Volume 3: Issue 1

Volume 3: Issue 1

On the Resolvent of a Non-Self-Adjoint Differential Operator in Hilbert Spaces

Author(s):  Reza Alizadeh and Ali Sameripour
DOI: 10.5281/zenodo.4595051
Keywords :  resolvent; distribution of eigenvalues; non-self-adjoint operators; elliptic operators
Cite this article:  Reza Alizadeh, & Ali Sameripour. (2021). On the Resolvent of a Non-Self-Adjoint Differential Operator in Hilbert Spaces. Online Mathematics Journal, 03(01), 1–7. DOI: 10.5281/zenodo.4595051. [BibTeX Export]

Abstract

The importance of studying non-self-adjoint differential operators is becoming more and more obvious to scientists. The non-self-adjoint operators appear in many branches of science. Today, these operators have many applications in kinetic theory and quantum mechanics to linearization of equations of mathematical physics. The spectrum of these operators is unstable and their resolvent is very unpredictable. In these operators, there is no general spectral theory and this causes problems in the study of these operators. In this paper, we consider a non-self-adjoint elliptic differential operator and study its resolvent.

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A Certain Subclass of Uniformly Convex Functions with Negative Coefficients Defined by Gegenbauer Polynomials

Author(s):  B. Venkateswarlu, P. Thirupathi Reddy, S. Sridevi, and Sujatha Tha
DOI: 10.5281/zenodo.4595454
Keywords :  analytic, coefficient bounds, extreme points, convolution, polynomials
Cite this article:  B. Venkateswarlu, P. Thirupathi Reddy, S. Sridevi, & Sujatha Tha. (2021). A Certain Subclass of Uniformly Convex Functions with Negative Coefficients Defined by Gegenbauer Polynomials. Online Mathematics Journal, 03(01), 8–18. DOI: 10.5281/zenodo.4595454. [BibTeX Export]

Abstract

In this paper, we introduce a new subclass of uniformly convex functions defined by Gegenbauer polynomials with negative coefficients. For functions in the class TS, we attain coefficient bounds, growth distortion properties, extreme points and radii of close-to-convexity, starlikeness and convexity. For this class, we also produced modified Hadamard product, convolution, and integral operators.

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Two-Parameter Generalization of the Collatz Function: Characterization of Terminal Cycles and Empirical Results

Author(s):  Boris B. Stefanov
DOI: 10.5281/zenodo.4609709
Keywords :  analytic, coefficient bounds, extreme points, convolution, polynomials
Cite this article:  Boris B. Stefanov. (2021). Two-Parameter Generalization of the Collatz Function: Characterization of Terminal Cycles and Empirical Results. Online Mathematics Journal, 03(01), 19–25. DOI: 10.5281/zenodo.4609709. [BibTeX Export]

Abstract

This paper proposes a new two-parameter generalization T_{b,s}(x) of the Z -> Z Collatz function T(x) and restates the eponymous conjecture in terms of the proposed function. The generalization obviates some of the conditions for emergence of terminal cycles for the Collatz T(x) function over the integers. The stopping behavior of the T_{b,s}(x) is qualitatively similar to that of the T(x). The paper presents theoretical discussion of the generalization and computational results on the terminal cycles and stopping times of T_{b,s}(x). The {1,2} cycle of T(x) is shown to be a case of coincidence of three independent cycle categories of T_{b,s}(x).

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