formulasearchengine - User contributions [en]

Representation theory of Hopf algebras

2014-09-19T10:59:22Z

131.174.90.41:

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Voltage multiplier

2014-02-04T10:19:55Z

131.174.28.142: /* Operation */

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Continuous stirred-tank reactor

2014-01-29T14:09:39Z

131.174.15.128:

{{cleanup|date=January 2009}}

'''Hidden Fields Equations (HFE)''' is a [[public key]] [[cryptosystem]] which was introduced at [[Eurocrypt]] in 1996 and proposed by {{fr icon}} [[:fr:Jacques Patarin|Jacques Patarin]] following the idea of the [[Matsumoto]] and [[Imai]] system. '''HFE''' is also known as HFE trapdoor function. It is based on [[polynomials]] over [[finite fields]] <math>\mathbb{F}_q </math> of different size to disguise the relationship between the [[private key]] and [[public key]]. '''HFE''' is in fact a family which consists of basic '''HFE''' and combinatorial versions of '''HFE'''. The HFE family of cryptosystems is based on the hardness of the problem of finding solutions to a system of multivariate [[quadratic equations]] (the so-called MQ problem) since it uses private [[affine transformations]] to hide the extension field and the private [[polynomials]]. Hidden Field Equations also have been used to construct digital signature schemes, e.g. Quartz and Sflash.<ref name="autogenerated2">[http://eprint.iacr.org/2001/029.pdf Christopher Wolf and Bart Preneel, Asymmetric Cryptography: Hidden Field Equations]</ref>

== Mathematical background ==
One of the central notions to understand how Hidden Field Equations work is to see that for two extension fields <math>\mathbb{F}_{q^n} </math> <math>\mathbb{F}_{q^m} </math> over the same base field <math>\mathbb{F}_q</math> one can interpret a system of <math>m</math> multivariate [[polynomials]] in <math>n</math> variables over <math>\mathbb{F}_q</math> as a function <math>\mathbb{F}_{q^n} \to \mathbb{F}_{q^m} </math> by using a suitable [[basis (linear algebra)|basis]] of <math>\mathbb{F}_{q^n} </math> over <math>\mathbb{F}_q</math>. In almost all applications the polynomials are quadratic, i.e. they have degree 2.<ref name="autogenerated1">[http://eprint.iacr.org/2001/029.pdf Nicolas T. Courtois On Multivariate Signature-only public key cryptosystems]</ref> We start with the simplest kind of polynomials, namely monomials, and show how they lead to quadratic systems of equations.

Let us consider a [[finite fields|finite field]] <math> \mathbb{F}_q</math>, where <math> q </math> is a power of 2, and an extension field <math>K</math>. Let <math>\beta_1,...,\beta_n</math> to be a [[basis (linear algebra)|basis]] of <math> K </math> as an <math>\mathbb{F}_q </math> [[vector space]]. Let <math> 0<h<q^n </math> such that <math> h=q^{\theta}+1 </math> for some <math>\theta</math> and [[Greatest common divisor of two polynomials|gcd]]<math> (h,q^n-1)=1 </math> and take a random element <math> u\in \mathbb{F}_{q^n}</math>. We represent <math>u</math> with respect to the basis as <math>u=(u_1,...,u_n)</math>. Define <math>v\in \mathbb{F}_{q^n}</math> by

:<math> v=u^{q^\theta} u \ \ \ \ (1)</math>

The condition [[Greatest common divisor of two polynomials|gcd]]<math> (h,q^n-1) =1 </math> is equivalent to requiring that the map <math> u \to u^h </math> on <math> K </math> is one to one and its inverse is the map <math> u \to u^{h'} </math> where <math> h' </math> is the multiplicative inverse of <math> h \ \bmod q^n-1 </math>. Choose two secret affine transformation, i.e. two invertible <math>n\times n</math> matrices <math> S=\{S_{ij}\} </math> and <math> T=\{T_{ij}\} </math> with entries in <math>\mathbb{F}_q </math> and two vectors <math> c=(c_1,...,c_n) </math> and <math> d=(d_1,...,d_n) </math> of length <math>n</math> over <math>\mathbb{F}_q </math> and define <math>x</math> and <math>y</math> via:

:<math> u=Sx+c \ \ \ \ v=Ty+d \ \ \ \ (2) </math>

Let <math> A^{(k)}={a_{ij}^{(k)}} </math> be the matrix of linear transformation in the basis <math>\beta_1,...,\beta_n</math> such that

:<math> \beta_{i}^{q^k}=\sum_{j=1}^{n} a_{ij}^{k}\beta_{j},\ \ a_{ij}^{k}\in\mathbb{F}_q</math>

for <math> 1\le i,k\le n </math>. Write all products of basis elements in terms of the basis, i.e.:

:<math> \beta_i\beta_j=\sum_{l=1}^{n}m_{ijl}\beta_{l},\ \ m_{ijl}\in\mathbb{F}_q</math>

for each <math> 1\le i,j\le n </math>. The system of <math> n </math> equations which is explicit in the <math> v_i </math> and quadratic in the <math> u_j </math> can be obtain by expanding (1) and equating to zero the coefficients of the <math> \beta_i </math>. By using the affine relations in (2) to replace the <math> u_j, v_i </math> with <math> x_k,y_l </math>, the system of <math> n </math> equations is [[linear]] in the <math> y_l </math> and of degree 2 in the <math> x_k </math>. Applying [[linear algebra]] it will give <math> n </math> explicit equations, one for each <math> y_l</math> as polynomials of degree 2 in the <math> x_k </math>.<ref name="autogenerated4">[http://eprint.iacr.org/2003/061.pdf Ilia Toli Hidden Polynomial Cryptosystems]</ref>

== Multivariate cryptosystem ==

The basic idea of the HFE family of using this as a multivariate [[cryptosystem]] is to build the secret key starting from a [[polynomial]] <math> P </math> in one unknown <math> x </math> over some [[finite field]] <math>\mathbb{F}_{q^n} </math> (normally value <math> q=2 </math> is used). This [[polynomial]] can be easily inverted over <math>\mathbb{F}_{q^n} </math>, i.e. it is feasible to find any solutions to the equation <math> P(x)=y </math> when such solution exist. The secret transformation either [[decryption]] and/or [[Digital signature|signature]] is based on this inversion. As explained above <math>P</math> can be identified with a system of <math>n</math> equations <math> (p_1,...,p_n) </math> using a fixed basis. To build a [[cryptosystem]] the [[polynomial]] <math> (p_1,...,p_n) </math> must be transformed so that the public information hides the original structure and prevents inversion. This is done by viewing the [[finite fields]] <math>\mathbb{F}_{q^n} </math> as a [[vector space]] over <math>\mathbb{F}_q </math> and by choosing two linear [[affine transformation]]s <math> S </math> and <math> T </math>. The triplet <math> (S,P,T) </math> constitute the private key. The private [[polynomial]] <math> P </math> is defined over <math>\mathbb{F}_{q^n} </math>.<ref name="autogenerated2"/><ref name="autogenerated3">[http://www.ssi.gouv.fr/fr/sciences/fichiers/lcr/fajo03.pdf Jean Charles Faugere and Antoine Joux, Algebraic Cryptanalysis of Hidden Field Equations (HFE) Cryptosystems Using Grobner Bases]</ref> The public key is <math> (p_1,...,p_n) </math>. Below is the diagram for MQ-trapdoor <math> (S,P,T) </math> in HFE
:<math>\text{input} x\to x=(x_1,...,x_n)\overset{\text{secret}: S}{\to}x'\overset{\text{secret}: P}{\to}y'\overset{\text{secret}: T}{\to}\text{output} y</math>

== HFE polynomial ==

The private [[polynomial]] <math> P </math> with degree <math> d </math> over <math> \mathbb{F}_{q^n} </math> is an element of <math> \mathbb{F}_{q^n}[x] </math>. If the terms of [[polynomial]] <math> P </math> have at most [[quadratic polynomial|quadratic]] terms over <math> \mathbb{F}_{q} </math> then it will keep the public polynomial small.<ref name="autogenerated2"/> The case that <math>P</math> consists of monomials of the form <math> x^{q^{s_i}+q^{t_i}}</math>, i.e. with 2 powers of <math>q</math> in the exponent
is the basic version of '''HFE''', i.e. <math> P </math> is chosen as

:<math> P(x)=\sum c_i x^{q^{s_i}+q^{t_i}} </math>

The degree <math> d </math> of the [[polynomial]] is also known as security parameter and the bigger its value the better for security since the resulting set of quadratic equations resembles a randomly chosen set of quadratic equations. On the other side large <math>d</math> slows down the deciphering. Since <math> P </math> is a [[polynomial]] of degree at most <math> d </math> the inverse of <math> P </math>, denoted by <math> P^{-1} </math> can be computed in <math> d^2(\ln d)^{O(1)} n^2 \mathbb{F}_q </math> operations.<ref>Nicolas T. Courtois, "The Security of Hidden Field Equations"</ref>

== Encryption and decryption ==
The public key is given by the <math>n</math> multivariate polynomials <math> (p_1,...,p_n) </math> over <math>\mathbb{F}_q</math>. It is thus necessary to transfer the message <math> M </math> from <math> \mathbb{F}_{q^n} \to \mathbb{F}_q^n </math> in order to encrypt it, i.e. we assume that <math> M </math> is a vector <math> (x_1,...,x_n)\in \mathbb{F}_q^n </math>. To encrypt message <math> M </math> we evaluate each <math>p_i</math> at <math>(x_1,...,x_n)</math>. The ciphertext is <math>(p_1(x_1,...,x_n), p_2(x_1,...,x_n), ... ,p_n(x_1,...,x_n))\in \mathbb{F}_q^n</math>.

To understand decryption let us express encryption in terms of <math> S, T, P </math>. Note that these are ''not'' available to the sender. By evaluating the <math>p_i</math> at the message we first apply <math> S </math>, resulting in <math> x' </math>. At this point <math> x' </math> is transferred from <math> \mathbb{F}{q^n} \to \mathbb{F}_{q^n} </math> so we can apply the private polynomial <math> P </math> which is over <math> \mathbb{F}_{q^n} </math> and this result is denoted by <math> y'\in \mathbb{F}_{q^n} </math>. Once again, <math> y' </math> is transferred to the vector <math> (y_1',...,y_n') </math> and the transformation <math> T </math> is applied and the final output <math> y\in \mathbb{F}_{q^n} </math> is produced from <math> (y_1,...,y_n)\in \mathbb{F}_q^n </math>.

To decrypt <math> y </math>, the above steps are done in reverse order. This is possible if the private key <math> (S,P,T) </math> is known. The crucial step in the deciphering is not the inversion of <math> S </math> and <math> T </math> but rather the computations of the solution of <math> P(x')=y' </math>. Since <math> P </math> is not necessary a bijection, one may find more than one solution to this inversion (there exist at most d different solutions <math> X'=(x_1',...,x_d')\in\mathbb{F}_{q^n} </math> since <math> P </math> is a polynomial of degree d). The redundancy denoted as <math> r </math> is added at the first step to the message <math> M </math> in order to select the right <math> M </math> from the set of solutions <math> X'</math>.<ref name="autogenerated2"/><ref name="autogenerated4"/><ref>[http://www.cryptosystem.net/hfe.pdf Jacques Patarin, Hidden Field Equations (HFE) and Isomorphic Polynomial (IP): two new families of asymmetric algorithm]</ref> The diagram below shows the basic HFE for encryption.
:<math>M\overset{+r}{\to}x\overset{\text{secret}: S}{\to}x'\overset{\text{secret}: P}{\to}y'\overset{\text{secret}: T}{\to}y</math>

== HFE variations ==

Hidden Field Equations has four basic variations namely '''+,-,v and f''' and it is possible to combine them in various way. The basic principle is the following:

:01. The '''+''' sign consists of linearity mixing of the public equations with some random equations.
:02. The '''-''' sign is due to Adi Shamir and intends to remove the redundancy 'r' of the public equations.
:03. The '''f''' sign consists of fixing some <math> f </math> input variables of the public key.
:04. The '''v''' sign is defined as a construction and sometimes quite complex such that the inverse of the function can be found only if some v of the variables called vinegar variables are fixed. This idea is due to Jacques Patarin.

The operations above preserve to some extent the trapdoor solvability of the function.

HFE- and HFEv are very useful in signature schemes as they prevent from slowing down the signature generation and also enhance the overall security of HFE whereas for [[encryption]] both HFE- and HFEv will lead to a rather slow [[decryption]] process so neither too many equations can be removed (HFE-) nor too many variables should be added (HFEv). Both HFE- and HFEv were used to obtain Quartz.

For encryption, the situation is better with HFE+ since the [[decryption]] process takes the same amount of time, however the public key has more equations than variables.<ref name="autogenerated2"/><ref name="autogenerated1"/>

== HFE attacks ==

There are two famous recent attacks on HFE:

01. Shamir-Kipnis: Recover the Private Key.

The key point of this attack is to recover the private key as sparse univariate polynomials over the extension field <math> \mathbb{F}_{q^n} </math>. The attack only works for basic HFE and fails for all its variations.

02. Faugere: Fast Gröbner Bases.

The idea of Faugere's attacks is to use fast algorithm to compute a [[Gröbner basis]] of the system of polynomial equations. Faugere broke the HFE challenge 1 in 96 hours in 2002 and in 2003 Faugere and Joux worked together on the security of HFE.<ref name="autogenerated2"/>

== References ==
{{Reflist}}
* [http://eprint.iacr.org/2002/138 Nicolas T. Courtouis, Magnus Daum and Patrick Felke, On the Security of HFE, HFEv- and Quartz]
* [http://www.win.tue.nl/~asidoren/HFE2004.pdf Andrey Sidorenko, Hidden Field Equations, EIDMA Seminar 2004 Technische Universiteit Eindhoven]
* Yvo G. Desmet, Public Key Cryptography-PKC 2003, ISBN 3-540-00324-X

==External links==
* [http://www.minrank.org/hfe/ Nicolas Courtois HFE page]
{{Cryptography navbox | public-key}}

[[Category:Public-key encryption schemes]]
[[Category:Finite fields]]
[[Category:Multivariate cryptography]]

Crystalline cohomology

2013-12-04T13:45:02Z

131.174.22.156: /* Crystalline cohomology */ Made link to Witt vectors

{{wiktionary|$|⠫|＄}}
'''$''' is the [[dollar sign|dollar]] or [[Dollar sign|peso]] currency sign, primarily used to represent currencies.

'''$''' may also refer to:

{{tocright}}

==Currency==
*[[Dollar]], used in many countries
*[[Peso]], used in many countries
*[[Brazilian real]], the currency of Brazil
*[[Nicaraguan córdoba]], the currency of Nicaragua
*[[Tongan paʻanga]], the currency of the Tonga
*[[Cape Verdean escudo]], the currency of Cape Verde
*[[Portuguese escudo]] (defunct), the currency of Portugal prior to the introduction of the Euro
*<math>\mathrm{S}\!\!\!\Vert</math>, the [[Cifrão]], a similar symbol

==Mathematics and computers==
* $, a [[sigil (computer programming)]] in computer programming
* A factory method for the JavaScript library [[JQuery#Usage_styles|JQuery]]
* A notation for the [[Superfactorial]] (''n$'')

==Music==
*[[$ (Mark Sultan album)|''$'' (Mark Sultan album)]], the second solo studio album by garage rock/doo-wop musician Mark Sultan
*[[Dollar$ (soundtrack)|''Dollar$'' (soundtrack)]], the soundtrack album to the 1971 Richard Brooks movie of the same name

==Television and film==
*''[[$h*! My Dad Says]]'', a CBS television comedy
*[[$ (film)|''$'' (film)]], also known as ''Dollars''

==See also==
*[[Dollar (disambiguation)]]
*[[Ֆ]]

{{dollar|state=expanded}}
{{peso|state=expanded}}

{{disambiguation}}

Universal composability

2013-08-27T11:44:24Z

131.174.142.205:

'''Absolute electrode potential''', in [[electrochemistry]], according to an [[IUPAC]] definition,<ref>[http://goldbook.iupac.org/A00022.html IUPAC Gold Book - absolute electrode potential]</ref> is the [[electrode potential]] of a [[metal]] measured with respect to a universal reference system (without any additional metal–solution interface).

==Definition==
According to a more specific definition presented by Trasatti,<ref> Sergio Trasatti, "The Absolute Electrode Potential: an Explanatory Note (Recommendations 1986)", International Union of Pure and Applied Chemistry,
Pure & AppL Chem., Vol. 58, No. 7, pp. 955–66, 1986. http://www.iupac.org/publications/pac/1986/pdf/5807x0955.pdf (pdf)</ref> the absolute electrode potential is the difference in electronic energy between a point inside the metal ([[Fermi level]]) of an [[electrode]] and a point outside the [[electrolyte]] in which the electrode is submerged (an electron at rest in vacuum).

This potential is difficult to determine accurately. For this reason, [[standard hydrogen electrode]] is typically used for reference potential. The absolute potential of the SHE is 4.44 ± 0.02 [[volt|V]] at 25 [[°C]]. Therefore, for any electrode at 25 °C:

:<math>E^M_{\rm{(abs)}} = E^M_{\rm{(SHE)}}+(4.44 \pm 0.02)\ {\mathrm V}</math>

where:
:{{mvar|E}} is electrode potential
:V is [[volt]]
:''M'' denotes the electrode made of metal M
:(abs) denotes the absolute potential
:(SHE) denotes the electrode potential relative to the standard hydrogen electrode.

A different definition for the absolute electrode potential (also known as absolute half-cell potential and single electrode potential) has also been discussed in the literature.<ref> Alan L. Rockwood, "Absolute half-cell thermodynamics: Electrode potential", Physical Review A, Vol 33, No. 1, pp. 554–59, 1986.</ref> In this approach, one first defines an isothermal absolute single-electrode process (or absolute half-cell process.) For example, in the case of a generic metal being oxidized to form a solution-phase ion, the process would be

:M(metal) → M+(solution) + {{subatomic particle|electron|link=yes}}(gas)

For the [[hydrogen]] electrode, the absolute half-cell process would be

:{{sfrac|1|2}}H2 (gas) → [[hydron (chemistry)|H+]](solution) + {{subatomic particle|electron}}(gas)

Other types of absolute electrode reactions would be defined analogously.

In this approach, all three species taking part in the reaction, including the electron, must be placed in thermodynamically well-defined states. All species, including the electron, are at the same temperature, and appropriate standard states for all species, including the electron, must be fully defined. The absolute electrode potential is then defined as the Gibbs free energy for the absolute electrode process. To express this in volts one divides the Gibb’s free energy by the negative of Faraday’s constant.

Rockwood's approach to absolute-electrode thermodynamics is easily expendable to other thermodynamic functions. For example, the absolute half-cell entropy has been defined as the entropy of the absolute half-cell process defined above.<ref>Alan L. Rockwood, "Absolute half-cell entropy", Physical Review A, vol. 36, No. 3, pp. 1525–26, 1987.</ref> An alternative definition of the absolute half-cell entropy has recently been published by Fang et al.<ref>Zheng Fang, Shaofen Wang, Zhenghua Zhang, and Guanzhou Qiu, "The electrochemical Peltier heat of the standard hydrogen electrode reaction", Thermochimica Acta, Vol. 473, pp. 40–44, 2008.</ref> who define it as the entropy of the following reaction (using the hydrogen electrode as an example):

:{{sfrac|1|2}}H2 (gas) → H+(solution) + {{subatomic particle|electron}}(metal)

This approach differs from the approach described by Rockwood in the treatment of the electron, i.e. whether it is placed in the gas phase or in the metal.

==Determination==
The basis for determination of the absolute electrode potential under the Trasatti definition is given by the equation:

:<math>E^M{\rm (abs)} = \phi^M + \Delta ^M_S \psi</math>

where:
:{{math|''E''''M''(abs)}} is the absolute potential of the electrode made of metal M
:<math>\phi^M</math> is the electron [[work function]] of metal M
:<math>\Delta ^M_S \psi</math> is the [[Volta potential|contact (Volta) potential]] difference at the metal(''M'')–solution(''S'') interface.

For practical purposes, the value of the absolute electrode potential of the standard hydrogen electrode is best determined with the utility of data for an [[ideally polarizable electrode|ideally-polarizable]] [[mercury (element)|mercury]] (Hg) electrode:

:<math>E^\ominus {\rm (H^+/H_2)(abs)} = \phi^{\rm{Hg}} + \Delta ^{\rm{Hg}} _S \psi^\ominus_{\sigma=0} - E^{\rm{Hg}}_{\sigma=0}\rm{(SHE)}</math>

where:
:<math>E^\ominus {\rm (H^+/H_2)(abs)}</math> is the absolute standard potential of the hydrogen electrode
:{{math|1=''σ'' = 0}} denotes the condition of the [[point of zero charge]] at the interface.

The types of physical measurements required under the Rockwood definition are similar to those required under the Trasatti definition, but they are used in a different way, e.g. in Rockwood's approach they are used to calculate the equilibrium vapor pressure of the electron gas. The numerical value for the absolute potential of the standard hydrogen electrode one would calculate under the Rockwood definition is sometimes fortuitously close to the value one would obtain under the Trasatti definition. This near-agreement in the numerical value depends on the choice of ambient temperature and standard states, and is the result of the near-cancellation of certain terms in the expressions. For example, if a standard state of one atmosphere ideal gas is chosen for the electron gas then the cancellation of terms occurs at a temperature of 296 K, and the two definitions give an equal numerical result. At 298.15 K a near-cancellation of terms would apply and the two approaches would produce nearly the same numerical values. However, there is no fundamental significance to this near agreement because it depends on arbitrary choices, such as temperature and definitions of standard states.

==See also==
* [[Standard electrode potential]]
* [[Galvani potential]]
* [[Electrochemical potential]]

==References==
{{Reflist}}

[[Category:Electrochemistry]]
[[Category:Potentials]]

AW*-algebra

2013-05-23T15:23:18Z

131.174.142.211: fixed incorrect statement to a correct one

In [[population ecology]], '''Moran's theorem''' (or the Moran effect) states that the time [[correlation]] of two separate populations of the same species is equal to the correlation between the environmental variabilities where they live.

The theorem is named after [[Pat Moran (statistician)|Pat Moran]], who stated it in a paper on the dynamics of the [[Canadian lynx]] populations.<ref>Moran, P. A. P. 1953. The statistical analysis of the Canadian lynx cycle. II. Synchronization and meteorology. Australian Journal of Zoology 1: 291-298.</ref> It has been used to explain the synchronization of widely dispersed populations. It has the important consequence for [[conservation biology|conservation ecology]] that [[Minimum viable population|viability]] of spatially structured populations is lower than one would expect from the local populations: it increases the probability that several local populations go extinct simultaneously.<ref>Jörgen Ripa, Theoretical Population Ecology and Evolution Group, [http://equation-of-the-month.blogspot.co.uk/2012/02/moran-effect.html Equation of the month: the Moran effect]</ref>

In its original form it stated: If the two populations have population dynamics given by
:<math>N_1(t+1)=f(N_1(t))+\epsilon_1(t)</math>
:<math>N_2(t+1)=f(N_2(t))+\epsilon_2(t)</math>
where <math>N_i</math> is the population size of population <math>i</math>, <math>f</math> is a linear renewal function updating the populations in the same way, and <math>\epsilon_i</math> the environmental variabilities. Then <math>\rho_{N_1,N_2}=\rho_{\epsilon_1,\epsilon_2}</math>.

The original form assumed a strictly linear structure, but this assumption can be weakened to allow for non-linear functions. It has been suggested that the term "Moran effect" should be used for systems that do not strictly follow the original description.<ref>Esa Ranta, Veijo Kaitala, Per Lundberg, Ecology of Populations, Cambridge University Press, 2006 p. 78</ref> In the general case the correlations will be lower, and the accuracy of the Moran description depends on whether the populations tend to converge to an equilibrium state (good accuracy for low variance variability) or tend to oscillate (eventual breakdown of the correlation).<ref>T. Royama 2005. Moran effect on nonlinear population processes. Ecological Monographs 75:277–293. http://dx.doi.org/10.1890/04-0770</ref>

It has been tested experimentally in a number of cases, such as variation of fruit production,<ref>Rosenstock, T. S., Hastings, A., Koenig, W. D., Lyles, D. J. and Brown, P. H. (2011), Testing Moran's theorem in an agroecosystem. Oikos, 120: 1434–1440. doi: 10.1111/j.1600-0706.2011.19360.x</ref> acorn production,<ref>Ecology. 2013 Jan;94(1):83-93.
Large-scale spatial synchrony and cross-synchrony in acorn production by two California oaks.
Koenig WD, Knops JM.</ref> bird populations<ref>SÆTHER, B.-E., ENGEN, S., GRØTAN, V., FIEDLER, W., MATTHYSEN, E., VISSER, M. E., WRIGHT, J., MØLLER, A. P., ADRIAENSEN, F., VAN BALEN, H., BALMER, D., MAINWARING, M. C., MCCLEERY, R. H., PAMPUS, M. and WINKEL, W. (2007), The extended Moran effect and large-scale synchronous fluctuations in the size of great tit and blue tit populations. Journal of Animal Ecology, 76: 315–325. doi: 10.1111/j.1365-2656.2006.01195.x</ref> and coral reef fishes.<ref>Ecology. 2007 Jan;88(1):158-69. Spatial synchrony in coral reef fish populations and the influence of climate. Cheal AJ, Delean S, Sweatman H, Thompson AA.</ref>

==References==
{{reflist}}

[[Category:Population ecology]]
[[Category:Demography]]

Tightness of measures

2013-05-21T13:27:49Z

131.174.142.211: /* Compact spaces */

'''AMS-LaTeX''' is a collection of
[[LaTeX]] document classes and packages developed for the [[American Mathematical Society]] (AMS). Its additions to LaTeX include the typesetting of multi-line and other mathematical statements, document classes, and fonts containing numerous mathematical symbols.<ref>{{cite book
| url = http://www.ctan.org/tex-archive/info/mil/mil.pdf
| title = Math into LaTeX
| author = George Gratzer
| year = 1996
| isbn = 0-8176-3805-9
| accessdate = 2007-10-08
}}</ref>

It has largely superseded the
plain [[TeX]] macro package
'''AMS-TeX'''. AMS-TeX was originally written by [[Michael Spivak]], and was used by the AMS from 1983 to 1985.

The following code of the
LaTeX2e produces the AMS-LaTeX logo ([[Image:AMS-LaTeX.svg|80px|\AmS-\LaTeX]]):

<source lang=latex>
%%% -- AMS-LaTeX_logo.tex -------
\documentclass{article}
\usepackage{amsmath}

\begin{document}
\AmS-\LaTeX
\end{document}
</source>

The package has a suite of facilities to format multi-line equations. For example, the following
code,

<source lang=latex>
\begin{align}
y &= (x+1)^2 \\
&= x^2+2x+1
\end{align}
</source>

causes the equals signs in the two lines to be aligned with one another, like this:

:<math>
\begin{align}
y &= (x+1)^2 \\
&= x^2+2x+1
\end{align}
</math>

AMS-LaTeX also includes many flexible commands for formatting and numbering theorems, lemmas, etc. For example, one may use the environment <tt>theorem</tt>

<source lang=latex>
\begin{theorem}[Pythagoras] Suppose $a\leq b\leq c$ are the side-lengths of a right triangle.\\ Then $a^2+b^2=c^2$.\end{theorem}
\begin{proof}. . . \end{proof}
</source>

to generate

<blockquote>
'''Theorem''' (''Pythagoras'') ''Suppose'' <math>a\leq b\leq c</math> ''are the side-lengths of a right triangle. '' ''Then'' <math>a^2+b^2=c^2</math>. 
'''Proof'''. . . □
</blockquote>

==See also==
* [[AMSRefs]]

==References==
<references />

==External links==
*[http://www.ams.org/tex/ AMS TeX Resources]
*[http://www.tex.ac.uk/cgi-bin/texfaq2html?label=AMSpkg TeX FAQ on AMS packages]
*[http://www.tex.ac.uk/cgi-bin/texfaq2html?label=AMSpkg TeX FAQ on AMS-TeX]

[[Category:TeX]]

{{TeX navbox}}
{{LaTeX navbox}}
{{compu-library-stub}}
{{digital-typography-stub}}

Rank product

2013-03-04T10:44:45Z

131.174.204.233:

{{otheruses4|exact categories in the sense of Quillen|exact categories in the sense of Barr|regular category|exact categories in the sense of Buchsbaum|abelian category}}
In [[mathematics]], an '''exact category''' is a concept of [[category theory]] due to [[Daniel Quillen]] which is designed to encapsulate the properties of [[short exact sequence]]s in [[abelian category|abelian categories]] without requiring that morphisms actually possess [[kernel (category theory)|kernels and cokernels]], which is necessary for the usual definition of such a sequence.

==Definition==
An exact category '''E''' is an [[additive category]] possessing a [[class (set theory)|class]] ''E'' of "short exact sequences": triples of objects connected by arrows
: <math>M' \to M \to M''\ </math>
satisfying the following axioms inspired by the properties of [[short exact sequence]]s in an [[abelian category]]:
* ''E'' is closed under isomorphisms and contains the canonical ("split exact") sequences:
::<math> M' \rightarrow M' \oplus M''\rightarrow M'';</math>
* Suppose <math>M \to M''</math> occurs as the second arrow of a sequence in ''E'' (it is an '''admissible epimorphism''') and <math>N \to M''</math> is any arrow in '''E'''. Then their [[pullback (category theory)|pullback]] exists and its projection to <math>N</math> is also an admissible epimorphism. [[Dual (category theory)|Dually]], if <math>M' \to M</math> occurs as the first arrow of a sequence in ''E'' (it is an '''admissible monomorphism''') and <math>M' \to N</math> is any arrow, then their [[pushout (category theory)|pushout]] exists and its coprojection from <math>N</math> is also an admissible monomorphism. (We say that the admissible epimorphisms are "stable under pullback", resp. the admissible monomorphisms are "stable under pushout".);
* Admissible monomorphisms are [[kernel (category theory)|kernel]]s of their corresponding admissible epimorphisms, and dually. The composition of two admissible monomorphisms is admissible (likewise admissible epimorphisms);
* Suppose <math>M \to M''</math> is a map in '''E''' which admits a kernel in '''E''', and suppose <math>N \to M</math> is any map such that the composition <math>N \to M \to M''</math> is an admissible epimorphism. Then so is <math>M \to M''.</math> Dually, if <math>M' \to M</math> admits a cokernel and <math>M \to N</math> is such that <math>M' \to M \to N</math> is an admissible monomorphism, then so is <math>M' \to M.</math>

Admissible monomorphisms are generally denoted <math>\rightarrowtail</math> and admissible epimorphisms are denoted <math>\twoheadrightarrow.</math> These axioms are not minimal; in fact, the last one has been shown by {{Harvard citations|txt=yes|last=Keller|first=Bernhard|year=1990}} to be redundant.

One can speak of an '''exact functor''' between exact categories exactly as in the case of [[exact functor]]s of abelian categories: an exact functor <math>F</math> from an exact category '''D''' to another one '''E''' is an [[additive functor]] such that if
:<math>M' \rightarrowtail M \twoheadrightarrow M''</math>
is exact in '''D''', then
:<math>F(M') \rightarrowtail F(M) \twoheadrightarrow F(M'')</math>
is exact in '''E'''. If '''D''' is a subcategory of '''E''', it is an '''exact subcategory''' if the inclusion functor is fully faithful and exact.

==Motivation==
Exact categories come from abelian categories in the following way. Suppose '''A''' is abelian and let '''E''' be any [[strictly full subcategory|strictly full]] additive subcategory which is closed under taking [[extension (algebra)|extension]]s in the sense that given an exact sequence
:<math>0 \to M' \to M \to M'' \to 0\ </math>
in '''A''', then if <math>M', M''</math> are in '''E''', so is <math>M</math>. We can take the class ''E'' to be simply the sequences in '''E''' which are exact in '''A'''; that is,
:<math>M' \to M \to M''\ </math>
is in ''E'' iff
:<math>0 \to M' \to M \to M'' \to 0\ </math>
is exact in '''A'''. Then '''E''' is an exact category in the above sense. We verify the axioms:
* '''E''' is closed under isomorphisms and contains the split exact sequences: these are true by definition, since in an abelian category, any sequence isomorphic to an exact one is also exact, and since the split sequences are always exact in '''A'''.
* Admissible epimorphisms (respectively, admissible monomorphisms) are stable under pullbacks (resp. pushouts): given an exact sequence of objects in '''E''',
::<math>0 \to M' \xrightarrow{f} M \to M'' \to 0,\ </math>
:and a map <math>N \to M''</math> with <math>N</math> in '''E''', one verifies that the following sequence is also exact; since '''E''' is stable under extensions, this means that <math>M \times_{M''} N</math> is in '''E''':
::<math>0 \to M' \xrightarrow{(f,0)} M \times_{M''} N \to N \to 0.\ </math>
* Every admissible monomorphism is the kernel of its corresponding admissible epimorphism, and vice-versa: this is true as morphisms in '''A''', and '''E''' is a full subcategory.
* If <math>M \to M''</math> admits a kernel in '''E''' and if <math>N \to M</math> is such that <math>N \to M \to M''</math> is an admissible epimorphism, then so is <math>M \to M''</math>: See {{Harvard citations|txt=yes|last=Quillen|year=1972}}.

Conversely, if '''E''' is any exact category, we can take '''A''' to be the category of [[exact functor|left-exact functor]]s from '''E''' into the category of [[abelian group]]s, which is itself abelian and in which '''E''' is a natural subcategory (via the [[Yoneda lemma|Yoneda embedding]], since Hom is left exact), stable under extensions, and in which a sequence is in ''E'' if and only if it is exact in '''A'''.

==Examples==
* Any abelian category is exact in the obvious way, according to the construction of [[#Motivation]].
* A less trivial example is the category '''Ab'''tf of [[torsion-free abelian group]]s, which is a strictly full subcategory of the (abelian) category '''Ab''' of all abelian groups. It is closed under extensions: if
::<math>0 \to A \to B \to C \to 0\ </math>
:is a short exact sequence of abelian groups in which <math>A, C</math> are torsion-free, then <math>B</math> is seen to be torsion-free by the following argument: if <math>b</math> is a torsion element, then its image in <math>C</math> is zero, since <math>C</math> is torsion-free. Thus <math>b</math> lies in the kernel of the map to <math>C</math>, which is <math>A</math>, but that is also torsion-free, so <math>b = 0</math>. By the construction of [[#Motivation]], '''Ab'''tf is an exact category; some examples of exact sequences in it are:
::<math>0 \to \mathbb{Z} \xrightarrow{\left(\begin{smallmatrix} 1 \\ 2 \end{smallmatrix}\right)} \mathbb{Z}^2 \xrightarrow{(-2, 1)} \mathbb{Z} \to 0,</math>
::<math>0 \to \mathbb{Q} \to \mathbb{R} \to \mathbb{R}/\mathbb{Q} \to 0,</math>
::<math>0 \to d\Omega^0(S^1) \to \Omega^1_c(S^1) \to H^1_{\text{dR}}(S^1) \to 0,</math>
:where the last example is inspired by [[de Rham cohomology]] (<math>\Omega^1_c(S^1)</math> and <math>d\Omega^0(S^1)</math> are the [[closed and exact differential forms]] on the [[circle group]]); in particular, it is known that the cohomology group is isomorphic to the real numbers. This category is not abelian.
* The following example is in some sense complementary to the above. Let '''Ab'''t be the category of abelian groups ''with'' torsion (and also the zero group). This is additive and a strictly full subcategory of '''Ab''' again. It is even easier to see that it is stable under extensions: if
::<math>0 \to A \to B \to C \to 0\ </math>
:is an exact sequence in which <math>A, C</math> have torsion, then <math>B</math> naturally has all the torsion elements of <math>A</math>. Thus it is an exact category; some examples of its exact sequences are
::<math>0 \to \mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}/4\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0,</math>
::<math>0 \to \mathbb{Z}/2\mathbb{Z} \xrightarrow{(1,0,0)} (\mathbb{Z}/2\mathbb{Z})^2 \oplus \mathbb{Z} \to (\mathbb{Z}/2\mathbb{Z}) \oplus \mathbb{Z} \to 0,</math>
::<math>0 \to (\mathbb{Z}/2\mathbb{Z}) \oplus \mathbb{Z} \to (\mathbb{Z}/2\mathbb{Z})^2 \oplus \mathbb{Z} \xrightarrow{(0,1,0)} \mathbb{Z}/2\mathbb{Z} \to 0,</math>
:where in the second example, the <math>(1,0,0)</math> means inclusion as the first summand, and in the last example, the <math>(0,1,0)</math> means projection onto the second summand. One interesting feature of this category is that it illustrates that the notion of cohomology does not make sense in general exact categories: for consider the "complex"
::<math>\mathbb{Z}/2\mathbb{Z} \xrightarrow{(1,0,0)} (\mathbb{Z}/2\mathbb{Z})^2 \oplus \mathbb{Z} \xrightarrow{(0,1,0)} \mathbb{Z}/2\mathbb{Z}</math>
:which is obtained by pasting the marked arrows in the last two examples above. The second arrow is an admissible epimorphism, and its kernel is (from the last example), <math>(\mathbb{Z}/2\mathbb{Z}) \oplus \mathbb{Z}</math>. Since the two arrows compose to zero, the first arrow [[mathematical jargon#factor through|factors through]] this kernel, and in fact the factorization is the inclusion as the first summand. Thus the quotient, if it were to exist, would have to be <math>\mathbb{Z}</math>, which is not actually in '''Ab'''t. That is, the cohomology of this complex is undefined.

==References==
* {{cite journal
| last = Keller
| first = Bernhard
| title = Chain complexes and stable categories
| year = 1990
| journal = [[Manuscripta Mathematica]]
| volume = 67
| pages = 379–417
| quote = Appendix A. Exact Categories
| doi = 10.1007/BF02568439
| ref = harv
}}

* {{Cite document
| last = Quillen
| first = Daniel
| authorlink = Daniel Quillen
| chapter = Higher algebraic K-theory: I
| title = Higher K-Theories
| year = 1972
| series = Lecture Notes in Mathematics
| publisher = Springer
| volume = 341
| doi = 10.1007/BFb0067053
| pages = 85–147
| ref = harv
| postscript = 
| isbn = 978-3-540-06434-3
}}

[[Category:Additive categories]]
[[Category:Homological algebra]]

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