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Topic: Kronecker delta

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Delta (letter)

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Fundamental theorem of calculus

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	PlanetMath: Kronecker delta
		It is a special case of the generalized Kronecker delta symbol.
		The delta symbol was first used in print by Kronecker in 1868[1].
		This is version 3 of Kronecker delta, born on 2002-01-04, modified 2006-10-28.
	planetmath.org /encyclopedia/KroneckerDelta.html (123 words)

	Leopold Kronecker - Wikipedia, the free encyclopedia
		Kronecker was a student and lifelong friend of Ernst Kummer.
		Kronecker wrote 1845 his dissertation, at the University of Berlin, on number theory, giving special formulation to units in certain algebraic number fields.
		Kronecker also contributed to the concept of continuity, reconstructing the form of irrational numbers in real numbers.
	en.wikipedia.org /wiki/Kronecker (271 words)

	Biography of Kronecker
		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.
		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.andrews.edu /~calkins/math/biograph/biokrone.htm (1213 words)

	Dirac delta function - Wikipedia, the free encyclopedia
		The Dirac delta function, sometimes referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere.
		The Dirac delta function was named after the Kronecker delta, since it can be used as a continuous analogue of the discrete Kronecker delta.
		The Dirac delta is not a function; but it can be usefully treated as a distribution, as well as a measure.
	en.wikipedia.org /wiki/Dirac_delta_function (911 words)

	The Dirac Delta Function (Site not responding. Last check: 2007-10-17)
		The Dirac delta function, invented by P.A.M. Dirac for his important formulations of quantum mechanics, is a continuous analog of the Kronecker delta.
		To appreciate the connection between the Kronecker delta and the delta function, it is important to recognize that subscripts are a kind of function.
		The Kronecker delta extracts a single element from an infinite sum, and the delta function extracts the value of a function at a single point from an integral.
	www.chm.uri.edu /urichm/chm532/delta/node4.html (558 words)

	Kronecker delta - Wikipedia, the free encyclopedia
		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.
		Kronecker Delta is also a German Lager brewed in the UK.
	en.wikipedia.org /wiki/Kronecker_delta (212 words)

	Kronecker, Leopold (Site not responding. Last check: 2007-10-17)
		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)

	PlanetMath: generalized Kronecker delta symbol
		we can write the generalized delta function as a determinant of traditional delta symbols.
		"generalized Kronecker delta symbol" is owned by matte.
		This is version 2 of generalized Kronecker delta symbol, born on 2003-03-22, modified 2003-03-23.
	planetmath.org /encyclopedia/GeneralizedKroneckerDeltaSymbol.html (147 words)

	Generalized Functions and Related Objects - Wolfram Mathematica
		Inserting a delta function in an integral effectively causes the integrand to be sampled at discrete points where the argument of the delta function vanishes.
		The indefinite integral of the delta function is the Heaviside theta function.
		Related to the multidimensional Dirac delta function are two integer functions: discrete delta and Kronecker delta.
	reference.wolfram.com /mathematica/tutorial/GeneralizedFunctionsAndRelatedObjects.html (348 words)

	MUG: Kronecker delta (4.11.95)
		Is it possible to define a Kronecker delta in a way that would allow Maple to compute infinite sums of expressions involving the delta?
		I think your "delta" procedure should also return 1 if the two arguments are equal, even if they aren't numeric.
		This just did not matter when I was investigating situations in which the arguments to the deltas always evaluated to specific integers in the end.
	www.math.rwth-aachen.de /mapleAnswers/html/66.html (516 words)

	ipedia.com: Dirac delta function Article (Site not responding. Last check: 2007-10-17)
		The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one.
		Technically speaking, the Dirac delta is not a function but a distribution — a mathematical expression that is well defined only when integrated.
		The Fourier transform of the Dirac delta is the constant function, and the convolution of δ with any distribution S yields S.
	www.ipedia.com /dirac_delta_function.html (534 words)

	Impulses (Site not responding. Last check: 2007-10-17)
		In this applet, the Kronecker delta function is delayed so that it is centered in the visible window.
		But unlike the Kronecker delta function, its value is infinite at zero.
		The Dirac delta function, of course, cannot be represented precisely, because its infinitely narrow width and infinite height are problematic for the computer.
	ptolemy.eecs.berkeley.edu /eecs20/week11/impulses.html (405 words)

	BME 456: Mathematical Preliminaries
		This is actually call the Kronecker delta and represents a second order tensor, that is equivalent to the identity matrix.
		The number of quantities, denoted as p, is equal to the dimension of the space in which we are working (d = 2 for 2D, d = 3 for 3D) raised to the power of the number of independent indices.
		Now, examining the above expression with the index notation, we see that essentially when the vector is summed with the Kronecker delta, the index that is repeated on the entity is replace with the free index on the Kronecker delta.
	www.engin.umich.edu /class/bme456/ch1mathprelim/bme456mathprelim.htm (5183 words)

	Kronecker delta -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-17)
		This property is similar to one of the main properties of the (Click link for more info and facts about Dirac delta function) Dirac delta function:
		For example, in (The part of algebra that deals with the theory of linear equations and linear transformation) linear algebra, the (A scalar matrix in which all of the diagonal elements are unity) identity matrix can be written as
		This is a more accurate way to notate the identity matrix, considered as a (Click link for more info and facts about linear mapping) linear mapping.
	www.absoluteastronomy.com /encyclopedia/k/kr/kronecker_delta.htm (265 words)

	Francisco Olivera / Dissertation / Appendix B: Mathematical Proofs
		In this Appendix, the mathematical proofs concerning whether or not the convolution of two probability density functions (p.d.f.'s) of a certain type results in a p.d.f.
		The proofs are applied to the following four statistical distributions: (1) Kronecker delta which represents the pure translation flow model, (2) exponential which represents the linear reservoir flow model, (3) gamma which represents the cascade of linear reservoirs flow model, and (4) first passage times which represents the diffusion wave flow model.
		The Kronecker delta distribution represents the pure translation flow model.
	www.crwr.utexas.edu /gis/gishyd98/library/olivera/appb/app_b.htm (362 words)

	The Mathematica Book Online: Advanced Mathematics in Mathematica \| Calculus
		The unit step function UnitStep[x] is effectively the indefinite integral of the delta function.
		The unit step function is often used in setting up piecewise continuous functions, and in representing signals and other quantities that become non-zero only beyond some point.
		The indefinite integral of the delta function is the unit step function.
	documents.wolfram.com /mathematica/book/section-3.5.13 (394 words)

	Kronecker Delta Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-10-17)
		Looking For kronecker delta - Find kronecker delta and more at Lycos Search.
		Find kronecker delta - Your relevant result is a click away!
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	encyclopedia.localcolorart.com /encyclopedia/Kronecker_delta (357 words)

	Differentiation with respect to N-arrays
		The Type II Kronecker delta array is involved in all differentiation where
		As shown in the previous section discussing the properties of the Kronecker deltas, the Type II Kronecker delta can be written as an array outer product of identity matrices.
		This is illustrated in Figure 23 for the six-mode differentiation symbol.
	pcf1.chembio.ntnu.no /~bka/research/diagram/node19.html (293 words)

	[No title] (Site not responding. Last check: 2007-10-17)
		Now I have put in the delta functions for both the independence of correlations in the components of the field (the Kronecker delta) and the lack of correlations in the time direction (which you need for the fluctuation-dissipation theorem).
		In fact, the units of a Dirac delta function are those of the reciprocal of the units of the arguement.
		One easy way to see this is to think about a volume integral over a delta function with vector argument, the result is unitless, therefore the Dirac delta function has units of 1/Volume.
	www.csit.fsu.edu /~novotny/micro/microx (341 words)

	Lecture 5 --- 6.837 Fall '01 (Site not responding. Last check: 2007-10-17)
		The most common (but not the only) way to generate the table values necessary to represent our function is to multiply the function by a sampling grid.
		A sampling grid is composed of periodically spaced Kronecker delta functions.
		The definiton of the 2-D Kronecker delta is:
	graphics.lcs.mit.edu /classes/6.837/F02/lectures/Lecture20/Slide05.html (46 words)

	N-mode Kronecker delta definitions (Site not responding. Last check: 2007-10-17)
		This Kronecker delta is used for summing of an index involving
		This Kronecker delta is used in array differentiation; here symbolized as the array
		which means that both are equal to the identity matrix and it is arbitrary which Kronecker delta is used.
	pcf1.chembio.ntnu.no /~bka/research/diagram/node7.html (177 words)

	Tensors and tensor algebra (Site not responding. Last check: 2007-10-17)
		The delta function has the same properties independent of any rotation.
		Note that it is not a scalar because a scalar is a single number whereas the delta function has several elements.
		Clearly the sum or difference of two tensors of the same rank is also a tensor, and similarly if one multiplies all elements of a tensor by a scalar it is still a tensor.
	astron.berkeley.edu /~jrg/ay202/node185.html (790 words)

	The Hutchinson Dictionary of Scientific Biography: Kronecker, Leopold (1823-1891)@ HighBeam Research (Site not responding. Last check: 2007-10-17)
		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.
	highbeam.com /library/doc0.asp?DOCID=1P1:99916670&... (170 words)

	Bra-ket notation - Conservapedia
		The kets may also represent a continuous set of states.
		In such circumstances (which would include, for example, a free particle), Dirac also found it necessary to develop what is known as the Dirac Delta Function as an analogue to the Kronecker Delta Function.
		This relation should be thought of as a strictly canonical for use in integration, since writing the Dirac Delta Function in the absence of an integral is somewhat dubious.
	www.conservapedia.com /Dirac_Notation (535 words)

	[No title]
		It includes routines for linear coordinate transforms, handling delta functions, moving the integration to the inside or outside of an expression and evaluating the integral for some or all of the coordinates.
		g[- + y, - - y], {x, y},..] 2 2 ipolyexpand moves ipoly to the inside of an expression as much as possible, removing delta functions.
		{j, 1, k}, {k, 1, 5}] ipolyexpand moves ipoly to the inside of an expression as much as possible, removing Kronecker delta functions.
	library.wolfram.com /infocenter/MathSource/3652/AboutFunctions.txt (475 words)

	Faith - Blogs
		I haven't been doing much of anything except work, and deadline is next Saturday night so you got another week of blessed silence from me approaching.
		The only console hooked up at home is a slim PS2 that the kids got for Christmas last year.
		05-21-2007, 10:54 PM by Ben to The Kronecker Delta
	edgecase.net /devsite/blogs (1118 words)

	Definition of delta - Merriam-Webster Online Dictionary
		Middle English deltha, from Greek delta, of Semitic origin; akin to Hebrew dāleth daleth
		1: the 4th letter of the Greek alphabet — see alphabet table2: something shaped like a capital Greek delta; especially : the alluvial deposit at the mouth of a river3: an increment of a variable —symbol Δ4: delta wave
		See a map of "delta" in the Visual Thesaurus
	www.m-w.com /cgi-bin/dictionary?book=Dictionary&va=delta (74 words)

	The Kronecker Delta Function Song
		But he did something else, and it equals one or zero--
		A little delta and a tiny m- n !
		If the subscripts are equal, the value is just one,
	www.haverford.edu /physics-astro/songs/kronecker.htm (64 words)

	► » DC Component of White Noise? (Site not responding. Last check: 2007-10-17)
		In this case does that mean that white noise must have
		PSD(w) = No/2(1 - d(0)) where d(w) is the Kronecker Delta?
		delta functions) will have a finite energy at any particular
	www.comp-one.net /DC-Component-of-White-Noise-7676279.html (2112 words)

	MySpace.com - Bombie Kronecker delta - UK - Alternative / Ambient / Other - www.myspace.com/kronecker
		MySpace.com - Bombie Kronecker delta - UK - Alternative / Ambient / Other - www.myspace.com/kronecker
		The people who helped me make the music are thanked; Chris drums, Jo violin, + Math keyboards.
		Bombie Kronecker delta's Latest Blog Entry [Subscribe to this Blog]
	www.myspace.com /kronecker (56 words)

	Quantum field theory - Wikipedia, the free encyclopedia
		where δ(x) stands for the Dirac delta function.
		This prevents us from postulating the same commutation relation as before.
		The way out is to replace the Kronecker delta with a Dirac delta function.
	en.wikipedia.org /wiki/Quantum_field_theory (3491 words)

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