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Encyclopedia results for Computational number theory

Computational number theory





Encyclopedia results for Computational number theory

  1. Computational number theory

    In mathematics , computational number theory , also known as algorithmic number theory , is the study of algorithm s for performing number theory number theoretic computation s. The best known problem in the field is integer factorization . See also Computational complexity of mathematical operations Sage Math Number Theory Library PARI GP Further reading Victor Shoup , A Computational Introduction to Number Theory and Algebra . Cambridge, 2005, ISBN 0 521 85 154 8 Henri Cohen number theorist Henri Cohen , A Course in Computational Algebraic Number Theory , Graduate Texts in Mathematics 138, Springer Verlag, 1993. Eric Bach and Jeffrey Shallit , Algorithmic Number Theory , volume 1 Efficient Algorithms . MIT Press, 1996, ISBN 0 262 02405 5 Richard Crandall and Carl Pomerance , Prime Numbers A Computational Perspective , Springer Verlag, 2001, ISBN 0 387 94777 9 Hans Riesel , Prime Numbers and Computer Methods for Factorization , second edition, Birkh user, 1994, ISBN 0 8176 3743 5, ISBN 3 7643 3743 5 Number theoretic algorithms Number theory footer Category Computational number theory Category Number theory Numtheory stub ar de Algorithmische Zahlentheorie fr Th orie algorithmique des nombres pl Algorytmiczna teoria liczb ...   more details



  1. Computational learning theory

    In theoretical computer science , computational learning theory is a mathematical field related to the analysis ... such as minimizing the number of mistakes made on new samples. In addition to performance bounds, computational learning theorists study the time complexity and feasibility of learning. In computational learning theory, a computation is considered feasible if it can be done in polynomial time . There are two ... learnability, Proc. 1st ACM Workshop on Computational Learning Theory, 1988 42 55. L. Pitt ... to Computational Learning Theory http www.santafe.edu shalizi reviews vapnik nature Review of The Nature ... Inference.htm Basics of Bayesian inference Category Computational learning theory Category Theoretical ... in negative results are Computational complexity P NP problem P &ne NP cryptography Cryptographic One way function s exist. There are several different approaches to computational learning theory. These differences are based on making assumptions about the inference principles used to generalize ... theory , proposed by Vladimir Vapnik Bayesian inference , arising from work first done by Thomas Bayes . Algorithmic learning theory , from the work of E. M. Gold. Online machine learning , from the work of Nick Littlestone. Computational learning theory has led to several practical algorithms. For example, PAC theory inspired boosting , VC theory led to support vector machine s, and Bayesian inference led to belief networks by Judea Pearl . See also Information theory References Surveys Angluin, D. 1992. Computational learning theory Survey and selected bibliography. In Proceedings of the Twenty Fourth Annual ACM Symposium on Theory of Computing May 1992 , pp. 351 369. http portal.acm.org ... to their probabilities. Theory of Probability and its Applications, 16 2 264 280, 1971. Feature ... Annual ACM Symposium on Theory of Computing, pages 433 444, New York. ACM. http citeseer.ist.psu.edu .... Journal of the ACM, 36 4 929 865, 1989. Probably approximately correct learning L. Valiant. A Theory ...   more details



  1. Number theory

    primes and solving simple diophantine equations . Number theory is a branch of pure mathematics ... . Questions in number theory are often best understood through the study of Complex analysis analytical ... number theoretic objects in some fashion analytic number theory . One may also study real numbers ... term for number theory is arithmetic by the early twentieth century, ref name HeathAr By 1921 ... by number theory . The word arithmetic is used by the general public to mean elementary calculations ..., op. cit., 199 200 . ref While Babylonian number theory or what survives of Babylonian mathematics ... of number theory, this means, by and large, Plato and Euclid , respectively. Plato had a keen ... he meant, in part, theorising on number, rather than what arithmetic or number theory have come ..., topics that belong unambiguously to number theory and are basic thereto Books VII to IX ... and studied in north Africa and Constantinople during his formative years, ca. 1175 1200 no number theory ... Xylander , 1575 . Early modern number theory Fermat Image Pierre de Fermat.png thumb right upright Pierre ... Weil Andr Weil, Number theory an approach through history from Hammurapi to Legendre , Birkh user, 1984, pp. 45 46. ref He wrote down nearly no proofs in number theory he had no models in the area. ref name Weil2 Weil, op. cit., p. 118. This was more so in number theory than in other areas remark ... Leonhard Euler.jpg thumb upright Leonhard Euler The interest of Leonhard Euler 1707 1783 in number theory ... of modern number theory, ref name Weilbirth Weil, op. cit., pp. 1 and 2. ref after Fermat ... Weil, op. cit., p. 2, and Varadarajan, op. cit., p. 37 ref Euler s work on number theory includes ... analytic number theory . In his work of sums of four squares, Partition function number theory Partition ... the use of what can be seen as analysis in particular, infinite series in number theory. Since ... of unity and number theory blockquote The theory of the division of the circle...which is treated in sec ...   more details



  1. Computational complexity theory

    Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those Complexity class classes to each other. In this context, a computational ... are also used, such as the amount of communication used in communication complexity , the number of logic gate gates in a circuit used in circuit complexity and the number of processors used in parallel computing . One of the roles of computational complexity theory is to determine the practical ... are analysis of algorithms and computability theory . A key distinction between analysis of algorithms and computational complexity theory is that the former is devoted to analyzing the amount of resources ... with the problem itself. In computational complexity theory, a problem refers to the abstract question ... is at most 10  km. For this reason, complexity theory addresses computational problems and not particular ... theory. A decision problem is a special type of computational problem whose answer is either ... title Computational Complexity A Modern Approach url http www.cs.princeton.edu theory complexity ... Theory of Computational Complexity publisher John Wiley & Sons year 2000 country US isbn 978 0 ... Theory Category Computational complexity theory Link FA de Link FA de ar ... ru simple Computational complexity theory sk Te ria zlo itosti sr ..., a computational problem consists of problem instances and solutions to these problem instances. For example, primality testing is the problem of determining whether a given number is prime number prime ... or no based on whether the number is prime or not. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes ..., imposing restrictions on the available resources is what distinguishes computational complexity ...   more details



  1. Computational group theory

    In mathematics , computational group theory is the study of group mathematics group s by means of computers. It is concerned with designing and analysing algorithm s and data structure s to compute information about groups. The subject has attracted interest because for many interesting groups including most of the sporadic groups it is impractical to perform calculations by hand. Important algorithms in computational group theory include the Schreier Sims algorithm for finding the order of a permutation group the Todd Coxeter algorithm and Knuth Bendix algorithm for coset enumeration the product replacement algorithm for finding random elements of a group Two important computer algebra system s CAS used for group theory are GAP computer algebra system GAP and Magma computer algebra system Magma . Historically, other systems such as CAS for character theory and Cayley computer algebra system Cayley a predecessor of Magma were important. Some achievements of the field include complete enumeration of List of small groups all finite groups of order less than 2000 computation of representations for all the sporadic groups References A http www.math.ohio state.edu akos notices.ps survey of the subject by kos Seress from Ohio State University , expanded from an article that appeared in the Notices of the American Mathematical Society is available online. There is also a http www.math.rutgers.edu sims publications survey.pdf survey by Charles Sims mathematician Charles Sims from Rutgers University and an http www.math.rwth aachen.de Joachim.Neubueser preprint.html older survey by Joachim Neub ser from RWTH Aachen . There are three books covering various parts of the subject Derek F. Holt, Bettina Eick, Bettina, Eamonn A. O Brien, Handbook of computational group theory , Discrete Mathematics and its Applications Boca Raton . Chapman & Hall CRC, Boca Raton, FL, 2005. ISBN 1 58488 ... Press, Cambridge, 2003. ISBN 0 521 66103 X. Category Computational group theory nl Computationele ...   more details



  1. Computational theory of mind

    Horst, Steven , 2005 http plato.stanford.edu entries computational mind The Computational Theory ... and is presumed by theorists of evolutionary psychology . The computational theory of mind is a philosophical concept that the mind functions as a computer or symbol manipulator. The theory is that the mind ... to get a specific output. The computational theory of mind claims that there are certain aspects of the mind that follow step by step processes to compute representations of the world. The computational .... The computational theory of mind is related to the representational theory of mind in that they both ... theory claims that all mental states are representations while the computational ... states are known as qualia . The computational theory of mind is also related to the language of thought ... of semantics. See below in semantics of mental states . Computer metaphor Computational theory ... ref Computational theory just uses some of the same principles as those found in digital computing. ref ... of the Computational Theory of Mind is the idea that thoughts are a form of computation, and a computation ... to form representations. Semantics of mental states The computational theory of mind ... really be said to understand it? This is essentially what the computational theory of mind presents ... theory ref that physical supervenience is not compatible with computational theory, using arguments ... version of a Computational Representational Theory of Thought. Alternative Theories Classical associationism ... Encyclopedia of Philosophy s entry on the computational theory http homepage.mac.com blinkcentral ... is not compatible with computational theory French Category theories of mind Category Philosophy ...Citations missing date July 2008 No footnotes date April 2009 In philosophy of mind philosophy , the computational theory of mind is the view that the human mind is an information processing system and that thinking is a form of computing . The theory was proposed in its modern form by Hilary Putnam ...   more details



  1. Computational and Mathematical Organization Theory

    Infobox journal title Computational and Mathematical Organization Theory cover editor Kathleen Carley Kathleen M. Carley , Zhiang Lin discipline Organizational studies Organization theory language former names abbreviation Comput. Math. Organ. Theory publisher Springer Science Business Media country frequency Quarterly history 1995 present openaccess license impact impact year website http www.springer.com business 26 management business for professionals journal 10588 link1 http www.springerlink.com content 1381 298X link1 name Online access link2 link2 name JSTOR OCLC 41558316 LCCN sn97047289 CODEN ISSN 1381 298X eISSN 1572 9346 Computational and Mathematical Organization Theory is a peer review Different styles of review double blind peer reviewed scientific journal in the field of Organizational studies organization theory . The journal is published quarterly by Springer Science Business Media . It was established in 1995 and initially published by Wolters Kluwer Kluwer . The founding Editor in chief editors were William A. Wallace organizational theorist William A. Wallace and Kathleen Carley . ref name first edition cite journal first1 Kathleen M. last1 Carley authorlink1 Kathleen Carley first2 William A. last2 Wallace authorlink2 William A. Wallace organizational theorist url http www.springerlink.com content 1381 298x 1 1 title Editorial month October year 1995 journal Computational and Mathematical Organization Theory format PDF ref Carley has continued as co editor in chief , a role she currently shares with Zhiang Lin . ref name about cite web url http www.springer.com business 26 management business for professionals journal 10588 title Computational and Mathematical Organization Theory accessdate 9 April 2011 ref Abstracting and indexing Computational and Mathematical Organization Theory is abstracted and indexed in Association for Computing Machinery ACM Digital Library , Cabell s , Computer Science Index , CSA database company CSA databases , Current Abstracts ...   more details



  1. Computational

    Computational number theory Computational ontology Computational origami Computational overhead ... Computational synthetic geometry Computational systems biology Computational theory of mind Computational ... Computational trust Computational universality Computational universe theory Computational vision Computational ...Wiktionary Computational may refer to Computer Computational algebra Computational Aeroacoustics Computational and Information Systems Laboratory Computational and Systems Neuroscience Computational archaeology Computational auditory scene analysis Computational biology Computational biomodeling Computational Center for Nanotechnology Innovations Computational chemical methods in solid state physics Computational chemistry Computational Chemistry Grid Computational Chemistry List Computational complexity of mathematical operations Computational complexity theory Computational creativity Computational cybernetics Computational Diffie Hellman assumption Computational Diffie Hellman problem Computational economics Computational electromagnetics Computational Engineering Computational epidemiology Computational epigenetics Computational epistemology Computational finance Computational fluid dynamics Computational forensics Computational formula for the variance Computational gene Computational genomics Computational geometry Computational Geometry Algorithms Library Computational geophysics Computational grid Computational group theory Computational hardness assumption Computational humor Computational indistinguishability Computational informatics Computational immunology Computational intelligence Computational Intelligence and Machine Learning Portal Computational Intensive Workload Computational intractability Computational irreducibility Computational journalism Computational learning theory Computational lexicology Computational linguistics Computational Linguistics journal Computational lithography Computational logic Computational magnetohydrodynamics Computational mathematics ...   more details



  1. Prime number theory

    Prime number theory may refer to Prime number Prime number theorem Number theory disambig Long comment to avoid being listed on short pages ...   more details



  1. List of number theory topics

    This is a list of number theory topics , by Wikipedia page. See also List of recreational number theory topics Topics in cryptography Factors Composite number Highly composite number Even and odd numbers ... Euler s criterion Legendre symbol Gauss s lemma number theory Congruence of squares Luhn formula Mod ... function number theory Integer partition Bell numbers Landau s function Pentagonal number theorem Bell series Lambert series Analytic number theory additive problems Twin prime Brun s constant ... square identity Lagrange s four square theorem Taxicab number Generalized taxicab number Cabtaxi number Schnirelmann density Sumset Landau Ramanujan constant Sierpinski number Seventeen or Bust Niven s constant Algebraic number theory See list of algebraic number theory topics Quadratic form s Unimodular ... Mahler s compactness theorem Mahler measure Effective results in number theory Mahler s theorem Sieve ... prime Woodall prime Prime pages Combinatorial number theory Covering system Small set combinatorics ... number theory Algorithmic number theory Residue number system Cunningham project Quadratic residuosity ... Prime Obsession Category Mathematics related lists Number theory Category Number theory ... Table of divisors Prime number , prime power Bonse s inequality Prime factor Table of prime factors Formula for primes Factorization RSA number Fundamental theorem of arithmetic Square free Square free integer Square free polynomial Square number Power of two Integer valued polynomial Fraction mathematics Fraction s Rational number Unit fraction Irreducible fraction in lowest terms Dyadic fraction Recurring decimal Cyclic number Farey sequence Ford circle Stern Brocot tree Dedekind sum Egyptian ... squares L function s Riemann zeta function Basel problem on 2 Hurwitz zeta function Bernoulli number Agoh Giuga conjecture Von Staudt Clausen theorem Dirichlet series Euler product Prime number theorem Prime counting function Offset logarithmic integral Legendre s constant Skewes number Bertrand ...   more details



  1. Algorithmic Number Theory Symposium

    Mathematics conferences Category Computational number theory Category Recurring events established ... Location University of Bordeaux Bordeaux , France Organizers Henri Cohen number theorist Henri Cohen ...   more details



  1. Effective results in number theory

    was just a small number effect, but small here included values of n up to a billion. The requirement of computability reflects on and contrasts with the approach used in analytic number theory ... explicit. The Siegel period Many of the principal results of analytic number theory that were proved ... number theory Category Diophantine equations fr R sultats effectifs en th orie des nombres ... on the Skewes number of 1933, these results were believed by some experts to be intrinsically ... of A , the so called implied constant , may also need to be made explicit, for computational ... Linfoot on the class number 1 problem ref H. Heilbronn, E. Linfoot, On the imaginary quadratic corpora of class number one. Quart. J. Math. Oxford Ser. 5 1934 , pp. 293&ndash 301. ref The 1935 result ... group s for some families of number fields grow and bounds for the best rational approximations to algebraic number s in terms of denominator s. These latter could be read quite directly as results on Diophantine equations, after the work of Axel Thue . The result used for Liouville number s in the proof .... The logic involved is closer to proof theory than to that of computability theory and computable function s. It is rather loosely conjectured that the difficulties may lie in the realm of computational complexity theory . Ineffective results are still being proved in the shape A or B , where ...   more details



  1. Probabilistic number theory

    Probabilistic number theory is a subfield of number theory , which explicitly uses probability to answer questions of number theory. One basic idea underlying it is that different prime number s are, in some serious sense, like independent random variables . This however is not an idea that has a unique useful formal expression. The founders of the theory were Paul Erd s , Aurel Wintner and Mark Kac during the 1930s, one of the most intense periods of investigation in analytic number theory . The Erd s Wintner theorem and the Erd s Kac theorem on additive function s were foundational results. See also analytic number theory areas of mathematics list of number theory topics list of probability topics mathematics probabilistic method probable prime References cite book title Introduction to Analytic and Probabilistic Number Theory author G rald Tenenbaum series Cambridge studies in advanced mathematics volume 46 publisher Cambridge University Press year 1995 isbn 0 521 41261 7 Category Number theory numtheory stub ...   more details



  1. Collaborative Computational Project Number 4

    Infobox Software name CCP4 developer CCLRC Daresbury Laboratory latest release version 6.1.3 latest release date release date and age 2010 06 01 df yes operating system UNIX , Linux , Apple Macintosh Mac , MS Windows programming language C programming language C , Fortran , Tcl , Python programming language Python genre X Ray Crystallography licence http www.ccp4.ac.uk ccp4license.php Various website http www.ccp4.ac.uk CCP4 The Collaborative Computational Project Number 4 in protein crystallography or CCP4 was set up in 1979 in the United Kingdom to support collaboration between researchers working in software development and assemble a comprehensive collection of software for structural biology . The CCP4 core team is located at the Research Complex at Harwell RCaH at Rutherford Appleton Laboratory RAL in Didcot , near Oxford, UK. CCP4 was originally supported by the UK Science and Engineering Research Council SERC , and is now supported by the Biotechnology and Biological Sciences Research Council BBSRC . The project is coordinated at CCLRC Daresbury Laboratory. The results of this effort gave rise to the CCP4 program suite, which is now distributed to academic and commercial users worldwide. Projects http www.ccp4.ac.uk ccp4i main.php CCP4i &mdash CCP4 Graphical User Interface http www.ysbl.york.ac.uk ccp4mg CCP4MG &mdash CCP4 Molecular Graphics Project http www.ccp4.ac.uk HAPPy HAPPy &mdash automated experimental phasing http www.ccp4.ac.uk MrBUMP MrBUMP &mdash automated Molecular Replacement http www.ebi.ac.uk msd srv prot int pistart.html PISA &mdash Protein Interfaces, Surfaces and Assemblies http www.mrc lmb.cam.ac.uk harry imosflm index.html MOSFLM GUI &mdash building a modern interface to MOSFLM See also CCP4 file format External links http www.ccp4.ac.uk CCP4 Main Web site http ccp4wiki.org ccp4wiki wiki index.php?title Main Page CCP4 Documentation wiki &mdash concentrates only on CCP4 http strucbio.biologie.uni konstanz.de ccp4wiki index.php Main Page C ...   more details



  1. Number Theory Foundation

    The Number Theory Foundation NTF is a non profit organization based in the United States which supports research and conferences in the field of number theory . The NTF funds the Selfridge prize which is awarded at the ANTS conference ANTS conferences. External links http www.math.uiuc.edu ntf NTF web site Category Number theory Category Non profit organizations based in the United States math stub ...   more details



  1. Intersection number (graph theory)

    In the mathematical field of graph theory , the intersection number of a graph math G V , E is the smallest number of elements in a representation of math G as an intersection graph of finite set s. Equivalently, it is the smallest number of clique graph theory cliques needed to cover all of the edges of math G . ref name gy06 citation title Graph Theory and its Applications first1 Jonathan L. last1 ... Fund. Math. volume 33 year 1945 pages 303 307 mr 0015448 . ref The intersection number of the graph is the smallest number math k such that there exists a representation of this type for which ... of a graph with a given number of elements is known as the intersection graph basis problem ... thumb A graph with intersection number four. The four shaded regions indicate cliques that cover all the edges of the graph. An alternative definition of the intersection number of a graph math G is that it is the smallest number of clique graph theory cliques in math G complete graph complete Glossary of graph theory Subgraphs subgraphs of math G that together cover all of the edges of math ... number is also sometimes called the edge clique cover number . ref citation title Sphericity, cubicity ... number at most math m , for each edge forms a clique and these cliques together cover all the edges. ref citation title Schaum s outline of theory and problems of graph theory first V. K. last Balakrishnan ... that every graph with math n vertices has intersection number at most math n sup 2 sup 4 . More strongly ... number equals the number of edges. ref name r85 An even tighter bound is possible when the number of edges is strictly greater than math n sup 2 sup 4 . Let p be the number of pairs of vertices that are not connected ... math t t &minus 1 p t t 1 . Then the intersection number of math G is at most math p t . ref name ... number of any math n vertex graph math G is at most math 2 e sup 2 sup d 1 sup 2 sup ln n , where math e is the e mathematical constant base of the natural logarithm and d is the degree graph theory ...   more details



  1. Unsolved Problems in Number Theory

    Unsolved Problems in Number Theory may refer to Unsolved problems in mathematics in the field of number theory . A book with this title by Richard K. Guy published by Springer Verlag First edition 1981, 161 pages, ISBN 0 387 90593 6 Second edition 1994, 285 pages, ISBN 0 387 94289 0 Third edition 2004, 438 pages, ISBN 0 387 20860 7 ISBN 13 978 0387208602 Books with a similar title include Solved and Unsolved Problems in Number Theory , by Daniel Shanks First edition, 1962 Second edition, 1978 Third edition, 1985, ISBN 0 8284 1297 9 Fourth edition, 1993 Old and New Unsolved Problems in Plane Geometry and Number Theory , by Victor Klee and Stan Wagon , 1991, ISBN 0 88385 315 9. mathdab ...   more details



  1. Crossing number (graph theory)

    number of crossings among all drawings of this graph, so the graph has crossing number cr G     3. In graph theory , the crossing number cr G of a graph G is the lowest number of edge ... the rectilinear crossing number is Complete complexity complete for the existential theory ... planar if and only if its crossing number is zero. The concept originated in Tur n s brick factory problem , in which P l Tur n asked to determine the crossing number of the complete bipartite graph ... journal J. Graph Theory volume 1 pages 7 9 year 1977 ref harv ref History Kazimierz Zarankiewicz ... number of a complete bipartite graph K sub m,n sub . The conjecture that this inequality is actually an equality is now known as Zarankiewicz Crossing Number Conjecture. The problem of determining the crossing number of the complete graph was first posed by Anthony Hill artist Anthony Hill ... originated a conjectural upper bound for this crossing number, which Richard Guy published in 1960 .... ref name pnas Cite journal title The minimum number of intersections in complete graphs first1 T.L. ... few graph families. In particular, except for a few initial cases, the crossing number of complete ... that, among all graphs with chromatic number n , the complete graph K sub n sub has the minimum number of crossings. That is, if the Guy Saaty conjecture on the crossing number of the complete graph is valid, every n chromatic graph has crossing number at least equal to the formula in the conjecture ... ref Variations Without further qualification, the crossing number allows drawings in which the edges may be represented by arbitrary curves the rectilinear crossing number requires all edges to be straight line segments, and may differ from the crossing number. In particular, the rectilinear crossing number of a complete graph is essentially the same as the minimum number of convex quadrilaterals ... . ref Cite journal title The rectilinear crossing number of a complete graph and Sylvester s four ...   more details



  1. Multiplicative number theory

    Multiplicative number theory is a subfield of analytic number theory that deals with prime numbers and with factorization and divisors . The focus is usually on developing approximate formulas for counting these objects in various contexts. The prime number theorem is a key result in this subject. The Mathematics Subject Classification for multiplicative number theory is 11Nxx. Scope Multiplicative number theory deals primarily in asymptotic estimates for arithmetic functions . Historically the subject has been dominated by the prime number theorem , first by attempts to prove it and then by improvements in the error term. The Dirichlet divisor problem that estimates the average order of the divisor function d n and Gauss s circle problem that estimates the average order of the number of representations of a number as a sum of two squares are also classical problems, and again the focus is on improving ... The methods belong primarily to analytic number theory , but elementary methods, especially sieve ... of multiplicative number theory. The distribution of prime numbers is closely tied to the behavior ... a number theory viewpoint and a complex analysis viewpoint. Standard texts A large part of analytic number theory deals with multiplicative problems, and so most of its texts contain sections on multiplicative number theory. These are some well known texts that deal specifically with multiplicative problems cite book last Davenport first Harold authorlink Harold Davenport title Multiplicative Number Theory edition 3rd edition publisher Springer location Berlin year 2000 isbn 9780387950976 cite ... mathematician Robert C. Vaughan title Multiplicative Number Theory I. Classical Theory publisher Cambridge University Press location Cambridge year 2005 isbn 9780521849036 See also Additive number theory Category Analytic number theory ... that there are an infinity of primes in each co prime residue class, and the prime number theorem ...   more details



  1. Infrastructure (number theory)

    field based key exchange protocol. Public key cryptography and computational number theory Warsaw, 2000 ... The infrastructure of a real quadratic field and its applications. Proceedings of the Number Theory ... of a real quadratic number field in terms of circular groups . It was also described by R. Schoof ref name schoof infrastructure1 R. J. Schoof Quadratic fields and factorization. Computational methods in number theory, Part II, 235&ndash 286, Math. Centre Tracts, 155, Math. Centrum, Amsterdam ... fractions and number theoretic computations. Number theory Winnipeg, Man., 1983 . Rocky Mountain J ... Arakelov class groups. English summary Algorithmic number theory lattices, number fields, curves and cryptography ... cubic function fields of unit rank 1. English summary Algorithmic number theory Portland, OR, 1998 ... for divisors. English summary Algorithmic number theory, 342&ndash 356, Lecture Notes in Comput ... Number Theory Category Algebra Category Algebraic structures ... the infrastructure of a Quadratic field real quadratic number field and applied his baby step giant ... H. W. Lenstra Jr. On the calculation of regulators and class numbers of quadratic fields. Number theory days, 1980 Exeter, 1980 , 123&ndash 150, London Math. Soc. Lecture Note Ser., 56, Cambridge ... and B. K. Schmid to certain Cubic field cubic number fields of Dirichlet s unit theorem unit rank ... the regulator and class number of a pure cubic field. Math. Comp. 41 1983 , no. 163, 235&ndash 286. MR 701638 ref ref name williams dueck G. W. Dueck, H. C. Williams Computation of the class number ... ref and by J. Buchmann and H. C. Williams to all number fields of unit rank one. ref name buchmann ... class of an algebraic number field of unit rank one. Math. Comp. 50 1988 , no. 182, 569&ndash 579. MR ... algorithm to compute the regulator of a number field of arbitrary unit rank. ref name buchmann habil ... habil.pdf PDF ref The first description of infrastructures in number fields of arbitrary unit ...   more details



  1. Journal of Number Theory

    Infobox Journal cover Image JNumberTh.jpg 250px discipline Mathematics abbreviation J Num. Th. publisher Elsevier country United States USA history 1969 website http www.math.ohio state.edu JNT ISSN 0022 314X The Journal of Number Theory ISSN 0022 314X , often abbreviated J. Number Theory or J. Num. Th. in bibliographies, is a mathematics journal that publishes a broad spectrum of original research in number theory . The journal was founded in 1969 by R.P. Bambah, P. Roquette, Arnold Ross A. Ross , A. Woods, and Hans Julius Zassenhaus H. Zassenhaus , under the auspices of Ohio State University . It is currently published by Elsevier , with 12 issues and 6 volumes per year. The editor in chief is Ohio State professor David Goss . External links http www.sciencedirect.com science journal 0022314X The Journal of Number Theory . Official web site at Elsevier. http www.math.ohio state.edu JNT JNT editorial office at Ohio State University. Ohio State University media Category Number theory Category Mathematics journals Category Publications established in 1969 Category Ohio State University Category Elsevier academic journals math journal stub ru Journal of Number Theory ...   more details



  1. International Journal of Number Theory

    Infobox journal title International Journal of Number Theory cover discipline Mathematics abbreviation editor Bruce C. Berndt, Ramdorai Sujatha, Michel Waldschmidt publisher World Scientific country history 2005 present frequency 8 year impact 0.318 impact year 2009 website http www.worldscinet.com ijnt ijnt.shtml ISSN 1793 0421 eISSN 1793 7310 OCLC 62161796 The International Journal of Number Theory was established in 2005 and is published by World Scientific . It covers number theory , encompassing areas such as analytic number theory , diophantine equation s, and modular form s. Abstracting and indexing The journal is abstracted and indexed in Zentralblatt MATH , Mathematical Reviews , Science Citation Index Science Citation Index Expanded , and Current Contents Physical, Chemical and Earth Sciences. According to the Journal Citation Reports , the journal s 2009 impact factor is 0.318, ranking it 233rd out of 255 journals in the category Mathematics . External links Official http www.worldscinet.com ijnt ijnt.shtml Category Publications established in 2005 Category Mathematics journals Category World Scientific academic journals Category English language journals ...   more details



  1. Additive number theory

    In number theory , the specialty additive number theory studies subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of Abelian group s and commutative semigroup s with an operation of addition. Additive number theory has close ties to combinatorial number theory and the geometry of numbers . Two principal objects of study are the sumset of two subsets math A math and math B math of elements from an Abelian group math G math , math A B a b a in A, b in B math , and the h fold sumset of math A math , math hA underset h underbrace A cdots A . math There are two main subdivisions listed below. Additive number theory The first is principally devoted to consideration of direct problems over typically the integers, that is, to determining which elements can be represented as a summand from math hA math , where math A math ... the spectrum of mathematics, including combinatorics, ergodic theory , analysis , graph theory , group theory , and linear algebraic and polynomial methods. See also Shapley Folkman lemma Multiplicative number theory References cite book author Henry Mann authorlink Henry Mann title Addition Theorems The Addition Theorems of Group Theory and Number Theory publisher http www.krieger publishing.com ... book title Additive Number Theory the Classical Bases volume 164 series Graduate Texts in Mathematics ... Additive Number Theory Inverse Problems and the Geometry of Sumsets volume 165 series Graduate Texts ... title Additive Number Theory urlname AdditiveNumberTheory Category Additive number theory pt Teoria ... number primes and Waring s problem which asks how large must math h math be to guarantee that math ... number is the sum of three primes, and so every sufficiently large even integer is the sum of four ... number of k th powers. In general, a set A of nonnegative integers is called a basis of order h if math ... question to be considered is how small can the number of representations of math n math as a sum ...   more details



  1. Algebra & Number Theory

    Algebra & Number Theory ISSN 1937 0652 is a peer reviewed mathematics journal published by the nonprofit organization Mathematical Sciences Publishers . ref http msp.org Mathematical Sciences Publishers ref It was launched on January 17, 2007 with the goal of providing an alternative to the current range of commercial specialty journals in algebra and number theory , an alternative of higher quality and much lower cost. ref http listserv.nodak.edu cgi bin wa.exe?A2 ind0701&L nmbrthry&T 0&P 2999 Announcement email to the NMBRTHRY email list ref The journal publishes original research articles in algebra and number theory , interpreted broadly, including algebraic geometry and arithmetic geometry , for example. ref http msp.berkeley.edu ant about journal about.html About the journal at the ANT website ref Issues are published both online and in print. Editorial board The Managing Editor is Bjorn Poonen of Massachusetts Institute of Technology MIT , and the Editorial Board Chair is David Eisenbud of University of California at Berkeley U. C. Berkeley . ref http msp.berkeley.edu ant about journal editorial.html Editorial board at the ANT website ref See also Jonathan Pila References reflist External links http jant.org Algebra & Number Theory http www.mathscipub.org Mathematical Sciences Publishers Category Mathematics journals Category Publications established in 2007 Category Mathematical Sciences Publishers academic journals ...   more details



  1. Analytic number theory

    Refimprove date September 2008 In mathematics , analytic number theory is a branch of number theory that uses ... number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique. Multiplicative number theory deals with the distribution ... number theory is concerned with the additive structure of the integers, such as Goldbach s conjecture ... number theory is the solution to Waring s problem . Developments within analytic number theory ... theory has in return been greatly influenced by the value placed in analytic number theory on quantitative upper and lower bounds. Another recent development is probabilistic number theory sfn Tenenbaum 1995 p 267 , which uses tools from probability theory to estimate the distribution of number ... number theory The great theorems and results within analytic number theory tend not to be exact structural ... examples illustrate. Multiplicative number theory Euclid showed that there are an infinite ... result in analytic number theory. Loosely speaking, it states that given a large number N , the number ... applications of analytic techniques to number theory, Dirichlet proved that any arithmetic progression ... is prime for some positive even k less than  16. Additive number theory One of the most important problems in additive number theory is Waring s problem , which asks whether it is possible, for any ... number theory is obtaining specific upper bounds on the error term  E r . It was shown ... zeta function , Number theory for the millenium, II Urbana, IL, 2000 pp.275 290, A K Peters, Natick .... Methods of analytic number theory Dirichlet series One of the most useful tools in multiplicative number theory are Dirichlet series, which are functions of a complex variable defined by an infinite ... . This was the beginning of analytic number theory. ref Iwaniec & Kowalski Analytic Number Theory ... conjecture is known as the Riemann Hypothesis and has many deep implications in number theory in fact ...   more details




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