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15 pages, 294 KB

Open AccessArticle

On Convergence of Toeplitz Quantization of the Sphere

by Yanlin Li, Mohamed Lemine H. Bouleryah and Akram Ali

Mathematics 2024, 12(22), 3565; https://doi.org/10.3390/math12223565 - 14 Nov 2024

Cited by 7 | Viewed by 1096

In this paper, we give an explicit expression of the Toeplitz quantization of a

C^{\infty}

smooth function on the sphere and show that the sequence of spectra of Toeplitz quantization of the function determines its decreasing rearrangement. We also use Toeplitz quantization [...] Read more.

In this paper, we give an explicit expression of the Toeplitz quantization of a

C^{\infty}

smooth function on the sphere and show that the sequence of spectra of Toeplitz quantization of the function determines its decreasing rearrangement. We also use Toeplitz quantization to prove a version of Szegö’s Theorem. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Its Applications, 2nd Edition)

14 pages, 319 KB

Open AccessArticle

Coefficient Inequalities for Multivalent Janowski Type q-Starlike Functions Involving Certain Conic Domains

by Muhammad Sabil Ur Rehman, Qazi Zahoor Ahmad, Isra Al-shbeil, Sarfraz Ahmad, Ajmal Khan, Bilal Khan and Jianhua Gong

Axioms 2022, 11(10), 494; https://doi.org/10.3390/axioms11100494 - 23 Sep 2022

Cited by 12 | Viewed by 1721

Abstract

In the current work, by using the familiar q-calculus, first, we study certain generalized conic-type regions. We then introduce and study a subclass of the multivalent q-starlike functions that map the open unit disk into the generalized conic domain. Next, we [...] Read more.

In the current work, by using the familiar q-calculus, first, we study certain generalized conic-type regions. We then introduce and study a subclass of the multivalent q-starlike functions that map the open unit disk into the generalized conic domain. Next, we study potentially effective outcomes such as sufficient restrictions and the Fekete–Szegö type inequalities. We attain lower bounds for the ratio of a good few functions related to this lately established class and sequences of the partial sums. Furthermore, we acquire a number of attributes of the corresponding class of q-starlike functions having negative Taylor–Maclaurin coefficients, including distortion theorems. Moreover, various important corollaries are carried out. The new explorations appear to be in line with a good few prior commissions and the current area of our recent investigation. Full article

(This article belongs to the Special Issue Special Functions and Polynomials: Theory, Practice, Applications, and Modeling)

20 pages, 332 KB

Open AccessArticle

Applications of Symmetric Conic Domains to a Subclass of q-Starlike Functions

by Shahid Khan, Nazar Khan, Aftab Hussain, Serkan Araci, Bilal Khan and Hamed H. Al-Sulami

Symmetry 2022, 14(4), 803; https://doi.org/10.3390/sym14040803 - 12 Apr 2022

Cited by 11 | Viewed by 2257

Abstract

In this paper, the theory of symmetric q-calculus and conic regions are used to define a new subclass of q-starlike functions involving a certain conic domain. By means of this newly defined domain, a new subclass of normalized analytic functions in [...] Read more.

In this paper, the theory of symmetric q-calculus and conic regions are used to define a new subclass of q-starlike functions involving a certain conic domain. By means of this newly defined domain, a new subclass of normalized analytic functions in the open unit disk E is given. Certain properties of this subclass, such as its structural formula, necessary and sufficient conditions, coefficient estimates, Fekete–Szegö problem, distortion inequalities, closure theorem and subordination results, are investigated. Some new and known consequences of our main results as corollaries are also highlighted. Full article

(This article belongs to the Special Issue Symmetry in Pure Mathematics and Real and Complex Analysis)

16 pages, 297 KB

Open AccessArticle

A Certain Subclass of Multivalent Analytic Functions Defined by the q-Difference Operator Related to the Janowski Functions

by Bo Wang, Rekha Srivastava and Jin-Lin Liu

Mathematics 2021, 9(14), 1706; https://doi.org/10.3390/math9141706 - 20 Jul 2021

Cited by 14 | Viewed by 2230

Abstract

A class of p-valent analytic functions is introduced using the q-difference operator and the familiar Janowski functions. Several properties of functions in the class, such as the Fekete–Szegö inequality, coefficient estimates, necessary and sufficient conditions, distortion and growth theorems, radii of [...] Read more.

A class of p-valent analytic functions is introduced using the q-difference operator and the familiar Janowski functions. Several properties of functions in the class, such as the Fekete–Szegö inequality, coefficient estimates, necessary and sufficient conditions, distortion and growth theorems, radii of convexity and starlikeness, closure theorems and partial sums, are discussed in this paper. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

18 pages, 370 KB

Open AccessArticle

Applications of Certain Conic Domains to a Subclass of q-Starlike Functions Associated with the Janowski Functions

by Bilal Khan, Hari Mohan Srivastava, Nazar Khan, Maslina Darus, Qazi Zahoor Ahmad and Muhammad Tahir

Symmetry 2021, 13(4), 574; https://doi.org/10.3390/sym13040574 - 31 Mar 2021

Cited by 27 | Viewed by 3266

Abstract

In our present investigation, with the help of the basic (or q-) calculus, we first define a new domain which involves the Janowski function. We also define a new subclass of the class of q-starlike functions, which maps the open unit [...] Read more.

In our present investigation, with the help of the basic (or q-) calculus, we first define a new domain which involves the Janowski function. We also define a new subclass of the class of q-starlike functions, which maps the open unit disk

U

, given by

U = \{z : z \in C and |z| < 1\},

onto this generalized conic type domain. We study here some such potentially useful results as, for example, the sufficient conditions, closure results, the Fekete-Szegö type inequalities and distortion theorems. We also obtain the lower bounds for the ratio of some functions which belong to this newly-defined function class and for the sequences of the partial sums. Our results are shown to be connected with several earlier works related to the field of our present investigation. Finally, in the concluding section, we have chosen to reiterate the well-demonstrated fact that any attempt to produce the rather straightforward

(p, q)

-variations of the results, which we have presented in this article, will be a rather trivial and inconsequential exercise, simply because the additional parameter p is obviously redundant. Full article

(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)

14 pages, 277 KB

Open AccessArticle

Q-Extension of Starlike Functions Subordinated with a Trigonometric Sine Function

by Saeed Islam, Muhammad Ghaffar Khan, Bakhtiar Ahmad, Muhammad Arif and Ronnason Chinram

Mathematics 2020, 8(10), 1676; https://doi.org/10.3390/math8101676 - 1 Oct 2020

Cited by 16 | Viewed by 2791

Abstract

The main purpose of this article is to examine the q-analog of starlike functions connected with a trigonometric sine function. Further, we discuss some interesting geometric properties, such as the well-known problems of Fekete-Szegö, the necessary and sufficient condition, the growth and [...] Read more.

The main purpose of this article is to examine the q-analog of starlike functions connected with a trigonometric sine function. Further, we discuss some interesting geometric properties, such as the well-known problems of Fekete-Szegö, the necessary and sufficient condition, the growth and distortion bound, closure theorem, convolution results, radii of starlikeness, extreme point theorem and the problem with partial sums for this class. Full article

(This article belongs to the Special Issue Complex Analysis and Geometric Function Theory)

25 pages, 8190 KB

Open AccessArticle

Permanent-Magnet SLM Drive System Using AMRRSPNNB Control System with DGWO

by Der-Fa Chen, Yi-Cheng Shih, Shih-Cheng Li, Chin-Tung Chen and Jung-Chu Ting

Energies 2020, 13(11), 2914; https://doi.org/10.3390/en13112914 - 6 Jun 2020

Cited by 2 | Viewed by 2374

Abstract

Because permanent-magnet synchronous linear motors (SLM) still exhibit nonlinear friction, ending effects and time-varying dynamic uncertainties, better control performances cannot be achieved by using common linear controllers. We propose a backstepping approach with three adaptive laws and a beating function to control the [...] Read more.

Because permanent-magnet synchronous linear motors (SLM) still exhibit nonlinear friction, ending effects and time-varying dynamic uncertainties, better control performances cannot be achieved by using common linear controllers. We propose a backstepping approach with three adaptive laws and a beating function to control the motion of permanent-magnet SLM drive systems that enhance the robustness of the system. In order to reduce greater vibration in situations with uncertainty actions in the aforementioned control systems, we propose an adaptive modified recurrent Rogers–Szego polynomials neural network backstepping (AMRRSPNNB) control system with three adaptive laws and reimbursed controller with decorated gray wolf optimization (DGWO), in order to handle external bunched force uncertainty, including nonlinear friction, ending effects and time-varying dynamic uncertainties, as well as to reimburse the minimal rebuild error of the reckoned law. In accordance with the Lyapunov stability, online parameter training method of the modified recurrent Rogers–Szego polynomials neural network (MRRSPNN) can be derived by utilizing an adaptive law. Furthermore, to help reduce error and better obtain learning fulfillment, the DGWO algorithm was used to change the two learning rates in the weights of the MRRSPNN. Finally, the usefulness of the proposed control system is validated by tested results. Full article

(This article belongs to the Special Issue Control and Monitoring of Permanent Magnet Synchronous Machines)

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28 pages, 10078 KB

Open AccessArticle

Mixed Modified Recurring Rogers-Szego Polynomials Neural Network Control with Mended Grey Wolf Optimization Applied in SIM Expelling System

by Der-Fa Chen, Yi-Cheng Shih, Shih-Cheng Li, Chin-Tung Chen and Jung-Chu Ting

Mathematics 2020, 8(5), 754; https://doi.org/10.3390/math8050754 - 9 May 2020

Cited by 3 | Viewed by 2370

Abstract

Due to a good ability of learning for nonlinear uncertainties, a mixed modified recurring Rogers-Szego polynomials neural network (MMRRSPNN) control with mended grey wolf optimization (MGWO) by using two linear adjusted factors is proposed to the six-phase induction motor (SIM) expelling continuously variable [...] Read more.

Due to a good ability of learning for nonlinear uncertainties, a mixed modified recurring Rogers-Szego polynomials neural network (MMRRSPNN) control with mended grey wolf optimization (MGWO) by using two linear adjusted factors is proposed to the six-phase induction motor (SIM) expelling continuously variable transmission (CVT) organized system for acquiring better control performance. The control system can execute MRRSPNN control with a fitted learning rule, and repay control with an evaluated rule. In the light of the Lyapunov stability theorem, the fitted learning rule in the MRRSPNN control can be derived, and the evaluated rule of the repay control can be originated. Besides, the MGWO by using two linear adjusted factors yields two changeable learning rates for two parameters to find two ideal values and to speed-up convergence of weights. Experimental results in comparisons with some control systems are demonstrated to confirm that the proposed control system can achieve better control performance. Full article

(This article belongs to the Special Issue Neural Networks and Learning Systems)

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13 pages, 241 KB

Open AccessArticle

Boas’ Formula and Sampling Theorem

by Tohru Morita and Ken-ichi Sato

Axioms 2015, 4(1), 71-83; https://doi.org/10.3390/axioms4010071 - 26 Jan 2015

Viewed by 4871

Abstract

In 1937, Boas gave a smart proof for an extension of the Bernstein theorem for trigonometric series. It is the purpose of the present note (i) to point out that a formula which Boas used in the proof is related with the Shannon [...] Read more.

In 1937, Boas gave a smart proof for an extension of the Bernstein theorem for trigonometric series. It is the purpose of the present note (i) to point out that a formula which Boas used in the proof is related with the Shannon sampling theorem; (ii) to present a generalized Parseval formula, which is suggested by the Boas’ formula; and (iii) to show that this provides a very smart derivation of the Shannon sampling theorem for a function which is the Fourier transform of a distribution involving the Dirac delta function. It is also shows that, by the argument giving Boas’ formula for the derivative f'(x) of a function f(x), we can derive the corresponding formula for f'''(x), by which we can obtain an upperbound of |f'''(x)+3R²f'(x)|. Discussions are given also on an extension of the Szegö theorem for trigonometric series, which Boas mentioned in the same paper. Full article

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