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

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

The Territorial Agenda 2030: Towards a common language? A review of a conceptual framework

The Territorial Agenda 2030 aims to provide multi-level strategic orientation to increase cohesion and overcome the 21st century pressing challenges. In multilingual contexts, the ideas and concepts communicated in such agendas must be clear and well-defined. In our study, we conducted a content analysis of the concepts of environment, inequality, justice, sustainability, territory and transition in contrast with former versions of this agenda. We found that, since 1983, the Territorial Agenda conceptual framework changed significantly in its meaning and semantic universe of reference.info:eu-repo/semantics/publishedVersio

The Constructive procedures in the spatial transformations of the surface elliptical hyperboloid of one sheet

Једнограни елиптички хиперболоид (ЈЕХ) је најопштији случај површи двоструке закривљености, из породице правоизводних површи, која настаје клизањем једне трансверзале–генератрисе, дуж три директрисе – мимоилазне праве простора. Дисертација даје одговор на другачији прилаз за постављање конструктивно–геометријског оквира површи у простор. Задавањем просторног шестотеменика – носача водиља површи, у различитим варијантама положаја, два пута по три изводнице, из различитих система изводница, у новом оквиру, развијен је низ конструктивних поступака, којима се генерише површ и одређују главни параметри ЈЕХ–а: средиште, главне ортогоналне осе и равни симетрије, стрикциона елипса и темена и утврђују њихове међусобне релације. Поступцима и научно заснованим методама пројективне синтетичке геометрије, нацртне геометрије и алатима компјутерске графике, у 3D окружењу (уз употребу софтвера Auto CAD), теза даје научни допринос, кроз решења – конструктивне поступке и теоретске закључке, у сфери геометрије правоизводне површи ЈЕХ–а и њених типова. У конструктивним поступцима просторних афиних трансформација једнограних хиперболоида, чије су водиље уписане у тростране призме, утврђене су инваријанте типова трансформација. На постављеним принципима иници–рана.су даља истраживања релација општих и специјалних типова ЈХ, у оквиру исте фамилије површи. Анализом свих типова пресека површи ЈЕХ–а са равнима, уз успостављање корелација са поларним карактеристикама површи, створена је теоријска база знања, корисна за практичну примену целе површи, или њених исечака, у поступцима геометријског 3D моделовања. Модели–структуре, са задатим правилним матрицама, креирани 3D поступцима, од исечака правог или косог типа ЈЕХ–а, представљају прототип за широку палету могућности примене у архитектонском обликовању кровних површина или целих објеката. Сличан, практично применљив, резултат дао је и део истраживања везан за продоре основних површи другог степена и ЈЕХ–а. Овакве комбинације површи, у поступцима просторног геометријског моделовања су скуп смерница, као део иницијативе и инспирације за пројектанте и градитеље. Резулатати истраживања су реализовани и кроз две апликације у софтверу Auto CAD (писану Аuto–lisp програмским језиком), која користи кружне пресеке ЈЕХ–а, за генерисање жичаног и површинског модела површи, са триангулисаном мрежом. Апликативност пресека површи по изводницама и кружницама, од значаја је за праксу грађења, која тежи једноставности извођења, а да при томе не умањује естетску вредност целокупне форме...Elliptic hyperboloid of one sheet (EHOS) is the general case of a double curved surface, from the family of ruled surfaces, which generates by ruling of a line–generator along three directing skewed lines, in the space. The dissertation gives an answer to a task – finding a new approach in creating constructive–geometry "boundaries" for this surface in the space settlement. A wide range of constructive procedures, for the determination and mutual relations of the main parameters of EHOS: center of the surface, orthogonal axes, planes of symmetry, minimal section curve (elliptical "throat") and its vertexes came out from the concept of hexagramic spatial "carrier" of two times three directors – skewed lines, in variety of their positions. The employment of procedures and scientifically based methods of Projective Synthetic Geometry, Descriptive Geometry and tools of computer graphics, in 3D surroundings (using Auto CAD software for engineers), are of importance for the scientific contributions of this dissertation, achieved solutions – constructive procedures and theoretical conclusions, in the sphere of geometry of ruled surface EHOS and its types. In the constructive procedures of spatial affine transformations of hyperboloids of one sheet, whose directing lines are "inscribed" in three sided prisms, the invariants of all the types of transformations are determined. Further investigations, concerning relations of general and special types of HOS, inside the same family of surfaces, are initiated by settled principles. The analysis of all the types of sections of EHOS, including the correlation to the polar properties of the surface, resulted with theoretical base, as an input for the practical application of "whole" surface or its fragments, in the procedures of 3D geometrical modeling. The models – structures, above regular patterns (polygons) are created by 3D procedures and tools, using fragments of straight or inclined EHOS. They represent the prototypes for the creative palette of possible architectural shapes of roofs or buildings. Very similar results came out of the part of research, concerning intersections of EHOS and basic second order surfaces (primitives): cone, cylinder and sphere. The "combinations" of such surfaces, in 3D geometrical modeling procedures, offered just some directions, like an initiation and inspiration for the architects and civil engineers..

On dinamic spiral patterns - polygonal frames inscribed in circular sections of quadric surfaces

Spiral forms, nowadays actual, especially in the area of architecture and design, were the inspiration point for an creative geometrical research. Acquainted with different approaches, present in practical and theoretical sense, from empiric creations to parametric modeling, we chose to explore the dynamic patterns which appear in spiral shapes generating process. Since the term "spiral" is directly connected to circles we aimed our investigation to quadric surfaces with circular sections, where inscribed polygons obtain the spiral form by "twisting". We observed the series of inscribed polygons as dynamical spiral patterns of scaled frames, according to the geometry of the basic quadric surface. This investigation includes surfaces: cone, sphere, ellipsoid and elliptic hyperboloid. Three types of regular polygons are here included: triangle, square and pentagon. 3D model presentation of dynamic spiral patterns is performed in engineering software Auto-CAD. While considering the possibilities in application of such creations (models), some optimal intersecting surfaces are discussed

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

Closed geodesics and billiards on quadrics related to elliptic KdV solutions

We consider algebraic geometrical properties of the integrable billiard on a quadric Q with elastic impacts along another quadric confocal to Q. These properties are in sharp contrast with those of the ellipsoidal Birkhoff billiards. Namely, generic complex invariant manifolds are not Abelian varieties, and the billiard map is no more algebraic. A Poncelet-like theorem for such system is known. We give explicit sufficient conditions both for closed geodesics and periodic billiard orbits on Q and discuss their relation with the elliptic KdV solutions and elliptic Calogero systemComment: 23 pages, Latex, 1 figure Postscrip

On p-Adic Sector of Adelic String

We consider construction of Lagrangians which are candidates for p-adic sector of an adelic open scalar string. Such Lagrangians have their origin in Lagrangian for a single p-adic string and contain the Riemann zeta function with the d'Alembertian in its argument. In particular, we present a new Lagrangian obtained by an additive approach which takes into account all p-adic Lagrangians. The very attractive feature of this new Lagrangian is that it is an analytic function of the d'Alembertian. Investigation of the field theory with Riemann zeta function is interesting in itself as well.Comment: 10 pages. Presented at the 2nd Conf. on SFT and Related Topics, Moscow, April 2009. Submitted to Theor. Math. Phy