Singular Points Of Real Sextic Curves I, 2010 George Fox University

#### Singular Points Of Real Sextic Curves I, David A. Weinberg, Nicholas J. Willis

*Faculty Publications - Department of Mathematics*

A complete classiﬁcation of the individual types of singular points is given for irreducible real sextic curves. This classiﬁcation is derived by using the computer algebra system Maple. There are 191 types of singular points for real irreducible sextic curves. We clarify that the classiﬁcation is based on computing just enough of the Puiseux expansion to separate the branches. A signiﬁcant portion of the proof consists of a sequence of large symbolic computations that can be done nicely using Maple.

Weak Hardy Space And Endpoint Estimates For Singular Integrals On Space Of Homogeneous Type, 2010 TÜBİTAK

#### Weak Hardy Space And Endpoint Estimates For Singular Integrals On Space Of Homogeneous Type, Yong Ding, Xinfeng Wu

*Turkish Journal of Mathematics*

We develop the theory of weak Hardy spaces H^{1,\infty} on space of homogeneous type. As some applications, we show that certain singular integral operators and fractional integral operators are bounded from H^{1,\infty} to L^{1,\infty} and L^{\frac{1}{1-\alpha},\infty}, respectively. We give also the endpoint estimates for Nagel and Stein's singular integrals studied in [10].

On Weakly M-Supplemented Primary Subgroups Of Finite Groups, 2010 TÜBİTAK

#### On Weakly M-Supplemented Primary Subgroups Of Finite Groups, Long Miao, Wolfgang Lempken

*Turkish Journal of Mathematics*

A subgroup H of a group G is said to be weakly M-supplemented in G if there exists a subgroup B of G provided that (1) G=HB, and (2) if H_1/H_G is a maximal subgroup of H/H_G, then H_1B=BH_1

Linear Koszul Duality, 2010 University of Massachusetts - Amherst

#### Linear Koszul Duality, I Mirkovic, S Riche

*Mathematics and Statistics Department Faculty Publication Series*

In this paper we construct, for *F*_{1} and *F*_{2} subbundles of a vector bundle *E*, a ‘Koszul duality’ equivalence between derived categories of _{m}-equivariant coherent(dg-)sheaves on the derived intersection , and the corresponding derived intersection . We also propose applications to Hecke algebras.

Stable Structures With High Topological Charge In Nonlinear Photonic Quasicrystals, 2010 University of Massachusetts - Amherst

#### Stable Structures With High Topological Charge In Nonlinear Photonic Quasicrystals, K Law, A Saxena, Pg Kevrekidis

*Mathematics and Statistics Department Faculty Publication Series*

Stable vortices with topological charges of 3 and 4 are examined numerically and analytically in photonic quasicrystals created by interference of five as well as eight beams, for cubic as well as saturable nonlinearities. Direct numerical simulations corroborate the analytical and numerical linear stability analysis predictions for such experimentally realizable structures.

Stability And Dynamics Of Matter-Wave Vortices In The Presence Of Collisional Inhomogeneities And Dissipative Perturbations, 2010 University of Massachusetts - Amherst

#### Stability And Dynamics Of Matter-Wave Vortices In The Presence Of Collisional Inhomogeneities And Dissipative Perturbations, S Middelkamp, Pg Kevrekidis

*Mathematics and Statistics Department Faculty Publication Series*

In this work, the spectral properties of a singly charged vortex in a Bose–Einstein condensate confined in a highly anisotropic (disc-shaped) harmonic trap are investigated. Special emphasis is placed on the analysis of the so-called anomalous (negative energy) mode of the Bogoliubov spectrum. We use analytical and numerical techniques to illustrate the connection of the anomalous mode to the precession dynamics of the vortex in the trap. Effects due to inhomogeneous interatomic interactions and dissipative perturbations motivated by finite-temperature considerations are explored. We find that both of these effects may give rise to oscillatory instabilities of the vortex, which are …

Multibreather And Vortex Breather Stability In Klein–Gordon Lattices: Equivalence Between Two Different Approaches, 2010 University of Massachusetts - Amherst

#### Multibreather And Vortex Breather Stability In Klein–Gordon Lattices: Equivalence Between Two Different Approaches, J Cuevas, V Koukouloyannis, Pg Kevrekidis

*Mathematics and Statistics Department Faculty Publication Series*

In this work, we revisit the question of stability of multibreather configurations, i.e., discrete breathers with multiple excited sites at the anti-continuum limit of uncoupled oscillators. We present two methods that yield quantitative predictions about the Floquet multipliers of the linear stability analysis around such exponentially localized in space, time-periodic orbits, based on the Aubry band method and the MacKay effective Hamiltonian method and prove that their conclusions are equivalent. Subsequently, we showcase the usefulness of the methods by a series of case examples including one-dimensional multi-breathers, and two-dimensional vortex breathers in the case of a lattice of linearly coupled …

Intrinsic Energy Localization Through Discrete Gap Breathers In One-Dimensional Diatomic Granular Crystals, 2010 University of Massachusetts - Amherst

#### Intrinsic Energy Localization Through Discrete Gap Breathers In One-Dimensional Diatomic Granular Crystals, G Theocharis, N Boechler, Pg Kevrekidis

*Mathematics and Statistics Department Faculty Publication Series*

We present a systematic study of the existence and stability of discrete breathers that are spatially localized in the bulk of a one-dimensional chain of compressed elastic beads that interact via Hertzian contact. The chain is diatomic, consisting of a periodic arrangement of heavy and light spherical particles. We examine two families of discrete gap breathers: (1) an unstable discrete gap breather that is centered on a heavy particle and characterized by a symmetric spatial energy profile and (2) a potentially stable discrete gap breather that is centered on a light particle and is characterized by an asymmetric spatial energy …

Collisional-Inhomogeneity-Induced Generation Of Matter-Wave Dark Solitons, 2010 University of Massachusetts - Amherst

#### Collisional-Inhomogeneity-Induced Generation Of Matter-Wave Dark Solitons, C Wang, Pg Kevrekidis

*Mathematics and Statistics Department Faculty Publication Series*

We propose an experimentally relevant protocol for the controlled generation of matter-wave dark solitons in atomic Bose–Einstein condensates (BECs). In particular, using direct numerical simulations, we show that by switching-on a spatially inhomogeneous (step-like) change of the s-wave scattering length, it is possible to generate a controllable number of dark solitons in a quasi-one-dimensional BEC. A similar phenomenology is also found in the two-dimensional setting of “disk-shaped” BECs but, as the solitons are subject to the snaking instability, they decay into vortex structures. A detailed investigation of how the parameters involved affect the emergence and evolution of solitons and vortices …

Ruelle-Lanford Functions For Quantum Spin Systems, 2010 University of Massachusetts - Amherst

#### Ruelle-Lanford Functions For Quantum Spin Systems, Y Ogata, L Rey-Bellet

*Mathematics and Statistics Department Faculty Publication Series*

We prove a large deviation principle for the expectation of macroscopic

observables in quantum (and classical) Gibbs states. Our proof is based

on Ruelle-Lanford functions [20, 34] and direct subadditivity arguments,

as in the classical case [23, 32], instead of relying on G¨artner-Ellis theorem,

and cluster expansion or transfer operators as done in the quantum case

in [21, 13, 27, 22, 16, 28]. In this approach we recover, expand, and unify

quantum (and classical) large deviation results for lattice Gibbs states. In

the companion paper [29] we discuss the characterization of rate functions

in terms of relative entropies.

Cells In Coxeter Groups I, 2010 University of Massachusetts - Amherst

#### Cells In Coxeter Groups I, M Belolipetsky, Pe Gunnells

*Mathematics and Statistics Department Faculty Publication Series*

No abstract provided.

Deterministic Equations For Stochastic Spatial Evolutionary Games, 2010 University of Massachusetts - Amherst

#### Deterministic Equations For Stochastic Spatial Evolutionary Games, Sh Hwang, Ma Katsoulakis, L Rey-Bellet

*Mathematics and Statistics Department Faculty Publication Series*

In this paper we investigate the approximation properties of the coarse-graining procedure applied to kinetic Monte Carlo simulations of lattice stochastic dynamics. We provide both analytical and numerical evidence that the hierarchy of the coarse models is built in a systematic way that allows for error control in both transient and long-time simulations. We demonstrate that the numerical accuracy of the CGMC algorithm as an approximation of stochastic lattice spin flip dynamics is of order two in terms of the coarse-graining ratio and that the natural small parameter is the coarse-graining ratio over the range of particle/particle interactions. The error …

Torsion In The Cohomology Of Congruence Subgroups Of Sl(4, Z) And Galois Representations, 2010 University of Massachusetts - Amherst

#### Torsion In The Cohomology Of Congruence Subgroups Of Sl(4, Z) And Galois Representations, A Ash, Pe Gunnells

*Mathematics and Statistics Department Faculty Publication Series*

We report on the computation of torsion in certain homology the-ories of congruence subgroups of SL(4, Z). Among these are the usual group cohomology, the Tate-Farrell cohomology, and the homology of the sharbly complex. All of these theories yield Hecke modules. We conjecture that the Hecke eigenclasses in these theories have attached Galois representations. The interpretation of our computations at the torsion primes 2,3,5 is explained. We provide evidence for our conjecture in the 15 cases of odd torsion that we found in levels 31.

Metaplectic Ice, 2010 University of Massachusetts - Amherst

#### Metaplectic Ice, B Brubaker, D Bump, G Chinta, S Friedberg, Pe Gunnells

*Mathematics and Statistics Department Faculty Publication Series*

We study spherical Whittaker functions on a metaplectic cover of GL(r + 1) over a nonarchimedean local field using lattice models from statistical mechanics. An explicit description of this Whittaker function was given in terms of Gelfand-Tsetlin patterns in [5, 17], and we translate this description into an expression of the values of the Whittaker function as partition functions of a six-vertex model. Properties of theWhittaker function may then be expressed in terms of the commutativity of row transfer matrices potentially amenable to proof using the Yang-Baxter equation. We give two examples of this: first, the equivalence of two different …

Best-Fit Quasi-Equilibrium Ensembles: A General Approach To Statistical Closure Of Underresolved Hamiltonian Dynamics, 2010 University of Massachusetts - Amherst

#### Best-Fit Quasi-Equilibrium Ensembles: A General Approach To Statistical Closure Of Underresolved Hamiltonian Dynamics, B Turkington, P Plechac

*Mathematics and Statistics Department Faculty Publication Series*

A new method of deriving reduced models of Hamiltonian dynamical systems

is developed using techniques from optimization and statistical estimation. Given

a set of resolved variables that define a model reduction, the quasi-equilibrium

ensembles associated with the resolved variables are employed as a family of trial

probability densities on phase space. The residual that results from submitting

these trial densities to the Liouville equation is quantified by an ensemble-averaged

cost function related to the information loss rate of the reduction. From an initial

nonequilibrium state, the statistical state of the system at any later time is estimated

by minimizing the …

Skyrmions, Rational Maps & Scaling Identities, 2010 Aristotle University of Thessaloniki

#### Skyrmions, Rational Maps & Scaling Identities, E. G. Charalampidis, T. A. Ioannidou, N. S. Manton

*Mathematics and Statistics Department Faculty Publication Series*

Starting from approximate Skyrmion solutions obtained using the rational map ansatz, improved approximate Skyrmions are constructed using scaling arguments. Although the energy improvement is small, the change of shape clarifies whether the true Skyrmions are more oblate or prolate.

Dark Solitons In Cigar-Shaped Bose-Einstein Condensates In Double-Well Potentials, 2010 University of Massachusetts - Amherst

#### Dark Solitons In Cigar-Shaped Bose-Einstein Condensates In Double-Well Potentials, S Middelkamp, G Theocharis, Pg Kevrekidis, Dj Frantzeskakis, P Schmelcher

*Mathematics and Statistics Department Faculty Publication Series*

We study the statics and dynamics of dark solitons in a cigar-shaped Bose-Einstein condensate confined in a double-well potential. Using a mean-field model with a noncubic nonlinearity, appropriate to describe the dimensionality crossover regime from one- to three-dimensional, we obtain branches of solutions in the form of single and multiple dark soliton states, and study their bifurcations and stability. It is demonstrated that there exist dark soliton states which do not have a linear counterpart and we highlight the role of anomalous modes in the excitation spectra. Particularly, we show that anomalous mode eigenfrequencies are closely connected to the characteristic …

Distribution Of Eigenfrequencies For Oscillations Of The Ground State In The Thomas-Fermi Limit, 2010 University of Massachusetts - Amherst

#### Distribution Of Eigenfrequencies For Oscillations Of The Ground State In The Thomas-Fermi Limit, Pg Kevrekidis, De Pelinovsky

*Mathematics and Statistics Department Faculty Publication Series*

In this work, we present a systematic derivation of the distribution of eigenfrequencies for oscillations of the ground state of a repulsive Bose-Einstein condensate in the semi-classical (Thomas-Fermi) limit. Our calculations are performed in one, two, and three-dimensional settings. Connections with the earlier work of Stringari, with numerical computations, and with theoretical expectations for invariant frequencies based on symmetry principles are also given.

Dynamics Of Dark–Bright Solitons In Cigar-Shaped Bose–Einstein Condensates, 2010 University of Massachusetts - Amherst

#### Dynamics Of Dark–Bright Solitons In Cigar-Shaped Bose–Einstein Condensates, S Middelkamp, J J. Chang, C Hammer, R Carretero-Gonzalez, Pg Kevrekidis

*Mathematics and Statistics Department Faculty Publication Series*

We explore the stability and dynamics of dark–bright (DB) solitons in two-component elongated Bose–Einstein condensates by developing effective one-dimensional vector equations and solving the three-dimensional Gross–Pitaevskii equations. A strong dependence of the oscillation frequency and of the stability of the DB soliton on the atom number of its components is found; importantly, the wave may become dynamically unstable even in the 1D regime. As the atom number in the dark-soliton-supporting component is further increased, spontaneous symmetry breaking leads to oscillatory dynamics in the transverse degrees of freedom. Moreover, the interactions of two DB solitons are investigated with an emphasis on …

Controlling The Transverse Instability Of Dark Solitons And Nucleation Of Vortices By A Potential Barrier, 2010 University of Massachusetts - Amherst

#### Controlling The Transverse Instability Of Dark Solitons And Nucleation Of Vortices By A Potential Barrier, Manjun M, R. Carretero-Gonzalez, Pg Kevrekidis

*Mathematics and Statistics Department Faculty Publication Series*

We study possibilities to suppress the transverse modulational instability (MI) of dark-soliton stripes in two-dimensional Bose-Einstein condensates (BEC’s) and self-defocusing bulk optical waveguides by means of quasi-one-dimensional structures. Adding an external repulsive barrier potential (which can be induced in BEC by a laser sheet, or by an embedded plate in optics), we demonstrate that it is possible to reduce the MI wave number band, and even render the dark-soliton stripe completely stable. Using this method, we demonstrate the control of the number of vortex pairs nucleated by each spatial period of the modulational perturbation. By means of the perturbation theory, …