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2014-08-18T13:48:58Z

AurelioChill:

In [[mathematics]], a [[partially ordered set]] P is said to have '''Knaster's condition upwards''' (sometimes '''property (K)''') if any [[uncountable]] subset ''A'' of ''P'' has an [[linked set|upwards-linked]] uncountable subset. Anologous definition applies to '''Knaster's condition downwards'''.

The property is named after [[Poles|Polish]] [[mathematician]] [[Bronisław Knaster]].

Knaster's condition implies [[ccc]], and it is sometimes used in conjunction with a weaker form of [[Martin's axiom]], where the ccc requirement is replaced with Knaster's condition. Not unlike ccc, Knaster's condition is also sometimes used as a property of a [[topological space]], in which case it means that the topology (as in, the family of all open sets) with [[inclusion]]{{Disambiguation needed|date=February 2012}} satisfies the condition.

Furthermore, assuming [[Martin's axiom|MA]](<math>\omega_1</math>), ccc implies Knaster's condition, making the two equivalent.

== References ==
*{{cite book | last=Fremlin | first=David H. | title=Consequences of Martin's axiom | publisher=[[Cambridge University Press]]| location=Cambridge | year=1984 | isbn=0-521-25091-9 | others=Cambridge tracts in mathematics, no. 84}}

[[Category:Order theory]]

{{Mathlogic-stub}}

Main Page

2014-08-18T13:45:35Z

AurelioChill:

In [[quantum information theory]], '''quantum discord''' is a measure of nonclassical correlations between two subsystems of a quantum system. It includes correlations that are due to [[quantum mechanics|quantum physical]] effects but do not necessarily involve [[quantum entanglement]].

The notion of quantum discord was introduced by Harold Ollivier and [[Wojciech H. Zurek]]<ref name="zurek-2000">Wojciech H. Zurek, ''Einselection and decoherence from an information theory perspective'', [[Annalen der Physik]] vol. 9, 855–864 (2000) [http://onlinelibrary.wiley.com/doi/10.1002/1521-3889(200011)9:11/12%3C855::AID-ANDP855%3E3.0.CO;2-K/abstract abstract]</ref><ref name="olliver-zurek-2001">Harold Ollivier and Wojciech H. Zurek, ''Quantum Discord: A Measure of the Quantumness of Correlations'', [[Physics Review Letters]] vol. 88, 017901 (2001) [http://prl.aps.org/abstract/PRL/v88/i1/e017901 abstract]</ref> and, independently by L. Henderson and [[Vlatko Vedral]].<ref>L. Henderson and V. Vedral: ''Classical, quantum and total correlations'', [[Journal of Physics A]] 34, 6899 (2001), {{doi|10.1088/0305-4470/34/35/315}} [http://iopscience.iop.org/0305-4470/34/35/315]</ref> Olliver and Zurek referred to it also as a measure of ''quantumness'' of correlations.<ref name="olliver-zurek-2001"/> From the work of these two research groups it follows that quantum correlations can be present in certain mixed [[separable states]];<ref name="giorda-paris-2010-P1">Paolo Giorda, Matteo G. A. Paris: ''Gaussian quantum discord'', quant-ph arXiv:1003.3207v2 (submitted on 16 Mar 2010, version of 22 March 2010) [http://arxiv.org/PS_cache/arxiv/pdf/1003/1003.3207v2.pdf#page=1 p. 1]</ref> In other words, separability alone does not imply the absence of quantum effects. The notion of quantum discord thus goes beyond the distinction which had been made earlier between entangled versus separable (non-entangled) quantum states.

== Definition and mathematical relations ==
[[Image:Entropy-mutual-information-relative-entropy-relation-diagram.svg|thumb|256px|right|Individual (H(X),H(Y)), joint (H(X,Y)), and conditional entropies for a pair of correlated subsystems X,Y with mutual information I(X; Y).]]
In mathematical terms, quantum discord is defined in terms of the [[quantum mutual information]]. More specifically, quantum discord is the difference between two expressions which each, in the [[classical limit]], represent the [[mutual information]]. These two expressions are:
:<math>I (A; B) = H (A) + H (B) - H (A,B)</math>
:<math>J (A; B) = H (A) - H (A|B)</math>
where, in the classical case, <math>H(A)</math> is the [[information entropy]], <math>H(A,B)</math> the [[joint entropy]] and <math>H(A|B)</math> the [[conditional entropy]], and the two expressions yield identical results. In the nonclassical case, the quantum physics analogy for the three terms are used – <math>S (\rho_A)</math> the [[von Neumann entropy]], <math>S(\rho)</math> the [[joint quantum entropy]] and <math>S (\rho_A|\rho_B)</math> the [[conditional quantum entropy]], respectively, for [[probability density function]] <math>\rho</Math>
:<math>I (\rho) = S (\rho_A) + S (\rho_B) - S (\rho)</math>
:<math>J_A (\rho) = S (\rho_B) - S (\rho_B|\rho_A)</math>
The difference between the two expressions <math>I(\rho) - J_A(\rho)</math> defines the basis-dependent quantum discord, which is asymmetrical in the sense that <math>\mathcal D_A (\rho)</math> can differ from <math>\mathcal D_B (\rho)</math>.<ref name="dakic-vedral-brukner">Borivoje Dakić, Vlatko Vedral, Caslav Brukner: ''Necessary and sufficient condition for nonzero quantum discord'', Phys. Rev. Lett., vol. 105, nr. 19, 190502 (2010), [http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.0190v2.pdf arXiv:1004.0190v2] (submitted 1 April 2010, version of 3 November 2010)</ref><ref>For a succinct overview see for ex [http://arxiv.org/PS_cache/arxiv/pdf/0809/0809.1723v2.pdf arXiv:0809.1723v2]</ref> <math>J</math> represents the part of the correlations that can be attributed to classical correlations and varies in dependence on the chosen [[Matrix factorization#Eigendecomposition|eigenbasis]]; therefore, in order for the quantum discord to reflect the purely nonclassical correlations independently of basis, it is necessary that <math>J</math> first be maximized over the set of all possible [[Projective Hilbert space|projective]] [[quantum measurement|measurements]] onto the eigenbasis:<ref>For a more detailed overview see for ex. ''Signatures of nonclassicality in mixed-state quantum computation'', [[Physical Review A]] vol. 79, 042325 (2009), {{doi|10.1103/PhysRevA.79.042325}} [http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3A0811.4003 arXiv:0811.4003] and see for ex. Wojciech H. Zurek: ''Decoherence and the transition from quantum to classical - revisited'', [http://arxiv.org/ftp/quant-ph/papers/0306/0306072.pdf#page=10 p. 11]</ref>
:<math>\mathcal D_A (\rho) = I (\rho) - \max_{\{\Pi_j^A\}} J_{\{\Pi_j^A\}} (\rho) = S (\rho_A) - S(\rho) + \min_{\{\Pi_j^A\}} S (\rho_{B | \{\Pi_j^A\}} ) </math>
Nonzero quantum discord indicates the presence of correlations that are due to [[Observable#Incompatibility of observables in quantum mechanics|noncommutativity of quantum operators]].<ref>Shunlong Luo: ''Quantum discord for two-qubit systems'', [[Physical Review A]], vol. 77, 042303 (2008) [http://pra.aps.org/abstract/PRA/v77/i4/e042303 abstract]</ref> For [[pure state]]s, the quantum discord becomes a measure of [[quantum entanglement]],<ref name="datta-et-al-2007-P4">Animesh Datta, Anil Shaji, Carlton M. Caves: ''Quantum discord and the power of one qubit'', arXiv:0709.0548v1 [quant-ph], 4 Sep 2007, [http://arxiv.org/PS_cache/arxiv/pdf/0709/0709.0548v1.pdf#page=4 p. 4]</ref> more specifically, in that case it equals the entropy of entanglement.<ref name="giorda-paris-2010-P1"/>

Vanishing quantum discord is a criterion for the [[pointer state]]s, which constitute preferred effectively classical states of a system.<ref name="olliver-zurek-2001"/> It could be shown that quantum discord must be non-negative and that states with vanishing quantum discord can in fact be identified with pointer states.<ref>Animesh Datta: ''A condition for the nullity of quantum discord'', [http://arxiv.org/PS_cache/arxiv/pdf/1003/1003.5256v2.pdf arXiv:1003.5256v2]</ref> Other conditions have been identified which can be seen in analogy to the [[Peres–Horodecki criterion]]<ref>Bogna Bylicka, Dariusz Chru´sci´nski: ''Witnessing quantum discord in 2 x N systems'', arXiv:1004.0434v1 [quant-ph], 3 April 2010</ref> and in relation to the [[von Neumann entropy#Properties|strong subadditivity of the von Neumann entropy]].<ref name="madhok-datta-2011">Vaibhav Madhok, Animesh Datta: ''Role of quantum discord in quantum communication'' [http://arxiv.org/abs/1107.0994v1 arXiv:1107.0994v1], (submitted 5 July 2011)</ref>

Efforts have been made to extend the definition of quantum discord to continuous variable systems, in particular to bipartite systems described by Gaussian states.<ref name="giorda-paris-2010-P1"/>

== Properties ==
Zurek provided a physical interpretation for discord by showing that it "determines the difference between the efficiency of quantum and classical Maxwell’s demons...in extracting work from collections of correlated quantum systems".<ref name="zurek-2003">W. H. Zurek: ''Quantum discord and Maxwell’s demons", [[Physical Review A]], vol. 67, 012320 (2003), [http://pra.aps.org/abstract/PRA/v67/i1/e012320 abstract]'</ref>

Discord can also be viewed In operational terms as an "entanglement consumption in an extended [[State-merging|quantum state merging]] protocol".<ref name="madhok-datta-2011"/><ref>D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, A. Winter: ''[http://arxiv.org/abs/1008.3205 Operational interpretations of quantum discord]'', quant-ph, arXiv:1008.3205</ref> Providing evidence for non-entanglement quantum correlations normally involves elaborate [[quantum tomography]] methods; however, in 2011, such correlations could be demonstrated experimentally in a room temperature nuclear magnetic resonance system, using [[chloroform]] molecules that represent a two-[[qubit]] quantum system.<ref>R. Auccaise, J. Maziero, L. C. Céleri, D. O. Soares-Pinto, E. R. deAzevedo, T. J. Bonagamba, R. S. Sarthour, I. S. Oliveira, R. M. Serra: ''Experimentally Witnessing the Quantumness of Correlations'', [[Physics Review Letters]], vol. 107, 070501 (2011) [http://prl.aps.org/abstract/PRL/v107/i7/e070501 abstract] ([http://arxiv.org/abs/1104.1596 arXiv:1104.1596])</ref><ref>Miranda Marquit: ''[http://www.physorg.com/news/2011-08-quantum-entanglement.html Quantum correlations – without entanglement]'', [[PhysOrg]], August 24, 2011</ref>

Quantum discord has been seen as a possible basis for the performance in terms of [[quantum computation]] ascribed to certain [[Quantum state#Mixed states|mixed-state]] quantum systems,<ref name="datta-et-al-2007-P1">Animesh Datta, Anil Shaji, [[Carlton M. Caves]]: ''Quantum discord and the power of one qubit'', arXiv:0709.0548v1 [quant-ph], 4 Sep 2007, [http://arxiv.org/PS_cache/arxiv/pdf/0709/0709.0548v1.pdf#page=1 p. 1]</ref> with a ''mixed quantum state'' representing a [[statistical ensemble]] of pure states (see [[quantum statistical mechanics]]).

Evidence has been provided for poignant differences between the properties of quantum entanglement and quantum discord. It has been shown that quantum discord is more resilient to [[Quantum decoherence#Dissipation|dissipative environments]] than is quantum entanglement. This has been shown for Markovian environments as well as for non-Markovian environments based on a comparison of the dynamics of discord with that of [[Concurrence (quantum computing)|concurrence]], where discord has proven to be more robust.<ref>See [http://arxiv.org/abs/0911.1096] as well as [http://arxiv.org/PS_cache/arxiv/pdf/0911/0911.1845v1.pdf] and citations therein</ref> It has been shown that, at least for certain models of a qubit pair which is in thermal equilibrium and form an [[open quantum system]] in contact with a [[Heat reservoir|heat bath]], the quantum discord increases with temperature in certain temperature ranges, thus displaying a behaviour that is quite in contrast with that of entanglement, and that furthermore, surprisingly, the classical correlation actually decreases as the quantum discord increases.<ref>T. Werlang, G. Rigolin: ''Thermal and magnetic discord in Heisenberg models'', [[Physical Review A]], vol. 81, no. 4 (044101) (2010), {{doi|10.1103/PhysRevA.81.044101}} [http://pra.aps.org/abstract/PRA/v81/i4/e044101 abstract], [http://arxiv.org/PS_cache/arxiv/pdf/0911/0911.3903v2.pdf fulltext (arXiv)]</ref> Nonzero quantum discord can persist even in the limit of one of the subsystems undergoing an infinite acceleration, whereas under this condition the quantum entanglement drops to zero due to the [[Unruh effect]].<ref>Animesh Datta: ''Quantum discord between relatively accelerated observers'', arXiv:0905.3301v1 [quant-ph] 20 May 2009, [http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.3301v1.pdf]</ref>

==Alternative measures==

An operational measure, in terms of distillation of local pure states, the ‘quantum deficit’.<ref>Jonathan Oppenheim, Michał Horodecki, Paweł Horodecki and Ryszard Horodecki:"Thermodynamical Approach to Quantifying Quantum Correlations" [[Physical Review Letters]] 89, 180402 (2002) [http://arxiv.org/abs/quant-ph/0112074]</ref> The one-way and zero-way versions were shown to be equal to the relative entropy of quantumness.<ref> Michał Horodecki, Paweł Horodecki, Ryszard Horodecki, Jonathan Oppenheim, Aditi Sen De, Ujjwal Sen, Barbara Synak-Radtke: "Local versus nonlocal information in quantum-information theory: Formalism and phenomena" [[Physical Review A]] 71, 062307 (2005) [http://arxiv.org/abs/quant-ph/0410090]</ref>

Other measures of nonclassical correlations include the measurement induced disturbance (MID) measure and the localized noneffective unitary (LNU) distance<ref>see for ex.: Animesh Datta, Sevag Gharibian: ''Signatures of non-classicality in mixed-state quantum computation'', [[Physical Review A]] vol. 79, 042325 (2009) [http://pra.aps.org/abstract/PRA/v79/i4/e042325 abstract], [http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3A0811.4003 arXiv:0811.4003]</ref> and various entropy-based measures.<ref>Matthias Lang, Anil Shaji, Carlton Caves: ''Entropic measures of nonclassical correlations'', American Physical Society, APS March Meeting 2011, March 21–25, 2011, [http://adsabs.harvard.edu/abs/2011APS..MARX29007L abstract #X29.007], [http://arxiv.org/abs/1105.4920 arXiv:1105.4920]</ref>

There exists a geometric measure of discord,<ref name="dakic-vedral-brukner"/> which obeys a factorization law<ref>Wei Song, Long-Bao Yu, Ping Dong, Da-Chuang Li, Ming Yang, Zhuo-Liang Cao: ''Geometric measure of quantum discord and the geometry of a class of two-qubit states'', [http://arxiv.org/abs/1112.4318v2 arXiv:1112.4318v2] (submitted on 19 December 2011, version of 21 December 2011)</ref>, can be put in relation to von Neumann measurements,<ref>S. Lu, S. Fu: ''Geometric measure of quantum discord'', Phys. Rev. A, vol. 82, no. 3, 034302 (2010)</ref> and a measure of ‘measurement-induced nonlocality’ (MIN).<ref>S. Luo and S. Fu: ''Measurement-Induced Nonlocality''], Phys. Rev. Lett. 106, 120401 (2011) ([http://prl.aps.org/abstract/PRL/v106/i12/e120401 abstract]). Cited after Guo-Feng Zhang, Heng Fan, Ai-Ling Ji, Wu-Ming Liu: ''Dynamics of geometric discord and measurement-induced nonlocality at finite temperature'', [http://arxiv.org/abs/1201.1949 arXiv:1201.1949] (submitted on 10 January 2012)</ref>

==References==
{{Reflist}}

[[Category:Quantum information science]]

Main Page

2014-08-18T10:06:58Z

AurelioChill:

In [[statistics]], the '''Jarque–Bera test''' is a [[goodness-of-fit]] test of whether sample data have the [[skewness]] and [[kurtosis]] matching a [[normal distribution]]. The test is named after [[Carlos Jarque]] and [[Anil K. Bera]]. The [[test statistic]] ''JB'' is defined as

: <math>
\mathit{JB} = \frac{n}{6} \left( S^2 + \frac14 (K-3)^2 \right)
</math>

where ''n'' is the number of observations (or degrees of freedom in general); ''S'' is the sample [[skewness]], and ''K'' is the sample [[kurtosis]]:

: <math>
S = \frac{ \hat{\mu}_3 }{ \hat{\sigma}^3 }
= \frac{\frac1n \sum_{i=1}^n (x_i-\bar{x})^3} {\left(\frac1n \sum_{i=1}^n (x_i-\bar{x})^2 \right)^{3/2}} ,
</math>
: <math>
K = \frac{ \hat{\mu}_4 }{ \hat{\sigma}^4 }
= \frac{\frac1n \sum_{i=1}^n (x_i-\bar{x})^4} {\left(\frac1n \sum_{i=1}^n (x_i-\bar{x})^2 \right)^2} ,
</math>

where <math>\hat{\mu}_3</math> and <math>\hat{\mu}_4</math> are the estimates of third and fourth [[central moment]]s, respectively, <math>\bar{x}</math> is the sample [[mean]], and
<math>\hat{\sigma}^2</math> is the estimate of the second central moment, the [[variance]].

If the data comes from a normal distribution, the ''JB'' statistic [[asymptotic analysis|asymptotically]] has a [[chi-squared distribution]] with two [[degrees of freedom (statistics)|degrees of freedom]], so the statistic can be used to [[statistical hypothesis testing|test]] the hypothesis that the data are from a [[normal distribution]]. The [[null hypothesis]] is a joint hypothesis of the skewness being zero and the [[excess kurtosis]] being zero. Samples from a normal distribution have an expected skewness of 0 and an expected excess kurtosis of 0 (which is the same as a kurtosis of 3). As the definition of ''JB'' shows, any deviation from this increases the JB statistic.

For small samples the chi-squared approximation is overly sensitive, often rejecting the null hypothesis when it is in fact true. Furthermore, the distribution of p-values departs from a uniform distribution and becomes a right-skewed uni-modal distribution, especially for small p-values. This leads to a large [[Type I error]] rate. The table below shows some p-values approximated by a chi-squared distribution that differ from their true alpha levels for small samples.

:{| class="wikitable"
|+Calculated p-value equivalents to true alpha levels at given sample sizes
! True α level !! 20 !! 30 !! 50 !! 70 !! 100
|-
! 0.1
| 0.307 || 0.252 || 0.201 || 0.183 || 0.1560
|-
! 0.05
| 0.1461 || 0.109 || 0.079 || 0.067 || 0.062
|-
! 0.025
| 0.051 || 0.0303 || 0.020 || 0.016 || 0.0168
|-
! 0.01
| 0.0064 || 0.0033 || 0.0015 || 0.0012 || 0.002
|}
(These values have been approximated by using [[Monte Carlo simulation]] in [[Matlab]])

In [[MATLAB]]'s implementation, the chi-squared approximation for the JB statistic's distribution is only used for large sample sizes (> 2000). For smaller samples, it uses a table derived from [[Monte Carlo simulations]] in order to interpolate p-values.<ref name="MathWorks">{{cite web|url=http://www.mathworks.com/access/helpdesk/help/toolbox/stats/jbtest.html|title=Analysis of the JB-Test in MATLAB|publisher=MathWorks|accessdate=May 24, 2009}}</ref>

==History==
Considering normal sampling, and √''β''1 and ''β''2 contours, {{harvtxt|Bowman|Shenton|1975}} noticed that the statistic ''JB'' will be asymptotically ''χ''2(2)-distributed; however they also noted that “large sample sizes would doubtless be required for the ''χ''2 approximation to hold”. Bowman and Shelton did not study the properties any further, preferring [[D’Agostino’s K-squared test]].

Around 1979, Anil Bera and [[Carlos Jarque]] while working on their dissertations on regression analysis, have applied the [[Lagrange multiplier principle]] to the [[Pearson distribution|Pearson family of distributions]] to test the normality of unobserved regression residuals and found that the ''JB'' test was asymptotically optimal (although the sample size needed to “reach” the asymptotic level was quite large). In 1980 the authors published a paper ({{harvnb|Jarque|Bera|1980}}), which treated a more advanced case of simultaneously testing the normality, [[homoscedasticity]] and absence of [[autocorrelation]] in the residuals from the [[linear regression model]]. The ''JB'' test was mentioned there as a simpler case. A complete paper about the JB Test was published in the ''International Statistical Review'' in 1987 dealing with both testing the normality of observations and the normality of unobserved regression residuals, and giving finite sample significance points.

==Jarque–Bera test in regression analysis==
According to Robert Hall, David Lilien, et al. (1995) when using this test along with multiple regression analysis the right estimate is:

: <math>
\mathit{JB} = \frac{n-k}{6} \left( S^2 + \frac14 (K-3)^2 \right)
</math>

where ''n'' is the number of observations and ''k'' is the number of regressors when examining residuals to an equation.

==References==
{{reflist}}
<HR>
{{refbegin}}
* {{cite journal
| first1 = K.O. | last1 = Bowman
| first2 = L.R. | last2 = Shenton
| title = Omnibus contours for departures from normality based on √''b''1 and ''b''2
| year = 1975
| journal = Biometrika
| volume = 62 | issue = 2
| pages = 243–250
| jstor = 2335355
| ref = CITEREFBowmanShenton1975
}}
* {{cite journal
| first1 = Carlos M. | last1 = Jarque | authorlink1 = Carlos Jarque
| first2 = Anil K. | last2 = Bera
| title = Efficient tests for normality, homoscedasticity and serial independence of regression residuals
| year = 1980
| journal = Economics Letters
| volume = 6 | issue = 3
| pages = 255–259
| doi = 10.1016/0165-1765(80)90024-5
| ref = CITEREFJarqueBera1980
}}
* {{cite journal
| first1 = Carlos M. | last1 = Jarque | authorlink1 = Carlos Jarque
| first2 = Anil K. | last2 = Bera
| title = Efficient tests for normality, homoscedasticity and serial independence of regression residuals: Monte Carlo evidence
| year = 1981
| journal = Economics Letters
| volume = 7 | issue = 4
| pages = 313–318
| doi = 10.1016/0165-1765(81)90035-5
| ref = CITEREFJarqueBera1981
}}
* {{cite journal
| first1 = Carlos M. | last1 = Jarque | authorlink1 = Carlos Jarque
| first2 = Anil K. | last2 = Bera
| title = A test for normality of observations and regression residuals
| year = 1987
| journal = International Statistical Review
| volume = 55 | issue = 2
| pages = 163–172
| jstor = 1403192
| ref = CITEREFJarqueBera1987
}}
* {{cite book
| first = | last = Judge
| coauthors = et al.
| title = Introduction and the theory and practice of econometrics
| year = 1988
| edition = 3rd
| pages = 890–892
}}
* {{cite book
| first1 = Robert E. | last1 = Hall
| first2 = David M. | last2 = Lilien
| coauthors = et al.
| title = EViews User Guide
| year = 1995
| pages = 141
}}
{{refend}}

== Implementations ==
* [http://www.alglib.net/statistics/hypothesistesting/jarqueberatest.php ALGLIB] includes implementation of the Jarque–Bera test in C++, C#, Delphi, Visual Basic, etc.
* [[gretl]] includes an implementation of the Jarque–Bera test
* [[R (programming language)|R]] includes implementations of the Jarque–Bera test: ''jarque.bera.test'' in package ''tseries'', for example, and ''jarque.test'' in package ''moments''.
* [[Matlab|MATLAB]] includes implementation of the Jarque–Bera test, the function "jbtest".
* [[Python (programming language)|Python]] [[statsmodels]] includes implementation of the Jarque–Bera test, "statsmodels.stats.stattools.py".

{{Statistics}}

{{DEFAULTSORT:Jarque-Bera test}}
[[Category:Normality tests]]

Main Page

2014-08-18T10:00:40Z

AurelioChill:

{{refimprove|note=all references are to the primary articles of Majumdar; there is no single reference which could support his bio|date=February 2011}}
{{Infobox scientist
|image = Sudhansu Datta Majumdar.jpg
|image_size = 150px
| name = Sudhansu Datta Majumdar
| birth_date = 1915
| birth_place = [[Sylhet]] ([[British India]], now [[Bangladesh]])
| death_date = 1997
| death_place = [[Calcutta]]
| residence = Calcutta
| citizenship = India
| nationality = India
| ethnicity =
| field = [[Physics]] / [[General Relativity]] /[[Electrodynamics]]/[[Quantum Physics]]/[[Group Theory]]
| work_institution = [[Calcutta University]], [[Indian Institute of Technology, Kharagpur]], [[Visva Bharati]], [[Shantiniketan]]
| alma_mater = [[Presidency College, Calcutta]]
| doctoral_advisor =
| doctoral_students =
| known_for = [[General Relativity]], [[Electrodynamics]], [[Spectroscopy]], [[Group Theory]]
| author_abbreviation_bot =
| author_abbreviation_zoo =
| prizes =
| religion =
| footnotes =
}}

'''Sudhansu Datta Majumdar''' (1915–1997) was an Indian physicist, and faculty member of the [[Indian Institute of Technology, Kharagpur]].

==Biography==

Born in 1915 in Sylhet (now in Bangladesh), Sudhansu Datta Majumdar had his education in Sylhet; [[Presidency College, Calcutta]], and University College of Science, Calcutta. In an academic career spanning several decades, he served in different capacities in various institutions. Beginning with a stint in the Palit Laboratory of Physics, [[Calcutta University]], from where he wrote the now famous Majumdar–Papapetrou paper,<ref name = majumdar>{{cite journal| last = Majumdar| first = S D| authorlink = | title = A Class of Exact Solutions of Einstein's Field Equations | year = 1947| journal = [[Physical Review]]| volume = 72| issue = 5| pages =390–398| doi = 10.1103/PhysRev.72.390|bibcode = 1947PhRv...72..390M }}</ref> he was appointed Lecturer in Physics in Calcutta University in 1951. Subsequently, he became a reader there in 1960. During 1956–57, he went to Cambridge University, United Kingdom, on an educational tour in order to interact with [[P. A. M. Dirac]]. In 1962, Majumdar obtained the rare honor of the degree of D.Sc. in Physics from Calcutta University, one of his thesis examiners being [[J.A. Wheeler]]. Three years later, in 1965, he joined [[IIT, Kharagpur]], as a Professor of Physics where he served till 1975. His last academic appointment was, as a Professor of Mathematics in Visva Bharati, Shantiniketan. In 1974, he was invited by [[Yeshiva University]], New York, to deliver a course of lectures. He visited the Mathematics Department, Monash University, Australia, between July and December, 1976. [[Calcutta Mathematical Society]] elected him as their president in 1980. The diverse areas in which he contributed substantially include --- [[General Relativity]], [[Electrodynamics]], [[Group Theory]] and [[Spectroscopy]]. He died in Calcutta in 1997.<ref name = memorial>{{cite journal | title = Memorial: Sudhansu Datta Majumdar (1915-1997)| journal = [[Ansatz (journal)|Ansatz]] | volume = 3 | url = http://www.phy.iitkgp.ernet.in/ansatz3/Memorial.html}}</ref>

==Majumdar–Papapetrou solution==

:''"Majumdar–Papapetrou solution" redirects to here.''

The phenomenon of static equilibrium for a system of point charges is well known in Newtonian theory, where the mutual gravitational and electrostatic forces can be balanced by fine-tuning the charge suitably with the particle masses. The corresponding generalisation, in the form of static solutions of the coupled, source-free Einstein-Maxwell equations, was discovered by Majumdar and Papapetrou independently in 1947.<ref name = majumdar>{{cite journal| last = Datta Majumdar| first = Sudhansu| authorlink = | title = A Class of Exact Solutions of Einstein's Field Equations| pages =390–398 | year = 1947 | journal = [[Physical Review]]| volume = 72| issue = 5| page = 390| doi = 10.1103/PhysRev.72.390|bibcode = 1947PhRv...72..390M }}</ref><ref name = papapetrou>{{cite journal| last = Papapetrou| first = A | authorlink = | title = | journal = [[Proceedings of Royal Irish Academy A]]| volume = 51| issue = | pages =191| year =1947}}</ref> These gravitational fields assume no spatial symmetry and also contain geodesics which are incomplete. While work continued on understanding these solutions better, a renewed interest in this metric was generated by the important observation of [[Werner Israel|Israel]] and Wilson in 1972 that static black-hole spacetimes with the mass being equal to the magnitude of the charge are of Majumdar–Papapetrou form. In the same year, it was shown by [[James Hartle|Hartle]] and [[Stephen Hawking|Hawking]]<ref name = hartle-hawking>{{cite journal| author= Hartle, James B.; and Hawking, Stephen| title = Solutions of the Einstein-Maxwell equations with many black holes| journal = [[Communications in Mathematical Physics]]| volume = 26| issue =2 | pages =87–101| year =1972 |doi=10.1007/BF01645696 |bibcode = 1972CMaPh..26...87H }}</ref> that these spacetimes can be analytically extended to electrovacuum black hole spacetimes with a regular domain of outer communication. They interpreted this as a system of charged black holes in equilibrium under their gravitational and electrical forces. Each one of these many black holes or the multi-black holes system has a spherical topology and hence is a fairly regular object. In a more recent development, the uniqueness of the metric was discussed by Heusler, Chrusciel and others. These and other aspects of the Majumdar–Papapetrou metric have attracted considerable attention on the classical side, as well as in the work and applications from the perspective of string theory. In particular, the mass equal to charge aspect of these models was used extensively in certain string theoretic considerations connected to black hole entropy and related issues.

==Majumdar–Papapetrou geometries==

Majumdar–Papapetrou geometries generalize axially symmetric solutions to Einstein-Maxwell equations found by [[Hermann Weyl]] to a completely nonsymmetric and general case. The line element is given by:
:<math>
ds^2 = -U(x,y,z)^{-2}dt^2 + U(x,y,z)^2 (dx^2 + dy^2 + dz^2),
</math>

where the only nonvanishing component of the vector potential <math>A_{\mu}\ </math> is the scalar potential <math> \Phi (x)\ </math>. The relation between the metric and the scalar field is given by
:<math>
\Phi(x) = A_{t}(x) = U^{-1}(x),
</math>

where the electrostatic field is normalized to unity at infinity. The source-free Einstein-Maxwell equations then reduce to the Laplace equation given by:
:<math>
\nabla ^2 U(x,y,z) = \frac{\partial^2 U}{\partial x^2} + \frac{\partial^2 U}{\partial y^2} + \frac{\partial^2 U}{\partial z^2} = 0,
</math>

where U(x,y,z) can be extended in spatial directions till one encounters a singularity or till U(x,y,z) vanishes.

It was later shown by Hartle and Hawking<ref name = hartle-hawking>{{cite journal| last = Hawking| first = Hartle| authorlink = | title = Solutions of the Einstein-Maxwell equations with many black holes| journal = [[Communications in Mathematical Physics]]| volume = 26| issue =2 | pages =87–101| year =1972 |doi=10.1007/BF01645696| last2 = Hawking| first2 = S. W. |bibcode = 1972CMaPh..26...87H }}</ref> that these solutions can be "glued" together to construct multi-blackhole solutions of charged blackholes. These charged blackholes are in static equilibrium with each other with the gravitational and the electrostatic forces canceling each other out. The Majumdar–Papapetrou solution, thus, can be seen as early example of [[Bogomol'nyi-Prasad-Sommerfield bound|BPS]] configuration where static equilibrium results due to the cancellation of opposing forces. Examples of such BPS configurations include [[cosmic strings]] (attractive gravitational force balances with the repulsive scalar force), [[Magnetic monopole|monopoles]], BPS configurations of [[D-branes]] (cancellation of NS-NS and RR forces, NS-NS being the gravitational force and RR being the generalization of the electrostatic force), etc.

==Electrodynamics of crystalline media and the Cherenkov Effect==

During the fifties, there was a resurgence of interest in the [[Cherenkov effect]] both in its experimental and theoretical aspects. Professor Majumdar was fascinated by the problem, because it was perhaps the only classical electrodynamical derivation that fetched Nobel prizes in a world dominated by the Quantum. As was usual with him, he approached the problem in an absolutely novel way.<ref name = sdm-cherenkov-1>{{cite journal| last = Majumdar| first = S D | authorlink = | title = Cherenkov Radiation in Anisotropic Media| journal = [[Proceedings of the Royal Society A]]| volume = 316 | issue = 1527| pages =525–537| year =1970 |doi=10.1098/rspa.1970.0094| last2 = Pal| first2 = R. }}</ref><ref name = sdm-cherenkov-2>{{cite journal| last = Majumdar| first = S D | authorlink = | title = Cherenkov Radiation in Biaxial Crystals – I| journal = [[Annals of Physics]]| volume = 76 | issue = 2| pages =419–427| year =1973| last2 = Pal| first2 = R. |bibcode = 1973AnPhy..76..419D |doi = 10.1016/0003-4916(73)90041-9 }}</ref><ref name = sdm-cherenkov-3>{{cite journal| last = Majumdar| first = S D | authorlink = | title = Cherenkov Radiation in Biaxial Crystals – II| journal = [[Annals of Physics]]| volume = 76 | issue = 2| pages =428–436| year =1973|bibcode = 1973AnPhy..76..428D |doi = 10.1016/0003-4916(73)90042-0 }}</ref> Instead of studying the Cherenkov radiation field in the rest frame of the medium through which the charged particle whizzes by, he decided to jump to the rest frame of the charge. The great advantage of this approach is that the electromagnetic field becomes static and can be described by just two scalar potentials, which was a totally new formulation of the problem. However, the flowing medium now acquires a complicated magneto-electric character. This however came as a blessing in disguise, because it led to a discovery in the electrodynamics of crystalline media. Majumdar found that a most general doubly anisotropic medium with tensor permittivity and tensor permeability with non-parallel principal axes could sometimes behave like an 'isotropic' or 'uniaxial' medium as far as the structure of the Fresnel wave surface is concerned. Armed with this insight and his new formulation of the problem, he derived, for the first time, a closed expression for the Cherenkov output in a biaxial crystal in terms of [[elliptic functions]].

His students and collaborators followed up his studies.<ref name = gps-cherenkov-1>{{cite journal| last = Sastry| first = G P | authorlink = | title = Cherenkov Ray Cones in Crystalline Media| journal = [[Proceedings of the Royal Society A]]| volume = 411 | issue = 1840| pages =35–47| year =1987| doi = 10.1098/rspa.1987.0052| last2 = Kumar| first2 = K.|bibcode = 1987RSPSA.411...35S }}</ref><ref name = gps-cherenkov-2>{{cite journal| last = Sastry| first = G P | authorlink = | title = Cherenkov Radiation in Spatially Dispersive Media| journal = [[Proceedings of the Royal Society A]]| volume = 374 | issue = 1759| pages =531–541| year =1981| doi = 10.1098/rspa.1981.0035| last2 = Chowdhury| first2 = D.|bibcode = 1981RSPSA.374..531S }}</ref> A major contribution that resulted was the prediction of a new phenomenon called The Cherenkov analogue of conical refraction. A surprising system of intersecting Cherenkov rings in a biaxial crystal at precisely defined particle energies was predicted. These rings were later found in the photographs taken by V.P. Zrelov at the Proton Synchrotron facility at [[Dubna]], [[Moscow]].

==Theory of group representations==

Professor Majumdar's work on group theory has its origins in one of his early papers on [[molecular spectroscopy]] where a novel method for deriving the [[Clebsch-Gordan series]] and coefficients of [[SU(2)]] was discussed. The new approach made it possible to establish a connection between the [[Clebsch-Gordan Coefficients]] (CGC) and the Gauss [[hypergeometric function]] which was eventually identified as the generating function of the CGC.<ref name = sdm-su3-1>{{cite journal| last = Majumdar| first = S D | authorlink = | title = On the representations of the group SU(3)| journal = [[Journal of Physics A]]| volume = 1| issue = 2| pages =203| year =1968| doi = 10.1088/0305-4470/1/2/304|bibcode = 1968JPhA....1..203M }}</ref><ref name = sdm-su3-2>{{cite journal| last = Majumdar| first = S D | authorlink = | title = Some results on the groups SU(2) and SU(3)| journal = [[Progress in Theoretical Physics]]| volume = 38| issue = 5| pages =1176| year =1967|doi=10.1143/PTP.38.1176 |bibcode = 1967PThPh..38.1176M }}</ref><ref name = sdm-su3-3>{{cite journal| last = Majumdar| first = S D | authorlink = | title = The Clebsch-Gordan coefficients of SU(3) and the orthogonalization problem| journal = [[Journal of Mathematical Physics]]| volume = 14| issue = 9| pages =1248| year =1973| doi = 10.1063/1.1666474|bibcode = 1973JMP....14.1248D }}</ref> The Majumdar form of the CGC of SU(2) has appeared in acclaimed textbooks. Barut and Wilson have extensively investigated the symmetry properties of the three non-trivial forms of the CGC, namely, the [[Wigner-Racah]], the van der Waerden and the Majumdar form. The success of the above approach for SU(2) inspired Majumdar to extend his method and obtain a similar reduction for SU(3). The SU(3) generators were expressed as differential operators in four independent variables. In terms of these, the eigenvalue equation of the quadratic [[Casimir operator]] became a partial differential equation in four independent variables, the polynomial solutions of which, form the bases of an irreducible representation of [[SU(3)]].

The forms of the new operators made apparent the fact that the basis states of an irreducible representation of SU(3)are linear combinations of the CG series of SU(2) with the same value of j, m and j1 – j2. Obtaining the SU(2) basis for SU(3) was thereby shown to be closely related to the theory of coupling of two angular momenta. The basic states of SU(3) were later used in deriving the matrix elements of finite transformations of SU(3). Simple analytic continuation of Majumdar's generating function of the SU(2) CGC was later understood to be the 'master function' for the solution of several problems of non-compact groups such as SU(1,1) and SL(2,C). The interpretation and domain of the complex variables, however, change from case to case. For example, in the representation theory of [[SL(2,C)]] these represent a pair of complex numbers i.e. spinors transforming according to the fundamental representation of SL(2,C) and the complex conjugate respectively. On the other hand, for the CG problem of SU(1,1), they transform according to two distinct SU(1,1) groups.

== References ==
{{Reflist}}

== External links ==
*[http://web.archive.org/web/20110721154816/http://www.phy.iitkgp.ernet.in/ansatz3/sdm.html The Genius Who Touched My Life , G P Sastry]
*[http://web.archive.org/web/20110721154846/http://www.phy.iitkgp.ernet.in/ansatz4/sdmdb.pdf An Homage to Sudhansu Datta Majumdar, D Basu]
*[http://dl.dropbox.com/u/2067224/Issue%20Archive/2007oct09.pdf The Life and Science of SDM, '''''The Scholars Avenue''''', Oct 10 2007]

{{Persondata 
| NAME = Datta Majumdar, Sudhanshu
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = Indian physicist
| DATE OF BIRTH = 1915
| PLACE OF BIRTH = Sylhet (British India, now Bangladesh)
| DATE OF DEATH = 1997
| PLACE OF DEATH = Calcutta
}}
{{DEFAULTSORT:Datta Majumdar, Sudhanshu}}
[[Category:Articles created via the Article Wizard]]
[[Category:1915 births]]
[[Category:1997 deaths]]
[[Category:Indian physicists]]
[[Category:People from Kolkata]]
[[Category:Bengali people]]
[[Category:University of Calcutta alumni]]
[[Category:Indian Institute of Technology Kharagpur faculty]]
[[Category:People from Sylhet]]
[[Category:Presidency University, Kolkata alumni]]
[[Category:University of Calcutta faculty]]
[[Category:Visva-Bharati University faculty]]
[[Category:Indian academics]]

Main Page

2014-08-18T06:02:33Z

AurelioChill:

'''Pseudomanifold''' is a special type of [[topological space]].
It looks like a manifold at most of the points, but may contain singularities.
For example, the cone of solutions of <math>z^2=x^2+y^2</math> forms a pseudomanifold.

[[File:Pinched torus.jpg|thumb|<center>A pinched torus</center>]]
A pseudomanifold can be considered as a [[combinatorial]] realisation of the general idea of a [[manifold]] with [[Mathematical singularity|singularities]].
The concepts of [[orientability]], orientation and [[degree of a mapping]] make sense for pseudomanifolds and moreover, within the combinatorial approach, pseudomanifolds form the natural domain of definition for these concepts.<ref>{{Citation|first=H.|last=Steifert|first2=W.|last2=Threlfall|title=Textbook of Topology|publisher=Academic Press Inc.|year=1980|ISBN=0-12-634850-2}}</ref><ref>{{Citation|first=H.|last=Spanier|title=Algebraic Topology|publisher=McGraw-Hill Education|year=1966|ISBN=0-07-059883-5}}</ref>

== Definition ==

A topological space ''X'' endowed with a [[Triangulation (topology)|triangulation]] ''K'' is an ''n''-dimensional pseudomanifold if the following conditions hold:<ref name="BRAS">{{cite journal |last1=Brasselet|first1=J. P.|year=1996 |title=Intersection of Algebraic Cycles |journal= Journal of Mathematical Sciences|publisher=Springer New York|volume= 82|issue= 5|pages=3625 − 3632|url=http://www.springerlink.com/content/ju28j2wqm174hx10}}</ref>

# {{nowrap|1=''X'' = {{!}}''K''{{!}}}} is the [[union (set theory)|union]] of all ''n''-[[simplex|simplices]].
# Every {{nowrap|1=(''n'' – 1)-simplex}} is a [[Euclidean_simplex#Faces|face]] of exactly two ''n''-simplices for ''n > 1''.
# For every pair of ''n''-simplices σ and σ' in ''K'', there is a [[sequence]] of ''n''-simplices {{nowrap|1=σ = σ0, σ1, …, σ''k'' = σ'}} such that the [[intersection (mathematics)|intersection]] {{nowrap|1=σ''i'' ∩ σ''i''+1}} is an {{nowrap|1=(''n'' − 1)-simplex}} for all ''i''.

=== Implications of the definition ===

*Condition 2 means that ''X'' is a '''non-branching''' [[simplicial complex]].<ref name="ANO">{{citeweb|url=http://eom.springer.de/p/p075720.htm|author=D. V. Anosov|title=Pseudo-manifold|accessdate=August 6, 2010}}</ref>
*Condition 3 means that ''X'' is a '''strongly connected''' simplicial complex.<ref name="ANO"/>

== Examples ==
*A [[Pinched Torus|pinched torus]] (see figure) is an example of an [[orientable surface|orientable]], [[compact surface|compact]] 2-dimensional pseudomanifold.<ref name="BRAS"/>

* Complex [[algebraic varieties]] (even with singularities) are examples of pseudomanifolds.<ref name="ANO"/>

* [[Thom space]]s of [[vector bundle]]s over triangulable [[compact manifold]]s are examples of pseudomanifolds.<ref name="ANO"/>

* Triangulable, [[compact space|compact]], [[connected space|connected]], [[homology manifold]]s over '''Z''' are examples of pseudomanifolds.<ref name="ANO"/>

== References ==

{{Reflist}}

[[Category:Topological spaces]]