A Course of Applied Mathematics for Engineers and Physicists attempts a synthesis between the various theoretical concepts with the tools and techniques useful to the engineer, aiming at an equilibrium between mathematical rigour and a practical point of view with applications in mind. THE MAIN TOPICS DISCUSSED ARE: • Linear and non linear Algebra • Topology (Topological and Metric spaces, Convexity Connexity, Orthogonal Projection in Hilbert spaces, etc.) • Integration (Lebesgue Integral, Analytic functions, Fourier and Laplace Transforms, Distributions) The focus is on the applications of the above in Linear and Nonlinear Optimization, Combinatorial Optimization (Graphs), Dynamical Programming, Approximations, Solutions of Integral and Differential equations.

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.

Undergraduate and Graduate Students of Mathematics, Physics & Engineering