Journal of Progressive Research in Mathematics 2021-05-03T06:22:09+00:00 Managing Editor Open Journal Systems Journal of Progressive Research in Mathematics Density Matrix Method Begins To Occupy Leading Position In Quantum Physics 2021-02-24T14:58:53+00:00 Boris V. Bondarev <h2>This article focuses on how quantum mechanics has been discovered and developed over the past century, and how its development has been hindered by many false theories. Some of these theories have come to light because a new tool has not yet been born that would ensure the existence of new theories. Other false quantum theories appeared due to the authors ignorance of the basics of quantum mechanics and are still taught to students.</h2> <h2>This work opens up new paths of quantum physics that start with the Lindblad equation for the statistical operator and the equation for the density matrix. The last equation is equivalent to the equation for the statistical operator.</h2> 2021-02-13T05:17:35+00:00 ##submission.copyrightStatement## Sum Formulas For Generalized Tetranacci Numbers: Closed Forms of the Sum Formulas ∑_{k=0}ⁿx^{k}W_{k} and ∑_{k=1}ⁿx^{k}W_{-k} 2021-02-19T14:58:31+00:00 Yüksel Soykan <p>In this paper, closed forms of the sum formulas ∑_{k=0}ⁿx^{k}W_{k} and ∑_{k=1}ⁿx^{k}W_{-k} for generalized Tetranacci numbers are presented. As special cases, we give summation formulas of Tetranacci, Tetranacci-Lucas, and other fourth-order recurrence sequences.</p> 2021-02-19T14:56:33+00:00 ##submission.copyrightStatement## Estimates of Fekete-Szego Functional of a Sub- class of Analytic and Bi-Univalent Functions by Means of Chebyshev Polynomials 2021-02-24T14:49:45+00:00 Rasheed Olawale Ayinla <p>In this work, a new subclass C( ; ; n; t) of analytic and bi-univalent functions is dened by subordination principle and investigated. The initial coecient bounds and the upper estimates of the Fekete-Szego functional were obtained using Chebyshev polynomials.</p> 2021-02-24T14:49:45+00:00 ##submission.copyrightStatement## Distributed Control for Non-Cooperative Systems Under Conjugation Conditions 2021-03-03T04:51:48+00:00 A. A. Alsaban H. M. Serag Lamiaa Mohamed Abdelrhman <p>In this paper, the distributed control for non-cooperative elliptic systems under conjugation conditions is established. First, the existence and uniqueness of the state for these systems with Dirichlet and conjugation conditions is proved, then the set of equations and inequalities that characterizes the distributed control of these systems is found. The non-cooperative Neumann systems with conjugation conditions is also discussed.</p> 2021-03-03T04:36:51+00:00 ##submission.copyrightStatement## Studies On the Recurrence Properties of Generalized Pentanacci Sequence 2021-04-06T09:47:47+00:00 Yüksel Soykan <p>In this paper, we investigate the recurrence properties of the generalized Pentanacci sequence and present how the generalized Pentanacci sequence at negative indices can be expressed by the sequence itself at positive indices.</p> 2021-04-06T09:47:47+00:00 ##submission.copyrightStatement## Study of the homology theory of Hecke Algebras 2021-04-28T11:05:38+00:00 Yousif Abdullah Alrashidi <p>In the ready product, we study the simplicial and cyclic homology of a unital -graded Hecke algebras &nbsp;over &nbsp;and consider a couple of properties of it. Along these lines, we given a relation between simplicial and cyclic homology of graded Hecke algebras.</p> 2021-04-28T11:05:38+00:00 ##submission.copyrightStatement## Solvability of Forward-Backward Stochastic Partial Differential Equations with Non-Lipschitz Coefficients 2021-04-29T16:08:42+00:00 Hong Yin <p>In this paper we study the solvability of a class of fully-coupled forward-backward stochastic partial differential equations (FBSPDEs). Lipschitz conditions are usually required for the well-posedness of such FBSPDEs. We showed that the Lipschitz conditions can actually be removed by the Yosida Approximation Scheme.</p> 2021-04-29T16:08:42+00:00 ##submission.copyrightStatement## Toeplitz Determinants for a Subclass of Analytic Functions 2021-05-03T06:22:09+00:00 Rasheed Olawale Ayinla Risikat Ayodeji Bello <p>A new subclass of analytic functions that generalizes some known subclasses of analytic functions was defined and investigated. The bounds for Toeplitz determinants of T<sub>2</sub>(2), T<sub>2</sub>(3), T<sub>3(1) </sub> and T<sub>3</sub>(2) were obtained.</p> 2021-05-03T05:54:38+00:00 ##submission.copyrightStatement##