Algorithmically random sequence - Revision history

en>CharlesHBennett: Changed intro to say algorithmic randomness can also be defined for sequences in bases other than two. See talk

2014-07-06T19:35:50Z

Changed intro to say algorithmic randomness can also be defined for sequences in bases other than two. See talk

← Older revision		Revision as of 21:35, 6 July 2014
Line 1:		Line 1:
	~~Andrew Simcox~~ is the ~~name his mothers~~ and ~~fathers gave him and he completely enjoys this name. Distributing production is how he tends to make a living~~. ~~Ohio~~ is where my ~~home~~ is ~~but my~~ spouse ~~desires us to transfer. To perform lacross is the thing~~ I ~~adore most of all~~.<br><br>Here is my ~~webpage psychic phone readings [~~[http://~~203~~.~~250~~.~~78.160~~/~~zbxe~~/~~?document_srl=1792908 similar web site]~~]		Alyson is the title people use to contact me and I believe it sounds fairly great when you say it. Credit authorising is where my main earnings arrives from. One of the things she enjoys most is canoeing and she's been performing it for fairly a while. My spouse and I reside in Mississippi and I love each day residing right here.<br><br>Here is my blog [http://modenpeople.co.kr/modn/qna/292291 real psychic]

en>BD2412: /* Properties and examples of Martin-Löf random sequences */Fixing links to disambiguation pages using AWB

2014-02-20T20:27:47Z

Properties and examples of Martin-Löf random sequences: Fixing links to disambiguation pages using AWB

← Older revision		Revision as of 22:27, 20 February 2014
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	~~A '''perfectly matched layer''' ('''PML''')~~ is ~~an artificial absorbing layer for [[wave equation]]s, commonly used to truncate computational regions in [[numerical method]]s to simulate problems with open boundaries, especially in~~ the ~~[[Finite-difference time-domain method\|FDTD]]~~ and ~~[[finite element method\|FE]] methods~~. ~~The key property of a PML that distinguishes it from an ordinary absorbing material is that it~~ is ~~designed so that waves incident upon the PML from a non-PML medium do not reflect at the interface—this property allows the PML~~ to ~~strongly absorb outgoing waves from the interior of~~ a ~~computational region without reflecting them back into the interior~~.		Andrew Simcox is the name his mothers and fathers gave him and he completely enjoys this name. Distributing production is how he tends to make a living. Ohio is where my home is but my spouse desires us to transfer. To perform lacross is the thing I adore most of all.<br><br>Here is my webpage psychic phone readings [[http://203.250.78.160/zbxe/?document_srl=1792908 similar web site]]

	PML was originally formulated by Berenger in 1994 for use with [[Maxwell's equations]], and since that time there have been several related reformulations of PML for both Maxwell's equations and for other wave equations. Berenger's original formulation is called a '''split-field PML''', because it splits the [[electromagnetic field]]s into two unphysical fields in the PML region. A later formulation that has become more popular because of its simplicity and efficiency is called '''uniaxial PML''' or '''UPML''' (Gedney, 1996), in which the PML is described as an artificial [[birefringence\|anisotropic]] absorbing material. Although both Berenger's formulation and UPML were initially derived by manually constructing the conditions under which incident [[plane wave]]s do not reflect from the PML interface from a homogeneous medium, ''both'' formulations were later shown to be equivalent to ~~a much more elegant and general approach: '''stretched-coordinate PML''' (Chew and Weedon, 1994; Teixeira and Chew, 1998)~~. ~~In particular, PMLs were shown to correspond to a [[coordinate transformation]] in which one (or more) coordinates are mapped to [[complex number]]s; more technically, this~~ is ~~actually an [[analytic continuation]] of~~ the wave equation into complex coordinates, replacing propagating (oscillating) waves by [[exponentially decaying]] waves. This viewpoint allows PMLs to be derived for inhomogeneous media such as [[waveguide]]s, as well as for other [[coordinate system]]s and wave equations.

	~~==Technical description==~~

	Specifically, for a PML designed to absorb waves propagating in the ''x'' direction, the following transformation is included in the wave equation. Wherever an ''x'' derivative <math>\partial/\partial x</math> appears in the wave equation, it is replaced by:
	~~:<math>\frac{\partial}{\partial x} \to \frac{1}{1 + \frac{i\sigma(x)}{\omega}} \frac{\partial}{\partial x}</math>~~
	~~where ω is the [[angular frequency]] and σ is some [[function (mathematics)\|function]]~~ of ~~''x''. Wherever σ is positive, propagating waves are attenuated because:~~
	~~:<math>e^{i(kx - \omega t)} \to e^{i(kx - \omega t) - \frac{k}{\omega} \int^x \sigma(x') dx'} ,</math>~~
	where we have taken a planewave propagating in the +''x'' direction (for <math>k > 0</math>) and applied the transformation (analytic continuation) to complex coordinates: <math>x \to x + \frac{i}{\omega} \int^x \sigma(x') dx'</math>, or equivalently <math>dx \to dx (1 + i\sigma/\omega)</math>. ~~The same coordinate transformation causes waves to attenuate whenever their ''x'' dependence is in the form~~ <~~math~~>~~e^{ikx}~~<~~/math~~> ~~for some [[propagation constant]] ''k'': this includes planewaves propagating at some angle with the ''x'' axis and also [[transverse mode]]s of a waveguide.~~

	~~The above coordinate transformation can be left as-~~is ~~in the transformed wave equations, or can be combined with the material description (e.g. the~~ [[permittivity]] and [[Permeability (electromagnetism)\|permeability]] in Maxwell's equations) to form a UPML description. Note also that the coefficient σ/ω depends upon frequency—this is so the attenuation rate is proportional to ''k''/ω, which is independent of frequency in a homogeneous material (not including [[material dispersion]], e.g. for [[vacuum]]) because of the [[dispersion relation]] between ω and ''k''. However, this frequency-dependence means that a [[time domain]] implementation of PML, e.g. in the [[FDTD]] method, is more complicated than for a frequency-independent absorber, and involves the [[auxiliary differential equation]] (ADE) approach (equivalently, ''i''/ω appears as an [[integral]] or [[convolution]] in time domain).

	Perfectly matched layers, in their original form, only attenuate propagating waves; purely [[evanescent waves]] (exponentially decaying fields) oscillate in the PML but do not decay more quickly. However, the attenuation of evanescent waves can also be accelerated by including a [[real number\|real]] coordinate stretching in the PML: this corresponds to making σ in the above expression a [[complex number]], where the imaginary part yields a real coordinate stretching that causes evanescent waves to decay more quickly.

	One caveat with perfectly matched layers is that they are only reflectionless for the ''exact'' wave equation. Once the wave equation is [[discretization\|discretized]] for simulation on a computer, some small numerical reflections appear. For this reason, the PML absorption coefficient σ is typically turned on gradually from zero (e.g. [[quadratic function\|quadratically]]) over a short distance on the scale of the [[wavelength]] of the wave.

	~~==When perfectly matched layers fail==~~

	Perfectly matched layers have shown their efficiency in a lot of situations : acoustic or electromagnetic waves, Schrödinger equation, etc. But PML fail for media where backward waves appear as in [[negative index metamaterials]] or in [[Plasma_(physics)\|plasma]], but also for anisotropic acoustic and elastic waves, for aeroacoustic waves etc.

	Backward waves are waves with opposite [[Group velocity\|group]] and [[phase velocity]]. The outgoing waves are the ones with a group velocity which are pointed to the exterior of the domain. But perfectly matched layers attenuate waves according to their phase velocity ''k''/ω and not their group velocity. So perfectly matched layers amplify exponentially backward waves instead of attenuating them (Bécache, Fauqueux and Joly, 2003).

	~~== References ==~~

	*{{cite journal \| author= J. Berenger \| title= A perfectly matched layer for the absorption of electromagnetic waves \| journal= Journal of Computational Physics \| year= 1994 \| volume= 114 \| pages= 185–200 \| doi= 10.1006/jcph.1994.1159 \| issue= 2 \| bibcode=1994JCoPh.114..185B}}
	*{{cite journal \| author= S.D. Gedney \| title= An anisotropic perfectly matched layer absorbing media for the truncation of FDTD latices\| journal= Antennas and Propagation, IEEE Transactions on \| year= 1996 \| volume= 44 \| pages= 1630–1639 \| doi= 10.1109/8.546249 \| issue= 12 \| bibcode=1996ITAP...44.1630G}}
	*{{cite journal \| author= W. C. Chew and W. H. Weedon \| title= A 3d perfectly matched medium from modified Maxwell's equations with stretched coordinates\| journal= Microwave Optical Tech. Letters \| year= 1994 \| volume= 7 \| pages= 599–604 \| doi= 10.1002/mop.4650071304 \| issue= 13 }}
	*{{cite journal \| author= F. L. Teixeira W. C. Chew \| title= General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media\| journal= IEEE Microwave and Guided Wave Letters \| year= 1998 \| volume= 8 \| pages= 223–225 \| doi= 10.1109/75.678571 \| issue= 6 }}
	*{{cite journal \| author= E. Bécache, S. Fauqueux and P. Joly\| title= Stability of perfectly matched layers, group velocities and anisotropic waves\| journal= Journal of Computational Physics \| year= 2003 \| volume= 188 \| pages= 399–433\| doi=10.1016/S0021-9991(03)00184-0 \| issue= 2}} [http://hal.archives-ouvertes.fr/docs/00/07/22/83/PDF/RR-4304.pdf]
	*{{cite book \| author=[[Allen Taflove]] and Susan C. Hagness \| title=Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. \| publisher=Artech House Publishers \| year=2005 \| isbn=1-58053-832-0 }}
	* S. G. Johnson, [http://~~math~~.~~mit~~.~~edu/~stevenj/18~~.~~369~~/~~pml.pdf Notes on Perfectly Matched Layers], online MIT course notes (Aug. 2007).~~
	*{{cite journal \| author= R. M. S. de Oliveira and C. L. S. S. Sobrinho\| title= UPML formulation for truncating conductive media in curvilinear coordinates\| journal= Numerical Algorithms \| year= 2007 \| volume= 46 \| pages= 295–319\| doi= 10.1007/~~s11075-007-9139-6 }} [http://download.springer.com/static/pdf/502/art%253A10.1007%252Fs11075-007-9139-6.pdf?auth66=1391108927_9cfae44a5698a56067166a18abd13095&ext=.pdf]~~

	~~==External links==~~
	*[http://www.youtube.com/watch?v=~~XcL9iEK0GDY Animation on the effects of PML (YouTube)]~~


	~~[[Category:Numerical differential equations]]~~
	~~[[Category:Partial differential equations]]~~
	~~[[Category:Wave mechanics~~]]

en>Jochen Burghardt: /* References */ added open-access url

2014-01-31T09:14:58Z

References: added open-access url

← Older revision		Revision as of 11:14, 31 January 2014
Line 1:		Line 1:
	~~Jayson Berryhill~~ is ~~how I~~'m called ~~and my spouse doesn~~'~~t like~~ it ~~at all~~. ~~She~~ is ~~really~~ ~~real psychic readings (~~[~~http~~:~~//165~~.~~132~~.39.93/xe/~~visitors~~/~~372912 http~~://~~165~~.~~132~~.39.93/]) ~~fond~~ of ~~caving~~ but ~~she doesn~~'~~t have~~ the ~~time recently~~. ~~Office supervising~~ is ~~exactly~~ where ~~my primary earnings comes from~~ but I'~~ve always wanted my own business~~. ~~Some time ago he chose to reside in North Carolina~~ and ~~he doesn~~'~~t strategy on changing it~~.~~<br><br>Feel free~~ to ~~visit my web real psychics~~ ([http://~~chungmuroresidence~~.~~com~~/xe/~~reservation_branch2~~/~~152663 mouse click the up coming webpage~~]) ~~blog~~ - [http://~~www~~.~~familysurvivalgroup~~.com/~~easy~~-~~methods~~-~~planting~~-~~looking-backyard~~/ ~~tarot card readings~~]		A '''perfectly matched layer''' ('''PML''') is an artificial absorbing layer for [[wave equation]]s, commonly used to truncate computational regions in [[numerical method]]s to simulate problems with open boundaries, especially in the [[Finite-difference time-domain method\|FDTD]] and [[finite element method\|FE]] methods. The key property of a PML that distinguishes it from an ordinary absorbing material is that it is designed so that waves incident upon the PML from a non-PML medium do not reflect at the interface—this property allows the PML to strongly absorb outgoing waves from the interior of a computational region without reflecting them back into the interior.

			PML was originally formulated by Berenger in 1994 for use with [[Maxwell's equations]], and since that time there have been several related reformulations of PML for both Maxwell's equations and for other wave equations. Berenger's original formulation is called a '''split-field PML''', because it splits the [[electromagnetic field]]s into two unphysical fields in the PML region. A later formulation that has become more popular because of its simplicity and efficiency is called '''uniaxial PML''' or '''UPML''' (Gedney, 1996), in which the PML is described as an artificial [[birefringence\|anisotropic]] absorbing material. Although both Berenger's formulation and UPML were initially derived by manually constructing the conditions under which incident [[plane wave]]s do not reflect from the PML interface from a homogeneous medium, ''both'' formulations were later shown to be equivalent to a much more elegant and general approach: '''stretched-coordinate PML''' (Chew and Weedon, 1994; Teixeira and Chew, 1998). In particular, PMLs were shown to correspond to a [[coordinate transformation]] in which one (or more) coordinates are mapped to [[complex number]]s; more technically, this is actually an [[analytic continuation]] of the wave equation into complex coordinates, replacing propagating (oscillating) waves by [[exponentially decaying]] waves. This viewpoint allows PMLs to be derived for inhomogeneous media such as [[waveguide]]s, as well as for other [[coordinate system]]s and wave equations.

			==Technical description==

			Specifically, for a PML designed to absorb waves propagating in the ''x'' direction, the following transformation is included in the wave equation. Wherever an ''x'' derivative <math>\partial/\partial x</math> appears in the wave equation, it is replaced by:
			:<math>\frac{\partial}{\partial x} \to \frac{1}{1 + \frac{i\sigma(x)}{\omega}} \frac{\partial}{\partial x}</math>
			where ω is the [[angular frequency]] and σ is some [[function (mathematics)\|function]] of ''x''. Wherever σ is positive, propagating waves are attenuated because:
			:<math>e^{i(kx - \omega t)} \to e^{i(kx - \omega t) - \frac{k}{\omega} \int^x \sigma(x') dx'} ,</math>
			where we have taken a planewave propagating in the +''x'' direction (for <math>k > 0</math>) and applied the transformation (analytic continuation) to complex coordinates: <math>x \to x + \frac{i}{\omega} \int^x \sigma(x') dx'</math>, or equivalently <math>dx \to dx (1 + i\sigma/\omega)</math>. The same coordinate transformation causes waves to attenuate whenever their ''x'' dependence is in the form <math>e^{ikx}</math> for some [[propagation constant]] ''k'': this includes planewaves propagating at some angle with the ''x'' axis and also [[transverse mode]]s of a waveguide.

			The above coordinate transformation can be left as-is in the transformed wave equations, or can be combined with the material description (e.g. the [[permittivity]] and [[Permeability (electromagnetism)\|permeability]] in Maxwell's equations) to form a UPML description. Note also that the coefficient σ/ω depends upon frequency—this is so the attenuation rate is proportional to ''k''/ω, which is independent of frequency in a homogeneous material (not including [[material dispersion]], e.g. for [[vacuum]]) because of the [[dispersion relation]] between ω and ''k''. However, this frequency-dependence means that a [[time domain]] implementation of PML, e.g. in the [[FDTD]] method, is more complicated than for a frequency-independent absorber, and involves the [[auxiliary differential equation]] (ADE) approach (equivalently, ''i''/ω appears as an [[integral]] or [[convolution]] in time domain).

			Perfectly matched layers, in their original form, only attenuate propagating waves; purely [[evanescent waves]] (exponentially decaying fields) oscillate in the PML but do not decay more quickly. However, the attenuation of evanescent waves can also be accelerated by including a [[real number\|real]] coordinate stretching in the PML: this corresponds to making σ in the above expression a [[complex number]], where the imaginary part yields a real coordinate stretching that causes evanescent waves to decay more quickly.

			One caveat with perfectly matched layers is that they are only reflectionless for the ''exact'' wave equation. Once the wave equation is [[discretization\|discretized]] for simulation on a computer, some small numerical reflections appear. For this reason, the PML absorption coefficient σ is typically turned on gradually from zero (e.g. [[quadratic function\|quadratically]]) over a short distance on the scale of the [[wavelength]] of the wave.

			==When perfectly matched layers fail==

			Perfectly matched layers have shown their efficiency in a lot of situations : acoustic or electromagnetic waves, Schrödinger equation, etc. But PML fail for media where backward waves appear as in [[negative index metamaterials]] or in [[Plasma_(physics)\|plasma]], but also for anisotropic acoustic and elastic waves, for aeroacoustic waves etc.

			Backward waves are waves with opposite [[Group velocity\|group]] and [[phase velocity]]. The outgoing waves are the ones with a group velocity which are pointed to the exterior of the domain. But perfectly matched layers attenuate waves according to their phase velocity ''k''/ω and not their group velocity. So perfectly matched layers amplify exponentially backward waves instead of attenuating them (Bécache, Fauqueux and Joly, 2003).

			== References ==

			*{{cite journal \| author= J. Berenger \| title= A perfectly matched layer for the absorption of electromagnetic waves \| journal= Journal of Computational Physics \| year= 1994 \| volume= 114 \| pages= 185–200 \| doi= 10.1006/jcph.1994.1159 \| issue= 2 \| bibcode=1994JCoPh.114..185B}}
			*{{cite journal \| author= S.D. Gedney \| title= An anisotropic perfectly matched layer absorbing media for the truncation of FDTD latices\| journal= Antennas and Propagation, IEEE Transactions on \| year= 1996 \| volume= 44 \| pages= 1630–1639 \| doi= 10.1109/8.546249 \| issue= 12 \| bibcode=1996ITAP...44.1630G}}
			*{{cite journal \| author= W. C. Chew and W. H. Weedon \| title= A 3d perfectly matched medium from modified Maxwell's equations with stretched coordinates\| journal= Microwave Optical Tech. Letters \| year= 1994 \| volume= 7 \| pages= 599–604 \| doi= 10.1002/mop.4650071304 \| issue= 13 }}
			*{{cite journal \| author= F. L. Teixeira W. C. Chew \| title= General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media\| journal= IEEE Microwave and Guided Wave Letters \| year= 1998 \| volume= 8 \| pages= 223–225 \| doi= 10.1109/75.678571 \| issue= 6 }}
			*{{cite journal \| author= E. Bécache, S. Fauqueux and P. Joly\| title= Stability of perfectly matched layers, group velocities and anisotropic waves\| journal= Journal of Computational Physics \| year= 2003 \| volume= 188 \| pages= 399–433\| doi=10.1016/S0021-9991(03)00184-0 \| issue= 2}} [http://hal.archives-ouvertes.fr/docs/00/07/22/83/PDF/RR-4304.pdf]
			*{{cite book \| author=[[Allen Taflove]] and Susan C. Hagness \| title=Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. \| publisher=Artech House Publishers \| year=2005 \| isbn=1-58053-832-0 }}
			* S. G. Johnson, [http://math.mit.edu/~stevenj/18.369/pml.pdf Notes on Perfectly Matched Layers], online MIT course notes (Aug. 2007).
			*{{cite journal \| author= R. M. S. de Oliveira and C. L. S. S. Sobrinho\| title= UPML formulation for truncating conductive media in curvilinear coordinates\| journal= Numerical Algorithms \| year= 2007 \| volume= 46 \| pages= 295–319\| doi= 10.1007/s11075-007-9139-6 }} [http://download.springer.com/static/pdf/502/art%253A10.1007%252Fs11075-007-9139-6.pdf?auth66=1391108927_9cfae44a5698a56067166a18abd13095&ext=.pdf]

			==External links==
			*[http://www.youtube.com/watch?v=XcL9iEK0GDY Animation on the effects of PML (YouTube)]


			[[Category:Numerical differential equations]]
			[[Category:Partial differential equations]]
			[[Category:Wave mechanics]]

178.255.11.6 at 16:27, 22 February 2012

2012-02-22T16:27:14Z

New page

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