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Topic: Characteristic (algebra)

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Irreducible polynomial

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	Characteristic (Site not responding. Last check: 2007-10-14)
		In abstract algebra, the characteristic of a ring R is defined to be the smallest positive integer n such that 1R+.
		In abstract algebra, the characteristic of a ring R is defined to be the smallest positive integer n such that 1
		Characteristic is also sometimes used as a piece of jargon in discussions of universals in metaphysics, often in the phrase 'distinguishing characteristics'.
	www.termsdefined.net /ch/characteristic.html (438 words)

	Eigenvectors (from algebra, linear) -- Encyclopædia Britannica (Site not responding. Last check: 2007-10-14)
		Algebra is fundamental not only to all further mathematics and statistics but to the natural sciences, computer science, economics, and business.
		Algebra is as useful as all the other branches of mathematics—to which it is closely related.
		The study of the algebra of real numbers and the recent recognition of the fundamental importance of the basic principles have led to the development of what is now called modern algebra or abstract algebra.
	www.britannica.com /eb/article-231055?tocId=231055 (671 words)

	Characteristic (algebra) -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-14)
		And yet another equivalent definition: the characteristic of R is the unique natural number n such that R contains a (Click link for more info and facts about subring) subring (Click link for more info and facts about isomorphic) isomorphic to the (Click link for more info and facts about factor ring) factor ring Z/nZ.
		Since the (A number of the form a+bi where a and b are real numbers and i is the square root of -1) complex numbers contain the rationals, their characteristic is 0.
		It is isomorphic either to the (An integer or a fraction) rational number field Q, or a finite field; the structure of the prime field and the characteristic each determine the other.
	www.absoluteastronomy.com /encyclopedia/c/ch/characteristic_(algebra).htm (600 words)

	Adela Vraciu's Research Interests (Site not responding. Last check: 2007-10-14)
		Characteristic p methods yield powerful results, which can usually be proved in the characteristic zero case as well, via a standard reduction to characteristic p.
		The understanding of this algebra structure is central to the applications and development of tight closure theory.
		In commutative algebra, linkage is defined as an equivalence relation on the set of ideals in a given Gorenstein ring.
	www.math.sc.edu /~vraciu/research.html (374 words)

	Clifford algebra - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-14)
		In characteristic not 2 the algebra Cℓ(V,Q) inherits a Z-grading from the canonical isomorphism with the exterior algebra Λ(V).
		classification of Clifford algebras is isomorphic to the algebra of 4 by 4 complex matrices.
		Paul Dirac when he was trying to write a relativistic first-order wave equation for the electron, and give an explicit isomorphism from the Clifford algebra to the algebra of complex matrices.
	evil-wire.luvfeed.org /cache/3771 (3294 words)

	Subject Index
		characteristic equation of a pencil of matrices 70
		characteristic polynomial of a finite rank-n geometric lattice 172
		characteristic polynomial of a linear recurring sequence 341
	www.elsevier.com /homepage/saj/523281/subject.htm (1223 words)

	Dear Stefan:
		This Hopf algebra is one of particular importance among several ``combinatorial Hopf algebras'' that have arisen recently in different contexts, like Gessel's Hopf algebra of quasi-symmetric functions and Loday-Ronco's Hopf algebra of planar binary trees.
		These Hopf algebras are also known as the restricted specializations of quantized enveloping algebras associated to the semisimple Lie algebra algebra g.
		In fact this Hopf algebra is the degree zero part of a larger one which is based on all the planar trees.
	condor.depaul.edu /~scatoiu/seminar/february2002/abstracts.html (1659 words)

	PlanetMath: alternative algebra
		An alternative algebra is a flexible algebra, provided that the algebra is not Boolean (characteristic
		Cross-references: Jordan identity, Jordan algebra, commutative, clear, associative, subalgebra, iff, Boolean, theorem, identities, algebra, flexible algebra, symmetric group, fixed, coordinates, associator, non-associative algebra
		This is version 5 of alternative algebra, born on 2004-10-10, modified 2005-06-22.
	planetmath.org /encyclopedia/AlternativeAlgebra.html (90 words)

	Commutative Algebra: Homological and Birational Theory
		For rings of positive characteristic, the Frobenius map, which sends an element of the ring to its pth power, where p is the characteristic of the ring, is an extremely powerful tool.
		Rees algebras are the rings in which integral dependence of ideals can be studied and they are the algebraic objects that appear in the process of resolution of singularities.
		Rees algebras have been used in Kawasaki's celebrated proof of the existence of Macaulifications, a weak form of resolution of singularities.
	www.pims.math.ca /birs/workshops/2004/04w5027 (1320 words)

	Álgebra Conmutativa y Aplicaciones (Site not responding. Last check: 2007-10-14)
		Any algebraic variety can be decomposed into a union of equidimensional varieties (that is, algebraic varieties whose irreducible components are all of the same dimension).
		The algorithm is based on a result which allows to obtain the Chow form of an equidimensional variety from a particular description of one of its fibres with respect to a finite morphism.
		There are different ways to describe algebraic varieties (i.e., the set of common zeroes of a set of multivariate polynomials).
	www.famaf.unc.edu.ar /vaq2001/pf/abstracts_alg-conm-aplic.html (1171 words)

	List of Publications
		To be published in the Handbook of Algebra (ELSVIER, Amsterdam, The Netherlands), Vol.
		Characteristic Polynomials of D-Modules and of Finitely Generated D-Field Extensions.
		Characteristic Polynomials of Inversive Difference Modules and Some Properties of Inversive Difference Dimension.
	faculty.cua.edu /levin/Publications.htm (923 words)

	New Simple Lie algebras: Melikian Algebras
		Later on several generalizations of the Witt algebra were constructed (by Jacobson, Zassenhaus, Frank, Kaplansky).
		Later on A. Premet [P1] proved that any non-classical simple Lie p-algebra over an algebraically closed field of characteristic p=5 with a torus of maximal dimension 2 is either of Cartan type or is isomorphic to the Melikian algebra L(1,1), which I constructed in [M1], [M2].
		In characteristic p=3 it is the simple 10 dimensional Kostrikin algebra L(1).
	www.cs.uwm.edu /~melikian/res/node2.html (341 words)

	Probability theory (Site not responding. Last check: 2007-10-14)
		Independent sequences of families of events, and their extension to algebras and sigma-algebras.
		Derivatives of a characteristic function and the moments of the random variable.
		Characterization of the recurrence by using the characteristic function.
	www.ma.ic.ac.uk /~grigor/prob.htm (410 words)

	Edwin H. Connell: Elements of Abstract and Linear Algebra
		It covers abstract algebra in general, but the focus is on linear algebra.
		The goal is to do the minimum amount of abstract algebra necessary to do the linear algebra, and to have material so basic that it is beneficial to students in computer science and the physical sciences.
		Teaching abstract algebra and linear algebra as separate courses results in a loss of synergy and a loss of momentum.
	www.math.miami.edu /~ec/book/author.html (1122 words)

	Linear Algebra - Preliminary lectures (Site not responding. Last check: 2007-10-14)
		It is generally the case that (input or output) data from an experiment is given in a discrete form (discrete measurements).
		Linear Algebra is then useful for solving problems in such a context with applications in topics such as Physics, Fluid Dynamics, Signal Processing and, more generally Numerical Analysis.
		The characteristic polynomial applies also to matrices and the following theorem is given.
	vision.unige.ch /~marchand/teaching/linalg (1819 words)

	Reduced norm in a central simple algebra (Site not responding. Last check: 2007-10-14)
		A neat invariant associated to a central simple algebra A/k is the reduced norm.
		Serre doesn't mention it because it is tangential to his aim of relating Br(k) with Galois cohomology, but it will help us give an explicit description of generalized quaternions (central simple algebras of dimension 4), and will let us give a cohomology-free proof of the triviality of the Brauer group of a finite field.
		But if A is a matrix algebra then A, considered its own left regular representation, is a direct sum of n copies of the standard representation of M
	www.math.harvard.edu /~elkies/M250.04/reduced.html (495 words)

	[No title]
		The associated primes of (0) in H ev are the quotients of the associated primes of Im(Tr Z/p) in F[V ] Z/p by Im(Tr Z/p).
		Then the natural map v i : S(W i) & A is G equivariant, where S(W) denotes the symmetric algebra on W, and, the induced map b(A G) Ä i=1 v G i : b(A G) Ä i=1 S(W i) & A is an epimorphism.
		By the definition of the symmetric algebra functor S() we have 14 INVARIANT THEORY AND THE KOSZUL COMPLEX an inclusion W i H S(W i) i and by definition of v i the triangle W i A S i S(W i) commutes.
	hopf.math.purdue.edu /SmithL/koszulii.txt (5191 words)

	Characteristic of a Ring (Site not responding. Last check: 2007-10-14)
		Investigation of the output characteristic of a rotating ring gas laser...
		IngentaConnect Characteristic Wavelengths of Irregular Structure in Saturn's B R...
		Appendix: Lifting Homomorphisms from Characteristic p to Characteristic 0...
	www.scienceoxygen.com /math/278.html (153 words)

	Modern Algebra II Lecture Notes, 02/11/99 (Site not responding. Last check: 2007-10-14)
		Examples: rings that have characteristic 0 include the ring Z of integers, the ring Q of rationals, the ring R of reals, and the ring C of complex numbers.
		Theorem 13.3, Characteristic of a Ring with Unity: Let R be a ring with unity 1.
		Theorem 13.4, Characteristic of an Integral Domain: The characteristic of an integral domain is 0 or a prime number.
	www.assumption.edu /Alfano/MAT352-SP99/Notes/021199.html (213 words)

	Passman's Abstracts (Site not responding. Last check: 2007-10-14)
		Let K[G] be the group algebra of a torsion group G over an infinite field K, and let U=U(G) denote its group of units.
		If K^t[G] is a twisted group algebra satisfying a nondegenerate multilinear generalized polynomial identity f=0, then we show that G has certain normal subgroups of finite index which can be viewed as being almost central.
		We also consider simple Artinian rings with involution, in characteristic not 2, and we determine those bounded Z-filtrations that are maximal subject to being stable under the action of the involution.
	math.wisc.edu /~passman/abstracts.html (3362 words)

	Predicates (Site not responding. Last check: 2007-10-14)
		Then L is called restricted if the Lie algebra (ad) L is closed under the p-th power map (which associates to a matrix its p-th power).
		Lie algebras of characteristic zero are never restricted.
		However, if L is a Lie algebra of characteristic p>0 then L need not have a Levi subalgebra but the function will always find one if one exists.
	www.umich.edu /~gpcc/scs/magma/text952.htm (310 words)

	[No title]
		NILPOTENCY IN THE HOMOTOPY OF SIMPLICIAL COMMUTATIVE ALGEBRAS JAMES M. Abstract.In this paper, we continue a study of simplicial commutative al* *gebras with finite Andr'e-Quillen homology, that was begun in [19].
		To be more specific, given such a simplicial algebra A having characteristic * 2, M. Andr'e [2] showed that the homotopy groups ßA have the structure of a divided * power algebra*.
		For simplicial supplemented `-algebras in characteristic 2, we will n* eed certain properties of this spectral sequence which can be described by combining the re *sults of [11, x6] with Proposition 1.5.
	hopf.math.purdue.edu /Turner/nilpotence.txt (2827 words)

	Linear Algebra Glossary (Site not responding. Last check: 2007-10-14)
		The characteristic equation of a (square) matrix A is the polynomial equation:
		Values of the scalar lambda which satisfy the characteristic equation are known as eigenvalues of the matrix.
		The characteristic polynomial of a companion matrix is, in fact, the polynomial that was used to define the matrix originally.
	www.csit.fsu.edu /~burkardt/papers/linear_glossary.html (13710 words)

	Algebra and Algorithms for Differential Systems
		Differential algebra is a generalization of classical commutative algebra.
		A difficult problem of differential algebra is to test whether two irreducible components defined by their characteristic sets are included one in the other.
		Elimination in differential algebra allows to go from state-space to input-output representation by eliminating the state variables.
	pauillac.inria.fr /algo/seminars/sem97-98/hubert.html (1506 words)

	Math 310 - Glossary of Linear Algebra Terms
		The algebraic multiplicity of an eigenvalue c of a matrix A is the number of times the factor (t-c) occurs in the characteristic polynomial of A.
		The characteristic polynomial of a n by n matrix A is the polynomial in t given by the formula det(A - tI).
		The column space of a matrix is the subspace spanned by the columns of the matrix considered as a set of vectors.
	www.math.uic.edu /math310/glossary.html (1060 words)

	GAP Manual: 71 The Specht Share Package
		As the (modular) representation theory of these algebras closely resembles that of the (modular) representation theory of the symmetric groups --- indeed, the later is a special case of the former --- many of the combinatorial tools from the representation theory of the symmetric group are included in the package.
		The ``modular'' representation theory of the Iwahori--Hecke algebras of type A was pioneered by Dipper and James [DJ1,DJ2]; here we briefly outline the theory, referring the reader to the references for details.
		is the Hecke algebras analogue of the characteristic of the field in the modular representation theory of finite groups.
	www.math.jussieu.fr /~jmichel/htm/CHAP071.htm (6740 words)

	A Note on the Characteristic Equation in the Max-Plus Algebra - De Schutter, De Moor (ResearchIndex)
		Abstract: In this paper we discuss the characteristic equation of a matrix in the maxplus algebra.
		We show that the derivation of Olsder and Roos is not entirely correct and we give the correct formulas for the coecients of this alternative version of the...
		A note on the characteristic equation in the max-plus algebra.
	citeseer.ist.psu.edu /deschutter97note.html (445 words)

	Algebra Exercises (Site not responding. Last check: 2007-10-14)
		If an eigenvalue has multiplicity two in the characteristic equation of M then M has two linearly independent eigenvectors corresponding to it.
		The highest power of the variable in the characteristic equation of an n by n matrix M has coefficient 1 or -1.
		Since the determinant of a matrix representing a transformation is independent of the basis, so is the characteristic function, and hence the eigenvalues.
	www-math.mit.edu /~djk/18_022/exercises/exercise01.html (805 words)

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