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Topic: Algebraic structure


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In the News (Sat 17 Aug 19)

  
  Guide to the Mathematics Subject Classification Scheme
Algebra is principally concerned with symmetry, patterns, discrete sets, and the rules for manipulating arithmetic operations; one might think of this as the outgrowth of arithmetic and algebra classes in primary and secondary school.
55: Algebraic topology is the study of algebraic objects attached to topological spaces; the algebraic invariants illustrate some of the rigidity of the spaces.
One might characterize algebra and geometry as the search for elegant conclusions from small sets of axioms; in analysis on the other hand the measure of success is more frequently the ability to hone a tool which could be applied throughout science.
www.math.niu.edu /~rusin/known-math/index/beginners.html   (5525 words)

  
 Encyclopedia :: encyclopedia : Algebraic topology   (Site not responding. Last check: 2007-10-19)
Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces.
The basic method now applied in algebraic topology is to investigate spaces via algebraic invariants, by mapping them, for example, to groups which have a great deal of manageable structure in a way that respects the relation of homeomorphism of spaces.
Beyond simplicial homology, which is defined only for simplicial complexes, one can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf cohomology to investigate the solvability of differential equations defined on the manifold in question.
www.hallencyclopedia.com /Algebraic_topology   (613 words)

  
 Matrix (mathematics) - Wikipedia, the free encyclopedia
Matrices are used to describe linear equations, keep track of the coefficients of linear transformations and to record data that depend on two parameters.
Matrices can be added, multiplied, and decomposed in various ways, marking them as a key concept in linear algebra and matrix theory.
M(n, C), the ring of complex square matrices, is a complex associative algebra.
en.wikipedia.org /wiki/Matrix_(mathematics)   (1664 words)

  
 Catalogue of Algebraic Systems.
Hermann Grassmann and the Prehistory of Universal Algebra by Desmond Fearnley-Sander
Algebraic Systems This is essentially a textbook on algebraic systems put together by Christer Blomqvist.
It is intended for undergraduate students taking an abstract algebra class at the junior/senior level, as well as for students taking their first graduate algebra course.
www.math.usf.edu /~eclark/algctlg   (669 words)

  
 The world's top Mathematics websites
Although mathematics itself is not usually considered a natural science, the specific structures that are investigated by mathematicians often have their origin in the natural sciences, most commonly in physics.
The study of structure starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra.
The modern fields of differential geometry and algebraic geometry generalize geometry in different directions: differential geometry emphasizes the concepts of functions, fiber bundles, derivatives, smoothness and direction, while in algebraic geometry geometrical objects are described as solution sets of polynomial equations.
www.websbiggest.com /dir-wiki.cfm/Mathematics   (1825 words)

  
 Resource: Learning Math: Patterns, Functions, and Algebra
Learning Math: Patterns, Functions, and Algebra is the first of five video- and Web-based mathematics courses for elementary and middle school teachers.
Patterns, Functions, and Algebra explores the “big ideas” in algebraic thinking, such as finding, describing, and using patterns; using functions to make predictions; understanding linearity and proportional reasoning; understanding non-linear functions; and understanding and exploring algebraic structure.
Begin to explore what it means to think algebraically and learn to use algebraic thinking skills to make sense of different situations.
www.learner.org /resources/series140.html   (732 words)

  
 Algebraic structure at opensource encyclopedia   (Site not responding. Last check: 2007-10-19)
In abstract algebra, an algebraic structure consists of a set together with one or more operations on the set which satisfy certain axioms.
In case there are no ambiguities, we usually identify the set with the algebraic structure.
Depending on the operations and axioms, the algebraic structures get their names.
www.wiki.tatet.com /Algebraic_structure.html   (462 words)

  
 Algebraic Structure of Complex Numbers
The theory of complex numbers can be developed wholy in algebraic terms, see, for example, Landau.
The possibility of embedding of the set R of reals into the set of complex numbers C, as defined by (1), is probably the single most important property of complex numbers.
For, without (1) and (2), the theory of complex numbers would not deliver the closure to the branch of algebra that drove much of its development, viz., the search for the roots of polynomial equations.
www.cut-the-knot.org /arithmetic/algebra/ComplexNumbers.shtml   (1242 words)

  
 Inside Reality - A Soup of Computing, Maths and Philosophy
In abstract algebra we must make distinctions based on the type of objects to which we apply operations, so addition for example would be defined differently for integers, rational numbers and vectors.
Sometimes the word algebra is used to mean a specific algebraic structure rather than the field of study.
This post considers the 'group', a type of algebraic structure which all readers are familliar with - whether or not you know it.
www.insidereality.net   (6160 words)

  
 Boolean logic - Facts, Information, and Encyclopedia Reference article
Boolean logic is named after George Boole, an English mathematician at University College Cork, who first defined an algebraic system of logic in the mid 19th century.
The algebra of sets will be used here as a way to introduce Boolean logic.
Mathematicians, engineers, and programmers often use + for OR and · for AND (since in some ways those operations are analogous to addition and multiplication in other algebraic structures and this notation makes it very easy to get sum of products form for people who are familiar with normal algebra).
www.startsurfing.com /encyclopedia/b/o/o/Boolean_logic.html   (1722 words)

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