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Topic: Euclidean norm

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	Euclidean space - Wikipedia, the free encyclopedia
		A Euclidean space is a particular metric space that enables the investigation of topological properties such as compactness.
		Euclidean space plays a part in the definition of a manifold which embraces the concepts of both Euclidean and non-Euclidean geometry.
		Euclidean n-space is the prototypical example of an n-manifold, in fact, a smooth manifold.
	en.wikipedia.org /wiki/Euclidean_space (739 words)

	Normed vector space - Wikipedia, the free encyclopedia
		A semi normed vector space is a 2-tuple (V,p) where V is a vector space and p a semi norm on V.
		A surjective isometry between the normed vector spaces V and W is called a isometric isomorphism, and V and W are called isometrically isomorphic.
		The definition of many normed spaces (in particular, Banach spaces) involves a seminorm defined on a vector space and then the normed space is defined as the quotient space by the subspace of elements of seminorm zero.
	en.wikipedia.org /wiki/Normed_vector_space (889 words)

	Euclidean space -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by (Greek geometer (3rd century BC)) Euclid.
		Euclidean space plays a part in the definition of a (A pipe that has several lateral outlets to or from other pipes) manifold which embraces the concepts of both (additional info and facts about Euclidean) Euclidean and (Geometry based on axioms different from Euclid's) non-Euclidean geometry.
		Euclidean n-space is the prototypical example of an n- (A pipe that has several lateral outlets to or from other pipes) manifold, in fact, a (additional info and facts about smooth manifold) smooth manifold.
	www.absoluteastronomy.com /encyclopedia/e/eu/euclidean_space.htm (1034 words)

	Norm (mathematics) - Encyclopedia, History, Geography and Biography
		In linear algebra, functional analysis and related areas of mathematics a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector.
		A vector space with a norm (semi-norm) is called a normed vector space (semi-normed vector space).
		Equivalent norms define the same notions of continuity and convergence and do not need to be distinguished for most purposes.
	www.arikah.net /encyclopedia/Norm_%28mathematics%29 (884 words)

	Encyclopedia: Euclidean space
		The Euclidean distance of two points x = (x1,...,xn) and y = (y1,...,yn) in Euclidean n-space is computed as It is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem.
		Norm Jump to: navigation, search In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions.
		In mathematics the Euclidean distance or Euclidean metric is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem.
	www.nationmaster.com /encyclopedia/Euclidean-space (2271 words)

	PlanetMath: Euclidean valuation (Site not responding. Last check: 2007-10-08)
		An Euclidean valuation is a function from non-zero elements to the non-negative integers
		Euclidean valuations are important because they let us define greatest common divisors and use Euclid's algorithm.
		This is version 3 of Euclidean valuation, born on 2002-05-27, modified 2002-11-15.
	planetmath.org /encyclopedia/EuclideanNorm2.html (96 words)

	PlanetMath: vector p-norm (Site not responding. Last check: 2007-10-08)
		The 1-norm is also called the taxicab metric (sometimes Manhattan metric) since the distance of two points can be viewed as the distance a taxi would travel on a city (horizontal and vertical movements).
		norm in function spaces is a generalization of these norms by using counting measure.
		Euclidean vector norm, vector Euclidean norm, vector 1-norm, vector 2-norm, vector infinity-norm, L^p metric, L^p
	planetmath.org /encyclopedia/Taxicab.html (229 words)

	Normed vector space
		In this case, the two norms define the same notions of continuity and convergence and do not need to be distinguished for most purposes.
		All norms on a finite-dimensional vector space V are equivalent.
		The dual V ' of a normed vector space V is the space of all continuous linear maps from V to the base field (the complexes or the reals) — such linear maps are called "functionals".
	www.sciencedaily.com /encyclopedia/normed_vector_space (899 words)

	Matrix Factorization and Matrix Norms
		denotes the Euclidean vector norm of the vector
		Several properties that hold for the spectral norm of a matrix were given in the lecture and you were asked to prove them as an un-graded homework problem.
		is the Euclidean vector norm of the vector
	www.ee.ucla.edu /~brien/Rec5_MatrixFactorizationAndNorms.htm (718 words)

	Math Forum - Ask Dr. Math
		We will first prove that D is a Euclidean domain for d = -1 or -2, when we take as function f the norm: f(x + yw) = N(x + yw) = x^2 - dy^2.
		However, it could still be the case that D would be a Euclidean domain, but with a different Euclidean function.
		So far, we have shown: * D is a Euclidean domain, with the norm as Euclidean function, for d = -1 or -2 * These are the only negative d for which D is a Euclidean domain with the norm as Euclidean function.
	mathforum.org /library/drmath/view/62284.html (1451 words)

	MATH2071: LAB #5: Norms, Errors and Condition Numbers (Site not responding. Last check: 2007-10-08)
		The spectral matrix norm is not vector-bound to any vector norm, but it "almost" is. This norm is useful because we often want to think about the behavior of a matrix as being determined by its largest eigenvalue, and it often is. But there is no vector norm for which it is always true that
		The norms of the matrix and its inverse exert some limits on the relationship between the forward and backward errors.
		These quantities depend on the vector norm used, they cannot be defined in cases where the divisor is zero, and they are problematic when the divisor is small.
	www.math.pitt.edu /~sussmanm/2071Spring04/lab05 (2831 words)

	Banach space (Site not responding. Last check: 2007-10-08)
		This is indeed a norm since continuous functions defined on a closed interval are bounded.
		The space is complete under this norm, and the resulting Banach space is denoted by C[a, b].
		R or the space of all distributions on R, are complete but are not normed vector spaces and hence not Banach spaces.
	www.brainyencyclopedia.com /encyclopedia/b/ba/banach_space.html (1374 words)

	Livid's Lividict - norm (Site not responding. Last check: 2007-10-08)
		The norm must be {homogeneous} and {symmetric} and fulfil the following condition: the shortest way to reach a point is to go straight toward it.
		Every {convex} symmetric {closed} surface surrounding point 0 introduces a norm by means of {Minkowski functional}; all vectors that end on the surface have the same norm then.
		The most popular norm is the {Euclidean norm}; it is calculated by summing up squares of all coordinates and taking the square root; this is the essence of {Pythagorus's theorem}.
	livid.3322.org /lividict/foldoc/norm.html (256 words)

	[No title]
		All the real quadratic fields with D Euclidean with respect to the absolute value of the norm have been known for some time, but until recently it was an open problem as to whether such a D existed which was Euclidean but not with respect to the norm.
		More precisely, Clark gave a completely explicit example of a modified norm with respect to which the ring of integers of that quadratic field is Euclidean (it was well known that it wasn't Euclidean wrt.
		Basically, the `best' valuation will map 0 to 0, all the units to 1, all the prime elements for which it makes sense to 2 (those for which all the prime residue classes contain units), and all the remaining prime elements to 3; its values for composites can then be determined inductively.
	www.math.niu.edu /~rusin/known-math/97/euclidean.domain (697 words)

	Norms of Vectors
		A vector norm is a single number which represents the size or `length' of a vector.
		Although only three vector norms are used, it is worth giving the axiomatic definition, since this is a starting point for matrix norms.
		These properties ensure that the norm is a length; (i) says that length is non-negative, (ii) gives scaling of length, while (iii) is the triangle inequality.
	www.maths.lancs.ac.uk /~gilbert/m306a/node5.html (248 words)

	Research Topic: Description (Site not responding. Last check: 2007-10-08)
		The Euclidean distance if widely used for that matter, because in two or three dimensions, it gives the actual distance as in the real world.
		In high dimensions, the Euclidean distances from a point to its nearest and farthest neigbours tend to be very similar and nearly meaningless.
		The Euclidean distance does not care about this intrinsic structure and goes measuring distances through "shortcuts" as illustrated on the figure (the red line is the Euclidean Distance between fl dots and the blue line is the geodesic distance along the 2D spiral built up by the datapoints, between the same fl dots.
	www.dice.ucl.ac.be /mlg/index.php?page=PartTop&WhichTop=5 (381 words)

	Complete Inner Product Space
		The conjugate of the square of the norm equals the square of the norm of the two terms reversed.
		Examine the square of the norm of the difference between two quaternions which is necessarily equal to or greater than zero.
		This is twice the square of the norms of the two separate components.
	world.std.com /~sweetser/quaternions/quantum/bracket/bracket.html (774 words)

	Maple worksheets for Abstract Algebra (Site not responding. Last check: 2007-10-08)
		The third worksheet, QuadraticEuclidean.mws, looks at the this same norm over a number of other quadratic extensions of the integers.
		It considers extensions where the norm is Euclidean as well as extensions where the norm is not Euclidean.
		Then the norm is Euclidean, the students use it in the Euclidean algorithm.
	euler.slu.edu /courseware/AbstractAlgebra/AlgebraOverview.html (739 words)

	NORM (Site not responding. Last check: 2007-10-08)
		The NORM function computes the norm of a vector or a two-dimensional array.
		Returns the Euclidean or infinity norm of a vector or an array.
		Euclidian Norm of A = 4.35890 Infinity Norm of B = 6.9907048
	idlastro.gsfc.nasa.gov /idl_html_help/N6.html (219 words)

	Matrix norm (Site not responding. Last check: 2007-10-08)
		A matrix norm is a norm on the vector space of all real or complex m-by-n matrices.
		These norms are used to measure the "sizes" of matrices, and allow to talk about limits of sequences and infinite series of matrices.
		The set of all n-by-n matrices, together with such a sub-multiplicative norm, is a Banach algebra.
	www.explainthis.info /ma/matrix-norm.html (381 words)

	Minimizing the Euclidean Condition Number
		It is the Euclidean norm case that has widespread application in robust control analyses.
		For example, it is used for integral controllability tests based on steady-state information, for the selection of sensors and actuators based on dynamic information, and for studying the sensitivity of stability to uncertainty in control systems.
		Minimizing the scaled Euclidean condition number has been an open question---researchers proposed approaches to solving the problem numerically, but none of the proposed numerical approaches guaranteed convergence to the true minimum.
	epubs.siam.org /sam-bin/dbq/article/23868 (188 words)

	Normed vector space (Site not responding. Last check: 2007-10-08)
		For more abstract vector spaces, a norm is a generalization of this idea.
		is a romboid[?], for the 2-norm (Euclidian norm) it is the well-known unit circle, while for the infinity norm it is a square.
		(Note that the Euclidean norm gives rise to the Euclidean distance in this fashion.) This turns the normed space into a metric space and allows to define notions such as continuity and convergence.
	www.city-search.org /no/normed-vector-space.html (1103 words)

	LVQ Algorithm (can use Euclidean or taxicab norm for match)
		Template matching can be based on Euclidean or taxicab distances; in the latter case, the new algorithm emulates "fuzzy" ART.
		Interestingly, this is analogous to the equivalence of template-error and correlation matching with Euclidean norms.
		What is more, explicit computation of the taxicab norms (4) requires only absolute-value computations, which are typically faster than the comparison operations needed to evaluate the expressions (5).
	members.aol.com /gatmkorn/mewart.htm (1263 words)

	ipedia.com: P-adic number Article (Site not responding. Last check: 2007-10-08)
		A definite meaning is given to these sums based on Cauchy sequences using the familiar Euclidean metric.
		However, the definition of a Cauchy sequence relies on the metric chosen and, by choosing a different one, numbers other than the real numbers can be constructed.
		It can be proved that each norm on Q is equivalent either to the Euclidean norm or to one of the p-adic norms for some prime p.
	www.ipedia.com /p_adic_number.html (1383 words)

	Physics at Minnesota:
		real8 The Euclidean* norm of the vector x, that is, the square root of the conjugated dot product of x with itself.
		SNRM2 and DNRM2 compute the Euclidean norm of a real vector; SCNRM2 and DZNRM2 compute the Euclidean norm of a complex vector.
		The Euclidean norm is the square root of the conjugated dot product of a vector with itself.
	www.physics.umn.edu /support/doc/cxml/dznrm2.3dxml.html?printer=yes& (303 words)

	LMS JCM (3) 336-355 (Site not responding. Last check: 2007-10-08)
		Abstract: In this paper we study number fields which are Euclidean with respect to functions that are different from the absolute value of the norm, namely weighted norms that depend on a real parameter c.
		We introduce the Euclidean minimum of weighted norms as the set of values of c for which the function is Euclidean, and we show that the Euclidean minimum may be irrational and not isolated.
		We also present computational results on Euclidean minima of cubic number fields, and present a list of norm-Euclidean complex cubic fields that we conjecture to be complete.
	www.lms.ac.uk /jcm/3/lms2000-011 (125 words)

	The PSLQ Algorithm
		One can show, on using a working precision that is only slightly higher than that of the input data, that these bound results obtained from computer runs are reliable.
		In the context of these exclusion bounds it is important to realize that, even for very ``similar'' numbers, the Euclidean norms of their relations can differ greatly.
		The lesson here is that, if we can, we should ``normalize'' our input so that the resulting integer relation is of low Euclidean norm.
	www.cecm.sfu.ca /~aszanto/IntegerRelations/fpsac97/node8.html (291 words)

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