Regular singular point - Revision history

en>PWilkinson: Link disambiguation

2014-07-06T10:12:20Z

Link disambiguation

← Older revision		Revision as of 12:12, 6 July 2014
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	~~They contact me Emilia~~. ~~Her spouse and her reside in~~ Puerto Rico but home std test she will have to [http://www.smithandkenworld.com/activity-posted-by-2455/ home std test kit] move 1 [http://xn--299ay03byycca57h.kr/zbxe/?document_srl=319947 home std test kit] day or an additional. Hiring is ~~her [http://Www.slideshare.net/realtrafficsource/celebrities-with-herpes std testing] at home day job now~~ and ~~she will not alter it whenever quickly~~. ~~His spouse doesn't like it the way he does but~~ what ~~he over the counter std test truly likes doing is to do aerobics~~ and ~~he's been doing it for fairly a whilst~~.<br><br>~~Feel free to visit~~ my ~~homepage ::~~ [http://~~product-dev.younetco~~.com/~~xuannth/fox_ios1/index.php~~?do=/~~profile~~-~~1539/info/ std testing at home]~~		Nice to satisfy you, my title is Numbers Held although I don't truly like being known as like that. Managing individuals is his profession. Puerto Rico is exactly where he and his wife reside. Doing ceramics is what my family members and I appreciate.<br><br>Check out my website ... std testing at home - [http://riddlesandpoetry.com/?q=node/19209 her latest blog] -

en>Wavelength: inserting 1 hyphen: —> "one-dimensional"—wikt:one-dimensional

2014-02-20T20:51:48Z

inserting 1 hyphen: —> "one-dimensional"—wikt:one-dimensional

← Older revision		Revision as of 22:51, 20 February 2014
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	~~{{Context\|date=October 2009}}~~		They contact me Emilia. Her spouse and her reside in Puerto Rico but home std test she will have to [http://www.smithandkenworld.com/activity-posted-by-2455/ home std test kit] move 1 [http://xn--299ay03byycca57h.kr/zbxe/?document_srl=319947 home std test kit] day or an additional. Hiring is her [http://Www.slideshare.net/realtrafficsource/celebrities-with-herpes std testing] at home day job now and she will not alter it whenever quickly. His spouse doesn't like it the way he does but what he over the counter std test truly likes doing is to do aerobics and he's been doing it for fairly a whilst.<br><br>Feel free to visit my homepage :: [http://product-dev.younetco.com/xuannth/fox_ios1/index.php?do=/profile-1539/info/ std testing at home]
	In [[statistical physics]], the '''Smoluchowski coagulation equation''' is an [[integrodifferential equation]] introduced by [[Marian Smoluchowski]] in a seminal 1916 publication,<ref>{{cite journal\|last=Smoluchowski\|first=Marian\|title=Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen\|journal=Physik. ~~Z.\|year=1916\|volume=17\|pages=557–571, 585–599\|bibcode=1916ZPhy...17..557S}}</ref> describing the [[time evolution]] of the [[number density]] of particles as they coagulate (~~in ~~this context "clumping together")~~ to ~~size ''x'' at time ''t''.~~

	~~==Equation==~~

	~~In the case when the sizes of the coagulated particles are [[continuous variable]]s, the equation involves an~~ [~~[integral]]~~:

	~~: <math>\frac{\partial n(x,t)}{\partial t}=\frac{1}{2}\int^x_0K(x~~-~~y,y)n(x~~-~~y,t)n(y,t)\,dy~~ - ~~\int^\infty_0K(x,y)n(x,t)n(y,t)\,dy.<~~/~~math>~~

	~~If ''dy'' is interpreted as a discrete [[List of integration and measure theory topics#Intuitive foundations\|measure~~]~~], i.e. when particles join in [[discrete variable\|discrete]] sizes, then the discrete form of the equation is a~~ [~~[summation]]~~:

	~~: <math>\frac{\partial n(x_i,t)}{\partial t}=\frac{1}{2}\sum^{i~~-~~1}_{j=1}~~
	~~K(x_i-x_j,x_j)n(x_i-x_j,t)n(x_j,t)~~ - ~~\sum^\infty_{j=1}K(x_i,x_j)n(x_i,t)n(x_j,t)~~.</~~math>~~

	~~==Coagulation kernel~~==

	~~The [[linear operator\|operator~~]~~], ''K'',~~ is ~~known as the coagulation [[integral kernel\|kernel]] and describes the rate at which particles of size ''x'' coagulate with particles of size ''y''.~~ [~~[Closed-form expression\|Analytic solutions]] to the equation exist when the kernel takes one of three simple forms~~:

	~~: <math>K = 1,\quad K = x + y, \quad K = xy,<~~/~~math>~~

	~~known as the [[constant (mathematics)\|constant]], [[Additive function\|additive]], and [[Multiplicative function\|multiplicative]] kernels respectively~~. ~~However, in most practical applications the kernel takes on a significantly more complex form, for example the free~~-~~molecular kernel which describes [[collision]]s in a dilute [[gas]]~~-~~[[Phase (matter)\|phase~~]~~] system,~~

	~~: <math>K = \sqrt{\frac{\pi k_b T}{2}}\left(\frac{1}{m(x)}+\frac{1}{m(y)}\right)^{1/2}\left(d(x)+d(y)\right)^2.</math>~~

	~~Generally the coagulation equations that result from such physically realistic kernels are~~ not ~~solvable, and as such,~~ it ~~is necessary to appeal to [[Numerical analysis\|numerical methods]]~~. ~~There exist well-established [[deterministic]] methods that can be used when there is only one particle property (~~'~~'x'') of interest,~~ the ~~two principal ones being~~ the ~~[[Method of moments (probability theory)\|method of moments]]~~ and ~~[[sectional method]]~~s. In the [[Multivariable calculus\|multi-variate]] case however, when two or more properties (such as size, shape, composition etc.) are introduced, special approximation methods that suffer less from [[curse of dimensionality]] has to be applied. For instance, approximation based on Gaussian [[radial basis functions]] has been ~~successfully applied to the two-dimensional coagulation equation~~.<~~ref~~>Predicting multidimensional distributive properties of hyperbranched polymer resulting from AB2 polymerization with substitution, cyclization and shielding, I. Kryven, P.D. Iedema, Polymer 54 (14), 3472–3484<~~/ref~~>

	~~When accuracy of the solution is not of primary importance,~~ [~~[Monte Carlo method\|stochastic particle (Monte~~-~~Carlo) methods]] are an attractive alternative~~.

	=~~=See also==~~
	*[[Einstein relation (kinetic theory)\|Einstein–Smoluchowski relation]]
	*[[Smoluchowski factor]]

	~~== References ==~~
	~~{{Reflist}}~~

	~~{{DEFAULTSORT:Smoluchowski Coagulation Equation}}~~
	~~[[Category:Differential equations]]~~
	~~[[Category:Polish physicists]~~]

en>Duoduoduo: wikilink

2013-08-31T14:47:30Z

wikilink

← Older revision		Revision as of 16:47, 31 August 2013
Line 1:		Line 1:
	~~The name of~~ the ~~writer~~ is ~~Figures~~. ~~For a while she~~'s ~~been in South Dakota~~. ~~Managing people~~ is ~~what I do~~ and the ~~salary has been truly fulfilling~~. He is ~~really fond~~ of ~~doing ceramics but he is struggling~~ to ~~find time for it.~~<br><br>~~my weblog~~ .~~.. over~~ the ~~counter std test~~ - [~~http~~://~~bikedance~~.~~com~~/~~blogs/post/15246 click through~~ the ~~next page~~],		{{Context\|date=October 2009}}
			In [[statistical physics]], the '''Smoluchowski coagulation equation''' is an [[integrodifferential equation]] introduced by [[Marian Smoluchowski]] in a seminal 1916 publication,<ref>{{cite journal\|last=Smoluchowski\|first=Marian\|title=Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen\|journal=Physik. Z.\|year=1916\|volume=17\|pages=557–571, 585–599\|bibcode=1916ZPhy...17..557S}}</ref> describing the [[time evolution]] of the [[number density]] of particles as they coagulate (in this context "clumping together") to size ''x'' at time ''t''.

			==Equation==

			In the case when the sizes of the coagulated particles are [[continuous variable]]s, the equation involves an [[integral]]:

			: <math>\frac{\partial n(x,t)}{\partial t}=\frac{1}{2}\int^x_0K(x-y,y)n(x-y,t)n(y,t)\,dy - \int^\infty_0K(x,y)n(x,t)n(y,t)\,dy.</math>

			If ''dy'' is interpreted as a discrete [[List of integration and measure theory topics#Intuitive foundations\|measure]], i.e. when particles join in [[discrete variable\|discrete]] sizes, then the discrete form of the equation is a [[summation]]:

			: <math>\frac{\partial n(x_i,t)}{\partial t}=\frac{1}{2}\sum^{i-1}_{j=1}
			K(x_i-x_j,x_j)n(x_i-x_j,t)n(x_j,t) - \sum^\infty_{j=1}K(x_i,x_j)n(x_i,t)n(x_j,t).</math>

			==Coagulation kernel==

			The [[linear operator\|operator]], ''K'', is known as the coagulation [[integral kernel\|kernel]] and describes the rate at which particles of size ''x'' coagulate with particles of size ''y''. [[Closed-form expression\|Analytic solutions]] to the equation exist when the kernel takes one of three simple forms:

			: <math>K = 1,\quad K = x + y, \quad K = xy,</math>

			known as the [[constant (mathematics)\|constant]], [[Additive function\|additive]], and [[Multiplicative function\|multiplicative]] kernels respectively. However, in most practical applications the kernel takes on a significantly more complex form, for example the free-molecular kernel which describes [[collision]]s in a dilute [[gas]]-[[Phase (matter)\|phase]] system,

			: <math>K = \sqrt{\frac{\pi k_b T}{2}}\left(\frac{1}{m(x)}+\frac{1}{m(y)}\right)^{1/2}\left(d(x)+d(y)\right)^2.</math>

			Generally the coagulation equations that result from such physically realistic kernels are not solvable, and as such, it is necessary to appeal to [[Numerical analysis\|numerical methods]]. There exist well-established [[deterministic]] methods that can be used when there is only one particle property (''x'') of interest, the two principal ones being the [[Method of moments (probability theory)\|method of moments]] and [[sectional method]]s. In the [[Multivariable calculus\|multi-variate]] case however, when two or more properties (such as size, shape, composition etc.) are introduced, special approximation methods that suffer less from [[curse of dimensionality]] has to be applied. For instance, approximation based on Gaussian [[radial basis functions]] has been successfully applied to the two-dimensional coagulation equation.<ref>Predicting multidimensional distributive properties of hyperbranched polymer resulting from AB2 polymerization with substitution, cyclization and shielding, I. Kryven, P.D. Iedema, Polymer 54 (14), 3472–3484</ref>

			When accuracy of the solution is not of primary importance, [[Monte Carlo method\|stochastic particle (Monte-Carlo) methods]] are an attractive alternative.

			==See also==
			*[[Einstein relation (kinetic theory)\|Einstein–Smoluchowski relation]]
			*[[Smoluchowski factor]]

			== References ==
			{{Reflist}}

			{{DEFAULTSORT:Smoluchowski Coagulation Equation}}
			[[Category:Differential equations]]
			[[Category:Polish physicists]]

en>Tkircher: /* Hermite differential equation */

2012-06-11T01:05:39Z

Hermite differential equation

New page

The name of the writer is Figures. For a while she's been in South Dakota. Managing people is what I do and the salary has been truly fulfilling. He is really fond of doing ceramics but he is struggling to find time for it.<br><br>my weblog ... over the counter std test - [http://bikedance.com/blogs/post/15246 click through the next page],