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Topic: Theory of invariants

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	Knot Theory Invariants: The HOMFLY Polynomial
		The publication of the Jones Polynomial excited the mathematical community to the point that new polynomial invariants were being created at a stupendous rate.
		One of the objectives of the time was to find a polynomial that generalized both the Alexander and Jones polynomial.
		The HOMFLY Polynomial serves as a more general polynomial invariant that covers the Alexander and Jones Polynomials.
	library.thinkquest.org /12295/data/Invariants/Articles/HOMFLY.html (284 words)

	Joseph Louis Lagrange
		The first volume contains a memoir on the theory of the propagation of sound; in this he indicates a mistake made by Newton, obtains the general differential equation for the motion, and integrates it for motion in a straight line.
		The theory of the potential was elaborated in a paper sent to Berlin in 1777.
		The book is divided into three parts: of these, the first treats of the general theory of functions, and gives an algebraic proof of Taylor's theorem, the validity of which is, however, open to question; the second deals with applications to geometry; and the third with applications to mechanics.
	www.sciencedaily.com /encyclopedia/joseph_louis_lagrange (3039 words)

	Joseph Louis de Lagrange (Site not responding. Last check: 2007-10-16)
		The first volumecontains a memoir on the theory of the propagation of sound; in this he indicates a mistake made by Newton, obtains the general differential equation for the motion, and integrates it for motion in a straight line.
		The second volume contains a long paper embodying the results of several memoirs in the first volume on the theory andnotation of the calculus of variations; and he illustrates its use by deducing the principle of least action, and by solutions of various problems in dynamics.
		The book is divided into three parts: of these, thefirst treats of the general theory of functions, and gives an algebraic proof of Taylor's theorem, the validity of which is, however, open to question; the second deals with applicationsto geometry; and the third with applications to mechanics.
	www.therfcc.org /joseph-louis-de-lagrange-23056.html (2954 words)

	Operations on Curves
		The Igusa invariants of the curve y^2 + h * y = f are equal to the Igusa invariants of the polynomial h^2 + 4 * f except in characteristic 2, where the latter are not defined.
		Igusa invariants are given by a sequence [ J_2, J_4, J_6, J_8, J_(10) ] of five elements of the coefficient ring of the polynomials defining the curve.
		The Igusa--Clebsch invariants of a polynomial are defined in terms of certain nice symmetric polynomials in its roots, and, in characteristic zero, the J-invariants may be obtained from the I-invariants by some simple homogeneous transformations.
	www.math.niu.edu /help/math/magmahelp/text1023.html (1103 words)

	IET, Inc.
		Invariants have been used successfully in computer vision to extrapolate from the existing information in an image to fill in the missing or obscured information.
		Standard invariants can, in principle, be a powerful tool for object recognition, particularly as keys to index into a database of object models, because they can be computed for any view and then matched to the 3-D object that produces that view.
		True invariants of an object under the action of a group of transformations compute sets of numerical values that are exactly equal for that object regardless of the transformation.
	www.iet.com /solutions3_cs_1.html (701 words)

	Directory - Science: Physics: Quantum Mechanics: Quantum Field Theory: Topological (Site not responding. Last check: 2007-10-16)
		The field theories to be used are combinatorially and algebraically defined, and the emphasis is on numerical computation and detection of counterexamples rather than general structure.
		New Results in Topological Field Theory and Abelian Gauge Theory · These are the lecture notes of a set of lectures delivered at the 1995 Trieste summer school in June.
		Geometry of 2D Topological Field Theory · These lecture notes are devoted to the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories.
	www.incywincy.com /default?p=210316 (465 words)

	Polynomial
		In 1824, Abel proved the striking result that there can be no general formula (involving only the arithmetical operations and radicals) for the roots of a polynomial of degree ≥ 5 in terms of its coefficients (see Abel-Ruffini theorem).
		This result marked the start of Galois theory which engages in a detailed study of relations among roots of polynomials.
		Knot Theory Invariants: The HOMFLY Polynomial - A brief article on the HOMFLY polynomial and how it is calculated.
	www.nebulasearch.com /encyclopedia/article/Polynomial.html (1987 words)

	General Linear Models (GLM)
		The contributions of the theory of algebraic invariants to the development of statistical theory and methods are numerous, but a simple example familiar to even the most casual student of statistics is illustrative.
		Tests of differences in least squares means have the important property that they are invariant to the choice of the coding of effects for categorical predictor variables (e.g., the use of the sigma-restricted or overparameterized model) and to the choice of the particular g2 inverse of X'X used to solve the normal equations.
		The general implication of the theory of estimability of linear functions is that hypotheses which cannot be expressed as linear combinations of the rows of X (i.e., the combinations of observed levels of the categorical predictor variables) are not estimable, and therefore cannot be tested.
	www.statsoft.com /textbook/stglm.html (13045 words)

	HelicityReprints
		The simplest such invariant is winding number -- the net number of times two strings in a braid wrap about each other.
		The mathematical literature on these invariants usually employs techniques from algebraic topology that may be unfamiliar to physicists and mathematicians in other disciplines.
		We describe the Kontsevich integral, and prove that it is invariant to deformations of the braid.
	www.ucl.ac.uk /~ucahmab/BraidReprints/BraidSummaries.htm (681 words)

	[No title] (Site not responding. Last check: 2007-10-16)
		The contributions of the theory of algebraic invariants to the development of statistical theory and methods are numerous, but a simple example familiar to even the most casual student of statistics is illustrative.
		Tests of differences in least squares means have the important property that they are invariant to the choice of the coding of effects for categorical predictor variables (e.g., the use of the sigma-restricted or overparameterized model) and to the choice of the particular g2 inverse of X'X used to solve the normal equations.
		The general implication of the theory of estimability of linear functions is that hypotheses which cannot be expressed as linear combinations of the rows of X (i.e., the combinations of observed levels of the categorical predictor variables) are not estimable, and therefore cannot be tested.
	ebd06.ebd.csic.es /statbook/stglm.html (13054 words)

	Finite Relativity Theory
		After coordinatizing the figure in a suitable manner, we find that the line diagrams are invariant under the group of 16 binary translations acting on the colored figure.
		For another sort of invariance of the colored figure, try applying a symmetry of the square to each of the set of four diagonally-divided squares from which the figure's entries are drawn, and observe the induced effect on the figure itself.
		The significance of the notion of invariance and its group-theoretic treatment for the issue of objectivity is explored in Born (1953), for example.
	finitegeometry.org /sc/16/finiterelat.html (845 words)

	General Spectral Theory (Site not responding. Last check: 2007-10-16)
		DC MetaData for: Spectral Theory for Periodic Schrödinger Operators with Reflect...
		Spectral theory and limit theorems for geometrically ergodic Markov processes, I...
		Localization in the Spectral Theory of Operators on Banach Spaces...
	www.scienceoxygen.com /math/647.html (147 words)

	Citebase - Vassiliev Invariants in the Context of Chern-Simons Gauge Theory
		Citebase - Vassiliev Invariants in the Context of Chern-Simons Gauge Theory
		Vassiliev Invariants in the Context of Chern-Simons Gauge Theory
		Emphasis is made on the analysis of the perturbative study of the theory and its connection to the theory of Vassiliev invariants.
	citebase.eprints.org /cgi-bin/citations?archiveID=oai:arXiv.org:hep-th/9812105 (1252 words)

	Powell's Books - Annals of Mathematics Studies #0134: Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds ...
		This recoupling theory is a q-deformation of the SU(2) spin networks of Roger Penrose.
		The recoupling theory is developed in a purely combinatorial and elementary manner.
		Calculations are based on a reformulation of the Kirillov-Reshetikhin shadow world, leading to expressions for all the invariants in terms of state summations on 2-cell complexes.
	www.powells.com /biblio?isbn=0691036403 (558 words)

	Open Directory - Science: Physics: Quantum Mechanics: Quantum Field Theory: Topological (Site not responding. Last check: 2007-10-16)
		Geometry of 2D Topological Field Theory - These lecture notes are devoted to the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories.
		New Results in Topological Field Theory and Abelian Gauge Theory - These are the lecture notes of a set of lectures delivered at the 1995 Trieste summer school in June.
		Topological Quantum Field Theory, a Progress Report - A brief introduction to Topological Quantum Field Theory as well as a description of recent progress made in the field is presented.
	dmoz.org /Science/Physics/Quantum_Mechanics/Quantum_Field_Theory/Topological (504 words)

	Diplomarbeit: Invariantentheorie endlicher und kompakter Gruppen (Site not responding. Last check: 2007-10-16)
		In this diploma thesis I investigate the theory of invariants from an algorithmic point of view.
		Starting from a combinatorial interpretation of the coefficients of this closed form a correspondence between secondary invariants and binary trees is given.
		Finally the theory of invariants under local unitary tranformations is applied to the very new topic of quantum error correcting codes.
	www.iqc.ca /~mroetteler/abstracts/diplom.html (171 words)

	Artificial Intelligence and Scientific Creativity
		Three important areas in the philosophy of scientific creativity were covered in the symposium, namely the advancement of science through the introduction and maintenance of inconsistencies, the transfer of ideas from a known domain to a new domain and the social aspects of scientific discovery.
		Inconsistencies can arise as empirical anomalies which the current theory cannot explain, or can be generated by the radical innovation of scientists who suggest a new theory to compete with the current theory in explaining the same phenomenon.
		Zytkow clarified this distinction at the AISB Symposium: a theory is a result of analysis, and describes a simple element of nature, such as gravity or electromagnetism.
	www.dai.ed.ac.uk /homes/simonco/papers/AISBQ99.html (5539 words)

	Konstantin Sergeevich Sibirsky -Introduction to the Algebraic Theory of Invariants of Differential Equations - Y Daniel ... (Site not responding. Last check: 2007-10-16)
		Konstantin Sergeevich Sibirsky -Introduction to the Algebraic Theory of Invariants of Differential Equations - Y Daniel Liang
		Introduction to the Algebraic Theory of Invariants of Differential Equations
		1: Introduction to the Algebraic Theory of Invariants of Differential Equations.
	www.bookzsearch.com /206416introduction_algebraic_theory_invariants_differential_equations.html (49 words)

	[No title] (Site not responding. Last check: 2007-10-16)
		Invariants and covariants of the symmetric group, a noncommutative version
		This is a survey talk on the noncommutative version of the classical theory of invariants and covariants of the symmetric group.
		A brief review of the clasical theory of invariants and covariants of the symmetric group will be included in this talk.
	www-math.mit.edu /~combin/abstracts/oct03/rosas.html (93 words)

	MAI: Lite Mat
		Sammanfattning: The present talk is a survey of the basic method and recent results in the theory of invariants of families of differential equations.
		The problem of invariants of differential equations can be dated back to Laplace's 1773 work, when young Laplace (he was 24) published his renowned method based on what is known today as the Laplace invariants h and k.
		In consequence, a simple unified approach was developed for calculation of invariants of algebraic and differential equations independent on the assumption of linearity of the equations.
	www.mai.liu.se /LiteMat/2004/v12-04 (705 words)

	Knot Theory Science, Directory (Site not responding. Last check: 2007-10-16)
		A Circular History of Knot Theory Starting with the flawed theory of Kelvin's knotted vortex to the work of Thurston, Jones and Witten, knot theory has circled back to its ancestral origins of theoretical physics.
		Knot Theory Invariants: The HOMFLY Polynomial A brief article on the HOMFLY polynomial and how it is calculated.
		Knot Theory Online This site is designed for mathematics students at the high school and college levels as an introduction to an area of mathematics seldom explored in the typical math classroom - the Theory of Knots.
	www.indiapolicyinstitute.org /aW5kXzI2OTQ4.aspx (653 words)

	Amazon.ca: Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds (AM-134): Books: Louis H. Kauffman,Sostenes ... (Site not responding. Last check: 2007-10-16)
		This recoupling theory is a q-deformation of the SU(2) spin networks of Roger Penrose.
		The recoupling theory is developed in a purely combinatorial and elementary manner.
		Calculations are based on a reformulation of the Kirillov-Reshetikhin shadow world, leading to expressions for all the invariants in terms of state summations on 2-cell complexes.
	www.amazon.ca /Temperley-Lieb-Recoupling-Theory-Invariants-3-Manifolds/dp/0691036403 (337 words)

	Algebraic Combinatorics Abstracts (Site not responding. Last check: 2007-10-16)
		The second is a nice connexion with the diagrammatics of topological quantum field theory and with Schur-Weyl duality.
		We will present a noncommutative version of the classical theory of invariants and coinvariants of the symmetric group, as well as a noncommutative version of a theorem of Chevalley that says that the ring of polynomials can be written as the tensor product of its invariants times its coinvariants.
		We discuss how the theory of oscillating tableaux, which originally arose in the representation theory of the symplectic and orthogonal groups and the Brauer algebra, can be used to obtain further results on crossings and nestings of matchings.
	www.lacim.uqam.ca /~biagioli/CMS/abstracts.html (1511 words)

	Yang-Mills Theory And Invariants Of Links - Polyak, Reshetikhin (ResearchIndex) (Site not responding. Last check: 2007-10-16)
		In these notes we discuss some aspects of 2D Yang-Mills theory and its relation to invariants of knots in circle bundles over surfaces.
		An easy proof is given that the partition function of the Y M 2 with Wilson loops can be regarded as the integral of a function over the moduli space of flat bundles over surface with respect to the symplectic volume form.
		15 Quantum invariants of knots and 3-manifolds (context) - Turaev - 1994
	citeseer.ist.psu.edu /polyak97yangmills.html (559 words)

	Finite Relativity Theory (Site not responding. Last check: 2007-10-16)
		The three line diagrams above result from the three partitions, into pairs of 2-element sets, of the 4-element set from which the entries of the bottom colored figure are drawn.
		After coordinatizing the figure in a suitable manner, we find that this set of three line diagrams is invariant under the group of 16 translations acting on the colored figure.
		For another sort of invariance of the colored figure, try applying a symmetry of the square to each of the set of four diagonally-divided squares from which the figure's entries are drawn, and observe
	m759.freeservers.com /pages/DTrelativprob.html (338 words)

	Cornell Math - Early History (Site not responding. Last check: 2007-10-16)
		Oliver taught the theories of functions and probablility in addition to non-Euclidean geometry; George Jones offered advanced work in analytic geometry of two and three dimensions - lines and surfaces of first and second orders in addition to a course in modern synthetic geometry; and Lucien Wait covered advanced work in calculus - differential calculus.
		For example, in Oliver's course on the theory of functions, Broit and Bonquet's Theorie des fonctions elliptiques and Halphen's Traité des fonctions elliptiques were used.
		In teaching his class on the general theory of algebraic curves and surfaces McMahon adopted George Salmon's Treatise on the Analytic Geometry of Three Dimensions and Treatise on Higher Plane Curves while he used Salmon's Modern Higher Algebra in his lectures on the theory of invariants and covariants.
	www.math.cornell.edu /~www/General/History/historyP5.html (1557 words)

	HR - Automatic Theory Formation In Pure Mathematics
		This will build on the theory behind the HR program, and enhance that theory with an understanding of how to make concepts which involve aspects from more than one domain.
		These are ubiquitous, for example, in group theory, prime numbers are needed to develop the theory of Sylow subgroups, in graph theory, numerical invariants are mainstream concepts.
		For example, in group theory, one can of course look for subgroups, but it is also common to look for sub-semigroups.
	www.dai.ed.ac.uk /homes/simonco/research/hr/future.html (602 words)

	"Algebras, representations and explicit invariants"
		One aim of representation theory is to understand the modules for an algebra, i.e.
		Thus, invariants which are preserved under derived equivalences are of particular interest.
		In this talk we present various notions of homological dimensions and discuss whether they are suitable as a measure for how complicated the representation theory of an algebra is. Some of these invariants are classical and well understood for commutative algebras in the context of algebraic geometry.
	www.maths.gla.ac.uk /events/seminars/mkabs.php?id=438 (205 words)

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