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Von Neumann algebra

2015-01-12T06:30:52Z

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== Hollister Store Mailand klatschen und schreien ihren Dank. ==

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== Nike Laufschuhe Damen Test wenn Sie habe genug ==

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Sieve of Eratosthenes

2015-01-11T19:52:04Z

138.246.2.135: /* Algorithm complexity */

== 805 ==

クレセントシティはまた、実質的な利益を、他のクレセントシティは、間違いなく、途方に暮れたままに損失で少しクレセントシティを買うためにこの時間は唯一の状態なかった、そして、最も重要なこと、それは広場を買ったファミリーに追加されるxiaojiaを怒ら市、無力な、心は一部の人の意図である、という考えは失望撤退しなければならなかった [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオ腕時計 gps]。 シティスクエアは、リリース後に販売される、その後、活動に加えて、列の完全なの温床となるよう考えているだろうが、再び恥ずかしさの味を味わって、彼はxiaojiaが今影響があってもこのステップに強いてきた期待していなかった [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html casio 腕時計スタンダード]。 日の返済期間は、まだ完了していない瞬間がやたらと汗をかいて気になる助けることができなかったが、ちょうどプラスの欄の下にBiは、最終的には見知らぬ人が、両側を切望しそこ運ばれた、一緒にバランスを考え出すために追加することができます到着するとしている雄弁全体午後代金引数、最後に見知らぬ人、20万安い価格で、プラスに完成した列は、ほぼその暗い顔、「色」を殺したい、心がいっぱいです
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== ゆっくりと持ち上げ ==

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== 含み笑いは、それは一瞬ためらった面であり、言った ==

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== 彼を止める」 ==

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Multidimensional scaling

2014-12-04T08:54:18Z

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Kubo formula

2013-12-12T14:03:20Z

138.246.2.247:

'''Multicanonical ensemble''' (also called '''multicanonical sampling''' or '''flat histogram''') is a sampling technique that uses the [[Metropolis–Hastings algorithm]] to compute integrals where the integrand has a rough landscape with multiple local minima. It samples states according to the inverse of the [[density of states]],<ref name=Berg/> which has to be known a priori or be computed using other techniques like the [[Wang and Landau algorithm]].<ref name=Landau/> Multicanonical sampling is an important technique for spin systems like the [[Ising model]] or [[spin glass]]es.<ref name=Berg/><ref name=Newmann/><ref name=Dayal/>

==Motivation==

In systems with a large number of degrees of freedom, like spin systems, [[Monte Carlo integration]] is required. In this integration, [[importance sampling]] and in particular the [[Metropolis algorithm]], is a very important technique.<ref name=Newmann/> However, the Metropolis algorithm samples states according to <math>\exp(-\beta E)</math> where beta is the inverse of the temperature. This means that an energy barrier of <math>\Delta E</math> on the energy spectrum is exponentially difficult to overcome.<ref name=Berg/> Systems with multiple local energy minima like the [[Potts model]] become hard to sample as the algorithm gets stuck in the system's local minima.<ref name=Newmann/> This motivates other approaches, namely, other sampling distributions.

==Overview==

Multicanonical ensemble uses the Metropolis–Hastings algorithm with a sampling distribution given by the inverse of the density of states of the system, contrary to the sampling distribution <math>\exp(-\beta E)</math> of the Metropolis algorithm.<ref name=Berg/> With this choice, on average, the number of states sampled at each energy is constant, i.e. it is a simulation with a "flat histogram" on energy. This leads to an algorithm for which the energy barriers are no longer difficult to overcome. Another advantage over the Metropolis algorithm is that the sampling is independent of the temperature of the system, which means that one simulation allows the estimation of thermodynamical variables for all temperatures (thus the name "multicanonical": several temperatures). This is a great improvement in the study of first order [[phase transition]]s.<ref name=Berg/>

The biggest problem in performing a multicanonical ensemble is that the density of states has to be known ''a priori''.<ref name=Landau/><ref name=Newmann/> One important contribution to multicanonical sampling was the [[Wang and Landau algorithm]], which asymptotically converges to a multicanonical ensemble while calculating the density of states during the convergence.<ref name=Landau/>

The multicanonical ensemble is not restricted to physical systems. It can be employed on abstract systems which have a cost function ''F''. By using the density of states with respect to F, the method becomes general for computing higher dimensional integrals or finding local minima.<ref name=Lee/>

==Motivation==

Consider a system and it phase-space <math>\Omega</math> characterized by a configuration <math>\boldsymbol{r}\in \Omega</math> and a "cost" function ''F'' from the system's phase-space to a one dimensional space <math>\Gamma</math>: <math>F(\Omega) = \Gamma = [\Gamma_\min, \Gamma_\max]</math>, the spectrum of ''F''.

{| class="wikitable1" width=300px
|-
|
{{show
|example:
|
The [[Ising model]] with ''N'' sites is an example of such a system; the phase-space is a discrete phase-space defined by all possible configurations of ''N'' spins <math>\boldsymbol{r} = (\sigma_1, \ldots, \sigma_i, \ldots, \sigma_N)</math> where <math>\sigma_i\in\{-1,1\}</math>. The cost function is the [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the system:

: <math>H(\boldsymbol{r}) = - \sum_{\langle i~j\rangle} J_{ij} (1 - \sigma_i \sigma_j),</math>

where <math><i,j></math> is the sum over neighborhoods and <math>J_{ij}</math> is the interaction matrix.

The energy spectrum is <math>\Gamma = [E_\min, E_\max]</math> which, in this case, depends on the particular <math>J_{ij}</math> used. If all <math>J_{ij}</math> are 1 (the ferromagnetic Ising model), <math>E_\min = 0</math> (e.g. all spins are 1.) and <math>E_\max = 2 D N </math> (half spins are up, half spins are down). Also notice that in this system, <math>\Gamma \in \mathbb{Z}</math>

}}
|}

The computation of an average quantity <math>\langle Q\rangle</math> over the phase-space requires the evaluation of an integral:

: <math>\langle Q\rangle = \frac{1}{V}\int_\Omega Q(\boldsymbol{r}) p(\boldsymbol{r}) \,d\boldsymbol{r}</math>

where <math>p(\boldsymbol{r})</math> is weight of each state per volume,

: <math>p(\boldsymbol{r}) = \frac{1}{V}\int_\Omega p(\boldsymbol{r}) \,d\boldsymbol{r}.</math>

The density of states in respect with ''F'' is given by

: <math>\rho(f) = \frac{1}{V}\int_\Omega \delta(f - F(\boldsymbol{r})) \,d\boldsymbol{r}</math>

which means that if both ''Q'' and ''p'' do not depend on the particular state but only on the particular F's value of the state <math>F(\boldsymbol{r}) = F_\boldsymbol{r}</math>,

: <math>Q(\boldsymbol{r}) = Q(F_\boldsymbol{r}), p(\boldsymbol{r}) = p(F_\boldsymbol{r}),</math>

the formula for <math>\langle Q\rangle</math> can be integrated over ''f'' by adding a [[dirac delta function]],

: <math>
\begin{align}
\langle Q\rangle & = \int_{\Gamma_\min}^{\Gamma_\max} \int_\Omega Q(F_\boldsymbol{r}) p(F_\boldsymbol{r}) \delta(f - F_\boldsymbol{r}) \,d\boldsymbol{r} \,df \\
& = \int_{\Gamma_\min}^{\Gamma_\max} Q(f) p(f) \int_\Omega \delta(f - F_\boldsymbol{r}) \,d\boldsymbol{r} \, df \\
& = \int_{\Gamma_\min}^{\Gamma_\max} Q(f) p(f) \rho(f) \, df \\
\end{align}
</math>

i.e. the knowledge of the density of states allows the computation of averages over ''F'' using a one-dimensional integral instead of a multidimensional integral as it is the projection of the number of states on <math>\Omega</math> in <math>\Gamma</math>.

{| class="wikitable1" width=300px
|-
|
{{show
|example:
|
A system in contact with a heat bath at inverse temperature <math>\beta</math> is a clear example for computing this kind of integral. For instance, the mean energy of the system is weighted by the [[Boltzmann factor]]:

:<math>\langle E\rangle = \frac{1}{V}\int_\Omega H(\boldsymbol{r}) \frac{e^{-\beta H(\boldsymbol{r})}}{Z} \, d\boldsymbol{r}</math>

where

: <math>Z = \frac{1}{V}\int_\Omega e^{-\beta H(\boldsymbol{r})} \, d\boldsymbol{r}.</math>

The density of states <math>\rho(E)</math> can used to compute <math>\langle E\rangle</math>:

: <math>\langle E\rangle = \int_{E_\min}^{E_\max} E \frac{e^{-\beta E}}{Z} \rho(E) \, dE </math>

}}
|}

Because the number of states can be high for systems with high number of degrees of freedom, an analytical expression can be hard to obtain and the computation of <math>\langle Q\rangle</math> can be expensive. Typically, because the problem is a multidimensional integral, [[Monte Carlo integration]] is normally employed. On the simplest formulation, the method chooses ''N'' uniform random states <math>\boldsymbol{r}_i\in \Omega</math>, and uses the [[estimator]]

:<math>\overline{Q}_N = \frac{1}{N}\sum_{i = 0}^N Q(\boldsymbol{r}_i) p(\boldsymbol{r}_i)</math>

for computing <math>\langle Q\rangle</math>: <math>\overline{Q}_N</math> converges almost surely to <math>\langle Q\rangle</math> by the [[Law_of_large_numbers#Strong_law|strong law of large numbers]]:

:<math>\lim_{N\rightarrow\infty} \overline{Q}_N = \langle Q\rangle.</math>

One typical problem of this convergence is that the variance of ''Q'' can be very high, which leads to a high computational effort to achieve reasonable results.

{| class="wikitable1" width=300px
|-
|
{{show
|example
|
On the previous example, the states that mostly contribute to the integral are the ones with low energy. If the states are sampled uniformly, on average, the number of states which are sampled with energy ''E'' is given by the density of states. This density of states can be centered far away from the energy's minima and thus the average can be difficult to obtain.

}}
|}

To improve this convergence, the [[Metropolis–Hastings algorithm]] was proposed. Generally, Monte Carlo methods' idea is to use [[importance sampling]] to improve the convergence of the estimator <math>\overline{Q}_N</math> by sampling states according to an arbitrary distribution <math>P(\boldsymbol{r})</math> (notice the capital ''P'', different from ''p''), and use the appropriate estimator:

: <math>\overline{Q}_N = \frac{1}{X}\sum_{i = 0}^N Q(\boldsymbol{r}_i) P^{-1}(\boldsymbol{r}_i) p(\boldsymbol{r}_i)</math>

where <math>X = \sum_{i = 0}^N P^{-1}(\boldsymbol{r}_i)</math>.

Notice that when ''P'' is a uniform distribution, this estimator equals the one used on a uniform sampling, as it should.

One important choice of ''P'' is to define an arbitrary temperature, and use it equals to [[Boltzmann factor]] with the energy being the cost function:

: <math>P(\boldsymbol{r}) = p_\mathrm{Boltzmann}(\boldsymbol{r}) = \frac{e^{-\beta F(\boldsymbol{r})}}{\int_\Omega \, d\boldsymbol{r} e^{-\beta F(\boldsymbol{r})}}</math>

I.e. the lower the cost function of a particular state, the more likely it is to be sampled.

Historically, this occurred because the [[Equation of State Calculations by Fast Computing Machines|original idea]]<ref name=Metropolis/> was exactly to use [[Metropolis–Hastings algorithm]] to compute averages on a system in contact with a heat bath where the weight is given by the Boltzmann factor. On these systems, the choice of states according to it led to a considerable improvement on studying physical systems.<ref name=Newmann/>

However, it is not true that the sampling distribution must equals the weight distribution. One reason for this is that [[Metropolis–Hastings algorithm]] fails to converge when the cost function has multiple minima.<ref name=Berg/> The reason is that the algorithm uses a random walk with local steps. I.e. the random walk normally performs steps whose energy difference is of order 1. This means that the computational cost to the algorithm leave a specific region with a local minimum exponentially increases with the cost function's value of the minimum. I.e. the deeper the minimum, the more time the algorithm spends there, and harder (exponentially with deep) it will leave.

This is the motivation to introduce a multicanonic ensemble. The idea is to avoid becoming stuck on local minima of the cost function by making them "invisible" to the sampling technique.

==Multicanonic ensemble==

A multicanonic ensemble is choosing the sampling distribution used in the importance sampling to be

: <math>P(\boldsymbol{r}) = \frac{1}{\rho(F(\boldsymbol{r}))}</math>

where

: <math> \rho(f) = \frac{1}{V}\int_\Omega \delta(F(\boldsymbol{r}) - f) \, d\boldsymbol{r}</math>

is the density of states of the system with respect to the cost function (''V'' is the phase-space volume). The consequence of this choice is that the probability to sample a state and it has cost function ''f'' is

: <math> P(f) = \frac{1}{f_\max - f_\min}\int_\Omega \delta(f - F(\boldsymbol{r})) P(\boldsymbol{r})\,d\boldsymbol{r} = \frac{1}{f_\max - f_\min}\frac{1}{V}\rho(f) \int_\Omega \delta(f - F(\boldsymbol{r})) \, d\boldsymbol{r} = \frac{1}{f_\max - f_\min} = \text{constant}</math>

which motivates the name "flat histogram". I.e. all costs are equally sampled, and thus there are no barriers. For systems in contact with a heat bath, there is another important advantage: because the sampling is independent of the temperature, one simulation is enough to study all temperatures (thus the name "multicanonic": several temperatures).

{| class="wikitable1" width=300px
|-
|
{{show
|example:
|
On the ferromagnetic [[Ising model]] with ''N'' sites (exemplified on previous section), the density of states can be analytically computed. In this case, a multicanonic ensemble can be used to compute any other quantity ''Q'' by sampling the system according to <math>P(\boldsymbol{r})</math> and using the proper estimator <math>\overline{Q}</math> defined on the previous section.

}}
|}

==Tunneling time and critical slowing down==

Like in any other Monte Carlo method, there are correlations of the samples being drawn from <math>P(\boldsymbol{r})</math>. A typical measurement of the correlation is the ''tunneling time''. The tunneling time is defined by the number of Markov steps (of the Markov chain) the simulation needs to perform a round-trip between the minimum and maximum of the spectrum of ''F''. One motivation to use the tunneling time is that when it crosses the spectra, it passes through the region of the maximum of the density of states, thus de-correlating the process. On the other hand using round-trips ensures that the system visits all the spectrum.

Because the histogram is flat on the variable ''F'', a multicanonic ensemble can be seen as a diffusion process (i.e. a random walk) on the one dimensional line of ''F'' values. [[Detailed balance]] of the process dictates that there is no [[stochastic drift|drift]] on the process.<ref name=Robert/> This implies that the tunneling time, in local dynamics, should scale as a diffusion process, and thus the tunneling time should scale quadratically with the size of the spectrum, ''N'':

:<math>\tau_{tt} \propto N^2</math>

However, in some systems (the Ising model being the most paradigmatic), the scaling suffers from critical slowing down: it is <math>N^{2+z}</math> where <math>z>0</math> depends on the particular system.<ref name=Dayal/>

Non-local dynamics were developed to improve the scaling to a quadratic scaling<ref name=Wolff/> (see [[wolff algorithm]]), beating the critical slowing down. However, it is still an open question whether there is a local dynamics that does not suffer from critical slowing down in spin systems like Ising model.

==References==

{{Reflist|
refs=
<ref name=Berg>{{Cite doi|10.1103/PhysRevLett.68.9}}</ref>
<ref name=Landau>{{Cite doi|10.1103/PhysRevLett.86.2050}}</ref>
<ref name=Newmann>{{cite book
|title= Monte Carlo Methods in Statistical Physics
|last1=Newmann |first1=M E J
|last2=Barkema |first2=G T
|year= 2002
|publisher= Oxford University Press
|location=USA
|isbn=0198517971
|url=
|page=
|pages=
|ref=
}}
</ref>
<ref name=Lee>{{Cite doi|10.1103/PhysRevE.50.R651}}</ref>
<ref name=Metropolis>{{Cite doi|10.1063/1.1699114}}</ref>
<ref name=Robert>{{cite book
|title=Monte Carlo statistical methods
|last1=Robert |first1=Christian |authorlink1=
|last2=Casella |first2=George |authorlink2=
|year= 2004
|publisher=Springer
|location=
|isbn=978-0-387-21239-5
|url=http://www.springer.com/statistics/statistical+theory+and+methods/book/978-0-387-21239-5
}}
</ref>
<ref name=Dayal>{{Cite doi|10.1103/PhysRevLett.92.097201}}</ref>
<ref name=Wolff>{{Cite doi|10.1103/PhysRevLett.62.361}}</ref>
}}

[[Category:Monte Carlo methods]]
[[Category:Statistical algorithms]]
[[Category:Computational physics]]

Orthogonal wavelet

2013-11-20T09:16:14Z

138.246.2.47: /* Vanishing moments, polynomial approximation and smoothness */

'''Nonlinear control''' is the area of [[control engineering]] specifically involved with systems that are [[nonlinear system|nonlinear]], [[time-variant system|time-variant]], or both. Many well-established analysis and design techniques exist for [[LTI system|linear time-invariant (LTI) systems]] (e.g., [[root-locus]], [[Bode plot]], [[Nyquist stability criterion|Nyquist criterion]], [[control theory|state-feedback]], [[Full state feedback|pole placement]]); however, one or both of the controller and the system under control in a general [[control system]] may not be an LTI system, and so these methods cannot necessarily be applied directly. Nonlinear control theory studies how to apply existing linear methods to these more general control systems. Additionally, it provides novel control methods that cannot be analyzed using LTI system theory. Even when LTI system theory can be used for the analysis and design of a controller, a nonlinear controller can have attractive characteristics (e.g., simpler implementation, increased speed, or decreased control energy); however, nonlinear control theory usually requires more rigorous [[mathematical analysis]] to justify its conclusions.

== Properties of nonlinear systems ==

Some properties of nonlinear dynamic systems are

* They do not follow the principle of [[Superposition principle|superposition]] (linearity and homogeneity).
* They may have multiple isolated equilibrium points.
* They may exhibit properties such as [[limit cycle]], [[Bifurcation theory|bifurcation]], [[Chaos theory|chaos]].
* Finite escape time: Solutions of nonlinear systems may not exist for all times.

== Analysis and control of nonlinear systems ==
There are several well-developed techniques for analyzing nonlinear feedback systems:

* [[Describing function]] method
* [[Phase plane method]]
* [[Lyapunov stability]] analysis
* [[Singular perturbation]] method
* [[Vasile M. Popov|Popov]] criterion (described in ''The Lur'e Problem'' below)
* [[Center manifold theorem]]
* [[Small-gain theorem]]
* [[Passivity analysis]]

Control design techniques for nonlinear systems also exist. These can be subdivided into techniques which attempt to treat the system as a linear system in a limited range of operation and use (well-known) linear design techniques for each region:
* [[Gain scheduling]]

Those that attempt to introduce auxiliary nonlinear feedback in such a way that the system can be treated as linear for purposes of control design:
* [[Feedback linearization]]

And [[Aleksandr Lyapunov|Lyapunov]] based methods:
* [[Lyapunov redesign]]
* [[Nonlinear damping]]
* [[Backstepping]]
* [[Sliding mode control]]

== Nonlinear feedback analysis – The Lur'e problem ==
[[File:Lur'e_Problem_Block.jpg|thumb|400px|Lur'e problem block diagram]]

An early nonlinear feedback system analysis problem was formulated by [[A. I. Lur'e]].
Control systems described by the Lur'e problem have a forward path that is linear and time-invariant, and a feedback path that contains a memory-less, possibly time-varying, static nonlinearity.

The linear part can be characterized by four matrices (A,B,C,D), while the nonlinear part is Φ(y) with <math>\frac{\Phi(y)}{y} \in [a,b],\quad a (a sector nonlinearity).

=== Absolute stability problem ===
Consider:
# (A,B) is controllable and (C,A) is observable
# two real numbers a, b with a<b, defining a sector for function Φ

The problem is to derive conditions involving only the transfer matrix H(s) and {a,b} such that x=0 is a globally uniformly asymptotically stable equilibrium of the system.
This is known as the Lur'e problem.
There are two well-known wrong conjections on absolute stability:
* The [[Aizerman's conjecture]]
* The [[Kalman's conjecture]].
There are counterexamples to Aizerman's and Kalman's conjectures such that nonlinearity belongs to the sector of linear stability and unique stable equilibrium coexists with a stable periodic solution -- [[hidden oscillation]].

There are two main theorems concerning the problem:
* The [[Circle criterion]]
* The [[Vasile M Popov|Popov]] criterion.
which give sufficient conditions of absolute stability.

==== Popov criterion ====

The sub-class of Lur'e systems studied by [[Vasile M Popov|Popov]] is described by:

<math>
\begin{matrix}
\dot{x}&=&Ax+bu \\
\dot{\xi}&=&u \\
y&=&cx+d\xi \quad (1)
\end{matrix} </math>

<math> \begin{matrix} u = -\phi (y) \quad (2) \end{matrix} </math>

where x ∈ Rn, ξ,u,y are scalars and A,b,c,d have commensurate dimensions. The nonlinear element Φ: R → R is a time-invariant nonlinearity belonging to ''open sector'' (0, ∞). This means that
:Φ(0) = 0, y Φ(y) > 0, ∀ y ≠ 0;

The transfer function from u to y is given by 
:<math> H(s) = \frac{d}{s} + c(sI-A)^{-1}b \quad \quad</math>

'''Theorem:'''
Consider the system (1)-(2) and suppose
# A is [[Routh Hurwitz stability criterion|Hurwitz]]
#(A,b) is controllable
#(A,c) is observable
#d>0 and
# Φ ∈ (0,∞)

then the system is [[globally asymptotically stable]] if there exists a number r>0 such that 
infω ∈ R Re[(1+jωr)h(jω)] > 0 .

Things to be noted:
* The Popov criterion is applicable only to autonomous systems
* The system studied by Popov has a pole at the origin and there is no direct pass-through from input to output
* The nonlinearity Φ must satisfy an open sector condition

== Theoretical results in nonlinear control ==
===Frobenius Theorem===
The [[Frobenius theorem (differential topology)|Frobenius theorem]] is a [[deep result]] in Differential Geometry. When applied to Nonlinear Control, it says the following: Given a system of the form

<math> \dot x = \sum_{i=1}^k f_i(x) u_i(t) \, </math>

where <math>x \in R^n</math>, <math>f_1, \dots, f_k</math> are vector fields belonging to a distribution <math>\Delta</math> and <math>u_i(t)</math> are control functions, the integral curves of <math>x</math> are restricted to a manifold of dimension <math>m</math> if span(<math>\Delta) = m</math> and <math>\Delta</math> is an [[Involution (mathematics)|involutive]] distribution.

==See also==
* [[Feedback passivation]]
* [[Phase-locked loop]]

== Further reading ==
{{refbegin}}
* A. I. Lur'e and V. N. Postnikov, "On the theory of stability of control systems," ''Applied mathematics and mechanics'', 8(3), 1944, (in Russian).
* M. Vidyasagar, ''Nonlinear Systems Analysis'', 2nd edition, Prentice Hall, Englewood Cliffs, New Jersey 07632.
* A. Isidori, ''Nonlinear Control Systems,'' 3rd edition, Springer Verlag, London, 1995.
* H. K. Khalil, ''Nonlinear Systems,'' 3rd edition, Prentice Hall, Upper Saddle River, New Jersey, 2002. ISBN 0-13-067389-7
* B. Brogliato, R. Lozano, B. Maschke, O. Egeland, "Dissipative Systems Analysis and Control", Springer Verlag, London, 2nd edition, 2007.
*{{cite journal
| author = Leonov G.A., Kuznetsov N.V.
| year = 2011
| title = Algorithms for Searching for Hidden Oscillations in the Aizerman and Kalman Problems
| journal = Doklady Mathematics
| volume = 84
| pages = 475–481
| url = http://www.math.spbu.ru/user/nk/PDF/2011-DAN-Absolute-stability-Aizerman-problem-Kalman-conjecture.pdf
| doi = 10.1134/S1064562411040120
| issue = 1}}
*{{cite journal
| author = Bragin V.O., Vagaitsev V.I., Kuznetsov N.V., Leonov G.A.
| year = 2011
| title = Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua's Circuits
| journal = Journal of Computer and Systems Sciences International
| volume = 50
| pages = 511–543
| url = http://www.math.spbu.ru/user/nk/PDF/2011-TiSU-Hidden-oscillations-attractors-Aizerman-Kalman-conjectures.pdf
| doi = 10.1134/S106423071104006X
| issue = 5}}
*{{cite journal
| author = Leonov G.A., Kuznetsov N.V.
| year = 2011
| title = Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems
| journal = IFAC Proceedings Volumes (IFAC-PapersOnline)
| volume = 18
| pages = 2494–2505
| url = http://www.math.spbu.ru/user/nk/PDF/2011-IFAC-Hidden-oscillations-control-systems-Aizerman-problem-Kalman.pdf
| doi = 10.3182/20110828-6-IT-1002.03315
| issue = 1
| series = Proceedings of the 18th IFAC World Congress
| editor1-last = Sergio
| editor1-first = Bittanti
| isbn = 9783902661937}}
{{refend}}

{{DEFAULTSORT:Nonlinear Control}}
[[Category:Nonlinear control| ]]

File talk:Probability-Probability plot, quality characteristic data.png

2013-08-27T13:32:26Z

138.246.2.177: ←Created page with 'The plot should be clipped to <math>[0:1]</math>. By definition, no points can be outside this range. --~~~~'

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2013-07-11T09:16:40Z

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The '''constellation model''' is a probabilistic, [[generative model]] for category-level object recognition in [[computer vision]]. Like other [[part-based models]], the constellation model attempts to represent an object class by a set of ''N'' parts under mutual geometric constraints. Because it considers the geometric relationship between different parts, the constellation model differs significantly from appearance-only, or "[[bag-of-words model|bag-of-words]]" representation models, which explicitly disregard the location of image features.

The problem of defining a generative model for object recognition is difficult. The task becomes significantly complicated by factors such as background clutter, occlusion, and variations in viewpoint, illumination, and scale. Ideally, we would like the particular representation we choose to be robust to as many of these factors as possible.

In category-level recognition, the problem is even more challenging because of the fundamental problem of intra-class variation. Even if two objects belong to the same visual category, their appearances may be significantly different. However, for structured objects such as cars, bicycles, and people, separate instances of objects from the same category are subject to similar geometric constraints. For this reason, particular parts of an object such as the headlights or tires of a car still have consistent appearances and relative positions. The Constellation Model takes advantage of this fact by explicitly modeling the relative location, relative scale, and appearance of these parts for a particular object category. Model parameters are estimated using an [[unsupervised learning]] algorithm, meaning that the visual concept of an object class can be extracted from an unlabeled set of training images, even if that set contains "junk" images or instances of objects from multiple categories. It can also account for the absence of model parts due to appearance variability, occlusion, clutter, or detector error.

== History ==

The idea for a "parts and structure" model was originally introduced by Fischler and Elschlager in 1973.<ref>[http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1672195 M. Fischler and R. Elschlager. ''The Representation and Matching of Pictoral Structures.'' (1973)]</ref> This model has since been built upon and extended in many directions. The Constellation Model, as introduced by Dr. Perona and his colleagues, was a probabilistic adaptation of this approach.

In the late '90s, Burl et al.<ref>
[ftp://vision.caltech.edu/pub/tech-reports-vision/IWAFGR95.ps.Z M. Burl, T. Leung, and P. Perona. ''Face Localization via Shape Statistics.'' (1995)]</ref><ref>[ftp://vision.caltech.edu/pub/tech-reports-vision/ICCV95-faces.ps.Z T. Leung, M. Burl, and P. Perona. ''Finding Faces in Cluttered Scenes Using Random Labeled Graph Matching.'' (1995)]</ref><ref>[ftp://vision.caltech.edu/pub/tech-reports-vision/CVPR96-recog.ps.gz M. Burl and P. Perona. ''Recognition of Planar Object Classes'' (1996)]</ref><ref>[http://www.vision.caltech.edu/publications/ECCV98-recog.pdf M. Burl, M. Weber, and P. Perona. ''A Probabilistic Approach to Object Recognition Using Local Photometry and Global Geometry'' (1998)]</ref> revisited the Fischler and Elschlager model for the purpose of face recognition. In their work, Burl et al. used manual selection of constellation parts in training images to construct a statistical model for a set of detectors and the relative locations at which they should be applied. In 2000, Weber et al. <ref>[http://www.vision.caltech.edu/publications/MarkusWeber-thesis.pdf M. Weber. ''Unsupervised Learning of Models for Object Recognition.'' PhD Thesis. (2000)]</ref><ref>[ftp://vision.caltech.edu/pub/tech-reports/FG00-recog.pdf M. Weber, W. Einhaeuser, M. Welling and P. Perona. ''Viewpoint-Invariant Learning and Detection of Human Heads.'' (2000)]</ref><ref name="weber_towards">[ftp://vision.caltech.edu/pub/tech-reports/CVPR00-recog.pdf M. Weber, M. Welling, and P. Perona. ''Towards Automatic Discovery of Object Categories.'' (2000)]</ref><ref name="weber_unsupervised">[ftp://vision.caltech.edu/pub/tech-reports/ECCV00-recog.pdf M. Weber, M. Welling and P. Perona. ''Unsupervised Learning of Models for Recognition.'' (2000)]</ref> made the significant step of training the model using a more unsupervised learning process, which precluded the necessity for tedious hand-labeling of parts. Their algorithm was particularly remarkable because it performed well even on cluttered and occluded image data. Fergus et al.<ref name="object_class_recognition">[ftp://vision.caltech.edu/pub/tech-reports/Fergus_CVPR03.pdf R. Fergus, P. Perona, and A. Zisserman. ''Object Class Recognition by Unsupervised Scale-Invariant Learning.'' (2003)]</ref><ref name="fergus_thesis">[http://cs.nyu.edu/~fergus/papers/fergus_thesis.pdf R. Fergus. ''Visual Object Category Recognition.'' PhD Thesis. (2005)]</ref> then improved upon this model by making the learning step fully unsupervised, having both shape and appearance learned simultaneously, and accounting explicitly for the relative scale of parts.

== The method of Weber and Welling et al.<ref name="weber_unsupervised" /> ==

In the first step, a standard [[interest point detection]] method, such as Harris [[corner detection]], is used to generate interest points. [[Image feature]]s generated from the vicinity of these points are then clustered using [[k-means]] or another appropriate algorithm. In this process of [[vector quantization]], one can think of the centroids of these clusters as being representative of the appearance of distinctive object parts. Appropriate [[feature detector]]s are then trained using these clusters, which can be used to obtain a set of candidate parts from images.

As a result of this process, each image can now be represented as a set of parts. Each part has a type, corresponding to one of the aforementioned appearance clusters, as well as a location in the image space.

=== Basic generative model ===

Weber & Welling here introduce the concept of ''foreground'' and ''background''. ''Foreground'' parts correspond to an instance of a target object class, whereas ''background'' parts correspond to background clutter or false detections.

Let ''T'' be the number of different types of parts. The positions of all parts extracted from an image can then be represented in the following "matrix,"

:<math>

X^o =
\begin{pmatrix}
x_{11},x_{12},{\cdots} ,x_{1N_1} \\
x_{21},x_{22},{\cdots} ,x_{2N_2} \\
\vdots \\
x_{T1},x_{T2},{\cdots} ,x_{TN_T}
\end{pmatrix}

</math>

where <math>N_i\,</math> represents the number of parts of type <math>i \in \{1,\dots,T\}</math> observed in the image. The superscript ''o'' indicates that these positions are ''observable'', as opposed to ''missing''. The positions of unobserved object parts can be represented by the vector <math>x^m\,</math>. Suppose that the object will be composed of <math>F\,</math> distinct foreground parts. For notational simplicity, we assume here that <math>F = T\,</math>, though the model can be generalized to <math>F > T\,</math>. A ''hypothesis'' <math>h\,</math> is then defined as a set of indices, with <math>h_i = j\,</math>, indicating that point <math>x_{ij}\,</math> is a foreground point in <math>X^o\,</math>. The generative probabilistic model is defined through the joint probability density <math>p(X^o,x^m,h)\,</math>.

=== Model details ===

The rest of this section summarizes the details of Weber & Welling's model for a single component model. The formulas for multiple component models<ref name="weber_towards" /> are extensions of those described here.

To parametrize the joint probability density, Weber & Welling introduce the auxiliary variables <math>b\,</math> and <math>n\,</math>, where <math>b\,</math> is a binary vector encoding the presence/absence of parts in detection (<math>b_i = 1\,</math> if <math>h_i > 0\,</math>, otherwise <math>b_i = 0\,</math>), and <math>n\,</math> is a vector where <math>n_i\,</math> denotes the number of ''background'' candidates included in the <math>i^{th}</math> row of <math>X^o\,</math>. Since <math>b\,</math> and <math>n\,</math> are completely determined by <math>h\,</math> and the size of <math>X^o\,</math>, we have <math>p(X^o,x^m,h) = p(X^o,x^m,h,n,b)\,</math>. By decomposition,

:<math>
p(X^o,x^m,h,n,b) = p(X^o,x^m|h,n,b)p(h|n,b)p(n)p(b)\,
</math>

The probability density over the number of background detections can be modeled by a [[Poisson distribution]],

:<math>
p(n) = \prod_{i=1}^T \frac{1}{n_i!}(M_i)^{n_i}e^{-M_i}
</math>

where <math>M_i\,</math> is the average number of background detections of type <math>i\,</math> per image.

Depending on the number of parts <math>F\,</math>, the probability <math>p(b)\,</math> can be modeled either as an explicit table of length <math>2^F\,</math>, or, if <math>F\,</math> is large, as <math>F\,</math> independent probabilities, each governing the presence of an individual part.

The density <math>p(h|n,b)\,</math> is modeled by

:<math>
p(h|n,b) =

\begin{cases}
\frac{1}{ \textstyle \prod_{f=1}^F N_f^{b_f}}, & \mbox{if } h \in H(b,n) \\
0, & \mbox{for other } h
\end{cases}

</math>

where <math>H(b,n)\,</math> denotes the set of all hypotheses consistent with <math>b\,</math> and <math>n\,</math>, and <math>N_f\,</math> denotes the total number of detections of parts of type <math>f\,</math>. This expresses the fact that all consistent hypotheses, of which there are <math>\textstyle \prod_{f=1}^F N_f^{b_f}</math>, are equally likely in the absence of information on part locations.

And finally,

:<math>
p(X^o,x^m|h,n) = p_{fg}(z)p_{bg}(x_{bg})\,
</math>

where <math>z = (x^ox^m)\,</math> are the coordinates of all foreground detections, observed and missing, and <math>x_{bg}\,</math> represents the coordinates of the background detections. Note that foreground detections are assumed to be independent of the background. <math>p_{fg}(z)\,</math> is modeled as a joint Gaussian with mean <math>\mu\,</math> and covariance <math>\Sigma\,</math>.

=== Classification ===

The ultimate objective of this model is to classify images into classes "object present" (class <math>C_1\,</math>) and "object absent" (class <math>C_0\,</math>) given the observation <math>X^o\,</math>. To accomplish this, Weber & Welling run part detectors from the learning step exhaustively over the image, examining different combinations of detections. If occlusion is considered, then combinations with missing detections are also permitted. The goal is then to select the class with maximum a posteriori probability, by considering the ratio

:<math>
\frac{p(C_1|X^o)}{p(C_0|X^o)} \propto \frac{\sum_h p(X^o,h|C_1)}{p(X^o,h_0|C_0)}
</math>

where <math>h_0\,</math> denotes the null hypothesis, which explains all parts as background noise. In the numerator, the sum includes all hypotheses, including the null hypothesis, whereas in the denominator, the only hypothesis consistent with the absence of an object is the null hypothesis. In practice, some threshold can be defined such that, if the ratio exceeds that threshold, we then consider an instance of an object to be detected.

=== Model learning ===

After the preliminary step of interest point detection, feature generation and clustering, we have a large set of candidate parts over the training images. To learn the model, Weber & Welling first perform a greedy search over possible model configurations, or equivalently, over potential subsets of the candidate parts. This is done in an iterative fashion, starting with random selection. At subsequent iterations, parts in the model are randomly substituted, the model parameters are estimated, and the performance is assessed. The process is complete when further model performance improvements are no longer possible.

At each iteration, the model parameters

:<math>
\Theta = \{\mu, \Sigma, p(b), M\}\,
</math>

are estimated using [[expectation maximization]]. <math>\mu\,</math> and <math>\Sigma\,</math>, we recall, are the mean and covariance of the joint Gaussian <math>p_{fg}(z)\,</math>, <math>p(b)\,</math> is the probability distribution governing the binary presence/absence of parts, and <math>M\,</math> is the mean number of background detections over part types.

=== M-step ===

EM proceeds by maximizing the likelihood of the observed data,

:<math>
L(X^o|\Theta) = \sum_{i=1}^I \log \sum_{h_i} \int p(X_i^o,x_i^m,h_i|\Theta)dx_i^m
</math>

with respect to the model parameters <math>\Theta\,</math>. Since this is difficult to achieve analytically, EM iteratively maximizes a sequence of cost functions,

:<math>
Q(\tilde{\Theta}|\Theta) = \sum_{i=1}^I E[\log p(X_i^o,x_i^m,h_i|\tilde{\Theta})]
</math>

Taking the derivative of this with respect to the parameters and equating to zero produces the update rules:

:<math>
\tilde{\mu} = \frac{1}{I} \sum_{i=1}^I E[z_i]
</math>

:<math>
\tilde{\Sigma} = \frac{1}{I} \sum_{i=1}^I E[z_iz_i^T] - \tilde{\mu}\tilde{\mu}^T
</math>

:<math>
\tilde{p}(\bar{b}) = \frac{1}{I} \sum_{i=1}^I E[\delta_{b,\bar{b}}]
</math>

:<math>
\tilde{M} = \frac{1}{I} \sum_{i=1}^I E[n_i]
</math>

=== E-step ===

The update rules in the M-step are expressed in terms of [[sufficient statistics]], <math>E[z]\,</math>, <math>E[zz^T]\,</math>, <math>E[\delta_{b,\bar{b}}]\,</math> and <math>E[n]\,</math>, which are calculated in the E-step by considering the posterior density:

:<math>
p(h_i,x_i^m|X_i^o,\Theta) = \frac{p(h_i,x_i^m,X_i^o|\Theta)}{\textstyle \sum_{h_i \in H_b} \int p(h_i,x_i^m,X_i^o|\Theta) dx_i^m}
</math>

==The method of Fergus et al.<ref name="object_class_recognition" />==

In Weber et al., shape and appearance models are constructed separately. Once the set of candidate parts had been selected, shape is learned independently of appearance. The innovation of Fergus et al. is to learn not only two, but three model parameters simultaneously: shape, appearance, and relative scale. Each of these parameters are represented by Gaussian densities.

=== Feature representation ===

Whereas the preliminary step in the Weber et al. method is to search for the locations of interest points, Fergus et al. use the detector of Kadir and Brady<ref>[http://www.springerlink.com/content/t45n2g8543574026/fulltext.pdf T. Kadir and M. Brady. ''Saliency, scale and image description.'' (2001)]</ref> to find salient regions in the image over both location (center) and scale (radius). Thus, in addition to location information <math>X\,</math> this method also extracts associated scale information <math>S\,</math>. Fergus et al. then normalize the squares bounding these circular regions to 11 x 11 pixel patches, or equivalently, 121-dimensional vectors in the appearance space. These are then reduced to 10-15 dimensions by [[principal component analysis]], giving the appearance information <math>A\,</math>.

=== Model structure ===

Given a particular object class model with parameters <math>\Theta\,</math>, we must decide whether or not a new image contains an instance of that class. This is accomplished by making a Bayesian decision,

:<math>
R = \frac{p(\mbox{Object}|X,S,A)}{p(\mbox{No object}|X,S,A)}
</math>

:<math>
= \frac{p(X,S,A|\mbox{Object})p(\mbox{Object})}{p(X,S,A|\mbox{No object})p(\mbox{No object})}
</math>

:<math>
\approx \frac{p(X,S,A|\Theta)p(\mbox{Object})}{p(X,S,A|\Theta_{bg})p(\mbox{No object})}
</math>

where <math>\Theta_{bg}</math> is the background model. This ratio is compared to a threshold <math>T\,</math> to determine object presence/absence.

The likelihoods are factored as follows:

:<math>
p(X,S,A|\Theta) = \sum_{h \in H} p(X,S,A,h|\Theta) =
</math>

:<math>
\sum_{h \in H} \underbrace{ p(A|X,S,h,\Theta) }_{\mbox{Appearance}} \underbrace{ p(X|S,h,\Theta) }_{\mbox{Shape}} \underbrace{ p(S|h,\Theta) }_{\mbox{Rel. Scale}} \underbrace{ p(h|\Theta) }_{\mbox{Other}}
</math>

=== Appearance ===

Each part <math>p\,</math> has an appearance modeled by a Gaussian density in the appearance space, with mean and covariance parameters <math>\Theta_p^{app} = \{c_p,V_p\}</math>, independent of other parts' densities. The background model has parameters <math>\Theta_{bg}^{app} = \{c_{bg},V_{bg}\}</math>. Fergus et al. assume that, given detected features, the position and appearance of those features are independent. Thus, <math>p(A|X,S,h,\Theta) = p(A|h,\Theta)\,</math>. The ratio of the appearance terms reduces to

:<math>
\frac{p(A|X,S,h,\Theta)}{p(A|X,S,h,\Theta_{bg})} = \frac{p(A|h,\Theta)}{p(A|h,\Theta_{bg})}
</math>

:<math>
= \prod_{p=1}^P \left ( \frac{G(A(h_p)|c_p,V_p)}{G(A(h_p)|c_{bg},V_{bg})} \right )^{b_p}
</math>

Recall from Weber et al. that <math>h\,</math> is the hypothesis for the indices of foreground parts, and <math>b\,</math> is the binary vector giving the occlusion state of each part in the hypothesis.

=== Shape ===

Shape is represented by a joint Gaussian density of part locations within a particular hypothesis, after those parts have been transformed into a scale-invariant space. This transformation precludes the need to perform an exhaustive search over scale. The Gaussian density has parameters <math>\Theta^{\mbox{shape}} = \{\mu,\Sigma\}\,</math>. The background model <math>\Theta_{bg}\,</math> is assumed to be a uniform distribution over the image, which has area <math>\alpha\,</math>. Letting <math>f\,</math> be the number of foreground parts,

:<math>
\frac{p(X|S,h,\Theta)}{p(X|S,h,\Theta_{bg})} = G(X(h)|\mu,\Sigma)\alpha^f
</math>

=== Relative scale ===

The scale of each part <math>p\,</math> relative to a reference frame is modeled by a Gaussian density with parameters <math>\Theta^{\mbox{scale}} = \{t_p,U_p\}\,</math>. Each part is assumed to be independent of other parts. The background model <math>\Theta_{bg}\,</math> assumes a uniform distribution over scale, within a range <math>r\,</math>.

:<math>
\frac{p(S|h,\Theta)}{p(S|h,\Theta_{bg})} = \prod_{p=1}^P G(S(h_p)|t_p,U_p)^{d_p} r^f
</math>

=== Occlusion and statistics of feature detection ===

:<math>
\frac{p(h|\Theta)}{p(h|\Theta_{bg})} = \frac{p_{\mbox{Poiss}}(n|M)}{p_{\mbox{Poiss}}(N|M)} \frac{1}{^nC_r(N,f)} p(b|\Theta)
</math>

The first term models the number of features detected using a [[Poisson distribution]], which has mean M. The second term serves as a "book-keeping" term for the hypothesis variable. The last term is a probability table for all possible occlusion patterns.

=== Learning ===

The task of learning the model parameters <math>\Theta = \{\mu,\Sigma,c,V,M,p(b|\Theta),t,U\}\,</math> is accomplished by [[expectation maximization]]. This is carried out in a spirit similar to that of Weber et al. Details and formulas for the E-step and M-step can be seen in the literature.<ref name="fergus_thesis" />

== Performance ==

The Constellation Model as conceived by Fergus et al. achieves successful categorization rates consistently above 90% on large datasets of motorbikes, faces, airplanes, and spotted cats.<ref>[http://www.vision.caltech.edu/html-files/archive.html R. Fergus and P. Perona. Caltech Object Category datasets. http://www.vision.caltech.edu/html-files/archive.html (2003)]</ref> For each of these datasets, the Constellation Model is able to capture the "essence" of the object class in terms of appearance and/or shape. For example, face and motorbike datasets generate very tight shape models because objects in those categories have very well-defined structure, whereas spotted cats vary significantly in pose, but have a very distinctive spotted appearance. Thus, the model succeeds in both cases. It is important to note that the Constellation Model does not generally account for significant changes in orientation. Thus, if the model is trained on images of horizontal airplanes, it will not perform well on, for instance, images of vertically oriented planes unless the model is extended to account for this sort of rotation explicitly.

In terms of computational complexity, the Constellation Model is very expensive. If <math>N\,</math> is the number of feature detections in the image, and <math>P\,</math> the number of parts in the object model, then the hypothesis space <math>H\,</math> is <math>O(N^P)\,</math>. Because the computation of sufficient statistics in the E-step of [[expectation maximization]] necessitates evaluating the likelihood for every hypothesis, learning becomes a major bottleneck operation. For this reason, only values of <math>P \le 6</math> have been used in practical applications, and the number of feature detections <math>N\,</math> is usually kept within the range of about 20-30 per image.

== Variations ==

One variation that attempts to reduce complexity is the star model proposed by Fergus et al.<ref>[http://www.robots.ox.ac.uk/%7Efergus/papers/fergus_cvpr05.pdf R. Fergus, P. Perona, and A. Zisserman. ''A Sparse Object Category Model for Efficient Learning and Exhaustive Recognition.'' (2005)]</ref> The reduced dependencies of this model allows for learning in <math>O(N^2P)\,</math> time instead of <math>O(N^P)\,</math>. This allows for a greater number of model parts and image features to be used in training. Because the star model has fewer parameters, it is also better at avoiding the problem of over-fitting when trained on fewer images.

== References ==

{{reflist}}

== External links ==
* [http://courses.ece.uiuc.edu/ece598/ffl/lecture7_ConstellationModel_shortversion.pdf L. Fei-fei. ''Object categorization: the constellation models''. Lecture Slides. (2005)] (link not working)

== See also ==

* [[Part-based models]]
* [[One-shot learning]]

[[Category:Learning in computer vision]]
[[Category:Probabilistic models]]

Talk:Condition number

2013-02-07T18:21:24Z

138.246.2.58: /* The maximum value (for nonzero b and e) is easily seen to be the product of the two operator norms: */

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Nonuniform sampling

2013-02-06T17:39:49Z

138.246.2.58:

{| class="wikitable"
|+
k22 figures in n dimensions
|-
!n
![[Four-dimensional space|4]]
![[Five-dimensional space|5]]
![[Six-dimensional space|6]]
![[Seven-dimensional space|7]]
![[Eight-dimensional space|8]]
|- align=center
![[Coxeter group|Coxeter group]]
|A22
|A5
|E6
|<math>{\tilde{E}}_{6}</math>=E6+
|E6++
|- align=center
![[Coxeter-Dynkin diagram|Coxeter diagram]]
|{{CDD|nodes|3ab|nodes_11}}
|{{CDD|nodes|3ab|nodes|split2|node_1}}
|{{CDD|nodes|3ab|nodes|split2|node|3|node_1}}
|{{CDD|nodes|3ab|nodes|split2|node|3|node|3|node_1}}
|{{CDD|nodes|3ab|nodes|split2|node|3|node|3|node|3|node_1}}
|- align=center
![[Coxeter notation|Symmetry]] (order)
|[[32,2,-1]] (72)
|[[32,2,0]] (1440)
|[[32,2,1]] (103,680)
|[[32,2,2]] (∞)
|[[32,2,3]] (∞)
|- align=center
!Graph
|[[File:3-3 duoprism.png|80px]]
|[[File:5-simplex t2.svg|80px]]
|[[File:Up 1 22 t0 E6.svg|80px]]
|∞
|∞
|- align=center
!Name
|[[duoprism|−122]]
|[[Birectified 5-simplex|022]]
|[[1 22 polytope|122]]
|[[2 22 honeycomb|222]]
|322
|}

Rank (linear algebra)

2012-07-16T11:17:59Z

138.246.2.117: /* Column rank = row rank or rk(A) = rk(AT) */

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