en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q5601715

2013-03-19T23:04:54Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q5601715

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	~~Surely the second option would be more beneficial for any website~~. ~~Medical word press themes give you the latest medical designs~~. ~~I thought about what would happen by placing a text widget in~~ the ~~sidebar beneath my banner ad, and so it went. Word~~ - ~~Press also provides protection against spamming~~, ~~as security is~~ a ~~measure issue~~. ~~After activating, you will find their website link and get the activation code from their website~~. ~~<br><br>~~		In [[mathematics]], '''Carathéodory's existence theorem''' says that an [[ordinary differential equation]] has a solution under relatively mild conditions. It is a generalization of [[Peano existence theorem\|Peano's existence theorem]]. Peano's theorem requires that the right-hand side of the differential equation is continuous, while Carathéodory's theorem shows existence of solutions (in a more general sense) for some discontinuous equations. The theorem is named after [[Constantin Carathéodory]].

	~~When you loved this post and you would want to receive more details about [http~~://~~xyz~~.ms/~~backup_plugin_780467 backup plugin~~] ~~kindly visit~~ the ~~web site~~. ~~Thus~~, it is ~~imperative that you must Hire Word~~ - ~~Press Developers who have~~ the ~~expertise and proficiency in delivering theme integration and customization services. When you write a new post~~, ~~you~~'~~ll see a small bar that goes across~~ the text ~~input area~~. ~~A Wordpress plugin is~~ a ~~software that you can install into your Wordpress site~~. ~~Now~~, ~~I want~~ to ~~anxiety~~ that not ~~every single query will be answered~~. ~~The biggest advantage~~ of ~~using a coupon or deal plugin~~ is ~~that it gives your readers~~ the ~~coupons~~ and ~~deals within minutes of them becoming available~~. <br><br>~~ve labored so hard to publish and put up on their website. Now if we talk about them one by one then -wordpress blog customization means customization~~ of your blog such as installation of wordpress on your server by wordpress developer which will help you to acquire the SEO friendly blog application integrated with your site design as well as separate blog administration panel for starting up your own business blog,which demands a experienced wordpress designer. ~~Are you considering getting your website redesigned. Our skilled expertise~~, ~~skillfulness and excellence have been well known all across the world. Websites using this content based strategy are always given top scores by Google~~. <br><br>~~If all else fails~~, ~~please leave a comment on this post with the issue~~(s) ~~you're having and help will be~~ on the ~~way. The SEOPressor Word - Press SEO Plugin works by analysing each page and post against your chosen keyword~~ (~~or keyword phrase~~) ~~and giving~~ a ~~score~~, ~~with instructions on how to improve it~~. ~~The templates are designed to be stand alone pages that have a different look and feel from~~ the ~~rest of your website. Word - Press is~~ the ~~most popular open source content management system~~ (~~CMS~~) in ~~the world today. Wordpress template~~ is ~~loaded with lots of prototype that unite graphic features and content area.~~ <br><br>~~Instead~~, ~~you can easily just include it with our bodies integration field in e~~ - ~~Panel. I don'~~t ~~want~~ that for ~~my visitors and I'm quite sure they don'~~t ~~either~~. ~~The days~~ of ~~spending a lot~~ of ~~time and money to have a website built are long gone~~. ~~Word~~ - ~~Press is an open source content management system which is easy to use~~ and ~~offers many user friendly features. As for performing online business, websites and blogs are the only medium that are available to interact with customers and Word~~ - ~~Press perform this work with the help of cross~~-~~blog communication tools, comments and full user registration plug~~-~~ins~~.		== Introduction ==
			Consider the differential equation
			:<math> y'(t) = f(t,y(t)) \, </math>
			with initial condition
			:<math> y(t_0) = y_0, \, </math>
			where the function ƒ is defined on a rectangular domain of the form
			:<math> R = \{ (t,y) \in \mathbf{R}\times\mathbf{R}^n \,:\, \|t-t_0\| \le a, \|y-y_0\| \le b \}. </math>
			Peano's existence theorem states that if ƒ is [[continuous function\|continuous]], then the differential equation has at least one solution in a neighbourhood of the initial condition.<ref>{{harvtxt\|Coddington\|Levinson\|1955}}, Theorem 1.2 of Chapter 1</ref>

			However, it is also possible to consider differential equations with a discontinuous right-hand side, like the equation
			:<math> y'(t) = H(t), \quad y(0) = 0, </math>
			where ''H'' denotes the [[Heaviside function]] defined by
			:<math> H(t) = \begin{cases} 0, & \text{if } t \le 0; \\ 1, & \text{if } t > 0. \end{cases} </math>
			It makes sense to consider the [[ramp function]]
			:<math> y(t) = \int_0^t H(s) \,\mathrm{d}s = \begin{cases} 0, & \text{if } t \le 0; \\ t, & \text{if } t > 0 \end{cases} </math>
			as a solution of the differential equation. Strictly speaking though, it does not satisfy the differential equation at <math>t=0</math>, because the function is not differentiable there. This suggests that the idea of a solution be extended to allow for solutions that are not everywhere differentiable, thus motivating the following definition.

			A function ''y'' is called a ''solution in the extended sense'' of the differential equation <math>y' = f(t,y)</math> with initial condition <math>y(t_0)=y_0</math> if ''y'' is [[absolute continuity\|absolutely continuous]], ''y'' satisfies the differential equation [[almost everywhere]] and ''y'' satisfies the initial condition.<ref>{{harvtxt\|Coddington\|Levinson\|1955}}, page 42</ref> The absolute continuity of ''y'' implies that its derivative exists almost everywhere.<ref>{{harvtxt\|Rudin\|1987}}, Theorem 7.18</ref>

			== Statement of the theorem ==
			Consider the differential equation
			:<math> y'(t) = f(t,y(t)), \quad y(t_0) = y_0, \, </math>
			with <math>f</math> defined on the rectangular domain <math> R=\{(t,y) \, \| \, \|t - t_0 \| \leq a, \|y - y_0\| \leq b\} </math>. If the function <math>f</math> satisfies the following three conditions:
			* <math>f(t,y)</math> is [[Continuous function\|continuous]] in <math>y</math> for each fixed <math>t</math>,
			* <math>f(t,y)</math> is [[measurable function\|measurable]] in <math>t</math> for each fixed <math>y</math>,
			* there is a [[Lebesgue integral\|Lebesgue-integrable]] function <math>m(t)</math>, <math>\|t - t_0\| \leq a </math>, such that <math>\|f(t,y)\| \leq m(t)</math> for all <math>(t, y) \in R </math>,
			then the differential equation has a solution in the extended sense in a neighborhood of the initial condition.<ref>{{harvtxt\|Coddington\|Levinson\|1955}}, Theorem 1.1 of Chapter 2</ref>

			== Notes ==
			<references/>

			== References ==
			* {{Citation \| last1=Coddington \| first1=Earl A. \| last2=Levinson \| first2=Norman \| title=Theory of Ordinary Differential Equations \| publisher=[[McGraw-Hill]] \| location=New York \| year=1955}}.
			* {{Citation \| last1=Rudin \| first1=Walter \| author1-link=Walter Rudin \| title=Real and complex analysis \| publisher=[[McGraw-Hill]] \| location=New York \| edition=3rd \| isbn=978-0-07-054234-1 \| mr=924157 \| year=1987}}.

			{{DEFAULTSORT:Caratheodory's existence theorem}}
			[[Category:Ordinary differential equations]]
			[[Category:Theorems in analysis]]

en>David Eppstein: split off perfectly orderable graph into a separate article (total length is ok but I think the subjects are too different from each other); add more sections

2012-08-09T01:03:14Z

split off perfectly orderable graph into a separate article (total length is ok but I think the subjects are too different from each other); add more sections

New page

Surely the second option would be more beneficial for any website. Medical word press themes give you the latest medical designs. I thought about what would happen by placing a text widget in the sidebar beneath my banner ad, and so it went. Word - Press also provides protection against spamming, as security is a measure issue. After activating, you will find their website link and get the activation code from their website. 

When you loved this post and you would want to receive more details about [http://xyz.ms/backup_plugin_780467 backup plugin] kindly visit the web site. Thus, it is imperative that you must Hire Word - Press Developers who have the expertise and proficiency in delivering theme integration and customization services. When you write a new post, you'll see a small bar that goes across the text input area. A Wordpress plugin is a software that you can install into your Wordpress site. Now, I want to anxiety that not every single query will be answered. The biggest advantage of using a coupon or deal plugin is that it gives your readers the coupons and deals within minutes of them becoming available. ve labored so hard to publish and put up on their website. Now if we talk about them one by one then -wordpress blog customization means customization of your blog such as installation of wordpress on your server by wordpress developer which will help you to acquire the SEO friendly blog application integrated with your site design as well as separate blog administration panel for starting up your own business blog,which demands a experienced wordpress designer. Are you considering getting your website redesigned. Our skilled expertise, skillfulness and excellence have been well known all across the world. Websites using this content based strategy are always given top scores by Google. If all else fails, please leave a comment on this post with the issue(s) you're having and help will be on the way. The SEOPressor Word - Press SEO Plugin works by analysing each page and post against your chosen keyword (or keyword phrase) and giving a score, with instructions on how to improve it. The templates are designed to be stand alone pages that have a different look and feel from the rest of your website. Word - Press is the most popular open source content management system (CMS) in the world today. Wordpress template is loaded with lots of prototype that unite graphic features and content area. Instead, you can easily just include it with our bodies integration field in e - Panel. I don't want that for my visitors and I'm quite sure they don't either. The days of spending a lot of time and money to have a website built are long gone. Word - Press is an open source content management system which is easy to use and offers many user friendly features. As for performing online business, websites and blogs are the only medium that are available to interact with customers and Word - Press perform this work with the help of cross-blog communication tools, comments and full user registration plug-ins.

Greedy coloring - Revision history

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q5601715

en>David Eppstein: split off perfectly orderable graph into a separate article (total length is ok but I think the subjects are too different from each other); add more sections