Finite operator calculus

Science & Math

Rota, Gian-Carlo

Mathematics

Academic Press

This is the third volume of the collected papers of George Pólya, one of the most influential mathematicians and teachers of our time. Volumes I (Singularities of Analytic Functions) and II (Location of Zeros) were published in 1974. Volume III contains 58 papers spanning Pólya's career (the earliest is from 1913, the latest from 1976) and covering a wide range of subjects in mathematical analysis and mathematical physics. The commentaries on these papers attest to the fertility and continued importance of Pólya's ideas in current mathematics. This volume is the twenty-second in the series Mathematicians of Our Time, edited by Gian-Carlo Rota.

George Pólya: Collected Papers, Vol. 3: Analysis (Mathematicians of Our Time)

Pólya, George

Hersh, Joseph

Mit Pr

It has been clear within the last ten years that combinatorial geometry, together with its order-theoretic counterpart, the geometric lattice, can serve to catalyze the whole field of combinatorial theory, and a major aim of this book, now available in a preliminary edition, is to present the theory in a form accessible to mathematicians working in disparate subjects.  Earlier studies have been one-sided or restricted in their point of view; they were motivated primarily by the desire to extend the classical theory of graphs, or were lattice-theoretic approaches confined to axiomatics and algebraic dependence. These approaches largely ignored the original geometric motivations.  The present work brings all these aspects together in order to emphasize the many-sidedness of combinatorial geometry, and to point up the unifying role it may well play in current developments in combinatorics and its applications.  The book defines the axiomatics of combinatorial geometry, describes a variety of geometrical examples, and discusses the notion of a strong map between geometries. In addition, there is a brief presentation of coordinatization theory and a sketch of two important lines of future work, the "critical problem" and matching theory. The full chapter titles are given below.  Contents: 1. Introduction. 2. Geometrics and Geometric Lattices. 3. Six Classical Examples. 4. Span, Bases, Bonds, Dependence, and Circuits. 5. Cryptomorphic Versions of Geometry. 6. Simplicial Geometries. 7. Semimodular Functions. 8. A Glimpse of Matching Theory. 9. Maps. 10. The Extension Theorem. 11. Orthogonality. 12. Factorization of Relatively Complemented Lattices. 13. Factorization of Geometries. 14. Connected Sets. 15. Representation. 16. The Critical Problem. 17. Bibliography.

On the Foundations of Combinatorial Theory: Combinatorial Geometries

Crapo, Henry

A carefully revised edition of the well-respected ODE text, whose unique treatment provides a smooth transition to critical understanding of proofs of basic theorems. First chapters present a rigorous treatment of background material; middle chapters deal in detail with systems of nonlinear differential equations; final chapters are devoted to the study of second-order linear differential equations. The power of the theory of ODE is illustrated throughout by deriving the properties of important special functions, such as Bessel functions, hypergeometric functions, and the more common orthogonal polynomials, from their defining differential equations and boundary conditions. Contains several hundred exercises. Prerequisite is a first course in ODE.

Ordinary Differential Equations

Birkhoff, Garrett

John Wiley&Sons

Gian-Carlo Rota was one of the most original and colourful mathematicians of the 20th century. His work on the foundations of combinatorics focused on the algebraic structures that lie behind diverse combinatorial areas, and created a new area of algebraic combinatorics. Written by two of his former students, this book is based on notes from his influential graduate courses and on face-to-face discussions. Topics include sets and valuations, partially ordered sets, distributive lattices, partitions and entropy, matching theory, free matrices, doubly stochastic matrices, Moebius functions, chains and antichains, Sperner theory, commuting equivalence relations and linear lattices, modular and geometric lattices, valuation rings, generating functions, umbral calculus, symmetric functions, Baxter algebras, unimodality of sequences, and location of zeros of polynomials. Many exercises and research problems are included, and unexplored areas of possible research are discussed. A must-have for all students and researchers in combinatorics and related areas.

Combinatorics: The Rota Way (Cambridge Mathematical Library)

Kung, Joseph P. S.

Yan, Catherine H.

Cambridge University Press

Introduction to Geometric Probability (Lezioni Lincee)

Here is the first modern introduction to geometric probability, also known as integral geometry, presented at an elementary level, requiring little more than first-year graduate mathematics. Klein and Rota present the theory of intrinsic volumes due to Hadwiger, McMullen, Santaló and others, along with a complete and elementary proof of Hadwiger's characterization theorem of invariant measures in Euclidean n-space. They develop the theory of the Euler characteristic from an integral-geometric point of view. The authors then prove the fundamental theorem of integral geometry, namely, the kinematic formula. Finally, the analogies between invariant measures on polyconvex sets and measures on order ideals of finite partially ordered sets are investigated. The relationship between convex geometry and enumerative combinatorics motivates much of the presentation. Every chapter concludes with a list of unsolved problems.

Klain, Daniel A.

Here is the first modern introduction to geometric probability, also known as integral geometry, presented at an elementary level, requiring little more than first-year graduate mathematics. Klein and Rota present the theory of intrinsic volumes due to Hadwiger, McMullen, Santaló and others, along with a complete and elementary proof of Hadwiger's characterization theorem of invariant measures in Euclidean n-space. They develop the theory of the Euler characteristic from an integral-geometric point of view. The authors then prove the fundamental theorem of integral geometry, namely, the kinematic formula. Finally, the analogies between invariant measures on polyconvex sets and measures on order ideals of finite partially ordered sets are investigated. The relationship between convex geometry and enumerative combinatorics motivates much of the presentation. Every chapter concludes with a list of unsolved problems.

Written by two of Gian-Carlo Rota's former students, this book is based on notes from his courses and on personal discussions with him. Topics include sets and valuations, partially ordered sets, distributive lattices, partitions and entropy, matching theory, free matrices, doubly stochastic matrices, Moebius functions, chains and antichains, Sperner theory, commuting equivalence relations and linear lattices, modular and geometric lattices, valuation rings, generating functions, umbral calculus, symmetric functions, Baxter algebras, unimodality of sequences, and location of zeros of polynomials. Many exercises and research problems are included, and unexplored areas of possible research are discussed. This book should be on the shelf of all students and researchers in combinatorics and related areas.

Written by two of Gian-Carlo Rota's former students, this book is based on notes from his courses and on personal discussions with him. Topics include sets and valuations, partially ordered sets, distributive lattices, partitions and entropy, matching theory, free matrices, doubly stochastic matrices, Moebius functions, chains and antichains, Sperner theory, commuting equivalence relations and linear lattices, modular and geometric lattices, valuation rings, generating functions, umbral calculus, symmetric functions, Baxter algebras, unimodality of sequences, and location of zeros of polynomials. Many exercises and research problems are included, and unexplored areas of possible research are discussed. This book should be on the shelf of all students and researchers in combinatorics and related areas.

This collection represents a unique undertaking in scientific publishing to honor Nick Metropolis, the last survivor of the World War II Manhattan Project in Los Alamos. In this volume, some of the leading scientists and humanists of our time have contributed essays related to their respective disciplines, exploring various aspects of future developments in science and society, philosophy, national security, nuclear power, pure and applied mathematics, physics and biology, particle physics, computing, and information science.

Essays on the Future: In Honor of Nick Metropolis

Hecker, Siegfried

History & Philosophy

Birkhauser

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Books by Rota, Gian-Carlo

See the best price to sell, buy, or rent books by Rota, Gian-Carlo. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Finite operator calculus

George Pólya: Collected Papers, Vol. 3: Analysis (Mathematicians of Our Time)

On the Foundations of Combinatorial Theory: Combinatorial Geometries

Ordinary Differential Equations

Combinatorics: The Rota Way (Cambridge Mathematical Library)

Introduction to Geometric Probability (Lezioni Lincee)

Introduction to Geometric Probability (Lezioni Lincee)

Combinatorics: The Rota Way (Cambridge Mathematical Library)

Combinatorics: The Rota Way (Cambridge Mathematical Library)

Essays on the Future: In Honor of Nick Metropolis