Algebraic number and Automorphism: Difference between pages

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In [[mathematics]], an '''algebraic number''' is a number that is a [[root of a function|root]] of a non-zero [[polynomial]] in one variable with [[rational number|rational]] coefficients (or equivalently—by clearing [[denominator]]s—with [[integer]] coefficients). Numbers such as ''[[Pi|{{pi}}]]'' that are not algebraic are said to be [[transcendental number|transcendental]]; [[almost all]] [[real number|real]] and [[complex number]]s are transcendental. (Here "almost all" has the sense "all but a [[countable set]]"; see Properties below.)
In [[mathematics]], an '''automorphism''' is an [[isomorphism]] from a [[mathematical object]] to itself. It is, in some sense, a [[symmetry]] of the object, and a way of [[map (mathematics)|mapping]] the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a [[group (mathematics)|group]], called the '''automorphism group'''. It is, loosely speaking, the [[symmetry group]] of the object.


==Examples==
==Definition==
*The [[rational number]]s, expressed as the quotient of two [[integer]]s ''a'' and ''b'', ''b'' not equal to zero, satisfy the above definition because <math>x = a/b</math> is the root of <math>bx-a</math>.<ref>Some of the following examples come from Hardy and Wright 1972:159–160 and pp. 178–179</ref>
The exact definition of an automorphism depends on the type of "mathematical object" in question and what, precisely, constitutes an "isomorphism" of that object. The most general setting in which these words have meaning is an abstract branch of mathematics called [[category theory]]. Category theory deals with abstract objects and [[morphism]]s between those objects.


*The [[quadratic surd]]s (irrational roots of a quadratic polynomial <math>ax^2 + bx + c</math> with integer coefficients <math>a</math>, <math>b</math>, and <math>c</math>) are algebraic numbers. If the quadratic polynomial is monic <math>(a = 1)</math> then the roots are [[quadratic integer]]s.
In category theory, an automorphism is an [[endomorphism]] (i.e. a [[morphism]] from an object to itself) which is also an [[isomorphism]] (in the categorical sense of the word).


*The [[constructible number]]s are those numbers that can be constructed from a given unit length using straightedge and compass and their opposites. These include all quadratic surds, all rational numbers, and all numbers that can be formed from these using the [[Arithmetic#Arithmetic operations|basic arithmetic operations]] and the extraction of square roots. (Note that by designating cardinal directions for 1, −1, <math>i</math>, and <math>-i</math>, complex numbers such as <math>3+\sqrt{2}i</math> are considered constructible.)
This is a very abstract definition since, in category theory, morphisms aren't necessarily functions and objects aren't necessarily sets. In most concrete settings, however, the objects will be sets with some additional structure and the morphisms will be functions preserving that structure.


*Any expression formed using any combination of the basic arithmetic operations and extraction of [[nth root|''n''th roots]] gives an algebraic number.
In the context of [[abstract algebra]], for example, a mathematical object is an [[algebraic structure]] such as a [[group (mathematics)|group]], [[ring (mathematics)|ring]], or [[vector space]]. An isomorphism is simply a [[bijective]] [[homomorphism]]. (The definition of a homomorphism depends on the type of algebraic structure; see, for example: [[group homomorphism]], [[ring homomorphism]], and [[linear operator]]).


*Polynomial roots that ''cannot'' be expressed in terms of the basic arithmetic operations and extraction of ''n''th roots (such as the roots of <math>x^5 - x + 1 </math>). This [[Abel–Ruffini theorem|happens with many]], but not all, polynomials of degree 5 or higher.
The [[identity morphism]] ([[identity mapping]]) is called the '''trivial automorphism''' in some contexts. Respectively, other (non-identity) automorphisms are called '''nontrivial automorphisms'''.


*[[Gaussian integer]]s: those complex numbers <math>a+bi</math> where both <math>a</math> and <math>b</math> are integers are also quadratic integers.
==Automorphism group==
If the automorphisms of an object ''X'' form a set (instead of a proper [[class (set theory)|class]]), then they form a [[group (mathematics)|group]] under [[Function composition|composition]] of [[morphism]]s. This group is called the '''automorphism group''' of ''X''. That this is indeed a group is simple to see:
* [[Closure (binary operation)|Closure]]: composition of two endomorphisms is another endomorphism.
* [[Associativity]]: composition of morphisms is ''always'' associative.
* [[Identity element|Identity]]: the identity is the identity morphism from an object to itself which exists by definition.
* [[Inverse element|Inverses]]: by definition every isomorphism has an inverse which is also an isomorphism, and since the inverse is also an endomorphism of the same object it is an automorphism.


*[[Trigonometric functions]] of [[rational number|rational]] multiples of <math>\pi</math> (except when undefined). For example, each of <math>\cos(\pi/7)</math>, <math>\cos(3\pi/7)</math>, <math>\cos(5\pi/7)</math> satisfies <math>8x^3 - 4x^2 - 4x + 1 = 0</math>. This polynomial is [[irreducible polynomial|irreducible]] over the rationals, and so these three cosines are ''conjugate'' algebraic numbers. Likewise, <math>\tan(3\pi/16)</math>, <math>\tan(7\pi/16)</math>, <math>\tan(11\pi/16)</math>, <math>\tan(15\pi/16)</math> all satisfy the irreducible polynomial <math>x^4 - 4x^3 - 6x^2 + 4x + 1</math>, and so are conjugate [[algebraic integers]].
The automorphism group of an object ''X'' in a category ''C'' is denoted Aut<sub>''C''</sub>(''X''), or simply Aut(''X'') if the category is clear from context.


*Some [[irrational number]]s are algebraic and some are not:
==Examples==
**The numbers <math>\sqrt{2}</math> and <math>\sqrt[3]{3}/2</math> are algebraic since they are roots of polynomials <math>x^2 - 2</math> and <math>8x^3 - 3</math>, respectively.
* In [[set theory]], an arbitrary [[permutation]] of the elements of a set ''X'' is an automorphism. The automorphism group of ''X'' is also called the [[symmetric group]] on ''X''.
**The [[golden ratio]] <math>\phi</math> is algebraic since it is a root of the polynomial <math>x^2 - x - 1</math>.
* In [[elementary arithmetic]], the set of [[integer]]s, '''Z''', considered as a group under addition, has a unique nontrivial automorphism: negation. Considered as a [[ring (mathematics)|ring]], however, it has only the trivial automorphism. Generally speaking, negation is an automorphism of any [[abelian group]], but not of a ring or field.
**The numbers [[Pi|<math>\pi</math>]] and [[e (mathematical constant)|<math>e</math>]] are not algebraic numbers (see the [[Lindemann–Weierstrass theorem]]);<ref>Also [[Liouville number|Liouville's theorem]] can be used to "produce as many examples of transcendentals numbers as we please," cf Hardy and Wright p. 161ff</ref> hence they are transcendental.
* A group automorphism is a [[group isomorphism]] from a group to itself. Informally, it is a permutation of the group elements such that the structure remains unchanged. For every group ''G'' there is a natural group homomorphism ''G'' → Aut(''G'') whose [[image (mathematics)|image]] is the group Inn(''G'') of [[inner automorphism]]s and whose [[kernel (algebra)|kernel]] is the [[center (group theory)|center]] of ''G''. Thus, if ''G'' has [[Trivial group|trivial]] center it can be embedded into its own automorphism group.<ref name=Pahl>


== {{anchor|Degree of an algebraic number}} Properties ==
{{cite book |url=http://books.google.com/?id=kvoaoWOfqd8C&pg=PA376 |page=376 |chapter=§7.5.5 Automorphisms |title=Mathematical foundations of computational engineering |edition=Felix Pahl translation |author=PJ Pahl, R Damrath |isbn=3-540-67995-2 |year=2001 |publisher=Springer}}
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[[File:Algebraicszoom.png|thumb|Algebraic numbers on the [[complex plane]] colored by degree. (red=1, green=2, blue=3, yellow=4)]]
*The set of algebraic numbers is [[countable set|countable]] (enumerable).<ref>Hardy and Wright 1972:160</ref>
*Hence, the set of algebraic numbers has [[Lebesgue measure]] zero (as a subset of the complex numbers), i.e. "[[Almost everywhere|almost all]]" complex numbers are not algebraic.
*Given an algebraic number, there is a unique [[monic polynomial]] (with rational coefficients) of least [[degree of a polynomial|degree]] that has the number as a root. This polynomial is called its [[minimal polynomial (field theory)|minimal polynomial]]. If its minimal polynomial has degree <math>n</math>, then the algebraic number is said to be of ''degree <math>n</math>''. An algebraic number of degree 1 is a [[rational number]]. A real algebraic number of degree 2 is a [[quadratic irrational]].
*All algebraic numbers are [[computable number|computable]] and therefore [[definable number|definable]] and [[arithmetical numbers|arithmetical]].
*The set of real algebraic numbers is [[linearly ordered]], countable, [[densely ordered]], and without first or last element, so is [[order-isomorphic]] to the set of rational numbers.


==The field of algebraic numbers==
</ref>
[[File:Algebraic number in the complex plane.png|thumb|Algebraic numbers colored by degree (blue=4, cyan=3, red=2, green=1). The unit circle in black.]]
* In [[linear algebra]], an endomorphism of a [[vector space]] ''V'' is a [[linear transformation|linear operator]] ''V'' → ''V''. An automorphism is an invertible linear operator on ''V''. When the vector space is finite-dimensional, the automorphism group of ''V'' is the same as the [[general linear group]], GL(''V'').
The sum, difference, product and quotient of two algebraic numbers is again algebraic (this fact can be demonstrated using the [[resultant]]), and the algebraic numbers therefore form a [[field (mathematics)|field]], sometimes denoted by '''A''' (which may also denote the [[adele ring]]) or <span style="text-decoration: overline;">'''Q'''</span>. Every root of a polynomial equation whose coefficients are ''algebraic numbers'' is again algebraic. This can be rephrased by saying that the field of algebraic numbers is [[algebraically closed field|algebraically closed]]. In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the [[algebraic closure]] of the rationals.
* A field automorphism is a [[bijection|bijective]] [[ring homomorphism]] from a [[field (mathematics)|field]] to itself. In the cases of the [[rational number]]s ('''Q''') and the [[real number]]s ('''R''') there are no nontrivial field automorphisms. Some subfields of '''R''' have nontrivial field automorphisms, which however do not extend to all of '''R''' (because they cannot preserve the property of a number having a square root in '''R'''). In the case of the [[complex number]]s, '''C''', there is a unique nontrivial automorphism that sends '''R''' into '''R''': [[complex conjugate|complex conjugation]], but there are infinitely ([[uncountable|uncountably]]) many "wild" automorphisms (assuming the [[axiom of choice]]).<ref>{{cite journal | last = Yale | first = Paul B. | journal = Mathematics Magazine | title = Automorphisms of the Complex Numbers | volume = 39 | issue = 3 |date=May 1966 | pages = 135–141 | url = http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/PaulBYale.pdf | doi = 10.2307/2689301 | jstor = 2689301}}</ref> Field automorphisms are important to the theory of [[field extension]]s, in particular [[Galois extension]]s. In the case of a Galois extension ''L''/''K'' the [[subgroup]] of all automorphisms of ''L'' fixing ''K'' pointwise is called the [[Galois group]] of the extension.
* In [[graph theory]] an [[graph automorphism|automorphism of a graph]] is a permutation of the nodes that preserves edges and non-edges. In particular, if two nodes are joined by an edge, so are their images under the permutation.
* For relations, see [[Isomorphism#A relation-preserving isomorphism|relation-preserving automorphism]].
** In [[order theory]], see [[order automorphism]].
* In [[geometry]], an automorphism may be called a [[motion (geometry)|motion]] of the space. Specialized terminology is also used:
** In [[metric geometry]] an automorphism is a self-[[isometry]]. The automorphism group is also called the [[isometry group]].
** In the category of [[Riemann surface]]s, an automorphism is a bijective [[biholomorphy|biholomorphic]] map (also called a [[conformal map]]), from a surface to itself. For example, the automorphisms of the [[Riemann sphere]] are [[Möbius transformation]]s.
** An automorphism of a differentiable [[manifold]] ''M'' is a [[diffeomorphism]] from ''M'' to itself. The automorphism group is sometimes denoted Diff(''M'').
** In [[topology]], morphisms between topological spaces are called [[Continuous function (topology)|continuous maps]], and an automorphism of a topological space is a [[homeomorphism]] of the space to itself, or self-homeomorphism (see [[homeomorphism group]]). In this example it is ''not sufficient'' for a morphism to be bijective to be an isomorphism.


==Related fields==
==History==
===Numbers defined by radicals===
One of the earliest group automorphisms (automorphism of a group, not simply a group of automorphisms of points) was given by the Irish mathematician [[William Rowan Hamilton]] in 1856, in his [[icosian calculus]], where he discovered an order two automorphism,<ref>{{Cite journal
All numbers that can be obtained from the integers using a [[finite set|finite]] number of integer [[addition]]s, [[subtraction]]s, [[multiplication]]s, [[division (mathematics)|division]]s, and taking ''n''th roots (where ''n'' is a positive integer) are algebraic. The converse, however, is not true: there are algebraic numbers that cannot be obtained in this manner. All of these numbers are solutions to polynomials of degree ≥5. This is a result of [[Galois theory]] (see [[Quintic equation]]s and the [[Abel–Ruffini theorem]]). An example of such a number is the unique real root of the polynomial {{nowrap|''x''<sup>5</sup> − ''x'' − 1}} (which is approximately 1.167304).
|title=Memorandum respecting a new System of Roots of Unity
|author=Sir William Rowan Hamilton
|author-link=William Rowan Hamilton
|url=http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian/NewSys.pdf
|journal=[[Philosophical Magazine]]
|volume=12
|year=1856
|pages=446
}}</ref> writing:
{{quotation|so that <math>\mu</math> is a new fifth root of unity, connected with the former fifth root <math>\lambda</math> by relations of perfect reciprocity.}}


===Closed-form number===
==Inner and outer automorphisms==
{{Main|Closed-form number}}
In some categories—notably [[group (mathematics)|groups]], [[ring (mathematics)|rings]], and [[Lie algebra]]s—it is possible to separate automorphisms into two types, called "inner" and "outer" automorphisms.
Algebraic numbers are all numbers that can be defined explicitly or implicitly in terms of polynomials, starting from the rational numbers. One may generalize this to "[[closed-form number]]s", which may be defined in various ways. Most broadly, all numbers that can be defined explicitly or implicitly in terms of polynomials, exponentials, and logarithms are called "elementary numbers", and these include the algebraic numbers, plus some transcendental numbers. Most narrowly, one may consider numbers ''explicitly'' defined in terms of polynomials, exponentials, and logarithms – this does not include algebraic numbers, but does include some simple transcendental numbers such as ''e'' or log(2).


==Algebraic integers==
In the case of groups, the [[inner automorphism]]s are the conjugations by the elements of the group itself. For each element ''a'' of a group ''G'', conjugation by ''a'' is the operation φ<sub>''a''</sub> : ''G''&nbsp;→&nbsp;''G'' given by <math> \varphi_a (g) = a g a^{-1} </math> (or ''a''<sup>−1</sup>''ga''; usage varies). One can easily check that conjugation by ''a'' is a group automorphism. The inner automorphisms form a [[normal subgroup]] of Aut(''G''), denoted by Inn(''G''); this is called [[Goursat's lemma]].
{{Main|Algebraic integer}}
[[Image:Leadingcoeff.png|thumb|Algebraic numbers colored by leading coefficient (red signifies 1 for an algebraic integer).]]
An '''[[algebraic integer]]''' is an algebraic number that is a root of  a polynomial with integer coefficients with leading coefficient 1 (a monic polynomial). Examples of algebraic integers are {{nowrap|5 + 13√{{overline|2}}}}, {{nowrap|2 − 6''i''}}, and {{nowrap|{{frac|1|2}}(1 + ''i''√{{overline|3}}).}} (Note, therefore, that the algebraic integers constitute a proper [[superset]] of the [[integer]]s, as the latter are the roots of monic polynomials {{nowrap|''x'' − ''k''}} for all {{nowrap|''k'' ∈ '''Z'''.)}}


The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a [[ring (algebra)|ring]]. The name ''algebraic integer'' comes from the fact that the only rational numbers that are algebraic integers are the integers, and because the algebraic integers in any [[algebraic number field|number field]] are in many ways analogous to the integers. If ''K'' is a number field, its [[ring of integers]] is the subring of algebraic integers in ''K'', and is frequently denoted as ''O<sub>K</sub>''. These are the prototypical examples of [[Dedekind domain]]s.
The other automorphisms are called [[outer automorphism]]s. The [[quotient group]] Aut(''G'')&nbsp;/&nbsp;Inn(''G'') is usually denoted by Out(''G''); the non-trivial elements are the cosets that contain the outer automorphisms.


==Special classes of algebraic number==
The same definition holds in any [[unital algebra|unital]] [[ring (mathematics)|ring]] or [[algebra over a field|algebra]] where ''a'' is any [[Unit (ring theory)|invertible element]]. For [[Lie algebra]]s the definition is slightly different.
*[[Gaussian integer]]
*[[Eisenstein integer]]
*[[Quadratic irrational]]
*[[Fundamental unit (number theory)|Fundamental unit]]
*[[Root of unity]]
*[[Gaussian period]]
*[[Pisot–Vijayaraghavan number]]
*[[Salem number]]


==Notes==
==See also==
{{Reflist}}
* [[endomorphism ring]]
* [[antiautomorphism]]
* [[Frobenius automorphism]]
* [[morphism]]
* [[characteristic subgroup]]


==References==
==References==
*{{Citation |last=Artin |first=Michael |author-link=Michael Artin |title=Algebra |publisher=[[Prentice Hall]] |isbn=0-13-004763-5 |mr=1129886 |year=1991}}
<!-- See [[Wikipedia:Footnotes]] for instructions. -->
*{{Citation |last1=Ireland |first1=Kenneth |last2=Rosen |first2=Michael |title=A Classical Introduction to Modern Number Theory |edition=Second |publisher=[[Springer-Verlag]] |location=Berlin, New York |series=Graduate Texts in Mathematics |isbn=0-387-97329-X |mr=1070716 |year=1990 |volume=84}}
<references />
*[[G.H. Hardy]] and [[E.M. Wright]] 1978, 2000 (with general index) ''An Introduction to the Theory of Numbers: 5th Edition'', Clarendon Press, Oxford UK, ISBN 0-19-853171-0
*{{Lang Algebra}}
*[[Øystein Ore]] 1948, 1988, ''Number Theory and Its History'', Dover Publications, Inc. New York, ISBN 0-486-65620-9 (pbk.)


{{Number Systems}}
==External links==
* [http://www.encyclopediaofmath.org/index.php/Automorphism ''Automorphism'' at Encyclopaedia of Mathematics]
* {{MathWorld | urlname=Automorphism | title = Automorphism}}


{{DEFAULTSORT:Algebraic Number}}
[[Category:Morphisms]]
[[Category:Algebraic numbers| ]]
[[Category:Abstract algebra]]
[[Category:Symmetry]]

Revision as of 23:12, 14 January 2014

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.

Definition

The exact definition of an automorphism depends on the type of "mathematical object" in question and what, precisely, constitutes an "isomorphism" of that object. The most general setting in which these words have meaning is an abstract branch of mathematics called category theory. Category theory deals with abstract objects and morphisms between those objects.

In category theory, an automorphism is an endomorphism (i.e. a morphism from an object to itself) which is also an isomorphism (in the categorical sense of the word).

This is a very abstract definition since, in category theory, morphisms aren't necessarily functions and objects aren't necessarily sets. In most concrete settings, however, the objects will be sets with some additional structure and the morphisms will be functions preserving that structure.

In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space. An isomorphism is simply a bijective homomorphism. (The definition of a homomorphism depends on the type of algebraic structure; see, for example: group homomorphism, ring homomorphism, and linear operator).

The identity morphism (identity mapping) is called the trivial automorphism in some contexts. Respectively, other (non-identity) automorphisms are called nontrivial automorphisms.

Automorphism group

If the automorphisms of an object X form a set (instead of a proper class), then they form a group under composition of morphisms. This group is called the automorphism group of X. That this is indeed a group is simple to see:

  • Closure: composition of two endomorphisms is another endomorphism.
  • Associativity: composition of morphisms is always associative.
  • Identity: the identity is the identity morphism from an object to itself which exists by definition.
  • Inverses: by definition every isomorphism has an inverse which is also an isomorphism, and since the inverse is also an endomorphism of the same object it is an automorphism.

The automorphism group of an object X in a category C is denoted AutC(X), or simply Aut(X) if the category is clear from context.

Examples

History

One of the earliest group automorphisms (automorphism of a group, not simply a group of automorphisms of points) was given by the Irish mathematician William Rowan Hamilton in 1856, in his icosian calculus, where he discovered an order two automorphism,[3] writing: 36 year-old Diving Instructor (Open water ) Vancamp from Kuujjuaq, spends time with pursuits for instance gardening, public listed property developers in singapore developers in singapore and cigar smoking. Of late took some time to go China Danxia.

Inner and outer automorphisms

In some categories—notably groups, rings, and Lie algebras—it is possible to separate automorphisms into two types, called "inner" and "outer" automorphisms.

In the case of groups, the inner automorphisms are the conjugations by the elements of the group itself. For each element a of a group G, conjugation by a is the operation φa : G → G given by (or a−1ga; usage varies). One can easily check that conjugation by a is a group automorphism. The inner automorphisms form a normal subgroup of Aut(G), denoted by Inn(G); this is called Goursat's lemma.

The other automorphisms are called outer automorphisms. The quotient group Aut(G) / Inn(G) is usually denoted by Out(G); the non-trivial elements are the cosets that contain the outer automorphisms.

The same definition holds in any unital ring or algebra where a is any invertible element. For Lie algebras the definition is slightly different.

See also

References

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    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

External links

  • Automorphism at Encyclopaedia of Mathematics


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