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| − | [[Category:Articles containing maths]][[Category:Glossary]][[Category:Philosophy]]
| + | *[[Distributed Space]] ''- Research on the kind of network space required by [[the project]]'' |
| − | *Organising some notes about spacetime fundamentals from pedia
| + | *[[Physical Space]] ''- notes on the physical structure of space'' |
| − | ;See also
| + | *[[Conceptual Space]] ''- About the space of perceived experience referred to by [[w:Taoism|Taoism]] and [[w:Advaita vedanta|Advaita vedanta]]'' |
| − | *[[e]] | + | *[[NodeSpace]] ''- The first attempt at designing a network space fot [[the project]]'' |
| − | *[[Conceptual Space]]
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| − | = Notes on the general structure =
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| − | Every [[w:brane|p-brane]] sweeps out a (p+1)-dimensional world-volume as it propagates through spacetime. The volume of an ''n''-dimensional hypersphere can be expressed as:
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| − | ::<math>V_n={\pi^{n/2}R^n\over (n/2)!}.</math>
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| − | Note that the [[w:Gamma function|Gamma function]] is required for odd dimensions and that its value cancels out the apparent fractional power of π in those cases.
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| − | Why does it cancel the fractional power? Because based on <math>\Gamma(1/2) = \sqrt{\pi}</math>, we find that the Gamma function for any half-integer can be described as:
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| − | :<math>(n+1/2)!=\sqrt{\pi}\times \prod_{k=0}^n {2k + 1 \over 2}.</math>
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| − | This allows the top and bottom π terms to reduce to a single integer power on the top.
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| − | [[w:Taylor's theorem|Taylor's theorem]] expresses a function ''f''(''x'') as a power series in ''x'', basically because the ''n''<sup>th</sup> derivative of ''x''<sup>''n''</sup> is ''n''!.
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| − | :Why is it that when we reduce the function down to degree-1 through differentiation it ends up as n!? --[[User:Nad|Nad]] 17:16, 26 Oct 2006 (NZDT)
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| − | = The mathematical concepts making up the structure =
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| − | ==== The fundamental theorem of algebra ====
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| − | In [[w:mathematics|mathematics]], the [[w:fundamental theorem of algebra|fundamental theorem of algebra]] states that every complex [[w:polynomial|polynomial]] <math>p(z)</math> in one variable and of [[w:degree of a polynomial|degree]] <math>n</math> ≥ <math>1</math> has some complex [[w:root (mathematics)|root]]. In other words, the [[w:field (mathematics)|field]] of [[w:complex number|complex number]]s is [[w:algebraically closed field|algebraically closed]] and therefore, as for any other algebraically closed field, the equation <math>p(z)=0</math> has <math>n</math> roots ([[w:multiple roots of a polynomial|not necessarily distinct]]).
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| − | The name of the theorem is now considered a [[w:misnomer|misnomer]] by many mathematicians, since it is an instance of [[w:mathematical analysis|analysis]] rather than [[w:algebra|algebra]].
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| − | == Algebraically closed field ==
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| − | In mathematics, a [[w:field (mathematics)|field]] <math>F</math> is said to be [[w:algebraically closed|algebraically closed]] if every polynomial in one variable of degree at least <math>1</math>, with [[w:coefficient|coefficient]]s in <math>F</math>, has a zero ([[w:root (mathematics)|root]]) in <math>F</math>.
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| − | == What is a ''field''? ==
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| − | In [[w:abstract algebra|abstract algebra]], a [[w:field|field]] is an [[w:algebraic structure|algebraic structure]] in which the operations of addition, subtraction, multiplication and [[w:division (mathematics)|division]] (except division by zero) may be performed, and the same rules hold which are familiar from the [[w:arithmetic|arithmetic]] of ordinary [[w:number|number]]s.
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| − | The mathematical discipline concerned with the study of fields is called [[w:field theory (mathematics)|field theory]].
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| − | == Abstract algebra and abstract structures ==
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| − | [[w:Abstract algebra|Abstract algebra]] is the field of mathematics that studies algebraic structures, such as [[w:group (mathematics)|groups]], [[w:ring (mathematics)|rings]], [[w:field (mathematics)|fields]], [[w:module (mathematics)|modules]], [[w:vector space]]s, and [[w:algebra over a field|algebras]]. Most authors nowadays simply write ''algebra'' instead of ''abstract algebra''.
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| − | The term ''abstract algebra'' now refers to the study of all algebraic structures, as distinct from the [[w:elementary algebra|elementary algebra]] taught in schools, which teaches the correct rules for manipulating formulas and algebraic expressions involving [[w:real numbers|real]] and [[w:complex number|complex number]]s, and unknowns. Elementary algebra can be taken as an informal introduction to the structures known as the [[w:real field|real field]] and [[w:commutative algebra|commutative algebra]].
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| − | Contemporary mathematics and [[w:mathematical physics|mathematical physics]] make intensive use of abstract algebra; for example, theoretical physics draws on [[w:Lie algebra|Lie algebra]]s. Fields such as [[w:algebraic number theory|algebraic number theory]], [[w:algebraic topology|algebraic topology]], and [[w:algebraic geometry|algebraic geometry]] apply algebraic methods to other areas of mathematics. [[w:Representation theory|Representation theory]], roughly speaking, takes the 'abstract' out of 'abstract algebra', studying the concrete side of a given structure; see [[w:model theory|model theory]].
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| − | Two mathematical fields that study the properties of algebraic structures viewed as a whole are
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| − | [[w:universal algebra|universal algebra]] and [[w:category theory|category theory]]. Algebraic structures, together with the associated [[w:homomorphism|homomorphism]]s, form [[w:category theory|categories]]. Category theory is a powerful formalism for studying and comparing different algebraic structures.
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| − | In [[w:universal algebra|universal algebra]], a branch of [[w:pure mathematics|pure mathematics]], an [[w:algebraic structure|algebraic structure]] consists of one or more [[w:set|set]]s [[w:Closure (mathematics)|closed]] under one or more [[w:operations|operations]], satisfying some [[w:axiom|axiom]]s. [[w:Abstract algebra|Abstract algebra]] is primarily the study of algebraic structures and their properties.
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| − | Abstractly, an "algebraic structure," is the collection of all possible [[w:model theory|model]]s of a given set of axioms. More concretely, an algebraic structure is any particular model of some set of axioms. For example, the [[w:monster group|monster group]] both "is" an algebraic structure in the concrete sense, and abstractly, "has" the group structure in common with all other [[w:group (mathematics)|group]]s. This article employs both meanings of "structure."
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| − | This definition of an algebraic structure should not be taken as restrictive. Anything that satisfies the axioms defining a structure is an instance of that structure, regardless of how many other axioms that instance happens to have. For example, all [[w:group (mathematics)|group]]s are also [[w:semigroup|semigroup]]s and [[w:magma (algebra)|magma]]s.
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| − | == Lies ==
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| − | A [[w:Lie algebra|Lie algebra]] (pronounced "Lee") is an algebraic structure whose main use is in studying geometric objects such as [[w:Lie group|Lie group]]s and differentiable [[w:manifold|manifold]]s. Lie algebras were introduced to study the concept of [[w:infinitesimal transformation|infinitesimal transformation]]s.
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| − | A Lie algebra is a type of an [[w:algebra over a field|algebra over a field]]; it is a [[w:vector space|vector space]] ''g'' over some field ''F'' together with a [[w:binary operation|binary operation]] [·, ·] : ''g'' × ''g'' → ''g'', called the ''Lie bracket''
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| − | A [[w:Lie group|Lie group]] is a continuous group, in the sense that the group elements have the [[w:topology|topology]] of a [[w:manifold|manifold]], and the group operations are [[w:continuous function (topology)|continuous function]]s of the elements. For example, the 2×2 [[w:real number|real]] [[w:invertible matrix|invertible matrices]]
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