In this paper the Clifford Algebra is introduced and proposed as analternative to Gibbs' vector algebra as a unifying language for geometricoperations on vectors. Firstly, the algebra is constructed using a quotientof the tensor algebra and then its most important properties are proved,including how it enables division between vectors and how it is connected tothe exterior algebra. Further, the Clifford algebra is shown to naturallyembody the complex numbers and quaternions, whereupon its strength indescribing rotations is highlighted. Moreover, the wedge product, is shown asa way to generalize the cross product and reveal the true nature ofpseudovectors as bivectors. Lastly, we show how replacing the cross productwith the wedge product, within the Clifford algebra, naturally leads tosimplifying Maxwell's equations to a single equation.

De grundläggande aritmetiska räknelagarna är centrala för elevers utveckling inom algebra. Det är därför viktigt att elever ges möjlighet att urskilja och utveckla förståelse för dessa. Genom denna kvalitativa intervjustudie undersöktes hur 16 elever i årskurs 2-5, utifrån en i förväg designad instruktionssekvens, resonerar om, generaliserar och använder den associativa egenskapen för addition. Studien visar att många elever faktiskt urskiljer den associativa egenskapen för addition genom arbetet med instruktionssekvensen, men att endast ett fåtal tillämpar denna egenskap vid beräkning. Studien visar även att flera elever efter att ha urskilt associativitet kan göra generaliseringar av egenskapen relaterad till addition eller subtraktion. Slutsatsen av studien är att en instruktionssekvens som erbjuder systematisk variation och upprepning av uttryck med samma struktur, möjliggör för elever att urskilja och beskriva associativitet. Urskiljandet möjliggörs även av att elever uppmanas att betrakta uttrycken från flera perspektiv och beskriva dem utifrån frågor om likheter och skillnader. / The fundamental properties of arithmetic is a central aspect for students´ understanding of algebra. Therefore, it is important that students get the opportunity to discern and understand these properties. In this qualitative study based on interviews I examine how 16 students in grades 2-5, reason, make generalizations and use the associative property of addition when working on a pre-designed sequence of instructions. The study shows that many students do discern the associative property of addition when working on the sequence of instructions, but only a few of them apply this property when calculating. The study also shows that several of the students are making generalizations about the associative property by relating it to addition or subtraction. The conclusion of the study is that a sequence of instructions that offers systematic variation and a reccurance of expressions based on the same structures, makes it possible for students to discern and describe associativity. The discernment is also made possible by prompting the students to view the expressions from different perspectives and describing the expressions in terms of similarities and differences.

Det övergripande syftet med detta arbete är att ta reda på gymnasieelevers uppfattningar och missuppfattningar när de ska läsa, tolka och lösa uppgifter med algebraiska symboler. Arbetet fokuserar på elevernas svårigheter vilka upplevs olika i olika sammanhang såsom till exempel generalisering av tal, okänt tal och variabel. För att få svar på detta, intervjuades åtta gymnasieelever från olika årskurser och med skiftande algebrakunskaper. Resultatet blev att elever uppvisar olika svårigheter med algebraiska symboler, vilket stämmer överens med den forskning som är genomförd inom området. Detta ses som en följd av övergången från konkreta beräkningar i aritmetik till abstrakta och strukturerade beräkningar i algebra. Svårigheterna beror bland annat bero på att algebraiska symboler och tecken kan byta roll i ett algebraiskt uttryck medan elever brukar tolka dessa symboler endast ur en synvinkel. Elever ska därmed vara medvetna om symbolernas olika placeringar och dess egentliga innebörd. Slutsatsen banar då väg för lärare att arbeta utifrån olika utgångspunkter. / <p>Matematik</p>

by Ng Siu-Hung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaves 80-81). / Introduction --- p.iii / Chapter 1 --- Distributive Modules --- p.1 / Chapter 1.1 --- Basic Definitions --- p.1 / Chapter 1.2 --- Distributive modules --- p.3 / Chapter 1.3 --- Direct sum of distributive modules --- p.9 / Chapter 1.4 --- Endomorphisms of a distributive module --- p.13 / Chapter 1.5 --- Distributive modules satisfying chain conditions --- p.20 / Chapter 2 --- Rings with distributive lattices of right ideals --- p.25 / Chapter 2.1 --- Rings of quotients of right D-rings --- p.25 / Chapter 2.2 --- Localization of right D-rings --- p.28 / Chapter 2.3 --- Reduced primary factorizations in right ND-rings --- p.31 / Chapter 2.4 --- ND-rings --- p.38 / Chapter 3 --- Distributive modules over commutative rings --- p.43 / Chapter 3.1 --- Multiplication modules --- p.43 / Chapter 3.2 --- Properties of distributive modules over commutative rings --- p.48 / Chapter 3.3 --- Distributive modules over arithematical rings --- p.52 / Chapter 4 --- Chinese Modules and Universal Chinese rings --- p.59 / Chapter 4.1 --- Introduction --- p.59 / Chapter 4.2 --- Chinese Modules and CRT modules --- p.61 / Chapter 4.3 --- Universal Chinese Rings --- p.65 / Chapter 4.4 --- Chinese modules over Noetherian domains --- p.70 / Chapter 4.5 --- Remarks on CRT modules --- p.77 / Bibliography --- p.80

The purpose of this section is to define a boolean algebra and to determine some of the important properties of it. A boolean algebra is a set B with two binary operations, join and meet, denoted by + and juxtaposition respectively, and a unary operation, complement ation, denoted by ', which satisfy the following axioms: (1) for all a,b ∑ B (that is, for all a,b elements of B) a + b = b + a and a b = b a, (the commutative laws), (2) for all a,b,c ∑ B, a + b c =(a + b) (a + b) and a (b + c) = a b + a c, (the distributive laws), (3) there exists 0 ∑B such that for each a ∑B, a + 0 = a, and there exists 1 ∑B such that for each a ∑ B, a 1 = a, (4) for each a∑B, a + a' = 1 and a a' = 0. If a + e = a for all a in B then 0 = 0 + e = e + 0 = e, so that there is exactly one element in B which satisfies the first half of axiom 3, namely 0. Similarly there is exactly one element in B which satisfies the second half of axiom 3, namely 1. The O and 1 as defined above will be called the distinguished elements.

There has been much interest in algebras which have a basis consisting of diagrams, which are multiplied in some natural diagrammatic way. Examples of these so-called diagram algebras include the partition, Brauer and Temperley-Lieb algebras. These three examples all have the property that the product of two diagram basis elements is always a scalar multiple of another basis element. Motivated by this observation, we find that these algebras are examples of twisted semigroup algebras. Such algebras are an obvious extension of twisted group algebras, which arise naturally in various contexts; examples include the complex numbers and the quaternions, considered as algebras over the real numbers. The concept of a cellular algebra was introduced in a famous paper of Graham and Lehrer; an algebra is called cellular if it has a basis of a certain form, in which case the general theory of cellular algebras allows us to easily derive information about the semisimplicity of the algebra and about its representation theory, even in the non-semisimple case. Many diagram algebras (including the above three examples) are known to be cellular. The aim of this thesis is to deduce the cellularity of these examples (and others) by proving a general result about the cellularity of twisted semigroup algebras. This will extend a recent result of East. In Chapters 2 and 3 we discuss semigroup theory and twisted semigroup algebras, and realise the above three examples as twisted semigroup algebras. Chapters 4 to 7 detail and extend slightly the theory of cellular algebras. In Chapter 8 we state and prove the main theorem, which shows that certain twisted semigroup algebras are cellular. Under the assumptions of the main theorem, we explore the cell representations of twisted semigroup algebras in Chapter 9. Finally in Chapter 10, we apply the theorem to various examples, including the three diagram algebras mentioned above.

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Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1997. / Includes bibliographical references. Also available on the Internet.

Given a holomorphic self-map of complex projective space of de-gree larger than one, we prove that there exists a finite collection oftotally invariant algebraic sets with the following property: given anypositive closed (1,1)-current of mass 1 with no mass on any element of this family, the sequence of normalized pull-backs of the current converges to the Green current. Under suitable geometric conditions on the collection of totally invariant algebraic sets, we prove a sharper equidistribution result. / <p>QC 20110530</p>

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