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In [[functional analysis]], the '''Friedrichs extension''' is a [[canonical form|canonical]] [[self-adjoint operator|self-adjoint]] extension of a non-negative densely defined [[self-adjoint operator#Symmetric operators|symmetric operator]]. It is named after the mathematician [[Kurt Friedrichs]]. This extension is particularly useful in situations where an operator may fail to be [[essentially self-adjoint]] or whose essential self-adjointness is difficult to show. 
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An operator ''T'' is non-negative if
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:<math> \langle \xi \mid T \xi \rangle \geq 0 \quad \xi \in \operatorname{dom}\ T </math>
  <li>[http://bbs.wabaka.com/forum.php?mod=viewthread&tid=99169 http://bbs.wabaka.com/forum.php?mod=viewthread&tid=99169]</li>
 
 
==Examples==
  <li>[http://web.zaiwww.com/news/html/?193226.html http://web.zaiwww.com/news/html/?193226.html]</li>
 
 
'''Example'''. Multiplication by a non-negative function on an ''L''<sup>2</sup> space is a non-negative self-adjoint operator.
  <li>[http://www.think-group.cn/VK/blog/article.php?type=blog&cid=7&itemid=1725387 http://www.think-group.cn/VK/blog/article.php?type=blog&cid=7&itemid=1725387]</li>
 
 
'''Example'''. Let ''U'' be an open set in '''R'''<sup>''n''</sup>. On ''L''<sup>2</sup>(''U'') we consider [[differential operator]]s of the form
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:<math> [T \phi](x) = -\sum_{i,j} \partial_{x_i} \{a_{i j}(x) \partial_{x_j} \phi(x)\} \quad x \in U, \phi \in  \operatorname{C}_0^\infty(U), </math>
</ul>
 
where the functions ''a''<sub>''i j''</sub> are infinitely differentiable real-valued functions on ''U''. We consider ''T'' acting on the dense subspace of infinitely differentiable complex-valued functions of compact support, in symbols
 
:<math> \operatorname{C}_0^\infty(U) \subseteq L^2(U). </math>
 
If for each ''x'' ∈ ''U'' the ''n'' &times; ''n'' matrix
 
:<math> \begin{bmatrix} a_{1 1}(x) & a_{1 2}(x) & \cdots & a_{1 n}(x) \\ a_{2 1}(x) & a_{2 2} (x) & \cdots & a_{2 n}(x) \\ \vdots & \vdots & \ddots & \vdots \\ a_{n 1}(x) & a_{n 2}(x) & \cdots & a_{n n}(x)  \end{bmatrix} </math>
 
is non-negative semi-definite, then ''T'' is a non-negative operator. This means (a) that the matrix is [[hermitian matrix|hermitian]] and
 
:<math> \sum_{i, j} a_{i j }(x) c_i \overline{c_j} \geq 0 </math>
 
for every choice of complex numbers ''c''<sub>1</sub>, ..., ''c''<sub>n</sub>. This is proved using [[integration by parts]].  
 
These operators are [[elliptic operator|elliptic]] although in general elliptic operators may not be non-negative. They are however bounded from below.
 
== Definition of Friedrichs extension ==
 
The definition of the Friedrichs extension is based on the theory of closed positive forms on Hilbert spaces.  
If ''T'' is non-negative, then
 
:<math> \operatorname{Q}(\xi, \eta) = \langle \xi \mid T \eta \rangle + \langle  \xi \mid \eta \rangle </math>
 
is a sesquilinear form on dom ''T'' and
 
:<math> \operatorname{Q}(\xi, \xi) = \langle \xi \mid T \xi\rangle + \langle  \xi \mid \xi \rangle \geq \|\xi\|^2.</math>
 
Thus Q defines an inner product on dom ''T''. Let ''H''<sub>1</sub> be the [[complete space|completion]] of dom ''T'' with respect to Q. ''H''<sub>1</sub> is an abstractly defined space; for instance its elements can be represented as [[equivalence class]]es of [[Cauchy sequence]]s of elements of dom ''T''.   It is not obvious that all elements in ''H''<sub>1</sub> can identified with elements of ''H''.  However, the following can be proved:
 
The canonical inclusion
 
:<math> \operatorname{dom}\ T \rightarrow H </math>
 
extends to an ''injective'' continuous map ''H''<sub>1</sub> → ''H''. We regard ''H''<sub>1</sub> as a subspace of ''H''.
 
Define an operator ''A'' by
 
: <math> \operatorname{dom}\ A = \{\xi \in H_1: \phi_\xi: \eta \mapsto \operatorname{Q}(\xi, \eta) \mbox{ is bounded linear.} \} </math>
 
In the above formula, ''bounded'' is relative to the topology on ''H''<sub>1</sub> inherited from ''H''. By the [[Riesz representation theorem]]  applied to the linear functional φ<sub>ξ</sub> extended to ''H'', there is a unique ''A'' ξ ∈ ''H'' such that
 
:<math> \operatorname{Q}(\xi,\eta) = \langle  A \xi \mid  \eta \rangle \quad \eta \in H_1 </math>
 
'''Theorem'''. ''A'' is a non-negative self-adjoint operator such that ''T''<sub>1</sub>=''A'' - I extends ''T''.
 
''T''<sub>1</sub> is the Friedrichs extension of ''T''.
 
== Krein's theorem on non-negative self-adjoint extensions ==
 
[[M. G. Krein]] has given an elegant characterization of all non-negative self-adjoint extensions of a non-negative symmetric operator ''T''.
 
If ''T'', ''S'' are non-negative self-adjoint operators, write
 
:<math> T \leq S </math>
 
if, and only if,
 
* <math> \operatorname{dom}(S^{1/2}) \subseteq \operatorname{dom}(T^{1/2}) </math>
 
* <math> \langle T^{1/2} \xi \mid T^{1/2} \xi \rangle \leq  \langle S^{1/2} \xi \mid S^{1/2} \xi \rangle \quad \forall \xi \in \operatorname{dom}(S^{1/2}) </math>
 
'''Theorem'''. There are unique self-adjoint extensions ''T''<sub>min</sub> and ''T''<sub>max</sub> of any non-negative symmetric operator ''T'' such that
 
:<math> T_{\mathrm{min}} \leq T_{\mathrm{max}}, </math>
 
and every non-negative self-adjoint extension ''S'' of  ''T'' is between ''T''<sub>min</sub> and ''T''<sub>max</sub>, i.e.
 
:<math> T_{\mathrm{min}} \leq S \leq T_{\mathrm{max}}. </math>
 
The Friedrichs extension of ''T'' is ''T''<sub>max</sub>.<ref>A. Alfonso and B. Simon, The Birman-Krein-Vishik theory of self-adjoint extensions of semibounded operators. J. Operator Theory 4 (1980), 251-270.</ref>
 
==See also==
* [[Energetic extension]]
* [[Extensions of symmetric operators]]
 
== Notes ==
{{Reflist}}
 
==References==
* [[Naum Akhiezer|N. I. Akhiezer]] and I. M. Glazman, ''Theory of Linear Operators in Hilbert Space'', Pitman, 1981.
 
[[Category:Operator theory]]

Revision as of 22:43, 25 February 2014

Penile Secrets Nike Geneve

We laugh a lot, never fight, never raise our voices at each other, always love each other's company, and say "I love you" dozens of times a day. ​​It never seems to get old. Not that we don't have disagreements, but we always respect one another. Was getting more cranky than usual, Atwood says. I Nike Geneve wasn sure who I was anymore. Stack Exchange was so much a part of me.

I guarantee you that bf4 will cost 2gb or more on ultra. If you want to future proof then definitely get more than 2gb of vram! Seriously the most intensive games right now are pushing up to 1.8gb. And about the cooler master : yea its kinda pricey but considering the fact that you wouldn't need to liquid cool right away would save you quite a bit of $$$Also it is a good idea to OC any cpu to at least 4.0ghz if you have a rea​​lly .

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You also still get the period damage ticks as you normally would. AE situationsFor low health, Vibram Five Fingers burst AE situations (say things that last less than 1015s or so), Starfall Astral Storm is actually pretty strong. If you can have mushrooms preplaced without interrupting your DPS, then do so otherwise, don't worry about them..

The good thing about the whole transformation system is that you can do it at any time, whether you're moving or not, and that gives you the opportunity to do some interesting stunts, or to get out of some Ralph Lauren Suisse sticky situations. Our favorite surprise trick was rolling in ball mode until it came to a rest on its side, then transforming so that it stands up while lashing its tail around. The Shell Shocker there's never been anything like it, so it definitely scores some kind of points for originality.

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