On Elliptical Billiards in the Lobachevsky Space and associated Geodesic Hierarchies

We derive Cayley's type conditions for periodical trajectories for the billiard within an ellipsoid in the Lobachevsky space. It appears that these new conditions are of the same form as those obtained before for the Euclidean case. We explain this coincidence by using theory of geodesically equivalent metrics and show that Lobachevsky and Euclidean elliptic billiards can be naturally considered as a part of a hierarchy of integrable elliptical billiards.Comment: 14 pages, to appear in Journal of Geometry and Physic

Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics

We study geometry of confocal quadrics in pseudo-Euclidean spaces of an arbitrary dimension $d$ and any signature, and related billiard dynamics. The goal is to give a complete description of periodic billiard trajectories within ellipsoids. The novelty of our approach is based on introduction of a new discrete combinatorial-geometric structure associated to a confocal pencil of quadrics, a colouring in $d$ colours, by which we decompose quadrics of $d+1$ geometric types of a pencil into new relativistic quadrics of $d$ relativistic types. Deep insight of related geometry and combinatorics comes from our study of what we call discriminat sets of tropical lines $\Sigma^+$ and $\Sigma^-$ and their singularities. All of that enable usto get an analytic criterion describing all periodic billiard trajectories, including the light-like ones as those of a special interest.Comment: 29 pages, 7 figure

New Examples of Systems of the Kowalevski Type

A new examples of integrable dynamical systems are constructed. An integration procedure leading to genus two theta-functions is presented. It is based on a recent notion of discriminantly separable polynomials. They have appeared in a recent reconsideration of the celebrated Kowalevski top, and their role here is analogue to the situation with the classical Kowalevski integration procedure.Comment: 17 page

Istraživačko-edukacijsko središte Prehrambeno-biotehnološkog fakulteta Sveučilišta u Zagrebu smješteno u Zadru

Godine 2006. osnovano je u okviru Prehrambeno-biotehnološkog fakulteta Sveučilišta u Zagrebu Međunarodno edukacijsko-istraživačko središte locirano u zgradi Fakulteta u Zadru. Za stavljanje u funkciju prostora koji je u to vrijeme bio devastiran i praktički neupotrebljiv osigurana su sredstva uz značajnu participaciju Grada Zadra i Zadarske županije čime je omogućena izgradnja i opremanje laboratorija. Od 2010. godine u Centru se kontinuirano i sustavno provode brojne znanstveno-istraživačke aktivnosti koje obuhvaćaju implementaciju različitih kategorija projekata financiranih iz nacionalnih izvora te najvećim dijelom iz fondova EU. Uspješnom implementacijom projekata ojačan je znanstveno istraživački kapacitet Centra te infrastruktura kroz nabavu sofisticirane tehnološke te analitičke opreme i laboratorijskih uređaja. Također, u Centru se provode i istraživanja vezana uz izradu završnih, diplomskih i doktorskih radova te surednja s gospodarstvom

Systems of Hess-Appel'rot Type and Zhukovskii Property

We start with a review of a class of systems with invariant relations, so called {\it systems of Hess--Appel'rot type} that generalizes the classical Hess--Appel'rot rigid body case. The systems of Hess-Appel'rot type carry an interesting combination of both integrable and non-integrable properties. Further, following integrable line, we study partial reductions and systems having what we call the {\it Zhukovskii property}: these are Hamiltonian systems with invariant relations, such that partially reduced systems are completely integrable. We prove that the Zhukovskii property is a quite general characteristic of systems of Hess-Appel'rote type. The partial reduction neglects the most interesting and challenging part of the dynamics of the systems of Hess-Appel'rot type - the non-integrable part, some analysis of which may be seen as a reconstruction problem. We show that an integrable system, the magnetic pendulum on the oriented Grassmannian $Gr^+(4,2)$ has natural interpretation within Zhukovskii property and it is equivalent to a partial reduction of certain system of Hess-Appel'rot type. We perform a classical and an algebro-geometric integration of the system, as an example of an isoholomorphic system. The paper presents a lot of examples of systems of Hess-Appel'rot type, giving an additional argument in favor of further study of this class of systems.Comment: 42 page

Determination of Phenolic Content and DPPH Radical Scavenging Activity of Functional Fruit Juices Fortified with Thymus serpyllum L. and Salvia officinalis L. Extracts

The objective of this study was to spectrophotometric determinate the total phenolic, flavonoid, hydroxycinnamic acid, and flavonol content of orange, pineapple, and apple juices fortified with wild thyme (Thymus serpyllum L.), Dalmatian sage (Salvia officinalis L.), and wild thyme-Dalmatian sage (3 : 1, v / v) extracts, and to evaluate their DPPH radical scavenging activity as a contribution to the development of a new functional beverage. The plant extracts addition increased the amount of phenolic compounds in fruit juices and improved their antioxidant properties. The highest concentrations of bioactive compounds and the greatest DPPH radical activity were obtained by adding Dalmatian sage extract to orange juice. Our study provides the novelty of fortifying fruit juices with wild thyme and Dalmatian sage extracts and offers significant potential for the creation of functional beverages

Systems of Hess-Appel'rot type

We construct higher-dimensional generalizations of the classical Hess-Appel'rot rigid body system. We give a Lax pair with a spectral parameter leading to an algebro-geometric integration of this new class of systems, which is closely related to the integration of the Lagrange bitop performed by us recently and uses Mumford relation for theta divisors of double unramified coverings. Based on the basic properties satisfied by such a class of systems related to bi-Poisson structure, quasi-homogeneity, and conditions on the Kowalevski exponents, we suggest an axiomatic approach leading to what we call the "class of systems of Hess-Appel'rot type".Comment: 40 pages. Comm. Math. Phys. (to appear