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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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The algebra of quaternions

in** Topics in Quaternion Linear Algebra**

August 2014; p ublished online October 2017 .

Chapter. Subjects: Algebra. 11390 words.

This chapter concerns (scalar) quaternions and the basic properties of quaternion algebra, with emphasis on solution of equations such as *axb* = *c* and *ax* − *xb* = *c*. It studies the Sylvester...

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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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Almost étale coverings

in** The p-adic Simpson Correspondence (AM-193)**

February 2016; p ublished online October 2017 .

Chapter. Subjects: Algebra. 20888 words.

This chapter explains Faltings' theory of almost étale extensions, a tool that has become essential in many questions in arithmetic geometry, even beyond *p*-adic Hodge theory. It begins with...

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Appendix: Real and complex canonical forms

in** Topics in Quaternion Linear Algebra**

August 2014; p ublished online October 2017 .

Chapter. Subjects: Algebra. 3421 words.

This chapter presents canonical forms for real and complex matrices and for pairs of real and complex matrices, or matrix pencils, with symmetries. All these forms are known, and most are...

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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...