Serdica Mathematical Journal

Volume 46, Number 4, 2020

C O N T E N T S

A B S T R A C T S

CYCLIC PENTAGONS AND HEXAGONS WITH INTEGER SIDES, DIAGONALS AND AREAS
Ajai Choudhry ajaic203@yahoo.com

2020 Mathematics Subject Classification: 11D41, 51M25, 51M99.
Key words: Cyclic polygons, cyclic pentagons, cyclic hexagons, Brahmagupta pentagons, Brahmagupta hexagons.

A polygon all of whose sides, diagonals and the area are given by integers is called a Heron polygon. A cyclic Heron polygon is called a Brahmagupta polygon. In this paper we obtain parametrized families of Brahmagupta pentagons and hexagons, that is, cyclic pentagons and hexagons with integer sides, diagonals and areas.

OSCILLATION THEOREMS FOR SECOND-ORDER NONLINEAR DIFFERENCE EQUATIONS WITH SEVERAL SUBLINEAR NEUTRAL TERMS
G. E. Chatzarakis gea.xatz@aspete.gr, geaxatz@otenet.gr,
R. Kanagasabapathi kanagasabapathi6@gmail.com,
S. Selvarangam selvarangam.9962@gmail.com,
E. Thandapani ethandapani@yahoo.co.in

2020 Mathematics Subject Classification: 39A10.
Key words: Second-order difference equation, sublinear neutral term, oscillation.

This paper presents new sufficient conditions for the oscillation of second-order difference equations with several sublinear neutral terms. The results obtained here extend and generalize the existing results in the literature. Several examples are provided to illustrate and highlight the importance and novelty of the main results.

2020 Mathematics Subject Classification: 62L20, 60F05, 62G07.
Key words: Stochastic approximation algorithm, averaging principle, strong convergence rate.

The aim of this paper is to establish a multivariate compact law of the iterated logarithm for the averaged version of stochastic approximation algorithms. This is achieved by studying the strong convergence rate of a two-time-scale stochastic approximation algorithm, the use of which generalizes the averaging principle of stochastic approximation algorithms. The general result obtained is then applied to the well-known averaged versions of Robbins-Monro's and Kiefer-Wolfowitz's algorithms.

EXAMPLES OF GROUP AMALGAMATIONS WITH~NONTRIVIAL QUASI-KERNELS
Nikolay A. Ivanov nivanov@fmi.uni-sofia.bg

2020 Mathematics Subject Classification: 22D25, 20E06 (Primary) 46L05, 43A07, 20E08 (Secondary).
Key words: C^*-simplicity, free product of groups with amalgamation, inner amenability.

We introduce some examples of group amalgamations motivated by the problems of C^*-simplicity and unique trace property. Moreover, we prove that our examples are not inner amenable and identify a relatively large, simple, normal subgroup in each one.

THE STRUCTURE OF U(F(C_n × D₈))
Sheere Farhat Ansari sheere_farhat@rediffmail.com,
Meena Sahai meena_sahai@hotmail.com

2020 Mathematics Subject Classification: Primary 16U60; Secondary 20C05.
Key words: Unit group, dihedral group, cyclic group.

Let D_n be the dihedral group of order n. The structures of the unit groups of the finite group algebras FD₈ and F(C₂ × D₈) over a field F of characteristic 2 are given in: L. Creedon, J. Gildea. The structure of the unit group of the group algebra F_2^kD₈. Canad. Math. Bull. 54, 2 (2011), 237−243 and J. Gildea. Units of the group algebra F_2^k(C₂ × D₈). J. Algebra Appl. 10, 4 (2011), 643−647, respectively. In this article, we establish the structure of the unit group of the group algebra F(C_n × D₈), n ≥ 1 over a finite field F of characteristic p containing q = p^k elements.