Amol Sasane Livres

Focusing on the mathematical foundations of general relativity, this book presents spacetime physics in a structured definition-theorem-proof format, catering specifically to advanced undergraduate mathematics students. It emphasizes precise mathematical definitions to clarify complex physical concepts, helping to navigate the counter-intuitive aspects of spacetime. Mathematicians will appreciate the rigorous approach that bridges abstract mathematics with physical theories, enhancing understanding of gravitation in a comprehensive manner.

This highly readable volume on optimization in function spaces is based on author Amol Sasane's lecture notes, which he developed over several years while teaching a course for third-year undergraduates at the London School of Economics. The classroom-tested text is written in an informal but precise style that emphasizes clarity and detail, taking students step by step through each subject.Numerous examples throughout the text clarify methods, and a substantial number of exercises provide reinforcement. Detailed solutions to all of the exercises make this book ideal for self-study. The topics are relevant to students in engineering and economics as well as mathematics majors. Prerequisites include multivariable calculus and basic linear algebra. The necessary background in differential equations and elementary functional analysis is developed within the text, offering students a self-contained treatment.

Model reduction is an important engineering problem in which one aims to replace an elaborate model by a simpler model without undue loss of accuracy. The accuracy can be mathematically measured in several possible norms and the Hankel norm is one such. The Hankel norm gives a meaningful notion of distance between two linear systems: roughly speaking, it is the induced norm of the operator that maps past inputs to future outputs. It turns out that the engineering problem of model reduction in the Hankel norm is closely related to the mathematical problem of finding solutions to the sub-optimal Nehari-Takagi problem, which is called „the sub-optimal Hankel norm approximation problem“ in this book. Although the existence of a solution to the sub-optimal Hankel norm approximation problem has been known since the 1970's, this book presents explicit solutions and, in particular, new formulae for several large classes of infinite-dimensional systems for the first time.

Focusing on essential concepts in Functional Analysis, this book serves as an introductory course for those familiar with calculus and linear algebra. It covers key topics such as Banach spaces, continuous linear transformations, and the geometry of Hilbert spaces, along with applications in differential equations, optimization, and physics. With 197 problems and detailed solutions, it is designed for both classroom use and self-study, making it a valuable resource for students seeking to deepen their understanding of the subject.