Adic metric space pdf

Pdf segmentation of images in padic and euclidean metrics. If p is prime then the padic numbers form a complete metric space containing the rational numbers it is a completion of the rational numbers and it is also a field. Note that because of the division problem, when p is not prime, the padic numbers are not a field, but only a ring. Pictures of ultrametric spaces, the padic numbers, and. We propose a mathematical model of the human memoryretrieval process based on dynamic systems over a metric space ofpadic numbers. A metric space is a set with a distance function or metric, dp, q defined over the. Ultrametricity an ultrametric space is a space endowed with an ultrametric distance, defined as a distance satisfying the inequality da,c. Pdf a stream processing approach to distance measurement of. The completion theorem 6 every metric space m, and in our context elds f, can be completed, i. The p adic topology on z is the metric topology with the p adic metric d. If v,k k is a normed vector space, then the condition du,v ku. With the p adic topology, z is an ultrametric space. If a subset of a metric space is not closed, this subset can not be sequentially compact. When the space is z, q,orr, we usually form such a picture by imagining points on the number line.

This metric space is complete in the sense that every cauchy sequence converges to a point in q p. This makes us wonder if the ring of padic numbers with their padic soft metric space is a s. This metric on mg,r is called the weilpetersson metric. The p adic integers form a subset of the set of all p adic numbers. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. Igusas theorem on the rationality of the zeta function 15 5. One norm that we are quite familiar with is the absolute value, jj. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are. In contrast, the padic extension arises from the use of the counterintuitive padic metric.

Proof we prove sequential compactness since we are in metric space. When studying a metric space, it is valuable to have a mental picture that displays distance accurately. X a, there is a sequence x n in a which converges to x. In this work we gave some definitions and properties about both metric and ultra metric norms. Pictures of ultrametric spaces, the p adic numbers, and valued fields jan e. The padic integers form a subset of the set of all padic numbers. Arbitrary intersectons of open sets need not be open. In a complete metric space, a sequence is convergent if and only if it is a cauchy. Metricandtopologicalspaces university of cambridge. First, however, we must develop language that we can use in constructing and describing the p adic metric. A stream processing approach to distance measurement of integers in padic metric space. Maxda,b,db,c 1 a, b and c are points of this ultrametric space, instead of the usual triangular inequality, characteristic of euclidean geometry.

Before delving into the connection between the collatz conjecture and p adic numbers, we must rigorously introduce the ring of p adic integers. A metric space consists of a set xtogether with a function d. This space mfis unique up to isometries, that is, if mf 0 is a complete metric space having mas a dense subspace, then mf 0 is isometric to mf. A subset is called net if a metric space is called totally bounded if finite net. We assume that two ideas are close if they have a sufficiently long initial segment in common. An ultrametric or nonarchimedean metric on a set x is function d. The analogues of open intervals in general metric spaces are the following. A rather trivial example of a metric on any set x is the discrete metric dx,y 0 if x. With the padic topology, z is an ultrametric space. On the other hand, this metric topology also is the locally convex topology given by the family of lattices. First, however, we must develop language that we can use in constructing and describing the padic metric. Each of the following is an example of a closed set. I highly recommend ne chapter ii for a detailed discussion of this topic.

Itisknownthat the canonical coordinates associated to the weilpetersson metric coincide with the socalled bers coordinates on m g,r the universal covering space of mg,r. Now, since the rst digit is among 0p 1, we can extract a subsequence fb igof fa igall sharing the same rst digit. Thanks for contributing an answer to mathematics stack exchange. Concretely, a p p adic integer x x may be written as a base p p expansion. The completion of q with respect to the padic metric is denoted by qp and is called the field of.

Most crucial property of this norm is that it satisfies the ultra metric triangle inequality. This metrics allowed for an implicit use of human visual. Weils measure and the relationship with rational points over. This was worked out by berthelot ber91, but is rather subtle.

V w between banach spaces is continuous for the norm topologies if and only if it is bounded i. Further, a metric space is compact if and only if each realvalued continuous function on it is bounded and attains its least and greatest values. Moreover, each o in t is called a neighborhood for each of their points. Let a be a dense subset of x and let f be a uniformly continuous from a into y. A metric space x, d is a topological space, with the topology being induced from the metric d.

Whereas in the adic world, there is a formal unit disc bered over a twopoint space spaz p, and its generic ber is simply the open. In contrast, the p adic extension arises from the use of the counterintuitive p adic metric. But avoid asking for help, clarification, or responding to other answers. Each compact metric space is complete, but the converse is false. Introduction to padic numbers an overview of ultrametric spaces and padic numbers. D such that, 1 m is complete with respect to the metric d. A crucial difference is that the reals form an archimedean field, while the p padic numbers form a nonarchimedean field. Eichlinghofen, the 28th august 2015 by gilles bellot tudortmunduniversity facultyofmathematics. In a complete metric space, a sequence is convergent if and only if it is a cauchy sequence. A metric space is a set xtogether with a metric don it, and we will use the notation x. Human memory as a p adic dynamic system springerlink. U nofthem, the cartesian product of u with itself n times.

Integers that are congruent modulo a high power of p have a di erence with a large p adic valuation, and hence are assigned a. Often, if the metric dis clear from context, we will simply denote the metric space x. Any normed vector space can be made into a metric space in a natural way. We also assume that this dynamic system is located in the subconscious and is. In calculus on r, a fundamental role is played by those subsets of r which are intervals. Pictures of ultrametric spaces, the padic numbers, and valued fields jan e. Paper 2, section i 4e metric and topological spaces. Then we call k k a norm and say that v,k k is a normed vector space. As for the limitpoints, having had a quick look at the definition that uniqueness.

Metric spaces joseph muscat2003 last revised may 2009 a revised and expanded version of these notes are now published by springer. Integers that are congruent modulo a high power of p have a di erence with a large padic valuation, and hence are assigned a. Here is a more mathematical way of saying all this. Each has a unique padic expansion with no negative powers of p. When we discuss probability theory of random processes, the underlying sample spaces and eld structures become quite complex. Y,d y is a metric space and open subsets of y are just the intersections with y of open subsets of x. It is well known that q equipped with the metric induced by the euclidean norm. Defn a subset c of a metric space x is called closed if its complement is open in x. Recall that a topological space is complete if every cauchy. The padic numbers are most simply a field extension of q, the rational. The latter in turn constitute an extension of the eld of rational numbers, analogous to the completion of the rationals by the real numbers with respect to the standard ordinary metric. The algorithm produces partitions of the p adic metric space having a very simple geometry. Metric spaces are generalizations of the real line, in which some of the theorems that hold for r. The goal of these notes is to construct a complete metric space.