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	Learn more about Complex number in the online encyclopedia. (Site not responding. Last check: 2007-10-21)
		Complex numbers were first introduced in connection with explicit formulas for the roots of cubic polynomials.
		The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be described by the transpose of the matrix corresponding to z.
		The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics.
	www.onlineencyclopedia.org /c/co/complex_number.html (1629 words)

	Complex number - Wikipedia, the free encyclopedia
		In mathematics, the term "complex" when used as an adjective means that the field of complex numbers is the underlying number field considered, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra.
		The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be described by the transpose of the matrix corresponding to z.
		The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics.
	en.wikipedia.org /wiki/Complex_number (3135 words)

	Complex number - Psychology Wiki (Site not responding. Last check: 2007-10-21)
		In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for one of the square roots of negative one (−1).
		In mathematics, the adjective "complex" means that the field of complex numbers is the underlying number field considered, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra.
		Given a complex number (a + bi) which is to be divided by another complex number (c + di) whose magnitude is non-zero, there are two ways to do this; in either case it is the same as multiplying the first by the multiplicative inverse of the second.
	psychology.wikia.com /wiki/Complex_number (3473 words)

	Complex representation - Wikipedia, the free encyclopedia
		In other words, the group elements are expressed as complex matrices, and the complex conjugate of a complex representation is a different, non-equivalent representation.
		For example, the N-dimensional fundamental representation of SU(N) for N greater than two is a complex representation whose complex conjugate is often called the antifundamental representation.
		A representation of a group by roots of unity is called the character group.
	en.wikipedia.org /wiki/Complex_representation (105 words)

	Complex representation -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		In other words, the ((chemistry) two or more atoms bound together as a single unit and forming part of a molecule) group elements are expressed as complex matrices, and the complex conjugate of a complex representation is a different, non-equivalent representation.
		For example, the N-dimensional (Click link for more info and facts about fundamental representation) fundamental representation of SU(N) for N greater than two is a complex representation whose complex conjugate is often called the (Click link for more info and facts about antifundamental representation) antifundamental representation.
		A representation of a group by (Click link for more info and facts about roots of unity) roots of unity is called the (Click link for more info and facts about character group) character group.
	www.absoluteastronomy.com /encyclopedia/C/Co/Complex_representation.htm (130 words)

	Complex number Article, Complexnumber Information (Site not responding. Last check: 2007-10-21)
		The existence of complex numbers was not completely accepted until the geometrical interpretation (see below) had beendescribed by Caspar Wessel in 1799 ;it was rediscovered several years later and popularized by CarlFriedrich Gauss, and as a result the theory of complex numbers received a notable expansion.
		The conjugate of the complex numberz corresponds to the transformation which rotates through the same angle as z but in the opposite direction,and scales in the same manner as z; this can be described by the transpose of the matrix corresponding to z.
		Complex numbers are used in signal analysis and other fields as aconvenient description for periodically varying signals.
	www.anoca.org /numbers/real/complex_number.html (2228 words)

	Maths - Complex Numbers - Martin Baker
		We can look at complex numbers as extensions or generalisations of other algebras (for example they extend real numbers and they are a subset of quaternions and clifford agebras).
		Complex numbers are two dimensional in that they contain two scalar values and they can represent points in 2D vector space.
		Is there any advantages in using complex numbers to represent the complete state of a solid object, in other words it has a state variable that includes both the position.
	www.euclideanspace.com /maths/algebra/realNormedAlgebra/complex/index.htm (1559 words)

	Complex Number (Site not responding. Last check: 2007-10-21)
		Every complex number can be represented in the form a + ib, where a and b are real numbers called the real part and the imaginary part of the complex number, respectively.
		Complex numbers were first introduced in connection with explicit formulas for the roots of cubic polynomials.
		In mathematics, the term "complex" when used as an adjective means that the field of complex numbers is the underlying number field considered.
	www.multifractals.com /complex.html (2716 words)

	PlanetMath: Galois representation
		is finite: by the normal basis theorem, it is merely a permutation representation on the normal basis.
		Galois representations play a fundamental role in algebraic number theory, as many objects and properties related to global fields and local fields may be determined by certain Galois representations and their properties.
		This is version 14 of Galois representation, born on 2003-02-18, modified 2007-04-09.
	planetmath.org /encyclopedia/GaloisRepresentation.html (779 words)

	Complex numbers
		Complex numbers were originally introduced in the seventeenth century to represent the roots of polynomials which could not be represented with real numbers alone.
		Because a complex number can always be separated into its real and complex parts, we can represent a complex number as a point on a two-dimensional plane.
		The real part of a complex number is the projection of the point onto the real axis, and the imaginary part of the number is the projection onto the imaginary axis.
	www.tina.com /course/16complex/complex.htm (1491 words)

	Complex Numbers (Site not responding. Last check: 2007-10-21)
		Points in the plane, and thus complex numbers, also have a polar coordinate representation (r, θ), where r is the distance of a point from the origin, and θ is the angle counterclockwise from the positive x axis to the line segment from the origin to the point.
		The sum is the complex number at the far corner of the parallelogram, represented by the arrow from the origin to that point.
		That is, to multiply complex numbers in polar form, we multiply the moduli (lengths) and add the arguments (angles).
	www.msci.memphis.edu /~simmonsj/c4302/notes/html/complex.html (774 words)

	[No title]
		Every complex number can be represented in the form x + iy, where x and y are real numbers called the real part and the imaginary part of the complex number respectively.
		The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis' De Algebra tractatus.
		A complex number can also be viewed as a point or a position vector on the two dimensional Cartesian coordinate system.
	en-cyclopedia.com /wiki/Complex_number (2111 words)

	Mental Representation
		Representational theories may thus be contrasted with theories, such as those of Baker (1995), Collins (1987), Dennett (1987), Gibson (1966, 1979), Reid (1764/1997), Stich (1983) and Thau (2002), which deny the existence of such things.
		In philosophy, recent debates about mental representation have centered around the existence of propositional attitudes (beliefs, desires, etc.) and the determination of their contents (how they come to be about what they are about), and the existence of phenomenal properties and their relation to the content of thought and perceptual experience.
		According to Kosslyn (1980, 1982, 1983), a mental representation is "quasi-pictorial" when every part of the representation corresponds to a part of the object represented, and relative distances between parts of the object represented are preserved among the parts of the representation.
	plato.stanford.edu /entries/mental-representation (7873 words)

	Area Entrance - Vector and Complex Numbers:
		The two webpages Complex Numbers and Trig for Today's Students and Distributive Law for Complex Numbers, one or both should be read first, and followed by the easy consequences B2 to B10.
		Complex No.Axioms Here is a summary of the set theoretic viewpoint or codifiction of complex numbers.
		This representation or decomposition of an arrow as the sum of horizontal and vertical components leads to a third method for arrow addition given by the addition of components.
	whyslopes.com /etc/ComplexNumbers (2812 words)

	The Nature of Complex Numbers
		Complex numbers are ordered pairs of real numbers for which multiplication is defined in a special way.
		The addition of complex numbers is very simple; the sum of two complex numbers is the complex numbers whose components are the sums of the components of the two numbers.
		A complex number as a point in a two dimensional plane can be completely characterized by a radius (the distance of the point from the origin) and an angle.
	www.sjsu.edu /faculty/watkins/complex.htm (515 words)

	All Elementary Mathematics - Study Guide - Algebra - Complex numbers...
		Addition. A sum of complex numbers a+ bi and c+ di is called a complex number (a+ c) + (b+ d) i. So, at addition of complex numbers their abscissas and ordinates are added separately.
		Then, a complex number a+ bi will be represented by point P with abscissa a and ordinate b (see figure).
		Modulus of a complex number is a length of vector OP, representing this complex number in a coordinate (complex) plane.
	www.bymath.com /studyguide/alg/sec/alg26.html (658 words)

	Chapter 30. Complex Numbers (Site not responding. Last check: 2007-10-21)
		Complex numbers are an extension of the ordinary numbers used in everyday math.
		Complex numbers shorten the equations used in DSP, and enable techniques that are difficult or impossible with real numbers alone.
		Unfortunately, complex techniques are very mathematical, and it requires a great deal of study and practice to use them effectively.
	www.dspguide.com /ch30.htm (182 words)

	advantage of using complex exponentials in DSP ???
		The concept of frequency response of the system is defined using Complex exponential because if the input of the system is a complex exponenetial, the output is also a complex exp. with the same freq.
		The complex H(s) is the transfer function or frequency response.
		But complex representation is highly important in EM for impedance matching.
	www.edaboard.com /ftopic98910.html (820 words)

	XML Schema Part 1: Structures
		See Complex Type Definition Details (§3.4) for the composition and schema-validation contributions of complex type definition schema components, XML Representation of Complex Type Definition Schema Components (§4.3.3) for the XML representation of complex type definitions and Complex Type Definition Constraints (§5.11) for constraints on complex type definition components as such.
		Complex type {name}s and {target namespace}s are provided for reference from instances (see xsi:type (§2.6.1)), and for use in the XML Representation of Schemas and Schema Components (§4) (specifically in element and attribute).
		A complex type for which {abstract} is true must not appear as the {type definition} of an Element Declaration (§2.2.2.1), and must not be referenced from an xsi:type (§2.6.1) attribute in an instance document; such abstract complex types can be used as {base type definition}s, but they are never used directly to validate element content.
	www.w3.org /TR/2000/WD-xmlschema-1-20000407 (10581 words)

	The Origin of Complex Numbers
		Complex analysis can roughly be thought of as the subject that applies the theory of calculus to imaginary numbers.
		In an 1831 paper, he produced a clear geometric representation of x+iy by identifying it with the point (x, y) in the coordinate plane.
		As time passed, mathematicians gradually refined their thinking, and by the end of the nineteenth century complex numbers were firmly entrenched.
	math.fullerton.edu /mathews/n2003/ComplexNumberOrigin.html (1569 words)

	Complex Numbers
		The usual definition of complex numbers is all numbers of the form a+bi, where a and b are real numbers and i, the imaginary unit, is a number such that its square is -1.
		Complex numbers did not come about from this example, but in connection with the solution to cubic equations.
		For years after Bombelli's work, many still thought complex numbers were a waste of time, but there were others who used complex numbers extensively and through their work, much more was discovered.
	www.und.edu /instruct/lgeller/complex.html (779 words)

	Introduction to PNNI (Site not responding. Last check: 2007-10-21)
		In contrast, when the simple node representation is used, remote peer groups can choose to communicate through the local peer group, but the remote group must rely on border nodes within the local peer group to determine the path within the local peer group.
		The disadvantage of complex node representation is that it adds to the size of the database in remote peer groups.
		Complex node representation also requires more processing resources on the LGN that represents a peer group as a complex node.
	www.cisco.com /univercd/cc/td/doc/product/wanbu/8850px45/rel5/ppg/pintro.htm (2935 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		We must therfore provide a representation for complex numbers and a create a set of methods for operating on that representation.
		One double is the real part of the complex number and the other double is the imaginary part of the complex number.
		Following the definition of the Complex struct is a set of function declarations that operate on Complex structs.
	www.visc.vt.edu /rhudson/ee2984/hwk5.doc (1311 words)

	Calculation of Cosmological Baryon Asymmetry in Grand Unified Gauge Models (1982)
		Real and complex representations appear in many models; pseudoreal representations are rare, since they must be used in a ``doubled'' form to allow construction of mass terms for scalar fields.
		Fermions are usually placed in complex representations; this prevents the possibility of group-invariant fermion mass terms (allowed by chiral symmetries) and avoids unobserved right-handed fermions coupled to the weak current.
		In general, the presence of a large vacuum expectation value for a Higgs field in a complex representation of the gauge group lowers the rank of the effective gauge symmetry by breaking at least one of the U(1) invariances associated with the Cartan subalgebra.
	www.stephenwolfram.com /publications/articles/cosmology/82-calculations/4/text.html (1763 words)

	PARKA-DB: A Scalable Knowledge Representation System
		In contrast to knowledge representation systems, the upper bound on the size of a database is not limited by the size of main memory, but rather by the size of external storage.
		They allow a complex structural representation of the data (knowledge) that allows inferencing and complex query evaluation to be performed.
		As a functional knowledge representation system, PARKA-DB is designed to support several applications including CaPER, a memory-intensive case-based reasoning system, ForMAT, a case-based logistics planning system developed by Mitre Corporation, and a set of medical informatics programs being developed at the University of Maryland (by Merwyn Taylor).
	www.cs.umd.edu /projects/plus/Parka/parka-db.html (844 words)

	A Representation of Complex Movement Sequences based on Hierarchical Spatio-temporal Correspondence for Imitation ... (Site not responding. Last check: 2007-10-21)
		Problems concerning the movement representation are twofold: (1) The movement characteristics of observed movements have to be transferred from the perceptual level to the level of generated actions.
		We present methods for the representation of complex movement sequences that addresses these questions in the context of the imitation learning of writing movements using a robot arm with human-like geometry.
		In this poster we focus on the acquisition of the movement representation (identification and segmentation of movement primitives, generation of new writing styles by spatio-temporal morphing).
	www.kyb.tuebingen.mpg.de /de/publication.html?publ=2076 (410 words)

	RA Fisher Associates/ course: Complex Variables
		This unique interactive course is designed to introduce real and imaginary numbers, and to show how their use has for over 100 years been the standard way of describing the subject of electromagnetic wave propagation.
		In the second half of this course we will identify the interrelationships between the complex index of refraction, the complex dielectric function, the complex propagation vector, the complex susceptibility, and the real absorption coefficient.
		Throughout the course, examples are shown which demonstrate the convenience of solving simple wave propagation problems in the complex representation.
	www.rafisher.com /math.htm (560 words)

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