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

Billiard algebra, integrable line congruences, and double reflection nets

The billiard systems within quadrics, playing the role of discrete analogues of geodesics on ellipsoids, are incorporated into the theory of integrable quad-graphs. An initial observation is that the Six-pointed star theorem, as the operational consistency for the billiard algebra, is equivalent to an integrabilty condition of a line congruence. A new notion of the double-reflection nets as a subclass of dual Darboux nets associated with pencils of quadrics is introduced, basic properies and several examples are presented. Corresponding Yang-Baxter maps, associated with pencils of quadrics are defined and discussed.Comment: 18 pages, 8 figure

Uticaj promene načina korišćenja zemljišta na strukturu livadske crnice u zapadnoj Srbiji

Changes in land use can significantly affect aggregate distribution and water stability of structural aggregates. This study was conducted in the Kolubara River Valley, Western Serbia, to determine the effects of land use changes on composition and water stability of aggregates in humus horizons (0-30 cm) of noncarbonated Gleyic Fluvisols. This study was conducted at nine sites, where each site contained two adjacent land uses of natural grassland and arable land which underwent crop rotation for >100 years. Soil samples were taken from depths of 0-10, 10-20 and 20-30 cm for each land use. When the grassland was converted into arable land, the content of the agronomically most valuable aggregates (0.25-10 mm) of cultivated soils for a depth of 0-30 cm was significantly reduced by 22-40%, while the percentage of cloddy aggregates (>10 mm) increased by 41-68%, compared to grassland. In addition, the long-term arable soil had significantly (p lt 0.05) lower aggregate stability, determined by wet sieving, than grassland. The lowest aggregate stability was found in aggregates > 3 mm. Their content is ≈ 2.3 times lower in arable soil (12.6%) than in grassland (28.6%) at a depth of 0-10 cm. In addition, meanweight diameters of dry and wetstable aggregates and structure coefficient showed significant differences between land use at a depth of 0-30 cm. The results showed that the conversion of natural grassland to arable land in the lowland ecosystems of Western Serbia degraded aggregate distribution and stability.Istraživanja su sprovedena u dolini reke Kolubare da bi se utvrdili efekti promene načina korišćenja zemljišta na agregatni sastav i vodootpornost strukturnih agregata u humusnom horizontu (0-30 cm) beskarbonatne livadske crnice. Izabrano je devet lokacija koje su na bliskom rastojanju imale površine pod prirodnom livadskom vegetacijom i oranice koje su stvorene pre više od 100 godina razoravanjem prirodnih livada. Nakon razoravanja prirodnih livada i njihovog pretvaranja u oranice, sadržaj agronomski najpovoljnijih strukturnih agregata (prečnika 0,25-10 mm) u oranicama na dubini 0-30 cm, značajno je smanjen, za 22-40%, dok je sadržaj grudvastih agregata (>10 mm) povećan za 41-68 %, u poređenju sa livadom. Pored toga, višegodišnje oranice imaju značajno (p lt 0,05) manju vodootpornost strukturnih agregata, određenih mokrim prosejavanjem, od livada. Najmanju vodootpornost pokazali su strukturni agregati prečnika >3 mm. Takođe, prosečni maseni prečnici suvih i vodootpornih agregata i koeficijent strukture pokazali su značajne razlike između različitih načina korišćenja zemljišta na dubini 0-30 cm. Razoravanje livadskih crnica pod prirodnom livadskom vegetacijom istraženog područja i njihova višegodišnja obrada doveli su do značajnih negativnih promena agregatnog sastava i smanjenja vodootpornosti strukturnih agregata

A joint occurrence of atypical behavioral lateralization and schizophrenia: coincidental or causative?

The association between atypical lateralization of hand preferences (still a stigmatized behavioral trait in some cultures) with schizophrenia has been studied over the four last decades, repeatedly showing an increased incidence of atypical lateralization of hand dominance in this population. However, no final verdict on the causal direction between these two phenomena has been given. Atypical hand preferences, at the phenotypic level, have been subject to diverse classifications – commonly as left-handedness, but increasingly often as ‘non-right-handedness’, a ‘pooled handedness’ category made of two more homogeneous classes, and finally ‘left- and mixed-handedness’. Research has identified many associations of atypical lateralization of hand preferences with structural and functional brain asymmetries, cognitive performance, and clinical features in the population of schizophrenia patients. In this article, we critically appraise this work and suggest that both complex phenotypes, atypical lateralization of hand dominance and schizophrenia, most likely have common neurodevelopmental and genetic origin

EKOLOGIJA IZMEĐU POLITIKE I EKONOMIJE

Modern trends of socio-economic and technological development have a feature of permanent and the quickest changes in the history of human civilization. The consequences of such development raise many issues of sustainable development of our society in general and in particular. Although ecology has the character of requirements for sustainable development, in the latest developments, the question of its fate is importantly allocated by problems of relations between politics and economics. With these problems are opened main questions of this paper - how to survive development, to differentiate the interests and satisfy the assumptions of survival? In these changes which are globalizing ecology has a common interest and permanent needs of people and their communities. However, it is intended that this paper analyzes the consequences for that interests of the relationship between politics and economics in the current dynamics of the overall development. Expected results should contribute to the harmony between the political and economic interests and the needs of people on the one hand and ecology on the other hand, all in the human community, and not only in the state as a social and political community.Suvremeni tijekovi društveno-ekonomskog i tehnološko-tehničkog razvoja imaju obilježje stalnih i, neusporedivo, najbržih promjena u povijesti ljudske civilizacije. Posljedice takvog razvoja otvorile su brojna pitanja održivog razvoja društva uopće, i našeg posebno. Iako ekologija pruža uvjete za održivi razvoj, u suvremenim društveno-ekonomskim procesima, pitanje njene sudbine i samih procesa bitno je opredjeljeno odnosima u neravnostranom trokutu kojeg čine: ekologija, politika i ekonomija. Ovim pitanjima, otvaraju se osnovni problemi ovog rada - kako preživjeti razvoj, kako diferencirati interese strana u trokutu, i, na kraju, zadovoljiti pretpostavke opstanka? U novim globalnim promjenama ekologija ima zajednički prostorni značaj i trajnu potrebu ljudi i njihovih zajednica. Međutim, moja namjera u ovom radu jeste, da ekologiju kao suvremeni civilizacijski problem pokušam sagledati sa aspekta odnosa politike i ekonomije, kao dominirajućih ljudskih djelatnosti u ovoj dinamici ukupnog razvoja. Očekivani rezultati, treba li bi doprinijeti uspostavljanju harmonije između ljudi i njihovih aktivnosti u odnosu prema ekološkim okvirima života u znanosti koja izučava mehanizme opstanka živih bića ili, samo, u ekologiji i u ljudskoj zajednici