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  • 1. Albeverio, Sergio
    et al.
    Cianci, Roberto
    Khrennikov, Andrei
    Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
    p-Adic valued quantization2009In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 1, no 2, p. 91-104Article, review/survey (Refereed)
    Abstract [en]

    This review covers an important domain of p-adic mathematical physics — quantum mechanics with p-adic valued wave functions. We start with basic mathematical constructions of this quantum model: Hilbert spaces over quadratic extensions of the field of p-adic numbers ℚ p , operators — symmetric, unitary, isometric, one-parameter groups of unitary isometric operators, the p-adic version of Schrödinger’s quantization, representation of canonical commutation relations in Heisenberg andWeyl forms, spectral properties of the operator of p-adic coordinate.We also present postulates of p-adic valued quantization. Here observables as well as probabilities take values in ℚ p . A physical interpretation of p-adic quantities is provided through approximation by rational numbers.

  • 2.
    Aref’eva, I. Ya.
    et al.
    Russian Academy of Sciences, Russia.
    Djordjevic, G. S.
    University of Niš, Serbia.
    Khrennikov, Andrei
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Kozyrev, S. V.
    Russian Academy of Sciences, Russia.
    Rakic, Z.
    University of Belgrade, Serbia.
    Volovich, I. V.
    Russian Academy of Sciences, Russia.
    p-Adic mathematical physics and B. Dragovich research2017In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 9, no 1, p. 82-85Article in journal (Refereed)
    Abstract [en]

    We present a brief review of some parts of p-adic mathematical physics related to the scientific work of Branko Dragovich on the occasion of his 70th birthday.

  • 3. Dragovich, B.
    et al.
    Khrennikov, Andrei
    Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
    Kozyrev, S. V.
    Volovich, I. V.
    On p-adic mathematical physics2009In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 1, no 1, p. 1-17Article, review/survey (Refereed)
    Abstract [en]

    A brief review of some selected topics in p-adic mathematical physics is presented.

  • 4.
    Dragovich, B.
    et al.
    University of Belgrade, Serbia.
    Khrennikov, Andrei
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Kozyrev, S. V.
    Russian Academy of Sciences, Russia.
    Volovich, I. V.
    Russian Academy of Sciences, Russia.
    Vasiliy Sergeevich Vladimirov (09.01.1923 – 03.11.2012)2013In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 5, no 1, p. 83-86Article in journal (Other academic)
    Abstract [en]

    We present a brief biographical review of the scientific work and achievements of Vasiliy Sergeevich Vladimirov on the occasion of his death on November 3, 2012.

  • 5.
    Dragovich, B.
    et al.
    University of Belgrade, Serbia;Serbian Academy of Sciences and Arts, Serbia.
    Khrennikov, Andrei
    Linnaeus University, Faculty of Technology, Department of Mathematics. National Research University of Information Technologies, Russia.
    Kozyrev, S. V.
    Steklov Mathematical Institute of the Russian Academy of Sciences, Russia.
    Volovich, I. V.
    Steklov Mathematical Institute of the Russian Academy of Sciences, Russia.
    Zelenov, E. I.
    Steklov Mathematical Institute of the Russian Academy of Sciences, Russia.
    p-Adic mathematical physics: the first 30 years2017In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 9, no 2, p. 87-121Article in journal (Refereed)
    Abstract [en]

    p-Adic mathematical physics is a branch of modern mathematical physics based on the application of p-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas including biology. This paper contains a brief review of main achievements in some selected topics of p-adic mathematical physics and its applications, especially in the last decade. Attention is mainly paid to developments with promising future prospects.

  • 6. Durand, Fabien
    et al.
    Fan, Ai-Hua
    Khrennikov, Andrei
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Lin, Dongdai
    Vivaldi, Franco
    Wang, Yuefei
    Vladimir Sergeevich Anashin2011In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 3, no 4, p. 359-362Article in journal (Other academic)
  • 7.
    Iurato, G.
    et al.
    University of Palermo, Italy.
    Khrennikov, Andrei
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    On the topological structure of a mathematical model of human unconscious2017In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 9, no 1, p. 78-81Article in journal (Refereed)
    Abstract [en]

    On the basis of two our previous works, in this paper, following Jacques Lacan psychoanalytic theory, we wish to outline some further remarks on the topological structure of a mathematical model of human unconscious.

  • 8.
    Iurato, G.
    et al.
    University of Palermo, Italy.
    Khrennikov, Andrei
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Murtagh, F.
    University of Derby, UK ; Goldsmiths University of London, UK.
    Formal foundations for the origins of human consciousness2016In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 8, no 4, p. 249-279Article in journal (Refereed)
    Abstract [en]

    In the framework of p-adic analysis (the simplest version of analysis on trees in which hierarchic structures are presented through ultrametric distance) applied to formalize psychic phenomena, we would like to propose some possible first hypotheses about the origins of human consciousness centered on the basic notion of time symmetry breaking as meant according to quantum field theory of infinite systems. Starting with Freud’s psychophysical (hydraulic) model of unconscious and conscious flows of psychic energy based on the three-orders mental representation, the emotional order, the thing representation order, and the word representation order, we use the p-adic (treelike) mental spaces to model transition from unconsciousness to preconsciousness and then to consciousness. Here we explore theory of hysteresis dynamics: conscious states are generated as the result of integrating of unconscious memories. One of the main mathematical consequences of our model is that trees representing unconscious and consciousmental states have to have different structures of branching and distinct procedures of clustering. The psychophysical model of Freud in combination with the p-adic mathematical representation gives us a possibility to apply (for a moment just formally) the theory of spontaneous symmetry breaking of infinite dimensional field theory, to mental processes and, in particular, to make the first step towards modeling of interrelation between the physical time (at the level of the emotional order) and psychic time at the levels of the thing and word representations. Finally, we also discuss some related topological aspects of the human unconscious, following Jacques Lacan’s psychoanalytic concepts.

  • 9.
    Iurato, Giuseppe
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    A possible p-adic Weber-Fechner law2018In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 10, no 4, p. 299-304Article in journal (Refereed)
    Abstract [en]

    From a simple extension of a previous formal pattern of unconscious-conscious interconnection based on the representation of mental entities by m-adic numbers through hysteresis phenomenology, a pattern which has been then used to work out a possible psychoanalytic model of human consciousness, we now argue on related simple derivations of p-adic Weber-Fechner laws of psychophysics.

  • 10.
    Iurato, Giuseppe
    et al.
    University of Palermo, Italy.
    Khrennikov, Andrei
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Hysteresis model of unconscious-conscious interconnection: Exploring dynamics on m-adic trees2015In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 7, no 4, p. 312-321Article in journal (Refereed)
    Abstract [en]

    In this brief note, we focus attention on a possible implementation of a basic hysteretic pattern (the Preisach one), suitably generalized, into a formal model of unconscious-conscious interconnection and based on representation of mental entities by m-adic numbers.

  • 11.
    Khrennikov, Andrei
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Cognitive processes of the brain: An ultrametric model of information dynamics in unconsciousness2014In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 6, no 4, p. 293-302Article in journal (Refereed)
    Abstract [en]

    We discuss differences in mathematical representations of the physical and mentalworlds. Following Aristotle, we present the mental space as discrete, hierarchic, and totally disconnectedtopological space. One of the basic models of such spaces is given by ultrametricspaces and more specially by m-adic trees. We use dynamical systems in such spaces to modelflows of unconscious information at different level of mental representation hierarchy, for “mentalpoints”, categories, and ideas. Our model can be interpreted as an unconventional computationalmodel: non-algorithmic hierarchic “computations” (identified with the process of thinking at theunconscious level).

  • 12.
    Khrennikov, Andrei
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Foundations of analysis on superspace — 1: Differential calculus2015In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 7, no 2, p. 96-110Article in journal (Refereed)
    Abstract [en]

    The recent experimental confirmation of the existence of Higgs boson stimulates theoretical research on supersymmetric models; in particular, mathematics of such modeling. Therefore we plan to present essentials of one special approach to “super-mathematics”, so called functional superanalysis (in the spirit of De Witt, Rogers, Vladimirov and Volovich, and the author of this review) in compact and clear form in series of review-type papers. This first paper is a review on super-differential calculus for the concrete model of the superspace (invented by Vladimirov and Volovich). In the next review we plan to present the integral supercalculus. The main distinguishing feature of functional superanalysis is that this is a real super-extension of analysis of Newton and Leibniz, opposite to algebraic models of Martin and Berezin. Here functions of commuting and anticommuting variables are no simply algebraic elements belonging to Grassmann algebras, but point-wise maps, from superspace into superspace. Finally, we remark that the first non-Archimedean physical model was based on invention by Vladimirov and Volovich of superspaces based on supercommutative Banach superalgebras over non-Archimedean (in particular, p-adic) fields. This model plays the basic role in theory of p-adic superstrings.

  • 13.
    Khrennikov, Andrei
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Modelling of psychological behavior on the basis of ultrametric mental space: Encoding of categories by balls2010In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 2, no 1, p. 1-20Article, review/survey (Refereed)
  • 14.
    Khrennikov, Andrei
    et al.
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Kozyrev, Sergei V.
    Genetic code and deformation of the 2-dimensional 2-adic metric2011In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 3, no 2, p. 165-168Article in journal (Refereed)
  • 15.
    Khrennikov, Andrei
    et al.
    Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
    Schumann, Andrew
    p-Adic physics, non-well-founded reality and unconventional computing2009In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 1, no 4, p. 297-306Article in journal (Refereed)
    Abstract [en]

    We consider perspectives of application of coinductive and corecursive methods of non-well-founded mathematics to modern physics, especially to adelic and p-adic quantum mechanics. We also survey perspectives of relationship between modern physics and unconventional computing.

  • 16.
    Khrennikov, Andrei
    et al.
    Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
    Shelkovich, V. M.
    Skopina, M.
    p-adic orthogonal wavelet bases2009In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 1, no 2, p. 145-156Article in journal (Refereed)
    Abstract [en]

    We describe all MRA-based p-adic compactly supported wavelet systems forming an orthogonal basis for L 2(ℚ p ).

  • 17.
    Khrennikov, Andrei
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Shelkovich, V
    St.-Petersburg State Architecture and Civil Engineering University.
    van der Walt, J H
    University of Pretoria.
    Measure-free viewpoint on p-adic and adelic wavelets2013In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 5, no 3, p. 204-217Article in journal (Refereed)
    Abstract [en]

    Recently there were presented new examples of MRAs based on p-adic and adelicwavelets. This is a note of the methodological nature. Its aim is embed all known examples MRAsinto the measure-free scheme based on abstract Hilbert space formalism. On the one hand, suchan embedding unifies the known examples of MRAs. On the other hand, the presented measure freescheme can be used for construction of new examples ofMRAs, especially in the infinite dimensionalcase. The last section of this paper is devoted to another abstract MRAs scheme based on Rieszspaces.We start paper with a review on p-adic and adelic wavelets.

  • 18.
    Khrennikov, Andrei
    et al.
    Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
    Shelkovich, Vladimir M.
    An infinite family of p-adic non-Haar wavelet bases and pseudo-differential operators2009In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 1, no 3, p. 204-216Article in journal (Refereed)
    Abstract [en]

    In this paper an infinite family of new compactly supported non-Haar p-adic wavelet bases in L 2 (Q n p )  is constructed. We also study the connections between wavelet analysis and spectral analysis of p-adic pseudo-differential operators. A criterion for a multidimensional p-adic wavelet to be an eigenfunction for a pseudo-differential operator is derived. We prove that these wavelets are eigenfunctions of the fractional operator. Since many p-adic models use pseudo-differential operators (fractional operator), these results can be intensively used in these models.

  • 19.
    Khrennikov, Andrei
    et al.
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Van der Walt, Jan Harm
    On topological extensions of Archimedean and non-Archimedean rings2011In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 3, no 4, p. 326-333Article in journal (Refereed)
  • 20.
    Klein, M.
    et al.
    Ohalo College Katzrin, Israel.
    Khrennikov, Andrei
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Recursion over Partitions2014In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 6, no 4, p. 303-309Article in journal (Refereed)
    Abstract [en]

    The paper demonstrates how to apply a recursion on the fundamental concept of number. We propose a generalization of the partitions of a positive integer n, by defining new combinatorial objects, namely sub-partitions. A recursive formula is suggested, designated to solve the associated enumeration problem. It is highlighted that sub-partitions provide a good language to study rooted phylogenetic trees.

  • 21.
    Lindahl, Karl-Olof
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Applied Algebraic Dynamics2010In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 2, no 4, p. 360-362Article in journal (Refereed)
  • 22.
    Lindahl, Karl-Olof
    Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
    Linearization in Ultrametric Dynamics in Fields of Characteristic Zero — Equal Characteristic Case2009In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 1, no 4, p. 307-316Article in journal (Refereed)
    Abstract [en]

    Let K be a complete ultrametric field of characteristic zero whose corresponding residue field k is also of characteristic zero. We give lower and upper bounds for the size of linearization disks for power series over K near an indifferent fixed point. These estimates are maximal in the sense that there exist examples where these estimates give the exact size of the corresponding linearization disc. Similar estimates in the remaining cases, i.e. the cases in which K is either a p-adic field or a field of prime characteristic, were obtained in various papers on the p-adic case [5, 18, 35, 42] later generalized in [28, 30], and in [29, 31] concerning the prime characteristic case.

  • 23.
    Lindahl, Karl-Olof
    et al.
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Zieve, Michael
    Michigan State University.
    On Hyperbolic Fixed Points in Ultrametric Dynamics2010In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 2, no 3, p. 232-240Article in journal (Refereed)
    Abstract [en]

    Let K be a complete ultrametric field. We give lower and upper bounds for the size of linearization discs for power series over K near hyperbolic fixed points. These estimates are maximal in the sense that there exist examples where these estimates give the exact size of the corresponding linearization disc. In particular, at repelling fixed points, the linearization disc is equal to the maximal disc on which the power series is injective.

  • 24.
    Pourhadi, Ehsan
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics. Iran University of Science and Technology, Iran.
    Khrennikov, Andrei
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    On the solutions of Cauchy problem for two classes of semi-linear pseudo-differential equations over p-adic field2018In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 10, no 4, p. 322-343Article in journal (Refereed)
    Abstract [en]

    Throughout this paper, using the p-adic wavelet basis together with the help of separation of variables and the Adomian decomposition method (as a scheme in numerical analysis) we initially investigate the solution of Cauchy problem for two classes of the first and second order of pseudo-differential equations involving the pseudo-differential operators such as Taibleson fractional operator in the setting of p-adic field.

  • 25.
    Svensson, Per-Anders
    et al.
    Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
    Nyqvist, Robert
    Department of Mathematics and Science, Blekinge Institute of Technology, Karlskrona, Sweden.
    On the Number of Equivalence Classes of Attracting Dynamical Systems2009In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 1, no 3, p. 264-270Article in journal (Refereed)
    Abstract [en]

    We study discrete dynamical systems of the kind h(x) = x + g(x), where g(x) is a monic irreducible polynomial with coefficients in the ring of integers of a p-adic field K. The dynamical systems of this kind, having attracting fixed points, can in a natural way be divided into equivalence classes, and we investigate whether something can be said about the number of those equivalence classes, for a certain degree of the polynomial g(x).

  • 26.
    Yurova Axelsson, Ekaterina
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    On recent results of ergodic property for p-adic dynamical systems2014In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 6, no 3, p. 235-257Article in journal (Refereed)
    Abstract [en]

    Theory of dynamical systems in fields of p-adic numbers is an important part of algebraic and arithmetic dynamics. The study of p-adic dynamical systems is motivated by their applications in various areas of mathematics, physics, genetics, biology, cognitive science, neurophysiology, computer science, cryptology, etc. In particular, p-adic dynamical systems found applications in cryptography, which stimulated the interest to nonsmooth dynamical maps. An important class of (in general) nonsmooth maps is given by 1-Lipschitz functions. In this paper we present a recent summary of results about the class of 1-Lipschitz functions and describe measure-preserving (for the Haar measure on the ring of p-adic integers) and ergodic functions. The main mathematical tool used in this work is the representation of the function by the van der Put series which is actively used in p-adic analysis. The van der Put basis differs fundamentally from previously used ones (for example, the monomial and Mahler basis) which are related to the algebraic structure of p-adic fields. The basic point in the construction of van der Put basis is the continuity of the characteristic function of a p-adic ball. Also we use an algebraic structure (permutations) induced by coordinate functions with partially frozen variables.

  • 27.
    Yurova, Ekaterina
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    On ergodicity of p-adic dynamical systems for arbitrary prime p2013In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 5, no 3, p. 239-241Article in journal (Refereed)
    Abstract [en]

    In this paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of 1-Lipschitz p-adic functions that are defined on (and valuated in) the space ℤ p of p-adic integers for any prime p. The conditions are stated in terms of coordinate representations of p-adic functions.

  • 28.
    Yurova, Ekaterina
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    On measure-preserving functions over ℤ3.2012In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 4, no 4, p. 326-335Article in journal (Refereed)
    Abstract [en]

    This paper is devoted to (discrete) p-adic dynamical systems, an important domain ofalgebraic and arithmetic dynamics [31]-[41], [5]-[8]. In this note we study properties of measurepreservingdynamical systems in the case p = 3. This case differs crucially from the case p = 2.The latter was studied in the very detail in [43]. We state results on all compatible functions whichpreserve measure on the space of 3-adic integers, using previous work of A. Khrennikov and authorof present paper, see [24]. To illustrate one of the obtained theorems we describe conditions for the3-adic generalized polynomial to be measure-preserving on Z3. The generalized polynomials withintegral coefficients were studied in [17, 33] and represent an important class of T-functions. Inturn, it is well known that T-functions are well-used to create secure and efficient stream ciphers,pseudorandom number generators.

  • 29.
    Yurova, Ekaterina
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Van der Put basis and p-adic dynamics2010In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 2, no 2, p. 175-178Article in journal (Other academic)
1 - 29 of 29
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