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Positive Polynomials by Alexander Prestel,Charles Delzell Pdf
Positivity is one of the most basic mathematical concepts, involved in many areas of mathematics (analysis, real algebraic geometry, functional analysis, etc.). The main objective of the book is to give useful characterizations of polynomials. Beyond basic knowledge in algebra, only valuation theory as explained in the appendix is needed.
Positive Polynomials in Control by Didier Henrion,Andrea Garulli Pdf
Positive Polynomials in Control originates from an invited session presented at the IEEE CDC 2003 and gives a comprehensive overview of existing results in this quickly emerging area. This carefully edited book collects important contributions from several fields of control, optimization, and mathematics, in order to show different views and approaches of polynomial positivity. The book is organized in three parts, reflecting the current trends in the area: 1. applications of positive polynomials and LMI optimization to solve various control problems, 2. a mathematical overview of different algebraic techniques used to cope with polynomial positivity, 3. numerical aspects of positivity of polynomials, and recently developed software tools which can be employed to solve the problems discussed in the book.
Moments, Positive Polynomials and Their Applications by Jean-Bernard Lasserre Pdf
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
Positive Polynomials and Sums of Squares by Murray Marshall Pdf
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.
Positive Polynomials, Convex Integral Polytopes, and a Random Walk Problem by David E. Handelman Pdf
Emanating from the theory of C*-algebras and actions of tori theoren, the problems discussed here are outgrowths of random walk problems on lattices. An AGL (d,Z)-invariant (which is a partially ordered commutative algebra) is obtained for lattice polytopes (compact convex polytopes in Euclidean space whose vertices lie in Zd), and certain algebraic properties of the algebra are related to geometric properties of the polytope. There are also strong connections with convex analysis, Choquet theory, and reflection groups. This book serves as both an introduction to and a research monograph on the many interconnections between these topics, that arise out of questions of the following type: Let f be a (Laurent) polynomial in several real variables, and let P be a (Laurent) polynomial with only positive coefficients; decide under what circumstances there exists an integer n such that Pnf itself also has only positive coefficients. It is intended to reach and be of interest to a general mathematical audience as well as specialists in the areas mentioned.
Resolving Markov Chains onto Bernoulli Shifts via Positive Polynomials by Brian Marcus,Selim Tuncel Pdf
The two parts of this monograph contain two separate but related papers. The longer paper in Part A obtains necessary and sufficient conditions for several types of codings of Markov chains onto Bernoulli shifts. It proceeds by replacing the defining stochastic matrix of each Markov chain by a matrix whose entries are polynomials with positive coefficients in several variables; a Bernoulli shift is represented by a single polynomial with positive coefficients, $p$. This transforms jointly topological and measure-theoretic coding problems into combinatorial ones. In solving the combinatorial problems in Part A, the work states and makes use of facts from Part B concerning $p DEGREESn$ and its coefficients. Part B contains the shorter paper on $p DEGREESn$ and its coefficients, and is independ
Positive Trigonometric Polynomials and Signal Processing Applications by Bogdan Dumitrescu Pdf
This book gathers the main recent results on positive trigonometric polynomials within a unitary framework. The book has two parts: theory and applications. The theory of sum-of-squares trigonometric polynomials is presented unitarily based on the concept of Gram matrix (extended to Gram pair or Gram set). The applications part is organized as a collection of related problems that use systematically the theoretical results.
Certificates of Positivity for Real Polynomials by Victoria Powers Pdf
This book collects and explains the many theorems concerning the existence of certificates of positivity for polynomials that are positive globally or on semialgebraic sets. A certificate of positivity for a real polynomial is an algebraic identity that gives an immediate proof of a positivity condition for the polynomial. Certificates of positivity have their roots in fundamental work of David Hilbert from the late 19th century on positive polynomials and sums of squares. Because of the numerous applications of certificates of positivity in mathematics, applied mathematics, engineering, and other fields, it is desirable to have methods for finding, describing, and characterizing them. For many of the topics covered in this book, appropriate algorithms, computational methods, and applications are discussed. This volume contains a comprehensive, accessible, up-to-date treatment of certificates of positivity, written by an expert in the field. It provides an overview of both the theory and computational aspects of the subject, and includes many of the recent and exciting developments in the area. Background information is given so that beginning graduate students and researchers who are not specialists can learn about this fascinating subject. Furthermore, researchers who work on certificates of positivity or use them in applications will find this a useful reference for their work.
Integers, Polynomials, and Rings by Ronald S. Irving Pdf
This book began life as a set of notes that I developed for a course at the University of Washington entitled Introduction to Modern Algebra for Tea- ers. Originally conceived as a text for future secondary-school mathematics teachers, it has developed into a book that could serve well as a text in an - dergraduatecourseinabstractalgebraoracoursedesignedasanintroduction to higher mathematics. This book di?ers from many undergraduate algebra texts in fundamental ways; the reasons lie in the book’s origin and the goals I set for the course. The course is a two-quarter sequence required of students intending to f- ?ll the requirements of the teacher preparation option for our B.A. degree in mathematics, or of the teacher preparation minor. It is required as well of those intending to matriculate in our university’s Master’s in Teaching p- gram for secondary mathematics teachers. This is the principal course they take involving abstraction and proof, and they come to it with perhaps as little background as a year of calculus and a quarter of linear algebra. The mathematical ability of the students varies widely, as does their level of ma- ematical interest.
Valued Fields by Antonio J. Engler,Alexander Prestel Pdf
Absolute values and their completions – such as the p-adic number fields – play an important role in number theory. Krull's generalization of absolute values to valuations made possible applications in other branches of mathematics. In valuation theory, the notion of completion must be replaced by that of "Henselization". This book develops the theory of valuations as well as of Henselizations, based on the skills of a standard graduate course in algebra.
Optimization of Polynomials in Non-Commuting Variables by Sabine Burgdorf,Igor Klep,Janez Povh Pdf
This book presents recent results on positivity and optimization of polynomials in non-commuting variables. Researchers in non-commutative algebraic geometry, control theory, system engineering, optimization, quantum physics and information science will find the unified notation and mixture of algebraic geometry and mathematical programming useful. Theoretical results are matched with algorithmic considerations; several examples and information on how to use NCSOStools open source package to obtain the results provided. Results are presented on detecting the eigenvalue and trace positivity of polynomials in non-commuting variables using Newton chip method and Newton cyclic chip method, relaxations for constrained and unconstrained optimization problems, semidefinite programming formulations of the relaxations and finite convergence of the hierarchies of these relaxations, and the practical efficiency of algorithms.