74 research outputs found
The structures of Hausdorff metric in non-Archimedean spaces
For non-Archimedean spaces and let and be the
ballean of (the family of the balls in ), the space of mappings from
to and the space of mappings from the ballen of to
respectively. By studying explicitly the Hausdorff metric structures related to
these spaces, we construct several families of new metric structures (e.g., ) on the corresponding spaces, and study their convergence,
structural relation, law of variation in the variable including
some normed algebra structure. To some extent, the class is a counterpart of the usual Levy-Prohorov metric in the
probability measure spaces, but it behaves very differently, and is interesting
in itself. Moreover, when is compact and is a complete
non-Archimedean field, we construct and study a Dudly type metric of the space
of valued measures on Comment: 43 pages; this is the final version. Thanks to the anonymous
referee's helpful comments, the original Theorem 2.10 is removed, Proposition
2.10 is stated now in a stronger form, the abstact is rewritten, the
Monna-Springer is used in Section 5, and Theorem 5.2 is written in a more
general for
p-Adic and Adelic Harmonic Oscillator with Time-Dependent Frequency
The classical and quantum formalism for a p-adic and adelic harmonic
oscillator with time-dependent frequency is developed, and general formulae for
main theoretical quantities are obtained. In particular, the p-adic propagator
is calculated, and the existence of a simple vacuum state as well as adelic
quantum dynamics is shown. Space discreteness and p-adic quantum-mechanical
phase are noted.Comment: 10 page
Phase transitions for -adic Potts model on the Cayley tree of order three
In the present paper, we study a phase transition problem for the -state
-adic Potts model over the Cayley tree of order three. We consider a more
general notion of -adic Gibbs measure which depends on parameter
\rho\in\bq_p. Such a measure is called {\it generalized -adic quasi Gibbs
measure}. When equals to -adic exponent, then it coincides with the
-adic Gibbs measure. When , then it coincides with -adic quasi
Gibbs measure. Therefore, we investigate two regimes with respect to the value
of . Namely, in the first regime, one takes for some
J\in\bq_p, in the second one . In each regime, we first find
conditions for the existence of generalized -adic quasi Gibbs measures.
Furthermore, in the first regime, we establish the existence of the phase
transition under some conditions. In the second regime, when we prove the existence of a quasi phase transition. It turns out that
if and \sqrt{-3}\in\bq_p, then one finds the existence
of the strong phase transition.Comment: 27 page
Ergodicity criteria for non-expanding transformations of 2-adic spheres
In the paper, we obtain necessary and sufficient conditions for ergodicity
(with respect to the normalized Haar measure) of discrete dynamical systems
on 2-adic spheres of radius
, , centered at some point from the ultrametric space of
2-adic integers . The map is
assumed to be non-expanding and measure-preserving; that is, satisfies a
Lipschitz condition with a constant 1 with respect to the 2-adic metric, and
preserves a natural probability measure on , the Haar measure
on which is normalized so that
Some aspects of the -adic analysis and its applications to -adic stochastic processes
In this paper we consider a generalization of analysis on -adic numbers
field to the case of -adic numbers ring. The basic statements, theorems
and formulas of -adic analysis can be used for the case of -adic analysis
without changing. We discuss basic properties of -adic numbers and consider
some properties of -adic integration and -adic Fourier analysis. The
class of infinitely divisible -adic distributions and the class of -adic
stochastic Levi processes were introduced. The special class of -adic CTRW
process and fractional-time -adic random walk as the diffusive limit of it
is considered. We found the asymptotic behavior of the probability measure of
initial distribution support for fractional-time -adic random walk.Comment: 18 page
T-functions revisited: New criteria for bijectivity/transitivity
The paper presents new criteria for bijectivity/transitivity of T-functions
and 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 -adic functions and to study
measure-preservation/ergodicity of these
p-Adic Mathematical Physics
A brief review of some selected topics in p-adic mathematical physics is
presented.Comment: 36 page
Linearization in ultrametric dynamics in fields of characteristic zero - equal characteristic case
Let be a complete ultrametric field of charactersitic zero whose
corresponding residue field is also of charactersitic zero. We give
lower and upper bounds for the size of linearization disks for power series
over near an indifferent fixed point. These estimates are maximal in the
sense that there exist exemples where these estimates give the exact size of
the corresponding linearization disc. Similar estimates in the remaning cases,
i.e. the cases in which is either a -adic field or a field of prime
characteristic, were obtained in various papers on the -adic case
(Ben-Menahem:1988,Thiran/EtAL:1989,Pettigrew/Roberts/Vivaldi:2001,Khrennikov:2001)
later generalized in (Lindahl:2009 arXiv:0910.3312), and in (Lindahl:2004
http://iopscience.iop.org/0951-7715/17/3/001/,Lindahl:2010Contemp. Math)
concerning the prime characteristic case
On hyperbolic fixed points in ultrametric dynamics
Let K be a complete ultrametric field. We give lower and upper bounds for the
size of linearization discs for power series over K near hyperbolic fixed
points. These estimates are maximal in the sense that there exist examples
where these estimates give the exact size of the corresponding linearization
disc. In particular, at repelling fixed points, the linearization disc is equal
to the maximal disc on which the power series is injective.Comment: http://www.springerlink.com/content/?k=doi%3a%2810.1134%2fS2070046610030052%2
Sparse p-Adic Data Coding for Computationally Efficient and Effective Big Data Analytics
We develop the theory and practical implementation of p-adic sparse coding of data. Rather than the standard, sparsifying criterion that uses the pseudo-norm, we use the p-adic
norm.We require that the hierarchy or tree be node-ranked, as is standard practice in agglomerative and other hierarchical clustering, but not necessarily with decision trees. In order to structure the data, all computational processing operations are direct reading of the data, or are bounded by a constant number of direct readings of the data, implying linear computational time. Through p-adic sparse data coding, efficient storage results, and for bounded p-adic norm stored data, search and retrieval are constant time operations. Examples show the effectiveness of this new approach to content-driven encoding and displaying of data
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