The Contribution of Simon Serovajsky to Mathematics

11 october, 2024

Simon Serovajsky is an outstanding mathematician whose work has had a significant impact on the fields of optimization, control theory, and mathematical modeling. His publications cover a wide range of complex topics, including counterexamples in optimal control theory, consistent models in mathematical physics, and the architecture of mathematics. S.Ya. Serovajsky's contributions not only shape modern mathematical thinking, but also serve as the basis for innovative solutions in various scientific and applied fields.

We have made a short review of the works of Simon Serovajsky:

1. Optimization. 100 Examples

"Optimization: 100 Examples" is a book devoted to the analysis of scenarios for which the use of well-known optimization methods encounter certain difficulties. Analysing such examples allows a deeper understanding of the features of these optimization methods, including the limits of their applicability. In this way, the book seeks to stimulate further development and understanding of the theory of optimal control. The study of the presented examples makes it possible to more effectively diagnose problems that arise in the practical solution of optimal control problems, and to find ways to overcome the difficulties that have arisen.

Features:

  • Vast collection of examples
  • Simple. accessible presentation
  • Suitable as a research reference for anyone with an interest in optimization and optimal control theory, including mathematicians and engineers
  • Examples differ in properties, i.e. each effect for each class of problems is illustrated by a unique example.

Information from: https://www.taylorfrancis.com/books/mono/10.1201/9781003398585/optimization-simon-serovajsky

Book details:

Optimization. 100 Examples. – New York: Chapman and Hall/CRC, 2024. – 556 p. – ISBN 9781032500072, eBook ISBN 9781003398585

DOI (Великобритания - США) https://doi.org/10.1201/9781003398585

2. Mathematical Modelling

Mathematical Modelling sets out the general principles of mathematical modelling as a means comprehending the world. Within the book, the problems of physics, engineering, chemistry, biology, medicine, economics, ecology, sociology, psychology, political science, etc. are all considered through this uniform lens.

The author describes different classes of models, including lumped and distributed parameter systems, deterministic and stochastic models, continuous and discrete models, static and dynamical systems, and more. From a mathematical point of view, the considered models can be understood as equations and systems of equations of different nature and variational principles. In addition to this, mathematical features of mathematical models, applied control and optimization problems based on mathematical models, and identification of mathematical models are also presented.

Features:

  • Each chapter includes four levels: a lecture (main chapter material), an appendix (additional information), notes (explanations, technical calculations, literature review) and tasks for independent work; this is suitable for undergraduates and graduate students and does not require the reader to take any prerequisite course, but may be useful for researchers as well
  • Described mathematical models are grouped both by areas of application and by the types of obtained mathematical problems, which contributes to both the breadth of coverage of the material and the depth of its understanding
  • Can be used as the main textbook on a mathematical modelling course, and is also recommended for special courses on mathematical models for physics, chemistry, biology, economics, etc.

Information from: https://www.routledge.com/Mathematical-Modelling/Serovajsky/p/book/9780367474300

Book details:

Mathematical Modelling. – Chapman and Hall/CRC, London, 2021. – 466 p. ISBN: 9780367474300 (hbk), ISBN: 9781032147871 (pbk), ISBN: 9781003035602 (ebk), DOI: 10.1201/9781003035602 (Великобритания)

https://www.routledge.com/Mathematical-Modelling/Serovajsky/p/book/9780367474300

3. Architecture of Mathematics

"Architecture of Mathematics" describes the logical structure of Mathematics from its foundations to its real-world applications. It describes the many interweaving relationships between different areas of mathematics and its practical applications, and as such provides unique reading for professional mathematicians and nonmathematicians alike. This book can be a very important resource both for the teaching of mathematics and as a means to outline the research links between different subjects within and beyond the subject.

Features:

  • All notions and properties are introduced logically and sequentially, to help the reader gradually build understanding.
  • Focusses on illustrative examples that explain the meaning of mathematical objects and their properties.
  • Suitable as a supplementary resource for teaching undergraduate mathematics, and as an aid to interdisciplinary research.
  • Forming the reader's understanding of Mathematics as a unified science, the book helps to increase his general mathematical culture.

Information from: https://www.amazon.com/Architecture-Mathematics-Simon-Serovajsky/dp/1138601055

Book details:

Architecture of Mathematics. – Chapman and Hall/CRC, London, 2020. – 394 p. ISBN-10: 1138601055, ISBN-13: 978-1138601055. (Великобритания) https://www.amazon.com/Architecture-Mathematics-Simon-Serovajsky/dp/1138601055

4. Sequential Models of Mathematical Physics

The equations of mathematical physics are the mathematical models of the large class of phenomenon of physics, chemistry, biology, economics, etc. In "Sequential Models of Mathematical Physics", the author considers the justification of the process of constructing mathematical models. The book seeks to determine the classic, generalized and sequential solutions, the relationship between these solutions, its direct physical sense, the methods of its practical finding, and its existence.

Features:

  • Describes a sequential method based on the construction of space completion, as well as its applications in number theory, the theory of distributions, the theory of extremum, and mathematical physics
  • Presentation of the material is carried out on the simplest example of a one-dimensional stationary heat transfer process; all necessary concepts and constructions are introduced and illustrated with elementary examples, which makes the material accessible to a wide area of readers
  • The solution of a specific mathematical problem is obtained as a result of the joint application of methods and concepts from completely different mathematical directions

Information from: https://www.routledge.com/Sequential-Models-of-Mathematical-Physics/Serovajsky/p/book/9781138601031

Book details:

Sequential models of mathematical physics. – CRC Press, Taylor & Francis Group. – Boca Raton, London, New York, 2019. – 266 р. ISBN 9781138601031 - CAT# K387948. (Великобритания) https://www.crcpress.com/Sequential-Models-of-Mathematical-Physics/Serovajsky/p/book/9781138601031

5. Optimization and Differentiation

"Optimization and Differentiation" is an introduction to the application of optimization control theory to systems described by nonlinear partial differential equations. As well as offering a useful reference work for researchers in these fields, it is also suitable for graduate students of optimal control theory.

Information from: https://www.routledge.com/Optimization-and-Differentiation/Serovajsky/p/book/9781498750936

Book details:

Optimization and Differentiation.  Chapman & Hall/CRC Monographs and Research Notes in Mathematics CRS Press. Taylor & Francis group. London, 2017. – 517 p. October 13, 2017, ISBN 9781498750936 (Великобритания)

https://www.crcpress.com/Optimization-and-Differentiation/Serovajsky/p/book/9781498750936

6. Counterexamples in the optimal control theory

This monograph deals with cases where optimal control either does not exist or is not unique, cases where optimality conditions are insufficient of degenerate, or where extremum problems in the sense of Tikhonov and Hadamard are ill-posed, and other situations. A formal application of classical optimisation methods in such cases either leads to wrong results or has no effect. The detailed analysis of these examples should provide a better understanding of the modern theory of optimal control and the practical difficulties of solving extremum problems.

Information from: https://www.degruyter.com/document/doi/10.1515/9783110915532/html#overview

Book details:

Counterexamples in the optimal control theory. – Brill Academic Press. Netherlands. Utrecht-Boston, 2004. – 176 p. ISBN 90-6764-400-5. (Нидерланды) https://www.amazon.com/Counterexamples-Optimal-Control-Ill-Posed-Problems/dp/9067644005

Counterexamples in the optimal control theory. – De Gruyter, 2011. – 175 p. ISBN: 9783110915532. (Германия) https://www.degruyter.com/document/doi/10.1515/9783110915532/html

 

Mukhambetkaliyev Murat,

Teacher of the Department of Mathematics

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