P-Adic Numbers Ultrametric Analysis and Applications最新文献

Abstract

In 1970, W. M. Schmidt [6] generalized Roth’s well-known theorem on rational approximation to a single algebraic irrational, to include simultaneous rational approximation for a given (n) algebraic irrationals. As no analogue of Roth’s theorem for algebraic irrational power series over a finite field exists, we will show that there is no analogue of Schmidt’s theorem for such (n) elements.

Abstract

In the present paper, we consider a (p)-adic Ising model on a Cayley tree. For this model, (p)-adic analogue of the notion of weakly periodic Gibbs measures is introduced. For some normal subgroup of the group representation of the Cayley tree, the existence of such Gibbs measures is proved. We also study fixed points and their behaviour of the mapping which coincides with weakly periodic quantities of the functional equation. Moreover, the boundedness of such kinds of measures is established, which yields the occurrence of a phase transition.

Abstract

Let (p) be a prime. For (din mathbb{N}), let (mathbb{Q}_p^d) be the standard (d)-dimensional p-adic Hilbert space. Let (m in mathbb{N}) and (text{Sym}^m(mathbb{Q}_p^d)) be the (p)-adic Hilbert space of symmetric m-tensors. We prove the following result. Let ({tau_j}_{j=1}^n) be a collection in (mathbb{Q}_p^d) satisfying (i) (langle tau_j, tau_jrangle =1) for all (1leq j leq n) and (ii) there exists (b in mathbb{Q}_p) satisfying (sum_{j=1}^{n}langle x, tau_jrangle tau_j =bx) for all ( x in mathbb{Q}^d_p.) Then

We call Inequality (0.1) as the (p)-adic version of Welch bounds obtained by Welch [IEEE Transactions on Information Theory, 1974]. Inequality (0.1) differs from the non-Archimedean Welch bound obtained recently by M. Krishna as one can not derive one from another. We formulate (p)-adic Zauner conjecture.

Abstract

In this article we apply a polyadic approach to obtain very explicit description of a novel kind of wavelets on the ring of finite adèles, (mathbb{A}_{f}), which are also eigenfunctions of a Vladimirov-type pseudodifferential operator on (L^2(mathbb{A}_{f})). As an accompaniment, we solve the Cauchy problem for a certain pseudodifferential equation.

Abstract

Let (K) be a finite extension of (mathbb{Q}_p) and let (G) be a (p)-adic Lie group. In this paper, we define the Iwasawa algebra (K[[G]]) and prove that it is isomorphic to the convolution algebra of compactly supported distributions on (G). This has important applications in the theory of admissible representations of (G) on (p)-adic Banach spaces. In particular, we prove the Frobenius reciprocity for continuous principal series representations.

Abstract

In this paper, we establish some sufficient conditions for the boundedness of rough Hardy-Littlewood operators on the (p)-adic local central Morrey, (p)-adic Morrey-Herz, and (p)-adic local block spaces with variable exponents.

Abstract

We estimate the degree of approximation by linear means of Vallée-Poussin type of Vilenkin-Fourier series in classical Lebesgue spaces and in a space of generalized continuous functions. These results generalize ones obtained by I. Blahota and G.Gat for means of Walsh-Fourier series.

Abstract

The approach to the theory of a relativistic random process is considered by the path integral method as Brownian motion taking into account the boundedness of speed. An attempt was made to build a relativistic analogue of the Wiener measure as a weak limit of finite-difference approximations. A formula has been proposed for calculating the probability particle transition during relativistic Brownian motion. Calculations were carried out by three different methods with identical results. Along the way, exact and asymptotic formulas for the volume of some parts and sections of an N-1-dimensional unit cube were obtained. They can have independent value.

Abstract

Let (q) be an odd prime, and let (T_{q}:mathbb{Z}rightarrowmathbb{Z}) be the Shortened (qx+1) map, defined by (T_{q}left(nright)=n/2) if (n) is even and (T_{q}left(nright)=left(qn+1right)/2) if (n) is odd. The study of the dynamics of these maps is infamous for its difficulty, with the characterization of the dynamics of (T_{3}) being an alternative formulation of the famous Collatz Conjecture. This series of papers presents a new paradigm for studying such arithmetic dynamical systems by way of a neglected area of ultrametric analysis which we have termed (left(p,qright))-adic analysis, the study of functions from the (p)-adics to the (q)-adics, where (p) and (q) are distinct primes. In this, the first paper, working with the (T_{q}) maps as a toy model for the more general theory, for each odd prime (q), we construct a function (chi_{q}:mathbb{Z}_{2}rightarrowmathbb{Z}_{q}) (the Numen of (T_{q})) and prove the Correspondence Principle (CP): (xinmathbb{Z}backslashleft{ 0right} ) is a periodic point of (T_{q}) if and only there is a (mathfrak{z}inmathbb{Z}_{2}backslashleft{ 0,1,2,ldotsright} ) so that (chi_{q}left(mathfrak{z}right)=x). Additionally, if (mathfrak{z}inmathbb{Z}_{2}backslashmathbb{Q}) makes (chi_{q}left(mathfrak{z}right)inmathbb{Z}), then the iterates of (chi_{q}left(mathfrak{z}right)) under (T_{q}) tend to (+infty) or (-infty).

Abstract

In this paper, the group structure of the (p)-adic ball and sphere are studied. The dynamical system of isometry defined on invariant sphere is investigated. We define the binary operations (oplus) and (odot) on a ball and sphere, respectively, and prove that these sets are compact topological abelian group with respect to the operations. Then we show that any two balls (spheres) with positive radius are isomorphic as groups. We prove that the Haar measure introduced in (mathbb Z_p) is also a Haar measure on an arbitrary balls and spheres. We study the dynamical system generated by the isometry defined on a sphere and show that the trajectory of any initial point that is not a fixed point is not convergent. We study ergodicity of this (p)-adic dynamical system with respect to normalized Haar measure reduced on the sphere. For (pgeq 3) we prove that the dynamical systems are not ergodic. But for (p=2) under some conditions the dynamical system may be ergodic.