Riemann–Siegel formula: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
en>Majora4
m Reverted edits by 108.126.121.39 (talk) to last version by Geometry guy
 
en>Yobot
m WP:CHECKWIKI error fixes / special characters in sortkey fixed using AWB (9440)
Line 1: Line 1:
In [[topology]], the '''pasting lemma''' is an important result which says that two continuous functions can be "glued together" to create another continuous function. The lemma is implicit in the use of [[piecewise function]]s.


The pasting lemma is crucial to the construction of the [[fundamental group]] of a topological space; it allows one to concatenate continuous paths to create a new continuous path.


by Nas, is very fitting and the film agrees with it. The next step is to visit your Word - Press blog dashboard. These templates are professionally designed and are also Adsense ready. If you are using videos on your site then this is the plugin to use. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed. <br><br>These folders as well as files have to copied and the saved. You may either choose to link only to the top-level category pages or the ones that contain information about your products and services. You are able to set them within your theme options and so they aid the search engine to get a suitable title and description for the pages that get indexed by Google. Furthermore, with the launch of Windows 7 Phone is the smart phone market nascent App. Many times the camera is following Mia, taking in her point of view in almost every frame. <br><br>Here are a few reasons as to why people prefer Word - Press over other software's. The nominee in each category with the most votes was crowned the 2010 Parents Picks Awards WINNER and has been established as the best product, tip or place in that category. After age 35, 18% of pregnancies will end in miscarriage. Storing write-ups in advance would have to be neccessary with the auto blogs. If you loved this posting and you would like to get much more details about [http://dinky.in/?WordpressBackup58266 wordpress dropbox backup] kindly take a look at the website. Websites using this content based strategy are always given top scores by Google. <br><br>The primary differences are in the plugins that I install, as all sites don't need all the normal plugins. I have compiled a few tips on how you can start a food blog and hopefully the following information and tips can help you to get started on your food blogging creative journey. Some examples of its additional features include; code inserter (for use with adding Google Analytics, Adsense section targeting etc) Webmaster verification assistant, Link Mask Generator, Robots. It supports backup scheduling and allows you to either download the backup file or email it to you. Make sure you have the latest versions of all your plugins are updated. <br><br>There is no denying that Magento is an ideal platform for building ecommerce websites, as it comes with an astounding number of options that can help your online business do extremely well. Mahatma Gandhi is known as one of the most prominent personalities and symbols of peace, non-violence and freedom. It's not a secret that a lion share of activity on the internet is takes place on the Facebook. Page speed is an important factor in ranking, especially with Google. For your information, it is an open source web content management system.
== Formal statement ==
 
Let <math>X,Y</math> be both closed (or both open) subsets of a topological space ''A'' such that <math>A = X \cup Y</math>, and let ''B'' also be a topological space. If <math>f: A \to B</math> is continuous when restricted to both ''X'' and ''Y'', then ''f'' is continuous.
 
This result allows one to take two continuous functions defined on closed (or open) subsets of a topological space and create a new one.
 
'''Proof''': if ''U'' is a closed subset of ''B'', then <math>f^{-1}(U )\cap X</math> and <math>f^{-1}(U )\cap Y</math> are both closed since the intersection of two closed sets is closed, and ''f'' restricted to both ''X'' and ''Y'' is continuous. Therefore, their union, <math>f^{-1}(U)</math> is also closed. A similar argument applies when ''X'' and ''Y'' are both open. <math>\Box</math>
 
The infinite analog of this result (where <math>A=X_1\cup X_2\cup X_3\cup\cdots</math>)is not true for closed <math>X_1, X_2, X_3\ldots</math>. It is, however, true if the <math>X_1, X_2, X_3\ldots</math> are open; this follows from the fact that an arbitrary union of open sets is open.
 
== References ==
* [[James Munkres|Munkres, James]]; ''Topology'', Prentice Hall; 2nd edition (December 28, 1999). ISBN 0-13-181629-2.
 
[[Category:Topology]]

Revision as of 09:53, 23 August 2013

In topology, the pasting lemma is an important result which says that two continuous functions can be "glued together" to create another continuous function. The lemma is implicit in the use of piecewise functions.

The pasting lemma is crucial to the construction of the fundamental group of a topological space; it allows one to concatenate continuous paths to create a new continuous path.

Formal statement

Let X,Y be both closed (or both open) subsets of a topological space A such that A=XY, and let B also be a topological space. If f:AB is continuous when restricted to both X and Y, then f is continuous.

This result allows one to take two continuous functions defined on closed (or open) subsets of a topological space and create a new one.

Proof: if U is a closed subset of B, then f1(U)X and f1(U)Y are both closed since the intersection of two closed sets is closed, and f restricted to both X and Y is continuous. Therefore, their union, f1(U) is also closed. A similar argument applies when X and Y are both open.

The infinite analog of this result (where A=X1X2X3)is not true for closed X1,X2,X3. It is, however, true if the X1,X2,X3 are open; this follows from the fact that an arbitrary union of open sets is open.

References

  • Munkres, James; Topology, Prentice Hall; 2nd edition (December 28, 1999). ISBN 0-13-181629-2.