Serdica Mathematical Journal

Volume 41, Number 4, 2015

This issue contains contributions to the
First International Conference Mathematics Days in Sofia
held from 7th to 10th of July, 2014.

C O N T E N T S

A B S T R A C T S

EVOLUTION EQUATIONS FOR THE STEFAN PROBLEM
Martin Lukarevski martin.lukarevski@ugd.edu.mk

2010 Mathematics Subject Classification: 35R35, 35B65, 35J70.
Key words: Stefan problem, evoulution equation, free boundary problem.

We study particular kind of Stefan problem and use the theory of abstract quasilinear evolution equations for its solution.

EPW SEXTICS AND HILBERT SQUARES OF K3 SURFACES
Atanas Iliev ailiev@snu.ac.kr,
Carlo Madonna carlo.madonna@uam.es

2010 Mathematics Subject Classification: 14J35, 14F05.
Key words: K3 surface, EPW sextic, Eisenbud-Popescu-Walter sextic.

We prove that the Hilbert square S^[2] of a very general primitively polarized K3 surface S of degree d(n) = 2(4n² + 8n + 5), n ≥ 1 is birational to a double Eisenbud-Popescu-Walter sextic. Our result implies a positive answer, in the case when r is even, to a conjecture of O'Grady: On the Hilbert square of a very general K3 surface of genus r² + 2, r ≥ 1 there is an antisymplectic birational involution. We explicitly give this involution on S^[2] in terms of the corresponding EPW polarization on it.

SUMMABILITY METHODS IN WEIGHTED APPROXIMATION TO DERIVATIVES OF FUNCTIONS
Nisa Küçük nkucuk@etu.edu.tr,
Oktay Duman oduman@etu.edu.tr

2010 Mathematics Subject Classification: 41A30, 42B08, 47B38.
Key words: Summability process, weighted approximation, almost convergence, Cesàro method.

In this paper, we use summability methods on the approximation to derivatives of functions by a family of linear operators acting on weighted spaces. This point of view enables us to overcome the lack of ordinary convergence in the approximation. To support this idea, at the end of the paper, we will give a sequence of positive linear operators obeying the arithmetic mean approximation (or, approximation with respect to the Cesàro method) although it is impossible in the usual sense. Some graphical illustrations are also provided.

KOSZUL DUALITY FOR LOCALLY CONSTANT FACTORIZATION ALGEBRAS
Takuo Matsuoka motogeomtop@gmail.com

2010 Mathematics Subject Classification: 55M05, 16E40, 57R56, 16D90.
Key words: Koszul duality, factorization algebra, topological chiral homology, topological quantum field theory, higher Morita category.

Generalizing Jacob Lurie's idea on the relation between the Verdier duality and the iterated loop space theory, we study the Koszul duality for locally constant factorization algebras. We formulate an analogue of Lurie's "nonabelian Poincaré duality" theorem (which is closely related to earlier results of Graeme Segal, of Dusa McDuff, and of Paolo Salvatore) in a symmetric monoidal stable infinity 1-category carefully, using John Francis' notion of excision. Its proof depends on our study of the Koszul duality for E_n-algebras in [12]. As a consequence, we obtain a Verdier type equivalence for factorization algebras by a Koszul duality construction.

CONVOLUTIONAL CALCULUS OF DIMOVSKI AND QR-REGULARIZATION OF THE BACKWARD HEAT PROBLEM
Emilia Bazhlekova e.bazhlekova@math.bas.bg

2010 Mathematics Subject Classification: 35C10, 35R30, 44A35, 44A40.
Key words: convolutional calculus, non-classical convolution, Duhamel principle, ill-posed problem, quasi-reversibility.

The final value problem for the heat equation is known to be ill-posed. To deal with this, in the method of quasi-reversibility (QR), the equation or the final value condition is perturbed to form an approximate well-posed problem, depending on a small parameter ε. In this work, four known quasi-reversibility techniques for the backward heat problem are considered and the corresponding regularizing problems are treated using the convolutional calculus approach developed by Dimovski (I. H. Dimovski, Convolutional Calculus, Kluwer, Dordrecht, 1990). For every regularizing problem, applying an appropriate bivariate convolutional calculus, a Duhamel-type representation of the solution is obtained. It is in the form of a convolution product of a special solution of the problem and the given final value function. A non-classical convolution with respect to the space variable is used. Based on the obtained representations, numerical experiments are performed for some test problems.

NOTES ON OPTIMALITY CONDITIONS USING NEWTON DIAGRAMS AND SUMS OF SQUARES
Yoshiyuki Sekiguchi yoshi-s@kaiyodai.ac.jp

2010 Mathematics Subject Classification: 90C46, 13J30, 14M25.
Key words: Polynomial optimization, Newton diagram, optimality conditions, sums of squares.

We consider relationships between optimality conditions using Newton diagrams and sums of squares of polynomials and power series.

ON THE POISSON-CHARLIER POLYNOMIALS
Nejla Özmen nejlaozmen06@gmail.com,
Esra Erkuş-Duman eduman@gazi.edu.tr

2010 Mathematics Subject Classification: 33C45.
Key words: Poisson-Charlier polynomials, recurrence relation, generating function, hypergeometric function, Lauricella functions.

In this paper the Poisson-Charlier polynomials are introduced. Some of their recurrence relations are presented. Various families of bilinear and bilateral generating functions for these polynomials are derived. Furthermore, some special cases of the results are presented in this study.

MEAN VALUE THEOREMS FOR ANALYTIC FUNCTIONS
Lubomir Markov lmarkov@barry.edu

2010 Mathematics Subject Classification: 30C15.
Key words: Complex Rolle's Theorem, Mean Value Theorem, Evard-Jafari Theorem, Flett's Mean Value Theorem, Davitt-Powers-Riedel-Sahoo Theorem.

We prove a sharper Evard-Jafari Theorem, various mean value theorems, and an improved version of the Davitt-Powers-Riedel-Sahoo Theorem.

FINITE TIME BLOW UP OF THE SOLUTIONS TO NONLINEAR KLEIN-GORDON EQUATION WITH ARBITRARY HIGH POSITIVE INITIAL ENERGY
N. Kutev,
N. Kolkovska natali@math.bas.bg,
M. Dimova mkoleva@math.bas.bg

2010 Mathematics Subject Classification: 35L05, 35L15.
Key words: Klein-Gordon equation, Blow up, Arbitrary high positive initial energy.

The global behaviour of the weak solutions of the Cauchy problem to nonlinear Klein-Gordon equation in Rⁿ×R⁺ is investigated. Finite time blow up of the solutions with arbitrary high positive initial energy is proved under general structural conditions on the initial data.

HARDY-TYPE INEQUALITIES WITH WEIGHTS
Alexander Fabricant,
Nikolai Kutev,
Tsviatko Rangelov rangelov@math.bas.bg

2010 Mathematics Subject Classification: 26D10.
Key words: Hardy inequality, weights, sharp estimates.

Hardy-type inequality with weights is derived in abstract form. Particular cases are presented to demonstrate the applicability of the method and to show generalizations of existing results. Sharpness of inequalities is proved and the results are illustrated with several examples.

BI-CHARACTERISTIC CURVES OF BODY AND SURFACE WAVES AND APPLICATION IN GEOPHYSICS
Georgi Boyadzhiev gpb@math.bas.bg

2010 Mathematics Subject Classification: 35L53, 35Q86, 86A15.
Key words: Strongly coupled linear hyperbolic systems, modelling of multi-layered solid body, applications in Geophysics.

In this paper is given a new approach to 3D medelling of elastic piecewise homogeneous media, in particular Earth crust and upper Mantle. The method is based on the principle of tomography with Earthquake as a source of the signal and at least three receiver stations on the surface.
The wave propagation in such media is described by a system of three strongly coupled hyperbolic equations with piece-wise constant coefitients. The characteristic set and bi-characteristic curves are computed in a homogeneous half-space with free boundary as well as the formulae of reflection and diffraction of the bi-characteristics on the internal boundaries of the media. Applications of the characteristic set and bi-characteristic curves for the inverse problem in geophysics and Earth modelling are given.