Discretization of semilinear differential equations with an exponential dichotomy

AbstractA structural stability result for one-step discretizations of semilinear differential equations with an exponential dichotomy in its linear part is presented and interpreted as a shadowing result. Estimates are given in terms of the stepsize and of the order of the discretization method chosen

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

Wehrlites from continental mantle monitor the passage and degassing of carbonated melts

Continental rifting has been linked to the thinning and destruction of cratonic lithosphere and to the release of enough CO2 to impact the global climate. This fundamental plate tectonic process facilitates the infiltration and mobilisation of smallvolume carbonated melts, which may interact with mantle peridotite to form wehrlite through the reaction: enstatite thorn dolomite (melt) = forsterite thorn diopside thorn CO2 (vapour). Application to mantle xenolith suites from various rifts and basins shows that 2.9 to 10.2 kg CO2 are released per 100 kg of wehrlite formed. For the Eastern Rift (Africa), this results in estimated CO2 fluxes of 6.5 +/- 4.1 Mt yr(-1), similar to estimates of mantle contributions based on surficial CO2 surveys. Thus, wehrlite-bearing xenolith suites can be used to monitor present and past CO2 mobility through the continental lithosphere, ultimately with diffuse degassing to the atmosphere. They may also reveal the CO2 flux in lithospheric provinces where carbonated melts or continent-scale rifts are not observed at the surface.This work and collaborationwere stimulated by an invitation to SA and GMY to present at the Deep Carbon Observatory’s Deep Carbon 2019: Launching the Next Decade of Deep Carbon Science meeting in Washington DC (USA), and by an Alexander von Humboldt Fellowship to GMY, which we gratefully acknowledge. It was written while SA was funded through German Research Foundation fellowship AU356/11

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

Generalized transversality conditions for the Hahn quantum variational calculus

We prove optimality conditions for generalized quantum variational problems with a Lagrangian depending on the free end-points. Problems of calculus of variations of this type cannot be solved using the classical theory

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