Greedy coloring: Difference between revisions
Jump to navigation
Jump to search
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 |
en>Addbot m Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q5601715 |
||
| Line 1: | Line 1: | ||
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]]. | |||
== 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]] | |||
Latest revision as of 00:04, 20 March 2013
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'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.
Introduction
Consider the differential equation
with initial condition
where the function ƒ is defined on a rectangular domain of the form
Peano's existence theorem states that if ƒ is continuous, then the differential equation has at least one solution in a neighbourhood of the initial condition.[1]
However, it is also possible to consider differential equations with a discontinuous right-hand side, like the equation
where H denotes the Heaviside function defined by
It makes sense to consider the ramp function
as a solution of the differential equation. Strictly speaking though, it does not satisfy the differential equation at , 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 with initial condition if y is absolutely continuous, y satisfies the differential equation almost everywhere and y satisfies the initial condition.[2] The absolute continuity of y implies that its derivative exists almost everywhere.[3]
Statement of the theorem
Consider the differential equation
with defined on the rectangular domain . If the function satisfies the following three conditions:
- is continuous in for each fixed ,
- is measurable in for each fixed ,
- there is a Lebesgue-integrable function , , such that for all ,
then the differential equation has a solution in the extended sense in a neighborhood of the initial condition.[4]
Notes
- ↑ Template:Harvtxt, Theorem 1.2 of Chapter 1
- ↑ Template:Harvtxt, page 42
- ↑ Template:Harvtxt, Theorem 7.18
- ↑ Template:Harvtxt, Theorem 1.1 of Chapter 2
References
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.