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  • 1. Anashin,, V
    et al.
    Khrennikov, Andrei
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Yurova, Ekaterina
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Using van der Put basis to determine if a 2-adic function is measure-preserving or ergodic w.r.t. Haar measure2011In: Advances in Non-Archimedean Analysis: 11th International Conference p-adic Funcional Analysis, July 5-9, 2010, Université Blaise Pascal, Clemont-Ferrand, France / [ed] Jesus Araujo-Gomez, Bertin Diarra and Alain Escassut, American Mathematical Society (AMS), 2011, Vol. 551, p. 33-38Chapter in book (Refereed)
  • 2.
    Anashin, V. S.
    et al.
    Moscow State University, Russia.
    Khrennikov, Andrei
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Yurova, Ekaterina
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Characterization of ergodicity of p-adic dynamical systems by using the van der Put basis.2011In: Doklady. Mathematics, ISSN 1064-5624, E-ISSN 1531-8362, Vol. 83, no 3, p. 306-308Article in journal (Refereed)
  • 3.
    Anashin, V. S.
    et al.
    Moscow State Univeristy.
    Khrennikov, Andrei
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Yurova, Ekaterina
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Ergodicity of dynamical systems on 2-adic spheres2012In: Doklady. Mathematics, ISSN 1064-5624, E-ISSN 1531-8362, Vol. 86, no 3, p. 843-845Article in journal (Refereed)
  • 4.
    Anashin, Vladimir
    et al.
    Moscow MV Lomonosov State Univ, Fac Computat Math & Cybernet, Moscow 119991, Russia.
    Khrennikov, Andrei
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Yurova, Ekaterina
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Ergodicity criteria for non-expanding transformations of 2-adic spheres2014In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 34, no 2, p. 367-377Article in journal (Refereed)
    Abstract [en]

    In the paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of discrete dynamical systems < f; S2-r (a)> on 2-adic spheres S2-r (a) of radius 2(-r), r >= 1, centered at some point a from the ultrametric space of 2-adic integers Z(2). The map f: Z(2) -> Z(2) is assumed to be non-expanding and measure-preserving; that is, f satisfies a Lipschitz condition with a constant 1 with respect to the 2-adic metric, and f preserves a natural probability measure on Z(2), the Haar measure mu(2) on Z(2) which is normalized so that mu(2)(Z(2)) = 1.

  • 5.
    Anashin, Vladimir
    et al.
    Lomonosov Moscow State University.
    Khrennikov, Andrei
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Yurova, Ekaterina
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    T-functions revisited: new criteria for bijectivity/transitivity2014In: Designs, Codes and Cryptography, ISSN 0925-1022, E-ISSN 1573-7586, Vol. 71, no 3, p. 383-407Article in journal (Refereed)
    Abstract [en]

    The paper presents new criteria for bijectivity/transitivity of T-functions and a fast knapsack-like algorithm of evaluation of a T-function. Our approach is based on non-Archimedean ergodic theory: Both the criteria and algorithm use van der Put series to represent 1-Lipschitz p-adic functions and to study measure-preservation/ergodicity of these.

  • 6.
    Khrennikov, Andrei
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Yurova, Ekaterina
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Automaton model of protein: Dynamics of conformational and functional states2017In: Progress in Biophysics and Molecular Biology, ISSN 0079-6107, E-ISSN 1873-1732, Vol. 130, no A, p. 2-14Article in journal (Refereed)
    Abstract [en]

    In this conceptual paper we propose to explore the analogy between ontic/epistemic description of quantum phenomena and interrelation between dynamics of conformational and functional states of proteins. Another new idea is to apply theory of automata to model the latter dynamics. In our model protein's behavior is modeled with the aid of two dynamical systems, ontic and epistemic, which describe evolution of conformational and functional states of proteins, respectively. The epistemic automaton is constructed from the ontic automaton on the basis of functional (observational) equivalence relation on the space of ontic states. This reminds a few approaches to emergent quantum mechanics in which a quantum (epistemic) state is treated as representing a class of prequantum (ontic) states. This approach does not match to the standard protein structure-function paradigm. However, it is perfect for modeling of behavior of intrinsically disordered proteins. Mathematically space of protein's ontic states (conformational states) is modeled with the aid of p-adic numbers or more general ultrametric spaces encoding the internal hierarchical structure of proteins. Connection with theory of p-adic dynamical systems is briefly discussed.

  • 7.
    Khrennikov, Andrei
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Yurova, Ekaterina
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Criteria of ergodicity for p-adic dynamical systems in terms of coordinate functions2014In: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 60, p. 11-30Article in journal (Refereed)
    Abstract [en]

    This paper is devoted to the problem of ergodicity of p-adic dynamical systems. We solved the problem of characterization of ergodicity and measure preserving for (discrete) p-adic dynamical systems for arbitrary prime p for iterations based on 1-Lipschitz functions. This problem was open since long time and only the case p = 2 was investigated in details. We formulated the criteria of ergodicity and measure preserving in terms of coordinate functions corresponding to digits in the canonical expansion of p-adic numbers. (The coordinate representation can be useful, e.g., for applications to cryptography.) Moreover, by using this representation we can consider non-smooth p-adic transformations. The basic technical tools are van der Put series and usage of algebraic structure (permutations) induced by coordinate functions with partially frozen variables. We illustrate the basic theorems by presenting concrete classes of ergodic functions. As is well known, p-adic spaces have the fractal (although very special) structure. Hence, our study covers a large class of dynamical systems on fractals. Dynamical systems under investigation combine simplicity of the algebraic dynamical structure with very high complexity of behavior.

  • 8.
    Khrennikov, Andrei
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Yurova, Ekaterina
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Criteria of measure-preserving for p-adic dynamical systems in terms of the van der Put basis2013In: Journal of Number Theory, ISSN 0022-314X, E-ISSN 1096-1658, Vol. 133, no 2, p. 484-491Article in journal (Refereed)
    Abstract [en]

    This paper is devoted to (discrete) p-adic dynamical systems, an important domain of algebraic and arithmetic dynamics. We consider the following open problem from theory of p-adic dynamical systems. Given continuous function f : Z(p) -> Z(p). Let us represent it via special convergent series, namely van der Put series. How can one specify whether this function is measure-preserving or not for an arbitrary p? In this paper, for any prime p, we present a complete description of all compatible measure-preserving functions in the additive form representation. In addition we prove the criterion in terms of coefficients with respect to the van der Put basis determining whether a compatible function f : Z(p) -> Z(p) preserves the Haar measure. (C) 2012 Elsevier Inc. All rights reserved.

  • 9.
    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.

  • 10.
    Yurova Axelsson, Ekaterina
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    On the representation of the genetic code by the attractors of 2-adic function2015In: Physica scripta. T, ISSN 0281-1847, Vol. 2015, no T 165, article id 014043Article in journal (Refereed)
    Abstract [en]

    The genetic code is a map which gives the correspondence between codons in DNA and amino acids. As a continuation of the study made by Khrennikov and Kozyrev on the genetic code, we consider a construction, where amino acids are associated to the attractors of some two-adic function. In this paper, we give an explicit form of representations for the standard nuclear and vertebrate mitochondrial genetics codes. To set these functions we use a van der Put representation. The usage of the van der Put series reduces the complexity of computation for explicit form of the functions for the genetic codes.

  • 11.
    Yurova Axelsson, Ekaterina
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    P-adic dynamical systems and van der Put basis technique2013Doctoral thesis, comprehensive summary (Other academic)
    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, e.g., in 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 thesis we restrict our study to 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.

    In this thesis, we present a description of 1-Lipschitz measure-preserving and ergodic functions for arbitrary prime p.

  • 12.
    Yurova Axelsson, Ekaterina
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    The ergodicity of 1-Lipschitz transformations on 2-adic spheres2014In: Valuation Theory in Interaction / [ed] A. Campillo, F.-V. Kuhlmann, B. Teissier, European Mathematical Society Publishing House, 2014, p. 596-599Conference paper (Refereed)
    Abstract [en]

    In this paper we present results about ergodicity of dynamical systems on 2-adic spheres for 1-Lipschitz maps f : Z(2) -> Z(2) announced in [8], and extension of Theorem 3 from [8] for the case of spheres of radii greater than 1/8. We propose a new approach to study ergodic properties of 1-Lipschitz transformations of 2-adic spheres. We use a representation of continuous functions f via its van der Put series. This technique allows us to go beyond the classes of smooth 1-Lipschitz transformations which were studied earlier.

  • 13.
    Yurova Axelsson, Ekaterina
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Khrennikov, Andrei
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Generalization of Hensel's lemma: Finding the roots of p-adic Lipschitz functions2016In: Journal of Number Theory, ISSN 0022-314X, E-ISSN 1096-1658, Vol. 158, p. 217-233Article in journal (Refereed)
    Abstract [en]

    In this paper we consider the problem of finding the roots of p-adic functions. In the case, where the function is defined by a polynomial with integer p-adic coefficients, using Hensel's lifting lemma helps us find the roots of the p-adic function.

    We generalize Hensel's lifting lemma for a wider class of p  -adic functions, namely, the functions which satisfy the Lipschitz condition with constant , in particular, the functions of this class may be non-differentiable. The paper also presents an iterative procedure for finding approximate (in p  -adic metric) values of the root of pα-Lipschitz functions, thus generalizing the p-adic analogue of Newton's method for such a class of functions.

  • 14.
    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.

  • 15.
    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.

  • 16.
    Yurova, Ekaterina
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    On the Injective Embedding of p-Adic Integers in the Cartesian Product of p Copies of Sets of 2-Adic Integers2019In: Analysis, Probability, Applications, and Computation / [ed] Karl‐Olof Lindahl, Torsten Lindström, Luigi G. Rodino, Joachim Toft, Patrik Wahlberg, Birkhäuser Verlag, 2019, p. 233-239Conference paper (Refereed)
    Abstract [en]

    We study an injective embedding of p-adic integers in the Cartesian product of p copies of sets of 2-adic integers. This embedding allows to explicitly specify any p-adic integer through p specially selected 2-adic numbers. This representation can be used in p-adic mathematical physics, for example, in justifying choice of the parameter p.

  • 17.
    Yurova, Ekaterina
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    On the sub-coordinate representation of p-adic functions2018In: Advances in Ultrametric Analysis / [ed] Alain Escassut, Cristina Perez-Garcia, Khodr Shamseddine, USA: American Mathematical Society (AMS), 2018, Vol. 704, p. 285-290Conference paper (Refereed)
    Abstract [en]

    In this paper we introduce a new way of representation of p-adic functions, namely, the sub-coordinate representation. The main feature of such representation is that the values of a function f are given in the canonical form of representation of p-adic number. In the sub-coordinate representation thefunction f is determined by a set of p-valued functions that map a set {0,1,...,p - 1} into itself, and by the order of these functions. As one of the applications ofthe sub-coordinate representation, we study a problem of generalization of Hensel's lifting lemma.

  • 18.
    Yurova, Ekaterina
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Study of ergodicity of p-adic dynamical systems with the aid of van der Put basis2011Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    The study of p-adic dynamical systems is motivated by their applications in various (and surprisingly diverse) areas of mathematics, e.g., in physics, genetics, biology, cognitive science, neurophysiology, computer science, cryptology, etc.

    In this thesis we use decomposition of a continuous function f : zp -> zp into a convergent van der Put series to determine whether f is 1-Lipschitz, measure-preserving and/or ergodic.

    The main mathematical tool used in this research is the representation of the function by the van der Put series, which are special convergent series from p-adic analysis.

    This is the first attempt to use the van der Put basis to examine the properties of (discrete) dynamical systems in fields of p-adic numbers. Note that the van der Put basis differs fundamentally from previously used ones, for example, the monomial and Mahler bases, which are related to the algebraic structure of p-adic fields.

    The van der Put basis is related to the zero dimensional topology of these fields (ultrametric structure), since it consists of characteristic functions of p-adic balls; i.e., the basic point in the construction of this basis is the continuity of the characteristic function of a p-adic ball.

  • 19.
    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)
  • 20.
    Yurova, Ekaterina
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Khrennikov, Andrei
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Description of (Fully) Homomorphic Cryptographic Primitives Within the p-Adic Model of Encryption2019In: Analysis, Probability, Applications, and Computation: Proceedings of the 11th ISAAC Congress, Växjö (Sweden) 2017 / [ed] Karl‐Olof Lindahl, Torsten Lindström, Luigi G. Rodino, Joachim Toft, Patrik Wahlberg, Cham: Birkhäuser Verlag, 2019, p. 241-248Conference paper (Refereed)
    Abstract [en]

    In this paper we consider a description of homomorphic and fully homomorphic cryptographic primitives in the p-adic model. This model describes a wide class of ciphers (including substitution ciphers, substitution ciphers streaming, keystream ciphers in the alphabet of p elements), but certainly not all. Homomorphic and fully homomorphic ciphers are used to ensure the credibility of remote computing, including cloud technology. Within considered p-adic model we describe all homomorphic cryptographic primitives with respect to arithmetic and coordinate-wise logical operations in the ring of p-adic integers ℤ p . We show that there are no fully homomorphic cryptographic primitives for each pair of the considered set of arithmetic and coordinate-wise logical operations on ℤ p.

  • 21.
    Yurova, Ekaterina
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Khrennikov, Andrei
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Subcoordinate Representation of p-adic Functions and Generalization of Hensel's Lemma2018In: Izvestiya. Mathematics, ISSN 1064-5632, E-ISSN 1468-4810, Vol. 82, no 3, p. 632-645Article in journal (Refereed)
    Abstract [en]

    In this paper we describe a new representation of p-adic functions, the so-called subcoordinate representation. The main feature of the subcoordinaterepresentation of a p-adic function is that the values of the function f are given in the canonical form of the representation of p-adic numbers. The function f itself is determined by a tuple of p-valued functions from the set {0, 1,..., p-1} into itself and by the order in which these functions are used to determine the values of f. We also give formulae that enable one to pass from the subcoordinate representation of a 1-Lipschitz function to its van der Put series representation. The effective use of the subcoordinate representation of p-adic functions is illustrated by a study of the feasibility of generalizing Hensel's lemma.

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