Vijay Gupta

This book presents an in-depth study on advances in constructive approximation theory with recent problems on linear positive operators. State-of-the-art research in constructive approximation is treated with extensions to approximation results on linear positive operators in a post quantum and bivariate setting. Methods, techniques, and problems in approximation theory are demonstrated with applications to optimization, physics, and biology. Graduate students, research scientists and engineers working in mathematics, physics, and industry will broaden their understanding of operators essential to pure and applied mathematics. Topics discussed include: discrete operators, quantitative estimates, post-quantum calculus, integral operators, univariate Gruss-type inequalities for positive linear operators, bivariate operators of discrete and integral type, convergence of GBS operators.

This book presents an in-depth study on advances in constructive approximation theory with recent problems on linear positive operators. State-of-the-art research in constructive approximation is treated with extensions to approximation results on linear positive operators in a post quantum and bivariate setting. Methods, techniques, and problems in approximation theory are demonstrated with applications to optimization, physics, and biology. Graduate students, research scientists and engineers working in mathematics, physics, and industry will broaden their understanding of operators essential to pure and applied mathematics. Topics discussed include: discrete operators, quantitative estimates, post-quantum calculus, integral operators, univariate Gruss-type inequalities for positive linear operators, bivariate operators of discrete and integral type, convergence of GBS operators.

From the world's great concert halls to Los Angeles's Skid Row, violinist Vijay Gupta's searing memoir of prodigy, ambition, collapse, and renewal reveals how music is not just performance but also survival, a lifeline of human connection when everything else falls away--for readers of Jeremy Denk's Every Good Boy Does Fine Hua Hsu's Stay True, and Patrick Bringley's All the Beauty in the World. By twenty-five, Vijay Gupta had lived several lifetimes: he played Carnegie Hall at eight, studied at Juilliard and Yale before most had finished high school, joined the Los Angeles Philharmonic at nineteen, gave a celebrated TED Talk seen by millions, and launched a nonprofit. But behind the accolades was estrangement, addiction, and a private unraveling. Restrung is Gupta's unflinching memoir of breaking apart and remaking a self. It begins with a boy raised between the strict devotion of Bengali immigrant parents and the ruthless demands of the conservatory. It follows him through the shimmering world of elite orchestras, into the depths of burnout and family collapse, and ultimately toward an unexpected reawakening--where he discovered that the music he'd spent his life studying was seen not as a curio of high culture or mere entertainment, but a lifeline of connection--most vividly in Skid Row, where people living through addiction, homelessness, and incarceration heard it as survival itself. There, audiences spoke to how they saw their own lives reflected in the stories of composers too often frozen into marble busts: the rage of Beethoven, the fragility of Schumann's mind, the alienation of Bart k, the plight of Handel--who wrote Messiah bankrupt, ill, and broken, yet transformed despair into an enduring Hallelujah. Pico Iyer, in his foreword, calls Restrung "a rich and astonishing feast of stories" told with "vivid precision, unflinching candor, and heart," a book that unsettles assumptions about success while illuminating how art restores not just audiences, but artists themselves.

From the world's great concert halls to Los Angeles's Skid Row, violinist Vijay Gupta's searing memoir of prodigy, ambition, collapse, and renewal reveals how music is not just performance but also survival, a lifeline of human connection when everything else falls away--for readers of Jeremy Denk's Every Good Boy Does Fine Hua Hsu's Stay True, and Patrick Bringley's All the Beauty in the World. By twenty-five, Vijay Gupta had lived several lifetimes: he played Carnegie Hall at eight, studied at Juilliard and Yale before most had finished high school, joined the Los Angeles Philharmonic at nineteen, gave a celebrated TED Talk seen by millions, and launched a nonprofit. But behind the accolades was estrangement, addiction, and a private unraveling. Restrung is Gupta's unflinching memoir of breaking apart and remaking a self. It begins with a boy raised between the strict devotion of Bengali immigrant parents and the ruthless demands of the conservatory. It follows him through the shimmering world of elite orchestras, into the depths of burnout and family collapse, and ultimately toward an unexpected reawakening--where he discovered that the music he'd spent his life studying was seen not as a curio of high culture or mere entertainment, but a lifeline of connection--most vividly in Skid Row, where people living through addiction, homelessness, and incarceration heard it as survival itself. There, audiences spoke to how they saw their own lives reflected in the stories of composers too often frozen into marble busts: the rage of Beethoven, the fragility of Schumann's mind, the alienation of Bart k, the plight of Handel--who wrote Messiah bankrupt, ill, and broken, yet transformed despair into an enduring Hallelujah. Pico Iyer, in his foreword, calls Restrung "a rich and astonishing feast of stories" told with "vivid precision, unflinching candor, and heart," a book that unsettles assumptions about success while illuminating how art restores not just audiences, but artists themselves.

This brief studies recent work conducted on certain exponential type operators and other integral type operators. It consists of three chapters: the first on exponential type operators, the second a study of some modifications of linear positive operators, and the third on difference estimates between two operators. It will be of interest to students both graduate and undergraduate studying linear positive operators and the area of approximation theory.

This book is a valuable resource for Graduate students and researchers interested in current techniques and methods within the theory of moments in linear positive operators and approximation theory. Moments are essential to the convergence of a sequence of linear positive operators. Several methods are examined to determine moments including direct calculations, recurrence relations, and the application of hypergeometric series. A collection of operators in the theory of approximation are investigated through their moments and a variety of results are surveyed with fundamental theories and recent developments. Detailed examples are included to assist readers understand vital theories and potential applications.

This book presents a systematic overview of approximation by linear combinations of positive linear operators, a useful tool used to increase the order of approximation. Fundamental and recent results from the past decade are described with their corresponding proofs. The volume consists of eight chapters that provide detailed insight into the representation of monomials of the operators Ln , direct and inverse estimates for a broad class of positive linear operators, and case studies involving finite and unbounded intervals of real and complex functions. Strong converse inequalities of Type A in terminology of Ditzian–Ivanov for linear combinations of Bernstein and Bernstein–Kantorovich operators and various Voronovskaja-type estimates for some linear combinations are analyzed and explained. Graduate students and researchers in approximation theory will find the list of open problems in approximation of linear combinations useful. The book serves as a reference for graduateand postgraduate courses as well as a basis for future study and development.

This book presents a systematic overview of approximation by linear combinations of positive linear operators, a useful tool used to increase the order of approximation. Fundamental and recent results from the past decade are described with their corresponding proofs. The volume consists of eight chapters that provide detailed insight into the representation of monomials of the operators Ln , direct and inverse estimates for a broad class of positive linear operators, and case studies involving finite and unbounded intervals of real and complex functions. Strong converse inequalities of Type A in terminology of Ditzian–Ivanov for linear combinations of Bernstein and Bernstein–Kantorovich operators and various Voronovskaja-type estimates for some linear combinations are analyzed and explained. Graduate students and researchers in approximation theory will find the list of open problems in approximation of linear combinations useful. The book serves as a reference for graduateand postgraduate courses as well as a basis for future study and development.

The study of linear positive operators is an area of mathematical studies with significant relevance to studies of computer-aided geometric design, numerical analysis, and differential equations. This book focuses on the convergence of linear positive operators in real and complex domains. The theoretical aspects of these operators have been an active area of research over the past few decades. In this volume, authors Gupta and Agarwal explore new and more efficient methods of applying this research to studies in Optimization and Analysis. The text will be of interest to upper-level students seeking an introduction to the field and to researchers developing innovative approaches.

The approximation of functions by linear positive operators is an important research topic in general mathematics and it also provides powerful tools to application areas such as computer-aided geometric design, numerical analysis, and solutions of differential equations. q-Calculus is a generalization of many subjects, such as hypergeometric series, complex analysis, and particle physics. ?? This monograph is an introduction to combining approximation theory and q-Calculus with applications, by using well- known operators. The presentation is systematic and the authors include a brief summary of the notations and basic definitions of q-calculus before delving into more advanced material. The many applications of q-calculus in the theory of approximation, especially on various operators, which includes convergence of operators to functions in real and complex domain? forms the gist of the book. This book is suitable for researchers and students in mathematics, physics and engineering, and for professionals who would enjoy exploring the host of mathematical techniques and ideas that are collected and discussed in the book.

The study of linear positive operators is an area of mathematical studies with significant relevance to studies of computer-aided geometric design, numerical analysis, and differential equations. This book focuses on the convergence of linear positive operators in real and complex domains. The theoretical aspects of these operators have been an active area of research over the past few decades. In this volume, authors Gupta and Agarwal explore new and more efficient methods of applying this research to studies in Optimization and Analysis. The text will be of interest to upper-level students seeking an introduction to the field and to researchers developing innovative approaches.

The approximation of functions by linear positive operators is an important research topic in general mathematics and it also provides powerful tools to application areas such as computer-aided geometric design, numerical analysis, and solutions of differential equations. q-Calculus is a generalization of many subjects, such as hypergeometric series, complex analysis, and particle physics. ?? This monograph is an introduction to combining approximation theory and q-Calculus with applications, by using well- known operators. The presentation is systematic and the authors include a brief summary of the notations and basic definitions of q-calculus before delving into more advanced material. The many applications of q-calculus in the theory of approximation, especially on various operators, which includes convergence of operators to functions in real and complex domain? forms the gist of the book. This book is suitable for researchers and students in mathematics, physics and engineering, and for professionals who would enjoy exploring the host of mathematical techniques and ideas that are collected and discussed in the book.

Revision with unchanged content. Rapid advances in information processing, communication and sensing technologies have enabled more and more devices to be provided with embedded processors, networking capabilities and sensors. For the field of estimation and control, it is now possible to consider an architecture in which many simple components communicate and cooperate to achieve a joint team goal. This distributed architecture promises much in terms of performance, reliability and simplicity of design. However, at the same time, it requires extending the traditional theories of control, communication and computation and, in fact, looking at a unified picture of the three fields. The chief idea explored in this book is the joint design of information flow and the control law. While traditional control design has concentrated on calculating the optimal control input by assuming a particular information flow between the components, our approach seeks to synthesize the optimal information flow along with the optimal control law that satisfies the constraints of the information flow. As we demonstrate in the book, the joint design of information flow and the optimal control input satisfying the constraints of that information flow yields large improvements in performance over simply trying to fit traditional design theories on distributed systems.