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Topic: Alternative algebra

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	Cayley-Dickson construction - Wikipedia, the free encyclopedia
		The algebras produced by this process are known as Cayley-Dickson algebras; since they extend the complex numbers, they are hypercomplex numbers.
		These algebras all have a notion of norm and conjugate, with the general idea being that the product of an element and its conjugate should equal the square of its norm.
		This algebra was discovered by Graves in 1844, and is called the octonions or the "Cayley numbers".
	en.wikipedia.org /wiki/Cayley-Dickson_construction (737 words)

	Station Information - Alternative algebra
		In abstract algebra, an algebra (or more generally a magma) is called alternative if the subalgebra generated by any two of its elements is associative.
		An equivalent definition is to require, for all x and y in an algebra A, that x(xy) = (xx)y and (xy)y = x(yy).
		Every associative algebra is obviously alternative, but so too are some non-associative algebras such as the octonions.
	www.stationinformation.com /encyclopedia/a/al/alternative_algebra.html (111 words)

	Alternative algebra (Site not responding. Last check: 2007-11-06)
		In abstract algebra, an algebra is called alternative if (''xx'')''y''=''x''(''xy'') and y(''xx'')=(''yx'')''x'' for all x and y in the algebra, that is, if the multiplication is alternative.
		Equivalently, an algebra is alternative if and only if the subalgebra generated by any two of its elements is associative.
		Alternativity in algebras is a condition inbetween associativity and power associativity.
	www.keywordmage.net /al/alternative-algebra.html (117 words)

	[No title]
		Alternative elements in the Cayley-Dickson algebras Guillermo Moreno Abstract n We describe the alternative elements in A n = R 2 the Cayley- Dickson algebras for n 4.
		Notice that since `e0' is normed, alternative and strongly alternative el- ement we have that a is normed with b, a alternate with b and a alternate strongly with b if the pure part of a is normed, alternate and strongly alter- nate with b.
		On the automorphisms of the Cayley-Dickson algebras and its derivations.
	hopf.math.purdue.edu /Moreno/moreno.txt (4675 words)

	Alternative algebra -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-06)
		Equivalently, an algebra is alternative if and only if the (Click link for more info and facts about subalgebra) subalgebra generated by any two of its elements is (Click link for more info and facts about associative) associative.
		Every (Click link for more info and facts about associative algebra) associative algebra is obviously alternative, but so too are some non-associative algebras such as the (Click link for more info and facts about octonion) octonions.
		Alternativity in algebras is a condition inbetween associativity and (Click link for more info and facts about power associativity) power associativity.
	www.absoluteastronomy.com /encyclopedia/a/al/alternative_algebra.htm (132 words)

	Bulletin - 3rd December 2004
		A simpler alternative to algebra for solving maths and science problems has been invented by a Sussex cognitive scientist.
		"Algebra is a fundamental part of maths, but its difficulty is that it is very abstract," points out Peter.
		His diagrams, which, for example, convert an electric circuit of light bulbs into boxes that are assembled like building blocks to demonstrate different properties of the circuit, were inspired by the geometric designs of early scientists such as Galileo.
	www.sussex.ac.uk /press_office/bulletin/03dec04/article3.shtml (408 words)

	Articles - Algebra over a field (Site not responding. Last check: 2007-11-06)
		Algebras can also more generally be defined over any commutative ring K: we need a module A over K and a bilinear multiplication operation which satisfies the same identities as above; then A is a K-algebra, and K is the base ring of A.
		When the algebra can be endowed with a metric, then the structure coefficients are written with upper and lower indices, so as to distinguish their transformation properties under coordinate transformations.
		A commutative algebra is one whose multiplication is commutative; an associative algebra is one whose multiplication is associative.
	lastring.com /articles/Algebra_over_a_field?... (1322 words)

	Alternative - Wikipedia, the free encyclopedia
		An alternative is one of two (or more) possible choices.
		Alternative medicine, including Traditional Chinese Medicine, Acupuncture, etc.
		Alternative lifestyle, sometimes used non-derogatorily to describe homosexuals
	en.wikipedia.org /wiki/Alternative (91 words)

	Alternative (Site not responding. Last check: 2007-11-06)
		Alternative Lifestyles Forum A place to discuss environmental issues, alternative community building, intentional communities, ecovillages, alternative education and parenting, alternative energy and sustainable resources.
		Alternative Press Index Extensive directory of links and information about alternative media and other resources.
		Alternative Training Distance learning courses in conventional medicine for alternative practitioners.
	www.serebella.com /encyclopedia/article-Alternative.html (272 words)

	Red Mountain Mathematics (Site not responding. Last check: 2007-11-06)
		Algebra 1 curriculum that will assist students in meeting state mathematics standards expectations.
		A fast paced third-year course that extends the concept of algebra and geometry, and prepares students for enrollment in pre calculus.
		Students will be exposed to an in depth study of algebraic, trigonometric, logarithmic, exponential functions, related graphing applications and an introduction to limits.
	www.mpsaz.org /rmhs/classes/mathematics (249 words)

	Cushing \| Academics \| Course Catalog \| Mathematics (Site not responding. Last check: 2007-11-06)
		Honors Algebra I is an alternative to Algebra I for talented and highly-motivated mathematics students.
		Honors Algebra II is an alternative to Algebra II for talented and highly-motivated mathematics students.
		Algebra II Modified-A is the first part of a two-year course sequence that is designed specifically for those students who need more time and individual attention in order to become proficient with the concepts of Algebra.
	www.cushing.org /academics/courses-math.shtml (1255 words)

	Citations: Introduction to non-associative algebra - Schafer (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Fix C to be the split Cayley algebra endowed with hyperbolic norm form n and canonical involution denoted by.
		Thus, the dimension obtained by a particular two generator construction of an alternative algebra should remain unchanged when the identities are replaced by the associative law.
		Algebraic framework In this section we formulate quantum spin chains in a general algebraic framework and define the sequence of charges with Catalan tree pattern.
	citeseer.ist.psu.edu /context/171730/0 (2054 words)

	Zorn (Site not responding. Last check: 2007-11-06)
		These are algebras in which (xy)z - x(yz) is an alternating function in the sense that it is zero whenever any two of x, y, z are equal or, put another way, any two dimensional subalgebra is associative.
		He studied the structure of semisimple alternative rings in 1932, proving that such a ring is a direct sum of simple alternative algebras which he classified.
		In Alternative rings and related questions I: existence of the radical published in 1941 Zorn considered the theory of the radical of an alternative ring.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Zorn.html (1490 words)

	Science Fair Projects - Power associativity
		An algebra (or more generally a magma) is said to be power-associative if the subalgebra generated by any element is associative.
		Every associative algebra is obviously power-associative, but so are all other alternative algebras (like the octonions, which are non-associative) and even some non-alternative algebras like the sedenions.
		For f a real polynomial in x, and for any a in such an algebra define f(a) to be the element of the algebra resulting from the obvious substitution of a into f.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Power-associativity (424 words)

	CRHS Math Scope and Sequence (Site not responding. Last check: 2007-11-06)
		This course is an alternative to Algebra 1 for students who are not college bound OR who do not intend to major in the mathematics or scientific fields.
		This course is an alternative to Plane Geometry for students who are not college bound OR who do not intend to major in the mathematics or scientific fields.
		Designed for students planning to enter college, Algebra II covers the sets of numbers, open sentences in one variable, systems of linear open sentences, factoring of polynomials, relations and functions, rational and irrational numbers, quadratic relations, exponential functions, logarithms, series and progressions, and matrices and determinants.
	cl.k12.md.us /CRHSmath/CRHSmath/Scope.html (909 words)

	Alternative article - Alternative Alternative medicine Traditional Chinese Medicine Acupuncture - What-Means.com (Site not responding. Last check: 2007-11-06)
		Alternative article - Alternative Alternative medicine Traditional Chinese Medicine Acupuncture - What-Means.com
		This is a disambiguation page; that is, one that points to other pages that might otherwise have the same name.
		Alternative article - Alternative definition - what means Alternative
	www.what-means.com /encyclopedia/Alternative (94 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		An algebra is alternative >iff its each subalgebra generated by two elements, is associative -- that is >Artin's lemma).
		It's a surprising fact that a dvision algebra over R is alternative iff it is normed.
		I believe that it's true that Hurwitz proved his result for normed algebras; I'm not sure who first noted the equivalence with alternative algebras; maybe it was Zorn, his name is linked to non-associative algebra.
	www.math.niu.edu /~rusin/known-math/98/div_alg_hist (437 words)

	ACCELERATION: ALGEBRA I IN GRADE 8
		The goal of offering Algebra I in grade 8 is to maintain the advantage of enrichment in mathematics for as many students as can benefit from it.
		However, since Algebra I is the basis for all further study in mathematics, it is important that students have the best possible experience in this course.
		All algebra courses taught in Grade 8 will be subject to ongoing review to assure that the class is a high school equivalent.
	www.aps.edu /aps/policy/Directives/ALGEBRA.html (1198 words)

	math lessons - Sedenion
		The sedenions form a 16-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the octonions.
		But in contrast to the octonions, the sedenions do not even have the property of being alternative.
		The sedenions have a multiplicative identity element 1 and multiplicative inverses, but they are not a division algebra.
	www.mathdaily.com /lessons/Sedenion (170 words)

	The ``Pi Is Rational'' Page :)
		Using The Fundamental Postulate and Fundamental Theorem of Alternative Algebra (26 October 1995)
		Using The Fundamental Postulate and Fundamental Theorem of Alternative Algebra
		Alternative algebra can be summarized as two statements, the Fundamental Postulate and Fundamental Theorem of Alternative Algebra.
	dse.webonastick.com /pi (1089 words)

	Teaching Algebra: alternative energies term papers, alternative energies research papers, alternative energies essays (Site not responding. Last check: 2007-11-06)
		A large category of algebra misconceptions has been documented in Algebra: Some Common Misconceptions under the category of "The Meaning of Letters." These relate to students' difficulty with the meaning of letter variables.
		Or they may think that there are: 1) "rules used to determine which number a letter stands for", 2) specific letters for specific values, 3) a letter can only have one value, 4) a letter has to be a natural number, and 5) forgetting the signs for the variables.
		Within a few clicks, you can find free abstracts for the perfect Alternative Energies term paper on “Teaching Algebra.” At alternativeenergiestermpapers.com, you can perform general searches for any paper topic, as well specific searches for Alternative Energies term papers that will instantly return results for abstracts contained in the alternativeenergiestermpapers.com database.
	www.alternativeenergiestermpapers.com /paper/456/teaching-algebra.html (374 words)

	Sedenion Basic Algebra
		A sedenion is a hypercomplex number constituted from 16 basal elements.
		Not a composition algebra : n(ST) is not necessarily equal to n(S) n(T), where the norm of S is defined as
		Its unique algebraic properties, its zero divisors, and the solutions to a general linear equation are found.
	www.geocities.com /zerodivisor/sbasicalgebra.html (163 words)

	The world's top Resources websites (Site not responding. Last check: 2007-11-06)
		Health and alternative medicine resource includes sections on disease and alternative treatments plus nutrition and weight loss issues.
		Resources to practical alternatives for maintaining or improving one's health and recovering from chronic illnesses.
		Forum for discussion of alternative treatments for children with developmental delays, brain injury and other related neurological and metabolic conditions.
	dirs.org /dir-wiki.cfm/Top/Health/Alternative/Resources (1096 words)

	Math Curriculum (Site not responding. Last check: 2007-11-06)
		This course continues the study of the structure of algebraic of algebra and to provide the foundation for applying these skills to other mathematical and scientific fields.
		Topics include algebraic structure; first-degree equations in one and two variables solved algebraically and graphically; systems of equations and inequalities; functions and relations; polynomial and rational expression; exponents and radicals; logarithms; complex numbers; conic sections; polynomial equations; sequence and series; permutations, combinations and probability; and matrices.
		It is NOT an alternative to Algebra II.
	www.sdhc.k12.fl.us /~GAITHER.HIGH/curriculum/math.htm (1213 words)

	Reviewing the textbook 'Glencoe Pre-Algebra' (1997; Glencoe)
		Marianne Jennings described some fuzzy math to readers of The Textbook Letter in her 1996 article "Rain-Forest Algebra and MTV Geometry." That article was Jennings's analysis of fuzzy-math books that her daughter had been forced to use in public schools in Arizona [see note 1, below].
		Glencoe's two expectations reflect a basic tenet of fuzzy math: The "reformers" who have promoted fuzzy math have explicitly renounced the idea that it is necessary for each student to acquire certain basic skills, at certain points in the math curriculum, before moving forward [note 5].
		The term alternative assessment is an established code-phrase for the process of creating the impression that backward students have succeeded when they actually (and predictably) have failed.
	www.textbookleague.org /102fuzz.htm (5427 words)

	week59
		If your algebra has inverses, you might try to multiply this equation by the inverse of x or y (whichever one isn't 0) to prove the other is 0, but this only works if the algebra is associative.
		From this, the quaternions form a division algebra and are associative, but they neither are commutative, equal their own conjugates, nor have characteristic 2.
		In light of the results about alternativity, my original attempt to prove that division of 2^n onions requires associativity of 2^(n-1) onions looks even more convincing, (since alternativity of 2^(n-1) onions can be included in the induction hypothesis), but it's still not valid.
	math.ucr.edu /home/baez/week59.html (4525 words)

	Kenston High School ~ Mathematics Courses (Site not responding. Last check: 2007-11-06)
		If credit is awarded to the student, the student's final grade in either Algebra or Geometry will be computed into the student's high school GPA, final class rank (if applicable), and also indicated on the student's official high school transcript at the end of their freshman year.
		Integrated Algebra and Integrated Geometry together are an alternative to Algebra and Geometry.
		This study in intermediate Algebra expands on the concepts studied in both first-year Algebra and Geometry.
	www.kenston.k12.oh.us /khs/guidance/math.htm (1166 words)

	Preliminaries
		An associative algebra has multiplicative inverses iff it is a division algebra.
		In Section 2.2 we shall construct algebras that have multiplicative inverses, but are not division algebras.
		An algebra is power-associative if the subalgebra generated by any one element is associative.
	math.ucr.edu /home/baez/octonions/node2.html (533 words)

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