formulasearchengine - User contributions [en]

Representer theorem

2014-10-16T17:40:45Z

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Membership function (mathematics)

2014-10-16T15:54:32Z

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2014-10-05T22:47:39Z

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2014-03-30T10:45:08Z

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2014-02-23T16:25:40Z

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2014-02-23T05:50:14Z

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2014-02-23T05:44:55Z

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Fine topology (potential theory)

2014-01-05T19:49:26Z

84.227.244.203:

In [[operator theory]], '''[[Mark Naimark|Naimark]]'s dilation theorem''' is a result that characterizes [[POVM|positive operator valued measures]]. It can be viewed as a consequence of [[Stinespring factorization theorem|Stinespring's dilation theorem]].

== Note ==

In the mathematical literature, one may also find other results that bear Naimark's name.

== Some preliminary notions ==

Let ''X'' be a [[Compact space|compact]] [[Hausdorff space]], ''H'' be a [[Hilbert space]], and ''L(H)'' the [[Banach space]] of [[bounded operator]]s on ''H''. A mapping ''E'' from the [[Borel σ-algebra]] on ''X'' to <math>L(H)</math> is called a '''operator-valued measure''' if it is weakly countably additive, that is, for any disjoint sequence of Borel sets <math>\{ B_i \}</math>, we have

:<math>
\langle E (\cup _i B_i) x, y \rangle = \sum_i \langle E (B_i) x, y \rangle
</math>

for all ''x'' and ''y''. Some terminology for describing such measures are:

* ''E'' is called ''regular'' if the scalar valued measure

:<math>
B \rightarrow \langle E (B) x, y \rangle
</math>

is a regular Borel measure, meaning all compact sets have finite total variation and the measure of a set can be approximated by those of open sets.

* ''E'' is called ''bounded'' if <math>|E| = \sup_B \|E(B) \| < \infty</math>.

* ''E'' is called ''positive'' if ''E(B)'' is a positive operator for all ''B''.

* ''E'' is called ''self-adjoint '' if ''E(B)'' is self-adjoint for all ''B''.

* ''E'' is called ''spectral'' if <math>E (B_1 \cap B_2) = E(B_1) E(B_2)</math>.

We will assume throughout that ''E'' is regular.

Let ''C(X)'' denote the abelian C*-algebra of continuous functions on ''X''. If ''E'' is regular and bounded, it induces a map <math>\Phi _E : C(X) \rightarrow L(H)</math> in the obvious way:

:<math>\langle \Phi _E (f) h_1 , h_2 \rangle = \int _X f d \langle E(B) h_1, h_2 \rangle</math>

The boundedness of ''E'' implies, for all ''h'' of unit norm

:<math>
\langle \Phi _E (f) h , h \rangle = \int _X f d \langle E(B) h, h \rangle \leq \| f \| \cdot |E| .
</math>

This shows <math>\; \Phi _E (f)</math> is a bounded operator for all ''f'', and <math>\Phi _E</math> itself is a bounded linear map as well.

The properties of <math>\Phi_E</math> are directly related to those of ''E'':

* If ''E'' is positive, then <math>\Phi_E</math>, viewed as a map between C*-algebras, is also positive.

* <math>\Phi_E</math> is a homomorphism if, by definition, for all continuous ''f'' on ''X'' and <math>h_1, h_2 \in H</math>,

:<math>
\langle \Phi_E (fg) h_1, h_2 \rangle = \int _X f \cdot g \; d \langle E(B) h_1, h_2 \rangle
= \langle \Phi_E (f) \Phi_E (g) h_1 , h_2 \rangle.
</math>

Take ''f'' and ''g'' to be indicator functions of Borel sets and we see that <math>\Phi _E</math> is a homomorphism if and only if ''E'' is spectral.

* Similarly, to say <math>\Phi_E</math> respects the * operation means

:<math>
\langle \Phi_E ( {\bar f} ) h_1, h_2 \rangle = \langle \Phi_E (f) ^* h_1 , h_2 \rangle.
</math>

The LHS is

:<math>
\int _X {\bar f} \; d \langle E(B) h_1, h_2 \rangle,
</math>

and the RHS is

:<math>
\langle h_1, \Phi_E (f) h_2 \rangle = \int _X {\bar f} \; d \langle E(B) h_2, h_1 \rangle
</math>

So, for all ''B'', <math>\langle E(B) h_1, h_2 \rangle = \langle E(B) h_2, h_1 \rangle</math>, i.e. ''E(B)'' is self adjoint.

* Combining the previous two facts gives the conclusion that <math>\Phi _E</math> is a *-homomorphism if and only if ''E'' is spectral and self adjoint. (When ''E'' is spectral and self adjoint, ''E'' is said to be a [[projection-valued measure]] or PVM.)

== Naimark's theorem ==

The theorem reads as follows: Let ''E'' be a positive ''L(H)''-valued measure on ''X''. There exists a Hilbert space ''K'', a bounded operator <math>V: K \rightarrow H</math>, and a self-adjoint, spectral ''L(K)''-valued measure on ''X'', ''F'', such that

:<math>\; E(B) = V F(B) V^*.</math>

=== Proof ===

We now sketch the proof. The argument passes ''E'' to the induced map <math>\Phi_E</math> and uses [[Stinespring factorization theorem|Stinespring's dilation theorem]]. Since ''E'' is positive, so is <math>\Phi_E</math> as a map between C*-algebras, as explained above. Furthermore, because the domain of <math>\Phi _E</math>, ''C(X)'', is an abelian C*-algebra, we have that <math>\Phi_E</math> is [[Choi's theorem on completely positive maps|completely positive]]. By Stinespring's result, there exists a Hilbert space ''K'', a *-homomorphism <math>\pi : C(X) \rightarrow L(K)</math>, and operator <math>V: K \rightarrow H</math> such that

:<math>\; \Phi_E(f) = V \pi (f) V^*.</math>

Since π is a *-homomorphism, its corresponding operator-valued measure ''F'' is spectral and self adjoint. It is easily seen that ''F'' has the desired properties.

== Finite dimensional case ==

In the finite dimensional case, there is a somewhat more explicit formulation.

Suppose now <math>X = \{1, \cdots, n \}</math>, therefore ''C(X)'' is the finite dimensional algebra <math>\mathbb{C}^n</math>, and ''H'' has finite dimension ''m''. A positive operator-valued measure ''E'' then assigns each ''i'' a positive semidefinite ''m X m'' matrix <math>E_i</math>. Naimark's theorem now says there
is a projection valued measure on ''X'' whose restriction is ''E''.

Of particular interest is the special case when <math>\; \sum _i E_i = I</math> where ''I'' is the identity operator. (See the article on [[POVM]] for relevant applications.) This would mean the induced map <math>\Phi _E</math> is unital. It can be assumed with no loss of generality that each <math>E_i</math> is a rank-one projection onto some <math>x_i \in \mathbb{C}^m</math>. Under such assumptions, the case <math>n < m</math> is excluded and we must have either:

1) <math>n = m</math> and ''E'' is already a projection valued measure. (Because <math>\sum _{i=1}^n x_i x_i^* = I </math> if and only if <math>\{ x_i\}</math> is an orthonormal basis.)
,or

2) <math>n > m</math> and <math>\{ E_i \}</math> does not consist of mutually orthogonal projections.

For the second possibility, the problem of finding a suitable PVM now becomes the following: By assumption, the non-square matrix

:<math> M = \begin{bmatrix} x_1 & \cdots x_n \end{bmatrix}</math>

is an isometry, i.e. <math>M M^* = I</math>. If we can find a <math>(n-m) \times n</math> matrix ''N'' where

:<math>U = \begin{bmatrix} M \\ N \end{bmatrix} </math>

is a ''n X n'' unitary matrix, the PVM whose elements are projections onto the column vectors of ''U'' will then have the desired properties. In principle, such a ''N'' can always be found.

==References==

*V. Paulsen, ''Completely Bounded Maps and Operator Algebras'', Cambridge University Press, 2003.

[[Category:Operator theory]]
[[Category:Measure theory]]
[[Category:Theorems in functional analysis]]

Pluripolar set

2014-01-05T18:30:40Z

84.227.244.203:

{{refimprove|date=November 2011}}
[[Image:Fibonacci, Elias Gamma, and Elias Delta encoding schemes.GIF|thumb|Fibonacci, Elias Gamma, and Elias Delta vs binary coding]]
[[Image:riceEncodingScheme.GIF|thumb|Rice with ''k'' = 2, 3, 4, 5, 8, 16 versus binary]]
In [[data compression]], a '''universal code''' for integers is a [[prefix code]] that maps the positive integers onto binary codewords, with the additional property that whatever the true [[probability distribution]] on integers, as long as the distribution is monotonic (i.e., ''p''(''i'') ≥ ''p''(''i'' + 1) for all positive ''i''), the [[expected value|expected]] lengths of the codewords are within a constant factor of the expected lengths that the [[optimal code]] for that probability distribution would have assigned. A universal code is ''asymptotically optimal'' if the ratio between actual and optimal [[expected value|expected]] lengths is bounded by a function of the [[information entropy]] of the code that, in addition to being bounded, approaches 1 as entropy approaches infinity.

In general, most prefix codes for integers assign longer codewords to larger integers. Such a code can be used to efficiently communicate a message drawn from a set of possible messages, by simply ordering the set of messages by decreasing probability and then sending the index of the intended message. Universal codes are generally not used for precisely known probability distributions, and no universal code is known to be optimal for any distribution used in practice.

A universal code should not be confused with [[universal source coding]], in which the data compression method need not be a fixed prefix code and the ratio between actual and optimal expected lengths must approach one. However, note that an asymptotically optimal universal code can be used on [[Independent identically-distributed random variables|independent identically-distributed sources]], by using increasingly large [[block code|blocks]], as a method of universal source coding.

== Universal and non-universal codes ==
These are some universal codes for integers; an asterisk ([[Asterisk|*]]) indicates a code that can be trivially restated in [[lexicographical order]], while a double dagger ([[‡]]) indicates a code that is asymptotically optimal:
* [[Elias gamma coding]] *
* [[Elias delta coding]] * ‡
* [[Elias omega coding]] * ‡
* [[Exponential-Golomb coding|Exp-Golomb coding]] *, which has Elias gamma coding as a special case. (Used in [[H.264/MPEG-4 AVC]])
* [[Fibonacci coding]]
* [[Levenstein coding]] * ‡, the original universal coding technique [http://www.compression.ru/download/articles/int/levenstein_1968_on_the_redundancy_and_delay.pdf]
* [[Byte coding]], also known as [[comma coding]], where a special bit pattern (with at least two bits) is used to mark the end of the code — for example, if an integer is encoded as a sequence of [[nibble]]s representing digits in [[base 15]] instead of the more natural [[base 16]], then the highest nibble value (i.e., a sequence of four ones in binary) can be used to indicate the end of the integer.

These are non-universal ones:

* [[unary coding]], which is used in Elias codes
* [[Golomb coding|Rice coding]], which is used in the [[FLAC]] [[audio codec]] and which has unary coding as a special case
* [[Golomb coding]], which has Rice coding and unary coding as special cases.

Their nonuniversality can be observed by noticing that, if any of these are used to code the [[Gauss–Kuzmin distribution]] or the [[Zeta distribution]] with parameter s=2, expected codeword length is infinite. For example, using unary coding on the Zeta distribution yields an expected length of

: <math>E(l) = \frac{6}{\pi^2} \sum_{l=1}^\infty \frac{1}{l} = \infty . \,</math>

On the other hand, using the universal Elias gamma coding for the Gauss–Kuzmin distribution results in an expected codeword length (about 3.51 bits) near entropy (about 3.43 bits)[http://scholar.google.com/scholar?cluster=13442560459874106744].

==Relationship to practical compression==
[[Huffman coding]] and [[arithmetic encoding]] (when they can be used) give at least as good, and often better compression than any universal code.

However, universal codes are useful when Huffman coding cannot be used — for example, when one does not know the exact probability of each message, but only knows the rankings of their probabilities.

Universal codes are also useful when Huffman codes are inconvenient. For example, when the transmitter but not the receiver knows the probabilities of the messages, Huffman coding requires an overhead of transmitting those probabilities to the receiver. Using a universal code does not have that overhead.

Each universal code, like each other self-delimiting (prefix) binary code, has its own "implied probability distribution" given by ''p''(''i'')=2-''l''(''i'') where ''l''(''i'') is the length of the ''i''th codeword and ''p''(''i'') is the corresponding symbol's probability. If the actual message probabilities are ''q''(''i'') and [[Kullback–Leibler divergence]] ''D''KL(''q''||''p'') is minimized by the code with ''l''(''i''), then the optimal Huffman code for that set of messages will be equivalent to that code. Likewise, how close a code is to optimal can be measured by this divergence. Since universal codes are simpler and faster to encode and decode than Huffman codes (which is, in turn, simpler and faster than [[arithmetic encoding]]), the universal code would be preferable in cases where ''D''KL(''q''||''p'') is sufficiently small.
[http://www.cs.tut.fi/~albert/Dev/pucrunch/]

For any [[geometric distribution]] (an exponential distribution on integers), a Golomb code is optimal. With universal codes, the implicit distribution is approximately a [[power law]] such as <math>1/n^2</math> (more precisely, a [[Zipf distribution]]).
For the [[Fibonacci code]], the implicit distribution is approximately <math>1/n^q</math>, with

:<math>q = 1/\log_2(\varphi) \simeq 1.44,</math>

where <math>\varphi</math> is the [[golden ratio]]. For the ternary [[comma code]] (i.e., encoding in base 3, represented with 2 bits per symbol), the implicit distribution is a power law with <math>q=1+\log_3(4/3) \simeq 1.26</math>. These distributions thus have near-optimal codes with their respective power laws.

==References==
* David J. C. MacKay. ''[http://www.inference.phy.cam.ac.uk/mackay/itila/book.html Information Theory, Inference, and Learning Algorithms]'' Cambridge: Cambridge University Press, 2003. ISBN 0-521-64298-1
* [http://www.ics.uci.edu/~dan/pubs/DC-Sec3.html]

==External links==
* [http://www.inference.phy.cam.ac.uk/mackay/itila/ On-line textbook: Information Theory, Inference, and Learning Algorithms], by [[David MacKay (scientist)|David MacKay]], has a chapter on codes for integers, including an accessible introduction to Elias codes.
* [http://www-lat.compression.ru/download/integers.html Кодирование целых чисел] has mostly English-language papers on universal and other integer codes.



{{Compression Methods}}

[[Category:Data compression]]
[[Category:Lossless compression algorithms]]

Zonal spherical harmonics

2012-09-09T11:49:29Z

84.227.246.199: Typo

{{multiple issues|
{{Underlinked|date=November 2013}}
{{refimprove|date=November 2013}}
}}

'''Cmin''' is a term used in pharmacokinetics usually refers to the minimum plasma concentration that a drug achieves in tested area after the drug has been administrated and prior to the administration of a second dose. Cmin is the opposite of [[CMax|Cmax]], which is the maximum concentration that a drug achieves after dosing.'''Cmin''' has direct relationships with other minimum concentrations such as [[Minimum inhibitory concentration|MIC]] and is required to be higher than those concentrations to achieve the minimum efficacy.

In most cases Cmin is directly measurable. At steady state the minimum plasma concentration can also be calculated using the following equation:<ref>http://www.pharmpress.com/files/docs/clinical_pharmacokinetics_samplechapter.pdf</ref>
<center><math>C_{p min}= \frac{SFD k_a}{V_d(k_a-k)}\times\{\frac{e^{-k\tau}}{1-e^{-k\tau}}-\frac{e^{-k_a\tau}}{1-e^{-k_a\tau}}\}</math></center>

Cmin is also a very important parameter in BA/BE studies, it is part of the pharmacokinetic information recommended for submission of IND.<ref>http://www.fda.gov/downloads/Drugs/.../Guidances/ucm070124.pdf</ref>

==References==
{{Reflist}}

[[Category:Pharmacokinetics]]

{{pharmacology-stub}}