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Cluj and related polynomials in tori
Mircea V. Diudea, Csaba L. Nagy, Petra Žigert, Sandi Klavžar, 2010, izvirni znanstveni članek

Ključne besede: cluj polynomial, couting polynomial, torus
Objavljeno: 31.05.2012; Ogledov: 909; Prenosov: 5
URL Celotno besedilo (0,00 KB)

On the application of a simple computational model for slender transversely cracked beams in buckling problems
Matjaž Skrinar, 2007, izvirni znanstveni članek

Opis: This paper discusses the implementation of a simplified computational model that is widely used for the computation of transverse displacements in transversely cracked slender beams into the Euler's elastic flexural buckling theory. Two alternatives are studied instead of solving the corresponding differential equations to obtain exact analytical expressions for the buckling load ▫$P_{cr}$▫ due to the complexity of this approach. The first approach implements wisely selected polynomials to describe the behavior of the structure, which allows the derivation of approximate expressions for the critical buckling load. Although the relevance of the results strongly depends on the proper prime selection of the polynomial, it is shown that the later upgrading of the polynomials can lead to even more reliable results. The second approach is a purely numerical approach and presents the geometrical stiffness matrix for a beam finite element with a transverse crack. To support the discussed approaches, numerical examples covering several structures with different boundary conditions are briefly presented. The results obtained with the presented approaches are further compared with the values from enormous 2D finite elements models, where a detailed description of the crack was achieved with the discrete approach. It is evident that the drastic difference in the computational effort is not reflected in the significant differences in the results between the models.
Ključne besede: columns, transverse cracks, stability problems, buckling load, computational model, polynomial solutions, finite element method, geometrical stiffness matrix
Objavljeno: 01.06.2012; Ogledov: 1113; Prenosov: 5
URL Celotno besedilo (0,00 KB)

Roots of cube polynomials of median graphs
Sandi Klavžar, Riste Škrekovski, Boštjan Brešar, 2003

Opis: Polinom kock ▫$c(G,X)$▫ grafa ▫$G$▫ je definiran z ▫$sum_{i ge 0}alpha_i(G)x^i$▫, kjer ▫$alpha_i(G)$▫ označuje število induciranih ▫$i$▫-kock v ▫$G$▫. Naj bo ▫$G$▫ medianski graf. Dokazano je, da je vsaka racionalna ničla polinoma ▫$c(G,x)$▫ oblike ▫$-frac{t+1}{t}$▫ za neko celo število ▫$t>0$▫ in da ima ▫$c(G,x)$▫ vedno realno ničlo na intervalu ▫$[-2,-1)$▫. Nadalje ima ▫$c(G,x)$▫ ▫$p$▫-kratno ničlo natanko tedaj, ko je ▫$G$▫ kartezični produkt ▫$p$▫ dreves istega reda. Grafi acikličnih kubičnih kompleksov so karakterizirani kot grafi za katere velja ▫$c(H,-2)=0$▫ za vsak 2-povezan konveksen podgraf ▫$H$▫.
Ključne besede: matematika, teorija grafov, polinom kock, koren, medianski graf, kartezični produkt grafov, mathematics, graph theory, cube polynomial, root, median graph, Cartesian product
Objavljeno: 10.07.2015; Ogledov: 182; Prenosov: 2
URL Celotno besedilo (0,00 KB)

Trace-positive polynomials, sums of hermitian squares and the tracial moment problem
Sabine Burgdorf, 2011, doktorska disertacija

Opis: A polynomial ▫$f$▫ in non-commuting variables is trace-positive if the trace of ▫$f(underline{A})$▫ is positive for all tuples ▫$underline{A}$▫ of symmetric matrices of the same size. The investigation of trace-positive polynomials and of the question of when they can be written as a sum of hermitian squares and commutators of polynomials are motivated by their connection to two famous conjectures: The BMV conjecture from statistical quantum mechanics and the embedding conjecture of Alain Connes concerning von Neumann algebras. First, results on the question of when a trace-positive polynomial in two non-commuting variables can be written as a sum of hermitian squares and commutators are presented. For instance, any bivariate trace-positive polynomial of degree at most four has such a representation, whereas this is false in general if the degree is at least six. This is in perfect analogy to Hilbert's results from the commutative context. Further, a partial answer to the Lieb-Seiringer formulation of the BMV conjecture is given by presenting some concrete representations of the polynomials ▫$S_{m,4}(X^2; Y^2)$▫ as a sum of hermitian squares and commutators. The second part of this work deals with the tracial moment problem. That is, how can one describe sequences of real numbers that are given by tracial moments of a probability measure on symmetric matrices of a fixed size. The truncated tracial moment problem, where one considers only finite sequences, as well as the tracial analog of the ▫$K$▫-moment problem are also investigated. Several results from the classical moment problem in Functional Analysis can be transferred to this context. For instance, a tracial analog of Haviland's theorem holds: A traciallinear functional ▫$L$▫ is given by the tracial moments of a positive Borel measure on symmetric matrices of a fixed size s if and only if ▫$L$▫ takes only positive values on all polynomials which are trace-positive on all tuples of symmetric ▫$s times s$▫-matrices. This result uses tracial versions of the results of Fialkow and Nie on positive extensions of truncated sequences. Further, tracial analogs of results of Stochel and of Bayer and Teichmann are given. Defining a tracial Hankel matrix in analogy to the Hankel matrix in the classical moment problem, the results of Curto and Fialkow concerning sequences with Hankel matrices of finite rank or Hankel matrices of finite size which admit a flat extension also hold true in the tracial context. Finally, a relaxation for trace-minimization of polynomials using sums of hermitian squares and commutators is proposed. While this relaxation is not always exact, the tracial analogs of the results of Curto and Fialkow give a sufficient condition for the exactness of this relaxation.
Ključne besede: matematika, algebra, polinomi s pozitivno sledjo, prosta algebra, nekomutativni polinom, centralna enostavna algebra, reducirana sled, polinomska identiteta, kvadratna forma, prosta pozitivnost, vsota hermitskih kvadratov, problem momentov, mathematics, algebra, free algebra, noncommutative polynomial, central simple algebra, (reduced) trace, polynomial identity, central polynomial, quadratic form, free positivity, sum of hermitian squares, (truncated) moment problem
Objavljeno: 10.07.2015; Ogledov: 445; Prenosov: 3
URL Celotno besedilo (0,00 KB)
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The study of isochronicity and critical period bifurcations on center manifolds of 3-dim polynomial systems using computer algebra
Brigita Ferčec, Matej Mencinger, 2013, objavljeni znanstveni prispevek na konferenci

Opis: Using the solution of the center-focus problem from [4], we present the investigation of isochronicity and critical period bifurcations of two families of cubic 3-dim systems of ODEs. Both cubic systems have a center manifold filled with closed trajectories. The presented study is performed using computer algebra systems Mathematica and Singular.
Ključne besede: polynomial systems of ODEs, center manifolds, isochronicity, bifurcation of critical periods, CAS
Objavljeno: 10.07.2015; Ogledov: 209; Prenosov: 2
URL Celotno besedilo (0,00 KB)
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On Groebner bases and their use in solving some practical problems
Matej Mencinger, 2013, izvirni znanstveni članek

Opis: Groebner basis are an important theoretical building block of modern (polynomial) ring theory. The origin of Groebner basis theory goes back to solving some theoretical problems concerning the ideals in polynomial rings, as well as solving polynomial systems of equations. In this article four practical applications of Groebner basis theory are considered; we use Groebner basis to solve the systems of nonlinear polynomial equations, to solve an integer programming problem, to solve the problem of chromatic number of a graph, and finally we consider an original example from the theory of systems of ordinary (polynomial) differential equations. For practical computations we use systems MATHEMATICA and SINGULAR .
Ključne besede: polynomial system of (differential) equations, integer linear programming, chromatic number of a graph, polynomial rings, Groebner basis, CAS systems
Objavljeno: 10.07.2015; Ogledov: 224; Prenosov: 6
URL Celotno besedilo (0,00 KB)
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Isochronicity and normal forms of polynomial systems of ODEs
Valery Romanovski, Maoan Han, 2012, objavljeni znanstveni prispevek na konferenci

Opis: We propose a generalization of the notion of isochronicity for real polynomial autonomous systems to the case of complex two dimensional systems of ODEs. We study the generalized problem in the case of a quadratic system and a system with homogeneous cubic nonlinearities. Main tools of the study are algorithms of computational algebra based on the Groebner basis theory.
Ključne besede: polynomial systems of ODE`s, isochronicity, linearizability, normal forms
Objavljeno: 10.07.2015; Ogledov: 186; Prenosov: 2
URL Celotno besedilo (0,00 KB)

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