Serdica Mathematical Journal

Volume 43, Number 2, 2017

C O N T E N T S

A B S T R A C T S

ON THE CONVERGENCE OF THE METROPOLIS-HASTINGS MARKOV CHAINS
Dimiter Tsvetkov dimiter.p.tsvetkov@gmail.com,
Lyubomir Hristov lyubomir.hristov@gmail.com,
Ralitsa Angelova-Slavova ralitsa.slavova@yahoo.com

2010 Mathematics Subject Classification: Primary: 60J05, 65C05; secondary 60J22.
Key words: Markov chain, Metropolis-Hastings algorithm, total variation distance.

In this paper we study Markov chains associated with the Metropolis-Hastings algorithm. We consider conditions under which the sequence of the successive densities of such a chain converges to the target density according to the total variation distance for any choice of the initial density. In particular we prove that the positiveness of the proposal density is enough for the chain to converge. The content of this work basically presents a stand alone proof that the reversibility along with the kernel positivity imply the convergence.

THE GULF BETWEEN THE CLIQUE NUMBER AND ITS UPPER ESTIMATE PROVIDED BY FRACTIONAL COLORING OF THE NODES
S. Szabó sszabo7@hotmail.com
B. Zaválnij. bogdan@ttk.pte.hu

2010 Mathematics Subject Classification: 05C15.
Key words: clique, maximum clique, independent set, vertex coloring, 3-clique free coloring, clique search algorithm.

It is well known that coloring the nodes can be used to establish upper bounds for the clique number of a graph which in turn can be used to speed up practical clique search algorithms. E. Balas and J. Xue suggested factional coloring while S. Szabó and B. Zaválnij suggested triangle free coloring of the nodes to get tighter bounds. The main result of this paper is that the gap between the clique number and the upper bound provided by the coloring scheme combining the fractional and triangle free colorings still can be arbitrarily large.

OSCILLATION CRITERIA OF SOLUTIONS OF THIRD-ORDER NEUTRAL DIFFERENTIAL EQUATIONS WITH CONTINUOUSLY DISTRIBUTED DELAY
O. Moaaz o_moaaz@mans.edu.eg,
E. M. Elabbasy emelabbasy@mans.edu.eg,
E. Shaaban ebtesamshaaban@asmarya.edu.ly

2010 Mathematics Subject Classification: 34K10, 34K11.
Key words: Oscillation, third-order, neutral delay, differential equations.

In this paper, a class of third-order neutral delay differential equations with continuously distributed delay is studied. Also, we establish new oscillation results for the third-order equation by using the integral averaging technique due to Philos. Our results essentially improve and complement some earlier publications. Examples are provided to illustrate new results.

CONVERGENCE ANALYSIS OF SEMI-IMPLICIT EULER METHOD FOR NONLINEAR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS OF NEUTRAL TYPE
M. Milev marianmilev2002@gmail.com,
O. Farkhondeh Rouz omid_farkhonde_7088@yahoo.com,
D. Ahmadian d.ahmadian@tabrizu.ac.ir

2010 Mathematics Subject Classification: 65C20, 60H35, 65C30.
Key words: Neutral stochastic delay differential equations, mean-square convergence, semi-implicit Euler method.

The main purpose of this paper is to study the convergence of numerical solutions to a class of neutral stochastic delay differential equations (NSDDEs) in Itô sense. The basic idea is to reformulate the original problem eliminating the dependence on the differentiation of the solution in the past values, which leads to a stochastic differential algebraic system. It is shown that the Semi-implicit Euler (SIE) method with two parameters θ and λ is mean-square convergent with order p = ½ for Lipschitz continuous coefficients of underlying NSDDEs. A nonlinear numerical example illustrates the theoretical results.

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JORDAN ELEMENTARY MAPS ON ALTERNATIVE DIVISION RINGS
Bruno Leonardo Macedo Ferreira brunoferreira@utfpr.edu.br

2010 Mathematics Subject Classification: 17A36, 17D05.
Key words: Jordan semi-isomorphism; alternative rings.

In this work we prove that if R and R′ are arbitrary alternative division rings, then under a mild condition every Jordan semi-isomorphism (M, M^*) of R × R′ is a isomorphism or an anti-isomorphism.

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A CHARACTERIZATION OF DUPIN HYPERSURFACES IN R⁵
Carlos M. C. Riveros carlos@mat.unb.br

2010 Mathematics Subject Classification: 53A07, 53C42.
Key words: Dupin hypersurfaces, Lie curvature, Laplace invariants, lines of curvature.

In this paper we study Dupin hypersurfaces in R⁵ parametrized by lines of curvature, with four distinct principal curvatures. We give a local characterization of this class of hypersurfaces in terms of the principal curvatures and four vector valued functions of one variable. We prove that these vectorial functions describe plane curves or points in R⁵. We show that the Lie curvature of these Dupin hypersurfaces is constant with some conditions on the Laplace invariants and the Möbius curvature, but some Möbius curvatures are constant along certain lines of curvature. We give explicit examples of such Dupin hypersurfaces.

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R. Dangovski rumenrd@mit.edu,
C. Lalov chavdar.lalov@gmail.com

2010 Mathematics Subject Classification: 05A15, 05C38, 05C81, 60G50, 82B41.
Key words: self-avoiding walks, connective constant, honeycomb lattice, asymptotic behaviour.

We study self-avoiding walks (SAWs) on restricted square lattices, more precisely on the lattice strips Z × {−1, 0, 1} and Z × {−1, 0, 1, 2}. We obtain the value of the connective constant for the Z × {−1, 0, 1} lattice in a new shorter way and deduce close bounds for the connective constant for the Z × {−1, 0, 1, 2} lattice. Moreover, for both lattice strips we find close lower and upper bounds for the number of SAWs of length n by using the connective constant. Additionally, we present a transformation of SAWs on the square lattice to a special kind of walks on the honeycomb lattice. By using H. Duminil-Copin and S. Smirnov's results for SAWs on the honeycomb lattice we present non-rigorous ways by which close bounds for the number of SAWs and for the connective constant of the non-restricted square lattice could eventually be obtained without the need of long computer computations.

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