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| Author | : Andrei Y. Khrennikov |
| Publisher | : |
| Total Pages | : 284 |
| Release | : 2014-01-15 |
| Genre | : |
| ISBN | : 9789401583572 |
| Author | : Andrei Y. Khrennikov |
| Publisher | : Springer |
| Total Pages | : 264 |
| Release | : 1994-10-31 |
| Genre | : Science |
| ISBN | : 0792331729 |
Numbers ... , natural, rational, real, complex, p-adic .... What do you know about p-adic numbers? Probably, you have never used any p-adic (nonrational) number before now. I was in the same situation few years ago. p-adic numbers were considered as an exotic part of pure mathematics without any application. I have also used only real and complex numbers in my investigations in functional analysis and its applications to the quantum field theory and I was sure that these number fields can be a basis of every physical model generated by nature. But recently new models of the quantum physics were proposed on the basis of p-adic numbers field Qp. What are p-adic numbers, p-adic analysis, p-adic physics, p-adic probability? p-adic numbers were introduced by K. Hensel (1904) in connection with problems of the pure theory of numbers. The construction of Qp is very similar to the construction of (p is a fixed prime number, p = 2,3,5, ... ,127, ... ). Both these number fields are completions of the field of rational numbers Q. But another valuation 1 . Ip is introduced on Q instead of the usual real valuation 1 . I· We get an infinite sequence of non isomorphic completions of Q : Q2, Q3, ... , Q127, ... , IR = Qoo· These fields are the only possibilities to com plete Q according to the famous theorem of Ostrowsky.
| Author | : Vasili? Sergeevich Vladimirov |
| Publisher | : World Scientific |
| Total Pages | : 350 |
| Release | : 1994 |
| Genre | : Science |
| ISBN | : 9789810208806 |
p-adic numbers play a very important role in modern number theory, algebraic geometry and representation theory. Lately p-adic numbers have attracted a great deal of attention in modern theoretical physics as a promising new approach for describing the non-Archimedean geometry of space-time at small distances.This is the first book to deal with applications of p-adic numbers in theoretical and mathematical physics. It gives an elementary and thoroughly written introduction to p-adic numbers and p-adic analysis with great numbers of examples as well as applications of p-adic numbers in classical mechanics, dynamical systems, quantum mechanics, statistical physics, quantum field theory and string theory.
| Author | : Andrei Y. Khrennikov |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 271 |
| Release | : 2013-03-09 |
| Genre | : Science |
| ISBN | : 9401583560 |
Numbers ... , natural, rational, real, complex, p-adic .... What do you know about p-adic numbers? Probably, you have never used any p-adic (nonrational) number before now. I was in the same situation few years ago. p-adic numbers were considered as an exotic part of pure mathematics without any application. I have also used only real and complex numbers in my investigations in functional analysis and its applications to the quantum field theory and I was sure that these number fields can be a basis of every physical model generated by nature. But recently new models of the quantum physics were proposed on the basis of p-adic numbers field Qp. What are p-adic numbers, p-adic analysis, p-adic physics, p-adic probability? p-adic numbers were introduced by K. Hensel (1904) in connection with problems of the pure theory of numbers. The construction of Qp is very similar to the construction of (p is a fixed prime number, p = 2,3,5, ... ,127, ... ). Both these number fields are completions of the field of rational numbers Q. But another valuation 1 . Ip is introduced on Q instead of the usual real valuation 1 . I· We get an infinite sequence of non isomorphic completions of Q : Q2, Q3, ... , Q127, ... , IR = Qoo· These fields are the only possibilities to com plete Q according to the famous theorem of Ostrowsky.
| Author | : Zoran Rakic |
| Publisher | : American Institute of Physics |
| Total Pages | : 392 |
| Release | : 2006-04-06 |
| Genre | : Mathematics |
| ISBN | : |
The subject of this conference was recent developments in p-adic mathematical physics and related areas. The field of p-Adic mathematical physics was conceived in 1987 as a result of attempts to find non-Archimedean approaches to space-time at the Planck scale as well as to strings. Since then, many applications of p-adic numbers and adeles in physics and related sciences have emerged. Some of them are p-adic and adelic string theory, p-adic and adelic quantum mechanics and quantum field theory, ultrametricity of spin glasses, biological and hierarchical systems, p-adic dynamical systems, p-adic probability theory, p-adic models of cognitive processes and cryptography, as well as p-adic and adelic cosmology.
| Author | : Alain Escassut |
| Publisher | : World Scientific |
| Total Pages | : 600 |
| Release | : 2015-11-27 |
| Genre | : Mathematics |
| ISBN | : 9814730122 |
' The book first explains the main properties of analytic functions in order to use them in the study of various problems in p-adic value distribution. Certain properties of p-adic transcendental numbers are examined such as order and type of transcendence, with problems on p-adic exponentials. Lazard''s problem for analytic functions inside a disk is explained. P-adic meromorphics are studied. Sets of range uniqueness in a p-adic field are examined. The ultrametric Corona problem is studied. Injective analytic elements are characterized. The p-adic Nevanlinna theory is described and many applications are given: p-adic Hayman conjecture, Picard''s values for derivatives, small functions, branched values, growth of entire functions, problems of uniqueness, URSCM and URSIM, functions of uniqueness, sharing value problems, Nevanlinna theory in characteristic p>0, p-adic Yosida''s equation. Contents: Ultrametric FieldsHensel LemmaSpherically Complete ExtensionsAnalytic ElementsPower and Laurent SeriesFactorization of Analytic ElementsDerivative of Analytic ElementsVanishing along a Monotonous FilterMaximum PrincipleQuasi-Invertible Analytic ElementsMeromorphic FunctionsThe Corona Problem on Ab(d(0,1‾))Applications to CurvesGrowth of the Derivative of an Entire FunctionRational Decomposition for Entire Functionsand other papers Readership: Graduate students and researchers interested in p-adic analysis. Keywords:p-Adic;Transcendental Numbers;Meromorphic;Nevalinna Theory'
| Author | : S. Albeverio |
| Publisher | : Cambridge University Press |
| Total Pages | : 369 |
| Release | : 2010-03-18 |
| Genre | : Mathematics |
| ISBN | : 0521148561 |
A wide-ranging 2010 survey of new and important topics in p-adic analysis for researchers and graduate students.
| Author | : Andrei Y. Khrennikov |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 279 |
| Release | : 2013-03-14 |
| Genre | : Science |
| ISBN | : 1402026609 |
This book provides an overview of the theory of p-adic (and more general non-Archimedean) dynamical systems. The main part of the book is devoted to discrete dynamical systems. It presents a model of probabilistic thinking on p-adic mental space based on ultrametric diffusion. Coverage also details p-adic neural networks and their applications to cognitive sciences: learning algorithms, memory recalling.
| Author | : Sergio Albeverio |
| Publisher | : World Scientific |
| Total Pages | : 466 |
| Release | : 2000 |
| Genre | : Science |
| ISBN | : 9789810244408 |
In October 1998 a conference was held in Lisbon to celebrate Ludwig Streit's 60th birthday. This book collects some of the papers presented at the conference as well as other essays contributed by the many friends and collaborators who wanted to honor Ludwig Streit's scientific career and personality.The contributions cover many aspects of contemporary mathematical physics. Of particular importance are new results on infinite-dimensional stochastic analysis and its applications to a wide range of physical domains.List of Contributors: S Albeverio, T Hida, L Accardi, I Ya Aref'eva, I V Volovich; A Daletskii, Y Kondratiev, W Karwowski, N Asai, I Kubo, H-H Kuo, J Beckers, Ph Blanchard, G F Dell'Antonio, D Gandolfo, M Sirugue-Collin, A Bohm, H Kaldass, D Boll, G Jongen, G M Shim, J Bornales, C C Bernido, M V Carpio-Bernido, G Burdet, Ph Combe, H Nencka, P Cartier, C DeWitt-Morette, H Ezawa, K Nakamura, K Watanabe, Y Yamanaka, R Figari, F Gesztesy, H Holden, R Gielerak, G A Goldin, Z Haba, M-O Hongler, Y Hu, B Oksendal, A Sulem, J R Klauder, C B Lang, V I Man'ko, H Ouerdiane, J Potthoff, E Smajlovic, M Rckner, E Scacciatelli, J L Silva, J Stochel, F H Szafraniec, L V zquez, D N Kozakevich, S Jimnez, V R Vieira, P D Sacramento, R Vilela Mendes, D Voln?, P Samek.
| Author | : Alain Escassut |
| Publisher | : World Scientific |
| Total Pages | : 559 |
| Release | : 2015-11-27 |
| Genre | : Mathematics |
| ISBN | : 9814730114 |
"The book first explains the main properties of analytic functions in order to use them in the study of various problems in p-adic value distribution. Certain properties of p-adic transcendental numbers are examined such as order and type of transcendence, with problems on p-adic exponentials. Lazard's problem for analytic functions inside a disk is explained. P-adic meromorphics are studied. Sets of range uniqueness in a p-adic field are examined. The ultrametric Corona problem is studied. Injective analytic elements are characterized. The p-adic Nevanlinna theory is described and many applications are given: p-adic Hayman conjecture, Picard's values for derivatives, small functions, branched values, growth of entire functions, problems of uniqueness, URSCM and URSIM, functions of uniqueness, sharing value problems, Nevanlinna theory in characteristic p>0, p-adic Yosida's equation."--
