formulasearchengine - User contributions [en]

Leibniz integral rule

2014-12-01T01:44:54Z

130.216.97.131: Change typo of = to + in formal statement of rule

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Pólya enumeration theorem

2014-11-24T02:13:45Z

130.216.48.29: Fixed broken anchor

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Inverse iteration

2014-11-10T09:27:05Z

130.216.30.132: /* Implementation options */

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2014-05-15T09:37:47Z

130.216.218.47: Fix unit formatting

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2014-02-18T00:50:13Z

130.216.209.170: /* Comparison of Knots and Control Points */

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2014-01-29T05:28:09Z

130.216.218.47: fix spelling fix

The 18th-century Swiss mathematician '''[[Leonhard Euler]]''' (1707–1783) is among the most prolific and successful mathematicians in the [[history of mathematics|history of the field]]. His seminal work had a profound impact in numerous areas of mathematics and he is widely credited for introducing and popularizing modern notation and terminology, particularly in [[analysis (mathematics)|analysis]].

==Mathematical notation==
Euler introduced much of the mathematical notation in use today, such as the notation ''f''(''x'') to describe a function and the modern notation for the [[trigonometric functions]]. He was the first to use the letter ''e'' for the base of the [[natural logarithm]], now also known as [[Euler's number]]. The use of the Greek letter [[pi (letter)|<math>\pi</math>]] to denote the [[pi|ratio of a circle's circumference to its diameter]] was also popularized by Euler (although it did not originate with him).<ref name="pi">{{cite web| url = http://www.stephenwolfram.com/publications/talks/mathml/mathml2.html| title = Mathematical Notation: Past and Future| accessdate = August 2006| last = Wolfram| first = Stephen}}</ref> He is also credited for inventing the notation ''[[Imaginary unit|i]]'' to denote <math>\sqrt{-1}</math>.<ref name="i">{{cite web| url = http://scienceworld.wolfram.com/biography/Euler.html| title = Euler, Leonhard (1707–1783)| accessdate = April 2007}}</ref>

==Complex analysis==
[[Image:Euler's formula.svg|thumb|180px|A geometric interpretation of Euler's formula]]
Euler made important contributions to [[complex analysis]]. He introduced the scientific notation. He discovered what is now known as [[Euler's formula]], that for any [[real number]] <math>\varphi</math>, the complex [[exponential function]] satisfies

:<math>e^{i\varphi} = \cos \varphi + i\sin \varphi. \,</math>

This has been called "The most remarkable formula in mathematics " by [[Richard Feynman]].
<ref name="Feynman">
{{cite book |last= Feynman|first= Richard|title= The Feynman Lectures on Physics: Volume I |chapter= Chapter 22: Algebra|page=10 |date=June 1970}}</ref> [[Euler's identity]] is a special case of this:
:<math>e^{i \pi} +1 = 0 \,.</math>

This identity is particularly remarkable as it involves ''e'', <math>\pi</math>, ''i'', 1, and 0, arguably the five most important constants in mathematics.

==Analysis==
The development of [[calculus]] was at the forefront of 18th century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Understanding the infinite was naturally the major focus of Euler's research. While some of Euler's proofs may not have been acceptable under modern standards of [[mathematical rigor|rigor]], his ideas were responsible for many great advances. First of all, Euler introduced the concept of a [[function (mathematics)|function]], and introduced the use of the [[exponential function]] and [[logarithms]] in analytic proofs

Euler frequently used the logarithm function as a tool in analysis problems, and discovered new ways by which they could be used. He discovered ways to express various logarithmic functions in terms of power series, and successfully defined logarithms for complex and negative numbers, thus greatly expanding the scope where logarithms could be applied in mathematics. Most researchers in the field long held the view that <math>\log (x) = \log (-x)</math> for any positive real <math>x</math> since by using the additivity property of logarithms <math> 2 \log (-x) = \log ((-x)^2) = \log (x^2) = 2 \log (x) </math>. In a 1747 letter to [[Jean Le Rond d'Alembert]], Euler defined the natural logarithm of −1 as <math>i\pi</math> a [[complex number|pure imaginary]].<ref name=Boyer>{{cite book|title = A History of Mathematics|last= Boyer|first=Carl B.|coauthors= Uta C. Merzbach|publisher= [[John Wiley & Sons]]|isbn= 0-471-54397-7|pages = 439–445|year = 1991}}</ref>

Euler is well known in analysis for his frequent use and development of [[power series]]: that is, the expression of functions as sums of infinitely many terms, such as

:<math>e = \sum_{n=0}^\infty {1 \over n!} = \lim_{n \to \infty}\left(\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \cdots + \frac{1}{n!}\right).</math>

Notably, Euler discovered the power series expansions for ''e'' and the [[inverse tangent]] function

:<math>\arctan z = \sum_{n=0}^\infty \frac {(-1)^n z^{2n+1}} {2n+1}.</math>

His daring (and, by modern standards, technically incorrect) use of power series enabled him to solve the famous [[Basel problem]] in 1735:<ref name="Basel">{{cite book| last = Wanner| first = Gerhard| coauthors = Harrier, Ernst | title = Analysis by its history| edition = 1st|date=March 2005| publisher = Springer| page = 62}}</ref>

:<math>\lim_{n \to \infty}\left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{n^2}\right) = \frac{\pi ^2}{6}.</math>

In addition, Euler elaborated the theory of higher transcendental functions by introducing the [[gamma function]] and introduced a new method for solving [[quartic equation]]s. He also found a way to calculate integrals with complex limits, foreshadowing the development of [[complex analysis]]. Euler invented the [[calculus of variations]] including its most well-known result, the [[Euler–Lagrange equation]].

Euler also pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, [[analytic number theory]]. In breaking ground for this new field, Euler created the theory of [[hypergeometric series]], [[q-series]], [[hyperbolic functions|hyperbolic trigonometric functions]] and the analytic theory of [[continued fractions]]. For example, he proved the [[infinitude of primes]] using the divergence of the harmonic series, and used analytic methods to gain some understanding of the way [[prime numbers]] are distributed. Euler's work in this area led to the development of the [[prime number theorem]].<ref name="analysis">{{cite book| last = Dunham| first = William| title = Euler: The Master of Us All | year = 1999| publisher =The Mathematical Association of America | chapter = 3,4 }}</ref>

==Number theory==
Euler's great interest in number theory can be traced to the influence of his friend in the St. Peterburg Academy, [[Christian Goldbach]]. A lot of his early work on number theory was based on the works of [[Pierre de Fermat]], and developed some of Fermat's ideas.

One focus of Euler's work was to link the nature of prime distribution with ideas in analysis. He proved that [[Proof that the sum of the reciprocals of the primes diverges|the sum of the reciprocals of the primes diverges]]. In doing so, he discovered the connection between Riemann zeta function and prime numbers, known as the [[Proof of the Euler product formula for the Riemann zeta function|Euler product formula for the Riemann zeta function]].

Euler proved [[Newton's identities]], [[Fermat's little theorem]], [[Fermat's theorem on sums of two squares]], and made distinct contributions to the [[Lagrange's four-square theorem]]. He also invented the [[totient function]] φ(n) which assigns to a positive integer n the number of positive integers less than n and coprime to n. Using properties of this function he was able to generalize Fermat's little theorem to what would become known as [[Euler's theorem]]. He further contributed significantly to the understanding of [[perfect numbers]], which had fascinated mathematicians since [[Euclid]]. Euler made progress toward the prime number theorem and conjectured the law of [[quadratic reciprocity]]. The two concepts are regarded as the fundamental theorems of number theory, and his ideas paved the way for [[Carl Friedrich Gauss]].<ref name="numbertheory">{{cite book| last = Dunham| first = William| title = Euler: The Master of Us All | year = 1999| publisher =The Mathematical Association of America | chapter = 1,4}}</ref>

==Graph theory and topology==
[[Image:Konigsberg bridges.png|frame|right|Map of Königsberg in Euler's time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges.]]
{{Seealso|Seven Bridges of Königsberg}}

In 1736 Euler solved, or rather proved unsolvable, a problem known as the seven bridges of Königsberg.<ref name="bridge">{{cite journal| last = Alexanderson| first = Gerald|date=July 2006| title = Euler and Königsberg's bridges: a historical view| journal = Bulletin of the American Mathematical Society| url = http://www.ams.org/bull/0000-000-00/S0273-0979-06-01130-X/S0273-0979-06-01130-X.pdf|format=PDF| doi = 10.1090/S0273-0979-06-01130-X| volume = 43| pages = 567| issue = 4}}</ref> The city of [[Königsberg]], [[Kingdom of Prussia]] (now Kaliningrad, Russia) is set on the [[Pregel]] River, and included two large islands which were connected to each other and the mainland by seven bridges. The question is whether it is possible to walk with a route that crosses each bridge exactly once, and return to the starting point.
Euler's solution of the Königsberg bridge problem is considered to be the first theorem of [[graph theory]]. In addition, his recognition that the key information was the number of bridges and the list of their endpoints (rather than their exact positions) presaged the development of [[topology]].<ref name="bridge"/>

[[Image:Euler GDR stamp.jpg|thumb|250px|left|This stamp of the former [[German Democratic Republic]] honoring Euler displaying his formula relating the number of faces, edges and vertices of a convex polyhedron.]]
Euler also made contributions to the understanding of [[planar graphs]]. He introduced a formula governing the relationship between the number of edges, vertices, and faces of a convex polyhedron. Given such a polyhedron, the alternating sum of vertices, edges and faces equals a constant: ''V'' − ''E'' + ''F'' = 2. This constant, χ, is the [[Euler characteristic]] of the plane. The study and generalization of this equation, specially by [[Augustin Louis Cauchy|Cauchy]]<ref name="Cauchy">{{cite journal|author=Cauchy, A.L.|year=1813|title=Recherche sur les polyèdres - premier mémoire|journal=[[Journal de l'École Polytechnique]]|volume= 9 (Cahier 16)|pages=66–86}}</ref> and Lhuillier,<ref name="Lhuillier">{{cite journal|author=Lhuillier, S.-A.-J.|title=Mémoire sur la polyèdrométrie|journal=Annales de Mathématiques|volume=3|year=1861|pages=169–189}}</ref> is at the origin of [[topology]]. Euler characteristic, which may be generalized to any [[topological space]] as the alternating sum of the [[Betti number]]s, naturally arises from [[homology (mathematics)|homology]]. In particular, it is equal to 2 − 2''g'' for a closed oriented [[surface]] with genus ''g'' and to 2 − ''k'' for a non-orientable surface with k crosscaps. This property led to the definition of [[rotation system]]s in [[topological graph theory]].

==Applied mathematics==
Some of Euler's greatest successes were in applying analytic methods to real world problems, describing numerous applications of [[Bernoulli number|Bernoulli's numbers]], [[Fourier series]], [[Venn diagram]]s, [[Euler number]]s, [[E (mathematical constant)|e]] and [[pi|π]] constants, continued fractions and integrals. He integrated [[Gottfried Leibniz|Leibniz]]'s [[derivative|differential calculus]] with Newton's [[Method of Fluxions]], and developed tools that made it easier to apply calculus to physical problems. In particular, he made great strides in improving [[numerical analysis|numerical approximation]] of integrals, inventing what are now known as the ''Euler approximations''. The most notable of these approximations are [[Euler method]] and the [[Euler–Maclaurin formula]]. He also facilitated the use of [[differential equation]]s, in particular introducing the [[Euler–Mascheroni constant]]:

:<math>\gamma = \lim_{n \rightarrow \infty } \left( 1+ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots + \frac{1}{n} - \ln(n) \right).</math>

One of Euler's more unusual interests was the application of mathematical ideas in [[music]]. In 1739 he wrote the ''Tentamen novae theoriae musicae,'' hoping to eventually integrate [[music theory]] as part of mathematics. This part of his work, however did not receive wide attention and was once described as too mathematical for musicians and too musical for mathematicians.<ref name="music">{{cite journal| author = Ronald Calinger | year = 1996| title = Leonhard Euler: The First St. Petersburg Years (1727–1741)| journal = Historia Mathematica| volume = 23| issue = 2| pages = 144–145| doi=10.1006/hmat.1996.0015}}</ref>

== Works ==
The works which Euler published separately are:
*''Dissertatio physica de sono'' (Dissertation on the physics of sound) (Basel, 1727, in quarto)
*''Mechanica, sive motus scientia analytice; expasita'' (St Petersburg, 1736, in 2 vols. quarto)
*''Einleitung in die Arithmetik'' (ibid., 1738, in 2 vols. octavo), in German and Russian
*''Tentamen novae theoriae musicae'' (ibid. 1739, in quarto)
*''Methodus inveniendi lineas curvas, maximi minimive proprietate gaudentes'' (Lausanne, 1744, in quarto)
**[[s:la:Methodus inveniendi/Additamentum II|Additamentum II]] ([[:Wikisource:Methodus inveniendi/Additamentum II|English translation]])
*''Theoria motuum planetarum et cometarum'' (Berlin, 1744, in quarto)
*''Beantwortung, &c.'' or Answers to Different Questions respecting Comets (ibid., 1744, in octavo)
*''Neue Grundsatze, &c.'' or New Principles of Artillery, translated from the English of Benjamin Robins, with notes and illustrations (ibid., 1745, in octavo)
*''Opuscula varii argumenti'' (ibid., 1746–1751, in 3 vols. quarto)
*''Novae et carrectae tabulae ad loco lunae computanda'' (ibid., 1746, in quarto)
*''Tabulae astronomicae solis et lunae'' (ibid., in quarto)
*''Gedanken, &c.'' or Thoughts on the Elements of Bodies (ibid. in quarto)
*''Rettung der gall-lichen Offenbarung, &c.'', Defence of Divine Revelation against Free-thinkers (ibid., 1747, in quarto)
*''Introductio in analysin infinitorum'' (Introduction to the analysis of the infinites)(Lausanne, 1748, in 2 vols. quarto)
*''Scientia navalis, seu tractatus de construendis ac dirigendis navibus'' (St Petersburg, 1749, in 2 vols. quarto)
*[[s:fr:Exposé concernant l’examen de la lettre de M. de Leibnitz|Exposé concernant l’examen de la lettre de M. de Leibnitz]] (1752, its [[s:Investigation of the letter of Leibniz|English translation]])
*''Theoria motus lunae'' (Berlin, 1753, in quarto)
*''Dissertatio de principio mininiae actionis, una cum examine objectionum cl. prof. Koenigii'' (ibid., 1753, in octavo)
*''Institutiones calculi differentialis, cum ejus usu in analysi Intuitorum ac doctrina serierum'' (ibid., 1755, in quarto)
*''Constructio lentium objectivarum, &c.'' (St Petersburg, 1762, in quarto)
*''Theoria motus corporum solidorum seu rigidorum'' (Rostock, 1765, in quarto)
*''Institutiones, calculi integralis'' (St Petersburg, 1768–1770, in 3 vols. quarto)
*''Lettres a une Princesse d'Allernagne sur quelques sujets de physique et de philosophie'' (St Petersburg, 1768–1772, in 3 vols. octavo)
*''Anleitung zur Algebra'' [http://web.mat.bham.ac.uk/C.J.Sangwin/euler/index.html Elements of Algebra] (ibid., 1770, in octavo); Dioptrica (ibid., 1767–1771, in 3 vols. quarto)
*''Theoria motuum lunge nova methodo pertr. arctata' (ibid., 1772, in quarto)
*''Novae tabulae lunares'' (ibid., in octavo); ''La théorie complete de la construction et de la manteuvre des vaisseaux'' (ibid., 1773, in octavo)
*''Eclaircissements svr etablissements en favour taut des veuves que des marts'', without a date
*''Opuscula analytica'' (St Petersburg, 1783–1785, in 2 vols. quarto). See [[Ferdinand Rudio|F. Rudio]], ''Leonhard Euler'' (Basel, 1884).

==See also==
* [[List of Leonhard Euler's namesakes]]

==References==
<references/>

{{DEFAULTSORT:Contributions Of Leonhard Euler To Mathematics}}
[[Category:History of mathematics]]
[[Category:Leonhard Euler]]

Probably approximately correct learning

2013-12-03T03:02:21Z

130.216.30.113: /* Definitions and terminology */ clarify example

{{about|topological algebras associated to topological groups|the purely algebraic case of discrete groups|group ring}}

In [[mathematics]], the '''group algebra''' is any of various constructions to assign to a [[locally compact group]] an [[operator algebra]] (or more generally a [[Banach algebra]]), such that representations of the algebra are related to representations of the group. As such, they are similar to the [[group ring]] associated to a discrete group.

== Group algebras of topological groups: ''Cc''(''G'')==
For the purposes of [[functional analysis]], and in particular of [[harmonic analysis]], one wishes to carry over the group ring construction to [[topological group]]s ''G''. In case ''G'' is a [[locally compact group|locally compact Hausdorff group]], ''G'' carries an essentially unique left-invariant countably additive [[Borel measure]] μ called [[Haar measure]]. Using the Haar measure, one can define a [[convolution]] operation on the space ''Cc''(''G'') of complex-valued continuous functions on ''G'' with [[compact support]]; ''Cc''(''G'') can then be given any of various [[norm (mathematics)|norm]]s and the [[completeness (order theory)|completion]] will be a group algebra.

To define the convolution operation, let ''f'' and ''g'' be two functions in ''Cc''(''G''). For ''t'' in ''G'', define

:<math> [f * g](t) = \int_G f(s) g \left (s^{-1} t \right )\, d \mu(s).</math>

The fact that ''f'' * ''g'' is continuous is immediate from the [[dominated convergence theorem]]. Also

:<math> \operatorname{Support}(f * g) \subseteq \operatorname{Support}(f) \cdot \operatorname{Support}(g) </math>

were the dot stands for the product in ''G''. ''Cc''(''G'') also has a natural [[involution (mathematics)|involution]] defined by:

:<math> f^*(s) = \overline{f(s^{-1})} \Delta(s^{-1}) </math>

where Δ is the [[Haar_measure#The_modular_function|modular function]] on ''G''. With this involution, it is a [[*-algebra]].

<blockquote>'''Theorem.''' With the norm:

:<math> \|f\|_1 := \int_G |f(s)| d\mu(s), </math>

''Cc''(''G'') becomes an involutive [[normed algebra]] with an [[approximate identity]].</blockquote>

The approximate identity can be indexed on a neighborhood basis of the identity consisting of compact sets. Indeed if ''V'' is a compact neighborhood of the identity, let ''fV'' be a non-negative continuous function supported in ''V'' such that

:<math> \int_V f_{V}(g)\, d \mu(g) =1.</math>

Then {''fV''}''V'' is an approximate identity. A group algebra has an identity, as opposed to just an approximate identity, if and only if the topology on the group is the [[discrete topology]].

Note that for discrete groups, ''Cc''(''G'') is the same thing as the complex group ring '''C'''[''G''].

The importance of the group algebra is that it captures the [[unitary representation]] theory of ''G'' as shown in the following

<blockquote>'''Theorem.''' Let ''G'' be a locally compact group. If ''U'' is a strongly continuous unitary representation of ''G'' on a Hilbert space ''H'', then

: <math> \pi_U (f) = \int_G f(g) U(g)\, d \mu(g)</math>

is a non-degenerate bounded *-representation of the normed algebra ''Cc''(''G''). The map

: <math> U \mapsto \pi_U</math>

is a bijection between the set of strongly continuous unitary representations of ''G'' and non-degenerate bounded *-representations of ''Cc''(''G''). This bijection respects unitary equivalence and [[strong containment]]. In particular, π''U'' is irreducible if and only if ''U'' is irreducible.</blockquote>

Non-degeneracy of a representation π of ''Cc''(''G'') on a Hilbert space ''H''π means that

:<math> \left \{\pi(f) \xi : f \in \operatorname{C}_c(G), \xi \in H_\pi \right \} </math>

is dense in ''H''π.

== The convolution algebra ''L''1(''G'') ==
It is a standard theorem of [[measure theory]] that the completion of ''Cc''(''G'') in the ''L''1(''G'') norm is isomorphic to the space [[Lp space|''L''1(''G'')]] of equivalence classes of functions which are integrable with respect to the [[Haar measure]], where, as usual, two functions are regarded as equivalent if and only if they differ only on a set of Haar measure zero.

<blockquote>'''Theorem.''' ''L''1(''G'') is a [[Banach *-algebra]] with the convolution product and involution defined above and with the ''L''1 norm. ''L''1(''G'') also has a bounded approximate identity.</blockquote>

== The group C*-algebra ''C*''(''G'') ==
Let '''C'''[''G''] be the [[group ring]] of a [[discrete group]] ''G''.

For a locally compact group ''G'', the group [[C*-algebra]] ''C*''(''G'') of ''G'' is defined to be the C*-enveloping algebra of ''L''1(''G''), i.e. the completion of ''Cc''(''G'') with respect to the largest C*-norm:

:<math> \|f\|_{C^*} := \sup_\pi \|\pi(f)\|,</math>

where π ranges over all non-degenerate *-representations of ''Cc''(''G'') on Hilbert spaces. When ''G'' is discrete, it follows from the triangle inequality that, for any such π, one has:

:<math> \pi (f) \leq \| f \|_1,</math>

hence the norm is well-defined.

It follows from the definition that ''C*''(''G'') has the following [[universal property]]: any *-homomorphism from '''C'''[''G''] to some '''B'''(''H'') (the C*-algebra of [[bounded operator]]s on some [[Hilbert space]] ''H'') factors through the [[inclusion map]]:

:<math>\mathbf{C}[G] \hookrightarrow C^*_{\text{max}}(G).</math>

=== The reduced group C*-algebra ''Cr*''(''G'') ===
The reduced group C*-algebra ''Cr*''(''G'') is the completion of ''Cc''(''G'') with respect to the norm

:<math> \|f\|_{C^*_r} := \sup \left \{ \|f*g\|_2: \|g\|_2 = 1 \right \},</math>

where

:<math> \|f\|_2 = \sqrt{\int_G |f|^2 d\mu}</math>

is the ''L''2 norm. Since the completion of ''Cc''(''G'') with regard to the ''L''2 norm is a Hilbert space, the ''Cr*'' norm is the norm of the bounded operator acting on ''L''2(''G'') by convolution with ''f'' and thus a C*-norm.

Equivalently, ''Cr*''(''G'') is the C*-algebra generated by the image of the left regular representation on ℓ2(''G'').

In general, ''Cr*''(''G'') is a quotient of ''C*''(''G''). The reduced group C*-algebra is isomorphic to the non-reduced group C*-algebra defined above if and only if ''G'' is [[amenable]].

== von Neumann algebras associated to groups ==
The group von Neumann algebra ''W*''(''G'') of ''G'' is the enveloping von Neumann algebra of ''C*''(''G'').

For a discrete group ''G'', we can consider the [[Hilbert space]] ℓ2(''G'') for which ''G'' is an [[orthonormal basis]]. Since ''G'' operates on ℓ2(''G'') by permuting the basis vectors, we can identify the complex group ring '''C'''[''G''] with a subalgebra of the algebra of [[bounded operator]]s on ℓ2(''G''). The weak closure of this subalgebra, ''NG'', is a [[von Neumann algebra]].

The center of ''NG'' can be described in terms of those elements of ''G'' whose [[conjugacy class]] is finite. In particular, if the identity element of ''G'' is the only group element with that property (that is, ''G'' has the [[infinite conjugacy class property]]), the center of ''NG'' consists only of complex multiples of the identity.

''NG'' is isomorphic to the [[hyperfinite type II-1 factor|hyperfinite type II1 factor]] if and only if ''G'' is [[countable]], [[amenable]], and has the infinite conjugacy class property.

==See also==
* [[Graph algebra]]
* [[Incidence algebra]]
* [[Path algebra]]
* [[Groupoid algebra]]

==References==
*J, Dixmier, ''C* algebras'', ISBN 0-7204-0762-1
*A. A. Kirillov, ''Elements of the theory of representations'', ISBN 0-387-07476-7
*L. H. Loomis, "Abstract Harmonic Analysis", ASIN B0007FUU30
*{{springer|id=G/g045230|title=Group algebra of a locally compact group|author=A.I. Shtern}}
{{PlanetMath attribution|id=3628|title=Group $C^*$-algebra}}

[[Category:Algebras]]
[[Category:Ring theory]]
[[Category:Unitary representation theory]]
[[Category:Harmonic analysis]]
[[Category:Lie groups]]

[[ja:群環]]

Overdetermined system

2013-07-17T02:06:45Z

130.216.25.148: /* Approximate solutions */

In [[computer science]], '''Hirschberg's algorithm''', named after its inventor, [[Dan Hirschberg]], is a [[dynamic programming]] [[algorithm]] that finds the optimal [[sequence alignment]] between two [[string (computer science)|string]]s. Optimality is measured with the [[Levenshtein distance]], defined to be the sum of the costs of insertions, replacements, deletions, and null actions needed to change one string into the other. Hirschberg's algorithm is simply described as a [[divide and conquer algorithm|divide and conquer]] version of the [[Needleman–Wunsch algorithm]].<ref>[http://www.csse.monash.edu.au/~lloyd/tildeAlgDS/Dynamic/Hirsch/ Hirschberg's algorithm]</ref> Hirschberg's algorithm is commonly used in [[computational biology]] to find maximal global alignments of [[DNA]] and [[protein]] sequences.

==Algorithm information==
Hirschberg's algorithm is a generally applicable algorithm for optimal sequence alignment. [[BLAST]] and [[FASTA]] are suboptimal [[Heuristic (computer science)|heuristics]]. If ''x'' and ''y'' are strings, where length(''x'') = ''n'' and length(''y'') = ''m'', the [[Needleman-Wunsch algorithm]] finds an optimal alignment in [[Big O Notation|O]](''nm'') time, using O(''nm'') space. Hirschberg's algorithm is a clever modification of the Needleman-Wunsch Algorithm which still takes O(''nm'') time, but needs only O(min{''n'',''m''}) space.<ref>http://www.cs.tau.ac.il/~rshamir/algmb/98/scribe/html/lec02/node10.html</ref>
One application of the algorithm is finding sequence alignments of DNA or protein sequences. It is also a space-efficient way to calculate the [[longest common subsequence problem|longest common subsequence]] between two sets of data such as with the common [[diff]] tool.

The Hirschberg algorithm can be derived from the Needleman-Wunsch algorithm by observing that:<ref>{{cite journal|author=Hirschberg, D. S.|title=A linear space algorithm for computing maximal common subsequences|journal=Communications of the ACM|volume=18|issue=6|year=1975|pages=341–343|doi=10.1145/360825.360861}}</ref>
# one can compute the optimal alignment score by only storing the current and previous row of the Needleman-Wunsch score matrix;
# if <math>(Z,W) = \operatorname{NW}(X,Y)</math> is the optimal alignment of <math>(X,Y)</math>, and <math>X = X^l + X^r</math> is an arbitrary partition of <math>X</math>, there exists a partition <math>Y^l + Y^r</math> of <math>Y</math> such that <math>\operatorname{NW}(X,Y) = \operatorname{NW}(X^l,Y^l) + \operatorname{NW}(X^r,Y^r)</math>.

== Algorithm description ==

<math>X_i</math> denotes the i-th character of <math>X</math>, where <math>1 . <math>X_{i:j}</math> denotes a substring of size <math>j-i+1</math>, ranging from i-th to the j-th character of <math>X</math>. <math>\operatorname{rev}(X)</math> is the reversed version of <math>X</math>.

<math>X</math> and <math>Y</math> are sequences to be aligned. Let <math>x</math> be a character from <math>X</math>, and <math>y</math> be a character from <math>Y</math>. We assume that <math>\operatorname{Del}(x)</math>, <math>\operatorname{Ins}(y)</math> and <math>\operatorname{Sub}(x,y)</math> are well defined integer-valued functions. These functions represent the cost of deleting <math>x</math>, inserting <math>y</math>, and replacing <math>x</math> with <math>y</math>, respectively.

We define <math>\operatorname{NWScore}(X,Y)</math>, which returns the last line of the Needleman-Wunsch score matrix <math>\mathrm{Score}(i,j)</math>:

'''function''' NWScore(X,Y)
Score(0,0) = 0
'''for''' j=1 '''to''' length(Y)
Score(0,j) = Score(0,j-1) + Ins(Yj)
'''for''' i=1 '''to''' length(X)
Score(i,0) = Score(i-1,0) + Del(Xi)
'''for''' j=1 '''to''' length(Y)
scoreSub = Score(i-1,j-1) + Sub(Xi, Yj)
scoreDel = Score(i-1,j) + Del(Xi)
scoreIns = Score(i,j-1) + Ins(Yj)
Score(i,j) = max(scoreSub, scoreDel, scoreIns)
'''end'''
'''end'''
'''for''' j=0 '''to''' length(Y)
LastLine(j) = Score(length(X),j)
'''return''' LastLine

Note that at any point, <math>\operatorname{NWScore}</math> only requires the two most recent rows of the score matrix. Thus, <math>\operatorname{NWScore}</math> can be implemented in <math>O(\operatorname{min}\{\operatorname{length}(X),\operatorname{length}(Y)\})</math> space.

The Hirschberg algorithm follows:

'''function''' Hirschberg(X,Y)
Z = ""
W = ""
'''if''' length(X) == 0 '''or''' length(Y) == 0
'''if''' length(X) == 0
'''for''' i=1 '''to''' length(Y)
Z = Z + '-'
W = W + Yi
'''end'''
'''else if''' length(Y) == 0
'''for''' i=1 '''to''' length(X)
Z = Z + Xi
W = W + '-'
'''end'''
'''end'''
'''else if''' length(X) == 1 '''or''' length(Y) == 1
(Z,W) = NeedlemanWunsch(X,Y)
'''else'''
xlen = length(X)
xmid = length(X)/2
ylen = length(Y)

ScoreL = NWScore(X1:xmid, Y)
ScoreR = NWScore(rev(Xxmid+1:xlen), rev(Y))
ymid = PartitionY(ScoreL, ScoreR)

(Z,W) = Hirschberg(X1:xmid, y1:ymid) + Hirschberg(Xxmid+1:xlen, Yymid+1:ylen)
'''end'''
'''return''' (Z,W)

In the context of Observation (2), assume that <math>X^l + X^r</math> is a partition of <math>X</math>. Function <math>\mathrm{PartitionY}</math> returns index <math>\mathrm{ymid}</math> such that <math>Y^l = Y_{1:\mathrm{ymid}}</math> and <math>Y^r = Y_{\mathrm{ymid}+1:\operatorname{length}(Y)}</math>. <math>\mathrm{PartitionY}</math> is given by

'''function''' PartitionY(ScoreL, ScoreR)
'''return''' [[arg max]] ScoreL + rev(ScoreR)

== Example ==

Let

<math>
\begin{align}
X &= \mathrm{AGTACGCA},\\
Y &= \mathrm{TATGC},\\
\operatorname{Del}(x) &= -2,\\
\operatorname{Ins}(y) &= -2,\\
\operatorname{Sub}(x,y) &= \begin{cases} +2, & \mbox{if } x = y \\ -1, & \mbox{if } x \neq y.\end{cases}
\end{align}
</math>.

The optimal alignment is given by

W = AGTACGCA
Z = --TATGC-

Indeed, this can be verified by backtracking its corresponding Needleman-Wunsch matrix:

'''T A T G C'''
'''0''' -2 -4 -6 -8 -10
'''A''' '''-2''' -1 0 -2 -4 -6
'''G''' '''-4''' -3 -2 -1 0 -2
'''T''' -6 '''-2''' -4 0 -2 -1
'''A''' -8 -4 '''0''' -2 -1 -3
'''C''' -10 -6 -2 '''-1''' -3 1
'''G''' -12 -8 -4 -3 '''1''' -1
'''C''' -14 -10 -6 -5 -1 '''3'''
'''A''' -16 -12 -8 -7 -3 '''1'''

One starts with the top level call to <math>\operatorname{Hirschberg}(\mathrm{AGTACGCA}, \mathrm{TATGC})</math>. The call to <math>\operatorname{NWScore}(\mathrm{AGTA},Y)</math> produces the following matrix:

'''T A T G C'''
0 -2 -4 -6 -8 -10
'''A''' -2 -1 0 -2 -4 -6
'''G''' -4 -3 -2 -1 0 -2
'''T''' -6 -2 -4 0 -2 -1
'''A''' -8 -4 0 -2 -1 -3

Likewise, <math>\operatorname{NWScore}(\operatorname{rev}(\mathrm{CGCA}), \operatorname{rev}(Y))</math> generates the following matrix:

'''C G T A T'''
0 -2 -4 -6 -8 -10
'''A''' -2 -1 -3 -5 -4 -6
'''C''' -4 0 -2 -4 -6 -5
'''G''' -6 -2 2 0 -2 -4
'''C''' -8 -4 0 1 -1 -3

Their last lines are respectively

ScoreL = [ -8 -4 0 -2 -1 -3 ]
ScoreR = [ -8 -4 0 1 -1 -3 ]

<tt>PartitionY(ScoreL, ScoreR) = 2</tt>, such that <math>X = \mathrm{AGTA} + \mathrm{CGCA}</math> and <math>Y = \mathrm{TA} + \mathrm{TGC}</math>.

The entire Hirschberg recursion (which we omit for brevity) produces the following tree:

(AGTACGCA,TATGC)
/ \
(AGTA,TA) (CGCA,TGC)
/ \ / \
(AG,) (TA,TA) (CG,TG) (CA,C)
/ \ / \
(T,T) (A,A) (C,T) (G,G)

The leaves of the tree contain the optimal alignment.

==See also==
* [[Needleman-Wunsch algorithm]]
* [[Smith Waterman algorithm]]
* [[Levenshtein distance]]
* [[Longest common subsequence problem|Longest Common Subsequence]]

==References==
{{reflist}}

{{DEFAULTSORT:Hirschberg's Algorithm}}
[[Category:Sequence alignment algorithms]]
[[Category:Bioinformatics algorithms]]
[[Category:Articles with example pseudocode]]
[[Category:Dynamic programming]]

Exceptional isomorphism

2012-12-10T23:18:08Z

130.216.48.51: /* Groups */ trivial group is not exceptional isomorphism

In mathematics, '''Gabriel's theorem''', proved by [[Pierre Gabriel]], classifies the [[quiver (mathematics)|quivers]] of finite type in terms of [[Dynkin diagram]]s.

==Statement==

A quiver is of '''finite type''' if it has only finitely many isomorphism classes of indecomposable representations. {{harvtxt|Gabriel|1972}} classified all quivers of finite type, and also their indecomposable representations. More precisely, Gabriel's theorem states that:

# A (connected) quiver is of finite type if and only if its underlying graph (when the directions of the arrows are ignored) is one of the [[ADE classification|ADE]] [[Dynkin diagram]]s: <math>A_n</math>, <math>D_n</math>, <math>E_6</math>, <math>E_7</math>, <math>E_8</math>.
# The indecomposable representations are in a one-to-one correspondence with the positive roots of the [[root system]] of the Dynkin diagram.

{{harvtxt|Dlab|Ringel|1973}} found a generalization of Gabriel's theorem in which all Dynkin diagrams of finite dimensional semisimple Lie algebras occur.

==References==

*{{Citation | last1=Bernšteĭn | first1=I. N. | last2=Gelfand | first2=I. M. | last3=Ponomarev | first3=V. A. | title=Coxeter functors, and Gabriel's theorem | doi=10.1070/RM1973v028n02ABEH001526 | mr=0393065 | year=1973 | journal=Russian mathematical surveys | issn=0042-1316 | volume=28 | issue=2 | pages=17–32}}
*{{Citation | last1=Dlab | first1=Vlastimil | last2=Ringel | first2=Claus Michael | title=On algebras of finite representation type | url=http://books.google.com/books?id=_JrnAAAAMAAJ | publisher=Department of Mathematics, Carleton Univ., Ottawa, Ont. | series=Carleton mathematical lecture notes, | mr=0347907 | year=1973 | volume=2}}
*{{Citation | last1=Gabriel | first1=Peter | title=Unzerlegbare Darstellungen. I | doi=10.1007/BF01298413 | mr=0332887 | year=1972 | journal=Manuscripta Mathematica | issn=0025-2611 | volume=6 | pages=71–103}}

[[Category:Theorems in representation theory]]

First-countable space

2012-08-11T01:09:58Z

130.216.46.49: Corrected to R\N from R\Q

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2012-08-08T04:48:06Z

130.216.108.133:

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2012-07-30T04:41:36Z

130.216.46.16: /* Properties */ Recoded into LaTeX

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2012-06-03T23:48:23Z

130.216.211.90: /* Ewald Construction */

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2012-05-16T04:44:35Z

130.216.56.123: /* Rings plus pi bonds formulation */

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