Dedekind section - Factbites

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Topic: Dedekind section

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	PlanetMath: Dedekind cuts
		Dedekind defines a point to produce the division of the real line if this point is either the least or greatest element of either one of the classes mentioned above.
		This is version 23 of Dedekind cuts, born on 2002-05-16, modified 2004-02-15.
		The Dedekind completeness property of real numbers, expressed as the supremum property, now becomes straightforward to prove.
	planetmath.org /encyclopedia/DedekindCuts.html (594 words)

	Encyclopedia: Richard Dedekind
		Dedekind was born in Braunschweig (Brunswick) the youngest of four children of Julius Levin Ulrich Dedekind.
		Dedekind received his doctorate in 1852 and he was Gauss's last student.
		In mathematics, a Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for any element x in S, if a is in A and x ≤ a, then x is in A as well) and B is...
	www.nationmaster.com /encyclopedia/Richard-Dedekind (2467 words)

	Dedekind
		Dedekind and Dirichlet soon became close friends and the relationship was in many ways the making of Dedekind, whose mathematical interests took a new lease of life with the discussions between the two.
		Dedekind's work was quickly accepted, partly because of the clarity with which he presented his ideas and partly since Heinrich Weber lectured to Hilbert on these topics at the University of Königsberg.
		Dedekind's brilliance consisted not only of the theorems and concepts that he studied but, because of his ability to formulate and express his ideas so clearly, he introduced a new style of mathematics that been a major influence on mathematicians ever since.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Dedekind.html (1962 words)

	20th WCP: The Model Theory Of Dedekind Algebras
		THEOREM 9: All uncountable homogeneous Dedekind algebras are quasi-characterizable.
		Each Dedekind algebra is associated with a cardinal value function called the confirmation signature which counts the number of configurations in each isomorphism type occurring in the decomposition of the algebra.
		A subalgebra, A, of the Dedekind algebra B is a small subalgebra of B provided the cardinality of the domain of A is strictly smaller than the cardinality of the domain of B. Let B be an infinite Dedekind algebra.
	www.bu.edu /wcp/Papers/Logi/LogiWeav.htm (3006 words)

	Richard Dedekind
		Dedekind's construction of the real numbers using `Dedekind cuts' was part of the effort of Dedekind, Cantor, and Weierstrass, and others to bring a rigor to analysis; earlier attempts such as those by Cauchy and Bolzano were hindered by a lack of understanding of irrational numbers.
		Dedekind was born in Brunswick, the birthplace of Gauss, and received his degree under Gauss at Göttingen.
		These contributions, called part of the aritmetization of analysis, illustrate Dedekind's arithmetic and algebraic viewpoint; as a professor he was probably the first to give lectures on Galois theory.
	www.mthcsc.wfu.edu /~kuz/Stamps/Dedekind/Dedekind.html (213 words)

	Was sind und was sollen die Zahlen?: Dedekind
		Dedekind notes that the observation of this property of infinite sets is not new, but using it as a definition is new.
		Dedekind now proceeds to give a rigorous treatment of the natural numbers, and this will be far more exacting than his cursory remarks of 1872 indicated.
		By November of 1858 Dedekind had resolved the issue by showing how to obtain the real numbers (along with their ordering and arithmetical operations) from the rational numbers by means of cuts in the rationals -- for then he could prove the above mentioned least upper bound property from simple facts about the rational numbers.
	www.math.uwaterloo.ca /~snburris/htdocs/scav/dedek/dedek.html (1123 words)

	Dedekind, (Julius Wilhelm) Richard
		Dedekind was born in Brunswick and studied at Göttingen.
		In 1872 he introduced the Dedekind cut (which divides a line of infinite length representing all real numbers) to define irrational numbers in terms of pairs of sequences of rational numbers.
		This led to the Dedekind cut, explained in his book Stetigkeit und irrationale Zahlen 1872.
	www.cartage.org.lb /en/themes/Biographies/MainBiographies/D/Dedekind/1.html (152 words)

	AllRefer.com - Julius Wilhelm Richard Dedekind (Mathematics, Biography) - Encyclopedia
		Dedekind studied at GOttingen under the German mathematician Carl Gauss and in 1852 received his doctorate there for a thesis on Eulerian integrals.
		Dedekind led the effort to formulate rigorous definitions of basic mathematical concepts.
		Perhaps his best-known contribution is the "Dedekind cut," whereby real numbers can be defined in terms of rational numbers.
	reference.allrefer.com /encyclopedia/D/Dedekind.html (222 words)

	PlanetMath: Dedekind domain
		This is version 12 of Dedekind domain, born on 2002-04-19, modified 2004-03-18.
		It is worth noting that the second clause above implies that the maximal length of a strictly increasing chain of prime ideals is 1, so the Krull dimension of any Dedekind domain is 1.
		In particular, the affine ring of an algebraic set is a Dedekind domain if and only if the set is
	planetmath.org /encyclopedia/DedekindDomain.html (168 words)

	PlanetMath: ideals in a Dedekind domain
		This is version 5 of ideals in a Dedekind domain, born on 2002-07-03, modified 2004-04-14.
		"ideals in a Dedekind domain" is owned by saforres.
	planetmath.org /encyclopedia/EveryIdealInADedekindDomainIsAFactorOfAPrincipalIdeal.html (67 words)

	Amazon.ca: Books: Essays on the Theory of Numbers
		Richard Dedekind (1831-1916) is recognized as one of the great pioneers in the logical and philosophical analysis of the foundations of mathematics.
		Dedekind was not successful in imposing his terminology on later mathematicians.
		Dedekind completed his doctoral studies under Gauss, was a friend of Cantor and Riemann, and worked under Dirichlet.
	www.amazon.ca /exec/obidos/ASIN/0486210103 (893 words)

	Richard Julius Wilhelm Dedekind
		Dedekind's accomplishment was to define irrational numbers in terms of rationals.
		Richard Dedekind was a German mathematician who was born in 1831 in Brunswick.
		Dedekind made many original and important contributions to the theory of algebraic numbers.
	www.engr.iupui.edu /~orr/webpages/cpt120/mathbios/rdedek.htm (843 words)

	Dedekind's Problem
		Dedekind's Problem is to determine the number M(n) of distinct monotone functions of n variables.
		Although Dedekind first considered this question in 1897, there is still no concise closed-form expression for M(n).
		Postscript: For the answer, see Progress on Dedekind's Problem.
	www.mathpages.com /home/kmath030.htm (336 words)

	Dedekind Sums: A Combinatorial-Geometric Viewpoint - Beck, Robins (ResearchIndex)
		In this expository paper we show that there is a common thread to many generalizations of Dedekind sums, namely through the study of lattice point enumeration of rational polytopes.
		In particular, there are some natural finite Fourier series which we call Fourier-Dedekind sums, and which form the building blocks of the number of partitions of an integer from a finite set of positive integers.
		Abstract: The literature on Dedekind sums is vast.
	citeseer.ist.psu.edu /292212.html (474 words)

	CNN/SI - World Swimming - South Africa joins 50 free record hunters - Sunday August 29, 1999 02:37 PM
		Dedekind said he was inspired by fellow South African Penny Heyns, who has set eight world records in the past six weeks, including four at the PanPacs in the last eight days.
		Dedekind, who studies in the United States, had his fare to Australia paid by South African Swimming, after it initially said it could not afford to support sending its swimmers to the meet.
		Dedekind's best performance lifted him into tied fifth on the alltime performers list, just behind compatriot Roland Schoeman who set a surprising 22.04 during the B final at this month's U.S. championships.
	sportsillustrated.cnn.com /more/swimming/news/1999/08/29/panpacs_sprint_ap (505 words)

	Dedekind Cuts
		A dedekind cut comprises two nonempty sets of rationals, l and r, such that each rational appears in exactly one of the two sets, and all the rationals in l (left) are less than all the rationals in r (right).
		Dedekind proved all sorts of nice properties, so that at the end of the day, the cuts form a field, namely the field of real numbers, with the rationals as a dense subfield.
		A dedekind cut is a natural way to describe sqrt(2).
	www.mathreference.com /top-ms,dcuts.html (663 words)

	Mercury - Around the greens - Dedekind pulls off North Coast win
		Dedekind, who is Royal Durban's league captain, has set his sights on making the KZN team in 2006.
		According to Dedekind, he simply could not hole a putt on the front nine, despite some excellent birdie chances.
		At Umhlali, Dedekind had to take the humble route, and a most unconventional putting grip to win the tournament.
	www.themercury.co.za /index.php?fSectionId=286&fArticleId=2438214 (579 words)

	cohen_models
		Dedekind proposed a set be defined to be infinite if it is bijective with a proper subset of itself and defined to be finite otherwise.
		So ZFC proves there are no Dedekind sets, and so far Dedekind sets are not yet ruled out in ZF from what I have written so far.
		In ZF a proof by induction shows inductive finite -> Dedekind finite.
	www.math.niu.edu /~rusin/known-math/99/cohen_models (1570 words)

	Dedekind, Richard
		Dedekind's study of Dirichlet's work led to his own study of algebraic number fields, as well as his introduction of ideals.
		12, 1916, was a German mathematician known for his study of CONTINUITY and definition of the real numbers in terms of Dedekind "cuts"; his analysis of the nature of number and mathematical induction, including the definition of finite and infinite sets; and his influential work in NUMBER THEORY, particularly in algebraic number fields.
		Dedekind also introduced such fundamental concepts as RINGS.
	euler.ciens.ucv.ve /English/mathematics/dedekind.html (121 words)

	Dedekind cuts
		Richard Dedekind, along with Bernhard Riemann was the last research student of Gauss.
		The first construction of the Real numbers from the Rationals is due to the German mathematician Richard Dedekind (1831 - 1916).
		His arithmetisation of analysis was his most important contribution to mathematics, but was not enthusiastically received by leading mathematicians of his day, notably Kronecker and Weierstrass.
	turnbull.mcs.st-and.ac.uk /~john/analysis/Lectures/A3.html (364 words)

	Dedekind, Richard -- Encyclopædia Britannica
		In work originating from discussions on the foundations of the infinitesimal and derivative calculus by Baron Augustin-Louis Cauchy and Karl Weierstrauss, Cantor and Richard Dedekind developed methods of dealing with the large, and in...
		Galois's work was both the culmination of a main line of algebra—solving equations by radical methods—and the beginning of a new line—the study of abstract structures.
		Although not fully recognized in his lifetime, his treatment of the ideas of the infinite and of what constitutes a real number continues to influence modern mathematics.
	www.britannica.com /eb/article-9029718 (836 words)

	Porträt - Dedekind
		Dedekind war einer des wesentlichen Wegbereiter der modernen strukturellen Auffassung der Algebra und algebraischen Zahlentheorie, die er von Grund auf erneuerte.
		Dedekind und Weierstraß nahmen die Diskussion über die irrationalen Zahlen und die Kontinuität neu auf.
		Dedekind überwand seine Schwierigkeiten in der Theorie der algebraischen Zahlen, indem er zum Unendlichen Zuflucht suchte; Kronecker suchte seine Schwierigkeiten im Bereich des Endlichen zu lösen.
	www.zahlenjagd.at /dedekind.html (224 words)

	Dedekind cuts - Physics Help and Math Help - Physics Forums
		One very simple example of a Dedekind cut is the set of all negative rational numbers.
		The definition of "Dedekind cut" is usually given in 3 parts.
		so we know that the negative of a dedekind cut X should be the solution of the equation X+Y = 0.
	www.physicsforums.com /showthread.php?threadid=43792 (668 words)

	What are the 'real numbers,' really?
		By a Dedekind cut we mean a pair of sets (A,B), where A and B are nonempty subsets of Q, A is the set of all lower bounds of B (in Q), and B is the set of all upper bounds of A (in Q).
		In particular, the decimal expansions, the Dedekind cuts, and the equivalence classes of Cauchy sequences, though they appear to be entirely different, all turn out to have the same arithmetic and algebraic structure -- they are really the "same" object.
		Let Q be the set of rational numbers; we assume that we already have a good understanding of those.
	www.cartage.org.lb /en/themes/Sciences/Mathematics/calculus/realnumbers/complete/complete.htm (673 words)

	Orðasafn: D
		Dedekind continuity axiom samfellufrumsenda Dedekinds, samfellufrumsetning Dedekinds, samfelldnifrumsenda Dedekinds, samfelldnifrumsetning Dedekinds.
	www.hi.is /~mmh/ord/safn/safnD.html (2064 words)

	sport.iafrica.com olympics news
		Dedekind swam his semifinal in 22,39, the ninth fastest time overall, and Schoeman was just behind in 22,41, the tenth fastest.
		There was disappointment for South Africa in the pool this morning as both Brendon Dedekind and Roland Schoeman both narrowly failed to qualify for the final of the men’s 50m freestyle final.
		American Gary Hall was the top qualifier with a time of 22,07 with Pieter van den Hoggenband second in 22,11.
	sport.iafrica.com /olympics/news/73117.htm (124 words)

	Richard Dedekind
		The definition of real numbers by ‘ Dedekind section ’ as his method is called, although accepted by mathematicians and used as the very foundation of the modern theory of functions, had to meet serious criticism which subsequently led to attempts at improvement by the logistic philosophers.
		The results of Dedekind, Frege, and Peano had covered in conjunction the whole filed of elementary pure mathematics,
		While Frege had given a philosophic analysis of the concept of number, the Italian mathematician Peano and his school (Formulaire de Mathématiques, 1895-1905), in the course of extensive researches in symbolic logic, had shown that all propositions concerning the natural numbers which are required in mathematics can be deduced from a set of five axioms.
	www.geocities.com /paultabaka/dedekind.html (866 words)

	Construction de Dedekind
		Dedekind veut donc compléter l'ensemble des nombres rationnels en lui ajoutant des éléments de telle sorte que le nouvel ensemble obtenu vérifie cette fameuse propriété, ceci tout en préservant l'ordre, les opérations et les structures déjà définies.
		Dedekind propose une construction de l'ensemble des nombres réels qui semble en apparence antinaturelle et pour certains inutilisable.
		D. On démontre alors facilement la propriété suivante qui était l'objectif de Dedekind.
	www.reunion.iufm.fr /recherche/irem/histoire/construction_de_dedekind.htm (426 words)

	Hunt South Africa with Dedekind Safaris
		Dedekind Safaris is well known for their outstanding nyala, plains game, buffalo and rhino hunting.
		Dedekind safaris is owned and operated by Mark and Telani Dedekind.
		Dedekind Safaris also offers hunting in Thabazimbi, a scenic 2 1/2 hours from Johannesburg airport.
	www.booktrail.com /huntsa (231 words)

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In the News (Sat 2 Jun 12)