The Natural Logarithm on Time Scales

We define an appropriate logarithm function on time scales and present its main properties. This gives answer to a question posed by M. Bohner in [J. Difference Equ. Appl. {\bf 11} (2005), no. 15, 1305--1306].Comment: 6 page

Delta-Nabla Optimal Control Problems

We present a unified treatment to control problems on an arbitrary time scale by introducing the study of forward-backward optimal control problems. Necessary optimality conditions for delta-nabla isoperimetric problems are proved, and previous results in the literature obtained as particular cases. As an application of the results of the paper we give necessary and sufficient Pareto optimality conditions for delta-nabla bi-objective optimal control problems.Comment: Preprint version of an article submitted 28-Nov-2009; revised 02-Jul-2010; accepted 20-Jul-2010; for publication in Journal of Vibration and Contro

Exponential localization in one-dimensional quasiperiodic optical lattices

We investigate the localization properties of a one-dimensional bichromatic optical lattice in the tight binding regime, by discussing how exponentially localized states emerge upon changing the degree of commensurability. We also review the mapping onto the discrete Aubry-Andre' model, and provide evidences on how the momentum distribution gets modified in the crossover from extended to exponentially localized states. This analysis is relevant to the recent experiment on Anderson localization of a noninteracting Bose-Einstein condensate in a quasiperiodic optical lattice [G. Roati et al., Nature 453, 895 (2008)].Comment: 13 pages, 6 figure

R-matrix approach to integrable systems on time scales

A general unifying framework for integrable soliton-like systems on time scales is introduced. The $R$-matrix formalism is applied to the algebra of $\delta$-differential operators in terms of which one can construct infinite hierarchy of commuting vector fields. The theory is illustrated by two infinite-field integrable hierarchies on time scales which are difference counterparts of KP and mKP. The difference counterparts of AKNS and Kaup-Broer soliton systems are constructed as related finite-field restrictions.Comment: 21 page

Correlation function of weakly interacting bosons in a disordered lattice

One of the most important issues in disordered systems is the interplay of the disorder and repulsive interactions. Several recent experimental advances on this topic have been made with ultracold atoms, in particular the observation of Anderson localization, and the realization of the disordered Bose-Hubbard model. There are however still questions as to how to differentiate the complex insulating phases resulting from this interplay, and how to measure the size of the superfluid fragments that these phases entail. It has been suggested that the correlation function of such a system can give new insights, but so far little experimental investigation has been performed. Here, we show the first experimental analysis of the correlation function for a weakly interacting, bosonic system in a quasiperiodic lattice. We observe an increase in the correlation length as well as a change in shape of the correlation function in the delocalization crossover from Anderson glass to coherent, extended state. In between, the experiment indicates the formation of progressively larger coherent fragments, consistent with a fragmented BEC, or Bose glass.Comment: 16 pages, 8 figure

Euler-Lagrange equations for composition functionals in calculus of variations on time scales

In this paper we consider the problem of the calculus of variations for a functional which is the composition of a certain scalar function $H$ with the delta integral of a vector valued field $f$, i.e., of the form $H(\int_{a}^{b}f(t,x^{\sigma}(t),x^{\Delta}(t))\Delta t)$. Euler-Lagrange equations, natural boundary conditions for such problems as well as a necessary optimality condition for isoperimetric problems, on a general time scale, are given. A number of corollaries are obtained, and several examples illustrating the new results are discussed in detail.Comment: Submitted 10-May-2009 to Discrete and Continuous Dynamical Systems (DCDS-B); revised 10-March-2010; accepted 04-July-201

Effect of uniaxial stress on ferroelectric behavior of (Bi1/2Na1/2)TiO3-based lead-free piezoelectric ceramics

Prior studies have shown that a field-induced ferroelectricity in ceramics with general chemical formula (1-x-y) (Bi1/2 Na1/2) TiO3 -x BaTiO3 -y (K0.5 Na0.5) NbO3 and a very low remanent strain can produce very large piezoelectric strains. Here we show that both the longitudinal and transverse strains gradually change with applied electric fields even during the transition from the nonferroelectric to the ferroelectric state, in contrast to known Pb-containing antiferroelectrics. Hence, the volume change and, in turn, the phase transition can be affected using uniaxial compressive stresses, and the effect on ferroelectricity can thus be assessed. It is found that the 0.94 (Bi1/2 Na1/2) TiO3 -0.05 BaTiO3 -0.01 (K0.5 Na0.5) NbO3 ceramic (largely ferroelectric), with a rhombohedral R3c symmetry, displays large ferroelectric domains, significant ferroelastic deformation, and large remanent electrical polarizations even at a 250 MPa compressive stress. In comparison, the 0.91 (Bi1/2 Na1/2) TiO3 -0.07 BaTiO3 -0.02 (K0.5 Na0.5) NbO3 ceramic (largely nonferroelectric) possesses characteristics of a relaxor ferroelectric ceramic, including a pseudocubic structure, limited ferroelastic deformation, and low remanent polarization. The results are discussed with respect of the proposed antiferroelectric nature of the nonferroelectric state.open291

The Hahn Quantum Variational Calculus

We introduce the Hahn quantum variational calculus. Necessary and sufficient optimality conditions for the basic, isoperimetric, and Hahn quantum Lagrange problems, are studied. We also show the validity of Leitmann's direct method for the Hahn quantum variational calculus, and give explicit solutions to some concrete problems. To illustrate the results, we provide several examples and discuss a quantum version of the well known Ramsey model of economics.Comment: Submitted: 3/March/2010; 4th revision: 9/June/2010; accepted: 18/June/2010; for publication in Journal of Optimization Theory and Application

Domain switching energies: Mechanical versus electrical loading in La-doped bismuth ferrite-lead titanate

The mechanical stress-induced domain switching and energy dissipation in morphotropic phase boundary (1 - x)(Bi(1-y)La(y))FeO(3)-xPbTiO(3) during uniaxial compressive loading have been investigated at three different temperatures. The strain obtained was found to decrease with increasing lanthanum content, although a sharp increase in strain was observed for compositions doped with 7.5 and 10 at. % La. Increased domain switching was found in compositions with decreased tetragonality. This is discussed in terms of the competing influences of the amount of domain switching and the spontaneous strain on the macroscopic behavior under external fields. Comparison of the mechanically and electrically dissipated energy showed significant differences, discussed in terms of the different microscopic interactions of electric field and stress.open10