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This is an online evolving book that collects and organizes various fundamental results in applied mathematics, with emphasized applications in stochastic system modeling, optimal decision making, statistical learning, and mathematical finance. The content stems from various courses I have taken in Johns Hopkins University, and my self-study notes during my PhD research on optimal control of stochastic systems. This book is intended for readers in applied math and engineering that are already familiar with the topic and want to have an in-depth refreshment and summary such that various dots in commonly used applied math methods can be connected.
| Count | |
|---|---|
| Definition | ~1050 |
| Theorem | ~1000 |
| Lemma | ~700 |
| Corollary | ~200 |
| Words | ~200,000 |
This book is aimed to strike a balance between math-type books and enginnering-type books. It has following features:
-Self-contained.
-Unnecessary technical results are avoided without sacrificing rigorness and depth.
-Concepts, theorems and discussions are developed with real-world applications in mind.
-Extract and study essential mathematical structures, properties and generalizations.
-Prefer high-level abstract concept and concise proof via abstract argument.
-Various comparisons and discussions on similar definitions and theorems.
-Essential sources are provided on each topic.
This book is organized as two major parts: fundamentals and essential tools.
The first part 'fundamentals' covers essential concepts and theorems in real analysis, linear algebra, matrix theory, functional analysis, probability theory, statistics, stochastic process and differential geometry.
The second part 'essential tools' covers tools from convexity analysis, optimizations, calculus of variation, optimal control and dynamical programming, differential equations, dynamical system, statistical learning and mathematical finance.
I want to thank many Professors in Johns Hopkins University for the courses they delivered and their valuable discussion with me: Daniel Robinson for nonlinear optimization, Andrea Prosperetti for advanced engineering maths, Gregory Chirikjian for stochastic process on manifolds and Lie group, Michael Kahadan for group representation and Fourier transform, James C. Spall for Stochastic optimization, Marin Kobilarov for optimal control and estimation, Suchi Saria for machine learning, Michael Dimitz for algorithms, Sean sun and Ari Turner for statistical mechanics, Gregory Eyink for dynamic systems and advanced parametrization methods, Amitabh Basu for convexity.
Finally, I want to thank friends and classmates at JHU and the library serives at JHU.
You are free to:
-Share:copy and redistribute the material in any medium or format
-Adapt:remix, transform, and build upon the material.
under the following terms:
-Attribution:You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
-NonCommercial: You may not use the material for commercial purposes.
-ShareAlike: If you remix, transform, or build upon the material, you must distribute your contributions under the same license as the original.
*The licensor cannot revoke these freedoms as long as you follow the license terms.
*licence are created via creative commons (https://creativecommons.org)