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.     

Reference-wave solutions for high-frequency fields propagating in random media

Ray theory plays an important role in determining the propagation properties of high-frequency fields and their statistical measures in complicated random environments. According to the ray approach, the field at the observer can be synthesized from a variety of field species arriving along multiple ray trajectories resulting from refraction and scattering from boundaries and from scattering centers embedded in the random medium. For computations of the statistical measures, it is desirable therefore to possess a solution for the high-frequency field propagating along an isolated ray trajectory. For this reason, a new reference-wave method was developed to provide an analytic solution of the parabolic-wave equation.     

Scale-free networks on lattices

We suggest a method for embedding scale-free networks, with degree distribution Pk approximately k(-lambda), in regular Euclidean lattices accounting for geographical properties. The embedding is driven by a natural constraint of minimization of the total length of the links in the system. We find that all networks with lambda>2 can be successfully embedded up to a (Euclidean) distance xi which can be made as large as desired upon the changing of an external parameter. Clusters of successive chemical shells are found to be compact (the fractal dimension is df=d), while the dimension of the shortest path between any two sites is smaller than 1: dmin=(lambda-2)/(lambda-1-1/d), contrary to all other known examples of fractals and disordered lattices.     

Surface plasmon sensor based on the enhanced light transmission through arrays of nanoholes in gold films

Arrays of nanoholes in a gold film were used to monitor the binding of organic and biological molecules to the metallic surface. This technique is particularly sensitive to surface binding events because it is based upon the resonant surface plasmon enhanced transmission through the array of nanoholes. The sensitivity was found to be 400 nm per refractive index unit, which is comparable to other grating-based surface plasmon resonance (SPR) devices. The array of nanoholes is well suited for dense integration in a sensor chip. Furthermore, the optical geometry is collinear, which simplifies the alignment with respect to the traditional Kretschmann (reflection) arrangement for SPR sensing.     

Spectral effects in cyclobutane derivatives : The bathochromic-hypochromic shifts in compounds related to verbenone

The bathochromic-hypochromic spectral shifts, observed in bicyclic enones that contain a 4- or a 3-membered ring, are discussed in molecular-mechanical and molecular-orbital terms. It is proposed that the effect, not only by the cyclopropane but also by the cyclobutane ring, is due to extension of the CCCO chromophore, and not to its mere distortion from “normal” geometry. The high-energy absorption of umbellulone (3), 220 nm in ethanol, is characterized as a second π-π^*. 2-Methylbicyclo[3.2.1]oct-2-en-4-one (7) has been synthesized, and its preparation and spectrum are described.     

Linear combination of hybrid orbitals: Acyclic alkanes

A LCHO SCF-CI semiempirical scheme is formulated for compounds of carbon and hydrogen, and far-UV transition energies computed for a number of acyclic alkanes. The type of spectral bands (CH CH*, etc.) and symmetries of subbands are assigned, and their dependence on conformation and branching discussed. It is argued that, for pentanes and higher alkanes, spectral quantities can be estimated from data on the lower alkanes.Multi-Conformational Compounds, part X. For part IX, see Ref. [1].     

Visibility and parallelotopes in projective space

It is shown that if is a finite set of points in an ordered d-dimensional projective space, d2 and if all the hyperplanes spanned by points of are considered, then every such a hyperplane is visible from at most 2^d distinct points of ; if a hyperplane bounds 2^d distinct residences, then it contains exactly d points of . These results have been proved previously for d=2 and d=3.This paper is a revised version of results presented at the Conference on Geometry held at the University of Haifa (Israel), March. 10–13 (1975), [6].     

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.     

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.     

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.