For further interesting examples of ultrametric spaces in the context of trees, see 5. We propose a mathematical model of the human memoryretrieval process based on dynamic systems over a metric space ofp adic numbers. Since qp is a complete separable metric space, the stochastic process yt. You have met or you will meet the concept of a normed vector space both in algebra and analysis courses.
On the one hand, this theory generalizes the fuchsian and bers uniformizations of complex hyperbolic curves. With the p adic topology, z is an ultrametric space. 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. A little padic functional analysis part 1 of 2 im going to start with a little bit of terminology and notation about nonarchimedean banach spaces trusting that you can.
The padic completion of q and hensels lemma contents. The padic absolute value defines a metric d p on q by setting. What is known about continuous representations of padic. Hey, so i am a senior in high school and i am writing a paper on the p adic metric and p adic numbers. The padic integers form a subset of the set of all p adic numbers. Topologically, the ring zp of padic integers provides a mess of coun terexamples to standard misconceptions in metric space topology. A metric space x, d is a topological space, with the topology being induced from the metric d. In a complete metric space, a sequence is convergent if and only if it is a cauchy. Segmentation of images in padic and euclidean metrics. Survey of p adic numbers catherine warner december 14, 2018 1 introduction the p adic numbers, denoted q pis a complete eld that can be constructed from the rationals similarly to the construction of the reals using cauchy sequences. One natural candidate that arises in this context is the notion of a metric more precisely, the notion of a real analytic k. We also assume that this dynamic system is located in the subconscious and is controlled by the conscious, which specifies. Arithmetic properties of the p adic logarithm c edric dion1 universit e laval in number theory, many important problems require a good understanding of the arithmetic properties of prime numbers. Construction of cpand extension of padic valuations to c mike hamburg october 10, 2004 1 introduction here we will derive the structure of the padic complex numbers, that is, an algebraically closed, topologically complete.
The p adic topology on z is the metric topology with the p adic metric d. We do not develop their theory in detail, and we leave the veri. Construction of cp and extension of padic valuations to c. The padic numbers are most simply a field extension of q, the rational. We begin by emphasizing the properties of a nonarchimedean norm and exploring the topology it induces. Theres a lot more where this came from, but we wont need the rest of it just yet. The ultrametric triangle inequality underlies many of the interesting. We provide an introduction to the p adic metric and its applications to algebra, analysis, topology, and number theory. For a fixed prime number p, a fraction of the form a p k, where a is a p adic integer and k is a nonnegative integer.
Assume v to be a hausdor locally convex kvector space. Introduction to p adic numbers an overview of ultrametric spaces and p adic numbers. The padic numbers holly green 24th september 2018 abstract this paper is the nal product from my place on imperial colleges urop programme. This property shows that in an ultra metric space all triangles are isosceles and legs are longer than. P adic numbers by paul james holloway a thesis submitted in partial ful lment of. Although there are many metrics on rn, d 2 is generally the one that is used unless explicit mention otherwise is made. We assume that two ideas are close if they have a sufficiently long initial segment in common. Euclidean or any of the padics ostrowskis theorem 1, it would be convenient to measure the distance between points inbakers map with a padic metric. These notes aim to provide a fast introduction to padic analysis assuming basic knowledge in algebra and analysis.
In fact, the construction of the p adics is a generalization of the construction of the reals. X, is called an ultrametric space, the corresponding metric is called an. An overview of ultrametric spaces and padic numbers. However, here we wish to relay another, deeper, relation between the cohomology. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. Metricandtopologicalspaces university of cambridge. Metric spaces a metric space is a set x that has a notion of the distance dx,y between every pair of points x,y. Thus the usual norm on q is archimedean, whilst the padic norms are all nonarchimedean.
As the 2 adic metric is the simplest i am going to focus on it and basically try to prove some of its properties. Eichlinghofen, the 28th august 2015 by gilles bellot tudortmunduniversity facultyofmathematics. This latter approach will allow to show that padic integers form a ring, denoted by zp. A set with a metric, such as d in the definition above, is called a metric space. For a metric space let us consider the space of all nonempty closed bounded subset of with the following metric. For continuous homomorphisms, these things are wellknown, going between additive and multiplicative structures being failrly straightforward using exp or log on the padic or real side. Human memory as a p adic dynamic system springerlink. Pictures of ultrametric spaces, the padic numbers, and. Some of this material is contained in optional sections of the book, but i will assume none of that and start from scratch. Like the rationals, but unlike the reals, the padics are totally disconnected. The field q p of padic numbers can then be defined as the completion of the metric space q, d p. Hensels lemma 10 acknowledgments references introduction the padics present a particularly rich subject.
It is an extension of q, so the temptation is to imagine q. Introduction to padic numbers an overview of ultrametric spaces and p adic numbers. It provides an introduction to the padic numbers and their applications, based loosely. X, d is a complete metric space if any cauchy sequence in x has a limit in x. In contrast, the p adic extension arises from the use of the counterintuitive p adic metric. The p adic integers form a subset of the set of all p adic numbers. This makes us wonder if the ring of padic numbers with their padic soft metric space is a s. Segmentation of images in padic and euclidean metrics article in doklady mathematics 643 november 2001 with 82 reads how we measure reads.
Notes on metric spaces these notes introduce the concept of a metric space, which will be an essential notion throughout this course and in others that follow. Riemann 1854 now it seems that the empirical notions on which the metric determinations of space are based, the concept of a solid body and a light ray, lose their validity in the in. Padic metric article about padic metric by the free. 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.
1390 1291 1280 82 487 231 829 1139 569 151 1148 1214 318 1044 231 1189 423 398 1452 1136 28 637 1489 1118 1371 1221 465 704 965 1493 463 210 1267 877 1181 651 36 1300 1053 977 768 1410 432 1308 1085 199 330 892