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Defined as solutions of linear differential or difference equations with polynomial coefficients, D-finite functions play an important role in various areas of mathematics. This book is a comprehensive introduction to the theory of these functions with a special emphasis on computer algebra algorithms for computing with them: algorithms for detecting relations from given data, for evaluating D-finite functions, for executing closure properties, for obtaining various kinds of “explicit” expressions, for factoring operators, and for definite and indefinite symbolic summation and integration are explained in detail. The book comes “with batteries included” in the sense that it requires no background in computer algebra as the relevant facts from this area are summarized in the beginning. This makes the book accessible to a wide range of readers, from mathematics students who plan to work themselves on D-finite functions to researchers who want to apply the theory to their own work. Hundreds of exercises invite the reader to apply the techniques in the book and explore further aspects of the theory on their own. Solutions to all exercises are given in the appendix. When algorithms for D-finite functions came up in the early 1990s, computer proofs were met with a certain skepticism. Fortunately, these times are over and computer algebra has become a standard tool for many mathematicians. Yet, this powerful machinery is still not as widely known as it deserves. This book helps to spread the word that certain tasks can be safely delegated to a computer algebra system, and also what the limitations of these techniques are.

Discrete mathematics. --- Differential equations. --- Computer science --- Discrete Mathematics. --- Differential Equations. --- Symbolic and Algebraic Manipulation. --- Mathematics. --- Differential equations, Linear. --- Equacions diferencials lineals

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This book contains the contributions resulting from the 6th Italian-Japanese workshop on Geometric Properties for Parabolic and Elliptic PDEs, which was held in Cortona (Italy) during the week of May 20–24, 2019. This book will be of great interest for the mathematical community and in particular for researchers studying parabolic and elliptic PDEs. It covers many different fields of current research as follows: convexity of solutions to PDEs, qualitative properties of solutions to parabolic equations, overdetermined problems, inverse problems, Brunn-Minkowski inequalities, Sobolev inequalities, and isoperimetric inequalities.

Mathematical analysis. --- Analysis (Mathematics). --- Mathematical optimization. --- Convex geometry . --- Discrete geometry. --- Applied mathematics. --- Engineering mathematics. --- Functional analysis. --- Analysis. --- Optimization. --- Convex and Discrete Geometry. --- Applications of Mathematics. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Engineering --- Engineering analysis --- Mathematical analysis --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Discrete mathematics --- Geometry --- Combinatorial geometry --- 517.1 Mathematical analysis --- Mathematics --- Geometric analysis --- Differential equations, Parabolic --- Differential equations, Elliptic --- Parabolic differential equations --- Parabolic partial differential equations --- Differential equations, Partial --- Geometric analysis PDEs (Geometric partial differential equations) --- Geometria analítica --- Equacions diferencials parabòliques --- Equacions diferencials el·líptiques --- Equacions diferencials el·líptiques lineals --- Equacions diferencials lineals --- Equacions en derivades parcials --- Equació de la calor --- Equacions de reacció-difusió --- Anàlisi geomètrica --- Geometria algebraica --- Coordenades --- Corbes