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

Doživljaj narodne glazbe kod djece

Through everyday exposure to language and music, individuals within a nation become sensitive to the melodic and rhythmical structure of their folk musical culture. It represents improvisational abilities of individuals and groups as well. Despite all changes, it indisputably maintains all characteristics of music parameters as inheritance of past ages. Due to its social role, it appears throughout everyone’s life – and it also represents an important part of children’s life. In the study that was carried out with two groups of children between six and nine years of age, we were interested in determining how children experience music making with elements of folk music and how it is possible to create the circumstances which can provide the spontaneity of folk music within the structured environment (like primary school). The research was designed as a phenomenological case study. This method allowed us to gather data which provided a deeper insight into the ways in which participants are able to play using elements of folk music and the way they feel while using such material. The results show that, contrary to the basic fact of spontaneity in folk music, 6 and 7 year old participants were not able to use music parameters to play with and had yet to learn how symbolic play works out in musical language on the basis of communication. The most natural way to bridge the gap between “learning” songs and experiencing individual musical expression in a manner of folk music in children seems to be creative work with lyrics in Slovene language. We also found that children develop social competences of a great value, when they are involved in symbolic play with folk music elements in the improvisational mode.Velik je dio kulture svakog naroda narodna glazba koja predstavlja improvizacijske sposobnosti pojedinaca i grupa. Unatoč svim promjenama, ona nesporno održava sve karakteristike glazbenih parametara kao nasljeđivanje prošlih vremena. U studiji koja je provedena sa skupinama djece u dobi od šest do osam godina, zanimalo nas je kako se dječja igra može oblikovati folklorno-glazbenim elementima. Sposobnosti grupnoga glazbenog stvaralaštva proučavane su kroz aktivnosti holističkoga glazbenog obrazovanja, što uključuje – baš kao i dječju narodnu glazbu – pokret, pjevanje, sviranje dječjih instrumenata i aktivnosti kreativnoga rada s različitim glazbenim elementima. Istraživanje je osmišljeno kao fenomenološka studija. Ta nam je metoda omogućila prikupljanje podataka koji su pružili dublji uvid u način na koji se sudionici osjećaju dok koriste elemente narodne glazbe. Rezultati pokazuju da se djeca, u suprotnosti s osnovnom činjenicom spontanosti u narodnoj glazbi, tek trebaju naučiti igrati s elementima narodne glazbe u glazbenom jeziku. Ritam se činio najjasnijim glazbenim parametrom, unutar kojeg su sudionici prvi put osjetili velik improvizacijski potencijal. Također smo otkrili da u igranju s elementima narodne glazbe postoji velika vrijednost razvoja socijalnih kompetencija

The truncated moment problem on curves y=q(x)y=q(x) and yxℓ=1yx^\ell=1

In this paper we study the bivariate truncated moment problem (TMP) on curves of the form $y=q(x)$, $q(x)\in \mathbb{R}[x]$, $\text{deg } q\geq 3$, and $yx^\ell=1$, $\ell\in \mathbb{N}\setminus\{1\}$. For even degree sequences the solution based on the number of moment matrix extensions was first given by Fialkow using the truncated Riesz-Haviland theorem and a sum-of-squares representations for polynomials, strictly positive on such curves. Namely, the upper bound on this number is quadratic in the degrees of the sequence and the polynomial determining a curve. We use a reduction to the univariate setting technique and improve Fialkow's bound to $\text{deg }q-1$ (resp. $\ell+1$) for curves $y=q(x)$ (resp. $yx^\ell=1$). This in turn gives analogous improvements of the degrees in the sum-of-squares representations referred to above. Moreover, we get the upper bounds on the number of atoms in the minimal representing measure, which are $k\;\text{deg }q$ (resp. $k(\ell+1)$) for curves $y=q(x)$ (resp. $yx^\ell=1$) for even degree sequences, while for odd ones they are $k\;\text{deg }q-\big\lceil\frac{\text{deg }q}{2} \big\rceil$ (resp. $k(\ell+1)-\big\lfloor\frac{\ell}{2} \big\rfloor+1$) for curves $y=q(x)$ (resp. $yx^\ell=1$). In the even case these are counterparts to the result by Riener and Schweighofer, which gives the same bound for odd degree sequences on all plane curves, while in the odd case it is a slight improvement of their bound in these special cases. Further on, we give another solution to the TMP on the curves studied based on the feasibility of a linear matrix inequality, corresponding to the univariate sequence obtained, and finally we solve concretely odd degree cases of the TMP on curves $y=x^\ell$, $\ell=2,3$, and add a new solvability condition to the even degree case on the curve $y=x^2$.Comment: 40 pages; To appear in Linear and Multilinear Algebr

The truncated Hamburger moment problems with gaps in the index set

In this article we solve four special cases of the truncated Hamburger moment problem (THMP) of degree $2k$ with one or two missing moments in the sequence. As corollaries we obtain, by using appropriate substitutions, the solutions to bivariate truncated moment problems of degree $2k$ for special curves. Namely, for the curves $y=x^3$ (first solved by Fialkow), $y^2=x^3$, $y=x^4$ where a certain moment of degree $2k+1$ is known and $y^3=x^4$ with a certain moment given. The main technique is the completion of the partial positive semidefinite matrix (ppsd) such that the conditions of Curto and Fialkow's solution of the THMP are satisfied. The main tools are the use of the properties of positive semidefinite Hankel matrices and a result on all completions of a ppsd matrix with one unknown entry, proved by the use of the Schur complements for $2\times 2$ and $3\times 3$ block matrices.Comment: 25 page