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
·
Preface (pp. i-ii)
·
Lukarevski, M.
Evolution equations for the Stefan problem
(pp. 333−342)
·
Iliev, A., C. Madonna.
EPW sextics and Hilbert squares of K3 surfaces
(pp. 343−354)
·
Küçük, N., O. Duman.
Summability methods in weighted approximation to derivatives of functions
(pp. 335−368)
·
Matsuoka, T.
Koszul duality for locally constant factorization algebras
(pp. 369−414)
·
Bazhlekova, E.
Convolutional calculus of Dimovski and QR-regularization of the backward heat problem
(pp. 415−430)
·
Sekiguchi, Y.
Notes on optimality conditions using Newton diagrams and sums of squares
(pp. 431−456)
·
Özmen, N., E. Erkuş-Duman.
On the Poisson-Charlier polynomials
(pp. 457−470)
·
Markov, L.
Mean value theorems for analytic functions
(pp. 471−480)
·
Kutev, N., N. Kolkovska, M. Dimova.
Finite time blow up of the solutions to nonlinear Klein-Gordon equation with arbitrary high positive initial energy
(pp. 481−492)
·
Fabricant, A., N. Kutev, Ts. Rangelov.
Hardy-type inequalities with weights
(pp. 493−512)
·
Boyadzhiev, G.
Bi-characteristic curves of body and surface waves and application in geophysics
(pp. 513−526)
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(4n^{2} + 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 r^{2} + 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 R^{n}×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.
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