Mathematical Institute SANU

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Musicki D.,University of Belgrade | Musicki D.,Mathematical Institute SANU
Theoretical and Applied Mechanics | Year: 2017

In this paper the extended Lagrangian formalism for the rheonomic systems (Dj. Mušicki, 2004), which began with the modification of the mechanics of such systems (V. Vujičić, 1987), is extended to the systems with variable mass, with emphasis on the corresponding energy relations. This extended Lagrangian formalism is based on the extension of the set of chosen generalized coordinates by new quantities, suggested by the form of nonstationary constraints, which determine the position of the frame of reference in respect to which these generalized coordinates refer. As a consequence, an extended system of the Lagrangian equations is formulated, accommodated to the variability of the masses of particles, where the additional ones correspond to the additional generalized coordinates. By means of these equations, the energy relations of such systems have been studied, where it is demonstrated that here there are four types of energy conservation laws. The obtained energy laws are more complete and natural than the corresponding ones in the usual Lagrangian formulation for such systems. It is demonstrated that the obtained energy laws, are in full accordance with the energy laws in the corresponding vector formulation, if they are expressed in terms of the quantities introduced in this formulation of mechanics. The obtained results are illustrated by an example: the motion of a rocket, which ejects the gasses backwards, while this rocket moves up a straight line on an oblique plane, which glides uniformly in a horizontal direction.

Cvetkovic D.,Mathematical Institute SANU | Rowlinson P.,University of Stirling | Stanic Z.,University of Belgrade | Yoon M.-G.,Gangneung - Wonju National University
Bulletin, Classe des Sciences Mathematiques et Naturelles, Sciences Mathematiques | Year: 2011

The eigenvalues of a graph are the eigenvalues of its adjacency matrix. An eigenvalue of a graph is called main if the corresponding eigenspace contains a vector for which the sum of coordinates is different from 0. Connected graphs in which all eigenvalues are mutually distinct and main have recently attracted attention in control theory.

Dragovic V.,University of Texas at Dallas | Dragovic V.,Mathematical Institute SANU | Kukic K.,University of Belgrade
Journal of Geometric Mechanics | Year: 2014

We classify the discriminantly separable polynomials of degree two in each of three variables, defined by a property that all the discriminants as polynomials of two variables are factorized as products of two polynomials of one variable each. Our classification is based on the study of structures of zeros of a polynomial component P of a discriminant. This classification is related to the classification of pencils of conics in a delicate way. We establish a relationship between our classification and the classification of integrable quad-equations which has been suggested recently by Adler, Bobenko, and Suris. ©American Institute of Mathematical Sciences.

Cvetkovic D.,Mathematical Institute SANU | Simic S.K.,Mathematical Institute SANU | Stanic Z.,University of Belgrade
Computers and Mathematics with Applications | Year: 2010

We consider the class of graphs each of whose components is either a path or a cycle. We classify the graphs from the class considered into those which are determined and those which are not determined by the adjacency spectrum. In addition, we compare the result with the corresponding results for the Laplacian and the signless Laplacian spectra. It turns out that the signless Laplacian spectrum performs the best, confirming some expectations from the literature. © 2010 Elsevier Ltd. All rights reserved.

Dragovic V.,Mathematical Institute SANU | Dragovic V.,University of Lisbon
Communications in Mathematical Physics | Year: 2010

A new view on the Kowalevski top and the Kowalevski integration procedure is presented. For more than a century, the Kowalevski 1889 case, has attracted full attention of a wide community as the highlight of the classical theory of integrable systems. Despite hundreds of papers on the subject, the Kowalevski integration is still understood as a magic recipe, an unbelievable sequence of skillful tricks, unexpected identities and smart changes of variables. The novelty of our present approach is based on our four observations. The first one is that the so-called fundamental Kowalevski equation is an instance of a pencil equation of the theory of conics which leads us to a new geometric interpretation of the Kowalevski variables w, x1, x2 as the pencil parameter and the Darboux coordinates, respectively. The second is observation of the key algebraic property of the pencil equation which is followed by introduction and study of a new class of discriminantly separable polynomials. All steps of the Kowalevski integration procedure are now derived as easy and transparent logical consequences of our theory of discriminantly separable polynomials. The third observation connects the Kowalevski integration and the pencil equation with the theory of multi-valued groups. The Kowalevski change of variables is now recognized as an example of a two-valued group operation and its action. The final observation is surprising equivalence of the associativity of the two-valued group operation and its action to the n = 3 case of the Great Poncelet Theorem for pencils of conics. © 2010 Springer-Verlag.

Fuji H.,Nagoya University | Gukov S.,California Institute of Technology | Gukov S.,Max Planck Institute For Mathematik | Stosic M.,University of Lisbon | And 4 more authors.
Journal of High Energy Physics | Year: 2013

We study singularities of algebraic curves associated with 3d N = 2 theories that have at least one global flavor symmetry. Of particular interest is a class of theories T K labeled by knots, whose partition functions package Poincaré polynomials of the S r -colored HOMFLY homologies. We derive the defining equation, called the super-A-polynomial, for algebraic curves associated with many new examples of 3d N = 2 theories T K and study its singularity structure. In particular, we catalog general types of singularities that presumably exist for all knots and propose their physical interpretation. A computation of super-A-polynomials is based on a derivation of corresponding superpolynomials, which is interesting in its own right and relies solely on a structure of differentials in S r -colored HOMFLY homologies. © 2013 SISSA.

Dragovi V.,Mathematical Institute SANU | Dragovi V.,University of Lisbon | Radnovi M.,Mathematical Institute SANU
Journal of Nonlinear Mathematical Physics | Year: 2012

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 integrability 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 properties and several examples are presented. Corresponding YangBaxter maps, associated with pencils of quadrics are defined and discussed. © V. Dragović and M. Radnović.

Brimberg J.,Royal Military College of Canada | Brimberg J.,University of Montréal | Mladenovic N.,University of Vallenciennes | Mladenovic N.,Mathematical Institute SANU | And 2 more authors.
Information Sciences | Year: 2015

The maximally diverse grouping problem requires finding a partition of a given set of elements into a fixed number of mutually disjoint subsets (or groups) in order to maximize the overall diversity between elements of the same group. In this paper we develop a new variant of variable neighborhood search for solving the problem. The extensive computational results show that our new heuristic significantly outperforms the current state of the art. Moreover, the best known solutions have been improved on 531 out of 540 test instances from the literature. © 2014 Elsevier Inc.

Belardo F.,Messina University | De Filippis V.,Messina University | Simic S.K.,Mathematical Institute SANU
Match | Year: 2011

Recently, in the book [A Combinatorial Approach to Matrix Theory and Its Applications, CRC Press (2009)] the authors proposed a combinatorial approach to matrix theory by means of graph theory. In fact, if A is a square matrix over any field, then it is possible to associate to A a weighted digraph G a, called Coates digraph. Through Ga (hence by graph theory) it is possible to express and prove results given for the matrix theory. In this paper we express the permanental polynomial of any matrix A in terms of permanental polynomials of some digraphs related to Ga.

Hedrih K.R.,Mathematical Institute SANU | Hedrih K.R.,University of Niš
International Journal of Nonlinear Sciences and Numerical Simulation | Year: 2010

Vibrations, mechanical energy, constraint reactions and power of reactive forces of a heavy mass particle moving along a rough curvilinear path with Coulomb-type friction are considered. A mathematical description is presented in a natural coordinate system of the corresponding line. Analytical expressions of kinetic, potential and total energies of the mass particle motion, as well as analytical expressions of the Coulomb type friction force are obtained. © Freund Publishing House Ltd.