Synchrotron radiation: Difference between revisions

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The exponent of (omega/omega_c) should be 1/2 not 2.
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{{About|a piecewise constant function|the unit step function|Heaviside step function}}
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In [[mathematics]], a [[function (mathematics)|function]] on the [[real number]]s is called a '''step function''' (or '''staircase function''') if it can be written as a [[finite set|finite]] [[linear combination]] of [[indicator function]]s of [[interval (mathematics)|interval]]s. Informally speaking, a step function is a [[piecewise]] [[constant function]] having only finitely many pieces.
[[Image:StepFunctionExample.png|thumb|right|250px|Example of a step function (the red graph). This particular step function is [[Continuous_function#Directional_and_semi-continuity|right-continuous]].]]
 
==Definition and first consequences==
A function <math>f: \mathbb{R} \rightarrow \mathbb{R}</math> is called a '''step function''' if it can be written as {{Citation needed|date=September 2009}}
 
:<math>f(x) = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}(x)\,</math> for all real numbers <math>x</math>
 
where <math>n\ge 0,</math> <math>\alpha_i</math> are real numbers, <math>A_i</math> are intervals, and <math>\chi_A\,</math> (sometimes written as <math>1_A</math>) is the [[indicator function]] of <math>A</math>:
 
:<math>\chi_A(x) =
\begin{cases}
1 & \mbox{if } x \in A, \\
0 & \mbox{if } x \notin A. \\
\end{cases}
</math>
 
In this definition, the intervals <math>A_i</math> can be assumed to have the following two properties:
 
# The intervals are [[disjoint set|disjoint]], <math>A_i\cap A_j=\emptyset</math> for <math>i\ne j</math>
# The [[union (set theory)|union]] of the intervals is the entire real line, <math>\cup_{i=0}^n A_i=\mathbb R.</math>
 
Indeed, if that is not the case to start with, a different set of intervals can be picked for which these assumptions hold. For example, the step function
 
: <math>f = 4 \chi_{[-5, 1)} + 3 \chi_{(0, 6)}\,</math>
 
can be written as
 
: <math>f = 0\chi_{(-\infty, -5)} +4 \chi_{[-5, 0]} +7 \chi_{(0, 1)} + 3 \chi_{[1, 6)}+0\chi_{[6, \infty)}.\,</math>
 
==Examples==
[[Image:Dirac distribution CDF.svg|325px|thumb|The [[Heaviside step function]] is an often used step function.]]
* A [[constant function]] is a trivial example of a step function. Then there is only one interval, <math>A_0=\mathbb R.</math>
* The [[Heaviside step function|Heaviside function]] ''H''(''x'') is an important step function. It is the mathematical concept behind some test [[Signal (electronics)|signals]], such as those used to determine the [[step response]] of a [[dynamical system (definition)|dynamical system]].
[[File:Rectangular function.svg|thumb|The [[rectangular function]], the next simplest step function.]]
* The [[rectangular function]], the normalized [[boxcar function]], is the next simplest step function, and is used to model a unit pulse.
 
=== Non-examples ===
* The [[integer part]] function is not a step function according to the definition of this article, since it has an infinite number of intervals. However, some authors define step functions also with an infinite number of intervals.<ref>for example see: {{Cite book | author=Bachman, Narici, Beckenstein | title=Fourier and Wavelet Analysis | publisher=Springer, New York, 2000 | isbn=0-387-98899-8 | chapter =Example 7.2.2}}</ref>
 
==Properties==
 
* The sum and product of two step functions is again a step function. The product of a step function with a number is also a step function. As such, the step functions form an [[algebra over a field|algebra]] over the real numbers.
* A step function takes only a finite number of values. If the intervals <math>A_i,</math> <math>i=0, 1, \dots, n,</math> in the above definition of the step function are disjoint and their union is the real line, then <math>f(x)=\alpha_i\,</math> for all <math>x\in A_i.</math>
* The [[Lebesgue integral]] of a step function <math>\textstyle f = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}\,</math> is <math>\textstyle \int \!f\,dx = \sum\limits_{i=0}^n \alpha_i \ell(A_i),\,</math> where <math>\ell(A)</math> is the length of the interval <math>A,</math> and it is assumed here that all intervals <math>A_i</math> have finite length. In fact, this equality (viewed as a definition) can be the first step in constructing the Lebesgue integral.<ref>{{Cite book | author=Weir, Alan J | authorlink= | coauthors= | title=Lebesgue integration and measure | date= | publisher=Cambridge University Press, 1973 | location= | isbn=0-521-09751-7 | unused_data=|chapter= 3}}</ref>
 
==See also==
*[[Simple function]]
*[[Piecewise defined function]]
*[[Sigmoid function]]
*[[Step detection]]
 
==References==
{{reflist}}
 
{{DEFAULTSORT:Step Function}}
[[Category:Special functions]]

Revision as of 00:13, 25 February 2014

Today I am sharing a dark experience with you because I am from this darkness today. Whoa! Google Chrome has crashed was the disaster which struck on me early this morning. My blood flow seemed to stop for a limited minutes. Next I regained my senses plus decided to fix Chrome crash.

Firstly, we should use your Antivirus or protection tool plus run a scan on your computer. It is possible which the computer is afflicted with virus or malware that slows down your computer. If there is nothing found in the scanning report, it can be the RAM which cause the problem.

Registry cleaning is significant because the registry will get crowded and messy when it is left unchecked. False entries send the operating program seeking files and directories which have lengthy ago been deleted. This takes time and utilizes valuable resources. So, a slowdown inevitably takes place. It is especially noticeable when you multitask.

If you feel you don't have enough funds at the time to upgrade, then the number one choice is to free up several area by deleting certain of the unwelcome files and folders.

Besides, when you may get a fix it utilities which could do the job for you effectively plus fast, then why not? There is 1 such system, RegCure that is surprisingly good and complete. It has features that other products do not have. It is the many recommended registry cleaner today.

The system is made and built for the purpose of helping we accomplish jobs plus not be pestered by windows XP error messages. When there are mistakes, what do we do? Some folks pull their hair and cry, whilst those sane ones have their PC repaired, whilst those absolutely wise ones research to have the mistakes fixed themselves. No, these errors were not also crafted to rob we off the income plus time. There are points to do to actually prevent this from happening.

Reboot PC - Simply reboot your PC to find if the error is gone. Frequently, rebooting the PC readjusts the internal settings and software plus hence fixes the issue. If it doesn't then move on to follow the instructions below.

Ally Wood is a expert software reviewer and has worked in CNET. Then she is working for her own review software company to give suggestions to the software creator and has completed deep test inside registry cleaner software. After reviewing the many popular registry cleaner, she has created complete review on a review site for we that could be accessed for free.

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