formulasearchengine - User contributions [en]

Device independent file format

2014-12-30T20:22:35Z

70.112.237.191: /* DVI-to-PDF converters */

My name is Luca Wicks. I life in Baulmes (Switzerland).<br><br>Review my webpage :: [http://hemorrhoidtreatmentfix.com/thrombosed-hemorrhoid-treatment thrombosed hemorrhoids treatment]

Phoenix Cluster

2014-01-01T09:49:06Z

70.112.97.77: Not exactly "recent" activity - what we're seeing is a snapshot of something occurring more than 5 billion years ago; reworded accordingly

'''Tom Sanders''' is an English mathematician, working on problems in additive combinatorics at the interface of harmonic analysis and analytic number theory.

Sanders studied mathematics at the [[University of Cambridge]], taking his Ph.D. in 2007 under the direction of [[Timothy Gowers]]. He held a Junior Research Fellowship at [[Christ's College, Cambridge|Christ's College]], Cambridge from 2006 until 2011, in addition to visiting fellowships at the [[Institute for Advanced Study]] in 2007, the [[MSRI]] in 2008, and the [[Mittag-Leffler Institute]] in 2009. Since 2011, he has held a Royal Society University Research Fellowship at the [[University of Oxford]], where he is also a Senior Research Fellow at the Mathematical Institute, and a Tutorial Fellow at [[St Hugh's College, Oxford|St Hugh's College]].

Among several striking results, he has improved the well-known theorem of [[Klaus Friedrich Roth]] on three-term arithmetic progressions,<ref>K.F. Roth, ''On certain sets of integers'', Journal of the London Mathematical Society, Volume 28 (1953) 104–109</ref> breaking the so-called logarithmic barrier. More precisely, he has shown that any subset of {1, 2, ..., N} of maximal cardinality containing no non-trivial three-term arithmetic progression is of size <math>O\left(\frac{N (\log \log N)^5}{\log N}\right)</math>.<ref>{{arxiv|1011.0104}}, [http://annals.math.princeton.edu/2011/174-1/p20].</ref>

In February 2011, he was awarded the [[Adams Prize]] (jointly with [[Harald Helfgott]]) for having "employed deep harmonic analysis to understand arithmetic progressions and answer long-standing conjectures in number theory".<ref>{{cite journal|last=(reprinted from a University of Cambridge announcement)|title=Helfgott and Sanders Awarded Adams Prize|journal=[[Notices of the American Mathematical Society]]|publisher=AMS|volume=58|issue=7|pages=966|url=http://www.ams.org/notices/201107/rtx110700966p.pdf}}</ref> In July 2012, he was awarded a Prize of the [[European Mathematical Society]] for his "fundamental results in additive combinatorics and harmonic analysis, which combine in a masterful way deep known techniques with the invention of new methods to achieve spectacular results."<ref>[http://www.6ecm.pl/en Announcement of the 2012 Prizes of the European Mathematical Society]</ref> In July 2013, he was awarded the [[Whitehead Prize]] of the [[London Mathematical Society]] for his "spectacular results in additive combinatorics and related areas"'', ''in particular "for his paper obtaining the best known upper bounds for sets of integers containing no 3-term arithmetic progressions, for his work dramatically improving bounds connected with Freiman's theorem on sets with small doubling, and for other results in additive combinatorics and harmonic analysis."<ref name=":0">http://www.lms.ac.uk/prizes/lms-prizes-2013</ref>
In September 2013, he was awarded the European Prize in Combinatorics. <ref>https://en.wikipedia.org/wiki/European_Prize_in_Combinatorics</ref>

==Notes==
{{Reflist|LMS Prizes 2013 = http://www.lms.ac.uk/prizes/lms-prizes-2013}}

==External links==
*[http://people.maths.ox.ac.uk/~sanders/ Homepage in Oxford]
*[http://arxiv.org/a/sanders_t_1 Papers on the arXiv]

{{Persondata 
|NAME= Sanders, Tom
|ALTERNATIVE NAMES=
|SHORT DESCRIPTION= British mathematician
|DATE OF BIRTH=
|PLACE OF BIRTH=
|DATE OF DEATH=
|PLACE OF DEATH=
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{{DEFAULTSORT:Sanders, Tom}}
[[Category:21st-century mathematicians]]
[[Category:English mathematicians]]
[[Category:Living people]]

Divisor summatory function

2013-12-14T01:01:24Z

70.112.11.57: /* Definition */ Very minor edit--just made the floor functions look nicer.

== Divisor summatory function==
This graph illustrates the [[divisor summatory function]] with the leading asymptotic terms subtracted. That is, it is a graph of

:<math>D(x)-x\log x - x(2\gamma-1)</math>

where <math>D(x)</math> is the divisor summatory function

:<math>D(x)=\sum_{n\le x} d(n)</math>

and <math>d(n)</math> is the [[divisor function]]. Here, <math>\gamma=0.577\ldots</math> is the [[Euler-Mascheroni constant]]. The graph extends up to <math>x=10^7</math>. The green lines that do not quite bound the picture are a graph of <math>\pm 2x^{7/22}</math> and give a general scale for the rate of growth for this function. Curiously, note that the function is not centered on the zero axis, but seems to be closer to the upper curve. Although the graph appears to be noisy, notice that it is less noisy than the corresponding graph for smaller ''n'', and is more noisy than a similar graph for larger ''n''.

== Licensing ==
Created by Linas Vepstas [[User:Linas]] 12 July 2006

{{GFDL|migration=relicense}}

{{Copy to Wikimedia Commons|bot=Fbot|priority=true}}

Observer effect (physics)

2013-12-11T23:49:56Z

70.112.12.124: Removed because this has nothing to do with the "observer effect", but is actually a mathematical certainty due to the uncertainty principle, i.e. having infinitely better instrumentation wouldn't change the result.

{{Unreferenced|date=August 2008}}

In [[probability theory]], the '''multidimensional Chebyshev's inequality''' is a generalization of [[Chebyshev's inequality]], which puts a bound on the probability of the event that a [[random variable]] differs from its [[expected value]] by more than a specified amount.

Let ''X'' be an ''N''-dimensional [[random vector]] with [[expected value]] <math>\mu=\mathbb{E} \left[ X \right] </math> and [[covariance matrix]]

: <math>V=\mathbb{E} \left[ \left(X - \mu \right) \left( X - \mu \right)^T \right]. \, </math>

If <math>V</math> is a [[positive-definite matrix]], for any [[real number]] <math>t>0</math>:
:<math>
\mathrm{Pr}\left( \sqrt{\left( X-\mu\right)^T \, V^{-1} \, \left( X-\mu\right) } > t \right) \le \frac{N}{t^2}
</math>

==Proof==
Since <math>V</math> is positive-definite, so is <math>V^{-1}</math>.
Define the random variable
:<math>
y = \left( X-\mu\right)^T \, V^{-1} \, \left( X-\mu\right) .
</math>
Since <math>y</math> is positive, [[Markov's inequality]] holds:
:<math>
\begin{align}\mathrm{Pr}\left( \sqrt{\left( X-\mu\right)^T \, V^{-1} \, \left( X-\mu\right) } > t\right) &= \mathrm{Pr}\left( \sqrt{y} > t\right)\\
&=\mathrm{Pr}\left( y > t^2 \right) \\
&\le \frac{\mathbb{E}[y]}{t^2} .\end{align}
</math>
Finally,
:<math>\begin{align}\mathbb{E}[y] &= \mathbb{E}[\left( X-\mu\right)^T \, V^{-1} \, \left( X-\mu\right)]\\
&=\mathbb{E}[ \mathrm{trace} ( V^{-1} \, \left( X-\mu\right) \, \left( X-\mu\right)^T )]\\
&= \mathrm{trace} ( V^{-1} V ) = N \end{align}.</math>

[[Category:Probabilistic inequalities]]
[[Category:Statistical inequalities]]

V-Cube 6

2013-10-01T02:53:44Z

70.112.203.223: /* Records */

{{Unreferenced|date=August 2010}}
Instead of a model-dependent treatment in terms of constituent quarks, hadrons
are represented by their interpolating quark currents taken at large virtualities.
The correlation function of these currents is introduced and treated in
the framework of the [[operator product expansion]] (OPE), where the short and
long-distance quark-gluon interactions are separated. The former are calculated
using QCD perturbation theory, whereas the latter are parametrized in
terms of universal vacuum condensates or light-cone distribution amplitudes.
The result of the QCD calculation is then matched, via dispersion relation, to
a sum over hadronic states. The sum rule obtained in this way allows to calculate
observable characteristics of the hadronic ground state. Inversely, the
parameters of QCD such as quark masses and vacuum condensate densities
can be extracted from sum rules which have experimentally known hadronic
parts. What is also very important, the interactions of quark-gluon currents
with QCD vacuum fields critically depend on the quantum numbers (spinparity,
flavor content) of these currents.
In [[quantum chromodynamics]], the [[color confinement|confining]] and strong coupling nature of the theory means that conventional perturbative techniques often fail to apply. The '''QCD sum rules''' (or '''[[Mikhail Shifman|Shifman]]–[[Arkady Vainshtein|Vainshtein]]–[[Valentine Zakharov Zakharov|Zakharov]] sum rules''') are a way of dealing with this. The idea is to work with gauge invariant operators and [[operator product expansion]]s of them. The vacuum to vacuum correlation function for the product of two such operators can be reexpressed as

:<math>\left\langle 0 | T\left\{ \mathcal{O}_1(x) \mathcal{O}_2(0) \right\} | 0 \right\rangle </math>
where we have inserted hadronic particle states on the right hand side.

==Correlation function of quark currents==

{{Empty section|date=February 2013}}

==See also==
*[[quantum chromodynamics]]
*[[Lattice QCD]]

==External links==
*[http://www.scholarpedia.org/article/Shifman-Vainshtein-Zakharov_sum_rule SVZ sum rules at Scholarpedia]
*{{cite journal | last = Shifman | first = Mikhail | authorlink = Mikhail Shifman | title = Snapshot of hadrons | journal = Prog.Theor.Phys.Suppl. | volume = 131 | pages = 1 | year = 1998 | doi = 10.1143/PTPS.131.1 | url = http://ptps.oxfordjournals.org/content/131/1 | format = [[arXiv|arXiv preprint]] at [http://arxiv.org/abs/hep-ph/9802214 hep-ph/9802214]|arxiv = hep-ph/9802214 |bibcode = 1998PThPS.131....1S }}

[[Category:Quantum chromodynamics]]

{{particle-stub}}