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Topic: Quotient algebra

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	Ratio article - Ratio algebra quotient benzene ring carbon hydrogen naphthalene oxygen - What-Means.com (Site not responding. Last check: 2007-10-21)
		In algebra, a ratio is the relationship between two quantities.
		It is expressed as the quotient of one magnitude divided by another, or as a relation between several variables.
		If a school has a twenty-to-one student-teacher ratio, that means that there are twenty times as many students as teachers.
	www.what-means.com /encyclopedia/Ratio (327 words)

	Encyclopedia: Derivative (Site not responding. Last check: 2007-10-21)
		Jump to: navigation, search In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist.
		In algebra, a vulgar fraction consists of one integer divided by a non-zero integer.
		Algebra is a branch of mathematics, which studies structure and quantity.
	www.nationmaster.com /encyclopedia/Derivative (5370 words)

	Affine Algebras (Quotient rings)
		Magma allows one to create the quotient ring of a multivariate polynomial ring P over a field by an ideal J of P. Such quotient rings are known as affine algebras.
		Given an ideal I of an affine algebra Q which is the quotient ring P/J, where P is a polynomial ring and J an ideal of P, return the ideal I' of P such that the image of I' under the natural epimorphism P -> Q is I. PreimageRing(I) : RngMPolRes -> RngMPol
		Given an element f of a finite dimensional affine algebra Q, return the representation matrix of f, which is a d by d matrix over the coefficient field of Q (where d is the dimension of Q) which represents f.
	www.math.colostate.edu /manuals/magma/htmlhelp/text407.html (1050 words)

	Kernel (algebra) : QuicklyFind Info
		In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective.
		Every Mal'cev algebra has a special neutral element (the zero vector in the case of vector spaces, the identity element in the case of groups, and the zero element in the case of rings or modules).
		The notion of ideal generalises to any Mal'cev algebra (as subspace in the case of vector spaces, normal subgroup in the case of groups, two-sided ring ideal in the case of rings, and submodule in the case of modules).
	www.quicklyfind.com /info/Kernel_(algebra).htm (1938 words)

	Quotient (Site not responding. Last check: 2007-10-21)
		For example, in the problem 6 ÷ 3, the quotient would be 2, while 6 would be called the dividend, and 3 the divisor.
		For example, the quotient of 13 ÷ 5 would be 2 while the remainder would be 3.
		In more abstract branches of mathematics, the word quotient is often used to describe sets, spaces, or algebraic structures whose elements are the equivalence classes of some equivalence relation on another set, space, or algebraic structure.
	www.worldhistory.com /wiki/Q/Quotient.htm (229 words)

	Quotient Term Algebra
		Identifying equivalent terms in the term algebra generates what is called the quotient term algebra or quotient term structure, and this aspect is worth developing in a little more detail.
		For the quotient term algebra, each member of the carrier(s) is an equivalence class of terms and we are at liberty to choose any member of an equivalence class to represent that class.
		An important property of the quotient term algebra is that this algebra is initial.
	scom.hud.ac.uk /staff/scomtlm/book/node285.html (340 words)

	Chapter 1Introduction (Site not responding. Last check: 2007-10-21)
		These ``quantum" enveloping algebras appeared in connection with problems in statistical mechanics, and later were shown to have interesting applications in conformal theory and topology.
		They proved that, associated to the quantized enveloping algebra of any simple Lie group at a primitive prime root of unity, there is a semisimple monoidal category with finite number of simple objects.
		This representation theory is very similar to the one of the corresponding Lie algebra in the sense that any integrable module is a direct sum of simple ones, and the simple modules have the same character formula as the ones of the Lie algebra of the same highest weight.
	www.math.vt.edu /quantum_topology/Docs/Bobtcheva_thesis2.html (1178 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		An extended example for the use of the quotient algebra in a Data Warehouse To show the use of the quotient algebra in a Data Warehouse, we use a small data warehouse schema example.
		The steps used to process the multidimensional query using quotient relations are: Restriction of stores which are located in California and partitioning by the attribute ST. EMBED Equation.3 The result of the quotient algebra operation R1 is given in the table 4.14.
		The criterion of the existence of a declarative query language is given by the quotient algebra and has been shown in illustrative examples in the paper.
	www.ifs.tuwien.ac.at /ifs/general_information/people/tjoa/pub_others/man_dolap98.doc (3603 words)

	Canonical Term Algebra and Reduced Expressions
		The individual terms which comprise such a subset are called canonical terms and algebras whose carrier sets consist of such a collection of canonical terms are referred to as canonical term algebras.
		It is not hard to see that the resulting algebra whose carrier set consists of these canonical terms will be isomorphic to the quotient term algebra and hence will itself be initial.
		Compare this with the quotient term algebra where the individual members of the carrier set are sets of values (equivalence classes).
	scom.hud.ac.uk /scomtlm/book/node286.html (324 words)

	QuickMath Automatic Math Solutions
		The tem 'algebra' is used for many things in mathematics, but in this section we'll just be talking about the sort of algebra you come across at high-school.
		Algebra is the branch of elementary mathematics which uses symbols to stand for unknown quantities.
		Although solving equations is really a part of algebra, it is such a big area that it has its own section in QuickMath.
	www.quickmath.com /www02/pages/modules/algebra/index.shtml (642 words)

	ipedia.com: Quantization (physics) Article (Site not responding. Last check: 2007-10-21)
		The classical theory is described using a spacelike foliation of spacetime with the state at each slice being described by an element of a symplectic manifold with the time evolution given by the symplectomorphism generated by a Hamiltonian function over the symplectic manifold.
		The quantum algebra of "operators" is a ℏ-deformation of the algebra of smooth functions over the symplectic space such that the leading term in the Taylor expansion over &hbar; of the commutator [A,B] is iℏ{A,B} where {.,.} is the Poisson bracket.
		Then, this quotient algebra is converted into a Poisson algebra by introducing a Poisson bracket derivable from the action called the Peierls bracket.
	www.ipedia.com /quantization__physics_.html (773 words)

	boolean ring (Site not responding. Last check: 2007-10-21)
		The quotient ring of a Boolean ring modulo a ring ideal corresponds to the factor algebra of the corresponding Boolean algebra modulo the corresponding order ideal.
		The quotient ring R/I of any Boolean ring R modulo any ideal I is again a Boolean ring.
		Every prime ideal P in a Boolean ring R is maximal: the quotient ring R/P is an integral domain and at the same time a Boolean ring, so it must be isomorphic to the field F
	www.yourencyclopedia.net /Boolean_ring (465 words)

	2.3.2 Algebra II -- Dr Stoy -- 16 TT (Site not responding. Last check: 2007-10-21)
		This course develops further some topics from a1 Linear Algebra, continues the study of rings begun in a3 Algebra I and also prepares the ground for b2 Algebra.
		Quotient Structures, which are studied and applied in a variety of contexts, provide a unifying theme running through the course.
		Quotient vector spaces and the associated homomorphism theorems.
	www.maths.ox.ac.uk /current-students/undergraduates/handbooks-synopses/2002/html/sect-a-02/node12.html (186 words)

	GAP Manual: 72 Vector Enumeration
		The algebra homomorphism, the isomorphic module for the matrix algebra, and the module homomorphism can be constructed as described in chapters Algebras and Modules.
		describe the finitely presented algebra, the quotient module it acts on, and the chosen generators names, i.e., the original structures for that VE was called.
		The quotient of a polynomial ring by the ideal generated by some polynomials will be finite-dimensional just when the polynomials have finitely many common roots in the algebraic closure of the ground ring.
	www.mcs.kent.edu /system/documentation/gap/CHAP072.htm (1272 words)

	[ref] 57 Algebras
		For an introduction into the concstruction of quotient algebras we refer to Chapter tut:algebras of the user's tutorial.
		If all involved algebras are matrix algebras, and either both are Lie algebras or both are associative then the result is again a matrix algebra of the appropriate type.
		The difference between an algebra homomorphism and an algebra-with-one homomorphism is that in the latter case, it is assumed that the identity of
	www.math.colostate.edu /manuals/gap/CHAP057.htm (3260 words)

	Discrete Maths at UBC
		Peak functions and Eulerian enumeration: In 1995 the complete duality of Q -- the Hopf algebra of quasisymmetric functions -- and NC -- the Hopf algebra of nonommutative symmetric functions -- was realised by Malvenuto and Reutenauer.
		Combinatorial Hopf Algebras: A Combinatorial Hopf algebra [CH-algebra] is a pair (H, z) where H is a graded connected Hopf algebra over a field F, and z: H -> F is a multiplicative functional.
		Now, in general, we can fix a partition \lambda of n, and ask for algebraic conditions on the coefficients of F so that the roots follow the pattern dictated by the parts in \lambda.
	www.math.ubc.ca /~steph/spring2003.html (922 words)

	[No title]
		Abstract.We apply the tools of stable homotopy theory to the study of mod- ules over the Steenrod algebra A; in particular, we study the (triangula ted) category Stable(A) of unbounded cochain complexes of injective comodules over A, the dual of A, in which the morphisms are cochain homotopy class* *es of maps.
		This material applies when is the dual of a group algebra, the dual of an enve* loping algebra, or the dual of the Steenrod algebra;* in these cases, Ext*(k; k) is the ordinary cohomology of with coefficients in k.
		Given a p-group G and an algebraically closed field k of characteristic p, t* hey define VG(k) to be the maximal ideal spectrum of the (graded) commutative Noe- therian ring H(G; k).
	hopf.math.purdue.edu /Palmieri/palmieri-steenrod.txt (9567 words)

	[No title]
		In other words, the Hopf algebras Hn and their duals may challenge the intuition of any readers (or authors) who are used to working with the Steenrod algebra.
		In this section, we show that after tensoring with F, each of these Hopf algebras is dual to a group algebra, we identify the groups, and we observe that the group associated to Hnjis torsion-free for each n and j.
		Bnjis dual to the group algebra of a finite group, and_the group_is precisely the group of points in the group scheme defined by Bnj Fp, where Fp is the algebraic closure of Fp: __ __ Pjn= Hom _Fp-alg(Bnj Fp, Fp).
	hopf.math.purdue.edu /Palmieri/quotient.txt (3601 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		A numerated Boolean algebra B is called recursively perfect, if a finite sequence of iterated quotients by the Frechet ideal consists of atomic Boolean algebras, except for the last in the sequence, which is a Xi-universal Boolean algebra over some class Xi of a hierarchy.
		A result obtained recently (jointly with S. Lempp and R. Solomon) states that the Lindenbaum algebra of the theory of the class M_fin is recursively perfect, namely, it is atomic, while its quotient algebra modulo the Frechet ideal is a Sigma^0_2-universal Boolean algebra.
		Some rough estimates show that the Lindenbaum algebras of the classes of models generated by the classes D, F, N, C, A, as well as these classes together with the class P are probably all recursively perfect (estimates of the algorithmic complexity of theories of some combinations of these classes are obtained in [2]).
	www.math.psu.edu /simpson/talks/cta/pere (464 words)

	Project-Team - galaad (Site not responding. Last check: 2007-10-21)
		During the process of resolution of polynomial systems, computing normal forms in an algebra quotient is usually done by exact methods.
		It consists of computing an implicit representation for an algebraic surface that is given parametrically in the projective space.
		It was conjectured in [50] that the quotient structure can be recover from these Bezoutian matrices by simple linear operations on their rows and columns.
	www.inria.fr /rapportsactivite/RA2003/galaad2003/module13.html (816 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)

	[No title]
		Thus the algebra ${\teneuf C}_0^X$ is obtained by choosing ${\cal A}$ equal to the closed linear space generated by functions of the form $\varphi\circ\pi_Y$ with $\varphi\in C_0(X/Y)$.
		The algebras $C_0(Y^)$ and $C_0(Y^)\otimes \teneuf K(Y_{\alpha}^{\bot})$ are denoted ${\mathbb T}(Y)$ and ${\cal T}(Y_{\alpha}^{\bot})$ in \cite{ABG}.
		Since the algebra ${\teneuf C}_0^X$ is generated by functions of the form $\varphi(Q)\psi(P)$ (or $\psi(P)\varphi(Q)$) with $\psi\in{\cal S}(X^*)$, it is easy to prove that $[\cchi(Q), T]\in{\teneuf K}(X)$ for all $T\in{\teneuf C}_0^X$.
	www.ma.hw.ac.uk /EJDE/conf-proc/04/d1/damak-tex (4929 words)

	Basic Algebra Calculators
		Algebra -- "Quickmath can expand, factor or simplify virtually any expression, cancel common factors within fractions, split fractions up into smaller ('partial') fractions and join two or more fractions together into a single fraction."
		Partial Fraction Expansion of a Rational Function -- "This tool performs a partial fraction expansion on a rational function", which is a quotient of polynomials.
		Math2.org -- Theorems, tables, identities, proofs and graphs for general math, algebra, geometry, trig, linear algebra, discrete math, statistics, calculus, and advanced topics.
	www.ifigure.com /math/algebra/algebra.htm (890 words)

	Technical Monograph PRG-118
		In particular, we show that the algebra of normal forms in a terminating system is a uniquely minimal covering of the term algebra.
		In the non-terminating case, the existence of this minimal covering is established in the completion of an ordered algebra formed by rewriting sequences.
		These include the existence of normal forms for arbitrary rewrite systems, and their uniqueness for confluent systems, in which case the algebra of normal forms is isomorphic to the canonical quotient algebra associated with the rules when seen as equations.
	web.comlab.ox.ac.uk /oucl/publications/monos/prg-118.html (159 words)

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