A Course of Applied Mathematics for Engineers and Physicists

Aiming at an equilibrium between mathematical rigour and a practical point of view with applications in mind, A Course of Applied Mathematics for Engineers and Physicists attempts to combine the various theoretical concepts with the tools and techniques useful to the engineer. The main topics discussed are Linear and non-linear Algebra, Topology (Topological and Metric spaces, Convexity Connexity, Orthogonal Projection in Hilbert spaces, etc.) and integration (Lebesgue Integral, Analytic functions, Fourier and Laplace Transforms, Distributions). The focus is on the applications of these to Linear and Nonlinear Optimization, Combinatorial Optimization (Graphs), Dynamical Programming, Approximations, Solutions of Integral and Differential equations.

Table of Contents:
Preface Linear (or Vector) Spaces Matrices – Linear Operators, Reductions Matrices – Endomorphisms, Reductions Canonical Form of a Matrix or of the Associated Endomorphism Linear and Bilinear Forms Hermitian Forms – Prehilbertian Spaces – Normed Vector Spaces Matrix Norms, Normal Self-adjoint-Unitary Operators and Numerical Analysis Appendix Topological Spaces Metric Spaces Connectivity Convexity and Applications Hilbert Spaces Orthogonal Projection Methods of Approximation and Optimization Compactness Appendix A Brief Introduction to Measure Theory and Lebesgue’s Integral The Fourier Integral and the Fourier Transformation The Laplace Transformation and Applications An Elementary Introduction to the Theory of Distributions Applications An Introduction to Analytic Functions I Cauchy Theorems An Introduction to Analytic Functions II Taylor and Laurent Series An Introduction to Analytic Functions III Zeros – Singularities – Poles An Introduction to Analytic Functions IV Applications Optimization Applications of Algebra and Topology General Introduction Linear Programming (the Simplex) A. The Algorithm of Dantzig The Simplex Algorithm B. Advanced Techniques and Applications B.1 Penalty The Simplex Algorithm. Advanced Techniques and Applications B.2 Duality The Simplex Algorithm. Advanced Techniques and Applications B.3 Integer Numbers’ Programming The method of “Cuts” Two Solved Problems as a Synthesis of the Earlier Presented Methods of Linear Programming Nonlinear Programming I Lagrange and Kuhn – Tucker Multipliers Nonlinear Programming II The Orthogonal Projection and the “Least Squares’ Approximation” Dynamic Programming – A Combinatorial Optimization Dynamic Programming – B. I Bellman’s Method of Optimization Dynamic Programming B. II. An Introduction to Optimal Control Theory Following Bellman Exercises – Applications of Bellman’s Method Appendix Index.

About the Author :
Marietta Manolessou: Head of Mathematics Department, E.I.S.T.I. International School of Information Technology, Avenue du Parc 95011, CERGY CEDEX FRANCE