##
ABSTRACT STRUCTURES

in** Category Theory**

May 2006; p ublished online September 2007 .

Chapter. Subjects: Algebra. 7867 words.

This chapter focuses on *abstract characterizations*. It is shown that one of the basic ways of giving abstract characterization is via a Universal Mapping Property (UMP). Epis and monos,...

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ADJOINTS

in** Category Theory**

May 2006; p ublished online September 2007 .

Chapter. Subjects: Algebra. 14096 words.

This chapter focuses on notion of adjoint functor, which applies everything that has been learned so far to unify and subsume all the different universal mapping properties encountered,...

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Algebra

October 2015; p ublished online October 2015 .

Book. Subjects: Algebra. 160 pages.

Algebra marked the beginning of modern mathematics, moving it beyond arithmetic, which involves calculations featuring given numbers, to problems in which some quantities are unknown. Now,...

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Algebra and the arithmetic of remainders

in** Algebra**

October 2015; p ublished online October 2015 .

Chapter. Subjects: Algebra. 4771 words.

‘Algebra and the arithmetic of remainders’ considers a new type of algebra, which is both an ancient topic and one that has found major contemporary application in Internet cryptography. It...

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The algebra of polynomials and cubic equations

in** Algebra**

October 2015; p ublished online October 2015 .

Chapter. Subjects: Algebra. 8772 words.

Polynomials are expressions of the form *p(x) = a _{0} + a_{1}x + a_{2}x^{2} + ... + a_{n}x^{n}
*; the number

*a*is the coefficient of

_{i}*x*,

_{i}*a*

_{0}is the constant...

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Algorithmic aspects

in** The Diophantine Frobenius Problem**

December 2005; p ublished online September 2007 .

Chapter. Subjects: Algebra. 11081 words.

This chapter is devoted to the computational aspects of the Frobenius number. After discussing a number of methods to solve **FP** when n = 3 (some of these procedures make use of diverse...

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Applications of the Frobenius number

in** The Diophantine Frobenius Problem**

December 2005; p ublished online September 2007 .

Chapter. Subjects: Algebra. 10973 words.

This chapter presents a number of applications of **FP** to a variety of problems. The complexity analysis of the Shellsort method was not well understood until J. Incerpi and R. Sedgewick...

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AUTOMORPHIC FORMS ON INNER FORMS OF <i>GL</i>(2)

in** Hilbert Modular Forms and Iwasawa Theory**

June 2006; p ublished online September 2007 .

Chapter. Subjects: Algebra. 58713 words.

This chapter is an exposition of the ideas and classical (and modern) results of the theory for Hilbert modular forms and quaternionic automorphic forms. It starts with a brief exposition...

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CATEGORIES

in** Category Theory**

May 2006; p ublished online September 2007 .

Chapter. Subjects: Algebra. 8344 words.

This chapter begins with a discussion of category theory, which can be defined as the mathematical study of (abstract) *algebras of functions*. The functions of sets, definitions of a...

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CATEGORIES OF DIAGRAMS

in** Category Theory**

May 2006; p ublished online September 2007 .

Chapter. Subjects: Algebra. 6900 words.

This chapter presents the proof for the Yoneda Lemma, which is probably the single most used result in category theory. It is interesting how often it comes up, especially in view of the...