Abstract: The present paper is a brief survey of properties of finite ultrametric spaces X and corresponding properties of the representing trees TX obtained by authors over the last six years. Some new results are also presented. In particular, a structural characteristic of the representing trees TX is found for the finite ultrametric spacesX which admit a ball-preserving mapping f: Y → Z for all nonempty Y ⊆ X and Z ⊆ Y. PubDate: 2019-01-01

Abstract: This work will be centered in commutative Banach subalgebras of the algebra of bounded linear operators defined on free Banach spaces of countable type. The main goal of this work will be to formulate a representation theorem for these operators through integrals defined by spectral measures type. In order to get this objective, we will show that, under special conditions, each one of these algebras is isometrically isomorphic to some space of continuous functions defined over a compact set. Then, we will identify such compact sets developing the Gelfand space theory in the non-Archimedean setting. This fact will allow us to define a measure which is known as spectral measure. As a second goal, we will formulate a matrix representation theorem for this class of operators in which the entries of the matrices will be integrals coming from scalar measures. PubDate: 2019-01-01

Abstract: In this paper we define a q-extension of Fubini numbers which we call q-Fubini numbers, and generalized q-Fubini numbers of order r. Using the p-adic Laplace transform and p-adic integration, we obtain these numbers as moments of appropriate p-adicmeasures. Then we establish some identities and congruences for these numbers. We establish also a relationship between generalized q-Fubini numbers of order r and q-Fubini numbers. Further, as done in previous works we introduce a concept of generalized q-Fubini numbers, attached to a continuous pℓℤp-invariant function ψ defined on ℤp. These numbers are also the moments of appropriate p-adic measures, we obtain identities and congruences which generalize those associated to q-Fubini numbers. PubDate: 2019-01-01

Abstract: We show that the Connes-Marcolli GL2-system can be represented on the Big Picture, a combinatorial gadget introduced by Conway in order to understand various results about congruence subgroups pictorially. In this representation the time evolution of the GL2-system is implemented by Conway’s distance between projective classes of commensurable lattices. We exploit these results in order to associate quantum statistical mechanical systems to congruence subgroups. This work is motivated by the study of congruence subgroups and their principal moduli in connection with monstrous moonshine. PubDate: 2019-01-01

Abstract: We show that any (1, 2)-rational function with a unique fixed point is topologically conjugate to a (2, 2)-rational function or to the function f(x) = ax/x2 + a. The case (2, 2) was studied in our previous paper, here we study the dynamical systems generated by the function f on the set of complex p-adic field ℂp. We show that the unique fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. We construct the corresponding Siegel disk of these dynamical systems. We determine a sufficiently small set containing the set of limit points. It is given all possible invariant spheres.We show that the p-adic dynamical system reduced on each invariant sphere is not ergodic with respect to Haar measure on the set of p-adic numbers ℚp.Moreover some periodic orbits of the system are investigated. PubDate: 2019-01-01

Abstract: This paper contains a brief review of a very diverse and vast scientific work of Andrei Yurievich Khrennikov on the occasion of his 60th birthday. PubDate: 2018-10-01

Abstract: In the present paper, we consider an interaction of the nearest-neighbors and next nearest-neighbors for the mixed type p-adic λ-Ising model with spin values {−1, +1} on the Cayley tree of order two.We obtained the uniqueness and existence of the p-adic quasi Gibbs measures for the model. Thereafter, as a main result, we proved the occurrence of phase transition for the p-adic λ-Ising model on the Cayley tree of order two. To establish the results, we employed some properties of p-adic numbers. Therefore, our results are not valid in the real case. PubDate: 2018-10-01

Abstract: For every finite ultrametric space X we can put in correspondence its representing tree TX. We found conditions under which the isomorphism of representing trees TX and TY implies the isometricity of ultrametric spaces X and Y having the same range of distances. PubDate: 2018-10-01

Abstract: In this outline of a program, based on rigorous renormalization group theory, we introduce new definitions which allow one to formulate precise mathematical conjectures related to conformal invariance as studied by physicists in the area known as higher-dimensional conformal bootstrap which has developed at a breathtaking pace over the last few years. We also explore a second theme, intimately tied to conformal invariance for random distributions, which can be construed as a search for very general first and second-quantized Kolmogorov-Chentsov Theorems. First-quantized refers to regularity of sample paths. Second-quantized refers to regularity of generalized functionals or Hida distributions and relates to the operator product expansion.We formulate this program in both the Archimedean and p-adic situations. Indeed, the study of conformal field theory and its connections with probability provides a golden opportunity where p-adic analysis can lead the way towards a better understanding of open problems in the Archimedean setting. Finally, we present a summary of progress made on a p-adic hierarchical model and point out possible connections to number theory. Parts of this article were presented in author’s talk at the 6th International Conference on p-adicMathematical Physics and its Applications,Mexico 2017. PubDate: 2018-10-01

Abstract: 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. PubDate: 2018-10-01

Abstract: We give a review of finite approximations of quantum systems, both in an Archimedean and a non-Archimedean setting. Proofs will generally be omitted. In the Appendix we present some numerical results. PubDate: 2018-10-01

Abstract: Yuri Manin’s approach to Zipf’s law (Kolmogorov complexity as energy) is applied to investigation of biological evolution. Model of constructive statistical mechanics where complexity is a contribution to energy is proposed to model genomics. Scaling laws in genomics are discussed in relation to Zipf’s law. This gives a model of Eugene Koonin’s Third Evolutionary Synthesis – physical model which should describe scaling in genomics. PubDate: 2018-10-01

Abstract: In this paper we apply Ax-Schanuel’s Theorem to the ultraproduct of p-adic fields in order to get some results towards algebraic independence of p-adic exponentials for almost all primes p. PubDate: 2018-10-01

Abstract: 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. PubDate: 2018-10-01

Abstract: We consider a new class of functions on the p-adic linear space ℚ p n for which a Fourier transform can be defined.We prove equalities of Parseval type, an inversion formula and a sufficient condition for a function to be represented as this Fourier transform. Also we give a sharp estimate of the L2(ℚ p n ) modulus of continuity in terms of Fourier transform generalizing the result of S. S. Platonov in the case n = 1. Finally we prove a generalization of this result and its converse for Lq(ℚ p n ) with appropriate q. PubDate: 2018-10-01

Abstract: We describe dynamical systems associated to (1 − 1)-rational functions on the field of p-adic numbers.We focus on sets of minimality of such systems. PubDate: 2018-07-01

Abstract: We discuss the properties of the continuations of real functions to the Levi-Civita field. In particular, we show that, whenever a function f is analytic on a compact interval [a, b] ⊂ ℝ, f and its analytic continuation f̅∞ satisfy the same properties that can be expressed in the language of real closed ordered fields. If f is not analytic, then this equivalence does not hold. These results suggest an analogy with the internal and external functions of nonstandard analysis: while the canonical continuations of analytic functions resemble internal functions, the continuations of non-analytic functions behave like external functions. Inspired by this analogy, we suggest some directions for further research. PubDate: 2018-07-01

Abstract: In this paper, we provide necessary and sufficient conditions for 1-Lipschitz functions that are uniformly differentiable mod p on ℤp to be measure-preserving, in Mahler’s expansion, and show that these conditions can be modified to guarantee the existence of a root of p-adic Lipschitz functions. PubDate: 2018-07-01

Abstract: This paper, which is a summary (in which considerable creative license has been taken) of the author’s talk at the sixth international conference on p-adic mathematical physics and its applications (CINVESTAV, Mexico City, October 2017), reviews some recent work connecting field theories defined on the p-adic numbers and ideas from the AdS/CFT correspondence. Some results are included, along with general discussion of the utility and interest of p-adic analogues of Lagrangian field theories, at least from the author’s perspective. A few challenges, shortcomings, and ideas for future work are also discussed. PubDate: 2018-07-01

Abstract: The differential phase shift quantum key distribution protocol is of high interest due to its relatively simple practical implementation. This protocol uses trains of coherent pulses and allows the legitimate users to resist individual attacks. In this paper, a new attack on this protocol is proposed which is based on the idea of information extraction from the part of each coherent state and then making decision about blocking the rest part depending on the amount of extracted information. PubDate: 2018-07-01