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Global Optimization, The Gaussian Ensemble, And Universal Ensemble Equivalence., M Costeniuc, Rs Ellis, H Touchette, B Turkington

Mathematics and Statistics Department Faculty Publication Series

Given a constrained minimization problem, under what conditions does there exist a related, unconstrained
problem having the same minimum points? This basic question in global optimization
motivates this paper, which answers it from the viewpoint of statistical mechanics. In this context, it
reduces to the fundamental question of the equivalence and nonequivalence of ensembles, which is
analyzed using the theory of large deviations and the theory of convex functions.
In a 2000 paper appearing in the Journal of Statistical Physics, we gave necessary and sufficient
conditions for ensemble equivalence and nonequivalence in terms of support and concavity
properties of the …

Smooth S-Cobordisms Of Elliptic 3-Manifolds, Wm Chen

Mathematics and Statistics Department Faculty Publication Series

The main result of this paper states that a symplectic s-cobordism of elliptic 3-manifolds is diffeomorphic to a product (assuming a canonical contact structure on the boundary). Based on this theorem, we conjecture that a smooth s-cobordism of elliptic 3-manifolds is smoothly a product if its universal cover is smoothly a product. We explain how the conjecture fits naturally into the program of Taubes of constructing symplectic structures on an oriented smooth 4-manifold with b⁺₂ ≥ 1 from generic self-dual harmonic forms. The paper also contains an auxiliary result of independent interest, which generalizes Taubes' theorem "SW ⇒ …

Nonconcave Entropies From Generalized Canonical Ensembles, M Costeniuc, Rs Ellis, H Touchette

Mathematics and Statistics Department Faculty Publication Series

It is well known that the entropy of the microcanonical ensemble cannot be calculated as the Legendre transform of the canonical free energy when the entropy is nonconcave. To circumvent this problem, a generalization of the canonical ensemble that allows for the calculation of nonconcave entropies was recently proposed. Here, we study the mean-field Curie-Weiss-Potts spin model and show, by direct calculations, that the nonconcave entropy of this model can be obtained by using a specific instance of the generalized canonical ensemble known as the Gaussian ensemble.

Generalized Canonical Ensembles And Ensemble Equivalence, M Costeniuc, Rs Ellis, H Touchette, B Turkington

Mathematics and Statistics Department Faculty Publication Series

This paper is a companion piece to our previous work [J. Stat. Phys. 119, 1283 (2005)], which introduced a generalized canonical ensemble obtained by multiplying the usual Boltzmann weight factor e^−βH of the canonical ensemble with an exponential factor involving a continuous function g of the Hamiltonian H. We provide here a simplified introduction to our previous work, focusing now on a number of physical rather than mathematical aspects of the generalized canonical ensemble. The main result discussed is that, for suitable choices of g, the generalized canonical ensemble reproduces, in the thermodynamic limit, all the microcanonical equilibrium properties …

Exponents For B-Stable Ideals, E Sommers, J Tymoczko

Mathematics and Statistics Department Faculty Publication Series

Let be a simple algebraic group over the complex numbers containing a Borel subgroup . Given a -stable ideal in the nilradical of the Lie algebra of , we define natural numbers which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types and some other types.

When , we recover the usual exponents of by Kostant (1959), and one of our conjectures reduces to a well-known factorization of the Poincaré polynomial of the Weyl group. The other conjecture reduces to a well-known result of Arnold-Brieskorn on the factorization of the characteristic …

Symmetry Breaking In Symmetric And Asymmetric Double-Well Potentials, G Theocharis, Pg Kevrekidis, Dj Frantzeskakis, P Schmelcher

Mathematics and Statistics Department Faculty Publication Series

Motivated by recent experimental studies of matter waves and optical beams in double-well potentials, we study the corresponding solutions of the nonlinear Schrödinger equation. Using a Galerkin-type approach, we obtain a detailed handle on the nonlinear solution branches of the problem, starting from the corresponding linear ones, and we predict the relevant bifurcations for both attractive and repulsive nonlinearities. The dynamics of the ensuing unstable solutions is also examined. The results illustrate the differences that arise between the steady states and the bifurcations emerging in symmetric and asymmetric double wells.

Metastability Within The Generalized Canonical Ensemble, H Touchette, M Costeniuc, Rs Ellis, B Turkington

Mathematics and Statistics Department Faculty Publication Series

We discuss a property of our recently introduced generalized canonical ensemble [M. Costeniuc, R.S. Ellis, H. Touchette, B. Turkington, The generalized canonical ensemble and its universal equivalence with the microcanonical ensemble, J. Stat. Phys. 119 (2005) 1283]. We show that this ensemble can be used to transform metastable or unstable (nonequilibrium) states of the standard canonical ensemble into stable (equilibrium) states within the generalized canonical ensemble. Equilibrium calculations within the generalized canonical ensemble can thus be used to obtain information about nonequilibrium states in the canonical ensemble.

Blowup With Small Bv Data In Hyperbolic Conservation Laws, Robin Young, Walter Szeliga

Mathematics and Statistics Department Faculty Publication Series

We construct weak solutions of 3×3 conservation laws which blow up in finite time. The system is strictly hyperbolic at every state in the solution, and the data can be chosen to have arbitrarily small total variation. This is thus an example where Glimm's existence theorem fails to apply, and it implies the necessity of uniform hyperbolicity in Glimm's theorem. Because our system is very simple, we can carry out explicit calculations and understand the global geometry of wave curves.

Nonlinearity Management In Higher Dimensions, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

In the present paper, we revisit nonlinearity management of the time-periodic nonlinear Schrödinger equation and the related averaging procedure. By means of rigorous estimates, we show that the averaged nonlinear Schrödinger equation does not blow up in the higher dimensional case so long as the corresponding solution remains smooth. In particular, we show that the H1 norm remains bounded, in contrast with the usual blow-up mechanism for the focusing Schrödinger equation. This conclusion agrees with earlier works in the case of strong nonlinearity management but contradicts those in the case of weak nonlinearity management. The apparent discrepancy is explained by …

High-Order-Mode Soliton Structures In Two-Dimensional Lattices With Defocusing Nonlinearity, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

While fundamental-mode discrete solitons have been demonstrated with both self-focusing and defocusing nonlinearity, high-order-mode localized states in waveguide lattices have been studied thus far only for the self-focusing case. In this paper, the existence and stability regimes of dipole, quadrupole, and vortex soliton structures in two-dimensional lattices induced with a defocusing nonlinearity are examined by the theoretical and numerical analysis of a generic envelope nonlinear lattice model. In particular, we find that the stability of such high-order-mode solitons is quite different from that with self-focusing nonlinearity. As a simple example, a dipole (“twisted”) mode soliton with adjacent excited sites which …

Dynamics And Manipulation Of Matter-Wave Solitons In Optical Superlattices, Mason A. Porter, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We study the existence and stability of bright, dark, and gap matter-wave solitons in optical superlattices. Then, using these properties, we show that (time-dependent) “dynamical superlattices” can be used to controllably place, guide, and manipulate these solitons. In particular, we use numerical experiments to displace solitons by turning on a secondary lattice structure, transfer solitons from one location to another by shifting one superlattice substructure relative to the other, and implement solitonic “path-following”, in which a matter wave follows the time-dependent lattice substructure into oscillatory motion.

Discrete Vector On-Site Vortices, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We study discrete vortices in coupled discrete nonlinear Schrödinger equations. We focus on the vortex cross configuration that has been experimentally observed in photorefractive crystals. Stability of the single-component vortex cross in the anti-continuum limit of small coupling between lattice nodes is proved. In the vector case, we consider two coupled configurations of vortex crosses, namely the charge-one vortex in one component coupled in the other component to either the charge-one vortex (forming a double-charge vortex) or the charge-negative-one vortex (forming a, so-called, hidden-charge vortex). We show that both vortex configurations are stable in the anti-continuum limit, if the parameter …

Radiationless Traveling Waves In Saturable Nonlinear Schrödinger Lattices, T.R.O Melvin, A R. Champneys, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

The long-standing problem of moving discrete solitary waves in nonlinear Schrödinger lattices is revisited. The context is photorefractive crystal lattices with saturable nonlinearity whose grand-canonical energy barrier vanishes for isolated coupling strength values. Genuinely localized traveling waves are computed as a function of the system parameters for the first time. The relevant solutions exist only for finite velocities.

Discrete Nonlinear Schrödinger Equations Free Of The Peierls–Nabarro Potential, S V. Dmitriev, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We derive a class of discrete nonlinear Schrödinger (DNLS) equations for general polynomial nonlinearity whose stationary solutions can be found from a reduced two-point algebraic problem. It is demonstrated that the derived class of discretizations contains subclasses conserving classical norm or a modified norm and classical momentum. These equations are interesting from the physical standpoint since they support stationary discrete solitons free of the Peierls–Nabarro potential. Focusing on the cubic nonlinearity we then consider a small perturbation around stationary soliton solutions and, solving corresponding eigenvalue problem, we (i) demonstrate that solitons are stable; (ii) show that they have two additional …

On A Notion Of Maps Between Orbifolds I. Function Spaces, Wm Chen

Mathematics and Statistics Department Faculty Publication Series

This is the first of a series of papers which is devoted to a comprehensive theory of maps between orbifolds. In this paper, we define the maps in the more general context of orbispaces, and establish several basic results concerning the topological structure of the space of such maps. In particular, we show that the space of such maps of C^r class between smooth orbifolds has a natural Banach orbifold structure if the domain of the map is compact, generalizing the corresponding result in the manifold case. Motivations and applications of the theory come from string theory and the …

On Anticyclotomic Ì-Invariants Of Modular Forms, R Pollack, T Weston

Mathematics and Statistics Department Faculty Publication Series

No abstract provided.

Standard Nearest-Neighbour Discretizations Of Klein–Gordon Models Cannot Preserve Both Energy And Linear Momentum, S V. Dmitriev, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We consider nonlinear Klein–Gordon wave equations and illustrate that standard discretizations thereof (involving nearest neighbours) may preserve either standardly defined linear momentum or standardly defined total energy but not both. This has a variety of intriguing implications for the 'non-potential' discretizations that preserve only the linear momentum, such as the self-accelerating or self-decelerating motion of coherent structures such as discrete kinks in these nonlinear lattices.

Skyrmion-Like Excitations In Dynamical Lattices, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We construct discrete analogs of Skyrmions in nonlinear dynamical lattices. The Skyrmion is built as a vortex soliton of a complex field, coupled to a dark radial soliton of a real field. Adjusting the Skyrmion ansatz to the lattice setting allows us to construct a baby-Skyrmion in two dimensions (2D) and extend it into the 3D case (1D counterparts of the Skyrmions are also found). Stability limits for these patterns are obtained analytically and verified numerically. The dynamics of unstable discrete Skyrmions is explored, and their stabilization by external potentials is discussed.

Singular Localization And Intertwining Functors For Reductive Lie Algebras In Prime Characteristic, R Bezrukavnikov, I Mirkovic, D Rumynin

Mathematics and Statistics Department Faculty Publication Series

In [BMR] we observed that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) {\em regular} central character can be identified with coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber. In the present paper we treat singular central characters.

The basic step is the Beilinson-Bernstein localization of modules with a fixed (generalized) central character $\lambda$ as sheaves on the partial flag variety corresponding to the singularity of $\lambda$. These sheaves are modules over a sheaf of algebras which is …

Solitary Waves In Discrete Media With Four-Wave Mixing, Rl Horne, Pg Kevrekidis, N Whitaker

Mathematics and Statistics Department Faculty Publication Series

In this paper, we examine in detail the principal branches of solutions that arise in vector discrete models with nonlinear intercomponent coupling and four wave mixing. The relevant four branches of solutions consist of two single mode branches (transverse electric and transverse magnetic) and two mixed mode branches, involving both components (linearly polarized and elliptically polarized). These solutions are obtained explicitly and their stability is analyzed completely in the anticontinuum limit (where the nodes of the lattice are uncoupled), illustrating the supercritical pitchfork nature of the bifurcations that give rise to the latter two, respectively, from the former two. Then …

Statics And Dynamics Of An Inhomogeneously Nonlinear Lattice, D Machacek, E Foreman, Q Hoq, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We introduce an inhomogeneously nonlinear Schrödinger lattice, featuring a defocusing segment, a focusing segment and a transitional interface between the two. We illustrate that such inhomogeneous settings present vastly different dynamical behavior in the vicinity of the interface than the one expected in their homogeneous counterparts. We analyze the relevant stationary states, as well as their stability, by means of perturbation theory and linear stability analysis. We find good agreement with the numerical findings in the vicinity of the anticontinuum limit. For larger values of the coupling, we follow the relevant branches numerically and show that they terminate at values …

Nonlinearity Management In Optics: Experiment, Theory, And Simulation, Martin Centurion, Mason Porter, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We conduct an experimental investigation of nonlinearity management in optics using femtosecond pulses and layered Kerr media consisting of glass and air. By examining the propagation properties over several diffraction lengths, we show that wave collapse can be prevented. We corroborate these experimental results with numerical simulations of the (2+1)-dimensional focusing cubic nonlinear Schrödinger equation with piecewise constant coefficients and a theoretical analysis of this setting using a moment method.

Modulational Instability In A Layered Kerr Medium: Theory And Experiment, Martin Centurion, Mason A. Porter, Ye Pu, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We present the first experimental investigation of modulational instability in a layered Kerr medium. The particularly interesting and appealing feature of our configuration, consisting of alternating glass-air layers, is the piecewise-constant nature of the material properties, which allows a theoretical linear stability analysis leading to a Kronig-Penney equation whose forbidden bands correspond to the modulationally unstable regimes. We find very good quantitative agreement between theoretical, numerical, and experimental diagnostics of the modulational instability. Because of the periodicity in the evolution variable arising from the layered medium, there are multiple instability regions rather than just one as in a uniform medium.

Fractional-Period Excitations In Continuum Periodic Systems, H E. Nistazakis, Mason A. Porter, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We investigate the generation of fractional-period states in continuum periodic systems. As an example, we consider a Bose-Einstein condensate confined in an optical-lattice potential. We show that when the potential is turned on nonadiabatically, the system explores a number of transient states whose periodicity is a fraction of that of the lattice. We illustrate the origin of fractional-period states analytically by treating them as resonant states of a parametrically forced Duffing oscillator and discuss their transient nature and potential observability.

Robert Macpherson And Arithmetic Groups, Pe Gunnells

Mathematics and Statistics Department Faculty Publication Series

We survey contributions of Robert MacPherson to the theory of arithmetic groups. There are two main areas we discuss: (i) explicit reduction theory for Siegel modular threefolds, and (ii) constructions of compactifications of locally symmetric spaces. The former is joint work with Mark McConnell, the latter with Lizhen Ji.

Geometry Of The Tetrahedron Space, E Babson, Pe Gunnells, R Scott

Mathematics and Statistics Department Faculty Publication Series

Let X be the space of all labeled tetrahedra in . In [E. Babson, P.E. Gunnells, R. Scott, A smooth space of tetrahedra, Adv. Math. 165(2) (2002) 285–312] we constructed a smooth symmetric compactification of X. In this article we show that the complement is a divisor with normal crossings, and we compute the cohomology ring .

Specializations Of One-Parameter Families Of Polynomials, F Hajir, S Wong

Mathematics and Statistics Department Faculty Publication Series

Let K be a number field, and suppose λ(x,t)∈K[x,t] is irreducible over K(t). Using algebraic geometry and group theory, we describe conditions under which the K-exceptional set of λ, i.e. the set of α∈K for which the specialized polynomial λ(x,α) is K-reducible, is finite. We give three applications of the methods we develop. First, we show that for any fixed n≥10, all but finitely many K-specializations of the degree n generalized Laguerre polynomial L n (t) (x) are K-irreducible and have Galois group S n . Second, we study specializations of the modular polynomial Φ n (x,t) (which vanishes on …

Exact Static Solutions For Discrete Φ4 Models Free Of The Peierls-Nabarro Barrier: Discretized First-Integral Approach, S V. Dmitriev, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

We propose a generalization of the discrete Klein-Gordon models free of the Peierls-Nabarro barrier derived in Spreight [Nonlinearity 12, 1373 (1999)] and Barashenkov et al. [Phys. Rev. E 72, 035602(R) (2005)], such that they support not only kinks but a one-parameter set of exact static solutions. These solutions can be obtained iteratively from a two-point nonlinear map whose role is played by the discretized first integral of the static Klein-Gordon field, as suggested by Dmitriev et al. [J. Phys. A 38, 7617 (2005)]. We then discuss some discrete ϕ4 models free of the Peierls-Nabarro barrier and identify for them the …

Equation-Free Dynamic Renormalization In A Glassy Compaction Model, L Chen, I G. Kevrekidis, Pg Kevrekidis

Mathematics and Statistics Department Faculty Publication Series

Combining dynamic renormalization with equation-free computational tools, we study the apparently asymptotically self-similar evolution of void distribution dynamics in the diffusion-deposition problem proposed by Stinchcombe and Depken [Phys. Rev. Lett. 88, 125701 (2002)]. We illustrate fixed point and dynamic approaches, forward as well as backward in time; these can be used to accelerate simulators of glassy dynamic phenomena.

On A Notion Of Maps Between Orbifolds Ii: Homotopy And Cw-Complex, Wm Chen

Mathematics and Statistics Department Faculty Publication Series

This is the second of a series of papers which is devoted to a comprehensive theory of maps between orbifolds. In this paper, we develop a basic machinery for studying homotopy classes of such maps. It contains two parts: (1) the construction of a set of algebraic invariants — the homotopy groups, and (2) an analog of CW-complex theory. As a corollary of this machinery, the classical Whitehead theorem (which asserts that a weak homotopy equivalence is a homotopy equivalence) is extended to the orbifold category.