P Adic Analysis

Constants with formulae of the form treated by D. Bailey, P. Borwein, and S. Plouffe (BBP formulae to a given base b) have interesting computational properties, such as allowing single dig- its in their base b expansion to be independently computed, and there are hints that they should be normal numbers, i.e., that their base b digits are randomly distributed.

The paper study counter-dependent pseudorandom number gen- erators based on m-variate (m > 1) ergodic mappings of the space of 2-adic integers Z 2 . The sequence of internal states of these generators is defined by the recurrence law x i+1 = H B i (x i ) mod 2 n , whereas their output sequence is z i =

The paper is the continuation of the previous paper on this topic. We first review some of the recent developments on 3x+1 problem. In the remaining part of the paper we prove the result announced at the end of the first paper. The proof uses the finite version of the method of mathematical induction. Index Terms-3x+1 problem, Collatz problem, p-adic numbers, 3-adic numbers, periods.

This is an introduction to $p$-adic geometry and $p$-adic analysis focusing on the theme of $p$-adic period mappings. We follow as closely as possible the development of the classical theory of complex period mappings, blending differential equations, group theory and geometry. The text ends with a detailed discussion of $p$-adic triangle groups.

Abstract: A simplified model of tachyon matter in classical and quantum mechanics is constructed. p-Adic path integral quantization of the model is considered. Recent results in using p-adic analysis, as well as perspectives of an adelic generalization, in the ...

The Black Hole is where sound, matter and energy come together in a beautifully synchronous fashion. It should be noted that the root is not a principle and only responds to the other two. It is the cosmic gene pool from which form is made but only when energy and sound act upon it. It remains even when the other two have ceased. The root contains the code for differentiation and replication. The actual black hole is a vortex of inconceivable rapidity of sound which carries the three forms of electricity.

A p-adic Schrödinger-type operator D[alpha]+VY is studied. D[alpha] ([alpha]>0) is the operator of fractional differentiation and is a singular potential containing the Dirac delta functions [delta]x concentrated on a set of points Y={x1,...,xn} of the field of p-adic numbers . It is shown that such a problem is well posed for [alpha]>1/2 and the singular perturbation VY is form-bounded for [alpha]>1. In the latter case, the spectral analysis of [eta]-self-adjoint operator realizations of D[alpha]+VY in is carried out.

For a prime p and a matrix A ∈ Z
n×n
, write A as A = p(A quo p)+
(A rem p) where the remainder and quotient operations are applied
element-wise. Write the p-adic expansion of A as A = A[0] + pA[1] +
p
2A[2] + · · · where each A[i] ∈ Z
n×n has entries between [0, p − 1].
Upper bounds are proven for the Z-ranks of A rem p, and A quo p.
Also, upper bounds are proven for the Z/pZ-rank of A[i]
for all i ≥ 0
when p = 2, and a conjecture is presented for odd primes.