Statistical mechanics of general discrete nonlinear Schr{\"o}dinger models: Localization transition and its relevance for Klein-Gordon lattices

We extend earlier work [Phys.Rev.Lett. 84, 3740 (2000)] on the statistical mechanics of the cubic one-dimensional discrete nonlinear Schrodinger (DNLS) equation to a more general class of models, including higher dimensionalities and nonlinearities of arbitrary degree. These extensions are physically motivated by the desire to describe situations with an excitation threshold for creation of localized excitations, as well as by recent work suggesting non-cubic DNLS models to describe Bose-Einstein condensates in deep optical lattices, taking into account the effective condensate dimensionality. Considering ensembles of initial conditions with given values of the two conserved quantities, norm and Hamiltonian, we calculate analytically the boundary of the 'normal' Gibbsian regime corresponding to infinite temperature, and perform numerical simulations to illuminate the nature of the localization dynamics outside this regime for various cases. Furthermore, we show quantitatively how this DNLS localization transition manifests itself for small-amplitude oscillations in generic Klein-Gordon lattices of weakly coupled anharmonic oscillators (in which energy is the only conserved quantity), and determine conditions for existence of persistent energy localization over large time scales.Comment: to be published in Physical Review

Energetically stable singular vortex cores in an atomic spin-1 Bose-Einstein condensate

We analyze the structure and stability of singular singly quantized vortices in a rotating spin-1 Bose-Einstein condensate. We show that the singular vortex can be energetically stable in both the ferromagnetic and polar phases despite the existence of a lower-energy nonsingular coreless vortex in the ferromagnetic phase. The spin-1 system exhibits energetic hierarchy of length scales resulting from different interaction strengths and we find that the vortex cores deform to a larger size determined by the characteristic length scale of the spin-dependent interaction. We show that in the ferromagnetic phase the resulting stable core structure, despite apparent complexity, can be identified as a single polar core with everywhere nonvanishing axially symmetric density profile. In the polar phase, the energetically favored core deformation leads to a splitting of a singly quantized vortex into a pair of half-quantum vortices that preserves the topology of the vortex outside the extended core region, but breaks the axial symmetry of the core. The resulting half-quantum vortices exhibit nonvanishing ferromagnetic cores.<br/

Culturally Responsive Teaching: Implications for Educational Justice

Educational justice is a major global challenge. In most underdeveloped countries, many students do not have access to education and in most advanced democracies, school attainment and success are still, to a large extent, dependent on a student’s social background. However, it has often been argued that social justice is an essential part of teachers’ work in a democracy. This article raises an important overriding question: how can we realize the goal of educational justice in the field of teaching? In this essay, I examine culturally responsive teaching as an educational practice and conclude that it is possible to realize educational justice in the field of teaching because in its true implementation, culturally responsive teaching conceptualizes the connection between education and social justice and creates the space needed for discussing social change in society

A Verified Certificate Checker for Finite-Precision Error Bounds in Coq and HOL4

Being able to soundly estimate roundoff errors of finite-precision computations is important for many applications in embedded systems and scientific computing. Due to the discrepancy between continuous reals and discrete finite-precision values, automated static analysis tools are highly valuable to estimate roundoff errors. The results, however, are only as correct as the implementations of the static analysis tools. This paper presents a formally verified and modular tool which fully automatically checks the correctness of finite-precision roundoff error bounds encoded in a certificate. We present implementations of certificate generation and checking for both Coq and HOL4 and evaluate it on a number of examples from the literature. The experiments use both in-logic evaluation of Coq and HOL4, and execution of extracted code outside of the logics: we benchmark Coq extracted unverified OCaml code and a CakeML-generated verified binary

Existence and Stability of Non-Trivial Scalar Field Configurations in Orbifolded Extra Dimensions

We consider the existence and stability of static configurations of a scalar field in a five dimensional spacetime in which the extra spatial dimension is compactified on an $S^1/Z_2$ orbifold. For a wide class of potentials with multiple minima there exist a finite number of such configurations, with total number depending on the size of the orbifold interval. However, a Sturm-Liouville stability analysis demonstrates that all such configurations with nodes in the interval are unstable. Nodeless static solutions, of which there may be more than one for a given potential, are far more interesting, and we present and prove a powerful general criterion that allows a simple determination of which of these nodeless solutions are stable. We demonstrate our general results by specializing to a number of specific examples, one of which may be analyzed entirely analytically.Comment: 23 pages, 7 figures, references added, factor of two corrected in kink energy definition, submitted to PR

Models for energy and charge transport and storage in biomolecules

Two models for energy and charge transport and storage in biomolecules are considered. A model based on the discrete nonlinear Schrodinger equation with long-range dispersive interactions (LRI's) between base pairs of DNA is offered for the description of nonlinear dynamics of the DNA molecule. We show that LRI's are responsible for the existence of an interval of bistability where two stable stationary states, a narrow, pinned state and a broad, mobile state, coexist at each value of the total energy. The possibility of controlled switching between pinned and mobile states is demonstrated. The mechanism could be important for controlling energy storage and transport in DNA molecules. Another model is offered for the description of nonlinear excitations in proteins and other anharmonic biomolecules. We show that in the highly anharmonic systems a bound state of Davydov and Boussinesq solitons can exist.Comment: 12 pages (latex), 12 figures (ps