Frequently asked questions and Operator monotone function: Difference between pages

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==Registration and content creation==
A function $f : [0, \infty) \to [0, \infty)$ is said to be an operator monotone function (complete Bernstein function, Nevanlinna-Pick function for the half-line) if $A \ge B \ge 0$ implies $f(A) \ge f(B) \ge 0$ for any self-adjoint matrices $A$, $B$. Many equivalent definitions can be given.<ref name="SSV"/>


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==Representation==
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A function $f$ is operator monotone if and only if
\[ f(z) = a z + b + \int_{(0, \infty)} \frac{z}{z + r} \, \frac{\rho(\mathrm d r)}{r} \]
for some $a, b \ge 0$ and a Radon measure $\rho$ such that $\int_{(0, \infty)} \min(r^{-1}, r^{-2}) \rho(\mathrm d r) < \infty$.


===How do I edit a page?===
==Relation to Bernstein functions==
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Operator monotone functions form a subclass of [[Bernstein function]]s. Namely, a Bernstein function $f$ is an operator monotone function if and only if the measure $\mu$ in the Bernstein representation of $f$:
\[ f(z) = a z + b + \int_{(0, \infty)} (1 - e^{-t z}) \mu(\mathrm d t) \]
has a [[completely monotone function|completely monotone]] density function. In this case
\[ \mu(\mathrm d t) = \left( \int_{(0, \infty)} e^{-t r} \rho(\mathrm d r) \right) \mathrm d t \]
This explains the name complete Bernstein functions.


===How do I create a new page?===
==Holomorphic extension==
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Every operator monotone function $f$ extends to a holomorphic function on $\C \setminus (-\infty, 0]$ such that
\begin{align*}
\Im f(z) & \ge 0 \qquad && \text{if } \Im z \ge 0 , \\
f(z) & \ge 0 \qquad && \text{if } \Im z = 0 , \\
\Im f(z) & \le 0 \qquad && \text{if } \Im z \le 0 .
\end{align*}
Conversely, any function $f$ with above properties is an operator monotone function.


===Where can I learn how to use a wiki?===
Functions with nonnegative imaginary part in the upper half-plane are often called Nevanlinna-Pick functions, or Pick functions.
Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.


===How do I write a bibliographical reference?===
==Operator monotone functions of the Laplacian==
Use the website http://zeteo.info/ to generate the references.
Operator monotone functions of the Laplacian are particularly regular examples of translation invariant non-local operators in $\R^n$. More precisely, $A = f(-\Delta)$ for an operator monotone $f$ if and only if
\[ -A u(x) = a \Delta u(x) + b u(x) + \int_{\R^n} (u(x + y) - u(x) - z \cdot \nabla u(x) \mathbf{1}_{|z| < 1}) k(z) \mathrm d z \]
for some $a, b \ge 0$ and $k(z)$ of the form
\begin{align*}
k(z) &= \int_0^\infty \int_0^\infty (4 \pi t)^{-n/2} e^{-|z|^2 / (4 t)} e^{-t r} \mathrm d t \rho(\mathrm d r)
\end{align*}


Look at the pages that are already created as an example of how to make the list of bibliography.
==References==
{{reflist|refs=
<ref name="SSV">{{Citation | last1=Schilling | first1=R. | last2=Song | first2=R. | last3=Vondraček | first3=Z. | title=Bernstein functions. Theory and Applications | year=2010 | publisher=de Gruyter, Berlin | series=Studies in Mathematics | volume=37 | url=http://dx.doi.org/10.1515/9783110215311 | doi=10.1515/9783110215311}}</ref>
}}


===Are there any rules on what we can write?===
{{stub}}
 
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Right now, in [[Mwiki:Current events|Current Events]] there is a '''to do''' list. Click on the links and edit the pages. The red links denote that there is a page needed that was not even started.
 
The organization of the wiki is not fully established. We may need some extra index pages or categories.
 
There are several pages already. But the wiki is still in a very premature state. Most pages need some more work. The idea of having a wiki is that no version of a page will ever be a final version. However, right now they make that very apparent.
 
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If this wiki project works well, it may become a massive reference which is always up to date. It can potentially be better than a book. For that we need several people involved and willing to edit the articles.
 
=== Isn't every wiki doomed to fail? ===
There are other scientific wiki projects online which can serve as examples: the [http://wiki.math.toronto.edu/DispersiveWiki/index.php/Main_Page dispersive wiki], the [http://www.wikiwaves.org/index.php/Main_Page water wave wiki] and also two quantum physics wikis called [http://qwiki.stanford.edu/index.php/Main_Page Qwiki] and [http://www.quantiki.org/wiki/Main_Page Quantiki].
 
The success of a wiki page depends on the contributions made by the users. The current experience with scientific wikis shows a questionable level of success. On the other hand, there are non-scientific wikis which are tremendously successful, for example: the [http://wikitravel.org/ travel guide wiki], the [http://harrypotter.wikia.com/wiki/Main_Page Harry Potter wiki], the [http://recipes.wikia.com/wiki/Recipes_Wiki recipes wiki], the [http://starwars.wikia.com/wiki/Main_Page Star Wars wiki], the [http://www.miwiki.net/ Monkey island wiki], the [http://baseball.wikia.com/ Baseball wiki] or the [http://www.mariowiki.com/ Super Mario wiki], among many others.
 
It would be interesting to understand this distinction of success between the non-scientific wikis and the scientific ones.
 
=== Why should I spend time writing on this wiki? ===
If you are a mathematician who has done some research in the area, you definitely want people to know about your results. If you write an easy to read reference in this wiki, that would help more people know about your work and how it is related with other results in the area. Just be careful not to overplay the importance of your own results (or you may be banned from editing again). The appropriate thing to do is to write about all the related results by other people as much as you write about yours. Also remember to follow the rules above in the [[#Are_there_any_rules_on_what_we_can_write|writing guidelines]].
 
If you are a student learning the subject, writing in this wiki may help you understand the topics better (especially if someone comes after you to correct you). Moreover, if you are learning the subject, you probably appreciate the existence of this wiki more than others and are willing to contribute back.
 
There is an interesting video about open science here [http://www.youtube.com/watch?v=DnWocYKqvhw]

Revision as of 05:14, 19 July 2012

A function $f : [0, \infty) \to [0, \infty)$ is said to be an operator monotone function (complete Bernstein function, Nevanlinna-Pick function for the half-line) if $A \ge B \ge 0$ implies $f(A) \ge f(B) \ge 0$ for any self-adjoint matrices $A$, $B$. Many equivalent definitions can be given.[1]

Representation

A function $f$ is operator monotone if and only if \[ f(z) = a z + b + \int_{(0, \infty)} \frac{z}{z + r} \, \frac{\rho(\mathrm d r)}{r} \] for some $a, b \ge 0$ and a Radon measure $\rho$ such that $\int_{(0, \infty)} \min(r^{-1}, r^{-2}) \rho(\mathrm d r) < \infty$.

Relation to Bernstein functions

Operator monotone functions form a subclass of Bernstein functions. Namely, a Bernstein function $f$ is an operator monotone function if and only if the measure $\mu$ in the Bernstein representation of $f$: \[ f(z) = a z + b + \int_{(0, \infty)} (1 - e^{-t z}) \mu(\mathrm d t) \] has a completely monotone density function. In this case \[ \mu(\mathrm d t) = \left( \int_{(0, \infty)} e^{-t r} \rho(\mathrm d r) \right) \mathrm d t \] This explains the name complete Bernstein functions.

Holomorphic extension

Every operator monotone function $f$ extends to a holomorphic function on $\C \setminus (-\infty, 0]$ such that \begin{align*} \Im f(z) & \ge 0 \qquad && \text{if } \Im z \ge 0 , \\ f(z) & \ge 0 \qquad && \text{if } \Im z = 0 , \\ \Im f(z) & \le 0 \qquad && \text{if } \Im z \le 0 . \end{align*} Conversely, any function $f$ with above properties is an operator monotone function.

Functions with nonnegative imaginary part in the upper half-plane are often called Nevanlinna-Pick functions, or Pick functions.

Operator monotone functions of the Laplacian

Operator monotone functions of the Laplacian are particularly regular examples of translation invariant non-local operators in $\R^n$. More precisely, $A = f(-\Delta)$ for an operator monotone $f$ if and only if \[ -A u(x) = a \Delta u(x) + b u(x) + \int_{\R^n} (u(x + y) - u(x) - z \cdot \nabla u(x) \mathbf{1}_{|z| < 1}) k(z) \mathrm d z \] for some $a, b \ge 0$ and $k(z)$ of the form \begin{align*} k(z) &= \int_0^\infty \int_0^\infty (4 \pi t)^{-n/2} e^{-|z|^2 / (4 t)} e^{-t r} \mathrm d t \rho(\mathrm d r) \end{align*}

References

  1. Schilling, R.; Song, R.; Vondraček, Z. (2010), Bernstein functions. Theory and Applications, Studies in Mathematics, 37, de Gruyter, Berlin, doi:10.1515/9783110215311, http://dx.doi.org/10.1515/9783110215311 

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