A note on a matrix version of the Farkas lemma

A linear polyomial non-negative on the non-negativity domain of finitely many linear polynomials can be expressed as their non-negative linear combination. Recently, under several additional assumptions, Helton, Klep, and McCullough extended this result to matrix polynomials. The aim of this paper is to study which of these additional assumptions are really necessary.Comment: 10 page

Judicial Checklist of Country of Origin Information and Due Process in the Light of EU (Fundamental) Law

The author identifies an important trend in deliberation of international standards of good practice in protection of refugees based on ad hoc networking and cooperation between judges and other experts under the auspices of the IARLJ. Since the legal questions in status determination procedures under EU law very often fall under the competence of the Court of Justice of the EU (CJEU), the author suggests that stakeholders in those projects must respect the constitutional structure of the EU by considering what kind of approach towards the standards of due process can be expected to be introduced by the CJEU. This is also relevant for the ongoing project of the IARLJ on the use of country-of-origin information and due process. While the minimum standards clauses under EU secondary law give a clear direction to the IARLJ's checklist as to where to build on a value added, it remains highly unclear what is the interpretative relevance of the Charter of the Fundamental Rights of the EU (the Charter) and the right to effective judicial protection. The author's short analysis of the recent case-law of the CJEU identifies two major, though different, approaches towards legal interpretation of the EU law based on the Charter. A combination of both – top-down and bottom-up - approaches puts the Charter into a specific perspective, where one cannot be expected to act under the legal circumstances of having EU constitutional law as a starting point, with the Charter being always placed at the top of legal argumentation. This will very probably also affect the “duty of cooperation” laid down in the Article 4(1) of the Qualifications Directive in relation to the use of country-of-origin information. In the conclusion, the author suggests that with its checklist on “best international practice”, the IARLJ might considerably improve legal uncertainties and minimum standards based on EU (fundamental) law

Non-negative spectral measures and representations of C*-algebras

Regular normalized W-valued spectral measures on a compact Hausdorff space X are in one-to-one correspondence with unital *-representations \rho:C(X)\to W, where W stands for a von Neumann algebra. In this paper we show that for every compact Hausdorff space X and every von Neumann algebras W_1,W_2 there is a one-to-one correspondence between unital *-representations \rho:C(X,W_1)\to W_2 and special B(W_1,W_2)-valued measures on X that we call non-negative spectral measures. Such measures are special cases of non-negative measures that we introduced in our previous paper in connection with moment problems for operator polynomials.Comment: 20 page

Depth Profile of Optically Recorded Patterns in Light-Sensitive Liquid Crystal Elastomers

We investigated nonlinear absorption and photobleaching processes in a liquid crystal elastomer (LCE) doped with light-sensitive azobenzene moiety. A conventional one-dimensional holographic grating was recorded in the material with the use of two crossed UV laser beams and the angular dependence of the diffraction efficiency in the vicinity of the Bragg peak was analyzed. These measurements gave information on the depth to which trans to cis isomerisation had progressed into the sample as a function of the UV irradiation time. Using a numerical model that takes into account the propagation of writing beams and rate equations for the local concentration of the absorbing trans conformer, we computed the expected spatial distribution of the trans and cis conformers and the shape of the corresponding Bragg diffraction peak for different irradiation doses. Due to residual absorption of the cis conformers the depth of the recording progresses logarithmically with time and is limited by the thermal relaxation from the cis to trans conformation.Comment: 19 pages (incl. figs), 6 figure

Moment problems for operator polynomials

We extend Haviland's theorem on the integral representation of positive linear functionals on usual (real multivariate) polynomials to the integral representation of positive linear maps on operator polynomials mapping into the space of operators. We also extend its special case of compact semialgebraic set studied by Schm\"udgen to matrix polynomials and show with a counterexample, that the extension to the operator polynomials fails in general

Liquid crystal films on curved surfaces: An entropic sampling study

The confining effect of a spherical substrate inducing anchoring (normal to the surface) of rod-like liquid crystal molecules contained in a thin film spread over it has been investigated with regard to possible changes in the nature of the isotropic-to-nematic phase transition as the sample is cooled. The focus of these Monte Carlo simulations is to study the competing effects of the homeotropic anchoring due to the surface inducing orientational ordering in the radial direction and the inherent uniaxial order promoted by the intermolecular interactions. By adopting entropic sampling procedure, we could investigate this transition with a high temperature precision, and we studied the effect of the surface anchoring strength on the phase diagram for a specifically chosen geometry. We find that there is a threshold anchoring strength of the surface below which uniaxial nematic phase results, and above which the isotropic fluid cools to a radially ordered nematic phase, besides of course expected changes in the phase transition temperature with the anchoring strength. In the vicinity of the threshold anchoring strength we observe a bistable region between these two structures, clearly brought out by the characteristics of the corresponding microstates constituting the entropic ensemble.Comment: 14 pages, 5 figure