Serdica Mathematical Journal

Volume 48, Number 4, 2022

C O N T E N T S

A B S T R A C T S

A STUDY OF THE MATRIX CLASSES \((\ell_\alpha, \ell_\alpha)\) AND \((\ell_\alpha, c)\), \(0 < \alpha \leq 1\)
Pinnangudi Narayanasubramanian Natarajan pinnangudinatarajan@gmail.com

2020 Mathematics Subject Classification: 40C05, 40H05.
Key words: \(A\)-transform, \(\alpha\)-norm, \(\alpha\)-normed linear space, convex, semigroup, convolution product.

In this paper, entries of sequences, infinite series and infinite matrices are real or complex numbers. The present paper is a continuation of [8] (P. N. Natarajan. Some properties of the matrix class \((\ell_\alpha, \ell_\alpha)\), \(0 < \alpha \leq 1\). Comment. Math. 60, no. 1−2 (2020), 23−36.), where we established some properties of the matrix class \((\ell_\alpha, \ell_\alpha)\), \(0 < \alpha \leq 1\). In this paper, we also record some properties of the class \((\ell_\alpha, c)\) and the sequence space \(\ell_\alpha\), \(0 < \alpha \leq 1\).

APPLICATION OF LIE SYMMETRIES TO SOLVING MODIFIED BLACK-SCHOLES EQUATION
Realeboha Ramoeletsi realeboharamoeletsi@gmail.com,
Bosiu Kaibe bosiukaibe@yahoo.com,
Ngaka J. Nchejane jn.nchejane@nul.ls

2020 Mathematics Subject Classification: 35-XX.
Key words: Lie point symmetries, modified Black-Scholes, financial mathematics, Lie algebra, invariant solution.

We perform Lie symmetry analysis on the modified Black-Scholes model described as a partial differential equation (PDE). As a result, a new complete Lie symmetry group and infinitesimal generators of the one-dimensional modified fractional Black-Scholes model are derived. Furthermore, we compute a family of exact invariant solutions that constitute the modified fractional Black-Scholes model using the associated infinitesimal generators and the corresponding similarity reduction equations. Using known solutions, more solutions are generated via group point transformations.

ALMOST PRIME IDEALS IN NONCOMMUTATIVE RINGS
Alaa Abouhalaka alaa1aclids@gmail.com,
Şehmus Fındık sfindik@cu.edu.tr

2020 Mathematics Subject Classification: 13A15, 17A01.
Key words: almost prime ideals, noncommutative rings.

A proper ideal \(P\) of a commutative ring with identity is an almost prime ideal if \(ab \in P{\setminus}P^2\) implies \(a \in P\) or \(b \in P\). In this paper we define almost prime ideals of a noncommutative ring, and provide some equivalent definitions. We also examine some cases such that all right ideals of a noncommutative ring are almost prime right ideals.

GENERALIZED PSEUDO-DIFFERENTIAL OPERATORS ASSOCIATED WITH SYMBOL CLASSES INVOLVING FRACTIONAL FOURIER TRANSFORM
Abhisekh Shekhar, abhi08.iitkgp@gmail.com,
Nawin Kumar Agrawal drnkumaragrawal@gmail.com

2020 Mathematics Subject Classification: 35S05, 46E10, 46E35, 46F05, 47G30, 46F12.
Key words: fractional Fourier transform, symbol classes, tempered distribution, pseudo-differential operators, Sobolev space.

Generalized pseudo-differential operators \(A(x,\Delta'_x)\) and \(\mathcal{A}(x,\Delta'_x)\) involving fractional Fourier transform (FrFT) associated with symbol-classes are defined and respective symbol classes are introduced. Product and commutators of the generalized pseudo-differential operators are investigated.

HARMONIOUS COLOURING OF LINE GRAPH OF COMMUTING AND NON-COMMUTING GRAPH OF \(D_{2n}\)
R. Divya divyaramakrishnan1224@gmail.com,
P. Chithra Devi chithradevi095@gmail.com

2020 Mathematics Subject Classification: 05C25, 05C76.
Key words: harmonious coloring, harmonious chromatic number, Dihedral group, line graph.

Harmonious coloring of graph \(G\) is a proper vertex coloring, where each pair of colors occurs at most on one pair of adjacent vertices. Minimum number of colors required for Harmonious coloring of \(G\) is the harmonious chromatic number, \(\chi_{H}\left(G\right)\). Here we determine the Harmonious chromatic number of the line graph of commuting graph and non-commuting graph of the dihedral group, \(D_{2n}\).

COEFFICIENT ESTIMATES FOR SOME GENERALIZED SUBCLASSES OF ANALYTIC FUNCTIONS WITH RESPECT TO SYMMETRIC AND CONJUGATE POINTS
Gagandeep Singh kamboj.gagandeep@yahoo.in,
Gurcharanjit Singh dhillongs82@yahoo.com

2020 Mathematics Subject Classification: 30C45, 30C50.
Key words: Univalent functions, close-to-convex functions, starlike functions with respect to symmetric points, subordination, conjugate points, coefficient bounds.

In this paper, we introduce certain unified subclasses of close-to-convex functions and quasi-convex functions with respect to symmetric and conjugate points in the unit disc \(E=\left\{z\in\mathbb{C}:\mid z \mid<1\right\}\) and establish the upper bounds of the first four coefficients for these classes. This study will work as a motivation for the other researchers in this field to study some more similar classes.

MONTE CARLO SIMULATION OF PRESENT VALUE OF CASH FLOWS
Reza Habibi rezahabibi2681@yahoo.com

2020 Mathematics Subject Classification: 60E05.
Key words: ARTA, cash flow, cause factor, infinite series, Monte Carlo, present value, stopping time.

This paper is concerned with financial application of Monte Carlo simulation of infinite series. In fact, present value of a cash flow is represented as infinite series. Hence, the Monte Carlo may be applied to approximate the present value. It is seen that the Monte Carlo method is sample approximation of the mean of cash flow computed at a stopping time. The stopping time, first, is considered as an up-crossing time point of sequence of uniform random variables and then it is replaced by up-crossing of macro-economic variables (called as factor) which brings the dependency structure to the problem which is modeled by auto-regressive to any things (ARTA) models. Here, two different Monte Carlo simulation methods are presented. A financial application is also given. Continuous time extensions of results are also proposed. Finally, a concluding remark section is given.