Quantum arithmetics provides a possible resolution of a long-lasting challenge of finding a mathematical justification for the canonical identification mapping p-adics to reals playing a key role in TGD in particular in p-adic mass calculations. p-Adic numbers have p-adic pinary expansions ∑ anp n satisfying an < p. of powers p n to be products of primes p1 < p satisfying an < p for ordinary p-...

We study Markovian and non-Markovian behaviour of stochastic processes generated by random dynamical systems on p-adic trees. In fact, such systems can be interpreted as stochastic neural networks operating on branches of the homogeneous p-adic tree (where p > 1 is a prime number). Key-Words: Random dynamical systems, p-adic numbers, Markovian property

New approach to p-adic and adelic strings, which takes into account that not only world sheet but also Minkowski space-time and string momenta can be p-adic and adelic, is formulated. p-Adic and adelic string amplitudes are considered within Feynman’s path integral formalism. The adelic Veneziano amplitude is calculated. Some discreteness of string momenta is obtained. Also, adelic coupling con...

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 re...

Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers, Q denotes the field of rational numbers, Qp denotes the field of p-adic rational numbers, C denotes the complex number field, and Cp denotes the completion of algebraic closure of Qp. Let νp be the normalized exponential valuation of Cp with |p|p = p−νp(p) = p−1. When one talks of q-exten...

We use a p-adic analogue of the analytic subgroup theorem of Wüstholz to deduce the transcendence and linear independence of some new classes of p-adic numbers. In particular we give p-adic analogues of results of Wüstholz contained in [20] and generalizations of results obtained by Bertrand in [3, 4].

Let p be a fixed odd positive integer. Throughout this paper Zp, Qp, C and Cp are respectively denoted as the ring of p-adic rational integers, the field of p-adic rational numbers, the complex numbers field and the completion of algebraic closure of Qp. The p-adic absolute value in Cp is normalized so that |p|p = 1/p. When one talks of q-extension, q is considered in many ways such as an indet...

In particular, the values at x = 0 are called Bernoulli numbers of order k, that is, Bn (0) = Bn k) (see [1, 2, 4, 5, 9, 10, 14]). When k = 1, the polynomials or numbers are called ordinary. The polynomials Bn (x) and numbers Bn were first defined and studied by Norlund [9]. Also Carlitz [2] and others investigated their properties. Recently they have been studied by Adelberg [1], Howard [5], a...

Let p be a fixed prime number. Throughout this paper, the symbols Z,Zp,Qp,C, and Cp will denote the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers, the complex number field, and the completion of the algebraic closure of Qp, respectively. Let N be the set of natural numbers and Z N ∪ {0}. Let vp be the normalized exponential valuation of Cp with |p|...

Using basic properties of p-adic numbers, we consider a simple new approach to describe main aspects of DNA sequence and genetic code. Central role in our investigation plays an ultrametric p-adic information space which basic elements are nucleotides, codons and genes. We show that a 5-adic model is appropriate for DNA sequence. This 5-adic model, combined with 2-adic distance, is also suitabl...