Inequalities for polynomials and their derivatives are very important in many areas of mathematics, as well as in other computational and applied sciences; in particular they play a fundamental role in approximation theory.&nbsp;Here, not only Extremal Problems and Inequalities of Markov-Bernstein Type for Algebraic Polynomials, but also ones for trigonometric polynomials and related functions, are treated in an integrated and comprehensive style in different metrics, both on general classes of polynomials and on important restrictive classes of polynomials. Primarily for graduate and PhD students, this book is useful for any researchers exploring problems which require derivative estimates. It is particularly useful for those studying inverse problems in approximation theory.

Extremal Problems and Inequalities of Markov-Bernstein Type for Algebraic Polynomials (Mathematical Analysis and its Applications)

Science & Math

Gardner, Robert B.

Govil, Narendra K.

Milovanović, Gradimir V.

Mathematics

Elsevier

The ideas of Elie Cartan are combined with the tools of Felix Klein and Sophus Lie to present in this book the only detailed treatment of the method of equivalence. An algorithmic description of this method, which finds invariants of geometric objects under infinite dimensional pseudo-groups, is presented for the first time. 
As part of the algorithm, Gardner introduces several major new techniques. In particular, the use of Cartan's idea of principal components that appears in his theory of Repere Mobile, and the use of Lie algebras instead of Lie groups, effectively a linear procedure, provide a tremendous simplification. One must, however, know how to convert from one to the other, and the author provides the Rosetta stone to accomplish this. In complex problems, it is essential to be able to identify natural blocks in group actions and not just individual elements, and prior to this publication, there was no reference to block matrix techniques.
The Method of Equivalence and Its Applications details ten diverse applications including Lagrangian field theory, control theory, ordinary differential equations, and Riemannian and conformal geometry.
This volume contains a series of lectures, the purpose of which was to describe the equivalence algorithm and to show, in particular, how it is applied to several pedagogical examples and to a problem in control theory called state estimation of plants under feedback. The lectures, and hence the book, focus on problems in real geometry. 
Special Features:
- only book to treat this subject in such depth and to include the algorithm, the use of principal components, and the use of infinitesimal analysis on the Lie algebra level.
Audience:
The reader should possess a background in calculus on manifolds and a familiarity with abstract linear and multilinear algebra. A first course on manifolds with an introduction to Lie groups and the ability to believe advanced results would be even more helpful. This volumes contains asides at various levels involving representation theory of Lie groups and Lie algebras, exterior differential systems, and Lie pseudo-groups.

The Method of Equivalence and Its Applications (CBMS-NSF Regional Conference Series in Applied Mathematics, No. 58)

Society for Industrial & Applied Mathematics

This book gives a treatment of exterior differential systems. It will in clude both the general theory and various applications. An exterior differential system is a system of equations on a manifold defined by equating to zero a number of exterior differential forms. When all the forms are linear, it is called a pfaffian system. Our object is to study its integral manifolds, i. e. , submanifolds satisfying all the equations of the system. A fundamental fact is that every equation implies the one obtained by exterior differentiation, so that the complete set of equations associated to an exterior differential system constitutes a differential ideal in the algebra of all smooth forms. Thus the theory is coordinate-free and computations typically have an algebraic character; however, even when coordinates are used in intermediate steps, the use of exterior algebra helps to efficiently guide the computations, and as a consequence the treatment adapts well to geometrical and physical problems. A system of partial differential equations, with any number of inde pendent and dependent variables and involving partial derivatives of any order, can be written as an exterior differential system. In this case we are interested in integral manifolds on which certain coordinates remain independent. The corresponding notion in exterior differential systems is the independence condition: certain pfaffian forms remain linearly indepen dent. Partial differential equations and exterior differential systems with an independence condition are essentially the same object.

Exterior Differential Systems (Mathematical Sciences Research Institute Publications)

Griffiths, P.A.

Chern, S.-S.

Bryant, Robert L. L.

Goldschmidt, Hubert L.

Springer

From the contents: Charles MORGAN: Canonical models and probabilistic semantics. - Francois LEPAGE: A many-valued probabilistic logic. - Piers RAWLING: The exchange paradox, finite additivity, and the principle of dominance. - Susan VINEBERG: The logical status of conditionalization and its role in confirmation. - Deborah MAYO: Science, error statistics, and arguing from error. - Mark N. LANCE: The best is the enemy of the good: Bayesian epistemology as a case study in unhelpful idealization. - Robert B. GARDNER & Michael C. WOOTEN: An application of Bayes' theorem to population genetics. - Peter D. JOHNSON, Jr.: Another look at group selection."

Logic, Probability and Science (Poznan Studies in the Philosophy of the Sciences and the Humanities, 71)

Shanks, Niall

History & Philosophy

Brill Academic Pub

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