Finitely Additive Measures and Relaxations of Extremal Problems

Finitely Additive Measures and Relaxations of Extremal Problems

Author: A.G. Chentsov

Publisher: Springer Science & Business Media

Published: 1996-09-30

Total Pages: 261

ISBN-13: 0306110385

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This monograph constructs correct extensions of extremal problems, including problems of multicriteria optimization as well as more general cone optimization problems. The author obtains common conditions of stability and asymptotic nonsensitivity of extremal problems under perturbation of a part of integral restrictions for finite and infinite systems of restrictions. Features include individual chapters on nonstandard approximation of finitely additive measures by indefinite integrals and constructions of attraction sets. Professor Chentsov illustrates abstract settings by providing examples of problems of impulse control, mathematical programming, and stochastic optimization.


Book Synopsis Finitely Additive Measures and Relaxations of Extremal Problems by : A.G. Chentsov

Download or read book Finitely Additive Measures and Relaxations of Extremal Problems written by A.G. Chentsov and published by Springer Science & Business Media. This book was released on 1996-09-30 with total page 261 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph constructs correct extensions of extremal problems, including problems of multicriteria optimization as well as more general cone optimization problems. The author obtains common conditions of stability and asymptotic nonsensitivity of extremal problems under perturbation of a part of integral restrictions for finite and infinite systems of restrictions. Features include individual chapters on nonstandard approximation of finitely additive measures by indefinite integrals and constructions of attraction sets. Professor Chentsov illustrates abstract settings by providing examples of problems of impulse control, mathematical programming, and stochastic optimization.

The Dual of L∞(X,L,λ), Finitely Additive Measures and Weak Convergence

The Dual of L∞(X,L,λ), Finitely Additive Measures and Weak Convergence

Author: John Toland

Publisher: Springer Nature

Published: 2020-01-03

Total Pages: 104

ISBN-13: 303034732X

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In measure theory, a familiar representation theorem due to F. Riesz identifies the dual space Lp(X,L,λ)* with Lq(X,L,λ), where 1/p+1/q=1, as long as 1 ≤ p∞. However, iL/isub∞/sub(X,L,λ)* cannot be similarly described, and is instead represented as a class of finitely additive measures./ppThis book provides a reasonably elementary account of the representation theory of iL/isub∞/sub(X,L,λ)*, examining pathologies and paradoxes, and uncovering some surprising consequences. For instance, a necessary and sufficient condition for a bounded sequence in iL/isub∞/sub(X,L,λ) to be weakly convergent, applicable in the one-point compactification of X, is given./ppWith a clear summary of prerequisites, and illustrated by examples including iL/isub∞/sub(bR/bsupn/sup) and the sequence space il/isub∞/sub, this book makes possibly unfamiliar material, some of which may be new, accessible to students and researchers in the mathematical sciences.


Book Synopsis The Dual of L∞(X,L,λ), Finitely Additive Measures and Weak Convergence by : John Toland

Download or read book The Dual of L∞(X,L,λ), Finitely Additive Measures and Weak Convergence written by John Toland and published by Springer Nature. This book was released on 2020-01-03 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt: In measure theory, a familiar representation theorem due to F. Riesz identifies the dual space Lp(X,L,λ)* with Lq(X,L,λ), where 1/p+1/q=1, as long as 1 ≤ p∞. However, iL/isub∞/sub(X,L,λ)* cannot be similarly described, and is instead represented as a class of finitely additive measures./ppThis book provides a reasonably elementary account of the representation theory of iL/isub∞/sub(X,L,λ)*, examining pathologies and paradoxes, and uncovering some surprising consequences. For instance, a necessary and sufficient condition for a bounded sequence in iL/isub∞/sub(X,L,λ) to be weakly convergent, applicable in the one-point compactification of X, is given./ppWith a clear summary of prerequisites, and illustrated by examples including iL/isub∞/sub(bR/bsupn/sup) and the sequence space il/isub∞/sub, this book makes possibly unfamiliar material, some of which may be new, accessible to students and researchers in the mathematical sciences.

Extensions and Relaxations

Extensions and Relaxations

Author: A.G. Chentsov

Publisher: Springer Science & Business Media

Published: 2013-03-14

Total Pages: 420

ISBN-13: 9401715270

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In this book a general topological construction of extension is proposed for problems of attainability in topological spaces under perturbation of a system of constraints. This construction is realized in a special class of generalized elements defined as finitely additive measures. A version of the method of programmed iterations is constructed. This version realizes multi-valued control quasistrategies, which guarantees the solution of the control problem that consists in guidance to a given set under observation of phase constraints. Audience: The book will be of interest to researchers, and graduate students in the field of optimal control, mathematical systems theory, measure and integration, functional analysis, and general topology.


Book Synopsis Extensions and Relaxations by : A.G. Chentsov

Download or read book Extensions and Relaxations written by A.G. Chentsov and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 420 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book a general topological construction of extension is proposed for problems of attainability in topological spaces under perturbation of a system of constraints. This construction is realized in a special class of generalized elements defined as finitely additive measures. A version of the method of programmed iterations is constructed. This version realizes multi-valued control quasistrategies, which guarantees the solution of the control problem that consists in guidance to a given set under observation of phase constraints. Audience: The book will be of interest to researchers, and graduate students in the field of optimal control, mathematical systems theory, measure and integration, functional analysis, and general topology.

Asymptotic Attainability

Asymptotic Attainability

Author: A.G. Chentsov

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 336

ISBN-13: 9401708053

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In this monograph, questions of extensions and relaxations are consid ered. These questions arise in many applied problems in connection with the operation of perturbations. In some cases, the operation of "small" per turbations generates "small" deviations of basis indexes; a corresponding stability takes place. In other cases, small perturbations generate spas modic change of a result and of solutions defining this result. These cases correspond to unstable problems. The effect of an unstability can arise in extremal problems or in other related problems. In this connection, we note the known problem of constructing the attainability domain in con trol theory. Of course, extremal problems and those of attainability (in abstract control theory) are connected. We exploit this connection here (see Chapter 5). However, basic attention is paid to the problem of the attainability of elements of a topological space under vanishing perturba tions of restrictions. The stability property is frequently missing; the world of unstable problems is of interest for us. We construct regularizing proce dures. However, in many cases, it is possible to establish a certain property similar to partial stability. We call this property asymptotic nonsensitivity or roughness under the perturbation of some restrictions. The given prop erty means the following: in the corresponding problem, it is the same if constraints are weakened in some "directions" or not. On this basis, it is possible to construct a certain classification of constraints, selecting "di rections of roughness" and "precision directions".


Book Synopsis Asymptotic Attainability by : A.G. Chentsov

Download or read book Asymptotic Attainability written by A.G. Chentsov and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this monograph, questions of extensions and relaxations are consid ered. These questions arise in many applied problems in connection with the operation of perturbations. In some cases, the operation of "small" per turbations generates "small" deviations of basis indexes; a corresponding stability takes place. In other cases, small perturbations generate spas modic change of a result and of solutions defining this result. These cases correspond to unstable problems. The effect of an unstability can arise in extremal problems or in other related problems. In this connection, we note the known problem of constructing the attainability domain in con trol theory. Of course, extremal problems and those of attainability (in abstract control theory) are connected. We exploit this connection here (see Chapter 5). However, basic attention is paid to the problem of the attainability of elements of a topological space under vanishing perturba tions of restrictions. The stability property is frequently missing; the world of unstable problems is of interest for us. We construct regularizing proce dures. However, in many cases, it is possible to establish a certain property similar to partial stability. We call this property asymptotic nonsensitivity or roughness under the perturbation of some restrictions. The given prop erty means the following: in the corresponding problem, it is the same if constraints are weakened in some "directions" or not. On this basis, it is possible to construct a certain classification of constraints, selecting "di rections of roughness" and "precision directions".

Measure Theory

Measure Theory

Author: Vladimir I. Bogachev

Publisher: Springer Science & Business Media

Published: 2007-01-15

Total Pages: 1075

ISBN-13: 3540345140

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This book giving an exposition of the foundations of modern measure theory offers three levels of presentation: a standard university graduate course, an advanced study containing some complements to the basic course, and, finally, more specialized topics partly covered by more than 850 exercises with detailed hints and references. Bibliographical comments and an extensive bibliography with 2000 works covering more than a century are provided.


Book Synopsis Measure Theory by : Vladimir I. Bogachev

Download or read book Measure Theory written by Vladimir I. Bogachev and published by Springer Science & Business Media. This book was released on 2007-01-15 with total page 1075 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book giving an exposition of the foundations of modern measure theory offers three levels of presentation: a standard university graduate course, an advanced study containing some complements to the basic course, and, finally, more specialized topics partly covered by more than 850 exercises with detailed hints and references. Bibliographical comments and an extensive bibliography with 2000 works covering more than a century are provided.

Conflict-Controlled Processes

Conflict-Controlled Processes

Author: A. Chikrii

Publisher: Springer Science & Business Media

Published: 2013-03-14

Total Pages: 419

ISBN-13: 9401711356

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This monograph covers one of the divisions of mathematical theory of control which examines moving objects functionating under conflict and uncertainty conditions. To identify this range of problems we use the term "conflict con trolled processes", coined in recent years. As the name itself does not imply the type of dynamics (difference, ordinary differential, difference-differential, integral, or partial differential equations) the differential games falI within its realms. The problems of search and tracking moving objects are also referred to the field of conflict controlled process. The contents of the monograph is confined to studying classical pursuit-evasion problems which are central to the theory of conflict controlled processes. These problems underlie the theory and are of considerable interest to researchers up to now. It should be noted that the methods of "Line of Sight", "Parallel Pursuit", "Proportional N avigation" ,"Modified Pursuit" and others have been long and well known among engineers engaged in design of rocket and space technology. An abstract theory of dynamic game problems, in its turn, is based on the methods originated by R. Isaacs, L. S. Pontryagin, and N. N. Krasovskii, and on the approaches developed around these methods. At the heart of the book is the Method of Resolving Functions which was realized within the class of quasistrategies for pursuers and then applied to the solution of the problems of "hand-to-hand", group, and succesive pursuit.


Book Synopsis Conflict-Controlled Processes by : A. Chikrii

Download or read book Conflict-Controlled Processes written by A. Chikrii and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 419 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph covers one of the divisions of mathematical theory of control which examines moving objects functionating under conflict and uncertainty conditions. To identify this range of problems we use the term "conflict con trolled processes", coined in recent years. As the name itself does not imply the type of dynamics (difference, ordinary differential, difference-differential, integral, or partial differential equations) the differential games falI within its realms. The problems of search and tracking moving objects are also referred to the field of conflict controlled process. The contents of the monograph is confined to studying classical pursuit-evasion problems which are central to the theory of conflict controlled processes. These problems underlie the theory and are of considerable interest to researchers up to now. It should be noted that the methods of "Line of Sight", "Parallel Pursuit", "Proportional N avigation" ,"Modified Pursuit" and others have been long and well known among engineers engaged in design of rocket and space technology. An abstract theory of dynamic game problems, in its turn, is based on the methods originated by R. Isaacs, L. S. Pontryagin, and N. N. Krasovskii, and on the approaches developed around these methods. At the heart of the book is the Method of Resolving Functions which was realized within the class of quasistrategies for pursuers and then applied to the solution of the problems of "hand-to-hand", group, and succesive pursuit.

Mathematical Reviews

Mathematical Reviews

Author:

Publisher:

Published: 2003

Total Pages: 1596

ISBN-13:

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Book Synopsis Mathematical Reviews by :

Download or read book Mathematical Reviews written by and published by . This book was released on 2003 with total page 1596 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Functional Differential Equations

Functional Differential Equations

Author:

Publisher:

Published: 2006

Total Pages: 680

ISBN-13:

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Book Synopsis Functional Differential Equations by :

Download or read book Functional Differential Equations written by and published by . This book was released on 2006 with total page 680 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Russian Mathematics

Russian Mathematics

Author:

Publisher:

Published: 2000

Total Pages: 532

ISBN-13:

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Book Synopsis Russian Mathematics by :

Download or read book Russian Mathematics written by and published by . This book was released on 2000 with total page 532 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Soochow Journal of Mathematics

Soochow Journal of Mathematics

Author:

Publisher:

Published: 2004

Total Pages: 560

ISBN-13:

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Book Synopsis Soochow Journal of Mathematics by :

Download or read book Soochow Journal of Mathematics written by and published by . This book was released on 2004 with total page 560 pages. Available in PDF, EPUB and Kindle. Book excerpt: