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From: Cheng Cosine on 14 Jun 2010 09:47 Hi: What does it mean by theoretically solving a problem? Does it mean that we formulate the problem well posed? Does it mean that we can write its solution implicitly in a form like y(x) = f(x) = integral (G*g), where G is the Green's function of the problem? But in the later case, how do we theoretically "solve" a nonlinear problem since there is no Green's function in nonlinear case? Thanks,
From: David Bernier on 14 Jun 2010 10:59 Cheng Cosine wrote: > Hi: > > What does it mean by theoretically solving a problem? Does it mean > that we formulate the problem well posed? Does it mean that we can > write its solution implicitly in a form like y(x) = f(x) = integral > (G*g), where G is the Green's function of the problem? But in the > later case, how do we theoretically "solve" a nonlinear problem since > there is no Green's function in nonlinear case? [...] Linearity makes it much easier to solve some boundary value problems. One example is the method using the Poisson kernel, < http://en.wikipedia.org/wiki/Poisson_integral_formula > to give an explicit formula for a solution to a problem with Dirichlet boundary conditions: < http://en.wikipedia.org/wiki/Dirichlet_boundary_condition > . f = convolution_integral (G, g) [ g -> boundary data, G-> "The" Green function or kernel for the unit disk, a square boundary, etc. and f = solution ]. In practice, I think kernels aren't too easy to give nice formulas for. Convolution products (or integrals) can be done efficiently through the Fast Fourier transform and its inverse, inverse Fast Fourier transform, the two operations being similar and done "fast". For non-linear problems, I don't know what methods are used in practice (finite element method?). The area of non-linear PDEs (say giving explicit formulas, or even existence questions) is very hard. The Ricci flow, used in the solution to the Poincare conjecture, is (I think) the name for an a priori hypothetical solution to a non-linear PDE. Non-linear PDEs occur naturally in physics. Terence Tao, who was a Fields medalist in 2006, does some work on non-linear PDEs (very complicated ...): < http://www.math.ucla.edu/~tao/Dispersive/wave.html > . David Bernier
From: spudnik on 14 Jun 2010 21:00 how about an "existence proof" of solvability? thus&so: time obviously doesn't bend, except in a subjective sense of living & dying, sleeping & waking ... it's too bad about Schroedinger's joke-cat, though ... Schroedinger's cat is dead; long-live Schroedinger's cat! the curvature of space was dyscovered with "synchronized sundials" by Aristarchus; it was measured in Alsace-Lorraine by Gauss, with his theodolite & trigonation ... for money! Dear Editor; It is apparent from the City ordinance, proposed to ban high-density polyethylene (HDPE) bags -- excepting take-out at restaurants -- that it will be a state-wide eco-tax. The "green fee" is slated to be 25 cents for any paper bag from the retailer, grocer or farmer at the market. This is unfortunate for two reasons, although, as I stated a year ago in Council, when it first came-up, the super-light-weight & super-inexpenseve bags (much less than the Staff Report was willing to concede) are so good at what they do, before they inevitably break-up & decompose (but , according to the apocrypha & studies of Heal the Bay etc., HDPEbagsR4ever) that coastal cities may be justified in a ban, to prevent stormdrain blockages. Firstly, just like with "hemp for haemarrhoids," it is not a panacea or much of an economic stop-gap, if only because "only criminals & baby-smotherers will have HDPE bags." Really, there are plenty of natural plastics; "plastic" is really an adjective, as in the plastic arts! Note also that even plant-derived plastic bags will be banned, although they are acknowledged to biodegrade. Secondly, a very small Carbon Tax would be much more realistic than simply allowing Waxman's CO2 cap & trade nostrum, of letting the abitrageurs & daytraders raise the price of our energy as much as they can in the "free market" -- with no provision whatever for government revenue (contrary to the slogan of "cap & tax" used by Tea Partiers, "Republicans," and the WSUrinal). As with the much-greater amount of materiel & energy that is required for the paper bags, we might do better to ban *low* density polypropolene bags at department & boutique stores, which are many times heavier than the HDPE bags. It is surprising that a fifth of the HDPE bags are recycled, considerng that a) they're only good for garbage, if they get dirty, and b) they are quite often re-used by folks; recycling them is an unsanitary joke, though composting might be educational fun. The retailers would get ten of the 25 cents, which seems to be a quite an incentive for the overhead. However, has anyone seen any analysis on the energy requirements for the "reusable" replacement, and their importation? --Sincerely, Brian H. --Stop BP's/Waxman's arbitragueur-daytripper's delight, cap&trade (Captain Tax in the feeble minds of Tea Partiers, "'republicans' R us," and the WSUrinal (and the latter just l o v e Waxman's '91 cap&trade bill !-)) http://wlym.com
From: William Elliot on 15 Jun 2010 01:38 On Mon, 14 Jun 2010, Cheng Cosine wrote: > What does it mean by theoretically solving a problem? Does it mean > that we formulate the problem well posed? Does it mean that we can > write its solution implicitly in a form like y(x) = f(x) = integral > (G*g), where G is the Green's function of the problem? But in the > later case, how do we theoretically "solve" a nonlinear problem since > there is no Green's function in nonlinear case? > It means that that it's been proven a solution exists and that there is no way of knowing what that solution is. For example, a well ordering of the reals or solutions for some differential equations, know to exist only because of general theorems. Practically by definition, the first non-recursive countable ordinal is inexpressible except for its defining property. Smaller ordinals can be explicitly expressed by finite formulas.
From: David Bernier on 15 Jun 2010 23:19
David Bernier wrote: > Cheng Cosine wrote: >> Hi: >> >> What does it mean by theoretically solving a problem? Does it mean >> that we formulate the problem well posed? Does it mean that we can >> write its solution implicitly in a form like y(x) = f(x) = integral >> (G*g), where G is the Green's function of the problem? But in the >> later case, how do we theoretically "solve" a nonlinear problem since >> there is no Green's function in nonlinear case? > [...] > > Linearity makes it much easier to solve some boundary value > problems. > > One example is the method using the Poisson kernel, > < http://en.wikipedia.org/wiki/Poisson_integral_formula > > to give an explicit formula for a solution > to a problem with Dirichlet boundary conditions: > < http://en.wikipedia.org/wiki/Dirichlet_boundary_condition > . Wikipedia has this: P_r (theta) = (1 - r^2)/(1 - 2r cos(theta) + r^2) , and indeed the function u, restricted to a fixed r, is P_r convolved with f, f being (say) f: [0, 2pi) -> R , R = the reals. Suppose we iterate convolving with P_(1/2), starting with f; then we get after n times: f*P_(1/2)*P_(1/2) ... P_(1/2), ^^^^ with n "P_(1/2)". P_0 == 1 identically, so [0, 2pi) is given measure 1. f*P_(1/2)*P_(1/2) is the solution for r = 1/2 of the Dirichlet problem with boundary values f*P_(1/2), and f*P_(1/2)*P_(1/2)*P_(1/2) is the solution for r = 1/2 of the Dirichlet problem with boundary values f*P_(1/2)*P_(1/2), so I think u on the circle of radius (1/2)^n is f*P_(1/2)*P_(1/2) ... P_(1/2) [ with n "P_(1/2)"s ], where one assumes that f is sufficiently well-behaved. In fluid dynamics, there is a method called potential flow, although I don't understand the physics too well: < http://en.wikipedia.org/wiki/Potential_flow > . David Bernier > f = convolution_integral (G, g) > [ g -> boundary data, G-> "The" Green function or kernel for > the unit disk, a square boundary, etc. and f = solution ]. > > In practice, I think kernels aren't too easy to give > nice formulas for. > > Convolution products (or integrals) can be done > efficiently through the Fast Fourier transform and > its inverse, inverse Fast Fourier transform, > the two operations being similar and done "fast". > > For non-linear problems, I don't know what methods > are used in practice (finite element method?). > > The area of non-linear PDEs (say giving > explicit formulas, or even existence questions) > is very hard. > > The Ricci flow, used in the solution to the Poincare > conjecture, is (I think) the name for an > a priori hypothetical solution to a non-linear PDE. > > Non-linear PDEs occur naturally in physics. > Terence Tao, who was a Fields medalist in 2006, > does some work on non-linear PDEs (very complicated ...): > < http://www.math.ucla.edu/~tao/Dispersive/wave.html > . > > David Bernier |