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\(p\)-adic mathematical physics: the first 30 years. (English) Zbl 1394.81009
The foundation of \(p\)-adic mathematical physics (PAMP) represents an important step towards a better understanding of the universe. Established in 1987, PAMP came to light motivated by important physical and mathematical modeling problems. Nonetheless, many PAMP applications emerged especially during the last 10 years.
Within their paper, the authors provide the reader with a captivating overview regarding the selected topic. A history perspective on PAMP’s development is given and connections with real world applications are precisely stated.
The article is structured in six main sections. The first section entitled \(Introduction\) discusses the genesis of \(p\)-adic mathematical physics, underlines motivations for studying such a field and shortly describes the structure of the paper. Necessary mathematical foundations depicted as notions and formulas are presented in Section 2, “Mathematical methods”. Section 3 discusses the main “Applications in physics”: \(p\)-adic strings, the quantization of the famous Riemann zeta-function, quantum mechanics, \(p\)-adic gravity and cosmology, \(p\)-adic stochastic processes, disordered systems and so on. Section 4 covers the “Applications in biology”, more precisely \(p\)-adic genetic code and bioinformation as well as \(p\)-adic models of cognition. Section 5, “Other applications”, offers information about how \(p\)-adic mathematical physics can be used in support of modern technologies’ development (topics like data mining, cryptography and information security are tackled). The authors conclude in Section 6. A comprehensive bibliography (consisting of 342 entries) is provided at the end of the paper aiming at readers interested in details regarding the field.
The concepts and properties discussed in each section are generally presented in a clear and accessible manner for graduate students, enthusiasts and more experienced readers (especially if interested in the applications of \(p\)-adic mathematical physics). The mathematical notations, definitions, theorems and relations mentioned throughout the paper are accompanied by explanations and carefully selected references.
As a side note regarding the structure of the article, we believe that a few more ingredients could have added completion. To give an example, a section including the authors’ predictions about the future development of the field would have been engaging for both enthusiasts and researchers.
To sum up, we recommend the readers interested in the history and applications of \(p\)-adic mathematical physics to consult this very well written overview and, maybe, contribute to a follow up.
As an overall conclusion, we stress that the current paper is a valuable work for both theoreticians and practitioners of the \(p\)-adic mathematical physics field.

MSC:
81-03 History of quantum theory
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
11F33 Congruences for modular and \(p\)-adic modular forms
83F05 Relativistic cosmology
92D20 Protein sequences, DNA sequences
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OEIS; darch
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References:
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