By Alain M. Robert

ISBN-10: 1441931503

ISBN-13: 9781441931504

Found on the flip of the twentieth century, p-adic numbers are often utilized by mathematicians and physicists. this article is a self-contained presentation of uncomplicated p-adic research with a spotlight on analytic subject matters. It bargains many positive factors not often handled in introductory p-adic texts similar to topological versions of p-adic areas within Euclidian house, a unique case of Hazewinkel’s sensible equation lemma, and a remedy of analytic parts.

**Read Online or Download A Course in p-adic Analysis (Graduate Texts in Mathematics) PDF**

**Similar mathematics books**

**Mathematical Methods for Physicists (6th Edition)**

This best-selling name presents in a single convenient quantity the fundamental mathematical instruments and methods used to unravel difficulties in physics. it's a important addition to the bookshelf of any severe scholar of physics or learn specialist within the box. The authors have placed significant attempt into revamping this re-creation.

**Duality for Nonconvex Approximation and Optimization**

During this monograph the writer provides the speculation of duality for nonconvex approximation in normed linear areas and nonconvex worldwide optimization in in the neighborhood convex areas. distinct proofs of effects are given, besides various illustrations. whereas the various effects were released in mathematical journals, this can be the 1st time those effects look in publication shape.

**Automatic Sequences (De Gruyter Expositions in Mathematics, 36)**

Automated sequences are sequences that are produced via a finite automaton. even though they don't seem to be random, they could glance as being random. they're advanced, within the feel of no longer being eventually periodic. they could additionally glance really advanced, within the feel that it will possibly now not be effortless to call the rule of thumb in which the series is generated; in spite of the fact that, there exists a rule which generates the series.

**AMERICAN WRITERS, Retrospective Supplement I**

This choice of severe and biographical articles covers amazing authors from the seventeenth century to the current day.

- Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry
- Principles of Differential and Integral Equations
- Mathematical Modelling of Biosystems
- Feedback Control Theory
- Ant Colony Optimization and Swarm Intelligence: 5th International Workshop, ANTS 2006, Brussels, Belgium, September 4-7, 2006. Proceedings
- Prufungstrainer Mathematik. Klausur- und Ubungsaufgaben mit vollstandigen Musterlosungen

**Additional info for A Course in p-adic Analysis (Graduate Texts in Mathematics)**

**Sample text**

0 '. a . 0 1 2 0- A projection of C x C 3. 1. Topological Groups Definition. A topological group is a group G equipped with a topology such that the map (x, y) H xy-t : G x G -* G is continuous. 18 L _ p-adic Numbers If G is a topological group, the inverse map x H x-1 is continuous (fix x = e in the continuous map (x, y) t-* xy-1) and hence a homeomorphism of order 2 of G. The translations x H ax (resp. , the inverse of x r- ax is x F+ a-tx). A subgroup of a topological group is a topological group for the induced topology.

Proposition, The following properties are equivalent: (1) P = 0 admits a solution in Zp. (ii) For each n > 0, P = 0 admits a solution in Z/ p" Z. (iii) For each n > 0, there are integers a", b" such that P(an,b,,)=- 0modp". PROOF. (iii) is a simple reformulation of (ii). Now for x = Et>o ai p' E Zp, define x" _ >t<" a; p' mod p" E Z/p'Z. Then if (x, y) E Zp x Zp, then P(xn, y,,) = P(x, y) mod p"ZP E ZP/p"ZP (= Z/p"Z), and hence (i) (i i ). Conversely, to prove (i i) (i) let us consider the finite sets X" ={(x,y)EZ/p"ZxZ/p"Z: P(x,y)=0}.

Let )), be an affine linear functional on V such that A. < l on E, A(v) = 1 for some v E E. Choose 1 = b - 1. Then v(0 1 i>o bi+l r>o so that the image F of %P is also contained in the convex hull of E: F C E = Ko. Moreover, by choice of the constant $, v >gE,>ob+l)=1. From the self-similarity representation of F we get a better approximation F=U $U+ b b) I ( 6 bv C VEX VEX b f. b) Iterating this inclusion in the self-similarity representation of F we get an even better approximation: I VE C 0-+b b, alb F= v +1 I VEX b uEE(Ob+ c) Eventually, this leads to a representation of the fractal F as the intersection of a decreasing sequence of compact sets K.