Integrability of maximal functions and Riesz potentials in Orlicz spaces of variable exponent

Our aim in this paper is to deal with Sobolev's type inequality, Hardy's type inequality and Trudinger's inequality for Riesz potentials of functions in Orlicz spaces of variable exponent. These results are based on the boundedness of maximal operators and so-called Hedberg's trick. Our methods can also be applied to the case of constant exponents with slight modifications.     

Menke points on the real line and their connection to classical orthogonal polynomials

We investigate the properties of extremal point systems on the real line consisting of two interlaced sets of points solving a modified minimum energy problem. We show that these extremal points for the intervals [−1,1], [0,∞) and (−∞,∞), which are analogues of Menke points for a closed curve, are related to the zeros and extrema of classical orthogonal polynomials. Use of external fields in the form of suitable weight functions instead of constraints motivates the study of “weighted Menke points” on [0,∞) and (−∞,∞). We also discuss the asymptotic behavior of the Lebesgue constant for the Menke points on [−1,1].     

Transformation from the nonautonomous to standard NLS equations

In this paper we show a systematical method to obtain exact solutions of the nonautonomous nonlinear Schrödinger (NLS) equation. An integrable condition is first obtained by the Painlevé analysis, which is shown to be consistent with that obtained by the Lax pair method. Under this condition, we present a general transformation, which can directly convert all allowed exact solutions of the standard NLS equation into the corresponding exact solutions of the nonautonomous NLS equation. The method is quite powerful since the standard NLS equation has been well studied in the past decades and its exact solutions are vast in the literature. The result provides an effective way to control the soliton dynamics. Finally, the fundamental bright and dark solitons are taken as examples to demonstrate its explicit applications.     

Influence of global features of a Bose-Einstein condensate on the vortex velocity

We study the way in which the geometry of the trapping potential affects the vortex velocity in a Bose-Einstein condensate confined by a toroidal trap. We calculate the vortex precession velocity through a simple relationship between such a velocity and the gradient of the numerically obtained vortex energy. We observe that our results correspond very closely to the velocity calculated through time evolution simulations. However, we find that the estimates derived from available velocity field formulas present appreciable differences. To resolve such discrepancies, we further study the induced velocity field, analyzing the effect of global features of the condensate on such a field and on the precession velocity.     

Optimized auxiliary basis sets for density fitted post-Hartree–Fock calculations of lanthanide containing molecules

Optimised auxiliary basis sets for lanthanide atoms (Ce to Lu) for four basis sets of the Karlsruhe error-balanced segmented contracted def2 - series (SVP, TZVP, TZVPP and QZVPP) are reported. These auxiliary basis sets enable the use of the resolution-of-the-identity (RI) approximation in post Hartree–Fock methods – as for example, second-order perturbation theory (MP2) and coupled cluster (CC) theory. The auxiliary basis sets are tested on an enlarged set of about a hundred molecules where the test criterion is the size of the RI error in MP2 calculations. Our tests also show that the same auxiliary basis sets can be used together with different effective core potentials. With these auxiliary basis set calculations of MP2 and CC quality can now be performed efficiently on medium-sized molecules containing lanthanides.     

The damped wave equation with unbounded damping

We analyze new phenomena arising in linear damped wave equations on unbounded domains when the damping is allowed to become unbounded at infinity. We prove the generation of a contraction semigroup, study the relation between the spectra of the semigroup generator and the associated quadratic operator function, the convergence of non-real eigenvalues in the asymptotic regime of diverging damping on a subdomain, and we investigate the appearance of essential spectrum on the negative real axis. We further show that the presence of the latter prevents exponential estimates for the semigroup and turns out to be a robust effect that cannot be easily canceled by adding a positive potential. These analytic results are illustrated by examples.     

Regularizing properties of space-periodic layer heat potentials and applications to boundary value problems in periodic domains

We introduce space-periodic layer heat potentials and we prove some regularizing properties in parabolic Schauder spaces defined on the boundary of infinite parabolic cylinders. Then, we show how to exploit these mapping properties for the space-periodic layer potentials in order to solve two initial-boundary value problems for the heat equation in an unbounded periodic domain.     

Positive and negative ions of the amino acid histidine formed in low‐energy electron collisions

Histidine is an aromatic amino acid crucial for the biological functioning of proteins and enzymes. When biological matter is exposed to ionising radiation, highly energetic particles interact with the surrounding tissue which leads to efficient formation of low‐energy electrons. In the present study, the interaction of low‐energy electrons with gas‐phase histidine is studied at a molecular level in order to extend the knowledge of electron‐induced reactions with amino acids. We report both on the formation of positive ions formed by electron ionisation and negative ions induced by electron attachment. The experimental data were complemented by quantum chemical calculations. Specifically, the free energies for possible fragmentation reactions were derived for the τ and the π tautomer of histidine to get insight into the structures of the formed ions and the corresponding neutrals. We report the experimental ionisation energy of (8.48 ± 0.03) eV for histidine which is in good agreement with the calculated vertical ionisation energy. In the case of negative ions, the dehydrogenated parent anion is the anion with the highest mass observed upon dissociative electron attachment. The comparison of experimental and computational results was also performed in view of a possible thermal decomposition of histidine during the experiments, since the sample was sublimated in the experiment by resistive heating of an oven. Overall, the present study demonstrates the effects of electrons as secondary particles in the chemical degradation of histidine. The reactions induced by those electrons differ when comparing positive and negative ion formation. While for negative ions, simple bond cleav ages prevail, the observed fragment cations exhibit partly restructuring of the molecule during the dissociation process.     

Stabilization of two-dimensional solitons and vortices against supercritical collapse by lattice potentials

It is known that optical-lattice (OL) potentials can stabilize solitons and solitary vortices against the critical collapse, generated by cubic attractive nonlinearity in the 2D geometry. We demonstrate that OLs can also stabilize various species of fundamental and vortical solitons against the supercritical collapse, driven by the double-attractive cubic-quintic nonlinearity (however, solitons remain unstable in the case of the pure quintic nonlinearity). Two types of OLs are considered, producing similar results: the 2D Kronig-Penney “checkerboard”, and the sinusoidal potential. Soliton families are obtained by means of a variational approximation, and as numerical solutions. The stability of all families, which include fundamental and multi-humped solitons, vortices of oblique and straight types, vortices built of quadrupoles, and supervortices, strictly obeys the Vakhitov-Kolokolov criterion. The model applies to optical media and BEC in “pancake” traps.     

Ab-initio effective potentials in crystal Hartree-Fock theory

Ab initio effective potential technique of Ewiget al is applicable directly to crystal Hartree-Fock formalism provided the effective potentials are suitably defined. Corresponding to every atom or every molecule by constructing a crystal one can assign a unit species configuration. The effective potentials in the crystal can be expressed as functions of lattice parameters in terms of integrals over the orbitals of this unit species. These expressions are in a more exact form than those usually employed in molecular calculations.