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Topic: Effective results in number theory

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	Class number problem for imaginary quadratic fields - Wikipedia, the free encyclopedia
		In mathematics, the Gauss class number problem (for imaginary quadratic fields), as is usually understood, is to provide for each n ≥ 1 a complete list of imaginary quadratic fields with class number n.
		The basic result that provided assurance that such a list would be finite did not, in the form proved around 1930, allow even in principle such a calculation.
		The general case awaited the discovery of Dorian Goldfeld that the class number problem could be connected to the L-functions of elliptic curves.
	www.wikipedia.org /wiki/Class_number_conjecture (344 words)

	Science Fair Projects - Diophantine approximation
		The smallness of the distance (in an absolute value sense) from the real number to be approximated and the rational number that approximates it is a crude measure of how good the approximation is. A subtler measure considers how good the approximation is by comparison to the size of the denominator.
		This result was improved by Axel Thue and others, leading in the end to a definitive theorem of Roth: the exponent in the theorem was reduced from n, the degree of the algebraic number, to any number greater than 2 (i.e.
		This showed that Diophantine approximation results were closely related to the general problem of cancellation in exponential sums, which occurs all over analytic number theory in the bounding of error terms.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Diophantine_approximation (554 words)

	Effective results in number theory (Site not responding. Last check: 2007-11-05)
		Many of the principal results of analytic numbertheory proved in the period 1900-1950 were in fact ineffective.
		These included lower bounds for class numbers (ideal class groups for some families of number fields grow); and boundsfor the best rational approximations to algebraic numbers in terms of denominators.
		The result used for Liouville numbers in the proof is effective in the way it applies the mean value theorem : but improvements (to what is now the Thue-Siegel-Roth theorem) were not.
	www.therfcc.org /effective-results-in-number-theory-178103.html (299 words)

	Class number problem (Site not responding. Last check: 2007-11-05)
		In mathematics, the Gauss class number problem(for imaginary quadratic fields, as is usually understood, is to provide for each n ≥ 1 acomplete list of imaginary quadratic fields with class number n.
		Thebasic result that provided assurance that such a list would be finite did not, in the form proved around 1930, allow even inprinciple such a calculation.
		That is because what entersthe analytic formula for the class number is not h, the class number, on its own — but hlog ε,where ε is a fundamental unit.
	www.therfcc.org /class-number-problem-355268.html (310 words)

	proof theory
		Proof theory, a form of metamathematics, studies the ways in which proofss are used in mathematics.
		For example, one may state that there is no proof theory for second-order logic, meaning that there is no syntactical calculus for this logic that simultaneously (1) is sound, and (2) is complete, and (3) admits a proof-checking algorithm.
		Proof theory, model theory, axiomatic set theory, and recursion theory are the so-called "four theories" of the foundations of mathematics.
	www.fact-library.com /proof_theory.html (357 words)

	Effective results in number theory: Definition and Links by Encyclopedian.com - All about Effective results in number ... (Site not responding. Last check: 2007-11-05)
		Effective results in number theory: Definition and Links by Encyclopedian.com - All about Effective results in number theory
		Many of the principal results of analytic number theory proved in the period 1900-1950 were in fact ineffective.
		These latter could be read quite directly as results on Diophantine equations, after work of Thue.
	www.encyclopedian.com /ef/Effective-results-in-number-theory.html (334 words)

	number theory and quantum mechanics (Site not responding. Last check: 2007-11-05)
		By far, the most active area of research linking QM and number theory is the work concerning the 'spectral interpretation' of the Riemann zeta zeros, suggesting a possible approach to the Riemann hypothesis involving quantum chaos.
		The number of pairwise mutually unbiased bases is at most N+1 and various constructions of N+1 such bases have been found when N is a power of a prime number.
		The number of such distinct bands is given by Euler's \phi-function which appears in the context of number theory.
	www.maths.ex.ac.uk /~mwatkins/zeta/qm-general.htm (5123 words)

	Big Game Lottery Results (Site not responding. Last check: 2007-11-05)
		Effective results in number theory 1: ion to the solution of Diophantine equation s, results in number theory have been scrutinised more t 3: s.
		These latter could be read quite directly as results on Diophantine equations, after the work of Axe 5: Later results, particularly of Alan BakerBaker, changed th 7: computational complexity theory.
		Results of the 2003 California recall 1: The following are the '''results of the 2003 California recall ''' election hel 16: ==Results == 28: == Certified election results ==
	www.daikaiju.com /edge/35023-big%20game%20lottery%20results.html (812 words)

	ColbyCosh.com
		In practice, the result is usually to create a classroom distraction for the majority of the kids and divert a lot of the teacher's time into dealing with one or two special needs children.
		Sure, there is a value in making my English 30 students write X number of academic essays a year, with an introduction, conclusion, and three body paragraphs, but at the same time there is a value in making those same students keep a creative writing portfolio that is evaluated at the end of the year.
		Number of male babies born in Quebec in 2004 who were given the name "Anakin", according to the provincial Régie des rentes: 3.
	www.colbycosh.com (12432 words)

	Robert Paterson's Radio Weblog
		It seems that a coalition's effectiveness (in the sense of its members' willingness to come to each other's aid) is directly related to the amount of time its members spend grooming each other Hence, the larger the group, the more time individuals devote to grooming with the members of their coalitionary clique.
		They explicitly state that when the number of individuals is much larger than this, it becomes difficult to control their behaviour by means of peer pressure alone (Hardin 1988).
		Terrien and Mills (1955), for example, found that the larger the organisation, the greater the number of control officials that is needed to ensure its smooth functioning.
	radio.weblogs.com /0107127/categories/.../2003/02/22.html (1694 words)

	A History of Fundamental Mathematics Research at Bell Labs
		In the mid 60's, scientists at the Math Center were among the pioneers of the modern theory of scheduling, particularly in the analysis of approximation algorithms and in multiprocessor scheduling (Ed Coffman, Ron Graham, David Johnson and Mike Garey).
		Stochastic planar matching theory (Coffman and Shor) was also developed as a technique for the analysis of algorithms in diverse problems of computer science and operations research.
		Despite this burst of activity in the theory of algorithms, there remains a large collection of important optimization problems for which no efficient algorithms are known.
	cm.bell-labs.com /cm/ms/departments/fm/history.html (1978 words)

	Algebraic Number Theory Archive (Site not responding. Last check: 2007-11-05)
		ANT-0342: 28 Mar 2002, On an Archimedean analogue of Tate's conjecture, by Dipendra Prasad and C.S.Rajan.
		ANT-0296: 8 Jun 2001, On the Iwasawa theory of p-adic Lie extensions, by Otmar Venjakob.
		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)

	[No title]
		The thesis might be rephrased as saying that the notion of effective or mechanical method in logic and mathematics is captured by Turing machines.
		The notion of "effective method" is intuitively clear but is not formally defined since it is not exactly clear what a "simple and precise instruction" is, and what exactly the "required intelligence to execute these instructions" is. (See for example effective results in number theory for cases well beyond the Euclidean algorithm.)
		Roger Penrose) have suggested that the human mind might be the result of quantum hypercomputation, although a lack of physical or neurological evidence renders this theory scientifically dubious.
	en-cyclopedia.com /wiki/Church-Turing_thesis (1055 words)

	UIUC Dept. of Mathematics Seminar Calendar (Site not responding. Last check: 2007-11-05)
		We show that the probability for the formation of bubbles is regulated by the number of AT base pairs over a critical length, which at physiological temperatures is of the order of the size of the bubble.
		This is a generalization of classical results of Freidlin and Wentzell, who considered the case when all the flow lines of the unperturbed flow are closed curves.
		Abstract: There are a number of settings in which a dynamic system coordinates local rules in order to effect global changes in the states of the system.
	torus.math.uiuc.edu /cal/math/cal?...&regexp=Number+Theory&use=Find (2129 words)

	math lessons - Thue-Siegel-Roth theorem
		In mathematics, the Thue-Siegel-Roth theorem, also known simply as Roth's theorem, is a foundational result in diophantine approximation to algebraic numbers.
		It is of a qualitative type, stating that a given algebraic number α may not have too many rational number approximations, that are 'very good'.
		By its nature, it was ineffective (see effective results in number theory); this is of particular interest since a major application of this type of result is to bounding the number of solutions of some diophantine equations.
	www.mathdaily.com /lessons/Thue-Siegel-Roth_theorem (373 words)

	Church-Turing thesis : The Church-Turing thesis (Site not responding. Last check: 2007-11-05)
		The notion of "effective method" is intuitively clear but isn't formally defined since it isn't exactly clear what a "simple and precise instruction" is, and what exactly the "required intelligence to execute these instructions" is. (See for example effective results in number theory for cases well beyond the Euclidean algorithm.)
		A few months earlier Alonzo Church had proven a similar result in A Note on the Entscheidungsproblem but he used the notions of recursive functions and Lambda-definable functions to formally describe effective computability.
		John Lucas[?] (and famously, Roger Penrose) have suggested that the human mind might be the result of quantum hypercomputation, although this proposition is epistemologically dubious.
	www.city-search.org /th/the-church-turing-thesis.html (1117 words)

	quantum field theory and number theory (Site not responding. Last check: 2007-11-05)
		My results...led to the conclusion that Feynman diagrams obtained from a field theory in even dimensions all evaluate to the same number-class up to the seven-loop level, the limit of computational ability at this time, although we believe this result to be true in general.
		Although nothing particularly new is added to number theory or supersymmetric QFT, and the claim about the PNT is spurious, this paper introduces a 'number theoretical QFT' (analogous to B.
		It is found that the variance, related to the second functional variation of the effective action, requires a further regularization and that the relative regularized variance turns out to be 2/N, where N is the number of the fields, thus being independent on the dimension D.
	www.maths.ex.ac.uk /~mwatkins/zeta/QFT.htm (4517 words)

	Legal Theory Blog
		One of the requirements for class certification is “adequacy of representation.”; Given the important role of class counsel, it is not surprising that both the representative party (the nominal plaintiff) and the counsel for the class must be capable of adequately representing the interests of the absent members of the class.
		Individual litigation may result in a greater individual award, but also greater attorney’s fees, and individual litigation may be riskier for plaintiffs, with some recovering more from their individual lawsuits but others recovering less or nothing at all.
		Although the industry's recent approach was controversial and resulted in major criticisms from legislators, academics, civil libertarians, consumer advocates, and university officials, the copyright holders' aggressive tactics are not new.
	lsolum.blogspot.com /2003_10_01_lsolum_archive.html (12057 words)

	REU Program in Number Theory, Summer 2002 (Site not responding. Last check: 2007-11-05)
		The REU Program in Number Theory is one of four REU Summer Programs offered in Summer 2002 by the University of Illinois.
		The UIUC number theory graduate program is probably the largest of its kind in the United States, and is ranked fourth in the nation according to a recent US News and World Report Survey, behind Princeton, Harvard, and Berkeley.
		More than a dozen UIUC faculty members work in number theory, with interests covering the full spectrum of number theory, and approximately two dozen graduate students are currently working on a thesis in number theory.
	www.math.uiuc.edu /~hildebr/reu/announce.html (373 words)

	Goal Number One: Chapter Summary
		Some important strategies to consider include forming a commission of non-partisan experts to evaluate the effects of the current drug control model and allowing cities and states greater flexibility to experiment with their own approaches to drug control.
		With the high number of new HIV and hepatitis infections, laws against the possession of clean needles are a virtual death sentence.
		Seven reports funded by the U.S. Government between 1991 and 1997 are unanimous in their conclusions that clean needle programs reduce HIV transmission, and none find that clean needle programs cause rates of drug use to increase.
	www.csdp.org /edcs/page20.htm (555 words)

	[No title]
		The aim of this talk is to survey several applications of numeration systems, first in discrete geometry, and second, in cryptology.
		Let G be a finitely generated multiplicative group contained in the complex numbers, and let a, b be non-zero complex numbers.
		We discuss its results and applications to number theory and differential equations.
	www.math.leidenuniv.nl /~desmit/ic/current.html (687 words)

	Citebase - Chern-Simons number diffusion in (1+1)-dimensional Higgs theory
		We review recent progress in the study of the anomalous baryon number non-conservation at high temperatures and in high energy collisions.
		Recent results on high temperature phase transitions are described, and applications to electroweak baryogenesis are considered.
		The 1+1 dimensional abelian Higgs model with fermions is a toy model for the theory of electroweak baryogenesis.
	citebase.eprints.org /cgi-bin/citations?archiveID=oai:arXiv.org:hep-ph/9407241 (1074 words)

	Frequently Asked Questions about Caffeine
		The variability in the amount of caffeine in a cup of coffee or tea is relatively large even if prepared by the same person using the same equipment and ingredients day after day.
		Cardiologist Arthur Klatsky said of the surprising results, ``This is not a fluke finding because our study was very large, involved a multiracial population, men, women, and examined closely numerous factors related to mortality such as alcohol consumption and smoking.''
		It's particularly effective in those who are not habitual users.
	db.uwaterloo.ca /~alopez-o/Coffee/caffaq.html (4181 words)

	Travel Guide - Online Reservation - Warsaw Accommodation (Site not responding. Last check: 2007-11-05)
		Traditionally, number theory mathematics concerned with the properties of integers and many open problems that are easily understood even by non-mathematicians.
		More generally, the field has come to be concerned with a wider class of problems that arose naturally from the study of integers.
		Number theory may be subdivided into several fields according to the methods used and the questions investigated.
	www.warsaw-hotel.info /poland-guide/Category:Number_theory (169 words)

	Alan Baker - MindSharer Article Archive (Site not responding. Last check: 2007-11-05)
		Alan Baker (born on August 19 1939) is a British mathematician.
		He is known for his work on effective methods in number theory, in particular those arising from transcendence theory.
		He was awarded the Fields Medal in 1970, at age 31.
	articles.mindsharer.com /html/Alan_Baker (98 words)

	Eas Results For Women (Site not responding. Last check: 2007-11-05)
		In most imperative computer programming languages, a for loop is a control structure which allows code to be executed iteratively.
		For loops, unlike while loops, are typically used when the number of iterations are known before enteringthe loop.
		The study offemale reproduction and reproductive organs is called gynaecology.
	www.witchware.com /File/18997-Eas.Results.For.Women.Html (674 words)

	Thomas R. Nicely's Home Page
		Code written primarily in C, and distributed asynchronously across available personal computers running under extended DOS or W*ndows, is employed to enumerate primes, prime gaps, prime constellations (twins, triplets, and quadruplets) and their reciprocal sums (to extrapolate estimates for the corresponding Brun's constants).
		Following are some websites of relevance to mathematics in general, and number theory in particular.
		PARI-GP, a software package for computer-aided number theory, including the ultraprecision libpari C libraries and the gp programmable interactive calculator.
	www.trnicely.net (2875 words)

	Mammogram Results (Site not responding. Last check: 2007-11-05)
		1) " Mammogram" -- In the context of Mammogram Results
		Mammography is the process of using low-dose X-rays (usually around 0.7 mSv) to examine the human breast.
		2) " Results" -- In the context of Mammogram Results
	www.lottery-news.net /dust13570-mammogram_results.html (485 words)

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