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	Julius Wihelm Richard Dedekind (Site not responding. Last check: 2007-10-08)
		Richard Dedekind attended school in Brunswick from the age of 7, and at this stage mathematics was not his main interest.
		Dedekind made a number of highly significant contributions to mathematics and his work would change the style of mathematics into what is familiar to us today.
		In the book Dedekind presented a logical theory of number and of complete induction, presented his principal conception of the essence of arithmetic, and dealt with the role of the complete system of real numbers in geometry in the problem of the continuity of space.
	www.stetson.edu /~efriedma/periodictable/html/Db.html (695 words)

	Richard Dedekind (Site not responding. Last check: 2007-10-08)
		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.
		Dedekind was among the first mathematicians who had accepted Cantor's work on the theory of infinite sets; other mathematicians didn't yet understand their ideas.
	www.encyclopedia-1.com /r/ri/richard_dedekind.html (884 words)

	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.
		He further notes that the completeness property, as he just phrased it, is deficient in the rationals, which motivates the definition of reals as cuts of rationals.
		This is version 23 of Dedekind cuts, born on 2002-05-16, modified 2004-02-15.
	planetmath.org /encyclopedia/DedekindCuts.html (594 words)

	20th WCP: The Model Theory Of Dedekind Algebras
		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.
		THEOREM 9: All uncountable homogeneous Dedekind algebras are quasi-characterizable.
	www.bu.edu /wcp/Papers/Logi/LogiWeav.htm (3006 words)

	Dedekind (Site not responding. Last check: 2007-10-08)
		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)

	Encyclopedia: Richard Dedekind (Site not responding. Last check: 2007-10-08)
		Julius Wilhelm Richard Dedekind (October 6, 1831 â“ February 12, 1916) was a German mathematician and Ernst Eduard Kummer's closest follower in arithmetic.
		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...
		Dedekind was among the first mathematicians who had accepted Cantor's work on the theory of infinite sets.
	www.nationmaster.com /encyclopedia/Richard-Dedekind (2467 words)

	Richard Dedekind -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		Dedekind was born in (A city in central Germany) Braunschweig (Brunswick) the youngest of four children of Julius Levin Ulrich Dedekind.
		Dedekind began teaching as Privatdozent in Göttingen and he gave courses on (A measure of how likely it is that some event will occur) probability and (The pure mathematics of points and lines and curves and surfaces) geometry.
		In the year 1874 he met (The official of a synagogue who conducts the liturgical part of the service and sings or chants the prayers intended to be performed as solos) Cantor in the Swiss city (A popular resort town in the Alps in west central Switzerland) Interlaken.
	www.absoluteastronomy.com /encyclopedia/R/Ri/Richard_Dedekind.htm (1315 words)

	Dedekind cut - Freepedia (Site not responding. Last check: 2007-10-08)
		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 closed upwards.
		For example it is shown that the typical Dedekind cut in the real numbers is either a pair with A the interval (-∞,a), in which case B must be [a,+∞); or a pair with A the interval (-∞,a], in which case B must be (a,+∞).
		The Dedekind cut is named after Richard Dedekind, who invented this construction in order to represent the real numbers as Dedekind cuts of the rational numbers.
	en.freepedia.org /Dedekind_completion.html (506 words)

	Was sind und was sollen die Zahlen?: Dedekind
		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.
		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.
		Dedekind notes that the observation of this property of infinite sets is not new, but using it as a definition is new.
	www.math.uwaterloo.ca /~snburris/htdocs/scav/dedek/dedek.html (1123 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)

	Richard Dedekind (Site not responding. Last check: 2007-10-08)
		Dedekind was born in Brunswick, the birthplace of Gauss, and received his degree under Gauss at Göttingen.
		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.
		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)

	Amazon.ca: Books: Essays on the Theory of Numbers (Site not responding. Last check: 2007-10-08)
		Richard Dedekind (1831-1916) is recognized as one of the great pioneers in the logical and philosophical analysis of the foundations of mathematics.
		Dedekind completed his doctoral studies under Gauss, was a friend of Cantor and Riemann, and worked under Dirichlet.
		Dedekind was not successful in imposing his terminology on later mathematicians.
	www.amazon.ca /exec/obidos/ASIN/0486210103 (893 words)

	Richard Julius Wilhelm Dedekind
		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.
		Dedekind's accomplishment was to define irrational numbers in terms of rationals.
	www.engr.iupui.edu /~orr/webpages/cpt120/mathbios/rdedek.htm (843 words)

	Dedekind, (Julius Wilhelm) Richard (Site not responding. Last check: 2007-10-08)
		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.
		Dedekind was born in Brunswick and studied at Göttingen.
		In 1858 he succeeded in producing a purely arithmetic definition of continuity and an exact formulation of the concept of the irrational number.
	www.cartage.org.lb /en/themes/Biographies/MainBiographies/D/Dedekind/1.html (152 words)

	Dedekind sums: a combinatorial-geometric viewpoint (Site not responding. Last check: 2007-10-08)
		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.
		Dedekind sums have enjoyed a resurgence of interest recently, from such diverse fields as topology, number theory, and combinatorial geometry.
		Using some simple generating functions, we show that generalized Dedekind sums are natural ingredients for such formulas.
	math.sfsu.edu /beck/papers/dedekind.html (187 words)

	CNN/SI - World Swimming - South Africa joins 50 free record hunters - Sunday August 29, 1999 02:37 PM (Site not responding. Last check: 2007-10-08)
		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.
		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.
	sportsillustrated.cnn.com /more/swimming/news/1999/08/29/panpacs_sprint_ap (505 words)

	Program Files\Netscape\Communicator\Program\dedekind
		Dedekind was then qualified as a university teacher and he began teaching at Göttingin giving courses on probability and geometry.
		Dedekind still learned courses on mathematics throughout this time by attending courses on abelian functions and elliptic functions.
		At this time Dedekind was thinking about how to teach differential and integral calculus when the thought of a 'Dedekind cut' came to him.
	www.andrews.edu /~calkins/math/biograph/199899/biodedek.htm (1430 words)

	PlanetMath: Dedekind domain
		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
		This is version 12 of Dedekind domain, born on 2002-04-19, modified 2004-03-18.
	planetmath.org /encyclopedia/DedekindDomain.html (168 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.
		At Umhlali, Dedekind had to take the humble route, and a most unconventional putting grip to win the tournament.
		According to Dedekind, he simply could not hole a putt on the front nine, despite some excellent birdie chances.
	www.themercury.co.za /index.php?fSectionId=286&fArticleId=2438214 (579 words)

	Generating functions and generalized Dedekind sums - Gessel (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		Abstract: We study sums of the form R(#), where R is a rational function and the sum is over all nth roots of unity # (often with # = 1 excluded).
		We call these generalized Dedekind sums, since the most well-known sums of this form are Dedekind sums.
		We note that one does not have to think of p fa;bg (n) as the lattice point count of a polytope to understand the proof of the...
	citeseer.ist.psu.edu /gessel96generating.html (498 words)

	Program Files\Netscape\Communicator\Program\dedexxx
		William Dedekind was born on October 6, 1831.
		Dedekind develoed the idea that both rational and irrational numbers could form a continuum(with no gaps) of real numbers, provided that the real numbers have a one-one relationship with points on a line.
		Dedekinds Cuts were developed as a way to prove a number rational or irrational.
	www.andrews.edu /~calkins/math/biograph/biodedek.htm (1253 words)

	[No title]
		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.
		The point is, Cohen's original 2nd model puts so many symmetries among the a_i,j1 and a_i,j2 's for example that any map from omega onto the Ui members is not symmetric and is left out of the final inner symmetric ZF model.
	www.math.niu.edu /~rusin/known-math/99/cohen_models (1570 words)

	Porträt - Dedekind (Site not responding. Last check: 2007-10-08)
		Dedekind und Weierstraß nahmen die Diskussion über die irrationalen Zahlen und die Kontinuität neu auf.
		Dedekind war einer des wesentlichen Wegbereiter der modernen strukturellen Auffassung der Algebra und algebraischen Zahlentheorie, die er von Grund auf erneuerte.
		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)

	Physics Help and Math Help - Physics Forums - Dedekind cuts
		I have no idea how Dedekind cuts work even after reading over the defn of it 3 or 4 times.
		One very simple example of a Dedekind cut is the set of all negative rational numbers.
		The set of all {-s} where s is in the set of all negative rational numbers is simply the set of all positive integers.
	www.physicsforums.com /printthread.php?t=43792 (592 words)

	Dedekind, Richard -- Encyclopædia Britannica
		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.
		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.
	www.britannica.com /eb/article-9029718 (836 words)

	Dedekind cuts (Site not responding. Last check: 2007-10-08)
		The first construction of the Real numbers from the Rationals is due to the German mathematician Richard Dedekind (1831 - 1916).
		Richard Dedekind, along with Bernhard Riemann was the last research student of Gauss.
		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)

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