Adic metric space pdf

The completion theorem describes the way in which a metric space can be completed and what completion means. It is well known that q equipped with the metric induced by the euclidean norm. When studying a metric space, it is valuable to have a mental picture that displays distance accurately. 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. A metric space x, d is a topological space, with the topology being induced from the metric d. Introduction to padic numbers an overview of ultrametric spaces and padic numbers. First, however, we must develop language that we can use in constructing and describing the p adic metric. Pdf segmentation of images in padic and euclidean metrics. A metric space is a set with a distance function or metric, dp, q defined over the. We assume that two ideas are close if they have a sufficiently long initial segment in common.

V w between banach spaces is continuous for the norm topologies if and only if it is bounded i. Human memory as a p adic dynamic system springerlink. 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 iff if then so thus on the other hand, let. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. In calculus on r, a fundamental role is played by those subsets of r which are intervals. The p adic topology on z is the metric topology with the p adic metric d. 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. An ultrametric or nonarchimedean metric on a set x is function d. We propose a mathematical model of the human memoryretrieval process based on dynamic systems over a metric space ofpadic numbers. This makes us wonder if the ring of padic numbers with their padic soft metric space is a s. Each of the following is an example of a closed set. Each has a unique padic expansion with no negative powers of p.

This metric space is complete in the sense that every cauchy sequence converges to a point in q p. First, however, we must develop language that we can use in constructing and describing the padic metric. If a subset of a metric space is not closed, this subset can not be sequentially compact. Proof we prove sequential compactness since we are in metric space. In jerzy browkins 1978 and 2000 papers on padic continued fractions, several algorithms for computing continued frac. Pictures of ultrametric spaces, the p adic numbers, and valued fields jan e. In contrast, the p adic extension arises from the use of the counterintuitive p adic metric. Introduction to p adic numbers an overview of ultrametric spaces and p adic numbers. Pictures of ultrametric spaces, the padic numbers, and valued fields jan e. The completion of q with respect to the padic metric is denoted by qp and is called the field of. I highly recommend ne chapter ii for a detailed discussion of this topic.

Before delving into the connection between the collatz conjecture and p adic numbers, we must rigorously introduce the ring of p adic integers. Paper 2, section i 4e metric and topological spaces. Any normed vector space can be made into a metric space in a natural way. Most crucial property of this norm is that it satisfies the ultra metric triangle inequality. Recall that a topological space is complete if every cauchy. When we discuss probability theory of random processes, the underlying sample spaces and eld structures become quite complex. 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. A metric space is a set xtogether with a metric don it, and we will use the notation x. If v is a normed space which is complete under its norm or rather, under the induced ultrametric dv,w kv. The padic completion of q and hensels lemma contents. Here is a more mathematical way of saying all this. A subset is called net if a metric space is called totally bounded if finite net. Each compact metric space is complete, but the converse is false. The padic numbers are most simply a field extension of q, the rational.

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. A stream processing approach to distance measurement of integers in padic metric space. Weils measure and the relationship with rational points over. 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. With the padic topology, z is an ultrametric space. D such that, 1 m is complete with respect to the metric d. Now, since the rst digit is among 0p 1, we can extract a subsequence fb igof fa igall sharing the same rst digit. On the other hand, this metric topology also is the locally convex topology given by the family of lattices. The algorithm produces partitions of the p adic metric space having a very simple geometry. Metricandtopologicalspaces university of cambridge. Moreover, each o in t is called a neighborhood for each of their points.

Metric spaces joseph muscat2003 last revised may 2009 a revised and expanded version of these notes are now published by springer. This is what allows the development of calculus on q p, and it is the interaction of this analytic and algebraic structure that gives the p adic number systems their power and utility. 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. The goal of these notes is to construct a complete metric space. Igusas theorem on the rationality of the zeta function 15 5. The padic integers form a subset of the set of all padic numbers. A metric space consists of a set xtogether with a function d. Metric spaces are generalizations of the real line, in which some of the theorems that hold for r. With the p adic topology, z is an ultrametric space. The completion theorem 6 every metric space m, and in our context elds f, can be completed, i. 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. The p adic integers form a subset of the set of all p adic numbers.

This is a null sequence since for big enough n, all x n are zero. The work of this paper will be further simplified by assuming that the process y. This metric on mg,r is called the weilpetersson metric. Integers that are congruent modulo a high power of p have a di erence with a large padic valuation, and hence are assigned a. Y,d y is a metric space and open subsets of y are just the intersections with y of open subsets of x. In contrast, the padic extension arises from the use of the counterintuitive padic metric. Defn a subset c of a metric space x is called closed if its complement is open in x. When the space is z, q,orr, we usually form such a picture by imagining points on the number line. This was worked out by berthelot ber91, but is rather subtle.

Often, if the metric dis clear from context, we will simply denote the metric space x. A crucial difference is that the reals form an archimedean field, while the p padic numbers form a nonarchimedean field. Then we call k k a norm and say that v,k k is a normed vector space. If v,k k is a normed vector space, then the condition du,v ku. U nofthem, the cartesian product of u with itself n times. 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. But avoid asking for help, clarification, or responding to other answers.

This metrics allowed for an implicit use of human visual. 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. We also assume that this dynamic system is located in the subconscious and is. In a complete metric space, a sequence is convergent if and only if it is a cauchy sequence. Let a be a dense subset of x and let f be a uniformly continuous from a into y. Note that because of the division problem, when p is not prime, the padic numbers are not a field, but only a ring. X, d is a complete metric space if any cauchy sequence in x has a limit in x. Concretely, a p p adic integer x x may be written as a base p p expansion. Pictures of ultrametric spaces, the padic numbers, and.

Thanks for contributing an answer to mathematics stack exchange. A set with a metric, such as d in the definition above, is called a metric space. Pdf a stream processing approach to distance measurement of. Arbitrary intersectons of open sets need not be open.

Eichlinghofen, the 28th august 2015 by gilles bellot tudortmunduniversity facultyofmathematics. 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. Ultrametricity an ultrametric space is a space endowed with an ultrametric distance, defined as a distance satisfying the inequality da,c. As for the limitpoints, having had a quick look at the definition that uniqueness. In a complete metric space, a sequence is convergent if and only if it is a cauchy.