formulasearchengine - User contributions [en]

Multivariate normal distribution

2014-03-01T16:46:59Z

145.97.197.247: /* Kullback–Leibler divergence */

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Direct image functor

2013-11-11T15:29:31Z

145.97.30.152: /* Higher direct images */

{{Redirect|Brill's theorem|the result in algebraic geometry|Brill–Noether theorem}}

[[File:Discriminant49CubicFieldFundamentalDomain.png|thumb|300px|right|A fundamental domain of the ring of integers of the field ''K'' obtained from '''Q''' by adjoining a root of ''x''3 − ''x''2 − 2''x'' + 1. This fundamental domain sits inside ''K'' &otimes;'''Q''''''R'''. The discriminant of ''K'' is 49 = 72. Accordingly, the volume of the fundamental domain is 7 and ''K'' is only [[Splitting of prime ideals in Galois extensions|ramified]] at 7.]]
In [[mathematics]], the '''discriminant of an [[algebraic number field]]''' is a numerical [[invariant (mathematics)|invariant]] that, loosely speaking, measures the size of the ([[ring of integers]] of the) algebraic number field. More specifically, it is related to the volume of the [[fundamental domain]] of the ring of integers, and it regulates which [[prime number|primes]] are [[Ramified prime#In algebraic number theory|ramified]].

The discriminant is one of the most basic invariants of a number field, and occurs in several important [[Analytic Number Theory|analytic]] formulas such as the [[functional equation (L-function)|functional equation]] of the [[Dedekind zeta function]] of ''K'', and the [[analytic class number formula]] for ''K''. An old theorem of [[Charles Hermite|Hermite]] states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an [[open problem]], and the subject of current research.<ref>{{harvnb|Cohen|Diaz y Diaz|Olivier|2002}}</ref>

The discriminant of ''K'' can be referred to as the '''absolute discriminant''' of ''K'' to distinguish it from the '''relative discriminant''' of an [[field extension|extension]] ''K''/''L'' of number fields. The latter is an [[Ideal (ring theory)|ideal]] in the ring of integers of ''L'', and like the absolute discriminant it indicates which primes are ramified in ''K''/''L''. It is a generalization of the absolute discriminant allowing for ''L'' to be bigger than '''Q'''; in fact, when ''L'' = '''Q''', the relative discriminant of ''K''/'''Q''' is the [[principal ideal]] of '''Z''' generated by the absolute discriminant of ''K''.

==Definition==
Let ''K'' be an algebraic number field, and let ''O''''K'' be its ring of integers. Let ''b''1, ..., ''b''''n'' be an [[integral basis]] of ''O''''K'' (i.e. a basis as a [[Module (mathematics)|'''Z'''-module]]), and let {σ1, ..., σ''n''} be the set of embeddings of ''K'' into the [[complex number]]s (i.e. [[injective]] [[ring homomorphism]]s ''K'' → '''C'''). The '''discriminant''' of ''K'' is the [[Square (algebra)|square]] of the [[determinant]] of the ''n'' by ''n'' [[Matrix (mathematics)|matrix]] ''B'' whose (''i'',''j'')-entry is σ''i''(''b''''j''). Symbolically,
:<math>\Delta_K=\left(\operatorname{det}\left(\begin{array}{cccc}
\sigma_1(b_1) & \sigma_1(b_2) &\cdots & \sigma_1(b_n) \\
\sigma_2(b_1) & \ddots & & \vdots \\
\vdots & & \ddots & \vdots \\
\sigma_n(b_1) & \cdots & \cdots & \sigma_n(b_n)
\end{array}\right)\right)^2.
</math>
 
Equivalently, the [[Field trace|trace]] from ''K'' to '''Q''' can be used. Specifically, define the [[Trace form#Trace form and discriminant|trace form]] to be the matrix whose (''i'',''j'')-entry is
'''Tr'''''K''/'''Q'''(''b''''i''''b''''j''). This matrix equals ''B''T''B'', so the discriminant of ''K'' is the determinant of this matrix.

==Examples==
*[[Quadratic fields|Quadratic number fields]]: let ''d'' be a [[square-free integer]], then the discriminant of <math>K=\mathbf{Q}(\sqrt{d})</math> is<ref name=MP130/>
:: <math>\Delta_K=\left\{\begin{array}{ll} d &\text{if }d\equiv 1\pmod 4 \\ 4d &\text{if }d\equiv 2,3\pmod 4. \\\end{array}\right.</math>
:An integer that occurs as the discriminant of a quadratic number field is called a [[fundamental discriminant]].<ref>Definition 5.1.2 of {{harvnb|Cohen|1993}}</ref>
*[[Cyclotomic field]]s: let ''n'' > 2 be an integer, let ζ''n'' be a [[Root of unity|primitive ''n''th root of unity]], and let ''K''''n'' = '''Q'''(ζ''n'') be the ''n''th cyclotomic field. The discriminant of ''K''''n'' is given by<ref>Proposition 2.7 of {{harvnb|Washington|1997}}</ref><ref name=MP130>{{citation | first1=Yu. I. | last1=Manin | authorlink1=Yuri I. Manin | first2=A. A. | last2=Panchishkin | title=Introduction to Modern Number Theory | series=Encyclopaedia of Mathematical Sciences | volume=49 | edition=Second | year=2007 | isbn=978-3-540-20364-3 | issn=0938-0396 | zbl=1079.11002 | page=130 }}</ref>
:: <math>\Delta_{K_n} = (-1)^{\varphi(n)/2} \frac{n^{\varphi(n)}}{\displaystyle\prod_{p|n} p^{\varphi(n)/(p-1)}}</math>
: where <math>\varphi(n)</math> is [[Euler's totient function]], and the product in the denominator is over primes ''p'' dividing ''n''.
*Power bases: In the case where the ring of integers has a [[power integral basis]], that is, can be written as ''O''''K'' = '''Z'''[α], the discriminant of ''K'' is equal to the [[discriminant]] of the [[Minimal polynomial (field theory)|minimal polynomial]] of α. To see this, one can chose the integral basis of ''O''''K'' to be ''b''1 = 1, ''b''2 = α, ''b''3 = ''α''2, ..., ''b''''n'' = ''α''''n''−1. Then, the matrix in the definition is the [[Vandermonde matrix]] associated to α''i'' = σ''i''(α), whose determinant squared is
:: <math>\prod_{1\leq i<j\leq n}(\alpha_i-\alpha_j)^2</math>
:which is exactly the definition of the discriminant of the minimal polynomial.
*Let ''K'' = '''Q'''(α) be the number field obtained by [[Adjunction (field theory)|adjoining]] a [[Root of a function|root]] α of the [[polynomial]] ''x''3 − ''x''2 − 2''x'' − 8. This is [[Richard Dedekind]]'s original example of a number field whose ring of integers does not possess a power basis. An integral basis is given by {1, α, α(α + 1)/2} and the discriminant of ''K'' is −503.<ref>{{harvnb|Dedekind|1878}}, pp. 30–31</ref><ref>{{harvnb|Narkiewicz|2004}}, p. 64</ref>
*Repeated discriminants: the discriminant of a quadratic field uniquely identifies it, but this is not true, in general, for [[degree of a number field|higher-degree]] number fields. For example, there are two [[isomorphism|non-isomorphic]] [[cubic field]]s of discriminant 3969. They are obtained by adjoining a root of the polynomial {{nobreak|''x''3 − 21''x'' + 28}} or {{nobreak|''x''3 − 21''x'' − 35}}, respectively.<ref>{{harvnb|Cohen|1993|loc=Theorem 6.4.6}}</ref>


==Basic results==
*'''Brill's theorem''':<ref>{{harvnb|Koch|1997|p=11}}</ref> The [[sign (mathematics)|sign]] of the discriminant is (−1)''r''2 where ''r''2 is the number of [[Algebraic number field#Archimedean places|complex places]] of ''K''.<ref>Lemma 2.2 of {{harvnb|Washington|1997}}</ref>
*A prime ''p'' ramifies in ''K'' if, and only if, ''p'' divides Δ''K'' .<ref>Corollary III.2.12 of {{harvnb|Neukirch|1999}}</ref>
*'''Stickelberger's theorem''':<ref>Exercise I.2.7 of {{harvnb|Neukirch|1999}}</ref>
:: <math>\Delta_K\equiv 0\text{ or }1 \pmod 4.</math>
*'''[[Minkowski's bound]]''':<ref>Proposition III.2.14 of {{harvnb|Neukirch|1999}}</ref> Let ''n'' denote the [[Degree of a field extension|degree]] of the extension ''K''/'''Q''' and ''r''2 the number of complex places of ''K'', then
:: <math>|\Delta_K|^{1/2}\geq \frac{n^n}{n!}\left(\frac{\pi}{4}\right)^{r_2} \geq \frac{n^n}{n!}\left(\frac{\pi}{4}\right)^{n/2}.</math>

*'''Minkowski's theorem''':<ref>Theorem III.2.17 of {{harvnb|Neukirch|1999}}</ref> If ''K'' is not '''Q''', then |Δ''K''| > 1 (this follows directly from the Minkowski bound).
*'''[[Hermite–Minkowski theorem]]''':<ref>Theorem III.2.16 of {{harvnb|Neukirch|1999}}</ref> Let ''N'' be a positive integer. There are only finitely many (up to isomorphisms) algebraic number fields ''K'' with |Δ''K''| < ''N''. Again, this follows from the Minkowski bound together Hermite's theorem (There are only finitely many algebraic number fields with prescribed discriminant).

==History==
[[File:Dedekind.jpeg|thumb|150px|right|Richard Dedekind showed that every number field possesses an integral basis, allowing him to define the discriminant of an arbitrary number field.<ref name=DedekindX>Dedekind's supplement X of the second edition of [[Peter Gustav Lejeune Dirichlet]]'s [[Vorlesungen über Zahlentheorie]] {{harv|Dedekind|1871}}</ref>]]
The definition of the discriminant of a general algebraic number field, ''K'', was given by Dedekind in 1871.<ref name=DedekindX/> At this point, he already knew the relationship between the discriminant and ramification.<ref>{{harvnb|Bourbaki|1994}}</ref>

Hermite's theorem predates the general definition of the discriminant with Charles Hermite publishing a proof of it in 1857.{{sfn|Hermite|1857}} In 1877, [[Alexander von Brill]] determined the sign of the discriminant.{{sfn|Brill|1877}} [[Leopold Kronecker]] first stated Minkowski's theorem in 1882,{{sfn|Kronecker|1882}} though the first proof was given by Hermann Minkowski in 1891.{{sfn|Minkowski|1891a}} In the same year, Minkowski published his bound on the discriminant.{{sfn|Minkowski| 1891b}} Near the end of the nineteenth century, [[Ludwig Stickelberger]] obtained his theorem on the residue of the discriminant modulo four.{{sfn|Stickelberger|1897}}<ref>All facts in this paragraph can be found in {{harvnb|Narkiewicz|2004|pp=59, 81}}</ref>

==Relative discriminant==
The discriminant defined above is sometimes referred to as the ''absolute'' discriminant of ''K'' to distinguish it from the '''relative discriminant''' Δ''K''/''L'' of an extension of number fields ''K''/''L'', which is an ideal in ''O''''L''. The relative discriminant is defined in a fashion similar to the absolute discriminant, but must take into account that ideals in ''O''''L'' may not be principal and that there may not be an integral basis of ''K''/''L''. Let {σ1, ..., σ''n''} be the set of embeddings of ''K'' into '''C''' which are the identity on ''L''. If ''b''1, ..., ''b''''n'' is any basis of ''K'' over ''L'', let ''d''(''b''1, ..., ''b''''n'') be the square of the determinant of the ''n'' by ''n'' matrix whose (''i'',''j'')-entry is σ''i''(''b''''j''). Then, the relative discriminant of ''K''/''L'' is the ideal generated by the ''d''(''b''1, ..., ''b''''n'') as {''b''1, ..., ''b''''n''} varies over all bases of ''K''/''L'' with the property that ''bi'' ∈ ''OK'' for all ''i''. Alternatively, the relative discriminant of ''K''/''L'' is the [[ideal norm|norm]] of the [[different ideal|different]] of ''K''/''L''.<ref name="NeukirchIII2">{{harvnb|Neukirch|1999|loc=§III.2}}</ref> When ''L'' = '''Q''', the relative discriminant Δ''K''/'''Q''' is the principal ideal of '''Z''' generated by the absolute discriminant Δ''K'' . In a [[tower of fields]] ''K''/''L''/''F'' the relative discriminants are related by

:<math>\Delta_{K/F} = \mathcal{N}_{L/F}\left({\Delta_{K/L}}\right) \Delta_{L/F}^{[K:L]}</math>

where <math>\mathcal{N}</math> denotes relative [[field norm|norm]].<ref>Corollary III.2.10 of {{harvnb|Neukirch|1999}} or Proposition III.2.15 of {{harvnb|Fröhlich|Taylor|1993}}</ref>

===Ramification===
The relative discriminant regulates the [[Tame ramification|ramification]] data of the field extension ''K''/''L''. A prime ideal ''p'' of ''L'' ramifies in ''K'' if, and only if, it divides the relative discriminant Δ''K''/''L''. An extension is unramified if, and only if, the discriminant is the unit ideal.<ref name="NeukirchIII2"/> The Minkowski bound above shows that there are no non-trivial unramified extensions of '''Q'''. Fields larger than '''Q''' may have unramified extensions, for example, for any field with [[class number (number theory)|class number]] greater than one, its [[Hilbert class field]] is a non-trivial unramified extension.

==Root discriminant==
The '''root discriminant''' of a number field, ''K'', of degree ''n'', often denoted rd''K'', is defined as the ''n''-th root of the absolute value of the (absolute) discriminant of ''K''.<ref name=voight2008>{{harvnb|Voight|2008}}</ref> The relation between relative discriminants in a tower of fields shows that the root discriminant does not change in an unramified extension. The existence of a [[class field tower]] provides bounds on the root discriminant: the existence of an infinite class field tower over '''Q'''(√-m) where ''m'' = 3·5·7·11·19 shows that there are infinitely many fields with root discriminant 2√''m'' ≈ 296.276.<ref name=koch181>{{harvnb|Koch|1997|pp=181–182}}</ref> If we let ''r'' and 2''s'' be the number of real and complex embeddings, so that ''n'' = ''r'' + 2''s'', put ''ρ'' = ''r''/''n'' and ''σ'' = 2''s''/''n''. Set ''α''(''ρ'', ''σ'') to be the infimum of rd''K'' for ''K'' with (''r', 2''s') = (''ρn'', ''σn''). We have<ref name=koch181/>

:<math> \alpha(\rho,\sigma) \ge 60.8^\rho 22.3^\sigma </math>

and on the assumption of the [[generalized Riemann hypothesis]]

:<math> \alpha(\rho,\sigma) \ge 215.3^\rho 44.7^\sigma . </math>

So we have ''α''(0,1) < 296.276. Martinet has shown ''α''(0,1) < 93 and ''α''(1,0) < 1059.<ref name=koch181/><ref>{{cite journal | zbl=0369.12007 | last=Martinet | first=Jacques | title=Tours de corps de classes et estimations de discriminants | language=French | journal=[[Inventiones Mathematicae]] | volume=44 | pages=65–73 | year=1978 }}</ref> {{harvnb|Voight|2008}} proves that for totally real fields, the root discriminant is > 14, with 1229 exceptions.

==Relation to other quantities==
*When embedded into <math>K\otimes_\mathbf{Q}\mathbf{R}</math>, the volume of the fundamental domain of ''O''''K'' is <math>\sqrt{|\Delta_K|}</math> (sometimes a different [[Measure (mathematics)|measure]] is used and the volume obtained is <math>2^{-r_2}\sqrt{|\Delta_K|}</math>, where ''r''2 is the number of complex places of ''K'').
*Due to its appearance in this volume, the discriminant also appears in the functional equation of the Dedekind zeta function of ''K'', and hence in the analytic class number formula, and the [[Brauer–Siegel theorem]].
*The relative discriminant of ''K''/''L'' is the [[Artin conductor]] of the [[regular representation]] of the [[Galois group]] of ''K''/''L''. This provides a relation to the Artin conductors of the [[character group|characters]] of the Galois group of ''K''/''L'', called the [[conductor-discriminant formula]].<ref>Section 4.4 of {{harvnb|Serre|1967}}</ref>

==Notes==
{{reflist|2}}

==References==

===Primary sources===
* {{Citation
| last=Brill
| first=Alexander von
| title=Ueber die Discriminante
| year=1877
| journal=Mathematische Annalen
| volume=12
| pages=87–89
| doi=10.1007/BF01442468
| url=http://gdz.sub.uni-goettingen.de/en/dms/load/toc/?PPN=PPN235181684_0012&DMDID=dmdlog8
| accessdate=2009-08-22
| issue=1
| mr=1509928
| jfm=09.0059.02
}}
* {{Citation
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| first=Richard
| author-link=Richard Dedekind
| title=Vorlesungen über Zahlentheorie von P.G. Lejeune Dirichlet
| url=http://gdz.sub.uni-goettingen.de/en/dms/load/toc/?PPN=PPN30976923X&DMDID=dmdlog1
| edition=2
| year=1871
| publisher=Vieweg
| accessdate=2009-08-05
}}
* {{Citation
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| title=Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Congruenzen
| url=http://gdz.sub.uni-goettingen.de/en/dms/load/toc/?PPN=PPN250442582_0023&DMDID=dmdlog10
| year=1878
| journal=Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen
| volume=23
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| issue=1
}}
*{{Citation
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| title=Extrait d'une lettre de M. C. Hermite à M. Borchardt sur le nombre limité d'irrationalités auxquelles se réduisent les racines des équations à coefficients entiers complexes d'un degré et d'un discriminant donnés
| journal=[[Crelle's Journal]]
| volume=53
| pages=182–192
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| url=http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN243919689_0053&DMDID=dmdlog14
| accessdate=2009-08-20
}}
*{{Citation
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| url=http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN243919689_0092&DMDID=dmdlog4
| journal=[[Crelle's journal]]
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| accessdate=2009-08-20
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}}
*{{Citation
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| url=http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN243919689_0107&DMDID=dmdlog25
| journal=Crelle's journal
| volume=107
| year=1891a
| pages=278–297
| accessdate=2009-08-20
| jfm=23.0212.01
}}
*{{Citation
| last=Minkowski
| first=Hermann
| author-link=Hermann Minkowski
| title=Théorèmes d'arithmétiques
| journal=[[Comptes rendus de l'Académie des sciences]]
| year=1891b
| volume=112
| pages=209–212
| jfm=23.0214.01
}}
*{{Citation
| last=Stickelberger
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| author-link=Ludwig Stickelberger
| contribution=Über eine neue Eigenschaft der Diskriminanten algebraischer Zahlkörper
| title=Proceedings of the First International Congress of Mathematicians, Zürich
| year=1897
| pages=182–193
| jfm=29.0172.03
}}

===Secondary sources===
* {{Bourbaki EHM}}
* {{Citation
| last=Cohen
| first=Henri
| author-link=Henri Cohen (number theorist)
| title=A Course in Computational Algebraic Number Theory
| publisher=[[Springer-Verlag]]
| location=Berlin, New York
| series=Graduate Texts in Mathematics
| isbn=978-3-540-55640-4
| year=1993
| volume=138
| mr=1228206
}}
* {{Citation
| last=Cohen
| first=Henri
| author-link=Henri Cohen (number theorist)
| last2=Diaz y Diaz
| first2=Francisco
| last3=Olivier
| first3=Michel
| contribution=A Survey of Discriminant Counting
| contribution-url=http://www.springerlink.com/content/2bl9qep1qb4k32vy/
| title=Algorithmic Number Theory, Proceedings, 5th International Syposium, ANTS-V, University of Sydney, July 2002
| editor-last=Fieker
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| editor2-last=Kohel
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| pages=80–94
| accessdate=2009-08-19
| mr=2041075
}}
* {{Citation
| last= Fröhlich
| first = Albrecht
| authorlink= Albrecht Fröhlich
| last2=Taylor
| first2=Martin
| authorlink2= Martin J. Taylor
| title=Algebraic number theory
| publisher=[[Cambridge University Press]]
| series=Cambridge Studies in Advanced Mathematics
| isbn=978-0-521-43834-6
| year=1993
| volume=27
| mr= 1215934
}}
* {{citation | first=Helmut | last=Koch | title=Algebraic Number Theory | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-63003-1 | zbl=0819.11044 | series=Encycl. Math. Sci. | volume=62 | edition=2nd printing of 1st}}
* {{Citation
| last=Narkiewicz
| first=Władysław
| title=Elementary and analytic theory of algebraic numbers
| edition=3
| year=2004
| publisher=Springer-Verlag
| location=Berlin
| series=Springer Monographs in Mathematics
| isbn=978-3-540-21902-6
| mr=2078267
}}
* {{Neukirch ANT}}
* {{Citation
| last=Serre
| first=Jean-Pierre
| author-link=Jean-Pierre Serre
| chapter=Local class field theory
| title=Algebraic Number Theory, Proceedings of an instructional conference at the University of Sussex, Brighton, 1965
| editor-last=Cassels
| editor-first=J. W. S.
| editor-link=J. W. S. Cassels
| editor2-last=Fröhlich
| editor2-first=Albrecht
| editor2-link=Albrecht Fröhlich
| publisher=Academic Press
| location=London
| isbn=0-12-163251-2
| year=1967
| mr=0220701
}}
* {{Citation
| first = John
| last = Voight
| contribution = Enumeration of totally real number fields of bounded root discriminant
| title = Algorithmic number theory. Proceedings, 8th International Symposium, ANTS-VIII, Banff, Canada, May 2008
| editor1-last=van der Poorten | editor1-first=Alfred J. | editor1-link=Alfred van der Poorten
| editor2-last=Stein
| editor2-first=Andreas
| pages = 268–281
| publisher = Springer-Verlag
| location=Berlin
| series=Lecture Notes in Computer Science
| volume=5011
| year = 2008
| arxiv = 0802.0194
| doi = 10.1007/978-3-540-79456-1_18
| isbn=978-3-540-79455-4| mr = 2467853 | zbl=1205.11125
}}
* {{Citation
| last=Washington
| first=Lawrence
| title=Introduction to Cyclotomic Fields
| edition=2 nd
| publisher=Springer-Verlag
| location=Berlin, New York
| series=Graduate Texts in Mathematics
| isbn=978-0-387-94762-4
| year=1997
| volume=83
| mr=1421575 | zbl=0966.11047
}}

==Further reading==
* {{Citation | last=Milne | first=James S. | author-link=James S. Milne | title=Algebraic Number Theory | year=1998 | url=http://www.jmilne.org/math/CourseNotes/ant.html | accessdate=2008-08-20}}

{{DEFAULTSORT:Discriminant Of An Algebraic Number Field}}
[[Category:Algebraic number theory]]

Marginal rate of substitution

2013-10-10T10:59:47Z

145.97.245.249: /* See also */

{{String theory}}

:''This article is about S-duality (strong–weak duality) in physics. For the mathematical S-duality (Spanier–Whitehead duality), see [[S-duality (homotopy theory)]].''

In [[theoretical physics]], '''S-duality''' is an equivalence of two physical theories, which may be either [[quantum field theory|quantum field theories]] or [[string theory|string theories]]. S-duality is useful for doing calculations in theoretical physics because it relates a theory in which calculations are difficult to a theory in which they are easier.<ref>Frenkel 2009, p.2</ref>

In quantum field theory, S-duality generalizes a well known fact from [[classical electrodynamics]], namely the [[invariant (physics)|invariance]] of [[Maxwell's equations]] under the interchange of [[electric field|electric]] and [[magnetic field]]s.<ref>Frenkel 2009, p.2</ref> One of the earliest known examples of S-duality in quantum field theory is [[Montonen-Olive duality]] which relates two versions of a [[quantum field theory]] called [[N=4 supersymmetric Yang-Mills theory]].<ref>Montonen and Olive 1977</ref> Recent work of [[Anton Kapustin]] and [[Edward Witten]] suggests that Montonen-Olive duality is closely related to a research program in mathematics called the [[geometric Langlands program]].<ref>Kapustin and Witten 2007</ref> Another realization of S-duality in quantum field theory is [[Seiberg duality]], which relates two versions of a theory called [[supersymmetric gauge theory|N=1 supersymmetric Yang-Mills theory]].<ref>Seiberg 1995</ref>

There are also many examples of S-duality in string theory. The existence of these [[string duality|string dualities]] implies that seemingly different formulations of string theory are actually physically equivalent. This led to the realization, in the mid-1990s, that all of the five consistent [[superstring theory|superstring theories]] are just different limiting cases of a single eleven-dimensional theory called [[M-theory]].<ref>Zwiebach 2009, p.325</ref>

==Overview==

In quantum field theory and string theory, a [[coupling constant]] is a number that controls the strength of interactions in the theory. For example, the strength of [[gravity]] is described by a number called [[Newton's constant]], which appears in [[Newton's law of gravity]] and also in the equations of [[Albert Einstein|Albert Einstein's]] [[general theory of relativity]]. Similarly, the strength of the [[electromagnetic force]] is described by a coupling constant, which is related to the charge carried by a single [[proton]].

To compute observable quantities in quantum field theory or string theory, physicists typically apply the methods of [[perturbation theory]]. In perturbation theory, quantities called [[probability amplitude]]s, which determine the probability for various physical processes to occur, are expressed as [[infinite series|sums of infinitely many terms]], where each term is proportional to a [[exponent|power]] of the coupling constant <math>g</math>:

: <math>A=A_0+A_1g+A_2g^2+A_3g^3+\dots</math>.

In order for such an expression to make sense, the coupling constant must be less than 1 so that the higher powers of <math>g</math> become negligibly small and the sum is finite. If the coupling constant is not less than 1, then the terms of this sum will grow larger and larger, and the expression gives a meaningless infinite answer. In this case the theory is said to be ''strongly coupled'', and one cannot use perturbation theory to make predictions.

For certain theories, S-duality provides a way of doing computations at strong coupling by translating these computations into different computations in a weakly coupled theory. S-duality is a particular example of a general notion of [[duality (physics)|duality]] in physics. The term ''duality'' refers to a situation where two seemingly different [[physical system]]s turn out to be equivalent in a nontrivial way. If two theories are related by a duality, it means that one theory can be transformed in some way so that it ends up looking just like the other theory. The two theories are then said to be ''dual'' to one another under the transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena.

S-duality is useful because it relates a theory with coupling constant <math>g</math> to an equivalent theory with coupling constant <math>1/g</math>. Thus it relates a strongly coupled theory (where the coupling constant <math>g</math> is much greater than 1) to a weakly coupled theory (where the coupling constant <math>1/g</math> is much less than 1 and computations are possible). For this reason, S-duality is called a '''strong-weak duality'''.

==S-duality in quantum field theory==

===A symmetry of Maxwell's equations===

In [[classical physics]], the behavior of the [[electric field|electric]] and [[magnetic field]] is described by a system of equations known as [[Maxwell's equations]]. Working in the language of [[vector calculus]] and assuming that no [[electric charge]]s or [[Electric current|currents]] are present, these equations can be written<ref>Griffiths 1999, p.326</ref>

:<math>\begin{align}
\nabla \cdot \mathbf{E} &= 0, \\
\nabla \cdot \mathbf{B} &= 0, \\
\nabla \times \mathbf{E} &= -\frac{\partial\mathbf B}{\partial t}, \\
\nabla \times \mathbf{B} &= \frac{1}{c^2} \frac{\partial \mathbf E}{\partial t}.
\end{align}</math>

Here <math>\mathbf{E}</math> is a [[Euclidean vector|vector]] (or more precisely a ''[[vector field]]'' whose magnitude and direction may vary from point to point in space) representing the electric field, <math>\mathbf{B}</math> is a vector representing the magnetic field, <math>t</math> is time, and <math>c</math> is the [[speed of light]]. The other symbols in these equations refer to the [[divergence]] and [[curl (mathematics)|curl]], which are concepts from [[vector calculus]].

An important property of these equations<ref>Griffiths 1999, p.327</ref> is their [[invariant (physics)|invariance]] under the transformation that simultaneously replaces the electric field <math>\mathbf{E}</math> by the magnetic field <math>\mathbf{B}</math> and replaces <math>\mathbf{B}</math> by <math>-1/c^2\mathbf{E}</math>:

:<math>\begin{align}
\mathbf{E} &\rightarrow\mathbf{B} \\
\mathbf{B} &\rightarrow -\frac{1}{c^2}\mathbf{E}.
\end{align}</math>

In other words, given a pair of electric and magnetic fields that [[vacuum solution|solve]] Maxwell's equations, it is possible to describe a new physical setup in which these electric and magnetic fields are essentially interchanged, and the new fields will again give a solution of Maxwell's equations. This situation is the most basic manifestation of S-duality in quantum field theory. Indeed, as we explain below, there are versions of S-duality that directly generalize this symmetry of Maxwell's equations in the framework of quantum field theory.

===Montonen-Olive duality===

{{main|Montonen–Olive duality}}

In quantum field theory, the electric and magnetic fields are unified into a single entity called the [[electromagnetic field]], and this [[field (physics)|field]] is described by a special type of quantum field theory called a [[gauge theory]] or [[Yang-Mills theory]]. In a gauge theory, the physical fields have a high degree of [[symmetry]] which can be understood mathematically using the notion of a [[Lie group]]. This Lie group is known as the [[gauge group]]. The electromagnetic field is described by a very simple gauge theory corresponding to the [[abelian group|abelian]] gauge group [[unitary group|U(1)]], but there are other gauge theories with more complicated [[non-abelian gauge theory|non-abelian gauge groups]].<ref>For an introduction to quantum field theory in general including the basics of gauge theory, see Zee 2010.</ref>

It is natural to ask whether there is an analog in gauge theory of the symmetry interchanging the electric and magnetic fields in Maxwell's equations. The answer was given in the late 1970s by [[Claus Montonen]] and [[David Olive]],<ref>Montonen and Olive 1977</ref> building on earlier work of [[Peter Goddard (physicist)|Peter Goddard]], [[Jean Nuyts]], and Olive.<ref>Goddard, Nuyts, and Olive 1977</ref> Their work provides an example of S-duality now known as [[Montonen-Olive duality]]. Montonen-Olive duality applies to a very special type of gauge theory called [[N=4 supersymmetric Yang-Mills theory]], and it says that two such theories may be equivalent in a certain precise sense.<ref>Frenkel 2009, p.2</ref> If one of the theories has a gauge group <math>G</math>, then the dual theory has gauge group <math>{^L}G</math> where <math>{^L}G</math> denotes the [[Langlands dual group]] which is in general different from <math>G</math>.<ref>Frenkel 2009, p.5</ref>

An important quantity in quantum field theory is complexified [[coupling constant]]. This is a [[complex number]] defined by the formula<ref>Frenkel 2009, p.12</ref>

:<math>\tau=\frac{\theta}{2\pi}+\frac{4\pi i}{g^2}</math>

where <math>\theta</math> is the [[theta angle]], a quantity appearing in the [[Lagrangian]] that defines the theory,<ref>Frenkel 2009, p.12</ref> and <math>g</math> is the [[coupling constant]]. For example, in the Yang-Mills theory that describes the electromagnetic field, this number <math>g</math> is simply the [[elementary charge]] <math>e</math> carried by a single [[proton]].<ref>Frenkel 2009, p.2</ref> In addition to exchanging the gauge groups of the two theories, Montonen-Olive duality transforms a theory with complexified coupling coupling constant <math>\tau</math> to a theory with complexified constant <math>-1/\tau</math>.<ref>Frenkel 2009, p.12</ref>

===Relation to the Langlands program===

[[Image:ECClines-3.svg|right|thumb|346px|The [[geometric Langlands correspondence]] is a relationship between abstract geometric objects associated to an [[algebraic curve]] such as the [[elliptic curve]]s illustrated above. ]]

{{main|Langlands program}}

In mathematics, the classical [[Langlands correspondence]] is a collection of results and conjectures relating [[number theory]] to the branch of mathematics known as [[representation theory]].<ref>Frenkel 2007</ref> Formulated by [[Robert Langlands]] in the late 1960s, the Langlands correspondence is related to important conjectures in number theory such as the [[Taniyama-Shimura conjecture]], which includes [[Fermat's last theorem]] as a special case.<ref>Frenkel 2007</ref>

In spite of its importance in number theory, establishing the Langlands correspondence in the number theoretic context has proved extremely difficult.<ref>Frenkel 2007</ref> As a result, some mathematicians have worked on a related conjecture known as the [[geometric Langlands correspondence]]. This is a geometric reformulation of the classical Langlands correspondence which is obtained by replacing the [[number fields]] appearing in the original version by [[function field of an algebraic variety|function fields]] and applying techniques from [[algebraic geometry]].<ref>Frenkel 2007</ref>

In a paper from 2007, [[Anton Kapustin]] and [[Edward Witten]] suggested that the geometric Langlands correspondence can be viewed as a mathematical statement of Montonen-Olive duality.<ref>Kapustin and Witten 2007</ref> Starting with two Yang-Mills theories related by S-duality, Kapustin and Witten showed that one can construct a pair of quantum field theories in two-dimensional [[spacetime]]. By analyzing what this [[dimensional reduction]] does to certain physical objects called [[D-branes]], they showed that one can recover the mathematical ingredients of the geometric Langlands correspondence.<ref>Aspinwall et al. 2009, p.415</ref> Their work shows that the Langlands correspondence is closely related to S-duality in quantum field theory, with possible applications in both subjects.<ref>Frenkel 2007</ref>

===Seiberg duality===

{{main|Seiberg duality}}

Another realization of S-duality in quantum field theory is [[Seiberg duality]], first introduced by [[Nathan Seiberg]] around 1995.<ref>Seiberg 1995</ref> Unlike Montonen-Olive duality, which relates two versions of the maximally supersymmetric gauge theory in four-dimensional spacetime, Seiberg duality relates less symmetric theories called [[supersymmetric gauge theory|N=1 supersymmetric gauge theories]]. The two N=1 theories appearing in Seiberg duality are not identical, but they give rise to the same physics at large distances. Like Montonen-Olive duality, Seiberg duality generalizes the symmetry of Maxwell's equations that interchanges electric and magnetic fields.

==S-duality in string theory==

[[File:StringTheoryDualities.jpg|thumb|350px|A diagram of string theory dualities. Yellow arrows indicate S-duality. Blue arrows indicate [[T-duality]].]]

Up until the mid 1990s, physicists working on [[string theory]] believed there were five distinct versions of the theory: [[type I string|type I]], [[type IIA string|type IIA]], [[type IIB string|type IIB]], and the two flavors of [[heterotic string]] theory ([[special orthogonal group|SO(32)]] and [[E8 (mathematics)|E8×E8]]). The different theories allow different types of strings, and the particles that arise at low energies exhibit different symmetries.

In the mid 1990s, physicists noticed that these five string theories are actually related by highly nontrivial dualities. One of these dualities is S-duality. The existence of S-duality in string theory was first proposed by [[Ashoke Sen]] in 1994.<ref>Sen 1994</ref> It was shown that [[type IIB string theory]] with the coupling constant <math>g</math> is equivalent via S-duality to the same string theory with the coupling constant <math>1/g</math>. Similarly, [[type I string theory]] with the coupling <math>g</math> is equivalent to the [[SO(32)]] [[heterotic string]] theory with the coupling constant <math>1/g</math>.

The existence of these dualities showed that the five string theories were in fact not all distinct theories. In 1995, at the string theory conference at [[University of Southern California]], [[Edward Witten]] made the surprising suggestion that all five of these theories were just different limits of a single theory now known as [[M-theory]].<ref>Witten 1995</ref> Witten's proposal was based on the observation that type IIA and E8×E8 heterotic string theories are closely related to a gravitational theory called eleven-dimensional [[supergravity]]. His announcement led to a flurry of work now known as the [[second superstring revolution]].

==See also==

* [[T-duality]]
* [[U-duality]]
* [[Mirror symmetry (string theory)|Mirror symmetry]]
* [[AdS/CFT correspondence]]

==Notes==

{{reflist|2}}

==References==

* {{cite book |editor1-first=Paul |editor1-last=Aspinwall |editor2-first=Tom |editor2-last=Bridgeland |editor3-first=Alastair |editor3-last=Craw |editor4-first=Michael |editor4-last=Douglas |editor5-first=Mark |editor5-last=Gross |editor6-first=Anton |editor6-last=Kapustin |editor7-first=Gregory |editor7-last=Moore |editor8-first=Graeme |editor8-last=Segal |editor9-first=Balázs |editor9-last=Szendröi |editor10-first=P.M.H. |editor10-last=Wilson |title=Dirichlet Branes and Mirror Symmetry |year=2009 |publisher=American Mathematical Society}}

* {{cite journal |last1=Frenkel |first=Edward |year=2007 |title=Lectures on the Langlands program and conformal field theory |journal=Frontiers in number theory, physics, and geometry II |publisher=Springer |pages=387–533 }}

* {{cite journal |last1=Frenkel |first=Edward |year=2009 |title=Gauge theory and Langlands duality |journal=Seminaire Bourbaki }}

* {{cite journal| title=Gauge theories and magnetic charge | last1=Goddard | first1=Peter |last2=Nuyts |first2=Jean | last3=Olive | first3=David | journal=Nuclear Physics B | volume=125| issue=1 | year=1977| pages= 1–28 }}

* {{cite book |last1=Griffiths |first1=David |title=Introduction to Electrodynamics |year=1999 |publisher=Prentice-Hall |location=New Jersey }}

* {{cite journal |last1=Kapustin |first1=Anton |last2=Witten |first2=Edward |title=Electric-magnetic duality and the geometric Langlands program |journal=Communications in Number Theory and Physics |volume=1 |issue=1 |pages=1–236 |year=2007 }}

* {{cite journal| title=Magnetic monopoles as gauge particles?| last1= Montonen| first1=Claus| last2=Olive| first2=David| journal=Physics Letters B | volume=72| issue=1| year=1977| pages= 117–120 }}

* {{cite journal |last=Seiberg |first=Nathan |year=1995 |title=Electric-magnetic duality in supersymmetric non-Abelian gauge theories |journal=Nuclear Physics B |volume=435 |issue=1 |pages=129-146 }}

* {{cite journal |last1=Sen |first1=Ashoke |year=1994 |title=Strong-weak coupling duality in four-dimensional string theory |journal=International Journal of Modern Physics A |volume=9 |issue=21 |pages=3707–3750 }}

* {{cite conference |title=Some problems of strong and weak coupling |last1=Witten |first1=Edward |date=March 13–18, 1995 |publisher=World Scientific |booktitle=Proceedings of Strings '95: Future Perspectives in String Theory }}

* {{cite journal |last1=Witten |first1=Edward |year=1995 |title=String theory dynamics in various dimensions |journal=Nuclear Physics B |volume=443 |issue=1 |pages=85–126 }}

* {{cite book |last=Zee |first=Anthony |title=Quantum Field Theory in a Nutshell, 2nd Edition |year=2010 |publisher=Princeton University Press}}

* {{cite book |last1=Zwiebach |first1=Barton |title=A First Course in String Theory |year=2009 |publisher=Cambridge University Press}}

{{DEFAULTSORT:S-Duality}}
[[Category:Quantum field theory]]
[[Category:String theory]]

Matrix addition

2013-09-13T09:33:33Z

145.97.197.247: /* See also */

{{More footnotes|date=November 2009}}

In [[linear algebra]], a '''symmetric matrix''' is a [[square matrix]] that is equal to its [[transpose]]. Formally, matrix ''A'' is symmetric if

:<math>A = A^{\top}.</math>

Because equal matrices have equal dimensions, only square matrices can be symmetric.

The entries of a symmetric matrix are symmetric with respect to the [[main diagonal]]. So if the entries are written as ''A'' = (''a''''ij''), then ''a''''ij'' = a''ji'', for all indices ''i'' and ''j''.

The following 3×3 matrix is symmetric:

:<math>\begin{bmatrix}
1 & 7 & 3\\
7 & 4 & -5\\
3 & -5 & 6\end{bmatrix}.</math>

Every square [[diagonal matrix]] is symmetric, since all off-diagonal entries are zero. Similarly, each diagonal element of a [[skew-symmetric matrix]] must be zero, since each is its own negative.

In linear algebra, a [[real number|real]] symmetric matrix represents a [[self-adjoint operator]] over a [[real number|real]] [[inner product space]]. The corresponding object for a [[complex number|complex]] inner product space is a [[Hermitian matrix]] with complex-valued entries, which is equal to its [[conjugate transpose]]. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.

== Properties ==

The finite-dimensional [[spectral theorem]] says that any symmetric matrix whose entries are [[real number|real]] can be [[diagonal matrix|diagonalized]] by an [[orthogonal matrix]]. More explicitly: For every symmetric real matrix ''A'' there exists a real orthogonal matrix ''Q'' such that ''D'' = ''Q''T''AQ'' is a [[diagonal matrix]]. Every symmetric matrix is thus, [[up to]] choice of an [[orthonormal basis]], a diagonal matrix.

Another way to phrase the spectral theorem is that a real ''n''×''n'' matrix ''A'' is symmetric if and only if there is an orthonormal basis of <math>\mathbb{R}^n</math> consisting of eigenvectors for ''A''.

Every real symmetric matrix is [[Hermitian matrix|Hermitian]], and therefore all its [[eigenvalues]] are real. (In fact, the eigenvalues are the entries in the diagonal matrix ''D'' (above), and therefore ''D'' is uniquely determined by ''A'' up to the order of its entries.) Essentially, the property of being symmetric for real matrices corresponds to the property of being Hermitian for complex matrices.

A complex symmetric matrix can be diagonalized using a unitary matrix: thus if ''A'' is a complex symmetric matrix, there is a unitary matrix ''U'' such that
''UAU''t is a diagonal matrix. This result is referred to as the '''Autonne–Takagi factorization'''. It was originally proved by Leon Autonne (1915) and [[Teiji Takagi]] (1925) and rediscovered with different proofs by several other mathematicians.<ref>{{harvnb|Horn|Johnson|2013|page=278}}</ref><ref>See:
*{{citation|first=L.|last= Autonne|title= Sur les matrices hypohermitiennes et sur les matrices unitaires|journal= Ann. Univ. Lyon|volume= 38|year=1915|pages= 1–77}}
*{{citation|first=T.|last= Takagi|title= On an algebraic problem related to an analytic theorem of Carathéodory and Fejér and on an allied theorem of Landau|journal= Japan. J. Math.|volume= 1 |year=1925|pages= 83–93}}
*{{citation|title=Symplectic Geometry|first=Carl Ludwig|last= Siegel|journal= American Journal of Mathematics|volume= 65|year=1943|pages=1-86|url= http://www.jstor.org/stable/2371774}}, Lemma 1, page 12
*{{citation|first=L.-K.|last= Hua|title= On the theory of automorphic functions of a matrix variable I–geometric basis|journal= Amer. J. Math.|volume= 66 |year=1944|pages= 470–488}}
*{{citation|first=I.|last= Schur|title= Ein Satz über quadratische formen mit komplexen koeffizienten|journal=Amer. J. Math.|volume=67|year=1945|pages=472–480}}
*{{citation|first1=R.|last1= Benedetti|first2=P.|last2= Cragnolini|title=On simultaneous diagonalization of one Hermitian and one symmetric form|journal= Linear Algebra Appl. |volume=57 |year=1984| pages=215–226}}
</ref> In fact the matrix ''B'' = ''A''*''A'' is Hermitian and non-negative, so there is a unitary matrix ''V'' such that ''V''*''BV'' is diagonal with non-negative real entries. Thus ''C'' = ''V''''t''''AV'' is complex symmetric with ''C''*''C'' real. Writing ''C'' = ''X'' + ''iY'' with ''X'' and ''Y'' real symmetric matrices, ''C''*''C'' = ''X''2 − ''Y''2 + i
(''XY'' − ''YX''). Thus ''XY'' = ''YX''. Since ''X'' and ''Y'' commute, there is a real orthogonal matrix ''W'' such that ''WXW''''t'' and ''WYW''''t'' are diagonal. Setting ''U'' = ''WV''''t'', the matrix ''UAU''''t'' is diagonal. Post-multiplying ''U'' by a diagonal matrix the diagonal entries can be taken to be non-negative. Since their squares are the eigenvalues of ''A''*''A'', they coincide with the [[singular value]]s of ''A''.

The sum and difference of two symmetric matrices is again symmetric, but this is not always true for the [[matrix multiplication|product]]: given symmetric matrices ''A'' and ''B'', then ''AB'' is symmetric if and only if ''A'' and ''B'' [[commutativity|commute]], i.e., if ''AB'' = ''BA''. So for integer ''n'', ''An'' is symmetric if ''A'' is symmetric. If ''A'' and ''B'' are ''n''×''n'' real symmetric matrices that commute, then there exists a basis of <math>\mathbb{R}^n</math> such that every element of the basis is an [[eigenvector]] for both ''A'' and ''B''.

If ''A''−1 exists, it is symmetric if and only if ''A'' is symmetric.

Let Mat''n'' denote the space of {{nowrap|1=''n'' × ''n''}} matrices. A symmetric ''n'' × ''n'' matrix is determined by ''n''(''n'' + 1)/2 scalars (the number of entries on or above the [[main diagonal]]). Similarly, a [[skew-symmetric matrix]] is determined by ''n''(''n'' − 1)/2 scalars (the number of entries above the main diagonal). If Sym''n'' denotes the space of {{nowrap|1=''n'' × ''n''}} symmetric matrices and Skew''n'' the space of {{nowrap|1=''n'' × ''n''}} skew-symmetric matrices then since {{nowrap|1=Mat''n'' = Sym''n'' + Skew''n''}} and {{nowrap|1=Sym''n'' ∩ Skew''n'' = {0}}}, i.e.
:<math> \mbox{Mat}_n = \mbox{Sym}_n \oplus \mbox{Skew}_n , </math>
where ⊕ denotes the [[Direct sum of modules|direct sum]]. Let {{nowrap|1=X ∈ Mat''n''}} then
:<math> X = \frac{1}{2}(X + X^{\top}) + \frac{1}{2}(X - X^{\top}) . </math>
Notice that {{nowrap|1=½(''X'' + ''X''T) ∈ Sym''n''}} and {{nowrap|1=½(''X'' − ''X''T) ∈ Skew''n''.}} This is true for every [[square matrix]] ''X'' with entries from any [[field (mathematics)|field]] whose [[characteristic (algebra)|characteristic]] is different from 2.

Any matrix [[Matrix congruence|congruent]] to a symmetric matrix is again symmetric: if ''X'' is a symmetric matrix then so is ''AXA''T for any matrix ''A''.


Denote with <math>\langle \cdot,\cdot \rangle</math> the standard [[inner product]] on '''R'''''n''. The real ''n''-by-''n'' matrix ''A'' is symmetric if and only if

:<math>\langle Ax,y \rangle = \langle x, Ay\rangle \quad \forall x,y\in\Bbb{R}^n.</math>

Since this definition is independent of the choice of [[basis (linear algebra)|basis]], symmetry is a property that depends only on the [[linear operator]] A and a choice of [[inner product]]. This characterization of symmetry is useful, for example, in [[differential geometry]], for each [[tangent space]] to a [[manifold]] may be endowed with an inner product, giving rise to what is called a [[Riemannian manifold]]. Another area where this formulation is used is in [[Hilbert space]]s.

A symmetric matrix is a [[normal matrix]].

== Decomposition ==
Using the [[Jordan normal form]], one can prove that every square real matrix can be written as a product of two real symmetric matrices, and every square complex matrix can be written as a product of two complex symmetric matrices.<ref>{{cite journal | first=A. J.|last= Bosch | title=The factorization of a square matrix into two symmetric matrices | journal=[[American Mathematical Monthly]] | year=1986 | volume=93 | pages=462–464 | doi=10.2307/2323471 | issue=6 | jstor=2323471}}</ref>

Every real [[non-singular matrix]] can be uniquely factored as the product of an [[orthogonal matrix]] and a symmetric [[positive definite matrix]], which is called a [[polar decomposition]]. Singular matrices can also be factored, but not uniquely.

[[Cholesky decomposition]] states that every real positive-definite symmetric matrix ''A'' is a product of a lower-triangular matrix ''L'' and its transpose, <math>A=L L^T</math>.
If the matrix is symmetric indefinite, it may be still decomposed as <math> P A P^T = L T L^T</math> where <math>P</math> is
a permutation matrix (arising from the need to [[Pivot element|pivot]]), <math>L</math> a lower unit triangular matrix, <math>T</math> a symmetric tridiagonal matrix, and <math>D</math>
a direct sum of symmetric 1×1 and 2×2 blocks.<ref>{{cite book | author=G.H. Golub, C.F. van Loan. | title=Matrix Computations | publisher=The Johns Hopkins University Press, Baltimore, London | year=1996}}</ref>

Every complex symmetric matrix ''A'' can be diagonalized, moreover the [[Eigendecomposition of a matrix#Symmetric_matrices|eigen decomposition]] takes a simpler form:
:<math>A = Q \Lambda Q^{\top} </math>
where ''Q'' is an [[unitary matrix]]. If A is real ''Q'' is an [[orthogonal matrix]], (the columns of which are [[eigenvectors]] of ''A''), and ''Λ'' is real and diagonal (having the [[eigenvalues]] of ''A'' on the diagonal). To see orthogonality, suppose <math>x</math> and <math>y</math> are eigenvectors corresponding to distinct eigenvalues <math>\lambda_1</math>, <math>\lambda_2</math>. Then
:<math>\lambda_1 \langle x,y \rangle = \langle Ax, y \rangle = \langle x, Ay \rangle = \lambda_2 \langle x, y \rangle</math>
so that if <math>\langle x, y \rangle \neq 0</math> then <math>\lambda_1 = \lambda_2</math>, a contradiction; hence <math>\langle x, y \rangle = 0</math>.

== Hessian ==

Symmetric real ''n''-by-''n'' matrices appear as the [[Hessian matrix|Hessian]] of twice continuously differentiable functions of ''n'' real variables.

Every [[quadratic form]] ''q'' on '''R'''''n'' can be uniquely written in the form ''q''('''x''') = '''x'''T''A'''''x''' with a symmetric ''n''-by-''n'' matrix ''A''. Because of the above spectral theorem, one can then say that every quadratic form, up to the choice of an orthonormal basis of '''R'''''n'', "looks like"
:<math>q(x_1,\ldots,x_n)=\sum_{i=1}^n \lambda_i x_i^2</math>
with real numbers λ''i''. This considerably simplifies the study of quadratic forms, as well as the study of the level sets {'''x''' : ''q''('''x''') = 1} which are generalizations of [[conic section]]s.

This is important partly because the second-order behavior of every smooth multi-variable function is described by the quadratic form belonging to the function's Hessian; this is a consequence of [[Taylor's theorem]].

== Symmetrizable matrix ==
An ''n''-by-''n'' matrix ''A'' is said to be '''symmetrizable''' if there exist an invertible [[diagonal matrix]] ''D'' and symmetric matrix ''S'' such that {{nowrap|1=''A'' = ''DS''.}}
The transpose of a symmetrizable matrix is symmetrizable, for {{nowrap|1=(''DS'')T = ''D−T(''DSD'')T.}} A matrix {{nowrap|1=''A'' = (''aij'')}} is symmetrizable if and only if the following conditions are met:
# <math>a_{ij} = 0</math> implies <math>a_{ji}=0</math> for all <math>1 \le i \le j \le n.</math>
# <math>a_{i_1i_2} a_{i_2i_3}\dots a_{i_ki_1} = a_{i_2i_1} a_{i_3i_2}\dots a_{i_1i_k}</math> for any finite sequence <math>(i_1, i_2, \dots, i_k).</math>

== See also ==
Other types of [[symmetry]] or pattern in square matrices have special names; see for example:

* [[Antimetric matrix]]
* [[Centrosymmetric matrix]]
* [[Circulant matrix]]
* [[Covariance matrix]]
* [[Coxeter matrix]]
* [[Hankel matrix]]
* [[Hilbert matrix]]
* [[Persymmetric matrix]]
* [[Skew-symmetric matrix]]
* [[Toeplitz matrix]]

See also [[symmetry in mathematics]].

== Notes ==
{{Reflist}}
==References==
*{{citation|last=Horn|first= Roger A.|last2= Johnson|first2= Charles R.|title= Matrix analysis|edition=2nd| publisher=Cambridge University Press|year= 2013|id= ISBN 978-0-521-54823-6}}

== External links ==
* {{springer|title=Symmetric matrix|id=p/s091680}}
* [http://farside.ph.utexas.edu/teaching/336k/Newton/node66.html A brief introduction and proof of eigenvalue properties of the real symmetric matrix]

[[Category:Matrices]]

Notation for theoretic scheduling problems

2013-05-14T14:04:57Z

145.97.237.159: Fixed math notation for examples of objective functions

{{good article}}
{{IPA notice}}
In [[linguistics]], a '''diaphoneme''' or '''diaphone''' is an abstract [[phonology|phonological]] unit that identifies a correspondence between related sounds of two or more [[variety (linguistics)|varieties]].<ref>{{Harvcoltxt|Crystal|2011}}</ref> For example, the vowel that constitutes the English word ''eye'' {{IPA|/aɪ/}} is pronounced differently depending on dialect ({{IPA|[aɪ̯]}} or {{IPA|[ʌɪ̯]}} in [[Received pronunciation|RP]] and [[General American]], {{IPA|[ae̯]}} or {{IPA|[əi̯]}} in [[Scottish English]], {{IPA|[ɑɪ̯]}} in [[Australian English]], {{IPA|[ɔɪ̯]}} in [[Irish English]], {{IPA|[aː]}} in [[South African English]], and {{IPA|[aː]}} or {{IPA|[əi̯]}} in [[Southern American English]]) but is considered the same by speakers.

'''Diaphonology''' studies the realization of diaphones across dialects, and is important if an [[orthography]] is to be adequate for more than one dialect of a language. In historical linguistics, it is concerned with the reflexes of an ancestral phoneme as a language splits into dialects, such as the modern realizations of [[Old English]] {{IPA|/oː/}}.

==Usage==
The term ''diaphone'' first appeared in usage by phoneticians like [[Daniel Jones (phonetician)|Daniel Jones]]<ref>{{Harvcoltxt|Collins|Mees|1999|p=326}}, citing {{Harvcoltxt|Jones|1932}}</ref> and [[Harold E. Palmer]].<ref>{{Harvcoltxt|Chao|1934|p=364}}, citing {{Harvcoltxt|Palmer|1931}}</ref><ref>{{Harvcoltxt|Chao|1946|p=12}} states that Jones credited Palmer with using the term in print first, though {{Harvcoltxt|Wells|1982|p=69}} citing {{Harvcoltxt|Jones|1962}}, credits Jones with pushing the concept.</ref> Jones, who was more interested in transcription and coping with dialectal variation<ref>{{Harvcoltxt|Wells|1982|p=69}}</ref> than with how [[#Cognitive reality|cognitively real]] the phenomenon is,<ref>{{Harvcoltxt|Twaddell|1935|p=29}}</ref> originally used ''diaphone'' to refer to the family of sounds that are realized differently depending on dialect but that speakers consider to be the same;<ref>{{Harvcoltxt|Chao|1946|p=12}}</ref> an individual dialect or speaker's realization of this diaphone was called a ''diaphonic variant''. Because of confusion related to usage, Jones later coined the term ''diaphoneme'' to refer to his earlier sense of ''diaphone'' (the class of sounds) and used ''diaphone'' to refer to the variants.<ref>This is how the terms are used in {{Harvcoltxt|Kurath|McDavid|1961}} (cited in {{Harvcoltxt|Wells|1982|p=70}}), {{Harvcoltxt|Moulton|1961|p=502}}, and {{Harvcoltxt|Jones|1950}}</ref><ref>{{Harvcoltxt|Shores|1984}} uses ''diaphone'' in this sense but uses ''phoneme'' in place of ''diaphoneme.''</ref><ref>{{Harvcoltxt|Trask|1996|p=111}}</ref>

A diaphonemic inventory is a specific [[diasystem]] (a term popularized by [[Uriel Weinreich]]) that superimposes dialectal contrasts to access all contrasts in all dialects that are included.<ref>{{Harvcoltxt|Stockwell|1959|p=262}}</ref> This consists of a shared core inventory<ref>{{Harvcoltxt|Stockwell|1959|p=262}}, citing {{Harvcoltxt|Hockett|1955|pp=18–22}}</ref> and, when accounting for contrasts not made by all dialects (whether they are historical contrasts that have [[phonological change#Merger|been lost]] or [[phonological change#Split|innovative ones]] not made in all varieties <ref>{{Harvcoltxt|Geraghty|1983|p=20}}</ref>), only as many contrasts as are needed.<ref>{{Harvcoltxt|Stockwell|1959|p=262}}</ref><ref>{{Harvcoltxt|Wells|1982|p=70}} refers to this distinction as "differences in phonetic realization" and "differences between accents", respectively.</ref> The diaphonemic approach gets away from the assumption that linguistic communities are homogeneous, allows multiple varieties to be described in the same terms (something important for situations where people have abilities in more than one variety),<ref>{{Harvcoltxt|Nagara|1972|pp=24, 47}}, citing {{Harvcoltxt|Weinreich|1953|pp=14, 29}}</ref> and helps in ascertaining where speakers make diaphonic identifications as a result of similarities and differences between the varieties involved.<ref>{{Harvcoltxt|Nagara|1972|p=47}}</ref>

The '''linguistic variable''', a similar<ref>{{Harvcoltxt|Pederson|1977|p=274}}</ref> concept presented by [[William Labov]], refers to features with variations that are referentially identical but carry social and stylistic meaning.<ref>{{Harvcoltxt|Lavandera|1978|pp=173, 177}}, pointing to {{Harvcoltxt|Labov|1966}}</ref> This could include phonological, as well as morphological and syntactic phenomena.<ref>{{Harvcoltxt|Wolfram|1991|p=24}}</ref> Labov also developed [[variable rules analysis]], with variable rules being those that all members of a speech community (presumably) possess but vary in the frequency of use.<ref>{{Harvcoltxt|Romaine|1981|p=96}}, pointing to {{Harvcoltxt|Labov|1969}}</ref> The latter concept met resistance from scholars for a number of reasons<ref>{{Harvcoltxt|Fasold|1991|pp=8–15}}</ref> including the argument from critics that knowledge of rule probabilities was too far from speakers' competence.<ref>See {{Harvcoltxt|Romaine|1981|pp=100–106}} for a more in-depth discussion about communicative competence in relation to variable rules.</ref> Because of these problems, use of variable rules analysis died down by the end of the 1980s.<ref>{{Harvcoltxt|Fasold|1991|p=17}}</ref> Nevertheless, the linguistic variable is still used in [[sociolinguistics]]. For Labov, grouping variants together was justified by their tendency to fluctuate between each other within the same set of words.<ref>{{Harvcoltxt|Wolfram|1991|pp=24–25}}, citing {{Harvcoltxt|Labov|1966|p=53}}</ref> For example, Labov presented the variants of the vowel of ''bad'' or ''dance'':
{| class="wikitable" align="center"
! Phonetic value !! Score
|-
| {{IPA|[ɪə]}} || 1
|-
| {{IPA|[ɛə]}} || 2
|-
| {{IPA|[æ̝]}} || 3
|-
| {{IPA|[æː]}} || 4
|-
| {{IPA|[aː]}} || 5
|-
| {{IPA|[ɑː]}} || 6
|}

The different phonetic values were assigned numerical values that were then used in an overall score index. '''Overdifferentiation''' is when phonemic distinctions from one's primary language are imposed on the sounds of the second system where they are not required; '''underdifferentiation''' of phonemes occurs when two sounds of the second system are not maintained because they are not present in the primary system.<ref>{{Harvcoltxt|Nagara|1972|p=277}}, citing {{Harvcoltxt|Weinreich|1953|p=18}}</ref>

==Dialectology==
Inspired by {{Harvcoltxt|Trubetzkoy|1931}}, [[Uriel Weinreich]] first advocated the use of diasystems in structural [[dialectology]], and suggested that such a system would represent a higher level of abstraction that can unite related dialects into a single description and transcription.<ref>{{Harvcoltxt|Weinreich|1954|pp=389–390}}</ref> While phonemic systems describe the speech of a single variety, diaphonemic systems can reflect the contrasts that aren't made by all varieties being represented. The way these differ can be shown in the name ''New York''. This word may be transcribed phonemically as {{IPA|/nuː ˈjɔrk/}} in American English, which does not allow {{IPA|/j/}} after {{IPA|/n/}}; in [[Received Pronunciation]], syllable-final {{IPA|/r/}} doesn't occur so this name would be transcribed {{IPA|/njuː ˈjɔːk/}} to reflect that pronunciation. A diaphonemic transcription such as {{IPA|/n'''j'''uː ˈjɔ'''r'''k/}} (with both the {{IPA|/j/}} and the {{IPA|/r/}}) would thus cover both dialects. Neither is described exactly, but both are derivable from the diaphonemic transcription.

The desire of building a diasystem to accommodate all English dialects, combined with a blossoming [[generative phonology]], prompted American dialectologists to attempt the construction of an "overall system" of English phonology by analyzing dialectal distinctions as differences in the ordering of phonological rules<ref>{{Harvcoltxt|Davis|1973|p=1}}</ref><ref>{{Harvcoltxt|Allen|1977|pp=169, 226}}</ref> as well as in the presence or absence of such rules.<ref>{{Harvcoltxt|Saporta|1965|pp=218–219}}</ref> {{Harvcoltxt|Bickerton|1973|p=641}} even went so far as to claim that principled description of interdialectal code-switching would be impossible without such rules.

An example of this concept is presented in {{Harvcoltxt|Saporta|1965|p=223}} with a phonological difference between [[Castilian Spanish|Castilian]] and Uruguayan Spanish:

{| class="wikitable"
|-
! Castilian !! Uruguayan !! Gloss
|-
| {{IPA|[ˈklase]}} || {{IPA|[ˈklase]}} || 'class'
|-
|{{IPA|[ˈklasɛs]}} || {{IPA|[ˈklasɛ]}} || 'classes'
|}

Without the use of ordered rules, Uruguayan Spanish could be interpreted as having two additional phonemes and morphophonemic vowel alternation with its plural marker. Attempting to construct a diasystem that encodes such a variety would thus represent all Spanish varieties as having seven vowel phonemes (with contrasts only in final position). Due to both varieties having closed allophones of [[mid vowel]]s in open syllables and open allophones in closed syllables, using ordered rules minimizes the differences so that the underlying form for both varieties is the same and Uruguayan Spanish simply has a subsequent rule that deletes {{IPA|/s/}} at the end of a syllable; constructing a diaphonemic system thus becomes a relatively straightforward process. {{Harvcoltxt|Saporta|1965|p=220}} suggests that the rules needed to account for dialectal differences, even if not [[#Cognitive reality|psychologically real]], may be ''historically'' accurate.

The nature of an overall system for English was controversial: the analysis in {{Harvcoltxt|Trager|Smith|1951}}<ref>{{Harvcoltxt|Allen|1977|p=224}} and {{Harvcoltxt|Stockwell|1959|p=258}} point to earlier works by these authors as approaching the same goal but in less detail</ref> was popular amongst American linguists for a time (in the face of criticism, particularly from [[Hans Kurath]]<ref>e.g. {{Harvcoltxt|Kurath|1957}}</ref>); James Sledd<ref>particularly in {{Harvcoltxt|Sledd|1966}}</ref> put forth his own diaphonemic system that accommodated [[Southern American English]]; both {{Harvcoltxt|Troike|1971}} and {{Harvcoltxt|Reed|1972}} modified the scheme of ''[[The Sound Pattern of English]]'' by focusing on the diaphoneme, believing that it could address neutralizations better than [[structural linguistics|structuralist]] approaches;<ref>{{Harvcoltxt|Allen|1977|p=227}}</ref> and ''The Pronunciation of English in the Atlantic States '' (''PEAS'') by [[Hans Kurath|Kurath]] and McDavid combined several dialects into one system transcribed in the IPA.<ref>{{Harvcoltxt|Allen|1977|p=224}}, citing {{Harvcoltxt|Kurath|McDavid|1961}}</ref>

{{Harvcoltxt|Weinreich|1954}} argued that {{Harvcoltxt|Trager|Smith|1951}} fell short in accurately representing dialects because their methodology involved attempting to create a diasystem before establishing the relevant component phonemic systems.<ref>{{Harvcoltxt|Weinreich|1954|p=395}}</ref> {{Harvcoltxt|Voegelin|1956|p=122}} argues a similar problem occurs in the study of [[Hopi language|Hopi]] where [[transfer of training]] leads phoneticians to fit features of a dialect under study into the system of dialects already studied.

Beginning with {{Harvcoltxt|Trubetzkoy|1931}} linguists attempting to account for dialectal differences have generally distinguished between three types:
*'''Phonological''': the phonemic inventories and [[phonotactic]] restrictions
*'''Phonetic''': how a given phoneme is realized phonetically (RP and Australian English, for example, have almost the same exact phoneme system but with wildly different realizations of the vowels<ref>See {{Harvcoltxt|Turner|1966}}, cited in {{Harvcoltxt|Wells|1970|p=245}}</ref>). This distinction covers differences in the range of allophonic variation.
*'''Incidence''': one phoneme rather than another occurs in a given word or group of words (such as ''grass'', which has the same vowel of ''farce'' in RP but not in GA.<ref>{{Harvcoltxt|Fudge|1969|p=272}}</ref>)

Wells<ref>{{Harvcoltxt|Wells|1970|pp=232, 240, 243, 245}}, {{Harvcoltxt|Wells|1982|p=73}}</ref> expanded on this by splitting up the phonological category into "systemic" differences (those of inventory) and "structural" differences (those of phonotactics).

In addition, Both Wells and Weinreich mention ''realizational overlap'', wherein the same phone (or a nearly identical one) corresponds to different phonemes, depending on accent.<ref>{{Harvcoltxt|Wells|1982|p=83}}</ref><ref>{{Harvcoltxt|Weinreich|1954|p=394}}</ref> Some examples:
*''Autistic'' in [[Canadian English]] overlaps with the way speakers of [[Received Pronunciation]] say ''artistic'': {{IPA|[ɑːˈtʰɪstɪk]}}
*''Impossible'' in [[General American]] overlaps with RP ''impassable'': {{IPA|[ɪmpɑːsəbl̩]}}
{{Harvcoltxt|Hankey|1965|p=229}} notes a similar phenomenon in [[Western Pennsylvania]], where {{IPA|[æɪ]}} occurs either as the vowel of ''ashes'' or as the vowel of ''tiger'' but no speaker merges the two vowels (i.e. a speaker who says {{IPA|[ˈæɪʃɪz]}} will not say {{IPA|[ˈtæɪɡɚ]}}).

Realizational overlap occurs between the three dialects of [[Huastec language|Huastec]], which have the same phonological system even though cognate words often do not have the same reflexes of this system. For example, while the Central and Potosino dialects both have ''ch'' and ''ts''-type sounds, the words they are found in are reversed:<ref>{{Harvcoltxt|Kaufman|2006|p=65}}</ref>

{| class="wikitable"
|+Huastec diaphonemes
! Diaphoneme !! colspan=2| word !! Otontepec !! Central !! [[San Luis Potosi|Potosino]]
|-align=center
|{{IPA|#tʃ}} || {{IPA|#tʃan}} || 'snake' ||{{IPA|tʃan}} ||{{IPA|tʃan}} ||{{IPA|tsan}}
|-align=center
|{{IPA|#tʃʼ}} || {{IPA|#tʃʼak}} || 'flea' ||{{IPA|tʃʼak}} ||{{IPA|tʃʼak}} ||{{IPA|tsʼak}}
|-align=center
| {{IPA|#tʲ}} || {{IPA|#tʲiθ}} || 'pigweed' || {{IPA|tʲiθ}} || {{IPA|tsiθ}} || {{IPA|tʃiθ}}
|-align=center
| {{IPA|#tʲʼ}} || {{IPA|#tʲʼitʲab}} || 'comb' ||{{IPA|tʲʼitʲab}} ||{{IPA|tsʼitsab}} ||{{IPA|tʃʼitʃab}}
|}

[[Yuen Ren Chao]] created a diaphonemic transcription of major [[Chinese languages]] and dialects, in both Latin and [[Chinese character]] versions, called "[[General Chinese]]". It originally (1927) covered the various [[Wu dialects]], but by 1983 had expanded to cover the major dialects of Mandarin, Yue, Hakka, and Min as well. Apart from a few irregularities, GC can be read equally well in any of those dialects, and several others besides.

[[Classical Arabic#Phonology|Qur'anic Arabic]] uses a diaphonemic writing system that indicates both the pronunciation in Mecca, the western dialect the Qur'an was written in, and that of eastern Arabia, the [[prestige language|prestige dialect]] of [[pre-Islamic poetry]]. For example, final {{IPA|*aj}} was pronounced something like {{IPA|[eː]}} in Mecca, and written ي {{IPA|/j/}}, while it had merged with {{IPA|[aː]}} in eastern Arabia and was written as ا {{IPA|/ʔ/}}. In order to accommodate both pronunciations, the basic letter of Meccan Arabic was used, but the diacritic was dropped: ى. Similarly, the glottal stop had been lost in Meccan Arabic in all positions but initially, so the Meccan letters were retained with the eastern glottal stop indicated with a diacritic [[hamza]].<ref>Catherine Bateson, 1967, ''Arabic Language Handbook'', p 75 ff.</ref><ref>Versteegh, 1997, ''The Arabic Language'', p 40 ff, 56 ff</ref>

==Bilingualism==
[[Einar Haugen]] expanded the diaphonic approach to the study of [[bilingualism]],<ref>{{Harvcoltxt|Nagara|1972|p=48}}, citing {{Harvcoltxt|Haugen|1954|p=11}}</ref> believing diaphones represented the process of interlingual identification<ref>{{Harvcoltxt|Nagara|1972|p=52}}</ref> wherein sounds from different languages are perceptually linked into a single category.<ref>{{Harvcoltxt|Riney|Takagi|1999|p=295}}</ref> Because interlingual identifications may happen between unrelated varieties, it is possible to construct a diasystem for many different language contact situations, with the appropriateness of such a construction depending on its purpose<ref>{{Harvcoltxt|Nagara|1972|p=49}}</ref> and its simplicity depending on how [[isomorphism (sociology)|isomorphic]] the phonology of the systems are. For example, the [[New Mexican Spanish|Spanish]] of Los Ojos (a small village in [[Rio Arriba County, New Mexico]]) and the local variety of [[North American English regional phonology#Western Dialect|Southwestern English]] are fairly isomorphic with each other<ref>{{Harvcoltxt|Oliver|1972|p=362}}</ref> so a diaphonic approach for such a [[language contact]] situation would be relatively straightforward. {{Harvcoltxt|Nagara|1972}} makes use of a diaphonic approach in discussing the phonology of the [[Hawaiian Pidgin|pidgin English]] used by [[Japanese diaspora#Americas|Japanese immigrants]] on [[sugar plantations in Hawaii|Hawaiian plantations]].<ref>{{Harvcoltxt|Nagara|1972|p=9}}</ref>

Both Haugen and Weinreich considered the use of phonemes beyond a single language to be inappropriate when phonemic systems between languages were incommensurable with each other.<ref>{{Harvcoltxt|Pulgram|1964|p=66}} reiterates this point when he says,
<blockquote>"Since each state is a system consisting of members solely defined by their mutual relations, any two non-identical systems must necessarily be incommensurable, for no element in one can be identified with any element in the other. ...structurally we cannot identify or even compare any Spanish vowel-phoneme with any Italian vowel-phoneme, because a member of a 5-vowel system is intrinsically different from a member of a 7-term system."</blockquote></ref> Similarly, {{Harvcoltxt|Shen|1952}}, argues that phonemic representations may lead to confusion when dealing with phonological [[language transfer|interference]] and {{Harvcoltxt|Nagara|1972|p=56}} remarks that narrow [[phonetic transcription]] can be cumbersome, especially when discussing other grammatical features like [[syntax]] and [[morphology (linguistics)|morphology]]. [[Allophone]]s, which phonemic systems don't account for, may be important in the process of interference and interlingual identifications.<ref>{{Harvcoltxt|Nagara|1972|p=54}}, citing {{Harvcoltxt|Shen|1959|p=7}} and {{Harvcoltxt|Haugen|1954|pp=10–11}}</ref>

===Borrowing===
Similarly, the term ''diaphone'' can be used in discussions of cognates that occur in different languages due to borrowing. Specifically, {{Harvcoltxt|Haugen|1956|pp=46, 67}} used the term to refer to phonemes that are equated by speakers cross-linguistically because of similarities in shape and/or distribution. For example, loanwords in [[Huave language|Huave]] having "diaphonic identification" with [[Spanish language|Spanish]] include ''àsét'' ('oil', from Spanish ''[[w:aceite#Spanish|aceite]]'') and ''kàwíy'' ('horse', from Spanish ''[[w:caballo#Spanish|caballo]]'').<ref>{{Harvcoltxt|Diebold|1961|p=107}}</ref> This perception of sameness with native phonology means that speakers of the borrower language (in this case, Huave) will hear new features from the loaner language (in this case, Spanish) as equivalent to features of their own<ref>{{Harvcoltxt|Silverman|1992|p=289}}</ref> and substitute in their own when reproducing them.<ref>{{Harvcoltxt|Haugen|1950|p=212}}</ref> In these interlanguage transfers, when phonemes or phonotactic constraints are too different, more extreme compromises may occur; for example, the English phrase ''[[Merry Christmas]]'', when borrowed into [[Hawaiian Language|Hawaiian]], becomes ''mele kalikimaka.''<ref>{{Harvcoltxt|Golston|Yang|2001|p=40}}</ref>

==Pidgins and creoles==
The process of diaphonic identification occurs when [[pidgins]] are fashioned; although lexical and morphosyntactic patterns are shared, speakers often use the phonological systems of their native language, meaning they must learn to recognize such diaphonic correspondences in the speech of others to facilitate the [[mutual intelligibility]] of a working pidgin.<ref>{{Harvcoltxt|Goodman|1967|p=44}}. The author notes (p.48) a parallel process with culturally defined gestures and offers the term ''gestural diamorph'' for this phenomenon.</ref> {{Harvcoltxt|Bailey|1971}} proposes that rule differences can be used to determine the distance a particular utterance has between a [[post-creole continuum]]'s acrolectal and basolectal forms. {{Harvcoltxt|Bickerton|1973|pp=641–642}} points out that mesolectal varieties often have features not derivable from such rules.

==Cognitive reality==
The status of panlectal and polylectal grammars<ref>For the purpose of this article, ''panlectal'' grammars are those that encode for all varieties of a language, while ''polylectal'' ones encode fewer than that.</ref> has been subject to debate amongst generative phonologists since the 1970s;<ref>{{Harvcoltxt|McMahon|1996|p=441}}, citing {{Harvcoltxt|Brown|1972}} and {{Harvcoltxt|Newton|1972}} as potentially beginning such debate.</ref><ref>{{Harvcoltxt|Trudgill|1983|p=29}}</ref> one of the foremost areas of contention in regards to diaphonemes and diasystems is whether they reflect the actual [[communicative competence|linguistic competence]] of speakers. William Labov, although warm to the construction of a panlectal grammar, argued that it should be based in speakers' linguistic competence.<ref>{{Harvcoltxt|Ornstein|Murphy|1974|p=156}}</ref> [[Peter Trudgill]] argues against the formation of diasystems that are not cognitively real<ref>{{Harvcoltxt|Trudgill|1974|p=135}}</ref> and implies<ref>{{Harvcoltxt|Trudgill|1983}}, chapter 1</ref> that polylectal grammars that are not part of native speakers' competence are illegitimate. Similarly, {{Harvcoltxt|Wolfram|1982|p=16}} cautions that polylectal grammars are only appropriate when they "result in claims about speaker-hearer's capabilities..."

Although no linguists claim that ''pan''lectal grammars have psychological validity,<ref>{{Harvcoltxt|Yaeger-Dror|1986|p=916}}</ref> and polylectal diasystems are much more likely to be cognitively real for bilingual and bidialectal speakers,<ref>{{Harvcoltxt|Wells|1982|p=72}}</ref> speakers of only one dialect or language may still be aware of the differences between their own speech and that of other varieties.<ref>{{Harvcoltxt|Weinreich|1954|p=390}}</ref>
Take, for example, the word ''house'', which is pronounced:
*{{IPA|[haʊ̯s]}} in [[Buffalo, New York|Buffalo]]
* {{IPA|[həʊ̯s]}} in [[Toronto]] and [[Washington, D.C.]]
* {{IPA|[hæʊ̯s]}} in [[Philadelphia]]
* {{IPA|[hɛʊ̯s]}} in [[Charlottesville, Virginia|Charlottesville]].
Native speakers are able to calibrate the differences and interpret them as being the same.<ref>{{Harvcoltxt|Smith|1967|p=311}}</ref> A similar issue occurs in [[Chinese language|Chinese]]. When a "general word," is shared across multiple [[mutual intelligibility|mutually unintelligible]] dialects, it is regarded as the same word even though it is pronounced differently depending on a speaker's region. Thus a speaker from [[Beijing]] and [[Nanking]] may pronounce 遍 ('throughout') differently, ({{IPA|[pjɛn˥˩]}} and {{IPA|[pjɛ̃˥˩]}}, respectively), though they still regard the differences as minor and due to unimportant accentual differences.<ref>{{Harvcoltxt|Chao|1946|p=12}}</ref> Because speakers aren't normally able to hear distinctions not made in their own dialect<ref>{{Harvcoltxt|Troike|1970|p=65}}, cited in {{Harvcoltxt|Campbell|1971|p=194}}</ref> (for example, a speaker from the [[Southern United States]] who [[Phonological history of English high front vowels#Pin–pen merger|does not distinguish between ''pin'' and ''pen'']] won't hear the distinction when it's produced by speakers of other dialects), speakers who ''can'' hear such a contrast but don't produce it may still possess the contrast as part of their linguistic repertoire.

In discussing contextual cues to vowel identifications in English, {{Harvcoltxt|Rosner|Pickering|1994}} note that controlling for dialect is largely unimportant for eliciting identifications when vowels are placed between consonants, possibly because the /CVC/ structure often forms lexical items that can aid in identification; identifying vowels in isolation, which don't often carry such lexical information, must be matched to the listener's set of vowel prototypes with less deviation than in consonantal contexts.<ref>{{Harvcoltxt|Rosner|Pickering|1994|p=325}}, pointing to data from {{Harvcoltxt|Verbrugge|Strange|Shankweiler|Edman|1976}}</ref> In the first chapter of {{Harvcoltxt|Trudgill|1983}}, [[Peter Trudgill]] makes the case that these semantic contexts form the basis of intelligibility across varieties and that the process is irregular and ''ad hoc'' rather than the result of any sort of rule-governed passive polylectal competence.<ref>{{Harvcoltxt|Trudgill|1983|p=10}}</ref>

{{Harvcoltxt|De Camp|1971}} argues that a child's language acquisition process includes developing the ability to accommodate for the different varieties they are exposed to (including ones they would not actually employ) and the social significance of their use.<ref>{{Harvcoltxt|Ornstein|Murphy|1974|p=152}}; the authors point to other creolists like Charles-James Bailey and [[Derek Bickerton]] as expanding on this concept in accounting for speaker variation.</ref><ref>A similar claim occurs in {{Harvcoltxt|Francis|1983|p=18}}</ref> {{Harvcoltxt|Wilson|Henry|1998|pp=17–18}} point out that there may be [[critical period]]s for this similar to [[critical period hypothesis|those for language learning]]. This competence in multiple varieties is arguably the primary vehicle of linguistic change.<ref>{{Harvcoltxt|Bickerton|1973|p=643}}</ref>

[[John C. Wells|John Wells]] argues that going past the common core creates difficulties that add greater complexity and falsely assume a shared underlying form in all accents:<ref>{{Harvcoltxt|Wells|1982|p=70}}</ref>
<blockquote>"Only by making the diaphonemic representation a rather remote, underlying form, linked to actual surface representations in given accents by a long chain of rules–only in this way could we resolve the obvious difficulties of the taxonomic diaphoneme."</blockquote>

Wells gives the example of ''straight'', ''late'' and ''wait'', which rhyme in most English varieties but, because some dialects make [[phonemic contrast]]s with the vowels of these words (specifically, in regions north of [[England]]<ref>These dialects pronounce the vowels as {{IPA|[ɛɪ(x)]}}, {{IPA|[eː]}}, and {{IPA|[ɛɪ]}}, respectively; the assumption being that words like ''straight'' have an underlying but unpronounced velar fricative</ref>), a panlectal transcription would have to encode this contrast despite it being absent for most speakers, making such a system "a linguist's construct"<ref>{{Harvcoltxt|Wells|1982|pp=69, 71}}</ref> and not part of the grammar present in any native speaker's mind (which is what adherents of such a system attempt to achieve).<ref>In a similar vein, {{Harvcoltxt|Wolfram|1991|p=25}} argues that the linguistic variable is essentially a ''socio''linguistic construct.</ref>
{{Harvcoltxt|Hall|1965|p=337}} argues that such constructs are appropriate but only when they are removed before the final formulation of grammatical analysis. Wells puts even more weight on the [[phonotactics|phonotactic]] difference between [[rhotic accent|rhotic]] and non-rhotic accents—the former have an underlying {{IPA|/r/}} in words like ''derby'' and ''star'' while the latter, arguably, do not<ref>{{Harvcoltxt|Wells|1982|pp=71, 76}}</ref>—and to the [[Phonological history of English high front vowels#Happy tensing|unstressed vowel of ''happy'']], which aligns phonetically with the vowel of {{smallcaps|kit}} in some varieties and that of {{smallcaps|fleece}} in others.<ref>{{Harvcoltxt|Wells|1982|p=76}}</ref>

Hans Kurath, particularly prominent in comparative analysis of British and American regional features,<ref>{{Harvcoltxt|Allen|1977|p=221}}</ref> makes the case that the systematic features of British and American English largely agree but for a handful of divergences, for example:<ref>{{Harvcoltxt|Kurath|1964}}, cited in {{Harvcoltxt|Allen|1977|p=221}}</ref>
*postvocalic {{IPA|/r/}}<ref>In {{Harvcoltxt|Kurath|1957|p=117}}, Kurath notes that the incidence of vowels before {{IPA|/r/}} varies considerably between dialects, which requires special attention but no serious difficulties.</ref>
*ingliding and upgliding varieties of {{IPA|/e/}}
*New England short {{IPA|/ɵ/}}
*coalescence of {{IPA|/ɑ/}} and {{IPA|/ɔ/}}
*variation of {{IPA|/ʊ/}} and {{IPA|/u/}} in a few lexical items
*the vowel of ''poor'', ''door'', and ''sure''
*variations in {{IPA|/aɪ/}} and {{IPA|/aʊ/}}

Despite downplaying the divergences, Kurath argued that there is no "total pattern" (a term from {{Harvcoltxt|Trager|Smith|1951}}) that can be imposed on all English dialects, nor of even American ones:<ref>{{Harvcoltxt|Kurath|1957|p=120}}</ref>
<blockquote>"The linguist must analyze the system of each dialect separately before he can know what systematic features are shared by all dialects, or by groups of dialects. He must distinguish between the systematic features and sporadic unsystematized features of each dialect, since every dialect has elements that are not built into the system. To regard unsystematized features as part of a 'system' and to impose an 'over-all pattern' are spurious notions that must be rejected.</blockquote>

The description of a cognitively real polylectal grammar came with {{Harvcoltxt|Trudgill|1974}}'s set of rules for the speech of [[Norwich]] that, presumably, could generate any possible output for a specific population of speakers and was psychologically real for such speakers<ref>{{Harvcoltxt|Bickerton|1975|p=302}}, pointing particularly to chapter 8 of {{Harvcoltxt|Trudgill|1974}}</ref> such that native residents who normally exhibited sound mergers (e.g. between the vowels of ''days'' and ''daze'') could accurately and consistently make the distinction if called upon to imitate older Norwich speakers.<ref>{{Harvcoltxt|Trudgill|1974|p=141}}</ref><ref>In {{Harvcoltxt|Trudgill|1983|pp=11–12, 45–46}}, the author points out that there are Norwich speakers who do not accurately imitate the speech of others, and present "hyperdialectisms" (similar to [[hypercorrection]]); such speakers may then be said to have a different, less polylectal, grammar than the one described in {{Harvcoltxt|Trudgill|1974}}</ref>

{{Harvcoltxt|Berdan|1977}} argues that comprehension across varieties, when it is found, isn't sufficient enough evidence for the claim that polylectal grammars are part of speakers' linguistic competence.<ref>cited in {{Harvcoltxt|Trudgill|1983|p=29}}</ref> {{Harvcoltxt|Ballard|1971}} argues that an extrapolated panlectal (or even broadly polylectal) grammar from "idiosyncratic" grammars, such as those found in {{Harvcoltxt|Trudgill|1974}}, would still not be part of speakers' linguistic competence;<ref>{{Harvcoltxt|Ballard|1971|p=267}}</ref> {{Harvcoltxt|Moulton|1985|p=566}} argues that attempting a polylectal grammar that encodes for a large number of dialects becomes too bizarre and that the traditional reconstructed proto-language is more appropriate for the stated benefits of polylectal grammars. {{Harvcoltxt|Bailey|1973|pp=27, 65}}, notable for advocating the construction of polylectal grammars, says that the generative rules of such grammars should be panlectal in the sense that they are ''potentially'' learned in the acquisition process, though no speaker should be expected to learn all of them.

Although question remains to their psychological reality, the usefulness of diaphonemes is shown in {{Harvcoltxt|Newton|1972|pp=19–23}} with the loss of the [[close front rounded vowel|front rounded vowel]] phoneme {{IPA|/y/}} in [[Greek language|Greek]] words like ξ'''ύ'''λο and κ'''οι'''λιά; this vowel merged with {{IPA|/i/}} in most words and {{IPA|/u/}} in the rest, though the distribution varies with dialect. A diasystem would thus have to present an additional underlying diaphoneme {{IPA|/y/}} with generative rules that account for the dialectal distribution.<ref>{{Harvcoltxt|Kazazis|1976|p=515}}</ref> Similarly, the diaphonemic system in {{Harvcoltxt|Geraghty|1983}} goes beyond the common core, marking contrasts that only appear in some varieties;<ref>{{Harvcoltxt|Geraghty|1983|pp=19, 20}}</ref> Geraghty argues that, because of [[Fijian traditions and ceremonies#Marriages|Fijian marriage customs]] that prompt exposure to other dialects, speakers may possess a diasystem that represents multiple dialects as part of their communicative competence.<ref>{{Harvcoltxt|Geraghty|1983|p=64}}</ref>

==Representation==
There are a number of ways diaphones are represented in literature. One way is through the IPA, this can be done with slashes, as if they are phonemes, or with other types of brackets:
* double slashes: {{IPA|//bɪt//}}<ref>e.g. {{Harvcoltxt|Trudgill|1974}}</ref>
* exclamation points: {{IPA|!bɪt!}}<ref>e.g. {{Harvcoltxt|Sledd|1966}}</ref>
* vertical bars: {{IPA|{{!}}bɪt{{!}}}}<ref>{{Harvcoltxt|Trask|1996|p=111}}</ref>
* curved brackets: {{IPA|{b.ɪ.t}}}<ref>e.g. {{Harvcoltxt|Smith|1967|p=312}}; this bracketing comes from Smith's interpretation that the diaphoneme represents [[morphophonemic]] correspondences rather than phonemic ones.</ref>

The concept does not necessitate the formation of a transcription system. Diaphones can instead be represented with double slashes.<ref>{{Harvcoltxt|Cadora|1970|pp=14, 18}}; similarly, double wavy lines (≈) indicate oppositions within a diasystem in the same way that wavy lines (~) indicate phonemic oppositions within an individual variety.</ref> This is the case, for example in {{Harvcoltxt|Orten|1991}} and {{Harvcoltxt|Weinreich|1954}} where diaphonemes are represented with bracketing:<ref>{{Harvcoltxt|Orten|1991|p=50}}</ref>

:<math>\bigg/ \bigg/ \frac{RP, GA \qquad \mathrm{k}}{SSE, KA \qquad \mathrm{k}~vs. \mathrm{x}} \bigg/ \bigg/</math>

In this scheme, [[Scottish Standard English]] and the accent of [[Kirkwall]] are shown to make a phonemic contrast between {{IPA|/k/}} and {{IPA|/x/}} while [[Received Pronunciation|RP]] and [[General American|GA]] are shown to possess only the former so that ''lock'' and ''[[loch]]'' are pronounced differently in the former group and identically in the latter.

Diaphonemic systems don't necessarily even have to utilize the [[International Phonetic Alphabet|IPA]]. Diaphones are useful in constructing a writing system that accommodates multiple dialects with different phonologies.<ref>{{Harvcoltxt|Jones|1950}}, cited in {{Harvcoltxt|Householder|1952|p=101}}</ref> Even in dialectology, diaphonemic transcriptions may instead be based on the language's orthography, as is the case with Lee Pederson's Automated Book Code designed for information from the ''Linguistic Atlas of the Gulf States''.<ref>{{Harvcoltxt|Pederson|1987|pp=48, 51}}</ref><ref>{{Harvcoltxt|Pederson|1989|p=54}}</ref> and the diaphonemic transcription system used by Paul Geraghty for related Fijian languages uses a modified Roman script.<ref>{{Harvcoltxt|Geraghty|1983|pp=19–22}}</ref>

==See also==
*[[Comparative method]]
*[[Diasystem]]
*[[Ernst Pulgram]]
*[[Morphophonology]]
*[[Robert A. Hall, Jr.]]

==Notes==
{{reflist|colwidth=30em}}

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|place=Oxford
|publisher=Oxford University Press
}}
*{{citation
|last=Saporta
|first=Sol
|year=1965
|title=Orderd rules, dialect differences, and historical processes
|journal=Language
|volume=41
|issue=2
|pages=218–224
}}
*{{citation
|last=Shen
|first=Yao
|year=1952
|title=Departures from strict phonemic representations
|journal=Language Learning
|volume=4
|issue=3–4
|pages=83–91
}}
*{{citation
|last=Shen
|first=Yao
|year=1959
|title=Some allophones can be important
|journal=Language learning
|volume=9
|issue=1
|pages=7–18
}}
*{{citation
|last=Shores
|first=David
|year=1984
|title=The stressed vowels of the speech of Tangier Island, Virginia
|journal=Journal of English Linguistics
|volume=17
|pages=37–56
}}
*{{citation
|last=Silverman
|first=Daniel
|year=1992
|title=Multiple scansions in loanword phonology: Evidence from Cantonese
|journal=Phonology
|volume=9
|issue=2
|pages=289–328
}}
*{{citation
|last=Sledd
|first=James H.
|year=1966
|title=Breaking, umlaut, and the southern drawl
|journal=Language
|volume=42
|issue=1
|pages=18–41
}}
*{{citation
|last=Smith
|first=Henry Lee, Jr.
|year=1967
|title=The concept of the morphophone
|journal=Language
|volume=43
|issue=1
|pages=306–341
}}
*{{citation
|last=Stockwell
|first=Robert
|year=1959
|title=Structural dialectology: A proposal
|journal=American Speech
|volume=34
|issue=4
|pages=258–268
}}
*{{citation
|last=Trager
|first=George L.
|authorlink=George L. Trager
|last2=Smith
|first2=Henry L, Jr.
|year=1951
|title=An outline of English structure
|series=Studies in Linguistics occasional papers
|volume=3
|place=Norman, OK
|publisher=Battenberg Press
}}
*{{citation
|last=Trask
|first=Robert L.
|authorlink=Larry Trask
|year=1996
|title=A Dictionary of Phonetics and Phonology
|place=London
|publisher=Routledge
}}
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|last=Troike
|first=Rudolph
|editor-last=James
|editor-first=E.
|year=1970
|title=Receptive competence, productive competence, and performance.
|series=Georgetown University Monograph Series on Languages and Linguistics
|volume=22
|pages=63–74
}}
*{{citation
|last=Troike
|first=Rudolph
|editor-last=Allen
|editor-first=Harold B.
|editor2-last=Underwood
|editor2-first=Gary N.
|year=1971
|chapter=Overall pattern and generative phonology
|title=Readings in American Dialectology
|place=New York
|publisher=Appleton-Century-Crofts
|pages=324–342
}}
*{{citation
|last=Trubetzkoy
|first=Nikolai
|authorlink=Nikolai Trubetzkoy
|year=1931
|title=Phonologie et géographie linguistique
|journal=Travaux du Cercle Linguistique de Prague
|volume=4
|pages=228–234
}}
*{{citation
|last=Trudgill
|first=Peter
|authorlink=Peter Trudgill
|year=1974
|title=The Social Differentiation of English in Norwich
|place=Cambridge
|publisher=Cambridge University Press
}}
*{{citation
|last=Trudgill
|first=Peter
|year=1983
|title=On Dialect: Social and Geographical Perspectives
|place=New York
|publisher=New York University Press
}}
*{{citation
|last=Turner
|first=G.W.
|year=1966
|title=The English in Australia and New England
|place=London
|publisher=Longmans
}}
*{{citation
|last=Twaddell
|first=W. Freeman
|year=1935
|title=On defining the phoneme
|journal=Language
|volume=11
|issue=1
|pages=5–62
}}
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|last4=Edman
|first4=Thomas R.
|year=1976
|title=Consonant environment specifies vowel identity
|journal=Journal of the Acoustical Society of America
|volume=60
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|pages=213–224
}}
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|first=C.F.
|year=1956
|title=Phonemicizing for dialect study: With reference to Hopi
|journal=Language
|volume=32
|issue=1
|pages=116–135
|doi=10.2307/410660
}}
*{{citation
|last=Weinreich
|first=Uriel
|authorlink=Uriel Weinreich
|year=1953
|chapter=Languages in contact, findings and problems
|title=Publications of the Linguistic Circle of New York
|volume=1
|place=New York
|publisher=Linguistic Circle of New York
}}
*{{citation
|last=Weinreich
|first=Uriel
|year=1954
|title=Is a structural dialectology possible?
|journal=Word
|volume=10
|pages=388–400
}}
*{{citation
|last=Wells
|first=John Christopher
|authorlink=John C. Wells
|year=1970
|title=Local accents in England and Wales
|journal=Journal of Linguistics
|volume=6
|issue=2
|pages=231–252
|doi=10.1017/S0022226700002632
}}
*{{citation
|last=Wells
|first=John Christopher
|year=1982
|title=Accents of English: An Introduction
|place=Cambridge
|publisher=Cambridge University Press
}}
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|last=Wilson
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|year=1998
|title=Parameter setting within a socially realistic linguistics
|journal=Language in Society
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}}
*{{citation
|last=Wolfram
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|authorlink=Walt Wolfram
|year=1982
|title=Language knowledge and other dialects
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}}
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|last=Wolfram
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}}
*{{citation
|last=Yaeger-Dror
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|year=1986
|title=[untitled review]
|journal=Language
|volume=62
|issue=4
|pages=917–923
}}
{{refend}}

==Further reading==
*{{citation
|last=Bailey
|first=Charles-James
|year=1970
|title=A new intonation theory to account for pan-English and idiom-particular patterns
|journal=Research on Language & Social Interaction
|volume=2
|issue=3
|pages=522–604
|ref=none
}}
*{{citation
|last=Bailey
|first=Charles-James N.
|editor-last=Stockwell
|editor-first=Robert P.
|editor2-last=Macaulay
|editor2-first=Ronald K.S.
|year=1972
|chapter=The integration of linguistic theory: Internal reconstruction and the comparative method in descriptive analysis
|title=Linguistic Change and Generative Theory
|place=Bloomington
|publisher=Indiana University Press
|pages=22–31
|ref=none
}}
*{{citation
|last=Francescato
|first=Giuseppe
|year=1959
|title=A case of coexistence of phonemic systems
|journal=Lingua
|volume=8
|pages=78–86
|ref=none
}}
*{{citation
|last=Halle
|first=Morris
|authorlink=Morris Halle
|year=1962
|title=Phonology in generative grammar
|journal=Word
|volume=18
|pages=54–72
|ref=none
}}
*{{citation
|last=Hausmann
|first=Robert B.
|year=1975
|title=Underlying representation in dialectology
|journal=Lingua
|volume=35
|issue=1
|pages=61–71
|ref=none
}}
*{{citation
|last=Hockett
|first=Charles
|authorlink=Charles F. Hockett
|year=1967
|title=The State of the Art
|place=The Hague
|publisher=Mouton
|ref=none
}}
*{{citation
|last=Keyser
|first=Samuel Jay
|year=1963
|title=[untitled review of ''The Pronunciation of English in the Atlantic States'']
|journal=Language
|volume=39
|issue=2
|pages=303–316
|ref=none
}}
*{{citation
|last=Siertsema
|first=B.
|year=1968
|title=Pros and cons of macro-phonemes in new orthographies: (Masaba spelling problems)
|journal=Lingua
|volume=21
|pages=429–442
|ref=none
}}

[[Category:Dialectology]]
[[Category:Phonology]]

Talk:Axiom of countability

2012-02-05T17:47:15Z

145.97.197.215: Removed my criticism, I was wrong (I confused 1st countable for 2nd countable and vice versa).

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Only Singapore residents and authorized persons should buy Landed 'residential property'as outlined within the Residential Properties Act. Foreigners are eligible to purchaseunits in condominiums or apartments which are not landed dwelling houses. Foreignerswho wish to purchase landed property in Singapore must first search the approval ofthe Controller of Residential Property. Your Lawyer's Position in a Property Purchase Pending completion of your purchase, your lawyer will lodge a caveat in opposition to the title to the property - this serves to notify the general public (and any third party interested within the property) that you've a sound curiosity or claim to the title of the property arising from the contract for the sale and buy. Property was at all times on his thoughts Property The Singapore Property Awards recognise excellence in actual property growth initiatives or particular person properties in terms of design, aesthetics, performance, contribution to the constructed setting and neighborhood at massive. It represents an excellent achievement which developers, professionals and property homeowners aspire to attain. It bestows upon the winner the correct to use the coveted award emblem recognised extensively all through the FIABCI network. Mr Brooke, for one, expects costs in city areas to rise 15 per cent throughout the subsequent two years, whereas he sees prices in outlying areas taking another 3-four years to get better. - 2002 April sixteen by Audrey Tan Singapore Enterprise Times Does anyone have any information /expertise of the property market in Singapore? My associate and I need to move there quickly and have seen a terrific place we like nevertheless we actually do not know if the market has reached its peak or whether it's alleged to proceed to rise. Keppel Land is Asia's premier house developer with world-class iconic waterfront residences at Keppel Bay and Marina Bay in Singapore. Hedges Park, a brand new condominium by Hong Leong Group with spectacular full amenities. Shaw Properties (1997) Pte Ltd Yeo Hiap Seng Restricted

Talk:Subbase

2011-08-18T15:53:09Z

145.97.197.215: answer to question 'subbase generated by (-infty,a)'

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