The Moment Problem
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| Author | : Konrad Schmüdgen |
| Publisher | : Springer |
| Total Pages | : 530 |
| Release | : 2017-11-09 |
| Genre | : Mathematics |
| ISBN | : 3319645463 |
This advanced textbook provides a comprehensive and unified account of the moment problem. It covers the classical one-dimensional theory and its multidimensional generalization, including modern methods and recent developments. In both the one-dimensional and multidimensional cases, the full and truncated moment problems are carefully treated separately. Fundamental concepts, results and methods are developed in detail and accompanied by numerous examples and exercises. Particular attention is given to powerful modern techniques such as real algebraic geometry and Hilbert space operators. A wide range of important aspects are covered, including the Nevanlinna parametrization for indeterminate moment problems, canonical and principal measures for truncated moment problems, the interplay between Positivstellensätze and moment problems on semi-algebraic sets, the fibre theorem, multidimensional determinacy theory, operator-theoretic approaches, and the existence theory and important special topics of multidimensional truncated moment problems. The Moment Problem will be particularly useful to graduate students and researchers working on moment problems, functional analysis, complex analysis, harmonic analysis, real algebraic geometry, polynomial optimization, or systems theory. With notes providing useful background information and exercises of varying difficulty illustrating the theory, this book will also serve as a reference on the subject and can be used for self-study.
| Author | : James Alexander Shohat |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 160 |
| Release | : 1943-12-31 |
| Genre | : Mathematics |
| ISBN | : 0821815016 |
The book was first published in 1943 and then was reprinted several times with corrections. It presents the development of the classical problem of moments for the first 50 years, after its introduction by Stieltjes in the 1890s. In addition to initial developments by Stieltjes, Markov, and Chebyshev, later contributions by Hamburger, Nevanlinna, Hausdorff, Stone, and others are discussed. The book also contains some results on the trigonometric moment problem and a chapter devoted to approximate quadrature formulas.
| Author | : Mark Grigorʹevich Kreĭn |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 430 |
| Release | : |
| Genre | : Mathematics |
| ISBN | : 9780821886717 |
In this book, an extensive circle of questions originating in the classical work of P. L. Chebyshev and A. A. Markov is considered from the more modern point of view. It is shown how results and methods of the generalized moment problem are interlaced with various questions of the geometry of convex bodies, algebra, and function theory. From this standpoint, the structure of convex and conical hulls of curves is studied in detail and isoperimetric inequalities for convex hulls are established; a theory of orthogonal and quasiorthogonal polynomials is constructed; problems on limiting values of integrals and on least deviating functions (in various metrics) are generalized and solved; problems in approximation theory and interpolation and extrapolation in various function classes (analytic, absolutely monotone, almost periodic, etc.) are solved, as well as certain problems in optimal control of linear objects.
| Author | : Jean-Bernard Lasserre |
| Publisher | : World Scientific |
| Total Pages | : 384 |
| Release | : 2010 |
| Genre | : Mathematics |
| ISBN | : 1848164467 |
1. The generalized moment problem. 1.1. Formulations. 1.2. Duality theory. 1.3. Computational complexity. 1.4. Summary. 1.5. Exercises. 1.6. Notes and sources -- 2. Positive polynomials. 2.1. Sum of squares representations and semi-definite optimization. 2.2. Nonnegative versus s.o.s. polynomials. 2.3. Representation theorems : univariate case. 2.4. Representation theorems : mutivariate case. 2.5. Polynomials positive on a compact basic semi-algebraic set. 2.6. Polynomials nonnegative on real varieties. 2.7. Representations with sparsity properties. 2.8. Representation of convex polynomials. 2.9. Summary. 2.10. Exercises. 2.11. Notes and sources -- 3. Moments. 3.1. The one-dimensional moment problem. 3.2. The multi-dimensional moment problem. 3.3. The K-moment problem. 3.4. Moment conditions for bounded density. 3.5. Summary. 3.6. Exercises. 3.7. Notes and sources -- 4. Algorithms for moment problems. 4.1. The overall approach. 4.2. Semidefinite relaxations. 4.3. Extraction of solutions. 4.4. Linear relaxations. 4.5. Extensions. 4.6. Exploiting sparsity. 4.7. Summary. 4.8. Exercises. 4.9. Notes and sources. 4.10. Proofs -- 5. Global optimization over polynomials. 5.1. The primal and dual perspectives. 5.2. Unconstrained polynomial optimization. 5.3. Constrained polynomial optimization : semidefinite relaxations. 5.4. Linear programming relaxations. 5.5. Global optimality conditions. 5.6. Convex polynomial programs. 5.7. Discrete optimization. 5.8. Global minimization of a rational function. 5.9. Exploiting symmetry. 5.10. Summary. 5.11. Exercises. 5.12. Notes and sources -- 6. Systems of polynomial equations. 6.1. Introduction. 6.2. Finding a real solution to systems of polynomial equations. 6.3. Finding all complex and/or all real solutions : a unified treatment. 6.4. Summary. 6.5. Exercises. 6.6. Notes and sources -- 7. Applications in probability. 7.1. Upper bounds on measures with moment conditions. 7.2. Measuring basic semi-algebraic sets. 7.3. Measures with given marginals. 7.4. Summary. 7.5. Exercises. 7.6. Notes and sources -- 8. Markov chains applications. 8.1. Bounds on invariant measures. 8.2. Evaluation of ergodic criteria. 8.3. Summary. 8.4. Exercises. 8.5. Notes and sources -- 9. Application in mathematical finance. 9.1. Option pricing with moment information. 9.2. Option pricing with a dynamic model. 9.3. Summary. 9.4. Notes and sources -- 10. Application in control. 10.1. Introduction. 10.2. Weak formulation of optimal control problems. 10.3. Semidefinite relaxations for the OCP. 10.4. Summary. 10.5. Notes and sources -- 11. Convex envelope and representation of convex sets. 11.1. The convex envelope of a rational function. 11.2. Semidefinite representation of convex sets. 11.3. Algebraic certificates of convexity. 11.4. Summary. 11.5. Exercises. 11.6. Notes and sources -- 12. Multivariate integration 12.1. Integration of a rational function. 12.2. Integration of exponentials of polynomials. 12.3. Maximum entropy estimation. 12.4. Summary. 12.5. Exercises. 12.6. Notes and sources -- 13. Min-max problems and Nash equilibria. 13.1. Robust polynomial optimization. 13.2. Minimizing the sup of finitely many rational cunctions. 13.3. Application to Nash equilibria. 13.4. Exercises. 13.5. Notes and sources -- 14. Bounds on linear PDE. 14.1. Linear partial differential equations. 14.2. Notes and sources
| Author | : Christian Madsbjerg |
| Publisher | : Harvard Business Review Press |
| Total Pages | : 224 |
| Release | : 2014-02-11 |
| Genre | : Business & Economics |
| ISBN | : 1422191907 |
Businesses need a new type of problem solving. Why? Because they are getting people wrong. Traditional problem-solving methods taught in business schools serve us well for some of the everyday challenges of business, but they tend to be ineffective with problems involving a high degree of uncertainty. Why? Because, more often than not, these tools are based on a flawed model of human behavior. And that flawed model is the invisible scaffolding that supports our surveys, our focus groups, our R&D, and much of our long-term strategic planning. In The Moment of Clarity, Christian Madsbjerg and Mikkel Rasmussen examine the business world’s assumptions about human behavior and show how these assumptions can lead businesses off track. But the authors chart a way forward. Using theories and tools from the human sciences—anthropology, sociology, philosophy, and psychology—The Moment of Clarity introduces a practical framework called sensemaking. Sensemaking’s nonlinear problem-solving approach gives executives a better way to understand business challenges involving shifts in human behavior. This new methodology, a fundamentally different way to think about strategy, is already taking off in Fortune 100 companies around the world. Through compelling case studies and their direct experience with LEGO, Samsung, Adidas, Coloplast, and Intel, Madsbjerg and Rasmussen will show you how to solve problems as diverse as setting company direction, driving growth, improving sales models, understanding the real culture of your organization, and finding your way in new markets. Over and over again, executives say the same thing after engaging in a process of sensemaking: “Now I see it . . .” This experience—the moment of clarity—has the potential to drive the entire strategic future of your company. Isn’t it time you and your firm started getting people right? Learn more about the innovation and strategy work of ReD Associates at: redassociates.com
| Author | : Murray Marshall |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 201 |
| Release | : 2008 |
| Genre | : Mathematics |
| ISBN | : 0821844024 |
The study of positive polynomials brings together algebra, geometry and analysis. The subject is of fundamental importance in real algebraic geometry when studying the properties of objects defined by polynomial inequalities. Hilbert's 17th problem and its solution in the first half of the 20th century were landmarks in the early days of the subject. More recently, new connections to the moment problem and to polynomial optimization have been discovered. The moment problem relates linear maps on the multidimensional polynomial ring to positive Borel measures. This book provides an elementary introduction to positive polynomials and sums of squares, the relationship to the moment problem, and the application to polynomial optimization. The focus is on the exciting new developments that have taken place in the last 15 years, arising out of Schmudgen's solution to the moment problem in the compact case in 1991. The book is accessible to a well-motivated student at the beginning graduate level. The objects being dealt with are concrete and down-to-earth, namely polynomials in $n$ variables with real coefficients, and many examples are included. Proofs are presented as clearly and as simply as possible. Various new, simpler proofs appear in the book for the first time. Abstraction is employed only when it serves a useful purpose, but, at the same time, enough abstraction is included to allow the reader easy access to the literature. The book should be essential reading for any beginning student in the area.
| Author | : Mark Grigorʹevich Kreĭn |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 282 |
| Release | : 1968 |
| Genre | : Functional analysis |
| ISBN | : 9780821886403 |
| Author | : Gene H. Golub |
| Publisher | : Princeton University Press |
| Total Pages | : 376 |
| Release | : 2009-12-07 |
| Genre | : Mathematics |
| ISBN | : 1400833884 |
This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part. Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization. This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.
| Author | : Melinda Gates |
| Publisher | : Flatiron Books |
| Total Pages | : 224 |
| Release | : 2019-04-23 |
| Genre | : Social Science |
| ISBN | : 1250313562 |
NEW YORK TIMES BESTSELLER “In her book, Melinda tells the stories of the inspiring people she’s met through her work all over the world, digs into the data, and powerfully illustrates issues that need our attention—from child marriage to gender inequity in the workplace.” — President Barack Obama “The Moment of Lift is an urgent call to courage. It changed how I think about myself, my family, my work, and what’s possible in the world. Melinda weaves together vulnerable, brave storytelling and compelling data to make this one of those rare books that you carry in your heart and mind long after the last page.” — Brené Brown, Ph.D., author of the New York Times #1 bestseller Dare to Lead “Melinda Gates has spent many years working with women around the world. This book is an urgent manifesto for an equal society where women are valued and recognized in all spheres of life. Most of all, it is a call for unity, inclusion and connection. We need this message more than ever.” — Malala Yousafzai "Melinda Gates's book is a lesson in listening. A powerful, poignant, and ultimately humble call to arms." — Tara Westover, author of the New York Times #1 bestseller Educated A debut from Melinda Gates, a timely and necessary call to action for women's empowerment. “How can we summon a moment of lift for human beings – and especially for women? Because when you lift up women, you lift up humanity.” For the last twenty years, Melinda Gates has been on a mission to find solutions for people with the most urgent needs, wherever they live. Throughout this journey, one thing has become increasingly clear to her: If you want to lift a society up, you need to stop keeping women down. In this moving and compelling book, Melinda shares lessons she’s learned from the inspiring people she’s met during her work and travels around the world. As she writes in the introduction, “That is why I had to write this book—to share the stories of people who have given focus and urgency to my life. I want all of us to see ways we can lift women up where we live.” Melinda’s unforgettable narrative is backed by startling data as she presents the issues that most need our attention—from child marriage to lack of access to contraceptives to gender inequity in the workplace. And, for the first time, she writes about her personal life and the road to equality in her own marriage. Throughout, she shows how there has never been more opportunity to change the world—and ourselves. Writing with emotion, candor, and grace, she introduces us to remarkable women and shows the power of connecting with one another. When we lift others up, they lift us up, too.
| Author | : Raúl E. Curto |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 73 |
| Release | : 1998 |
| Genre | : Mathematics |
| ISBN | : 0821808699 |
In this book, the authors develop new computational tests for existence and uniqueness of representing measures $\mu$ in the Truncated Complex Moment Problem: $\gamma {ij}=\int \bar zizj\, d\mu$ $(0\le i+j\le 2n)$. Conditions for the existence of finitely atomic representing measures are expressed in terms of positivity and extension properties of the moment matrix $M(n)(\gamma )$ associated with $\gamma \equiv \gamma {(2n)}$: $\gamma {00}, \dots ,\gamma {0,2n},\dots ,\gamma {2n,0}$, $\gamma {00}>0$. This study includes new conditions for flat (i.e., rank-preserving) extensions $M(n+1)$ of $M(n)\ge 0$; each such extension corresponds to a distinct rank $M(n)$-atomic representing measure, and each such measure is minimal among representing measures in terms of the cardinality of its support. For a natural class of moment matrices satisfying the tests of recursive generation, recursive consistency, and normal consistency, the existence problem for minimal representing measures is reduced to the solubility of small systems of multivariable algebraic equations. In a variety of applications, including cases of the quartic moment problem ($n=2$), the text includes explicit contructions of minimal representing measures via the theory of flat extensions. Additional computational texts are used to prove non-existence of representing measures or the non-existence of minimal representing measures. These tests are used to illustrate, in very concrete terms, new phenomena, associated with higher-dimensional moment problems that do not appear in the classical one-dimensional moment problem.
