P-adic number - Factbites

P-adic number - Factbites
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Topic: P-adic number

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	P-adic number - Wikipedia, the free encyclopedia
		of p-adic numbers is an extension field of the rational numbers.
		Thus e is a member of the algebraic closure of p-adic numbers for all p.
		As metric spaces, both the p-adic integers and the p-adic numbers are complete.
	en.wikipedia.org /wiki/P-adic_number (2349 words)

	Math Forum - Ask Dr. Math
		The p-adic integers can be thought of as writing out integers in base p, but you can have infinitely many digits to the left of the decimal (and none on the right; but the rational p-adic numbers can have finitely many digits on the right of the decimal).
		As you would expect, a rational p-adic number is one p-adic integer divided by another (nonzero) p-adic integer.
		If you look at the integer mod p, then mod p^2, and p^3, and so on, then a certain digit a_i is going to come out at each step, until p^n gets bigger than your number, at which point the coefficients all become zero.
	mathforum.org /library/drmath/view/65286.html (976 words)

	The decomposition of the spacetime
		Two essential properties of the p-adic numbers are involved, namely the ultrametricity of the p-adic topology and the nondeterminism of the p-adic differential equations.
		Hence it seems that p-adic numbers are an essential part of the definition of TGD rather than mere approximative treatment of the theory at low energy limit.
		The number N(p) of sheets in the region D(p) of CH(red) is expected to give rise to coherent amplification of the matrix element by a factor N^2(p) if individual amplitudes have the same phase.
	www.saunalahti.fi /jawap/colors/string/spin.html (4033 words)

	TGD diary
		The observation is that the continuation to various p-adic numbers fields and their extensions for the fermionic determinant can be simply done by allowing only the eigenvalues which belong to the extension of rationals involved and solve field equations for the resulting Kähler function.
		A further thread in the evolution of ideas were number theoretical vision about physics which emerged gradually from an observation that elementary particle mass spectrum could be understood in terms of p-adic thermodynamics for super-conformal invariant system with Hamiltonian replaced by the generator of scaling.
		Later emerged the vision about physics as a generalized number theory obtained by algebraically continuing rational physics to various number fields with p-adic space-time sheets interpreted as space-time correlates of intentionality.
	matpitka.blogspot.com (8283 words)

	Quantum time, the relationship between geometrical time and psychological time
		Infinite p-adic primes are forced by the requirement that p-adic prime increases in a statistical sense and that the number of quantum jumps already occurred is infinite (assuming finite number of quantum jumps and therefore first quantum jump, one encounters the problem of deciding what was the first quantum history).
		The simplest common sense expectation a =about L_p =about 10^(11) years forcing the recent value of p to be finite, is not consistent with the idea that the p-adic prime p characterizing the entire e Universe is infinite.
		Contrary to this, the infinite size of 3-space together with p-adic length scale hypothesis suggests that the p-adic prime p associated with the entire universe is infinite so that a=about L_p propto sqrt(p) should be infinite.
	www.saunalahti.fi /jawap/colors/string/time.html (11336 words)

	p-adic Numbers
		Hensel (around 1897) was the first to associate with each prime number p a number system which he called the p-adic numbers.
		Fact -These properties turn the set of p-adic numbers into a metric space in which the operations of addition, subtraction, multiplication and multiplicative inversion (when defined) are continuous.
		A p-adic number whose power series has no negative powers of p is called a p-adic integer.
	wwwmaths.anu.edu.au /DoM/thirdyear/MATH3301/p_adics.html (737 words)

	Algebraic Number Theory Archive
		ANT-0295: 8 Jun 2001, On the structure theory of the Iwasawa algebra of a p-adic Lie group, by Otmar Venjakob.
		math.NT/0304377: 24 Apr 2003, Theory of Generalized Bernoulli-Hurwitz Numbers for the Algebraic Functions of Cyclotomic Type, by Yoshihiro Ônishi.
		ANT-0334: 13 Feb 2002, Reduction of abstract homomorphisms of lattices mod p and rigidity, by Chandrashekhar Khare and Dipendra Prasad.
	front.math.ucdavis.edu /ANT (12251 words)

	Algebraic P-adic Numbers
		Let w be an integer that is not a square mod p, and w does not have a square root in the p-adic numbers.
		This time we are looking for a q(x) that has no solution in the p-adic numbers.
		In the last section we defined the p-adic numbers as the completion of the rationals Q, using the p-adic topology.
	www.mathreference.com /id-val,palg.html (1282 words)

	P-adic Numbers
		In constructing P-adic Numbers in the form of Cauchy sequences, it is imperative that you note that a P-adic integer is when the P-adic Absolute Value of x is less than or equal to 1.
		P-adic Numbers are a new set of numbers that are unusual but very exciting because they unite number theory, algebra, analysis, and topology.
		An example of this is the 5-adic expansion of the number 405.
	www.willamette.edu /~zizza/Courses/SeniorSeminar/G2.1/padic.html (856 words)

	p-Adic numbers
		The number of the appearences might be a finite number in the p-adic topology and have well defined real counterpart obtained by the canonical correspondence.
		The image of a p-adic number is always finite and the map is continuous.
		It is obvious that p-adic numbers are not well ordered since infinitely many p-adic numbers have same norm.
	www.physics.helsinki.fi /~matpitka/padic.html (802 words)

	P-adic Numbers
		Thus a sequence of rationals that is cauchy, using the valuation metric, represents a p-adic number.
		Let p be a prime and let r be the ring of rational numbers that do not have p in the denominator.
		This is a valuation ring, and it produces the p-adic topology.
	www.mathreference.com /id-val,padic.html (1200 words)

	p-adic and adelic physics
		The fluctuations of the number field could appear on the Planck length, in particular in the gravitational collapse or near the cosmological singularity.
		Phase-locking and prime number theory were proposed as the unifying concepts for understanding the optimal synchronization of clocks and their 1/f frequency noise.
		I.V. Volovich, "Number theory as the ultimate physical theory", Preprint CERN-TH (Abstract) "At the Planck scale doubt is cast on the usual notion of space-time and one cannot think about elementary particles.
	www.maths.ex.ac.uk /~mwatkins/zeta/physics7.htm (7331 words)

	p-adic numbers
		Sarah Sumner - Investigating transcendence in the field of p-adic numbers...
		Topics in Number Theory : p-adic numbers, p-adic analysis, and applications...
		P-adic number - definition of P-adic number in Encyclopedia...
	www.scienceoxygen.com /math/318.html (142 words)

	week218
		This lets us study rational numbers "locally" at the prime p using p-adic numbers, just as we can study a rational function locally at a point using its Laurent series.
		This technique, of "explicitly" generating abelian extensions of a number field, is called "explicit class field theory" and, other than the (non-trivial) contribution by Shimura and Taniyama where they used higher-dimensional abelian varieties to push the analogy slightly further, it's still a big mystery.
		The degree of H over K is called the class number of K and, totally amazingly, it is related to the special value of an L-function.
	math.ucr.edu /home/baez/week218.html (6121 words)

	Rachdi, Monsan: Spectral density estimation for p-adic stationary processes
		Generalized functions over the field of p-adic numbers.
		Annales mathématiques Blaise Pascal, 5 no. 1 (1998), p.
	www.numdam.org /numdam-bin/item?id=AMBP_1998__5_1_25_0 (75 words)

	Find in a Library: Introduction to p-adic analytic number theory
		Find in a Library: Introduction to p-adic analytic number theory
		WorldCat is provided by OCLC Online Computer Library Center, Inc. on behalf of its member libraries.
	worldcatlibraries.org /wcpa/ow/5717f4f559214e43a19afeb4da09e526.html (56 words)

	p-Adics
		for p-Adic numbers so all of theoretical physics can be reconsidered using p-Adic numbers.
		(2)No physical interpretation of the p-adic string, for finite p, is known.
		The p-Adic studies in my research were an expedition into pure mathematics.
	www.physics.unc.edu /~frampton/padics2.html (341 words)

	Hida: On the search of genuine $p$-adic modular $L$-functions for $GL(n)$. With a correction to: On $p$-adic $L$-functions of $GL(2)\times{}GL(2)$ over totally real fields
		Kato, L-functions and Tamagawa numbers of motives, Progress in Math.
		Hida, On Selmer groups of adjoint modular Galois representations, Number Theory, Paris, LMS lecture notes series,
	www.numdam.org /numdam-bin/item?id=MSMF_1996_2_67__R1_0 (894 words)

	LjSEEK.com: LiveJournal Blogs Search Engine
		We've implemented number of new features in this time and started few new projects.
		Huh, we're late with update this month, also instead of description of new nice features we've implemented you've got downtime warning in the start of October.
		About 3 months have passed since our last news update.
	www.answers-zone.com /article/P-adic_number (374 words)

	P-adic number - Wikipedia, the free encyclopedia
		of p-adic numbers is an extension field of the rational numbers.
		The p-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory.
		The simplest introduction to p-adic numbers is to consider 10-adic integers, which are simply strings of digits in which you allow an infinite number of digits to the left, for example, the number...9999, and then do arithmetic with such numbers as usual.
	en.wikipedia.org /wiki/P-adic_numbers (2377 words)

	P-adic number - Wikipedia, the free encyclopedia
		The reals and the p-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general algebraic number fields, in an analogous way.
		For each prime number p, the p-adic number system extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems.
		of p-adic numbers is an extension field of the rational numbers.
	en.wikipedia.org /wiki/P-adic_number (2053 words)

	Math Forum - Ask Dr. Math
		The p-adic integers can be thought of as writing out integers in base p, but you can have infinitely many digits to the left of the decimal (and none on the right; but the rational p-adic numbers can have finitely many digits on the right of the decimal).
		As you would expect, a rational p-adic number is one p-adic integer divided by another (nonzero) p-adic integer.
		If you look at the integer mod p, then mod p^2, and p^3, and so on, then a certain digit a_i is going to come out at each step, until p^n gets bigger than your number, at which point the coefficients all become zero.
	mathforum.org /library/drmath/view/65286.html (976 words)

	P-adic number article - P-adic number prime number field rational numbers number theory Helmut Hasse - What-Means.com
		The reals and the p-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general algebraic number fields, in an analogous way.
		They have been used to solve several problems in number theory, many of them using Helmut Hasse's local-global principle, which roughly states that an equation can be solved over the rational numbers if and only if it can be solved over the real numbers and over the p-adic numbers for every prime p.
		If p is a fixed prime number, then any integer can be written as a p-adic expansion (usually referred to as writing the number in "base p") in the form
	www.what-means.com /encyclopedia/P-adic_number (976 words)

	PlanetMath: $p$-adic integers
		(Number theory :: Algebraic number theory: local and $p$-adic fields :: Miscellaneous)
		Cross-references: infinite prime, finite prime, finite extension, number field, independent, absolute value, subring, subset, topological field, field, valuations, equivalence class, product topology, topological spaces, finite, infinite, subspace, closed, compact, characterization, isomorphism, algebraic, maps, projection maps, inverse limit, fraction field, metric, induced, divides, metric induced by the norm, completion, ring, integers, prime
		(Field theory and polynomials :: Topological fields :: Formally $p$-adic fields)
	planetmath.org /encyclopedia/PAdicIntegers.html (976 words)

	Exact Solution of Linear Equation Systems over Rational Numbers by Parallel p-Adic Arithmetic - Limongelli, Pirastu (ResearchIndex)
		Limongelli, C. and Pirastu, R.: Exact solution of linear equation systems over rational number by parallel p-adic arithmetic.
		@techreport{ limongelli94exact, author = "Roberto {PIRASTU} Carla {LIMONGELLI}", title = "{Exact Solution of Linear Equation Systems over Rational Numbers by Parallel p-Adic Arithmetic}", number = "94-25", address = "Johannes Kepler University, Linz, Austria", year = "1994", url = "citeseer.ist.psu.edu/limongelli94exact.html" }
		The rationals are represented by truncated p-adic expansion.
	citeseer.ist.psu.edu /limongelli94exact.html (976 words)

	p-Adic numbers
		The number of the appearences might be a finite number in the p-adic topology and have well defined real counterpart obtained by the canonical correspondence.
		If one expresses the p-adic number as an infinite power series in the pinary basis
		p-Adic topology is defined by the p-adic norm.
	www.physics.helsinki.fi /~matpitka/padic.html (802 words)

	Alexander Grothendieck
		Perhaps Grothendieck's deepest single accomplishment is the invention of the Ã©tale and l-adic cohomology theories, which explain an observation of AndrÃ© Weil 's, that there is a deep connection between the topological characteristics of a variety and its diophantine (number theoretic) properties.
		For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex number s.
		Weil realized that to prove such a connection one needed a new cohomology theory, but neither he nor any other expert saw how to do this until such a theory was found by Grothendieck.
	www.bambooweb.com /articles/a/l/Alexander_Grothendieck.html (802 words)

	Citations: with an application to the Entscheidung problems - Turing, real (ResearchIndex)
		The main issue in the present work is to study the analogous problems in the case of p adic numbers and verify which ones carry over and which ones fail and the reason for the failures.
		a real or p adic number is viewed as infinite object, and is understood as the limit of finite objects, namely rational numbers.
		In fact, in terms of computational complexity, the fact that quintic equations may not have radical expressions for their roots is largely irrelevant; it simply rules out one mode of expression.
	citeseer.ist.psu.edu /context/272810/0 (1145 words)

	Algebraic Number Theory Archive
		ANT-0171: 26 Feb 1999, Polarizations on abelian varieties and self-dual ell-adic representations of inertia groups, by Alice Silverberg and Yuri G. Zarhin.
		ANT-0087: 1 Dec 1997, Degeneration of the l-adic Eisenstein symbol and of the elliptic polylog, by Annette Huber and Guido Kings.
		ANT-0267: 27 Nov 2000, On an analogue for number fields of a conjecture of de Jong on F_q[[t]]-analytic extensions of function fields, by Gebhard Boeckle.
	front.math.ucdavis.edu /ANT (12251 words)

	From PID to P-adic
		In the case of the p-adic numbers, p is a specific prime, and the terms of the series are integers, except perhaps for a finite number of fractions at the start.
		When r is the integers, and p is a prime number, t is the fractions whose denominators are not divisible by p.
		Let r be a pid, and let p generate a maximal ideal in r, hence p acts as the base for the valuation.
	www.mathreference.com /id-val,pid.html (1400 words)

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