Search results for "Ultrametric space"
showing 10 items of 16 documents
Existence of doubling measures via generalised nested cubes
2012
Working on doubling metric spaces, we construct generalised dyadic cubes adapting ultrametric structure. If the space is complete, then the existence of such cubes and the mass distribution principle lead into a simple proof for the existence of doubling measures. As an application, we show that for each $\epsilon>0$ there is a doubling measure having full measure on a set of packing dimension at most $\epsilon$.
Diffusion processes with ultrametric jumps
2007
Abstract In the theory of spin glasses the relaxation processes are modelled by random jumps in ultrametric spaces. One may argue that at the border of glassy and nonglassy phases the processes combining diffusion and jumps may be relevant. Using the Dirichlet form technique we construct a model of diffusion on the real line with jumps on the Cantor set. The jumps preserve the ultrametric feature of a random process on unit ball of 2-adic numbers.
Generators of Random Processes in Ultrametric Spaces and Their Spectra
2009
The L 2(\( \mathbb{S} \)) space of square integrable functions on an ultrametric space \( \mathbb{S} \) has rather specific structure. As a consequence in a natural way there appear in L 2(\( \mathbb{S} \)) the operators of which unitary counterparts in L 2(ℝn) would be difficult to construct. Such class of self-adjoint operators emerge from theory of random processes on ultrametric spaces. In this paper we collect known material on spectral properties of the generators of random processes on \( \mathbb{S}_B \) an ultrametric space of sequences. (The set of p-adic numbers is a subset of \( \mathbb{S}_B \).) Then we discuss structure of the eigenspaces of the generators.
Information and hierarchical structure in financial markets
1999
I investigate the information content present in the time series of stock prices of a portfolio of stocks traded in a financial market. By investigating the correlation coefficient between pairs of stocks I provide a working definition of a generalized distance between the stocks of the portfolio. This generalized distance is used to obtain an ultrametric distance matrix between the stocks. The ultrametric structure of the portfolio investigated has associated a taxonomy which is meaningful from an economic point of view.
Capabilities of Ultrametric Automata with One, Two, and Three States
2016
Ultrametric automata use p-adic numbers to describe the random branching of the process of computation. Previous research has shown that ultrametric automata can have a significant decrease in computing complexity. In this paper we consider the languages that can be recognized by one-way ultrametric automata with one, two, and three states. We also show an example of a promise problem that can be solved by ultrametric integral automaton with three states.
On the Hierarchy Classes of Finite Ultrametric Automata
2015
This paper explores the language classes that arise with respect to the head count of a finite ultrametric automaton. First we prove that in the one-way setting there is a language that can be recognized by a one-head ultrametric finite automaton and cannot be recognized by any k-head non-deterministic finite automaton. Then we prove that in the two-way setting the class of languages recognized by ultrametric finite k-head automata is a proper subclass of the class of languages recognized by (k + 1)-head automata. Ultrametric finite automata are similar to probabilistic and quantum automata and have only just recently been introduced by Freivalds. We introduce ultrametric Turing machines an…
On a Conjecture by Christian Choffrut
2017
It is one of the most famous open problems to determine the minimum amount of states required by a deterministic finite automaton to distinguish a pair of strings, which was stated by Christian Choffrut more than thirty years ago. We investigate the same question for different automata models and we obtain new upper and lower bounds for some of them including alternating, ultrametric, quantum, and affine finite automata.
Counting with Probabilistic and Ultrametric Finite Automata
2014
We investigate the state complexity of probabilistic and ultrametric finite automata for the problem of counting, i.e. recognizing the one-word unary language \(C_n=\left\{ 1^n \right\} \). We also review the known results for other types of automata.
Fixed point theory for cyclic weak ϕ-contraction in fuzzy metric spaces
2012
In this paper, we introduce cyclic weak $\phi-$contractions in fuzzy metric spaces and utilize the same to prove some results on existence and uniqueness of fixed point in fuzzy metric spaces. Some related results are also proved besides furnishing illustrative examples.
SPACES OF SMALL METRIC COTYPE
2010
Naor and Mendel's metric cotype extends the notion of the Rademacher cotype of a Banach space to all metric spaces. Every Banach space has metric cotype at least 2. We show that any metric space that is bi-Lipschitz equivalent to an ultrametric space has infinimal metric cotype 1. We discuss the invariance of metric cotype inequalities under snowflaking mappings and Gromov-Hausdorff limits, and use these facts to establish a partial converse of the main result.