Petak, 02. mart 2007. u 14h, sala 2 MI SANU:

Jozef Kratica, Mathematical Institute, Serbian Academy of Sciences and Arts Zorica Stanimirovi?, Du~Zan To~Zi?, Vladimir Filipovi?, University of Belgrade, Faculty of Mathematics
GENETIC ALGORITHMS FOR SOLVING HUB LOCATION PROBLEMS

Abstract: Hub networks are used in transportation and telecommunications systems (airline, ground transportation, postal delivery systems, computer networks, etc). Instead of serving every destination node (facility) from an origin node with a direct link, a hub network provides services via a specified set of hub nodes (hubs). According to the characteristics of a particular hub network there are several kinds of hub location problems. If the number of hub nodes is fixed to p, we are dealing with p-hub location problems. Each non-hub node can be allocated either to one (single allocation scheme) or more hubs (multiple allocation scheme). Fixed costs for locating hubs may also be assumed. Here we give a survey of the results provided by applying different genetic algorithms for solving previously mentioned hub location problems. In every case, selected genetic algorithm approach achieves all previously known optimal solutions and achieves the best known solutions on large-scale instances.

Petak, 09. mart 2007. u 14h, sala 2, SANU, BG:

Gradimir Milovanovi?, Elektronski fakultet, Ni~Z
NONSTANDARD GAUSSIAN QUADRATURE RULES USING FUNCTION DERIVATIVES

Recently Bojanov & Petrov (2001, 2003, 2005) and Milovanovi\'c & Cvetkovi\'c (2004, 2005, 2006) investigated Gaussian quadrature rules of the form \[ \int p(x) w(x) d x=\sum_{k=1}^n \frac{\sigma_k}{\int_{I_k} w(x) d x}\int_{I_k} p(x) w(x) d x, \] exact for all $p\in {\mathcal{P}}_{2n-1}$, where $w$ is the weight strictly positive function supported on the bounded or unbounded interval of the real line. In this talk we extend further investigation of the nonstandard quadrature rules. Namely, for $m\in\mathbb{N}$, we investigate quadrature rules of the form \[ \int p d\mu=\sum_{k=1}^n w_k p^{(m)}(x_k), \] where $\mu$ is finite positive Borel measure supported on the real line, with all polynomials being $\mu$-integrable, exact on the polynomial space \[ {\mathcal{P}}_{2n+m-1}^\lambda=\Bigl\{p~\Bigm|~p\in {\mathcal{P}}_{2n+m-1},~\lambda\in {\mathbb{R}},~p^{(k)}(\lambda)=0,~k=0,1,\ldots,m-1\Bigr\}. \] Here $\mathcal{P}_m$ is a space of algebraic polynomials of degree at most $m$. We present results on existence, regularity, convergence and numerical construction of the mentioned quadrature rules. Also we give an application of the mentioned quadrature rules to some Cauchy problems for ODE. We prove that crucial properties of mentioned quadrature rules are connected with the following transform \[ w(t)=\frac{1}{(m-1)!}\int (x-t)^{m-1} \psi(\lambda,x;t) d\mu(x), \] of the measure $d\mu$, where \[ \psi(\lambda,x;t)=\left\{\begin{array}{cc} \chi_{(\lambda,x)}(t),& \lambdax,\end{array}\right. \] where $\chi_A$ is the characteristic function of the set $A$. For some specific cases we give explicit expressions of the function $w$.

Petak, 16. mart 2007. sala 2 MI SANU:

Slobodan Simi?, Matemati?ki institut SANU
NEKI NOVI I STARI REZULTATI O NAJMANJOJ SOPSTVENOJ VREDNOSTI GRAFA

Sadr~^aj: Najmanja sopstvena vrednost grafa je najmanja sopstvena vrednost njegove matrice susedstva. U literaturi postoji veliki broj rezultata o najve?oj sopstvenoj vrednosti grafa (tj. njegovom spektralnom radijusu, odnosno indeksu). Najmanja sopstvena vrednost je znatno manje pro?avana. Cilj ovog predavanja je da se uka~^e na postoje?e rezultate, kao i da se prika~^u neki najnoviji rezultati koji su u procesu pripreme za publikovanje.

Petak, 23. mart 2007. u 14h, sala 2, MI SANU BGD:

Prof. Joachim Toft,Växjö Universitet, Sweden
REGULARITY AND WAVE-FRONT SET PROPERTIES OF DISTRIBUTION KERNELS TO POSITIVE OPERATORS

Abstract: We discuss regularity of distribution kernels to positive operators and to distributions which are positive with respect to some non-commutative convolution. These discussions are based on studying different types of wave-front sets (especially wave-front sets with respect to smoothness, quasi-analyticity and other related classes) of such distributions. We use the observation that distributions which satisfy such positivity conditions can be used to form different types of semi-scalar products. We are then able to apply Cauchy-Schwarz inequality together with some estimates from microlocal analysis to prove that distribution kernels of positive operators are in some sense most irregular along the diagonal. In particular, if such kernels are smooth (analytic) along the diagonal, then they are smooth (analytic) everywhere. We apply these properties to distributions which are positive with respect to some non-commutative convolution, and prove that such elements are as most irregular at the origin. Consequently, if such element is smooth (analytic) at the origin, then it is smooth (analytic) everywhere. In particular, we are able to recover and extend some results on positive definite functions and distributions by Bochner, Yoshino and others.

Petak, 30. mart 2007. u 14h, sala 718, MF BG:

Milo~Z Kurili?, Departman za matematiku i informatiku, PMF, Novi Sad
IGRE NA BOOLEOVIM ALGEBRAMA

Sadr~^aj: Uop~Ztavaju?i poznatu Banach-Mazurovu igru na realnoj pravi, Jech je 1984. godine uveo razne igre tipa "Beli se?e, Crni bira" (cut-and-choose) koje se u prebrojivo mnogo koraka igraju na proizvoljnoj kompletnoj Booleovoj algebri. Pokazalo se da osobine Booleovih algebri vezane za teoriju igara (npr. postojanje pobedni?kih strategija igra?a) imaju interesantne algebarske, kombinatorne i "forcing" interpretacije. Bi?e dat prikaz Jechovih igara, kao i raznih njihovih upo~Ztenja. Tako?e ?e biti prikazan rad M. Kurili?a i Borisa ~Jobota u kojem je uvedena igra koja opisuje kolaps kontinuuma na $\omega$.

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