# Caspian Journal of Mathematical Sciences Volume:9 Issue: 2, Summer Autumn 2020

• تاریخ انتشار: 1399/08/27
• تعداد عناوین: 15
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• Majied Dehghan, Saeid Ostadbashi * Pages 159-169
Let \$A\$ be a Banach algebra, \$Omega(A)\$ be the character space of \$A\$ and \$alphainOmega(A)\$. In this paper, we examine the characteristics of \$alpha\$-projective (injective) \$A\$-modules and demonstrate that these character-based \$A\$-modules also satisfy well-known classical homological properties on Banach \$A\$-modules.
Keywords: ‎Banach algebra‎, ‎\$A\$‎- ‎module‎, ‎Banach \$A\$‎- ‎module‎, ‎projective, injective modules
• Yogesh J. Bagul * Pages 170-181
It is pointed out that, one of the results in the recently published article, ’On the Iyengar-Madhava Rao-Nanjundiah inequality and it’s hyperbolic version’ [3] by J´ozsef S´andor is logically incorrect and new corrected result with it’s proof is presented.
Keywords: Iyengar-Madhava Rao-Nanjundiah inequality, logically incorrect, mathematical mistake, corrected version
• Shayesteh Rezaei *, Mostafa Hassanlou Pages 182-190
‎Let \$Omega_X\$ be a bounded‎, ‎circular and strictly convex domain in a complex Banach space \$X\$‎, ‎and \$mathcal{H}(Omega_X)\$ be the space of all holomorphic functions from \$Omega_X\$ to \$mathbb{C}\$‎. ‎The growth space \$mathcal{A}^nu(Omega_X)\$ consists of all \$finmathcal{H}(Omega_X)\$‎ ‎such that \$\$|f(x)|leqslant C nu(r_{Omega_X}(x)),quad xin Omega_X,\$\$‎ ‎for some constant \$C>0\$‎, ‎whenever \$r_{Omega_X}\$ is the Minkowski‎ ‎functional on \$Omega_X\$ and \$nu‎ :‎[0,1)rightarrow(0,infty)\$‎ ‎is a nondecreasing‎, ‎continuous and unbounded function‎. ‎For complex Banach spaces \$X\$ and \$Y\$‎ ‎and a holomorphic map \$varphi:Omega_XrightarrowOmega_Y\$‎, ‎put‎ ‎\$C_varphi( f)=fcirc varphi,finmathcal{H}(Omega_Y)\$‎. ‎We characterize those \$varphi\$ for which the composition operator‎ ‎\$ C_varphi:mathcal{A}^{omega}(Omega_Y)rightarrowmathcal{A}^{nu}(Omega_X)\$ is a bounded or compact operator‎.
Keywords: composition operator‎, ‎growth space‎, ‎circular domain
• AliReza Moniri Hamzekolaee *, Yahya Talebi Rostami, Zahra Heidari Pages 191-198

Lifting modules and their various generalizations as some main concepts in module theory have been studied and investigated extensively in recent decades. Some authors tried to present some homological aspects of lifting modules and -supplemented modules. In this work, we shall present a homological approach to -supplemented modules via fully invariant submodules. Lifting modules and H-supplemented modules with respect to images of a xed fully invariant submodule of a module where investigated in rst author's last works. We intend here to introduce and study a module M such that '(F) has a supplement as a direct summand for every endomorphism ' of M where F is a xed fully invariant submodule of M.

Keywords: oplus-supplemented module, E-oplus-supplemented module, I, F -lifting module, F -oplus-supplemented module, endomorphisms ring
• Abasalt Bodaghi *, Mohammad Maghsoudi Pages 199-209
In this article, we introduce the multi-\$m\$-Jensen mappings and characterize them as a single equation. Using a fixed point theorem, we study the generalized Hyers-Ulam stability for such mappings. As a consequence, we show that every multi-\$m\$-Jensen mappings (under some conditions) is hyperstable.
Keywords: Banach space, multi-m-Jensen mapping, Hyers-Ulam stability
• Mehdi Nodehi, Morteza Norouzi *, OmidReza Dehghan Pages 210-223

In this paper, we define topological hyperrings and study their basic concepts which supported by illustrative examples. We show some differences between topological rings and topological hyperrings. Also, by the fundamental relation \$Gamma^{*}\$, we indicate the role of complete parts (saturated subsets) and complete hyperrings in topological hyperrings and specially we show that if every (closed) open subset is a complete part in a topological complete hyperring then its fundamental ring is a topological ring. Finally, we study the quotient topology induced by \$Gamma^{*}\$-relation on an associated Krasner hyperring obtained by a ring and show that it is isomorphic to a quotient space of the ring by its ideals.

Keywords: Topological hyperring, \$Gamma^{*}\$-relation, complete part, complete hyperring
• Anar Nabiev *, Rauf Amirov Pages 224-242
In the present work, under some di¤erentiability conditions on the potential functions , we rst reduce the inverse scattering problem (ISP) for the polynomial pencil of the Scroedinger equation to the corresponding ISP for the generalized matrix Scrödinger equation . Then ISP will be solved in analogy of the Marchenko method. We aim to establish an e¤ective algorithm for uniquely reconstructing of the potential functions of the equation in that case when there is no a discrete spectrum
Keywords: One dimensional Schrödinger equation, scattering data, Jost solution, direct problem, inverse scattering problem
• Oghovese Ogbereyivwe * Pages 243-255
In this paper, weight function and composition technique is utilized to speeds up the convergence order and increase the efficiency of an existing quadrature based iterative method. This results in the proposition of its improved form from a two-point quadrature based method of convergence order ρ = 3 with efficiency index EI = 1:3161 to a three-point method of convergence order ρ = 8 with EI = 1:5157 at the cost of one additional function evaluation. The method is used to approximate the solution of some nonlinear equations and the results generated are compared with that of some existing methods. Numerical results shows that method developed herein is very efficient in approximation of solution of nonlinear equations.
Keywords: Nonlinear equation, Quadrature formula, Iterative method, weight function
• Goubi Mouloud * Pages 256-265
In this work we study numbers and polynomials generated by two type of composition of generating functions and get their explicit formulae. Furthermore we state an improvementof the composita formulae's given in [6] and [3], using the new composita formula's we construct a variety of combinatorics identities. This study go alone to de ne new family of generalized Bernoulli polynomials which include Hermite-Bernoulli polynomials introduced by G. Dattoli and al [1].
Keywords: Generating function, composition of generating func- tions, composita, Faa di Bruno formula
• Fatemeh Joveini, Mozhgan Akbari * Pages 266-283
In this paper, the Schwarz boundary value problem (BVP) for the inhomogeneous Cauchy-Riemann equation in a triangle is investigated explicitly. Firstly, by the technique of parquetingreflection and the Cauchy-Pompeiu representation formula a modified Cauchy-Schwarz representation formula is obtained. Then, the solution of the Schwarz BVP is explicitly solved. In particular, the boundary behaviors at the corner points are considered.
Keywords: Schwarz problem, Cauchy-Pompeiu formula, Cauchy-Schwarz represen- tation, Triangle
‎We consider the oscillator group equipped with‎ ‎a biinvariant Lorentzian metric‎. ‎Some geometrical properties of this space and the harmonicity properties of left-invariant vector fields on this space are determined‎. ‎In some cases‎, ‎all these vector fields are critical points for the energy functional‎ ‎restricted to vector fields‎. ‎Left-invariant vector fields defining harmonic maps are also classified‎, ‎and the energy of these vector‎ ‎fields is explicitly calculat‎e‎d.
Keywords: Oscillator group‎‎, ‎Harmonic vector fields‎, ‎Harmonic maps
• O. Baghani *, D. Vivek, K Kanagarajan Pages 294-304
This paper investigates the solvability, existence and uniqueness of solutions for a class of nonlinear fractional hybrid differential equations with Hilfer fractional derivative in a weighted normed space. The main result is proved by means of a fixed point theorem due to Dhage. An example to illustrate the results is included.
Keywords: Hybrid fractional differential equations, Initial value problem, Hilfer fractional derivative, fixed point theorem, Existence
• Imed Basdouri *, Ammar Derbali, Mohamed Elkhames Chraygui Pages 305-320
We consider the aff(1)-module structure on the spaces of bilinear diﬀerential operators acting on the spaces of weighted densities. We compute the ﬁrst diﬀerential cohomology of the Lie superalgebra aff(1) with coeﬃcients in space Dλ,ν;µ of bilinear diﬀerential operators acting on weighted densities. We study also the super analogue of this problem getting the same results.
Keywords: Contact geometry, diﬀerential operators, Lie (super)algebra
• Amirahmad Khajehnasiri *, Mostafa Safavi, Jalal Banar Pages 321-339
In this article, we apply the operational matrix to find the numerical solution of two- dimensional nonlinear Volterra integro-differential equation (2DNVIDE). Form this prospect, two-dimensional shifted Legendre functions (2DSLFs) has been presented for integration, product as well as differentiation. This method converts 2DNVIDE to an algebraic system of equations, so the numerical solution of 2DNVIDE is computable. The effectiveness and accuracy of the method were examined with some examples as well. The results and comparison with other methods have shown a remarkable performance.
Keywords: Volterra integro-differential equation, Two-dimensional Legendre polynomials, Operational matrix, Error estimation