De Sitter–Schwarzschild metric: Difference between revisions

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In [[mathematics]] &mdash; specifically, in [[geometric measure theory]] &mdash; '''spherical measure''' ''&sigma;''<sup>''n''</sup> is the &ldquo;natural&rdquo; [[Borel measure]] on the [[n-sphere|''n''-sphere]] '''S'''<sup>''n''</sup>.  Spherical measure is often normalized so that it is a [[probability measure]] on the sphere, i.e. so that ''&sigma;''<sup>''n''</sup>('''S'''<sup>''n''</sup>)&nbsp;=&nbsp;1.


==Definition of spherical measure==


There are several ways to define spherical measure. One way is to use the usual &ldquo;round&rdquo; or &ldquo;[[arclength]]&rdquo; [[metric space|metric]] ''&rho;''<sub>''n''</sub> on '''S'''<sup>''n''</sup>;  that is, for points ''x'' and ''y'' in '''S'''<sup>''n''</sup>, ''&rho;''<sub>''n''</sub>(''x'',&nbsp;''y'') is defined to be the (Euclidean) angle that they subtend at the centre of the sphere (the origin of '''R'''<sup>''n''+1</sup>).  Now construct ''n''-dimensional [[Hausdorff measure]] ''H''<sup>''n''</sup> on the metric space ('''S'''<sup>''n''</sup>,&nbsp;''&rho;''<sub>''n''</sub>) and define
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:<math>\sigma^{n} = \frac{1}{H^{n}(\mathbf{S}^{n})} H^{n}.</math>
One could also have given '''S'''<sup>''n''</sup> the metric that it inherits as a subspace of the Euclidean space '''R'''<sup>''n''+1</sup>;  the same spherical measure results from this choice of metric.
 
Another method uses [[Lebesgue measure]] ''&lambda;''<sup>''n''+1</sup> on the ambient Euclidean space '''R'''<sup>''n''+1</sup>:  for any measurable subset ''A'' of '''S'''<sup>''n''</sup>, define ''&sigma;''<sup>''n''</sup>(''A'') to be the (''n''&nbsp;+&nbsp;1)-dimensional volume of the &ldquo;wedge&rdquo; in the ball '''B'''<sup>''n''+1</sup> that it subtends at the origin.  That is,
:<math>\sigma^{n}(A) := \frac{1}{\alpha(n + 1)} \lambda^{n + 1} ( \{ t x \mid x \in A, t \in [0, 1] \} ),</math>
where
:<math>\alpha(m) := \lambda^{m} (\mathbf{B}_{1}^{m} (0)).</math>
 
The fact that all these methods define the same measure on '''S'''<sup>''n''</sup> follows from an elegant result of Christensen:  all these measures are obviously [[uniformly distributed measure|uniformly distributed]] on '''S'''<sup>''n''</sup>, and any two uniformly distributed Borel regular measures on a separable metric space must be constant (positive) multiples of one another.  Since all our candidate ''&sigma;''<sup>''n''</sup>&rsquo;s have been normalized to be probability measures, they are all the same measure.
 
==Relationship with other measures==
 
The relationship of spherical measure to Hausdorff measure on the sphere and Lebesgue measure on the ambient space has already been discussed.
 
Spherical measure has a nice relationship to [[Haar measure]] on the [[orthogonal group]].  Let O(''n'') denote the orthogonal group [[group action|acting]] on '''R'''<sup>''n''</sup> and let ''&theta;''<sup>''n''</sup> denote its normalized Haar measure (so that ''&theta;''<sup>''n''</sup>(O(''n''))&nbsp;=&nbsp;1).  The orthogonal group also acts on the sphere '''S'''<sup>''n''&minus;1</sup>.  Then, for any ''x''&nbsp;&isin;&nbsp;'''S'''<sup>''n''&minus;1</sup> and any ''A''&nbsp;&sube;&nbsp;'''S'''<sup>''n''&minus;1</sup>,
:<math>\theta^{n}(\{ g \in \mathrm{O}(n) \mid g(x) \in A \}) = \sigma^{n - 1}(A).</math>
 
In the case that '''S'''<sup>''n''</sup> is a [[topological group]] (that is, when ''n'' is 0, 1 or 3), spherical measure ''&sigma;''<sup>''n''</sup> coincides with (normalized) Haar measure on '''S'''<sup>''n''</sup>.
 
==Isoperimetric inequality==
 
There is an [[isoperimetric inequality]] for the sphere with its usual metric and spherical measure (see Ledoux &amp; Talagrand, chapter 1):
 
If ''A''&nbsp;&sube;&nbsp;'''S'''<sup>''n''&minus;1</sup> is any Borel set and ''B''&sube;&nbsp;'''S'''<sup>''n''&minus;1</sup> is a ''&rho;''<sub>''n''</sub>-ball with the same ''&sigma;''<sup>''n''</sup>-measure as ''A'', then, for any ''r''&nbsp;&gt;&nbsp;0,
:<math>\sigma^{n}(A_{r}) \geq \sigma^{n}(B_{r}),</math>
where ''A''<sub>''r''</sub> denotes the &ldquo;inflation&rdquo; of ''A'' by ''r'', i.e.
:<math>A_{r} := \{ x \in \mathbf{S}^{n} \mid \rho_{n}(x, A) \leq r \}.</math>
In particular, if ''&sigma;''<sup>''n''</sup>(''A'')&nbsp;&ge;&nbsp;&frac12; and ''n''&nbsp;&ge;&nbsp;2, then
:<math>\sigma^{n}(A_{r}) \geq 1 - \sqrt{\frac{\pi}{8}} \, \exp \left( - \frac{(n - 1) r^{2}}{2} \right).</math>
 
==References==
 
* {{cite journal
| last = Christensen
| first = Jens Peter Reus
| title = On some measures analogous to Haar measure
| journal = Mathematica Scandinavica
| volume = 26
| year = 1970
| pages = 103&ndash;106
| issn = 0025-5521
}} {{MathSciNet|id=0260979}}
* {{ cite book
| last1 = Ledoux
| first1 = Michel
| last2 = Talagrand | first2 = Michel | author2-link = Michel Talagrand
| title = Probability in Banach spaces
| publisher = Springer-Verlag
| location = Berlin
| year = 1991
| pages = xii+480
| isbn = 3-540-52013-9
}} {{MathSciNet|id=1102015}} (See chapter 1)
* {{cite book
| last = Mattila
| first = Pertti
| title = Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability
| series = Cambridge Studies in Advanced Mathematics No. 44
| publisher = Cambridge University Press
| location = Cambridge
| year = 1995
| pages = xii+343
| isbn = 0-521-46576-1
}} {{MathSciNet|id=1333890}} (See chapter 3)
 
[[Category:Measures (measure theory)]]

Latest revision as of 20:30, 4 October 2014


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Sync all of your notes across all the PCs and devices you use

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Microsoft has again given a new momentum to its potentials and hardships that are engaged in creating a good customer base across all the IT sections. This note-taking app is here to help you while keeping a record of all your thoughts and discoveries. OneNote serves you as your very own digital notebook on which you can easily record your thoughts, important events, ideas, discussions, reminders and many such things. You can capture your immediate reactions, thoughtful ideas and sayings, a list of important things to be done, or the plain old note that you are likely to forget. Owing to your electronic gadget, your notes will now travel with you no matter where you go, or at what time you access.
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Share notes via email, Evernote, and other services
4. Springpad
Introduced with the basic agenda of getting organized and getting inspired, Springpad is an ideal solution for users who have just started using a note-taking app. The app is a free personal organizer that helps you to save and organize tasks, recipes, notes, movies and more. It's easy-to-use interface and less complex features make it a popular choice of beginners. The app gives you full freedom to access or save your notes from anywhere and on any device.
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Create notes at home and at work

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Save details, schedules, suggestions and even tickets while travelling to another destination

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Manage your personal or work to-do lists and set reminder alerts.
5. Simplenote
The award-winning Simplenote is another handy solution to keep notes, manage lists, record ideas, and much more. Its immense integration with cloud computing allows your notes to stay in sync with all of your devices for free. The app is widely accepted for its speed and efficiency that ensure the safe and proper record taking process. All you need to do is open it, write thoughts, save, and you're done. You can search the collection of notes instantly by using a simple search term and keep them organized with tags and pins.
The above mentioned are some of the majorly accepted note-taking apps used by Android users located around the world. Google Keep, Catch notes, Any do, Lecture Notes, Note Anytime, and Write by Stylus Labs are some of the other such apps. Many of you might realize the need of having a special app to take notes, but you will love the comfort once you'll download them and start using it. You will need a computer, Smartphone or a tablet to fulfill your needs related to entertainment, work, information and others, but this app will help to stay organized.

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