suitable generating functions for vortices of alternating signs in the realm of quasi-two-dimensional Bose–Einstein condensates in the large density (so-called Thomas–Fermi) limit, where the vortices can be treated as effective particles. In addition to the vortex–vortex interaction included in earlier fluid dynamics constructions of such functions, the vortices here precess around the center of the trap. This results in the generating functions of the vortices of positive charge and of negative charge satisfying a modified, so-called, Tkachenko differential equation. From that equation, we reconstruct collinear few-vortex equilibria obtained in earlier work, as well as extend …

Dynamics Of Vortex Dipoles In Anisotropic Bose-Einstein Condensates, Roy H. Goodman, Panos Kevrekidis, R. Carretero-Gonzalez

Panos Kevrekidis

We study the motion of a vortex dipole in a Bose--Einstein condensate confined to an anisotropic trap. We focus on a system of ODEs describing the vortices' motion, which is in turn a reduced model of the Gross--Pitaevskii equation describing the condensate's motion. Using a sequence of canonical changes of variables, we reduce the dimension and simplify the equations of motion. We uncover two interesting regimes. Near a family of periodic orbits known as guiding centers, we find that the dynamics is essentially that of a pendulum coupled to a linear oscillator, leading to stochastic reversals in the overall direction …

Canonical Bases For Cluster Algebras, Mark Gross, Paul Hacking, Sean Keel, Maxim Kontesevich

Paul Hacking

In GHK11, Conjecture 0.6, the first three authors conjectured the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral tropical points of the mirror. Further, the structure constants for the multiplication rule in this basis should be given by counting broken lines (certain combinatorial objects, morally the tropicalisations of holomorphic discs). Here we prove the conjecture in the case of cluster varieties, where the statement is a more precise form of the Fock-Goncharov dual basis conjecture, FG06, Conjecture 4.3. In particular, under …

Scattering Of Matter-Waves In Spatially Inhomogeneous Environments, F. Tsitoura, P. Kruger, Panos Kevrekidis, D. J. Frantzeskakis

Panos Kevrekidis

We study scattering of quasi-one-dimensional matter waves at an interface of two spatial domains, one with repulsive and one with attractive interatomic interactions. It is shown that the incidence of a Gaussian wave packet from the repulsive to the attractive region gives rise to generation of a soliton train. More specifically, the number of emergent solitons can be controlled, e.g., by the variation of the amplitude or the width of the incoming wave packet. Furthermore, we study the reflectivity of a soliton incident from the attractive region to the repulsive one. We find the reflection coefficient numerically and employ analytical …

Moduli Of Surfaces With An Anti-Canonical Cycle, Mark Gross, Paul Hacking, Sean Keel

Paul Hacking

We prove a global torelli theorem for pairs (Y,D) where Y is a smooth projective rational surface and D ∈ |−Ky | is a cycle of rational curves, as conjectured by Friedman in 1984. In addition, we construct natural universal families for such pairs.

Solitons And Vortices In Two-Dimensional Discrete Nonlinear Schrodinger Systems With Spatially Modulated Nonlinearity, Panos Kevrekidis

Panos Kevrekidis

We consider a two-dimensional (2D) generalization of a recently proposed model [Gligorić et al., Phys. Rev. E 88, 032905 (2013)], which gives rise to bright discrete solitons supported by the defocusing nonlinearity whose local strength grows from the center to the periphery. We explore the 2D model starting from the anticontinuum (AC) limit of vanishing coupling. In this limit, we can construct a wide variety of solutions including not only single-site excitations, but also dipole and quadrupole ones. Additionally, two separate families of solutions are explored: the usual “extended” unstaggered bright solitons, in which all sites are excited in the …

Somewhat Stochastic Matrices, Branko Ćurgus, Robert I. Jewett

Mathematics Faculty Publications

The standard theorem for stochastic matrices with positive entries is generalized to matrices with no sign restriction on the entries. The condition that column sums be equal to 1 is kept, but the positivity condition is replaced by a condition on the distances between columns.

Arithmetical Rank Of The Edge Ideals Of Some N-Cyclic Graphs With A Common Edge, Guangjun Zhu, Feng Shi, Yan Gu

Turkish Journal of Mathematics

In this paper, we present some lower bounds and upper bounds on the arithmetical rank of the edge ideals of some n-cyclic graphs with a common edge. For some special n-cyclic graphs with a common edge, we prove that the arithmetical rank equals the projective dimension of the corresponding quotient ring.

When An Idea Comes, Write It Down Right Away: Mathematical Justification Of Vladimir Smirnov's Advice, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Among several advices to students, Vladimir Smirnov, a renowned Russian mathematician, suggested that when an idea comes, it is better to write it down right away. In this paper, we provide a quantitative justification for this advice.