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Submanifold

2014-03-01T18:28:40Z

137.205.17.168: /* Immersed submanifolds */

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Magnitude (astronomy)

2014-02-27T13:44:39Z

137.205.124.72: Defined the terms (m and M) in the apparent/absolute magnitude equation.

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Circular polarization

2014-02-18T17:58:00Z

137.205.238.226: /* Left/right handedness conventions {{Anchor|Left/Right}} */

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Speedup

2014-01-29T14:50:41Z

137.205.160.45: Removed irrelevant citation

[[File:Conjugate gradient illustration.svg|right|thumb|A comparison of the convergence of [[gradient descent]] with optimal step size (in green) and conjugate vector (in red) for minimizing a quadratic function associated with a given linear system. Conjugate gradient, assuming exact arithmetic, converges in at most ''n'' steps where ''n'' is the size of the matrix of the system (here ''n''=2).]]
In [[mathematics]], the '''conjugate gradient method''' is an [[algorithm]] for the [[numerical solution]] of particular [[system of linear equations|systems of linear equations]], namely those whose matrix is [[symmetric matrix|symmetric]] and [[positive-definite matrix|positive-definite]]. The conjugate gradient method is often implemented as an [[iterative method|iterative algorithm]], applicable to [[sparse matrix|sparse]] systems that are too large to be handled by a direct implementation or other direct methods such as the [[Cholesky decomposition]]. Large sparse systems often arise when numerically solving [[partial differential equation]]s or optimization problems.

The conjugate gradient method can also be used to solve unconstrained [[Mathematical optimization|optimization]] problems such as [[energy minimization]]. It was developed by [[Magnus Hestenes]] and [[Eduard Stiefel]].<ref>{{cite web|last=Straeter|first=T. A.|title=On the Extension of the Davidon-Broyden Class of Rank One, Quasi-Newton Minimization Methods to an Infinite Dimensional Hilbert Space with Applications to Optimal Control Problems|url=http://hdl.handle.net/2060/19710026200|work=NASA Technical Reports Server|publisher=NASA|accessdate=10 October 2011}}</ref>

The [[biconjugate gradient method]] provides a generalization to non-symmetric matrices. Various [[nonlinear conjugate gradient method]]s seek minima of nonlinear equations.


==Description of the method==

Suppose we want to solve the following [[system of linear equations]]
:'''Ax''' = '''b'''
for the vector '''x''' where the known ''n''-by-''n'' matrix '''A''' is [[Symmetric matrix|symmetric]] (i.e. '''A'''T = '''A'''), [[positive definite matrix|positive definite]] (i.e. '''x'''T'''Ax''' > 0 for all non-zero vectors '''x''' in '''R'''''n''), and [[real number|real]], and '''b''' is known as well.

We denote the unique solution of this system by <math>\scriptstyle \mathbf{x}_*</math>.

==The conjugate gradient method as a direct method==

We say that two non-zero vectors ''u'' and ''v'' are [[Inner automorphism|conjugate]] (with respect to ''A'') if

:<math> \mathbf{u}^\mathrm{T} \mathbf{A} \mathbf{v} = 0. </math>

Since '''A''' is symmetric and positive definite, the left-hand side defines an [[inner product space|inner product]]

:<math> \langle \mathbf{u},\mathbf{v} \rangle_\mathbf{A} := \langle \mathbf{A} \mathbf{u}, \mathbf{v}\rangle = \langle \mathbf{u}, \mathbf{A}^\mathrm{T} \mathbf{v}\rangle = \langle \mathbf{u}, \mathbf{A}\mathbf{v} \rangle = \mathbf{u}^\mathrm{T} \mathbf{A} \mathbf{v}. </math>

Two vectors are conjugate if they are orthogonal with respect to this inner product.
Being conjugate is a symmetric relation: if '''u''' is conjugate to '''v''', then '''v''' is conjugate to '''u'''.

Suppose that <math>P=\{\mathbf{p}_k: \forall i\neq k \in [1,n] \subset \mathbb{N}, \langle\mathbf{p}_i,\mathbf{p}_k\rangle_{A}=0 \}</math> is a set of ''n'' mutually conjugate directions. Then <math>P</math> is a [[basis (linear algebra)|basis]] of <math>\mathbb{R}^n</math>, so within <math>P</math> we can expand the solution <math>\mathbf{x}_*</math> of <math>\mathbf{Ax} = \mathbf{b}</math>:

:<math> \mathbf{x}_* = \sum^{n}_{i=1} \alpha_i \mathbf{p}_i</math>

and we see that

:<math> \mathbf{b}=\mathbf{A}\mathbf{x}_* = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i.</math>

For any <math>\mathbf{p}_k \in P</math>,

:<math> \mathbf{p}_k^\mathrm{T} \mathbf{b}=\mathbf{p}_k^\mathrm{T} \mathbf{A}\mathbf{x}_* = \sum^{n}_{i=1} \alpha_i\mathbf{p}_k^\mathrm{T} \mathbf{A} \mathbf{p}_i=\alpha_k\mathbf{p}_k^\mathrm{T} \mathbf{A} \mathbf{p}_k.</math>

(because <math>\forall i\neq k, p_i, p_k</math> are mutually conjugate)

:<math> \alpha_k = \frac{\mathbf{p}_k^\mathrm{T} \mathbf{b}}{\mathbf{p}_k^\mathrm{T} \mathbf{A} \mathbf{p}_k} = \frac{\langle \mathbf{p}_k, \mathbf{b}\rangle}{\,\,\,\langle \mathbf{p}_k, \mathbf{p}_k\rangle_\mathbf{A}} = \frac{\langle \mathbf{p}_k, \mathbf{b}\rangle}{\,\,\,\|\mathbf{p}_k\|_\mathbf{A}^2}. </math>

This result is perhaps most transparent by considering the inner product defined above.

This gives the following method for solving the equation '''Ax''' = '''b''': find a sequence of ''n'' conjugate directions, and then compute the coefficients <math>\scriptstyle \alpha_k</math>.

==The conjugate gradient method as an iterative method==

If we choose the conjugate vectors '''p'''''k'' carefully, then we may not need all of them to obtain a good approximation to the solution <math>\scriptstyle \mathbf{x}_*</math>. So, we want to regard the conjugate gradient method as an iterative method. This also allows us to approximately solve systems where ''n'' is so large that the direct method would take too much time.

We denote the initial guess for <math>\scriptstyle \mathbf{x}_*</math> by '''x'''0. We can assume without loss of generality that '''x'''0 = 0 (otherwise, consider the system '''Az''' = '''b''' − '''Ax'''0 instead). Starting with '''x'''0 we search for the solution and in each iteration we need a metric to tell us whether we are closer to the solution <math>\scriptstyle \mathbf{x}_*</math> (that is unknown to us). This metric comes from the fact that the solution <math>\scriptstyle \mathbf{x}_*</math> is also the unique minimizer of the following [[quadratic function]]; so if f('''x''') becomes smaller in an iteration it means that we are closer to <math>\scriptstyle \mathbf{x}_*</math>.
:<math> f(\mathbf{x}) = \frac12 \mathbf{x}^\mathrm{T} \mathbf{A}\mathbf{x} - \mathbf{x}^\mathrm{T} \mathbf{b} , \quad \mathbf{x}\in\mathbf{R}^n. </math>



This suggests taking the first basis vector '''p'''0 to be the negative of the gradient of ''f'' at '''x''' = '''x'''0. The gradient of ''f'' equals '''Ax'''0−'''b'''. Starting with a "guessed solution" '''x'''0 (we can always guess that <math>\scriptstyle \mathbf{x}_*</math> is '''0''' and set '''x'''0 to '''0''' if we have no reason to guess for anything else), this means we take '''p'''0 = '''b'''−'''Ax'''0. The other vectors in the basis will be conjugate to the gradient, hence the name ''conjugate gradient method''.

Let '''r'''''k'' be the [[residual (numerical analysis)|residual]] at the ''k''th step:
:<math> \mathbf{r}_k = \mathbf{b} - \mathbf{Ax}_k. \, </math>
Note that '''r'''''k'' is the negative gradient of ''f'' at '''x''' = '''x'''''k'', so the [[gradient descent]] method would be to move in the direction '''r'''''k''. Here, we insist that the directions '''p'''''k'' be conjugate to each other. We also require that the next search direction be built out of the current residue and all previous search directions, which is reasonable enough in practice.

The conjugation constraint is an orthonormal-type constraint and hence the algorithm bears resemblance to [[Gram–Schmidt process|Gram-Schmidt orthonormalization]].

This gives the following expression:
:<math> \mathbf{p}_{k} = \mathbf{r}_{k} - \sum_{i < k}\frac{\mathbf{p}_i^\mathrm{T} \mathbf{A} \mathbf{r}_{k}}{\mathbf{p}_i^\mathrm{T}\mathbf{A} \mathbf{p}_i} \mathbf{p}_i </math>
(see the picture at the top of the article for the effect of the conjugacy constraint on convergence). Following this direction, the next optimal location is given by
:<math> \mathbf{x}_{k+1} = \mathbf{x}_k + \alpha_k \mathbf{p}_k </math>
with
:<math> \alpha_{k} = \frac{\mathbf{p}_k^\mathrm{T} \mathbf{b}}{\mathbf{p}_k^\mathrm{T} \mathbf{A} \mathbf{p}_k} = \frac{\mathbf{p}_k^\mathrm{T} (\mathbf{r}_{k-1}+\mathbf{Ax}_{k-1})}{\mathbf{p}_{k}^\mathrm{T} \mathbf{A} \mathbf{p}_{k}} = \frac{\mathbf{p}_{k}^\mathrm{T} \mathbf{r}_{k-1}}{\mathbf{p}_{k}^\mathrm{T} \mathbf{A} \mathbf{p}_{k}}, </math>
where the last equality holds because '''p'''''k'' and '''x'''''k-1'' are conjugate.

===The resulting algorithm===

The above algorithm gives the most straightforward explanation of the conjugate gradient method. Seemingly, the algorithm as stated requires storage of all previous searching directions and residue vectors, as well as many matrix-vector multiplications, and thus can be computationally expensive. However, a closer analysis of the algorithm shows that '''r'''''k+1'' is conjugate to '''p'''''i'' for all ''i < k'' (can be proved by induction, for example), and therefore only '''r'''''k'', '''p'''''k'', and '''x'''''k'' are needed to construct '''r'''''k+1'', '''p'''''k+1'', and '''x'''''k+1''. Furthermore, only one matrix-vector multiplication is needed in each iteration.

The algorithm is detailed below for solving '''Ax''' = '''b''' where '''A''' is a real, symmetric, positive-definite matrix. The input vector '''x'''0 can be an approximate initial solution or '''0'''. It is a different formulation of the exact procedure described above.

:<math>
\begin{align}
& \mathbf{r}_0 := \mathbf{b} - \mathbf{A x}_0 \\
& \mathbf{p}_0 := \mathbf{r}_0 \\
& k := 0 \\
& \hbox{repeat} \\
& \qquad \alpha_k := \frac{\mathbf{r}_k^\mathrm{T} \mathbf{r}_k}{\mathbf{p}_k^\mathrm{T} \mathbf{A p}_k} \\
& \qquad \mathbf{x}_{k+1} := \mathbf{x}_k + \alpha_k \mathbf{p}_k \\
& \qquad \mathbf{r}_{k+1} := \mathbf{r}_k - \alpha_k \mathbf{A p}_k \\
& \qquad \hbox{if } r_{k+1} \hbox{ is sufficiently small then exit loop} \\
& \qquad \beta_k := \frac{\mathbf{r}_{k+1}^\mathrm{T} \mathbf{r}_{k+1}}{\mathbf{r}_k^\mathrm{T} \mathbf{r}_k} \\
& \qquad \mathbf{p}_{k+1} := \mathbf{r}_{k+1} + \beta_k \mathbf{p}_k \\
& \qquad k := k + 1 \\
& \hbox{end repeat} \\
& \hbox{The result is } \mathbf{x}_{k+1}
\end{align}
</math>
This is the most commonly used algorithm. The same formula for <math>\beta_k</math> is also used in the Fletcher–Reeves [[nonlinear conjugate gradient method]].

====Computation of alpha and beta====
In the algorithm, <math>\alpha_k</math> is chosen such that <math>\mathbf{r}_{k+1}</math> is orthogonal to <math>\mathbf{r}_k</math>. The denominator is simplified from

<math> \alpha_k = \frac{\mathbf{r}_k^\mathrm{T} \mathbf{r}_k}{\mathbf{r}_k^\mathrm{T} \mathbf{A p}_k} = \frac{\mathbf{r}_k^\mathrm{T} \mathbf{r}_k}{\mathbf{p}_k^\mathrm{T} \mathbf{A p}_k} </math>

since <math>\mathbf{r}_k = \mathbf{p}_k-\mathbf{\beta}_{k-1}\mathbf{p}_{k-1}</math>. The <math>\beta_k</math> is chosen such that <math>\mathbf{p}_{k+1}</math> is conjugated to <math>\mathbf{p}_k</math>. Initially, <math>\beta_k</math> is

<math>\beta_k = - \frac{\mathbf{r}_{k+1}^\mathrm{T} A \mathbf{p}_k}{\mathbf{p}_k^\mathrm{T} A \mathbf{p}_k}</math>

using <math>\mathbf{r}_k = \mathbf{r}_{k-1} - \alpha_{k-1} A \mathbf{p}_{k-1}</math> and equivalently <math> A \mathbf{p}_{k-1} = \frac{1}{\alpha_{k-1}} (\mathbf{r}_{k-1} - \mathbf{r}_k)</math>, the numerator of <math>\beta_k</math> is rewritten as

<math> \mathbf{r}_{k+1}^\mathrm{T} A \mathbf{p}_k = \frac{1}{\alpha_k} \mathbf{r}_{k+1}^\mathrm{T} (\mathbf{r}_k - \mathbf{r}_{k+1}) = - \frac{1}{\alpha_k} \mathbf{r}_{k+1}^\mathrm{T} \mathbf{r}_{k+1} </math>

because <math>\mathbf{r}_{k+1}</math> and <math>\mathbf{r}_k</math> are orthogonal by design. The denominator is rewritten as

<math> \mathbf{p}_k^\mathrm{T} A \mathbf{p}_k = (\mathbf{r}_k + \beta_{k-1} \mathbf{p}_{k-1})^\mathrm{T} A \mathbf{p}_k = \frac{1}{\alpha_k} \mathbf{r}_k^\mathrm{T} (\mathbf{r}_k - \mathbf{r}_{k+1}) = \frac{1}{\alpha_k} \mathbf{r}_k^\mathrm{T} \mathbf{r}_k </math>

using that the search directions <math>\mathbf{p}_k</math> are conjugated and again that the residuals are orthogonal. This gives the <math>\beta</math> in the algorithm after cancelling <math>\alpha_k</math>.

====Example code in Matlab====
<source lang="matlab">
function [x] = conjgrad(A,b,x)
r=b-A*x;
p=r;
rsold=r'*r;

for i=1:10^(6)
Ap=A*p;
alpha=rsold/(p'*Ap);
x=x+alpha*p;
r=r-alpha*Ap;
rsnew=r'*r;
if sqrt(rsnew)<1e-10
break;
end
p=r+rsnew/rsold*p;
rsold=rsnew;
end
</source>

===Numerical example===

To illustrate the conjugate gradient method, we will complete a simple example.

Considering the linear system '''Ax''' = '''b''' given by

:: <math>\mathbf{A} \mathbf{x}= \begin{bmatrix}
4 & 1 \\
1 & 3 \end{bmatrix}\begin{bmatrix}
x_1 \\
x_2 \end{bmatrix} = \begin{bmatrix}
1 \\
2 \end{bmatrix}
,</math>

we will perform two steps of the conjugate gradient method beginning with the initial guess

:: <math>\mathbf{x}_0 =
\begin{bmatrix}
2 \\
1 \end{bmatrix}
</math>

in order to find an approximate solution to the system.

====Solution====
For reference, the exact solution is
:: <math>
\mathbf{x} =
\begin{bmatrix} \frac{1}{11} \\\\ \frac{7}{11} \end{bmatrix}
</math>
Our first step is to calculate the residual vector '''r'''0 associated with '''x'''0. This residual is computed from the formula '''r'''0 = '''b''' - '''Ax'''0, and in our case is equal to

:: <math>
\mathbf{r}_0 =
\begin{bmatrix} 1 \\ 2 \end{bmatrix} -
\begin{bmatrix} 4 & 1 \\ 1 & 3 \end{bmatrix}
\begin{bmatrix} 2 \\ 1 \end{bmatrix} =
\begin{bmatrix}-8 \\ -3 \end{bmatrix}.

</math>

Since this is the first iteration, we will use the residual vector '''r'''0 as our initial search direction '''p'''0; the method of selecting '''p'''k will change in further iterations.

We now compute the scalar α0 using the relationship

::<math>
\alpha_0 = \frac{\mathbf{r}_0^\mathrm{T} \mathbf{r}_0}{\mathbf{p}_0^\mathrm{T} \mathbf{A p}_0} =
\frac{\begin{bmatrix} -8 & -3 \end{bmatrix} \begin{bmatrix} -8 \\ -3 \end{bmatrix}}{ \begin{bmatrix} -8 & -3 \end{bmatrix} \begin{bmatrix} 4 & 1 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} -8 \\ -3 \end{bmatrix} } =
\frac{73}{331}.
</math>

We can now compute '''x'''1 using the formula

::<math>
\mathbf{x}_1 = \mathbf{x}_0 + \alpha_0\mathbf{p}_0 =
\begin{bmatrix} 2 \\ 1 \end{bmatrix} + \frac{73}{331} \begin{bmatrix} -8 \\ -3 \end{bmatrix} = \begin{bmatrix} 0.2356 \\ 0.3384 \end{bmatrix}.
</math>

This result completes the first iteration, the result being an "improved" approximate solution to the system, '''x'''1. We may now move on and compute the next residual vector '''r'''1 using the formula

::<math>
\mathbf{r}_1 = \mathbf{r}_0 - \alpha_0 \mathbf{A} \mathbf{p}_0 =
\begin{bmatrix} -8 \\ -3 \end{bmatrix} - \frac{73}{331} \begin{bmatrix} 4 & 1 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} -8 \\ -3 \end{bmatrix} = \begin{bmatrix} -0.2810 \\ 0.7492 \end{bmatrix}.
</math>

Our next step in the process is to compute the scalar β0 that will eventually be used to determine the next search direction '''p'''1.

::<math>
\beta_0 = \frac{\mathbf{r}_1^\mathrm{T} \mathbf{r}_1}{\mathbf{r}_0^\mathrm{T} \mathbf{r}_0} =
\frac{\begin{bmatrix} -0.2810 & 0.7492 \end{bmatrix} \begin{bmatrix} -0.2810 \\ 0.7492 \end{bmatrix}}{\begin{bmatrix} -8 & -3 \end{bmatrix} \begin{bmatrix} -8 \\ -3 \end{bmatrix}} = 0.0088.
</math>

Now, using this scalar β0, we can compute the next search direction '''p'''1 using the relationship

::<math>
\mathbf{p}_1 = \mathbf{r}_1 + \beta_0 \mathbf{p}_0 =
\begin{bmatrix} -0.2810 \\ 0.7492 \end{bmatrix} + 0.0088 \begin{bmatrix} -8 \\ -3 \end{bmatrix} = \begin{bmatrix} -0.3511 \\ 0.7229 \end{bmatrix}.
</math>

We now compute the scalar α1 using our newly-acquired '''p'''1 using the same method as that used for α0.

::<math>
\alpha_1 = \frac{\mathbf{r}_1^\mathrm{T} \mathbf{r}_1}{\mathbf{p}_1^\mathrm{T} \mathbf{A p}_1} =
\frac{\begin{bmatrix} -0.2810 & 0.7492 \end{bmatrix} \begin{bmatrix} -0.2810 \\ 0.7492 \end{bmatrix}}{ \begin{bmatrix} -0.3511 & 0.7229 \end{bmatrix} \begin{bmatrix} 4 & 1 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} -0.3511 \\ 0.7229 \end{bmatrix} } =
0.4122.
</math>

Finally, we find '''x'''2 using the same method as that used to find '''x'''1.

::<math>
\mathbf{x}_2 = \mathbf{x}_1 + \alpha_1 \mathbf{p}_1 =
\begin{bmatrix} 0.2356 \\ 0.3384 \end{bmatrix} + 0.4122 \begin{bmatrix} -0.3511 \\ 0.7229 \end{bmatrix} = \begin{bmatrix} 0.0909 \\ 0.6364 \end{bmatrix}.
</math>

The result, '''x'''2, is a "better" approximation to the system's solution than '''x'''1 and '''x'''0. If exact arithmetic were to be used in this example instead of limited-precision, then the exact solution would theoretically have been reached after ''n'' = 2 iterations (''n'' being the order of the system).

==Convergence properties of the conjugate gradient method==
The conjugate gradient method can theoretically be viewed as a direct method, as it produces the exact solution after a finite number of iterations, which is not larger than the size of the matrix, in the absence of [[round-off error]]. However, the conjugate gradient method is unstable with respect to even small perturbations, e.g., most directions are not in practice conjugate, and the exact solution is never obtained. Fortunately, the conjugate gradient method can be used as an [[iterative method]] as it provides monotonically improving approximations <math>\mathbf{x}_{k}</math> to the exact solution, which may reach the required tolerance after a relatively small (compared to the problem size) number of iterations. The improvement is typically linear and its speed is determined by the [[condition number]] <math>\kappa(A)</math> of the system matrix <math>A</math>: the larger is <math>\kappa(A)</math>, the slower the improvement.<ref name=saad1996iterative>{{cite book|last=Saad|first=Yousef|title=Iterative methods for sparse linear systems|year=2003|publisher=Society for Industrial and Applied Mathematics|location=Philadelphia, Pa.|isbn=978-0-89871-534-7|pages=195|edition=2nd ed. --}}</ref>

If <math>\kappa(A)</math> is large, [[preconditioning]] is used to replace the original system <math>\mathbf{A x}-\mathbf{b} = 0</math> with <math>\mathbf{M}^{-1}(\mathbf{A x}-\mathbf{b}) = 0</math> so that <math>\kappa(\mathbf{M}^{-1}\mathbf{A})</math> gets smaller than <math>\kappa(\mathbf{A})</math>, see below.

==The preconditioned conjugate gradient method==
{{See also|Preconditioner}}
In most cases, [[preconditioning]] is necessary to ensure fast convergence of the conjugate gradient method. The preconditioned conjugate gradient method takes the following form:

:<math>\mathbf{r}_0 := \mathbf{b} - \mathbf{A x}_0</math>
:<math>\mathbf{z}_0 := \mathbf{M}^{-1} \mathbf{r}_0</math>
:<math>\mathbf{p}_0 := \mathbf{z}_0</math>
:<math>k := 0 \, </math>
:'''repeat'''
::<math>\alpha_k := \frac{\mathbf{r}_k^\mathrm{T} \mathbf{z}_k}{\mathbf{p}_k^\mathrm{T} \mathbf{A p}_k}</math>
::<math>\mathbf{x}_{k+1} := \mathbf{x}_k + \alpha_k \mathbf{p}_k</math>
::<math>\mathbf{r}_{k+1} := \mathbf{r}_k - \alpha_k \mathbf{A p}_k</math>
::'''if''' '''r'''''k''+1 is sufficiently small '''then''' exit loop '''end if'''
::<math>\mathbf{z}_{k+1} := \mathbf{M}^{-1} \mathbf{r}_{k+1}</math>
::<math>\beta_k := \frac{\mathbf{z}_{k+1}^\mathrm{T} \mathbf{r}_{k+1}}{\mathbf{z}_k^\mathrm{T} \mathbf{r}_k}</math>
::<math>\mathbf{p}_{k+1} := \mathbf{z}_{k+1} + \beta_k \mathbf{p}_k</math>
::<math>k := k + 1 \, </math>
:'''end repeat'''
:The result is '''x'''''k''+1

The above formulation is equivalent to applying the conjugate gradient method without preconditioning to the system{{ref label|agonizing_pain|1|^}}
:<math>\mathbf{E}^{-1}\mathbf{A}(\mathbf{E}^{-1})^\mathrm{T}\mathbf{\hat{x}}=\mathbf{E}^{-1}\mathbf{b}</math>
where <math>\mathbf{EE}^\mathrm{T}=\mathbf{M}</math> and <math>\mathbf{\hat{x}}=\mathbf{E}^\mathrm{T}\mathbf{x}</math>.

The preconditioner matrix '''M''' has to be symmetric positive-definite and fixed, i.e., cannot change from iteration to iteration.
If any of these assumptions on the preconditioner is violated, the behavior of the preconditioned conjugate gradient method may become unpredictable.

An example of a commonly used [[preconditioner]] is the [[incomplete Cholesky factorization]].

==The flexible preconditioned conjugate gradient method==

In numerically challenging applications, sophisticated preconditioners are used, which may lead to variable preconditioning, changing between iterations. Even if the preconditioner is symmetric positive-definite on every iteration, the fact that it may change makes the arguments above invalid, and in practical tests leads to a significant slow down of the convergence of the algorithm presented above. Using the [[nonlinear conjugate gradient method|Polak–Ribière]] formula

::<math>\beta_k := \frac{\mathbf{z}_{k+1}^\mathrm{T} \left(\mathbf{r}_{k+1}-\mathbf{r}_{k}\right)}{\mathbf{z}_k^\mathrm{T} \mathbf{r}_k}</math>

instead of the [[nonlinear conjugate gradient method|Fletcher–Reeves]] formula

::<math>\beta_k := \frac{\mathbf{z}_{k+1}^\mathrm{T} \mathbf{r}_{k+1}}{\mathbf{z}_k^\mathrm{T} \mathbf{r}_k}</math>

may dramatically improve the convergence in this case.<ref>{{cite journal |doi=10.1137/S1064827597323415 |title=Inexact Preconditioned Conjugate Gradient Method with Inner-Outer Iteration |year=1999 |last1=Golub |first1=Gene H. |last2=Ye |first2=Qiang |journal=SIAM Journal on Scientific Computing |volume=21 |issue=4 |pages=1305}}</ref> This version of the preconditioned conjugate gradient method can be called<ref>{{cite journal|doi=10.1137/S1064827599362314|title=Flexible Conjugate Gradients|year=2000|last1=Notay|first1=Yvan|journal=SIAM Journal on Scientific Computing|volume=22|issue=4|pages=1444}}</ref> '''flexible,''' as it allows for variable preconditioning. The implementation of the flexible version requires storing an extra vector. For a fixed preconditioner, <math>\mathbf{z}_{k+1}^\mathrm{T} \mathbf{r}_{k}=0,</math> so both formulas for <math>\beta_k</math> are equivalent in exact arithmetic, i.e., without the [[round-off error]].

The mathematical explanation of the better convergence behavior of the method with the [[nonlinear conjugate gradient method|Polak–Ribière]] formula is that the method is '''locally optimal''' in this case, in particular, it converges not slower than the locally optimal steepest descent method.<ref>{{cite journal|doi=10.1137/060675290|title=Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning|year=2008|last1=Knyazev|first1=Andrew V.|last2=Lashuk|first2=Ilya|journal=SIAM Journal on Matrix Analysis and Applications|volume=29|issue=4|pages=1267}}</ref>

==The conjugate gradient method vs. the locally optimal steepest descent method==

In both the original and the preconditioned conjugate gradient methods one only needs to always set <math>\beta_k := 0</math> in order to turn them into locally optimal, using the [[line search]], [[steepest descent]] methods. With this substitution, vectors <math>\mathbf{p}</math> are always the same as vectors <math>\mathbf{z}</math>, so there is no need to store vectors <math>\mathbf{p}</math>. Thus, every iteration of these [[steepest descent]] methods is a bit cheaper compared to that for the conjugate gradient methods. However, the latter converge faster, unless a (highly) variable [[preconditioner]] is used, see above.

==Derivation of the method==
{{main|Derivation of the conjugate gradient method}}

The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction method for optimization, and variation of the [[Arnoldi iteration|Arnoldi]]/[[Lanczos iteration|Lanczos]] iteration for [[eigenvalue]] problems. Despite differences in their approaches, these derivations share a common topic—proving the orthogonality of the residuals and conjugacy of the search directions. These two properties are crucial to developing the well-known succinct formulation of the method.

==Conjugate gradient on the normal equations==
The conjugate gradient method can be applied to an arbitrary ''n''-by-''m'' matrix by applying it to [[normal equations]] '''A'''T'''A''' and right-hand side vector '''A'''T'''b''', since '''A'''T'''A''' is a symmetric [[Positive-definite matrix#Negative-definite.2C_semidefinite_and_indefinite_matrices|positive-semidefinite]] matrix for any '''A'''. The result is conjugate gradient on the normal equations (CGNR).

: '''A'''T'''Ax''' = '''A'''T'''b'''

As an iterative method, it is not necessary to form '''A'''T'''A''' explicitly in memory but only to perform the matrix-vector and transpose matrix-vector multiplications. Therefore CGNR is particularly useful when ''A'' is a [[sparse matrix]] since these operations are usually extremely efficient. However the downside of forming the normal equations is that the [[condition number]] κ('''A'''T'''A''') is equal to κ2('''A''') and so the rate of convergence of CGNR may be slow and the quality of the approximate solution may be sensitive to roundoff errors. Finding a good [[preconditioner]] is often an important part of using the CGNR method.

Several algorithms have been proposed (e.g., CGLS, LSQR). The LSQR algorithm purportedly has the best numerical stability when '''A''' is ill-conditioned, i.e., '''A''' has a large [[condition number]].

==See also==
* [[Biconjugate gradient method]] (BiCG)
* [[Conjugate residual method]]
* [[Nonlinear conjugate gradient]] method
* [[Iterative_method#Linear_systems|Iterative method. Linear systems]]
* [[Preconditioning]]
* [[Belief_propagation#Gaussian_belief_propagation_.28GaBP.29|Gaussian Belief Propagation]]
* [[Krylov subspace]]

==Notes==
{{Reflist}}

==References==
The conjugate gradient method was originally proposed in
*{{cite journal|last = [[Magnus Hestenes|Hestenes]]|first = [[Magnus Hestenes|Magnus R.]]|coauthors = [[Eduard Stiefel|Stiefel, Eduard]]|title = Methods of Conjugate Gradients for Solving Linear Systems|journal = Journal of Research of the National Bureau of Standards|volume = 49|issue = 6|date=December 1952|url = http://nvl.nist.gov/pub/nistpubs/jres/049/6/V49.N06.A08.pdf|format=PDF}}
Descriptions of the method can be found in the following text books:
* {{cite book|first=Kendell A. |last=Atkinson |year=1988|title=An introduction to numerical analysis|edition=2nd |chapter= Section 8.9|publisher= John Wiley and Sons| isbn= 0-471-50023-2}}
* {{cite book|first=Mordecai|last= Avriel |year=2003|title=Nonlinear Programming: Analysis and Methods|publisher= Dover Publishing| isbn= 0-486-43227-0}}
* {{cite book|first1=Gene H. |last1=Golub |first2= Charles F.|last2= Van Loan|title=Matrix computations|edition=3rd|chapter= Chapter 10|publisher= Johns Hopkins University Press| isbn =0-8018-5414-8}}
* {{cite book|first=Yousef|last= Saad|title=Iterative methods for sparse linear systems|edition=2nd |chapter=Chapter 6
|publisher= SIAM| isbn= 978-0-89871-534-7}}

==External links==
* {{springer|title=Conjugate gradients, method of|id=p/c025030}}
* [http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf An Introduction to the Conjugate Gradient Method Without the Agonizing Pain] by Jonathan Richard Shewchuk.
* [http://www-users.cs.umn.edu/~saad/books.html Iterative methods for sparse linear systems] by Yousef Saad
* [http://www.stanford.edu/group/SOL/software/lsqr.html LSQR: Sparse Equations and Least Squares] by Christopher Paige and Michael Saunders.
* [http://www.onmyphd.com/?p=conjugate.gradient.method Derivation of fast implementation of conjugate gradient method and interactive example]

{{Numerical linear algebra}}

{{DEFAULTSORT:Conjugate Gradient Method}}
[[Category:Numerical linear algebra]]
[[Category:Gradient methods]]
[[Category:Articles with example pseudocode]]

Rank–nullity theorem

2014-01-29T11:41:48Z

137.205.238.248:

[[File:Disc integration.svg|thumb|right|170px]]
{{Calculus |Integral}}

'''Disk integration''', also known in [[integral calculus]] as the '''disk method''', is a means of calculating the [[volume]] of a [[solid of revolution]] of a solid-state material when [[Integral|integrating]] along the [[axis of revolution]]. This method models the resulting three-dimensional shape as a "stack" of an infinite number of disks of varying radius and infinitesimal thickness. It is also possible to use the same principles with "washers" instead of "disks" (the "'''washer method'''") to obtain "hollow" solids of revolutions.

== Definition ==

===Function of ''x''===
If the function to be revolved is a function of <math>x</math>, the following integral represents the volume of the solid of revolution:

: <math>\pi \int_a^b [R(x)]^2\ \mathrm{d}x</math>

where <math>R(x)</math> is the distance between the function and the axis of rotation. This works only if the [[axis of rotation]] is horizontal (example: <math>y=3</math> or some other constant).

===Function of ''y''===
If the function to be revolved is a function of <math>y</math>, the following integral will obtain the volume of the solid of revolution:

: <math>\pi \int_c^d [R(y)]^2\ \mathrm{d}y</math>

where <math>R(y)</math> is the distance between the function and the axis of rotation. This works only if the [[axis of rotation]] is vertical (example: <math>x=4</math> or some other constant).

===Washer method===
To obtain a "hollow" solid of revolution (the "washer method"), the procedure would be to take the volume of the inner solid of revolution and subtract it from the volume of the outer solid of revolution. This can be calculated in a single integral similar to the following:

: <math>\pi \int_a^b (\left[R_O(x)\right]^2 - \left[R_I(x)\right]^2) \mathrm{d}x</math>

where <math>R_O(x)</math> is the function that is farthest from the axis of rotation and <math>R_I(x)</math> is the function that is closest to the axis of rotation. One should take caution not to evaluate the square of the difference of the two functions, but to evaluate the difference of the squares of the two functions.

: <math> [R_O(x)]^2 - [R_I(x)]^2\ \not\equiv \; [R_O(x) - R_I(x)]^2</math>

''The above formula only works for revolutions about the x-axis.''

To rotate about any horizontal axis, simply subtract from that axis each formula:

if <math>h</math> is the value of a horizontal axis, then the volume =

: <math>\pi \int_a^b ([h-R_O(x)]^2 - [h-R_I(x)]^2)\, \mathrm{d}x.</math>

For example, to rotate the region between <math> y=-2x+x^2 </math> and <math> y=x </math>

along the axis <math>y=4</math>, one would integrate as follows:

: <math>\pi \int_0^3 ([4-(-2x+x^2)]^2 - [4-x]^2)\, \mathrm{d}x.</math>

The bounds of integration are the zeros of the first equation minus the second. Note that when you integrate along an axis other than the <math>x</math>, the further axis may not be that obvious. In the previous example, even though <math>y=x</math> is further up than <math> y=-2x+x^2 </math>, it is the inner axis since it is closer to <math>y=4</math>

The same idea can be applied to both the y-axis and any other vertical axis. You simply must solve each equation for <math>x</math> before you insert them into the integration formula.

== See also ==
* [[Solid of revolution]]
* [[Shell integration]]

==References==
{{Refbegin|normalfont=yes}}
*{{cite web|title=Volumes of Solids of Revolution|url=http://www.cliffsnotes.com/study_guide/topicArticleId-39909,articleId-39907.html|work=CliffsNotes.com|accessdate=July 12, 2013}}
*{{MathWorld|title=Method of Disks|urlname=MethodofDisks}} {{accessdate|July 12, 2013}}
*Frank Ayres, Elliott Mendelson. ''Schaum's outlines: Calculus''. McGraw-Hill Professional 2008, ISBN 978-0-07-150861-2. pp. 244–248 ({{Google books|Ag26M8TII6oC|online copy|page=244}}{{accessdate|July 12, 2013}})
{{Refend}}

[[Category:Integral calculus]]
[[Category:Volume]]

Nth root

2013-12-12T14:34:08Z

137.205.238.132:

{{Signs of death}}

'''Algor mortis''' ([[Latin]]: ''algor''—coldness; ''mortis''—of death) is the reduction in body temperature following [[death]]. This is generally a steady decline until matching [[ambient temperature]], although external factors can have a significant influence.

== Use in forensic science and crime solving ==

A measured rectal temperature can give some indication of the time of death. Although the [[heat conduction]] which leads to body cooling follows an [[exponential decay]] curve, it can be approximated as a linear process: 2° Celsius during the first hour and 1° Celsius per hour until the body nears ambient temperature.

The '''Glaister equation'''<ref>http://www.fmap.archives.gla.ac.uk/DC403/DC403_page.htm</ref><ref>{{cite book|title=Forensic Medicine|author=Guharaj, P. V.|chapter=Cooling of the body (algor mortis)|year=2003|edition=2nd|pages=61–62|publisher=Longman Orient|location=Hyderabad|url=http://books.google.com/books?isbn=8125024883}}</ref> estimates the hours elapsed since death as a [[linear function]] of the [[rectal temperature]]:

:<math>
\frac{98.4\,^{\circ}{\rm F} - \text{rectal temperature in Fahrenheit}}{1.5}
</math>


As [[decomposition]] occurs the internal body temperature tends to rise again.

== Factors affecting the rate of algor mortis ==
Generally, temperature change is considered an inaccurate means of determining time of death, as the rate of change is affected by several key factors including:<ref>http://www.crimewatchcanada.com/index.php?option=com_content&view=article&id=79:changes-to-a-body-after-death&catid=41:january-february-2010-issue-76&Itemid=53</ref>
* Stability or fluctuation of the ambient temperature.
* The thermal conductivity of the surface on which a body lies.
* Diseases or drugs which increase body temperature raising the starting temperature of the corpse at the time of death
* The existence of a "temperature plateau",<ref>{{cite journal|last=Kaliszan|first=M.|title=Verification of the exponential model of body temperature decrease after death in pigs|journal=Experimental Physiology|date=20 May 2005|volume=90|issue=5|pages=727–738|doi=10.1113}}</ref> a highly variable period of time in which the body does not cool.

==References==
{{reflist}}
* {{cite book | first=Richard| last=Saferstein| year=2004| title=Criminalistics An Introduction to Forensic Science| edition=8th| publisher=Pearson Prentice Hall| isbn=0-13-113706-9 }}
* Karen T. Taylor, "Forensic art and illustration", CRC Press, 2000, ISBN 0-8493-8118-5, p. 308
* Robert G. Mayer, "Embalming: history, theory, and practice", McGraw-Hill Professional, 2005, ISBN 0-07-143950-1, p. 106
* Calixto Machado, "Brain death: a reappraisal", Springer, 2007, ISBN 0-387-38975-X, pp. 73–74

==External links==
* [http://www.arrakis.es/~jacoello/date.pdf Standards Employed to Determine Time of Death]

{{DEFAULTSORT:Algor Mortis}}
[[Category:Signs of death]]
[[Category:Latin medical phrases]]

{{med-sign-stub}}
{{death-stub}}
{{Latin-vocab-stub}}

Inner measure

2013-11-02T13:46:17Z

137.205.0.119: small edit - poor grammar

Structural analysis is mainly concerned with finding out the behaviour of a physical structure when subjected to force. This action can be in the form of load due to the weight of things such as people, furniture, wind, snow, etc. or some other kind of excitation such as an earthquake, shaking of the ground due to a blast nearby, etc. In essence all these loads are dynamic, including the self-weight of the structure because at some point in time these loads were not there. The distinction is made between the dynamic and the static analysis on the basis of whether the applied action has enough acceleration in comparison to the structure's natural frequency. If a load is applied sufficiently slowly, the inertia forces (Newton's second law of motion) can be ignored and the analysis can be simplified as static analysis.
'''Structural dynamics''', therefore, is a type of [[structural analysis]] which covers the behaviour of [[structure]]s subjected to [[Dynamics (physics)|dynamic]] (actions having high acceleration) loading. Dynamic loads include people, wind, waves, traffic, [[earthquake]]s, and blasts. Any structure can be subjected to dynamic loading. Dynamic analysis can be used to find dynamic [[Displacement (vector)|displacements]], time history, and [[modal analysis]].

A dynamic analysis is also related to the inertia forces developed by a structure when it is excited by means of dynamic loads applied suddenly (e.g., wind blasts, explosion, earthquake).

A [[statics|static]] load is one which varies very slowly. A dynamic load is one which changes with time fairly quickly in comparison to the structure's natural frequency. If it changes slowly, the structure's response may be determined with static analysis, but if it varies quickly (relative to the structure's ability to respond), the response must be determined with a dynamic analysis.

Dynamic analysis for simple structures can be carried out manually, but for complex structures [[finite element analysis]] can be used to calculate the mode shapes and frequencies.

==Displacements==

A dynamic load can have a significantly larger effect than a static load of the same magnitude due to the structure's inability to respond quickly to the loading (by deflecting). The increase in the effect of a dynamic load is given by the [[dynamic amplification factor]] (DAF):

:<math> DAF = \frac{{u_{max}}}{{u_{static}}}</math>

where u is the deflection of the structure due to the applied load.

Graphs of dynamic amplification factors vs non-dimensional [[rise time]] (tr/T) exist for standard loading functions (for an explanation of rise time, see '''time history analysis''' below). Hence the DAF for a given loading can be read from the graph, the static deflection can be easily calculated for simple structures and the dynamic deflection found.

==Time history analysis==

A {{Not a typo|full time}} history will give the response of a structure over time during and after the application of a load. To find the {{Not a typo|full time}} history of a structure's response, you must solve the structure's [[equation of motion]].

===Example===

[[Image:Mass spring.svg|200px|right|Single degree of freedom system: simple mass spring model]]

A simple single [[Degrees of freedom (physics and chemistry)|degree of freedom]] [[system]] (a [[mass]], M, on a [[Spring (device)|spring]] of [[stiffness]], k for example) has the following equation of motion:

:<math>M{\ddot{x}} + kx = F(t)</math>
:
where <math>\ddot{x}</math> is the acceleration (the double [[derivative]] of the displacement) and x is the displacement.

If the loading F(t) is a [[Heaviside step function]] (the sudden application of a constant load), the solution to the equation of motion is:

:<math>x = \frac{{F_0}}{{k}}[1 - cos{(\omega t)}]</math>
:

where <math>\omega = \sqrt{\frac{{k}}{{M}}}</math> and the fundamental natural frequency, <math>f = \frac {\omega}{2\pi}</math>.

The static deflection of a single degree of freedom system is:

:<math>x_{static} = \frac{{F_0}}{{k}}</math>
:
so you can write, by combining the above formulae:

:<math>x = x_{static}[1 - cos(\omega t)]</math>

This gives the (theoretical) time history of the structure due to a load F(t), where the false assumption is made that there is no [[damping]].

Although this is too simplistic to apply to a real structure, the Heaviside Step Function is a reasonable model for the application of many real loads, such as the sudden addition of a piece of furniture, or the removal of a prop to a newly cast concrete floor. However, in reality loads are never applied instantaneously - they build up over a period of time (this may be very short indeed). This time is called the [[rise time]].

As the number of degrees of freedom of a structure increases it very quickly becomes too difficult to calculate the time history manually - real structures are analysed using [[non-linear]] [[finite element analysis]] software.

==Damping==

Any real structure will dissipate energy (mainly through friction). This can be modelled by modifying the DAF

:<math>DAF = 1 + e^{-c\pi}</math>
:
where <math>c=\frac{{\text {Damping Coefficient}}}{{\text{Critical Damping Coefficient}}}</math> and is typically 2%-10% depending on the type of construction:

* Bolted steel ~6%
* Reinforced concrete ~ 5%
* Welded steel ~ 2%
* Brick masonry ~ 10%

Generally damping would be ignored for non-transient events (such as wind loading or crowd loading), but would be important for transient events (for example, an [[impulse (physics)|impulse]] load such as an earthquake loading or bomb blast).

==Modal analysis==

A [[modal analysis]] calculates the frequency [[Normal mode|modes]] or natural frequencies of a given system, but not necessarily its full-time history response to a given input. The natural frequency of a system is dependent only on the [[stiffness]] of the structure and the [[mass]] which participates with the structure (including self-weight). It is not dependent on the load function.

It is useful to know the modal frequencies of a structure as it allows you to ensure that the frequency of any applied periodic loading will not coincide with a modal frequency and hence cause [[resonance]], which leads to large [[oscillations]].

The method is:

# Find the natural modes (the shape adopted by a structure) and natural frequencies
# Calculate the response of each mode
# Optionally superpose the response of each mode to find the full modal response to a given loading

===Energy method===

It is possible to calculate the frequency of different mode shape of system manually by the [[Energy principles in structural mechanics|energy method]]. For a given mode shape of a multiple degree of freedom system you can find an "equivalent" mass, stiffness and applied force for a single degree of freedom system. For simple structures the basic mode shapes can be found by inspection, but it is not a conservative method. Rayleigh's principle states:

"The frequency ω of an arbitrary mode of vibration, calculated by the energy method, is always greater than - or equal to - the fundamental frequency ωn."

For an assumed mode shape <math>\bar{u}(x)</math>, of a structural system with mass M; stiffness, EI ([[Young's modulus]], E, multiplied by the [[second moment of area]], I); and applied force, F(x):

:<math>\text{Equivalent mass, } M_{eq} = \int{M\bar{u}^2} du </math>
:
:<math>\text{Equivalent stiffness, }k_{eq} = \int{EI \bigg(\frac{{d^2\bar{u}}}{{dx^2}} \bigg)^2}dx</math>
:
:<math>\text{Equivalent force, }F_{eq} = \int{F\bar{u}}dx</math>
:
then, as above:

:<math>\omega = \sqrt{\frac{{k_{eq}}}{{M_{eq}}}}</math>

===Modal response===

The complete modal response to a given load F(x,t) is <math>v(x,t)=\sum{u_n(x,t)}</math>. The summation can be carried out by one of three common methods:

* Superpose complete time histories of each mode (time consuming, but exact)
* Superpose the maximum amplitudes of each mode (quick but conservative)
* Superpose the square root of the sum of squares (good estimate for well-separated frequencies, but unsafe for closely spaced frequencies)

To superpose the individual modal responses manually, having calculated them by the energy method:

Assuming that the rise time tr is known (T = 2π/ω), it is possible to read the DAF from a standard graph. The static displacement can be calculated with <math>u_{static}=\frac{F_{1,eq}}{k_{1,eq}}</math>. The dynamic displacement for the chosen mode and applied force can then be found from:

:<math>u_{max} = u_{static}DAF</math>

==Modal participation factor==

For real systems there is often mass participating in the [[forcing function (differential equations)|forcing function]] (such as the mass of ground in an [[earthquake]]) and mass participating in [[inertia]] effects (the mass of the structure itself, Meq). The [[modal participation factor]] Γ is a comparison of these two masses. For a single degree of freedom system Γ = 1.

:Γ <math> = \frac{\sum{M_n\bar{u}_n}}{\sum{M_n\bar{u}_n^2}}</math>

==External links==
* [http://sites.google.com/site/dyssolve/ DYSSOLVE: Dynamic System Solver] - An encrypted-source, lightweight, free-of-charge software that can be used to solve basic structural dynamics problems.
* [http://structdynviblab.mcgill.ca/ Structural Dynamics and Vibration Laboratory of McGill University]
* [http://frame3dd.sourceforge.net Frame3DD open source 3D structural dynamics analysis program]
* [http://www.noisestructure.com/products/FRF.php Frequency response function from modal parameters]
* [http://vibrationdata.wordpress.com/category/structural-dynamics/ Structural Dynamics Tutorials & Matlab scripts]

[[Category:Structural analysis]]
[[Category:Dynamics]]

Pound–Rebka experiment

2013-06-02T11:53:36Z

137.205.21.20:

In [[mathematics]], a [[subset]] <math>A</math> of a [[Polish space]] <math>X</math> is '''universally measurable''' if it is [[measurable set|measurable]] with respect to every [[complete measure|complete]] [[probability measure]] on <math>X</math> that measures all [[Borel set|Borel]] subsets of <math>X</math>. In particular, a universally measurable set of [[real number|real]]s is necessarily [[Lebesgue measurable]] (see [[#Finiteness condition]]) below.

Every [[analytic set]] is universally measurable. It follows from [[projective determinacy]], which in turn follows from sufficient [[large cardinal]]s, that every [[projective set]] is universally measurable.

==Finiteness condition==
The condition that the measure be a [[probability measure]]; that is, that the measure of <math>X</math> itself be 1, is less restrictive than it may appear. For example, Lebesgue measure on the reals is not a probability measure, yet every universally measurable set is Lebesgue measurable. To see this, divide the real line into countably many intervals of length 1; say, ''N''0=<nowiki>[0,1)</nowiki>, ''N''1=<nowiki>[1,2)</nowiki>, ''N''2=<nowiki>[-1,0)</nowiki>, ''N''3=<nowiki>[2,3)</nowiki>, ''N''4=<nowiki>[-2,-1)</nowiki>, and so on. Now letting μ be Lebesgue measure, define a new measure ν by
: <math>\nu(A)=\sum_{i=0}^\infty \frac{1}{2^{n+1}}\mu(A\cap N_i)</math>
Then easily ν is a probability measure on the reals, and a set is ν-measurable if and only if it is Lebesgue measurable. More generally a universally measurable set must be measurable with respect to every [[sigma-finite]] measure that measures all Borel sets.

== Example contrasting with Lebesgue measurability ==
Suppose <math>A</math> is a subset of [[Cantor space]] <math>2^\omega</math>; that is, <math>A</math> is a set of infinite [[sequence]]s of zeroes and ones. By putting a binary point before such a sequence, the sequence can be viewed as a [[real number]] between 0 and 1 (inclusive), with some unimportant ambiguity. Thus we can think of <math>A</math> as a subset of the interval <nowiki>[0,1]</nowiki>, and evaluate its [[Lebesgue measure]]. That value is sometimes called the '''coin-flipping measure''' of <math>A</math>, because it is the [[probability]] of producing a sequence of heads and tails that is an element of <math>A</math>, upon flipping a fair coin infinitely many times.

Now it follows from the [[axiom of choice]] that there are some such <math>A</math> without a well-defined Lebesgue measure (or coin-flipping measure). That is, for such an <math>A</math>, the probability that the sequence of flips of a fair coin will wind up in <math>A</math> is not well-defined. This is a pathological property of <math>A</math> that says that <math>A</math> is "very complicated" or "ill-behaved".

From such a set <math>A</math>, form a new set <math>A'</math> by performing the following operation on each sequence in <math>A</math>: Intersperse a 0 at every even position in the sequence, moving the other bits to make room. Now <math>A'</math> is intuitively no "simpler" or "better-behaved" than <math>A</math>. However, the probability that the sequence of flips of a fair coin will wind up in <math>A'</math> ''is'' well-defined, for the rather silly reason that the probability is zero (to get into <math>A'</math>, the coin must come up tails on every even-numbered flip).

For such a set of sequences to be ''universally'' measurable, on the other hand, an arbitrarily ''biased'' coin may be used--even one that can "remember" the sequence of flips that has gone before--and the probability that the sequence of its flips ends up in the set, must be well-defined. Thus the <math>A'</math> described above is ''not'' universally measurable, because we can test it against a coin that always comes up tails on even-numbered flips, and is fair on odd-numbered flips.

== References ==
* {{citation
| author = Alexander Kechris
| title = Classical Descriptive Set Theory
| publisher = Springer
| series = Graduate Texts in Mathematics
| volume = 156
| year = 1995
| isbn = 0-387-94374-9
}}
* {{citation
| author = Nishiura Togo
| title = Absolute Measurable Spaces
| publisher= Cambridge University Press
| year= 2008
| isbn = 0-521-87556-0
}}

[[Category:Descriptive set theory]]
[[Category:Determinacy]]
[[Category:Measure theory]]

Periodic travelling wave

2013-05-30T11:34:06Z

137.205.56.75: /* Applications of periodic travelling waves */

{{Infobox enzyme
| Name = Sulfoacetaldehyde dehydrogenase (acylating)
| EC_number = 1.2.1.81
| CAS_number =
| IUBMB_EC_number = 1/2/1/81
| GO_code =
| image =
| width =
| caption =
}}
'''Sulfoacetaldehyde dehydrogenase (acylating)''' ({{EC number|1.2.1.81}}, ''SauS'') is an [[enzyme]] with system name ''2-sulfoacetaldehyde:NADP+ oxidoreductase (CoA-acetylating)''.<ref>{{cite journal | title = Sulfoacetate is degraded via a novel pathway involving sulfoacetyl-CoA and sulfoacetaldehyde in Cupriavidus necator H16 |author = Weinitschke, S., Hollemeyer, K., Kusian, B., Bowien, B., Smits, T.H. and Cook, A.M. |journal = J. Biol. Chem. |year = 2010 |volume = 285 |pages = 35249–35254 |pmid = 20693281 |doi=10.1074/jbc.M110.127043 |issue=46 |pmc=2975148}}</ref> This enzyme [[catalysis|catalyses]] the following [[chemical reaction]]

: 2-sulfoacetaldehyde + CoA + NADP+ <math>\rightleftharpoons</math> sulfoacetyl-CoA + NADPH + H+

The enzyme is involved in degradation of [[sulfoacetate]].

== References ==
{{reflist}}

== External links ==
* {{MeshName|Sulfoacetaldehyde+dehydrogenase+(acylating)}}

[[Category:EC 1.2.1]]

Locally profinite group

2012-12-11T16:04:18Z

137.205.127.174: corrected typo in introduction

In the branch of [[abstract algebra]] called [[ring theory]], the '''double centralizer theorem''' can refer to any one of several similar results. These results concern the [[centralizer and normalizer|centralizer of a subring]] ''S'' of a ring ''R'', denoted '''C'''''R''(''S'') in this article. It is always the case that '''C'''''R''('''C'''''R''(''S'')) contains ''S'', and a double centralizer theorem gives conditions on ''R'' and ''S'' that guarantee that '''C'''''R''('''C'''''R''(''S'')) is ''equal'' to ''S''.

==Statements of the theorem==

===Motivation===
The centralizer of a subring ''S'' of ''R'' given by

:<math>\mathrm{C}_R(S)=\{r\in R \mid rs=sr \text{ for all } s\in S\}.\,</math>

Clearly '''C'''''R''('''C'''''R''(''S'')) &supe; ''S'', but it is not always the case that one can say the two sets are equal. The double centralizer theorems give conditions under which one can conclude that equality occurs.

There is another special case of interest. Let ''M'' be a right ''R'' module and give ''M'' the natural left ''E''-module structure, where ''E'' is End(''M''), the ring of endomorphisms of the abelian group ''M''. Every map ''m''''r'' given by ''m''''r''(''x'') = ''xr'' creates an additive endomorphism of ''M'', that is, an element of ''E''. The map ''r'' → ''m''''r'' is a ring homomorphism of ''R'' into the ring ''E'', and we denote the image of ''R'' inside of ''E'' by ''R''''M''. It can be checked that the [[kernel (algebra)|kernel]] of this canonical map is the [[annihilator (ring theory)|annihilator]] Ann(''M''''R''). Therefore by an [[Isomorphism_theorem#First_isomorphism_theorem_2|isomorphism theorem]] for rings, ''R''''M'' is isomorphic to the quotient ring ''R''/Ann(''M''''R''). Clearly when ''M'' is a [[faithful module]], ''R'' and ''R''''M'' are isomorphic rings.

So now ''E'' is a ring with ''R''''M'' as a subring, and '''C'''''E''(''R''''M'') may be formed. By definition one can check that '''C'''''E''(''R''''M'') = End(''M''''R''), the ring of ''R'' module endomorphisms of ''M''. Thus if it occurs that '''C'''''E''('''C'''''E''(''R''''M'')) = ''R''''M'', this is the same thing as saying '''C'''''E''(End(''M''''R'')) = ''R''''M''.

===Central simple algebras===
Perhaps the most common version is the version for [[central simple algebra]]s, as it appears in {{harv|Knapp|2007|loc=p.115}}:

'''Theorem''': If ''A'' is a finite dimensional central simple algebra over a field ''F'' and ''B'' is a simple subalgebra of ''A'', then '''C'''''A''('''C'''''A''(''B'')) = ''B'', and moreover the dimensions satisfy

:<math>\mathrm{dim}_F(B)\cdot\mathrm{dim}_F(\mathrm{C}_A(B))=\mathrm{dim}_F(A).\,</math>

===Artinian rings===
The following generalized version for [[Artinian ring]]s (which include finite dimensional algebras) appears in {{harv|Isaacs|2009|loc=p.187}}. Given a [[simple module|simple]] ''R'' module ''U''''R'', we will borrow notation from the above motivation section including ''R''''U'' and ''E''=End(''U''). Additionally, we will write ''D''=End(''U''''R'') for the subring of ''E'' consisting of ''R''-homomorphisms. By [[Schur's lemma]], ''D'' is a [[division ring]].

'''Theorem''': Let ''R'' be a right Artinian ring with a simple right module ''U''''R'', and let ''R''''U'', ''D'' and ''E'' be given as in the previous paragraph. Then
:<math>R_U=\mathrm{C}_E(\mathrm{C}_E(R_U))\,</math>.

;Remarks:
* In this version, the rings are chosen with the intent of proving the [[Jacobson density theorem]]. Notice that it only concludes that a particular subring has the centralizer property, in contrast to the central simple algebra version.
* Since algebras are normally defined over commutative rings, and all the involved rings above may be noncommutative, it's clear that algebras are not necessarily involved.
* If ''U'' is additionally a [[faithful module]], so that ''R'' is a right [[primitive ring]], then ''R''''U'' is ring isomorphic to ''R''.

===Polynomial identity rings===
In {{harv|Rowen|1980|loc=p.154}}, a version is given for [[polynomial identity ring]]s. The notation Z(''R'') will be used to denote the [[center of a ring]] ''R''.

'''Theorem''': If ''R'' is a [[simple ring|simple]] polynomial identity ring, and ''A'' is a simple Z(''R'') subalgebra of ''R'', then '''C'''''R''('''C'''''R''(''A'')) = ''A''.

;Remarks
* This version can be considered to be "between" the central simple algebra version and the Artinian ring version. This is because simple polynomial identity rings are Artinian,<ref>They are full matrix rings over polynomial identity division rings, according to {{harvtxt|Rowen|1980|page=151}}</ref> but unlike the Artinian version, the conclusion still refers to all central simple subrings of ''R''.

==Double centralizer property==
{{main|Balanced module}}

A module ''M'' is said to have the ''[[balanced module|double centralizer property]]'' or to be a ''[[balanced module]]'' if '''C'''''E''('''C'''''E''(''R''''M'')) = ''R''''M'', where ''E'' = End(''M'') and ''R''''M'' are as given in the motivation section. In this terminology, the Artinian ring version of the double centralizer theorem states that simple right modules for right Artinian rings are balanced modules.

== Notes ==

{{Reflist}}

==References==

*{{citation |last=Isaacs|first= I. Martin |title=Algebra: a graduate course |series=Graduate Studies in Mathematics |volume=100 |note=Reprint of the 1994 original |publisher=American Mathematical Society |place=Providence, RI |year=2009 |pages=xii+516 |isbn=978-0-8218-4799-2 |mr=2472787}}

*{{citation |last=Knapp|first=Anthony W. |title=Advanced algebra |series=Cornerstones |publisher=Birkhäuser Boston Inc. |place=Boston, MA |year=2007 |pages=xxiv+730 |isbn=978-0-8176-4522-9 |mr=2360434}}

*{{citation |last=Rowen|first=Louis Halle |title=Polynomial identities in ring theory |series=Pure and Applied Mathematics |volume=84 |publisher=Academic Press Inc. [Harcourt Brace Jovanovich Publishers] |place=New York |year=1980 |pages=xx+365 |isbn=0-12-599850-3|mr=576061}}



[[Category:Theorems in abstract algebra]]
[[Category:Ring theory]]

First-hitting-time model

2012-07-30T14:52:57Z

137.205.57.153: Nope, this is not a Cox model.

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Universal coefficient theorem

2012-06-19T13:00:29Z

137.205.56.204: Better bracket symbols in a formula

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Chernoff's distribution

2012-05-21T14:53:20Z

137.205.146.203:

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Coronal seismology

2012-05-10T09:27:29Z

137.205.160.3:

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Method of matched asymptotic expansions

2012-04-13T16:32:30Z

137.205.236.106: /* Outer and inner solutions */

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Residual-resistance ratio

2011-12-03T20:10:22Z

137.205.17.121:

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P-wave modulus

2011-02-27T17:27:48Z

137.205.222.193:

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