Steady-state vortex-screening solitons formed in biased photorefractive media

We report the observation of steady-state photorefractive vortex-screening solitons. As a singly charged circular vortex nested on a broad beam propagates through a biased strontium barium niobate crystal, it self-traps in both transverse dimensions despite the inherent anisotropy of the photorefractive nonlinearity. When the vortex beam is a doughnut-shaped narrow beam, it breaks up into two elongated slices (with a self-defocusing nonlinearity) or into two focused filaments (with a self-focusing nonlinearity). We demonstrate the optical guidance of a probe beam in a circular waveguide induced by the self-trapped vortex.     

Lyapunov Exponent and Density of States of a One-Dimensional Non-Hermitian Schrödinger Equation

We calculate, using numerical methods, the Lyapunov exponent (E) and the density of states (E) at energy E of a one-dimensional non-Hermitian Schrödinger equation with off-diagonal disorder. For the particular case we consider, both (E) and (E) depend only on the modulus of E. We find a pronounced maximum of (|E|) at energy E=2/ , which seems to be linked to the fixed point structure of an associated random map. We show how the density of states (E) can be expanded in powers of E. We find (|E|)=(1/ ²)+(4/3 ³) |E|²+. This expansion, which seems to be asymptotic, can be carried out to an arbitrarily high order.     

Self-similarity and fractals in soliton-supporting systems

We describe a principle that can be used to generate self-similarity and fractals in almost any nonlinear system in nature that supports solitons, given that some proper nonadiabatic conditions are met. We illustrate our idea on a particular optics example that also theoretically demonstrates fractals in nonlinear optics.     

Effective chiral magnetic currents, topological magnetic charges, and microwave vortices in a cavity with an enclosed ferrite disk

In microwaves, a TE-polarized rectangular-waveguide resonator with an inserted thin ferrite disk gives an example of a nonintegrable system. The interplay of reflection and transmission at the disk interfaces together with the material gyrotropy effect gives rise to whirlpool-like electromagnetic vortices in the proximity of the ferromagnetic resonance. Based on numerical simulation, we show that a character of microwave vortices in a cavity can be analyzed by means of consideration of equivalent magnetic currents. Maxwell equations allows introduction of a magnetic current as a source of the electromagnetic field. Specifically, we found that in such nonintegrable structures, magnetic gyrotropy and geometrical factors leads to the effect of symmetry breaking resulting in effective chiral magnetic currents and topological magnetic charges. As an intriguing fact, one can observe precessing behavior of the electric-dipole polarization inside a ferrite disk.     

Observation of soliton tunneling phenomena and soliton ejection

We study, theoretically and experimentally, the nonlinear dynamics of a wave packet launched inside a trap potential. Increasing the power of the wave packet transforms its dynamics from linear tunneling through a potential barrier, to soliton tunneling, and eventually, above a well-defined threshold, to the ejection of a soliton from the potential trap.     

Continuum Dynamics on Manifolds: Application to Elasticity of Residually-Stressed Bodies

This paper is concerned with the dynamics of continua on differentiable manifolds. We present a covariant derivation of the equations of motion, viewing motion as a curve in an infinite-dimensional Banach manifold of embeddings of a body manifold in a space manifold. Our main application is the motion of residually-stressed elastic bodies, where the residual stresses result from a geometric incompatibility between body and space manifolds. We then study a particular example of elastic vibrations of a two-dimensional curved annulus embedded in a sphere.     

A Tutorial on the Cross-Entropy Method

The cross-entropy (CE) method is a new generic approach to combinatorial and multi-extremal optimization and rare event simulation. The purpose of this tutorial is to give a gentle introduction to the CE method. We present the CE methodology, the basic algorithm and its modifications, and discuss applications in combinatorial optimization and machine learning.     

Crossed modules as homotopy normal maps

In this note we consider crossed modules of groups (N→G, G→Aut(N)), as a homotopy version of the inclusion N⊂G of a normal subgroup. Our main observation is a characterization of the underlying map N→G of a crossed module in terms of a simplicial group structure on the associated bar construction. This approach allows for “natural” generalizations to other monoidal categories, in particular we consider briefly what we call “normal maps” between simplicial groups.     

Overcoming intuitive interference in mathematics: insights from behavioral, brain imaging and intervention studies

It is well known that many students encounter difficulties when solving problems in mathematics. Research indicates that some of these difficulties may stem from intuitive interference with formal/logical reasoning. Our research aims at deepening the understanding of these difficulties and their underlying reasoning mechanisms to help students overcome them. For this purpose we carried out behavioral, brain imaging and intervention studies focusing on a previously demonstrated obstacle in mathematics education. The literature reports that many students believe that shapes with a larger area must have a larger perimeter. We measured the accuracy of responses, reaction time, and neural correlates (by fMRI) while participants compared the perimeters of geometrical shapes in two conditions: (1) congruent, in which correct response was in line with intuitive reasoning (larger area–larger perimeter) and (2) incongruent, in which the correct answer was counterintuitive. In the incongruent condition, accuracy dropped and reaction time for correct responses was longer than in the congruent condition. The congruent condition activated bilateral parietal brain areas, known to be involved in the comparison of quantities, while correctly answering the incongruent condition activated bilateral prefrontal areas known for their executive control over other brain regions. The intervention, during which students’ attention was drawn to the relevant variable, increased accuracy in the incongruent condition, while reaction times increased in both congruent and incongruent conditions. The findings of the three studies point to the importance of control mechanisms in overcoming intuitive interference in mathematics. Overall, it appears that adding a cognitive neuroscience perspective to mathematics education research can contribute to a better understanding of students’ difficulties and reasoning processes. Such information is important for educational research and practice.