formulasearchengine - User contributions [en]

Cat state

2014-09-23T17:10:54Z

155.41.83.198:

== Mbt Winkel Rotterdam Spreken van logeerpartijtjes ==

Maar het was echt een fantastische afwerking, omdat hij jongens gedrapeerd over hem gehad. Natuurlijk, de Dynamo wist gewoon verwelken en in plaats daarvan opgevoerd tot de druk op zoek naar de gelijkmaker, die bijna kwam in de 80e minuut .. Hoe op te lossen: Controleer de spelling en geval van het element en attribuut, (Vergeet XHTML is alleen kleine letters) en / of controleer dat ze beide zijn toegestaan in het gekozen type document, en / of het gebruik van CSS in plaats van dit attribuut. Vanwege een motorfiets klein formaat, vele malen het ziet er verder weg dan het eigenlijk is. En het [http://www.ardovlam.be/intranet/contactok.asp?m=3-Mbt-Winkel-Rotterdam Mbt Winkel Rotterdam] kan ook moeilijk zijn om de snelheid van de motorfiets nauwkeurig te beoordelen. Er zijn veel mensen bereiken geld verdienen gta 3 zeer snel doelen en doelstellingen Make Money Gta moet je daar heb ik verkeerd gegaan en wat naar u terug kunt komen MLM blog en heeft niets om een hoge kwaliteit middelen zou ook willen gaan. Bedankt voor de reactie, aardbewoner. Ja, natuurlijk vroeg ik de dokter, ik had gekregen een relevant antwoord dat ik niet zou'' t nodig hebben gehad om te posten. "Je moet jezelf in hun schoenen, loop een week in hun schoenen, en voel de pijn die ze voelen elke dag en u hen zou kunnen behandelen met een beetje meer respect," zegt ze. Het is niet een diepgaande artikel, maar heeft betrekking op het proces van slipgieten vrij goed. Ik kan niet instaan voor de juistheid van het artikel, hoewel, in termen van de lengte van de tijd die nodig is om een stuk te vormen, omdat deze artikelen hebben tegenstrijdige adviezen en ik niet weet wat ik moet geloven. NPR heeft een interview met de Sydney Morning Herald Malcom Knox, die voor het eerst het verhaal dat de in het boek beschreven gebeurtenissen nooit plaatsgevonden brak. Knox brengt een uitstekend punt, dat ik niet denken heeft genoeg aandacht gekregen: Khouri had beweerd dat maar liefst 5000 vrouwen werden het slachtoffer van eerwraak in Jordanië, waardoor het klinkt als onderdeel van de mainstream cultuur, terwijl het in feite het nummer is een paar tientallen gevallen, meestal op de rand. Spreken van logeerpartijtjes, hebben we ook een ooggetuigenverslag van de CEO PR stunt voor [http://www.kaasbistro.be/includes/curiosa.asp?new=77-New-Balance-Schoenen-Dames New Balance Schoenen Dames] dakloosheid bij 201 Portage gisteravond. Podcast: De media spreekt over Harry Wie terwijl Gerrard praat over FASD; WRHA artsen krijgen [http://www.metallink.be/flash/produkten.asp?m=16-Michael-Kors-Tas-Slangenprint Michael Kors Tas Slangenprint] politicalCanada Day was geweldig, nu weer aan het werk geladen [http://www.ilpastaiolo.be/Test_site/OLd/Slide/slidepasta.asp?k=24-Oakley-Store-Antwerpen Oakley Store Antwerpen] vrijdag Nighthawk op CJOBNorth eindigen liberale kandidaat zegt campagne vuile trucs "pesten" Public Trustee eindelijk vorken over Knizat goed na Dr<ul>

<li>[http://www.pierre-le-jeune.info/spip.php?article3 http://www.pierre-le-jeune.info/spip.php?article3]</li>

<li>[http://www.histoirepassion.eu/spip.php?article1078/ http://www.histoirepassion.eu/spip.php?article1078/]</li>

<li>[http://enseignement-lsf.com/spip.php?article64#forum18465057 http://enseignement-lsf.com/spip.php?article64#forum18465057]</li>

<li>[http://www.garanhunsnegocios.com/index.php?page=item&id=84150 http://www.garanhunsnegocios.com/index.php?page=item&id=84150]</li>

<li>[http://www.vedbuzz.com/index.php?page=item&id=142824 http://www.vedbuzz.com/index.php?page=item&id=142824]</li>

</ul>

Johnson–Lindenstrauss lemma

2013-10-20T17:52:27Z

155.41.81.126: /* Lemma */

'''Episodic tremor and slip (ETS)''' is a [[seismology|seismological]] phenomenon observed in some [[subduction|subduction zones]]. These events are characterized by non-[[earthquake]] seismic rumbling, or tremor, and [[slow earthquake|slow slip]] along the plate interface. Slow slip events are distinguished from earthquakes by their propagation speed and [[hypocenter|focus]]. In slow slip events, there is an apparent reversal of crustal motion, although the fault motion remains consistent with the direction of subduction. ETS events themselves are imperceptible to human beings and do not cause damage.<ref>{{cite web|title=Episodic Tremor and Slip beneath Vancouver Island|url=http://earthquakescanada.nrcan.gc.ca/pprs-pprp/re/ETS-eng.php|publisher=Natural Resources Canada|accessdate=17 June 2011}}</ref>

==Discovery==
[[Image:Cascadia earthquake sources.png|thumb|upright=1.75|Structure of the Cascadia subduction zone. The Juan de Fuca Plate is subducting northeastward under the North American Plate.]]

Nonvolcanic, episodic tremor was first identified in southwest Japan in 2002.<ref>{{cite journal |last1=Obara |first1=Kazushige |last2= |first2= |year=2002 |title=Nonvolcanic Deep Tremor Associated with Subduction in Southwest Japan |journal=Science |volume=296 |issue= |pages=1679–1681 |publisher= |doi= |url= |accessdate=1 June 2013}}</ref> Shortly after, the [[Geological Survey of Canada]] coined the term "episodic tremor and slip" to characterize observations of [[Global Positioning Satellite|GPS]] measurements in the [[Vancouver Island]] area.<ref>{{cite web|title=Geodynamics – Episodic Tremor and Slip (ETS)|url=http://gsc.nrcan.gc.ca/geodyn/ets_e.php|publisher=Natural Resources Canada – Geological Survey of Canada|accessdate=17 June 2011}}
</ref> Vancouver Island lies in the eastern, North American region of the [[Cascadia subduction zone]]. ETS events in Cascadia were observed to reoccur cyclically with a period of approximately 14 months.<ref name="Rogers2003">Rogers, G., and H. Dragert (2003), "Episodic tremor and slip on the Cascadia subduction zone: the chatter of silent slip", ''Science'', ''300'' (5627), 1942, [http://dx.doi.org/10.1126/science.1084783 doi:10.1126/science.1084783].
</ref> Analysis of measurements led to the successful prediction of ETS events in following years (e.g., 2003, 2004, 2005, and 2007). In Cascadia, these events are marked by about two weeks of 1 to 10 Hz seismic trembling and non-earthquake ("aseismic") slip on the plate boundary equivalent to a [[moment magnitude|magnitude]] 7 earthquake. (Tremor is a weak seismological signal only detectable by very sensitive seismometers.) Recent episodes of tremor and slip in the Cascadia region have occurred [[strike and dip|down-dip]] of the region ruptured in the [[1700 Cascadia earthquake]].

Since the initial discovery of this seismic mode in the Cascadia region, slow slip and tremor have been detected in other subduction zones around the world, including Japan and Mexico.<ref name="Liu2009">Liu, Y., and J. Rice (2009), Slow slip predictions based on granite and gabbro friction data compared to GPS measurements in Northern Cascadia, ''Journal of Geophysical Research'', ''114'', B09,407, [http://dx.doi.org/10.1029/2008JB006142 doi:10.1029/2008JB006142].</ref>
Slow slip is not accompanied by tremor in the [[Hikurangi Trench|Hikurangi Subduction Zone]].<ref name="Delahaye2009">Delahaye, E., J. Townend, M. Reyners, and G. Rogers (2009), "Microseismicity but no tremor accompanying slow slip in the Hikurangi subduction zone, New Zealand", ''Earth and Planetary Science Letters'', ''277'' (1-2), 21–28, [http://dx.doi.org/10.1016/j.epsl.2008.09.038 doi:10.1016/j.epsl.2008.09.038].
</ref>

Every five years a year-long quake of this type occurs beneath the New Zealand capital, Wellington. It was first measured in 2003, and has reappeared in 2008 and 2013.<ref>{{cite news| url= http://www.3news.co.nz/Silent-quake-gently-rocks-Wellington/tabid/1160/articleID/299393/Default.aspx|work=3 News NZ |title= 'Silent' quake gently rocks Wellington| date=May 28, 2013}}</ref>

==Characteristics==

===Slip behaviour===
[[File:Displacement of Albert Head GPS Station, Victoria, British Columbia, 2005-2012.svg|thumb|upright=2|right|GPS measurements from [[Victoria, British Columbia]] show periodic reversals in crustal deformation in the North American region of the Cascadia Subduction Zone.]]

In the Cascadia subduction zone, the [[Juan de Fuca Plate]], a relic of the ancient [[Farallon Plate]], is actively subducting eastward underneath the [[North American Plate]]. The boundary between the Juan de Fuca and North American plates is generally "locked" due to interplate friction. A GPS marker on the surface of the North American plate above the locked region will trend eastward as it is dragged by the subduction process. [[Geodesy|Geodetic]] measurements show periodic reversals in the motion (i.e., westward movement) of the overthrusting North American Plate.<ref name="Rogers2003" /> During these reversals, the GPS marker will be displaced to the west over a period of days to weeks. Because these events occur over a much longer duration than earthquakes, they are termed "slow slip events".

Slow slip events have been observed to occur in the Cascadia, Japan, and Mexico subduction zones.<ref name="Liu2009" /> Unique characteristics of slow slip events include periodicity on timescales of months to years, [[hypocentre|focus]] near or [[Strike and dip|down-dip]] of the locked zone, and [[Strike and dip|along-strike]] propagation of 5 to 15 km/d.<ref name="Liu2009" /> In contrast, a typical earthquake rupture velocity is 70 to 90% of the [[S-wave]] velocity, or approximately 3.5 km/s.

Because slow slip events occur in subduction zones, their relationship to [[megathrust]] earthquakes is of economic, human, and scientific importance. The seismic hazard posed by ETS events is dependent on their focus. If the slow slip event extends into the [[Shear_zone#General_introduction|seismogenic zone]], accumulated stress would be released, decreasing the risk of a catastrophic earthquake.<ref name="Rubinstein2010">Rubinstein, J., Shelly, D., Ellsworth, W. (2010), "Non-volcanic Tremor: A Window into the Roots of Fault Zones", in ''New Frontiers in Integrated Solid Earth Sciences'', edited by S. Cloetingh and J. Negendank, pp. 287--314, Springer Science+Business Media B.V., [http://dx.doi.org/10.1007/978-90-481-2737-5_8 doi:10.1007/978-90-481-2737-5_8].</ref><ref name="Kostoglodov2003">Kostoglodov, V., S. Singh, J. Santiago, S. Franco, K. Larson, A. Lowry, and R. Bilham (2003), "A large silent earthquake in the Guerrero seismic gap, Mexico", ''Geophysical Research Letters'', ''30'' (15), 1807, [http://dx.doi.org/10.1029/2003GL017219 doi:10.1029/2003GL017219].</ref> However, if the slow slip event occurs down-dip of the seismogenic zone, it may "load" the region with stress.<ref name="Rubinstein2010" /><ref name="Brudzinski2007a">Brudzinski, M., Cabral-Cano, E., Correa-Mora, F., Demets, C., Márquez-Azúa, B. (2007), "Slow slip transients along the Oaxaca subduction segment from 1993 to 2007", ''Geophysical Journal International'', ''171'' (2), 523--538, [http://dx.doi.org/10.1111/j.1365-246X.2007.03542.x doi:10.1111/j.1365-246X.2007.03542.x].</ref> The probability of a great earthquake ([[moment magnitude]] <math>M_w \geq 8.0</math>) occurring has been suggested to be 30 times greater during an ETS event than otherwise,<ref name="Mazotti2004">Mazzoti, S. and J. Adams (2004), "Variability of near-term probability for the next great earthquake on the Cascadia subduction zone", ''Bulletin of the Seismological Society of America'', ''94'', 1954--1959, [http://dx.doi.org/10.1785/012004032 doi:10.1785/012004032].</ref> but more recent observations have shown this theory to be simplistic.<ref name="Beroza2011">Beroza, G. C. and S. Ide (2011), "Non-Volcanic Tremor and Slow Earthquakes", ''Annu. Rev. Earth Planet. Sci.'', ''39'', 271-296.</ref> One factor is that tremor occurs in many segments at different times along the a plate boundary; another factor is that rarely have tremor and large earthquakes been observed to correlate in timing .<ref name="Brudzinski2007b">Brudzinski, M. and Allen, R. (2007), "Segmentation in episodic tremor and slip all along Cascadia", ''Geology'', ''35'' (10), 905--910, [http://dx.doi.org/10.1130/G23740A.1 doi:10.1130/G23740A.1].</ref>

===Tremor===
Slow slip events are frequently linked to non-volcanic seismological "rumbling", or tremor. Tremor is distinguished from earthquakes in several key respects: frequency, duration, and origin. [[Seismic wave]]s generated by earthquakes are high-frequency and short-lived. These characteristics allow seismologists to determine the [[hypocentre]] of an earthquake using [[Seismic_wave#Usefulness_of_P_and_S_waves_in_locating_an_event|first-arrival methods]]. In contrast, tremor signals are weak and extended in duration.<ref name="Shelly2007">Shelly, D., Berozal, G. and Ide, S (2007), "Non-volcanic tremor and low-frequency earthquake swarms", ''Nature'', ''446'', 305-307, [http://dx.doi.org/10.1038/nature05666 doi:10.1038/nature05666].</ref> Furthermore, while earthquakes are caused by the rupture of [[Fault (geology)|faults]], tremor is generally attributed to underground movement of fluids (magmatic or hydrothermal).<ref name="Schwartz2007">Schwartz, S., and J. Rokosky (2007), "Slow slip events and seismic tremor at circum-Pacific subduction zones", ''Reviews of Geophysics'', ''45'' (3), [http://dx.doi.org/10.1029/2006RG000208 doi:10.1029/2006RG000208].</ref> As well as in subduction zones, tremor has been detected in transform faults such as the [[San Andreas Fault|San Andreas]].<ref name="Nadeau2004">Nadeau, R. and Dolenc, D. (2004), "Nonvolcanic Tremors Deep Beneath the San Andreas Fault", ''Science'', ''307'' (5708), [http://dx.doi.org/10.1126/science.1107142 doi:10.1126/science.1107142].</ref>

In both the Cascadia and [[Nankai Trough|Nankai]] subduction zones, slow slip events are directly associated with tremor.<ref name="Rogers2003" /><ref name="Obara2004">Obara, K., H. Hirose, F. Yamamizu, and K. Kasahara (2004), "Episodic slow slip events accompanied by non-volcanic tremors in southwest Japan subduction zone", ''Geophysical Research Letters'', ''31'' (23), L23,602, [http://dx.doi.org/10.1029/2004GL020848 doi:10.1029/2004GL020848].</ref> In the Cascadia subduction zone, slip events and seismological tremor signals are temporally coincident,<ref name="Rogers2003" /> but this relationship does not extend to the Mexican subduction zone.<ref name="Payero2008">Payero, J., V. Kostoglodov, N. Shapiro, T. Mikumo, A. Iglesias, X. Pérez-Campos, and R. Clayton (2008), "Nonvolcanic tremor observed in the Mexican subduction zone", ''Geophysical Research Letters'', ''35'' (7), [http://dx.doi.org/10.1029/2007GL032877 doi:10.1029/2007GL032877]</ref> Furthermore, this association is not an intrinsic characteristic of slow slip events. In the [[Hikurangi Trough|Hikurangi Subduction Zone]], New Zealand, episodic slip events are associated with distinct, [[Fault (geology)#Dip-slip_faults|reverse-faulted]] microearthquakes.<ref name="Delahaye2009" />

Two types of tremor have been identified: one associated with geodetic deformation (as described above), and one associated with 5 to 10 second bursts excited by distant earthquakes. The second type of tremor has been detected worldwide; for example, it has been triggered in the San Andreas Fault by the [[2002 Denali earthquake]] and in Taiwan by the [[2001 Kunlun earthquake]].<ref name="Peng2008a">Peng, Z., Vidale, J., Creager, K., Rubinstein, J., Gomberg, J., and Bodin, P. (2008), "Strong tremor near Parkfield, CA, excited by the 2002 Denali Fault earthquake", ''Geophysical Research Letters'', ''35'', L23305,5, [http://dx.doi.org/10.1029/2008GL036080 doi:10.1029/2008GL036080].</ref><ref name="Peng2008b">Peng, Z., Chao, K. (2008), "Non-volcanic tremor beneath the Central Range in Taiwan triggered by the 2001 <math>M_w</math> 7.8 Kunlun earthquake", ''Geophysical Journal International'' ''175'' (2), [http://dx.doi.org/10.1111/j.1365-246X.2008.03886.x doi:10.1111/j.1365-246X.2008.03886.x].</ref>

==Geological interpretation==
Tremor is commonly associated with the underground movement of magmatic or hydrothermal fluids.<ref name="Schwartz2007" /> As a plate is subducted into the mantle, it loses water from its [[Porosity|porespace]] and due to phase changes of hydrous minerals (such as [[amphibole]]). It has been proposed that this liberation of water generates a [[supercritical fluid]] at the plate interface, lubricating plate motion.<ref name="Obara2002">Obara, K. (2002), "Nonvolcanic deep tremor associated with subduction in southwest Japan", ''Science'', ''296'' (5573), 1679, [http://dx.doi.org/10.1126/science.1070378 doi:10.1126/science.1070378].</ref> This supercritical fluid may open fractures in the surrounding rock, and that tremor is the seismological signal of this process.<ref name="Obara2002" /> Mathematical modelling has successfully reproduced the periodicity of episodic tremor and slip in the Cascadia region by incorporating this dehydration effect.<ref name="Liu2007">Liu, Y., and J. Rice (2007), "Spontaneous and triggered aseismic deformation transients in a subduction fault model", ''Journal of Geophysical Research'', ''112'' (10.1029) [http://dx.doi.org/10.1029/2007JB004930 doi:10.1029/2007JB004930].</ref> In this interpretation, tremor may be enhanced where the subducting oceanic crust is young, hot, and wet as opposed to older and colder.

However, alternative models have also been proposed. Tremor has been demonstrated to be influenced by tides or variable fluid flow through a fixed volume.<ref name="Rubinstein2010" /><ref name="Watanabe2007">Watanabe, T., Y. Hiramatsu, and K. Obara (2007), "Scaling relationship between the duration and the
amplitude of non-volcanic deep low-frequency tremors", ''Geophysical Research Letters'', ''34'' (7) [http://dx.doi.org/10.1029/2007GL029391 doi:10.1029/2007GL029391].</ref> Tremor has also been attributed to shear slip at the plate interface.<ref name="Rogers2003" />

==See also==
* [[Geodynamics]]
* [[Plate tectonics]]
* [[Seismology]]
* [[Slow earthquake]]

==References==
{{reflist}}

==External links==
* [http://www.nrcan-rncan.gc.ca/com/index-eng.php Natural Resources Canada]

[[Category:Geophysics]]
[[Category:Seismology and earthquake terminology]]

Cellular approximation theorem

2013-09-20T19:02:50Z

155.41.82.240: /* Idea of proof */ spurious comma

In [[numerical analysis]], the '''Shanks transformation''' is a [[non-linear]] [[series acceleration]] method to increase the [[rate of convergence]] of a [[sequence]]. This method is named after [[Daniel Shanks]], who rediscovered this sequence transformation in 1955. It was first derived and published by R. Schmidt in 1941.<ref>Weniger (2003).</ref>

{{Quote box
| quote=One can calculate only a few terms of a [[perturbation theory|perturbation expansion]], usually no more than two or three, and almost never more than seven. The resulting series is often slowly convergent, or even divergent. Yet those few terms contain a remarkable amount of information, which the investigator should do his best to extract. This viewpoint has been persuasively set forth in a delightful paper by Shanks (1955), who displays a number of amazing examples, including several from [[fluid mechanics]].
| source= [[Milton Van Dyke|Milton D. Van Dyke]] (1975) ''Perturbation methods in fluid mechanics'', p. 202.
| width= 60%
| align= right
}}

==Formulation==

For a sequence <math>\left\{a_m\right\}_{m\in\mathbb{N}}</math> the series

:<math>A = \sum_{m=0}^\infty a_m\,</math>

is to be determined. First, the partial sum <math>A_n</math> is defined as:

:<math>A_n = \sum_{m=0}^n a_m\,</math>

and forms a new sequence <math>\left\{A_n\right\}_{n\in\mathbb{N}}</math>. Provided the series converges, <math>A_n</math> will approach in the limit to <math>A</math> as <math>n\to\infty.</math>
The Shanks transformation <math>S(A_n)</math> of the sequence <math>A_n</math> is defined as<ref name=BenderOrszag368>Bender & Orszag (1999), pp. 368–375.</ref><ref name=VanDyke>Van Dyke (1975), pp. 202–205.</ref>

:<math>S(A_n) = \frac{A_{n+1}\, A_{n-1}\, -\, A_n^2}{A_{n+1}-2A_n+A_{n-1}}</math>

and forms a new sequence. The sequence <math>S(A_n)</math> often converges more rapidly than the sequence <math>A_n.</math>
Further speed-up may be obtained by repeated use of the Shanks transformation, by computing <math>S^2(A_n)=S(S(A_n)),</math> <math>S^3(A_n)=S(S(S(A_n))),</math> etc.

Note that the non-linear transformation as used in the Shanks transformation is of similar form as used in [[Aitken's delta-squared process]]. But while Aitken's method operates on the coefficients <math>\left\{a_m\right\}</math> of the original sequence, the Shanks transformation operates on the partial sums <math>A_n.</math>

==Example==

[[Image:Shanks transformation.svg|thumb|400px|right|Absolute error as a function of <math>n</math> in the partial sums <math>A_n</math> and after applying the Shanks transformation once or several times: <math>S(A_n),</math> <math>S^2(A_n)</math> and <math>S^3(A_n).</math> The series used is <math>\scriptstyle 4\left(1-\frac13+\frac15-\frac17+\frac19-\cdots\right),</math> which has the exact sum <math>\pi.</math>]]
As an example, consider the slowly convergent series<ref name=VanDyke/>

:<math> 4 \sum_{k=0}^\infty (-1)^k \frac{1}{2k+1} = 4 \left( 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots \right) </math>

which has the exact sum ''π'' ≈ 3.14159265. The partial sum <math>A_6</math> has only one digit accuracy, while six-figure accuracy requires summing about 400,000 terms.

In the table below, the partial sums <math>A_n</math>, the Shanks transformation <math>S(A_n)</math> on them, as well as the repeated Shanks transformations <math>S^2(A_n)</math> and <math>S^3(A_n)</math> are given for <math>n</math> up to 12. The figure to the right shows the absolute error for the partial sums and Shanks transformation results, clearly showing the improved accuracy and convergence rate.

{| class="wikitable" style="text-align:center; width:40%"
|-
! width="10% | <math>n</math>
! width="20% | <math>A_n</math>
! width="20% | <math>S(A_n)</math>
! width="20% | <math>S^2(A_n)</math>
! width="20% | <math>S^3(A_n)</math>
|-
| 0 || 4.00000000 || — || — || —
|-
| 1 || 2.66666667 || 3.16666667 || — || —
|-
| 2 || 3.46666667 || 3.13333333 || 3.14210526 || —
|-
| 3 || 2.89523810 || 3.14523810 || 3.14145022 || 3.14159936
|-
| 4 || 3.33968254 || 3.13968254 || 3.14164332 || 3.14159086
|-
| 5 || 2.97604618 || 3.14271284 || 3.14157129 || 3.14159323
|-
| 6 || 3.28373848 || 3.14088134 || 3.14160284 || 3.14159244
|-
| 7 || 3.01707182 || 3.14207182 || 3.14158732 || 3.14159274
|-
| 8 || 3.25236593 || 3.14125482 || 3.14159566 || 3.14159261
|-
| 9 || 3.04183962 || 3.14183962 || 3.14159086 || 3.14159267
|-
| 10 || 3.23231581 || 3.14140672 || 3.14159377 || 3.14159264
|-
| 11 || 3.05840277 || 3.14173610 || 3.14159192 || 3.14159266
|-
| 12 || 3.21840277 || 3.14147969 || 3.14159314 || 3.14159265
|}

The Shanks transformation <math>S(A_1)</math> already has two-digit accuracy, while the original partial sums only establish the same accuracy at <math>A_{24}.</math> Remarkably, <math>S^3(A_3)</math> has six digits accuracy, obtained from repeated Shank transformations applied to the first seven terms <math>A_0</math>, ... , <math>A_6.</math> As said before, <math>A_n</math> only obtains 6-digit accuracy after about summing 400,000 terms.

==Motivation==

The Shanks transformation is motivated by the observation that — for larger <math>n</math> — the partial sum <math>A_n</math> quite often behaves approximately as<ref name=BenderOrszag368/>

:<math>A_n = A + \alpha q^n, \,</math>

with <math>|q|<1</math> so that the sequence converges [[transient (oscillation)|transient]]ly to the series result <math>A</math> for <math>n\to\infty.</math>
So for <math>n-1,</math> <math>n</math> and <math>n+1</math> the respective partial sums are:

:<math>A_{n-1} = A + \alpha q^{n-1} \quad , \qquad A_n = A + \alpha q^n \qquad \text{and} \qquad A_{n+1} = A + \alpha q^{n+1}.</math>

These three equations contain three unknowns: <math>A,</math> <math>\alpha</math> and <math>q.</math> Solving for <math>A</math> gives<ref name=BenderOrszag368/>

:<math>A = \frac{A_{n+1}\, A_{n-1}\, -\, A_n^2}{A_{n+1}-2A_n+A_{n-1}}.</math>

In the (exceptional) case that the denominator is equal to zero: then <math>A_n=A</math> for all <math>n.</math>

==Generalized Shanks transformation==

The generalized ''k''th-order Shanks transformation is given as the ratio of the [[determinant]]s:<ref name=BenderOrszag389>Bender & Orszag (1999), pp. 389–392.</ref>
:<math>
S_k(A_n)
= \frac{
\begin{vmatrix}
A_{n-k} & \cdots & A_{n-1} & A_n \\
\Delta A_{n-k} & \cdots & \Delta A_{n-1} & \Delta A_{n} \\
\Delta A_{n-k+1} & \cdots & \Delta A_{n} & \Delta A_{n+1} \\
\vdots & & \vdots & \vdots \\
\Delta A_{n-1} & \cdots & \Delta A_{n+k-2} & \Delta A_{n+k-1} \\
\end{vmatrix}
}{
\begin{vmatrix}
1 & \cdots & 1 & 1 \\
\Delta A_{n-k} & \cdots & \Delta A_{n-1} & \Delta A_{n} \\
\Delta A_{n-k+1} & \cdots & \Delta A_{n} & \Delta A_{n+1} \\
\vdots & & \vdots & \vdots \\
\Delta A_{n-1} & \cdots & \Delta A_{n+k-2} & \Delta A_{n+k-1} \\
\end{vmatrix}
},
</math>
with <math>\Delta A_p = A_{p+1} - A_p.</math> It is the solution of a model for the convergence behaviour of the partial sums <math>A_n</math> with <math>k</math> distinct transients:

:<math>A_n = A + \sum_{p=1}^k \alpha_p q_p^n.</math>

This model for the convergence behaviour contains <math>2k+1</math> unknowns. By evaluating the above equation at the elements <math>A_{n-k}, A_{n-k+1}, \ldots, A_{n+k}</math> and solving for <math>A,</math> the above expression for the ''k''th-order Shanks transformation is obtained. The first-order generalized Shanks transformation is equal to the ordinary Shanks transformation: <math>S_1(A_n)=S(A_n).</math>

The generalized Shanks transformation is closely related to [[Padé approximant]]s and [[Padé table]]s.<ref name=BenderOrszag389/>

==See also==
*[[Aitken's delta-squared process]]
*[[Rate of convergence]]
*[[Richardson extrapolation]]
*[[sequence transformation]]

==Notes==
{{reflist}}

==References==
*{{citation | first=D. | last=Shanks | authorlink=Daniel Shanks | year=1955 | title=Non-linear transformation of divergent and slowly convergent sequences | journal=Journal of Mathematics and Physics | volume = 34 | pages=1–42 }}
*{{citation | first=R. | last=Schmidt | title=On the numerical solution of linear simultaneous equations by an iterative method | journal=Philosophical Magazine | volume=32 | year=1941 | pages=369–383 }}
*{{citation | first=M.D. | last=Van Dyke | authorlink=Milton Van Dyke | title=Perturbation methods in fluid mechanics | publisher=Parabolic Press | year=1975 | edition=annotated | isbn=0-915760-01-0 }}
*{{citation | first1=C.M. | last1=Bender | authorlink1=Carl M. Bender | first2=S.A. | last2=Orszag | authorlink2=Steven A. Orszag | title=Advanced mathematical methods for scientists and engineers | publisher=Springer | year=1999 | isbn=0-387-98931-5 }}
*{{cite arxiv | author=Weniger, E.J. | title=Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series| eprint=math.NA/0306302 | year=2003 | version=v1 }}

[[Category:Numerical analysis]]
[[Category:Asymptotic analysis]]

Max-plus algebra

2013-04-09T20:17:34Z

155.41.24.25: /* Useful enhancement elements */

{{Uniform polypeton db|Uniform polypeton stat table|hop}}
In [[geometry]], a 6-[[simplex]] is a [[Duality (mathematics)|self-dual]] [[Regular polytope|regular]] [[6-polytope]]. It has 7 [[vertex (geometry)|vertices]], 21 [[Edge (geometry)|edge]]s, 35 triangle [[Face (geometry)|faces]], 35 [[Tetrahedron|tetrahedral]] [[Cell (mathematics)|cells]], 21 [[5-cell]] 4-faces, and 7 [[5-simplex]] 5-faces. Its [[dihedral angle]] is cos−1(1/6), or approximately 80.41°.

== Alternate names ==

It can also be called a '''heptapeton''', or '''hepta-6-tope''', as a 7-[[facet (geometry)|facetted]] polytope in 6-dimensions. The [[5-polytope#A note on generality of terms for n-polytopes and elements|name]] ''heptapeton'' is derived from ''hepta'' for seven [[Facet (mathematics)|facets]] in [[Greek language|Greek]] and [[Peta-|''-peta'']] for having five-dimensional facets, and ''-on''. Jonathan Bowers gives a heptapeton the acronym '''hop'''.<ref>Klitzing, (x3o3o3o3o3o - hop)</ref>

== Coordinates ==

The [[Cartesian coordinate]]s for an origin-centered regular heptapeton having edge length 2 are:

:<math>\left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm1\right)</math>
:<math>\left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)</math>
:<math>\left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)</math>
:<math>\left(\sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)</math>
:<math>\left(\sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)</math>
:<math>\left(-\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>

The vertices of the ''6-simplex'' can be more simply positioned in 7-space as permutations of:
: (0,0,0,0,0,0,1)

This construction is based on [[Facet (geometry)|facets]] of the [[7-orthoplex]].

== Images ==

{{6-simplex Coxeter plane graphs|t0|150}}

== Related uniform 6-polytopes ==
The regular 6-simplex is one of 35 [[Uniform 6-polytope#The A6 .5B3.2C3.2C3.2C3.2C3.5D family .286-simplex.29|uniform 6-polytopes]] based on the [3,3,3,3,3] [[Coxeter group]], all shown here in A6 [[Coxeter plane]] [[orthographic projection]]s.

{{Heptapeton family}}

==Notes ==
{{reflist}}

== References==
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
** Coxeter, ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
** '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
*{{KlitzingPolytopes|polypeta.htm|6D uniform polytopes (polypeta)|x3o3o3o3o - hix}}

== External links ==
*{{GlossaryForHyperspace | anchor=Simplex | title=Simplex }}
* [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
{{Polytopes}}

[[Category:6-polytopes]]

Schur orthogonality relations

2012-07-17T20:18:25Z

155.41.66.179:

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Talk:Time constant

2010-12-07T17:10:13Z

155.41.24.252: /* Time constant in electrophysiology */

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