Ultrametric Research Articles

Ultrametric subsets with large Hausdorff dimension

It is shown that for every $\e\in (0,1)$, every compact metric space $(X,d)$ has a compact subset $S\subseteq X$ that embeds into an ultrametric space with distortion $O(1/\e)$, and $$\dim_H(S)\ge (1-\e)\dim_H(X),$$ where $\dim_H(\cdot)$ denotes Hausdorff dimension. The above $O(1/\e)$ distortion estimate is shown to be sharp via a construction based on sequences of expander graphs.

Tree-like structure of eternal inflation: A solvable model

In this paper we introduce a simple discrete stochastic model of eternal inflation that shares many of the most important features of the continuum theory as it is now understood. The model allows us to construct a multiverse and rigorously analyze its properties. Although simple and easy to solve, it has a rich mathematical structure underlying it. Despite the discreteness of the space-time the theory exhibits an unexpected non-perturbative analog of conformal symmetry that acts on the boundary of the geometry. The symmetry is rooted in the mathematical properties of trees, p-adic numbers, and ultrametric spaces; and in the physical property of detailed balance. We provide self-contained elementary explanations of the unfamiliar mathematical concepts, which have have also appeared in the study of the p-adic string. The symmetry acts on a huge collection of very low dimensional "multiverse fields" that are not associated with the usual perturbative degrees of freedom. They are connected with the late-time statistical distribution of bubble-universes in the multiverse. The conformal symmetry which acts on the multiverse fields is broken by the existence of terminal decays - to hats or crunches - but in a particularly simple way. We interpret this symmetry breaking as giving rise to an arrow of time. The model is used to calculate statistical correlations at late time and to discuss the measure problem. We show that the natural cutoff in the model is the analog of the so-called light-cone-time cutoff. Applying the model to the problem of the cosmological constant, we find agreement with earlier work.

The Ultra-Quasi-Metrically Injective Hull of a T 0-Ultra-Quasi-Metric Space

Jointly with E. Kemajou, in previous work we constructed the so-called Isbell-hull of a T0-quasi-metric space. In this article we continue these investigations by presenting a similar construction in the category of T0-ultra-quasi-metric spaces and contracting maps. Comparable studies in the area of ultra-metric spaces have been conducted before by Bayod and Martinez-Maurica.

Cavity approach to sphere packing in Hamming space

In this paper we study the hard sphere packing problem in the Hamming space by the cavity method. We show that both the replica symmetric and the replica symmetry breaking approximations give maximum rates of packing that are asymptotically the same as the lower bound of Gilbert and Varshamov. Consistently with known numerical results, the replica symmetric equations also suggest a crystalline solution, where for even diameters the spheres are more likely to be found in one of the subspaces (even or odd) of the Hamming space. These crystalline packings can be generated by a recursive algorithm which finds maximum packings in an ultrametric space. Finally, we design a message passing algorithm based on the cavity equations to find dense packings of hard spheres. Known maximum packings are reproduced efficiently in nontrivial ranges of dimensions and number of spheres.

Tree-valued resampling dynamics Martingale problems and applications

The measure-valued Fleming–Viot process is a diffusion which models the evolution of allele frequencies in a multi-type population. In the neutral setting the Kingman coalescent is known to generate the genealogies of the “individuals” in the population at a fixed time. The goal of the present paper is to replace this static point of view on the genealogies by an analysis of the evolution of genealogies. We encode the genealogy of the population as an (isometry class of an) ultra-metric space which is equipped with a probability measure. The space of ultra-metric measure spaces together with the Gromov-weak topology serves as state space for tree-valued processes. We use well-posed martingale problems to construct the tree-valued resampling dynamics of the evolving genealogies for both the finite population Moran model and the infinite population Fleming–Viot diffusion. We show that sufficient information about any ultra-metric measure space is contained in the distribution of the vector of subtree lengths obtained by sequentially sampled “individuals”. We give explicit formulas for the evolution of the Laplace transform of the distribution of finite subtrees under the tree-valued Fleming–Viot dynamics.

Möbius maps between ultrametric spaces are local similarities

We show that Mobius maps between ultrametric spaces are local similarities. The proof is based on the notions of chordal ultrametric and normalization, which we introduce in this paper.

Competitive clustering algorithms based on ultrametric properties

We propose in this paper two new competitive unsupervised clustering algorithms: the first algorithm deals with ultrametric data, it has a computational cost of O(n). The second algorithm has two strong features: it is fast and flexible on the processed data type as well as in terms of precision. The second algorithm has a computational cost, in the worst case, of O(n2), and in the average case, of O(n). These complexities are due to exploitation of ultrametric distance properties. In the first method, we use the order induced by an ultrametric in a given space to demonstrate how we can explore quickly data proximity. In the second method, we create an ultrametric space from a sample data, chosen uniformly at random, in order to obtain a global view of proximities in the data set according to the similarity criterion. Then, we use this proximity profile to cluster the global set. We present an example of our algorithms and compare their results with those of a classic clustering method.

Kaleidoscopical Configurations in $G$-Spaces

Let $G$ be a group and $X$ be a $G$-space with the action $G\times X\rightarrow X$, $(g,x)\mapsto gx$. A subset $F$ of $X$ is called a kaleidoscopical configuration if there exists a coloring $\chi:X\rightarrow C$ such that the restriction of $\chi$ on each subset $gF$, $g\in G$, is a bijection. We present a construction (called the splitting construction) of kaleidoscopical configurations in an arbitrary $G$-space, reduce the problem of characterization of kaleidoscopical configurations in a finite Abelian group $G$ to a factorization of $G$ into two subsets, and describe all kaleidoscopical configurations in isometrically homogeneous ultrametric spaces with finite distance scale. Also we construct $2^{\mathfrak c}$ (unsplittable) kaleidoscopical configurations of cardinality $\mathfrak c$ in the Euclidean space $\mathbb{R}^n$.

Trees, Ultrametrics, and Noncommutative Geometry

Noncommutative geometry is used to study the local geometry of ultrametric spaces and the geometry of trees at infinity. Connes’s example of the noncommutative space of Penrose tilings is interpreted as a non-Hausdorff orbit space of a compact, ultrametric space under the action of its local isometry group. This is generalized to compact, locally rigid, ultrametric spaces. The local isometry types and the local similarity types in those spaces can be analyzed using groupoid C∗-algebras. The concept of a locally rigid action of a countable group Γ on a compact, ultrametric space by local similarities is introduced. It is proved that there is a faithful unitary representation of Γ into the germ groupoid C∗-algebra of the action. The prototypical example is the standard action of Thompson’s group V on the ultrametric Cantor set. In this case, the C∗-algebra is the Cuntz algebra O2 and representations originally due to Birget and Nekrashevych are recovered. End spaces of trees are sources of ultrametric spaces. Some connections are made between locally rigid, ultrametric spaces and a concept in the theory of tree lattices of Bass and Lubotzky.

Fixed Point and Common Fixed Point Theorems for Cyclic Quasi-Contractions in Metric and Ultrametric Spaces

In this paper, we prove introduce some fixed point theorems for quasi-contraction under the cyclical conditions. Then, we point out that a common fixed point extension is also applicable via our earlier results equipped together with a weaker cyclical properties, namely a co-cyclic representation. Examples are as well provided along this paper.

Majorizing Measures and Ultrametric Spaces

Talagrand's proof of the sufficiency of existence of a majorizing measure for the sample boundedness of processes with bounded increments used a contraction from a certain ultrametric space. We give a short proof of existence of such an ultrametric using

Diameter and diametrical pairs of points in ultrametric spaces

Let F(X) be the set of finite nonempty subsets of a set X. We have found the necessary and sufficient conditions under which for a given function τ: F(X) → ℝ there is an ultrametric on X such that τ(A) = diamA for every A ∈ F(X). For finite nondegenerate ultrametric spaces (X, d) it is shown that X together with the subset of diametrical pairs of points of X forms a complete k-partite graph, k ⩾ 2, and, conversely, every finite complete k-partite graph with k ⩾ 2 can be obtained by this way. We use this result to characterize the finite ultrametric spaces (X, d) having the minimal card{(x, y): d(x, y) = diamX, x, y ∈ X} for given card X.

Hierarchical condensation near phase equilibrium

A novel mechanism of new phase formation is studied both experimentally and theoretically in the example of quasi-equilibrium stationary condensation in an ion–plasma sputterer. Copper condensates are obtained to demonstrate that a specific network structure is formed as a result of self-assembly in the course of deposition. The fractal pattern related is inherent in the phenomena of diffusion limited aggregation. Condensate nuclei are shown to form statistical ensemble of hierarchically subordinated objects distributed in ultrametric space. The Langevin equation and the Fokker–Planck equation related are found to describe stationary distribution of thermodynamic potential variations at condensation. Time dependence of the formation probability of branching structures is found to clarify the experimental situation.

Directed Binary Hierarchies and Directed Ultrametrics

Directed binary hierarchies have been introduced in order to give a graphical reduced representation of a family of association rules. This type of structure extends the classical binary hierarchical classification in a very specific way. In this paper an accurate formalization of this new structure is studied. A directed hierarchy is defined as a set of ordered pairs of subsets of the initial individual set satisfying specific conditions. A new notion of directed ultrametricity is studied. The main result consists in establishing a bijective correspondence between a directed ultrametric space and a directed binary hierarchy. Finally, an algorithm is proposed in order to transform a directed ultrametric structure into a graphical representation associated with a directed binary hierarchy.

Methods and applications of ultrametric and p-adic analysis: From wavelet theory to biophysics

This paper is devoted to new methods of p-adic and ultrametric analysis and their applications in physics and molecular biology. Although the basic definitions are given in the text, it is recommended that the reader be familiar with the material of the first two chapters in Vladimirov, Volovich and Zelenov’s book p-Adic Analysis andMathematical Physics [1]. Our purpose in this paper is to present some new methods and applications of p-adic and ultrametric analysis developed from 19941 till 2007. The overlap of material between this paper and [1] is minimized. In particular, we do not discuss p-adic quantum mechanics and string theory. This paper covers several topics of p-adic and ultrametric analysis and applications. Most of the results given here are new and have been presented earlier only in journal articles. The first topic is purely mathematical; it is related to the development of the theory of ultrametric wavelets and the study of it relationship to the spectral analysis of ultrametric pseudodifferential operators. The role of ultrametric pseudodifferential operators (such as the Vladimirov operator of p-adic fractional differentiation) in ultrametric analysis is similar to that played by the differentiation operator in real analysis. The use of wavelet bases makes it possible to calculate spectra of ultrametric pseudodifferential operators. In general, in the ultrametric case, wavelet theory is substantially simpler and more powerful and convenient than in the real case. We consider here both the p-adic case and the case of general locally compact ultrametric spaces. An important place in the paper is occupied by a discussion of applications of ultrametric analysis. We discuss applications to the theory of spin glasses, dynamics of macromolecules, and genetics. A common feature of the applications of ultrametric analysis considered here is that they describe collective effects in complex systems. The main destination of physics is describing phenomena based on first principles (physical laws). In many cases, such a description involves difficulties. Thus, it is important to develop various approximations relying on patterns of the effective behavior of systems under consideration in various regimes. A well-known example is the passage from a microscopic description of a many-particle system to a kinetic and, then, a hydrodynamic description [2]. Another example is the stochastic limit of quantum theory, in which the dynamics of a quantum system interacting with the environment is approximated by some quantum random process [3]. In this paper, we consider three examples of application to complex systems, where an important role in the description of a system is played by the ultrametric specifics. In general, complexity, which is often discussed in applications to biological systems (as well as in different contexts), does not lead to farreaching conclusions by itself. The examples considered in what follows show that, as supposed in [1], ultrametric analysis makes it possible to give a simple description of some complex systems, for which methods of real analysis are essentially insufficient. Certainly, we do not claim that ultrametric methods

Nested Hoare Triples and Frame Rules for Higher-order Store

Separation logic is a Hoare-style logic for reasoning about programs with heap-allocated mutable data structures. As a step toward extending separation logic to high-level languages with ML-style general (higher-order) storage, we investigate the compatibility of nested Hoare triples with several variations of higher-order frame rules. The interaction of nested triples and frame rules can be subtle, and the inclusion of certain frame rules is in fact unsound. A particular combination of rules can be shown consistent by means of a Kripke model where worlds live in a recursively defined ultrametric space. The resulting logic allows us to elegantly prove programs involving stored code. In particular, using recursively defined assertions, it leads to natural specifications and proofs of invariants required for dealing with recursion through the store.

A semantic model for graphical user interfaces

We give a denotational model for graphical user interface (GUI) programming using the Cartesian closed category of ultrametric spaces. The ultrametric structure enforces causality restrictions on reactive systems and allows well-founded recursive definitions by a generalization of guardedness. We capture the arbitrariness of user input (e.g., a user gets to decide the stream of clicks she sends to a program) by making use of the fact that the closed subsets of an ultrametric space themselves form an ultrametric space, allowing us to interpret nondeterminism with a "powerspace" monad. Algebras for the powerspace monad yield a model of intuitionistic linear logic, which we exploit in the definition of a mixed linear/non-linear domain-specific language for writing GUI programs. The non-linear part of the language is used for writing reactive stream-processing functions whilst the linear sublanguage naturally captures the generativity and usage constraints on the various linear objects in GUIs, such as the elements of a DOM or scene graph. We have implemented this DSL as an extension to OCaml, and give examples demonstrating that programs in this style can be short and readable.

On a Class of Markov Semigroups on Discrete Ultra-Metric Spaces

We consider a discrete ultra-metric space $\left( X,d\right) $ with a measure m and construct in a natural way a symmetric Markov semigroup $ \left\{ P_{t}\right\} _{t\geq 0}$ in $L^{2}\left( X,m\right) $ and the corresponding Markov process $\left\{ \mathcal{X}_{t}\right\} $ . We prove upper and lower bounds of its transition function and its Green function, give a criterion for the transience, and estimate its moments.

Characterizing the Cantor bi-cube in asymptotic categories

We present characterizations of metric spaces that are micro-, macro- or bi-uniformly equivalent to the extended Cantor set EC = \{\sum_{i=-n}^\infty\frac{2x_i}{3^i} \mid n\in\mathbb{N} ,\ (x_i)_{i\in\mathbb{Z}}\in\{0,1\}^\mathbb{Z}\} \subset\mathbb{R} , which is bi-uniformly equivalent to the Cantor bi-cube 2^{<\mathbb{Z}}=\{(x_i)_{i\in\mathbb{Z}}\in \{0,1\}^\mathbb{Z}\mid there exists n such that x_i=0 for all i\ge n\} endowed with the metric d((x_i),(y_i))= max _{i\in\mathbb{Z}}2^i|x_i-y_i| . The characterizations imply that any two (uncountable) proper isometrically homogeneous ultrametric spaces are coarsely (and bi-uniformly) equivalent. This implies that any two countable locally finite groups endowed with proper left-invariant metrics are coarsely equivalent. For the proof of these results we develop a technique of towers which may be of independent interest.

A metric between quasi-isometric trees

It is known that PQ-symmetric maps on the boundary characterize the quasi-isometry type of visual hyperbolic spaces, in particular, of geodesically complete R \mathbb {R} -trees. We define a map on pairs of PQ-symmetric ultrametric spaces which characterizes the branching of the space. We also show that when the ultrametric spaces are the corresponding end spaces, this map defines a metric between rooted geodesically complete simplicial trees with minimal vertex degree 3 in the same quasi-isometry class. Moreover, this metric measures how far the trees are from being rooted isometric.