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Topic: Leopold Kronecker

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	Kronecker. Who is Kronecker? What is Kronecker? Where is Kronecker? Definition of Kronecker. Meaning of Kronecker.
		Leopold Kronecker (1823-1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on "whole numbers", saying, "God made the natural numbers; all else is the work of man" (Bell 1986, p.
		Kronecker was a student and lifelong friend of Ernst Kummer.
		Kronecker wrote his 1845 dissertation, at the University of Berlin, on number theory, giving special formulation to units in certain algebraic number fields.
	www.knowledgerush.com /kr/encyclopedia/Kronecker (236 words)

	lkronecker (Site not responding. Last check: 2007-11-05)
		Kronecker entered the University of Berlin in 1841 and completed his Ph.D dissertation in 1845 on the units in a certain ring.
		Kronecker was one of the early advocates of the abstract approach to algebra.
		Kronecker died on December 29, 1891, at the age of 68.
	www.geocities.com /cheetofrito4_16/lkronecker.html (292 words)

	PS Wiki Encyclopedia (Site not responding. Last check: 2007-11-05)
		Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on "whole numbers", saying, "God made the integers; all else is the work of man" (Bell 1986, p.
		In his 1853 memoir on the algebraic solvability of equations, Kronecker extended the work of Ã‰variste Galois on the theory of equations.
		In analysis, Kronecker rejected the formulation of a continuous, nowhere differentiable function by his colleague, Karl Weierstrass.
	70.84.119.226 /~puresear/PSWiki/index.php?title=Leopold_Kronecker (245 words)

	Kronecker
		Kronecker's primary contributions were in the theory of equations.
		Kronecker was taught mathematics at school by Kummer and it was due to him that Kronecker became interested in mathematics.
		Kronecker was not to become a professor until Kummer retired in 1883.
	library.wolfram.com /examples/quintic/people/Kronecker.html (374 words)

	Kronecker (Site not responding. Last check: 2007-11-05)
		Kronecker was taught mathematics at Liegnitz Gymnasium by Kummer, and it was due to Kummer that Kronecker became interested in mathematics.
		We have already indicated that Kronecker's primary contributions were in the theory of equations and higher algebra, with his major contributions in elliptic functions, the theory of algebraic equations, and the theory of algebraic numbers.
		So Kronecker was consistent in his arguments and his beliefs, but many mathematicians, proud of their hard earned results, felt that Kronecker was attempting to change the course of mathematics and write their line of research out of future developments.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Kronecker.html (2082 words)

	Biography of Kronecker
		Leopold Kronecker (1823-1891) was born on December 7, 1823, in Liegnitz, Germany.
		Kronecker was taught mathematics at the Liegnitz Gymnasium by Kummer, and it was due to him that Kronecker became interested in mathematics.
		Kronecker's primary contributions were in the theory of equations and higher algebra, with his major contributions in elliptic functions, the theory of algebraic equations, and the theory of algebraic numbers.
	www.andrews.edu /~calkins/math/biograph/biokrone.htm (1213 words)

	Kronecker, Leopold (Site not responding. Last check: 2007-11-05)
		Kronecker was born at Liegnitz (now Legnica, Poland), and studied at Berlin, Bonn, and Breslau.
		Kronecker was obsessed with the idea that all branches of mathematics (apart from geometry and mechanics) should be treated as parts of arithmetic.
		The Kronecker delta is denoted by rs, for which rs = 0 when r s and rs = 1 when r = s, where r, s are 1, 2, 3,...
	www.cartage.org.lb /en/themes/Biographies/MainBiographies/k/Kronecker/1.html (167 words)

	Kronecker.htm
		Leopold Kronecker was born to Isidor Kronecker, a successful business man, and Johanna Prausnitzer, who also came from a wealthy family.
		Kronecker believed that mathematics should deal only with finite numbers and with a finite number of operations.
		It appears that, from the early 1870s, Kronecker was opposed to the use of irrational numbers, upper and lower limits, and the Bolzano-Weierstrass theorem, because of their non-constructive nature.
	www.cse.ohio-state.edu /~brinkmei/math/Kronecker.htm (228 words)

	Kronecker delta: (Site not responding. Last check: 2007-11-05)
		In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.
		The Kronecker delta is used in many areas of mathematics.
		This is a more accurate way to notate the identity matrix, considered as a linear mapping.
	wikipedia.openfun.org /wiki/Kronecker_delta (188 words)

	Search Results for Kronecker
		Kummer proposed Kronecker for election to the Berlin Academy in 1860, and the proposal was seconded by Borchardt and Weierstrass.
		Kronecker explained his programme based on studying only mathematical objects which could be constructed with a finite number of operation from the integers in Uber den Zahlbergriff in 1887.
		Kronecker never addressed the meeting, however, since his wife was seriously injured in a climbing accident in the late summer and died shortly afterwards.
	www-history.mcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=Kronecker&CONTEXT=1 (3707 words)

	Leopold Kronecker Biography / Biography of Leopold Kronecker World of Mathematics Biography
		Kronecker was born into a well-to-do family, and through inheritance and business interests, he remained independently wealthy throughout his life.
		Kronecker was extremely vocal in his views on number theory and what he perceived to be the invalidity of the use of anything other than finite numbers in number theory.
		Upon Kummer's retirement in 1883, Kronecker was appointed to the chair of mathematics in Berlin, his first official teaching position, and to the position of codirector of Berlin's mathematical seminar.
	www.bookrags.com /biography-leopold-kronecker-wom (570 words)

	The Leopold Language (Site not responding. Last check: 2007-11-05)
		Leopold is a simple programming language, with syntax loosely based on C.
		When a Leopold program is called from the command line, the OS will initialize this value parameter from a single integer that is part of the command line.
		The word print is a Leopold keyword; the expression may be any arithmetic expression; the semi-colon at the end is required.
	www.cs.colorado.edu /~main/theory/leo/leo.html (1352 words)

	The Hutchinson Dictionary of Scientific Biography: Kronecker, Leopold (1823-1891)@ HighBeam Research (Site not responding. Last check: 2007-11-05)
		German mathematician, skilled in many branches of the subject but pre-eminent in none, who is remembered chiefly for the 'Kronecker delta'.
		Kronecker was born in Liegnitz (now Legnica, Poland) on 7 December 1823 and attended secondary school there, where he was taught by the outstanding mathematician Ernst Kummer.
		Kummer remained a lifelong friend and exercised great influence upon Kronecker, especially in interesting him in number theory.
	www.highbeam.com /library/doc0.asp?DOCID=1P1:99916670&refid=ip_encyclopedia_hf (170 words)

	Leopold I -- Encyclopædia Britannica
		The Hungarian violinist Leopold Auer was especially renowned as a teacher.
		A year later his father, Leopold III, abdicated after an unsettled reign, and it was Baudouin's task to restore confidence in the monarchy (see Leopold, Kings of Belgium).
		The son of a German prince, Leopold married the heiress to the British throne, Princess Charlotte, but she died a year later.
	www.britannica.com /eb/article-9047847 (746 words)

	Kronecker, Leopold -- Encyclopædia Britannica
		Kronecker acquired a passion for number theory from Ernst Kummer, his instructor in mathematics at the Liegnitz Gymnasium, and earned his doctor's degree at the University of Berlin with…;
		Known to the general public through his three motion pictures, including Walt Disney's ‘Fantasia' (1940), Leopold Stokowski was a conductor famous for his flamboyant showmanship and the rich sonorities he evoked from an orchestra.
		The leading 19th-century German historian, Leopold von Ranke was a founder of the modern school of history—a champion of objectivity based on source materials rather than on legend and tradition.
	www.britannica.com /eb/article-9046275?&query=weierstrass, (761 words)

	Kronecker delta - Wikipedia, the free encyclopedia
		This function takes the value 1 if and only if all the upper indices match the corresponding lower one, and the value zero otherwise.
		Kronecker Delta is also a German Lager brewed in the UK.
		This page was last modified 11:55, 26 September 2005.
	en.wikipedia.org /wiki/Kronecker_delta (212 words)

	Finitism - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-05)
		In the philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps.
		(Most constructivists, in contrast, allow a countably infinite number of steps.) The most famous proponent of finitism was Leopold Kronecker, who said:
		Even stronger than finitism is ultrafinitism (also known as ultraintuitionism), associated primarily with Alexander Esenin-Volpin.
	www.encyclopedia-online.info /Finitism (129 words)

	Leopold - Prince Leopold, Founder of Spiritual Analysis (Site not responding. Last check: 2007-11-05)
		Biography of Leopold Löwenheim (1878-1957) Leopold Löwenheim's father was a mathematics teacher and he revised and edited his father's unfinished work
		Biography of Leopold Kronecker (1823-1891) Leopold Kronecker's parents were well off, his father, Isidor Kronecker, being a successful business man
		Nathan Leopold and Richard Loeb, two very rich and genius-level young men, cold bloodedly planned the perfect murder of a child for the thrill of getting away
	www.beedb.com /bed/leopold.html (271 words)

	Kronecker delta - Wikpedia (Site not responding. Last check: 2007-11-05)
		The Kronecker delta has the so-called sifting property that for $j\in\mathbb Z :$
		For example, in linear algebra, the identity matrix can be written as $\delta_{ij}\, while if it is considered as a tensor, the$ Kronecker tensor, it can be written $\delta^j_i with a contravariant index j.$
		This function takes the value 1 if and only if all the upper indices match the corresponding lower one, taking the value zero otherwise.
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Kronecker_delta (237 words)

	Re: Leopold Kronecker (Site not responding. Last check: 2007-11-05)
		In Reply to: Leopold Kronecker, for Lee posted by DickT on February 01, 2003 at 13:39:58:
		Kronecker objected to the idea of non-contructive existence theorems.
		There have been some mathematicians since then who agree with him.
	superstringtheory.com /forum/geomboard/messages3/78.html (150 words)

	leopold_kronecker - OneLook Dictionary Search
		We found 6 dictionaries with English definitions that include the word leopold kronecker:
		Tip: Click on the first link on a line below to go directly to a page where "leopold kronecker" is defined.
		Kronecker, Leopold : Columbia Encyclopedia, Six Edition [home, info]
	www.onelook.com /?w=leopold_kronecker&loc=resrd (88 words)

	The Mathematics Genealogy Project - Leopold Kronecker
		Click here to see the students listed in chronological order.
		According to our current on-line database, Leopold Kronecker has 10 students and 1485 descendants.
		If you have additional information or corrections regarding this mathematician, please use the update form.
	genealogy.math.ndsu.nodak.edu /html/id.phtml?id=17982 (99 words)

	Leopold Kronecker History Summary (Site not responding. Last check: 2007-11-05)
		death · theory · love · mathematician ·; mathematics · journal · wealthy · refused · colleagues · infinite number · irrational numbers · leopold · elliptic functions ·; transcendental numbers · kronecker
		Independently wealthy, Kronecker did not have to work and pursued mathematics for the love of the field.
		Although he made significant contributions to mathematical research, he fell out of favor with his colleagues when he refused to accept infinite numbers, transcendental numbers, irrational numbers, and other similar concepts.
	www.bookrags.com /history/sciencehistory/leopold-kronecker-scit-05123.html (114 words)

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