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The Problem of Moments

Author : James Alexander Shohat

Publisher : American Mathematical Soc.

Page : 160 pages

File Size : 33,2 MB

Release : 1943-12-31

Category : Mathematics

ISBN : 0821815016

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Book Description

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.



The Moment Problem

Author : Konrad Schmüdgen

Publisher : Springer

Page : 530 pages

File Size : 31,51 MB

Release : 2017-11-09

Category : Mathematics

ISBN : 3319645463

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Book Description

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.



Moments, Positive Polynomials and Their Applications

Author : Jean-Bernard Lasserre

Publisher : World Scientific

Page : 384 pages

File Size : 38,66 MB

Release : 2010

Category : Mathematics

ISBN : 1848164467

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Book Description

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



The Problem of Moments

Author : James Alexander Shohat

Publisher : American Mathematical Society(RI)

Page : 168 pages

File Size : 37,60 MB

Release : 1950

Category : Mathematics

ISBN :

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Book Description

Presents the development of the classical problem of moments for the first 50 years, after its introduction by Stieltjes in the 1890s. This book discusses the initial developments by Stieltjes, Markov, and Chebyshev, and later contributions by Hamburger, Nevanlinna, Hausdorff, and Stone.



Analytic Theory of Continued Fractions

Author : Hubert Stanley Wall

Publisher : Courier Dover Publications

Page : 449 pages

File Size : 25,3 MB

Release : 2018-05-16

Category : Mathematics

ISBN : 0486830446

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Book Description

One of the most authoritative and comprehensive books on the subject of continued fractions, this monograph has been widely used by generations of mathematicians and their students. Dr. Hubert Stanley Wall presents a unified theory correlating certain parts and applications of the subject within a larger analytic structure. Prerequisites include a first course in function theory and knowledge of the elementary properties of linear transformations in the complex plane. Some background in number theory, real analysis, and complex analysis may also prove helpful. The two-part treatment begins with an exploration of convergence theory, addressing continued fractions as products of linear fractional transformations, convergence theorems, and the theory of positive definite continued fractions, as well as other topics. The second part, focusing on function theory, covers the theory of equations, matrix theory of continued fractions, bounded analytic functions, and many additional subjects.



Moments of Impact

Author : Chris Ertel

Publisher : Simon and Schuster

Page : 272 pages

File Size : 36,51 MB

Release : 2014-02-11

Category : Business & Economics

ISBN : 1451697627

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Book Description

Two leading experts on designing strategic conversations unveil a simple, creative process that allows teams to tackle their most challenging issues. In our fast-changing world, leaders are increasingly confronted by messy, multifaceted challenges that require collaboration to resolve. But the standard methods for tackling these challenges—meetings packed with data-drenched presentations or brainstorming sessions that circle back to nowhere—just don’t deliver. Great strategic conversations generate breakthrough insights by combining the best ideas of people with different backgrounds and perspectives. In this book, two experts “crack the code” on what it takes to design creative, collaborative problem-solving sessions that soar rather than sink. Drawing on decades of experience as innovation strategists—and supported by cutting-edge social science research, dozens of real-life examples, and interviews with well over 100 thought leaders, executives, and fellow practitioners— they unveil a simple, creative process that leaders and their teams can use to unlock solutions to their most vexing issues. The book also includes a “Starter Kit” full of tools and tips for putting the book’s core principles into practice.



Great Moments in Mathematics Before 1650

Author : Howard Eves

Publisher : American Mathematical Soc.

Page : 270 pages

File Size : 38,80 MB

Release : 1983-12-31

Category : Mathematics

ISBN : 1614442142

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Book Description

Great Moments in Mathematics: Before 1650 is the product of a series of lectures on the history of mathematics given by Howard Eves. He presents here, in chronological order, 20 ``great moments in mathematics before 1650'', which can be appreciated by anyone who enjoys mathematics. These wonderful lectures could be used as the basis of a course on the history of mathematics but can also serve as enrichment to any mathematics course. Included are lectures on the Pythagorean Theorem, Euclid's Elements, Archimedes (on the sphere), Diophantus, Omar Khayyam, and Fibonacci.



The Power of Moments

Author : Chip Heath

Publisher : Simon and Schuster

Page : 320 pages

File Size : 22,75 MB

Release : 2017-10-03

Category : Business & Economics

ISBN : 1501147765

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Book Description

The New York Times bestselling authors of Switch and Made to Stick explore why certain brief experiences can jolt us and elevate us and change us—and how we can learn to create such extraordinary moments in our life and work. While human lives are endlessly variable, our most memorable positive moments are dominated by four elements: elevation, insight, pride, and connection. If we embrace these elements, we can conjure more moments that matter. What if a teacher could design a lesson that he knew his students would remember twenty years later? What if a manager knew how to create an experience that would delight customers? What if you had a better sense of how to create memories that matter for your children? This book delves into some fascinating mysteries of experience: Why we tend to remember the best or worst moment of an experience, as well as the last moment, and forget the rest. Why “we feel most comfortable when things are certain, but we feel most alive when they’re not.” And why our most cherished memories are clustered into a brief period during our youth. Readers discover how brief experiences can change lives, such as the experiment in which two strangers meet in a room, and forty-five minutes later, they leave as best friends. (What happens in that time?) Or the tale of the world’s youngest female billionaire, who credits her resilience to something her father asked the family at the dinner table. (What was that simple question?) Many of the defining moments in our lives are the result of accident or luck—but why would we leave our most meaningful, memorable moments to chance when we can create them? The Power of Moments shows us how to be the author of richer experiences.



Method of Moments for 2D Scattering Problems

Author : Christophe Bourlier

Publisher : John Wiley & Sons

Page : 122 pages

File Size : 19,12 MB

Release : 2013-08-05

Category : Computers

ISBN : 1118648684

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Book Description

Electromagnetic wave scattering from randomly rough surfaces in the presence of scatterers is an active, interdisciplinary area of research with myriad practical applications in fields such as optics, acoustics, geoscience and remote sensing. In this book, the Method of Moments (MoM) is applied to compute the field scattered by scatterers such as canonical objects (cylinder or plate) or a randomly rough surface, and also by an object above or below a random rough surface. Since the problem is considered to be 2D, the integral equations (IEs) are scalar and only the TE (transverse electric) and TM (transverse magnetic) polarizations are addressed (no cross-polarizations occur). In Chapter 1, the MoM is applied to convert the IEs into a linear system, while Chapter 2 compares the MoM with the exact solution of the field scattered by a cylinder in free space, and with the Physical Optics (PO) approximation for the scattering from a plate in free space. Chapter 3 presents numerical results, obtained from the MoM, of the coherent and incoherent intensities scattered by a random rough surface and an object below a random rough surface. The final chapter presents the same results as in Chapter 3, but for an object above a random rough surface. In these last two chapters, the coupling between the two scatterers is also studied in detail by inverting the impedance matrix by blocks. Contents 1. Integral Equations for a Single Scatterer: Method of Moments and Rough Surfaces. 2. Validation of the Method of Moments for a Single Scatterer. 3. Scattering from Two Illuminated Scatterers. 4. Scattering from Two Scatterers Where Only One is Illuminated. Appendix. Matlab Codes. About the Authors Christophe Bourlier works at the IETR (Institut d’Electronique et de Télécommunications de Rennes) laboratory at Polytech Nantes (University of Nantes, France) as well as being a Researcher at the French National Center for Scientific Research (CNRS) on electromagnetic wave scattering from rough surfaces and objects for remote sensing applications and radar signatures. He is the author of more than 160 journal articles and conference papers. Nicolas Pinel is currently working as a Research Engineer at the IETR laboratory at Polytech Nantes and is about to join Alyotech Technologies in Rennes, France. His research interests are in the areas of radar and optical remote sensing, scattering and propagation. In particular, he works on asymptotic methods of electromagnetic wave scattering from random rough surfaces and layers. Gildas Kubické is in charge of the “Expertise in electroMagnetism and Computation” (EMC) laboratory at the DGA (Direction Générale de l’Armement), French Ministry of Defense, where he works in the field of radar signatures and electromagnetic stealth. His research interests include electromagnetic scattering and radar cross-section modeling.



Matrices, Moments and Quadrature with Applications

Author : Gene H. Golub

Publisher : Princeton University Press

Page : 376 pages

File Size : 21,28 MB

Release : 2009-12-07

Category : Mathematics

ISBN : 1400833884

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Book Description

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.