mm-226 === Subject: Adaptive-mesh methods: how?Originator: twomack@cark.greenend.org.uk ([193.201.200.170])I can understand a process where you construct a mesh, construct thediscretised solution for that mesh, then look at places where thediscretised solution appears to have problems and make the mesh finerin those places, then recalculate the solution for the better mesh andcontinue until happy.But I keep reading about processes wch seem to change meshes inmid-stream, and I can't see how that can work; even if all you'redoing is integrating a system with respect to time, and change meshcorresponds to reducing the time-step h once you start running intotrouble, I can't see how you can justify taking an approximatesolution obtained with a large time-step and using it as the base forintegrating with a smaller time-step; you're following very accuratelya trajectory through the wrong point.Is it that these methods work for well-behaved systems where a well-behaved system is one where === Adaptive-mesh methods: how? I can understand a process where you construct a mesh, construct the> discretised solution for that mesh, then look at places where the> discretised solution appears to have problems and make the mesh finer> in those places, then recalculate the solution for the better mesh and> continue until happy. But I keep reading about processes wch seem to change meshes in> mid-stream, and I can't see how that can work; even if all you're> doing is integrating a system with respect to time, and change mesh> corresponds to reducing the time-step h once you start running into> trouble, I can't see how you can justify taking an approximate> solution obtained with a large time-step and using it as the base for> integrating with a smaller time-step; you're following very accurately> a trajectory through the wrong point. Is it that these methods work for well-behaved systems where a well-> behaved system is one where nearby trajectories stay nearby to the> end?If you are integrating a diffeq, when the predictor/corrector showstoo big a difference, you back up and proceed with a smaller mesh.If you are integrating a known function, then an adaptive scheme iseasy, either iterative or recursive. Forth and C versions of both willbe found on http://www.phys.virginia.edu/classes/551.jvn.fall01/CiSE_progs /Cprogs.htm-- Julian V. NobleProfessor Emeritus of ^^^^^^^^^^^^^^^^^^http://galileo.phys.virginia.edu/~jvn/ Science knows only one commandment: contribute to science. -- === how? I can understand a process where you construct a mesh, construct the>discretised solution for that mesh, then look at places where the>discretised solution appears to have problems and make the mesh finer>in those places, then recalculate the solution for the better mesh and>continue until happy.But I keep reading about processes wch seem to change meshes in>mid-stream, and I can't see how that can work; even if all you're>doing is integrating a system with respect to time, and change mesh>corresponds to reducing the time-step h once you start running into>trouble, I can't see how you can justify taking an approximate>solution obtained with a large time-step and using it as the base for>integrating with a smaller time-step; you're following very accurately>a trajectory through the wrong point.Is it that these methods work for well-behaved systems where a well->behaved system is one where nearby trajectories stay nearby to the>end?Tom If you are integrating a diffeq, when the predictor/corrector shows> too big a difference, you back up and proceed with a smaller mesh. If you are integrating a known function, then an adaptive scheme is> easy, either iterative or recursive. Forth and C versions of both will> be found on http://www.phys.virginia.edu/classes/551.jvn.fall01/CiSE_progs /Cprogs.htm> Forgot to mention that in solving elliptic pde's on a grid, you can obtain asolution first on a coarse grid (say, by relaxation or over-relaxation) andthen refine the grid, using the first solution as starting values for iter-ation on the finer grid. See Numerical Recipes, 2nd ed for various schemesthat do ts.-- Julian V. NobleProfessor Emeritus ^^^^^^^^^^^^^^^^^^http://galileo.phys.virginia.edu/~jvn/ Science knows only one commandment: contribute to science. -- === a meshOriginator: twomack@cark.greenend.org.uk ([193.201.200.170])I'm trying to find the vibrational modes of a strangely-shaped 2Dobject (OK, a circle for now, but it will get much more strangelyshaped later), by looking for the eigenvectors of an approximateLaplacian.So I have a mesh of triangles in 2D, and I know values of my functionat the vertices. For any particular triangle, I can find values ofdf/dx and df/dy by looking at the equation ax+by+c of the triangle[x1,y1,f(x1,y1)], [x2,y2,f(x2,y2)] and [x3,y3,f(x3,y3)]; ts is onelinear equation with constant coefficients, and a and b are thenestimates for the derivative on that triangle.But I want to estimate derivatives at the vertices. Should I justaverage the [a,b] estimates that I get for each of the trianglessharing the vertex? In numerical analysis, in ts sort of context,there always seems to be some clever weighting you can do to getbetter approximations than you might expect; does that hold here, too?I'm planning to get the ÔLaplacian matrix' as M_x*M_x + M_y*M_y, whereM_x and M_y are the matrices corresponding to obtaining df/dx anddf/dy by the averaging-per-triangle-estimates above; is ts in factlunacy? I note that the Taylor-series approximate second derivativef'' approx 2h^(-2) f(x-h) - 2f(x) + f(x+h)is _not_ quite the same as applying f' approx 1/2h === find derivatives on a mesh> I'm trying to find the vibrational modes of a strangely-shaped 2D> object (OK, a circle for now, but it will get much more strangely> shaped later), by looking for the eigenvectors of an approximate> Laplacian. So I have a mesh of triangles in 2D, and I know values of my function> at the vertices. For any particular triangle, I can find values of> df/dx and df/dy by looking at the equation ax+by+c of the triangle> [x1,y1,f(x1,y1)], [x2,y2,f(x2,y2)] and [x3,y3,f(x3,y3)]; ts is one> linear equation with constant coefficients, and a and b are then> estimates for the derivative on that triangle. But I want to estimate derivatives at the vertices. Should I just> average the [a,b] estimates that I get for each of the triangles> sharing the vertex? In numerical analysis, in ts sort of context,> there always seems to be some clever weighting you can do to get> better approximations than you might expect; does that hold here, too? I'm planning to get the ÔLaplacian matrix' as M_x*M_x + M_y*M_y, where> M_x and M_y are the matrices corresponding to obtaining df/dx and> df/dy by the averaging-per-triangle-estimates above; is ts in fact> lunacy? I note that the Taylor-series approximate second derivative f'' approx 2h^(-2) f(x-h) - 2f(x) + f(x+h) is _not_ quite the same as applying f' approx 1/2h (f(x+h)-f(x-h))> twice, but is similar. TomOne method I tried for a similar problem (plane stress elasticity using3-node Turner triangles) 30+ years ago is based on the following reasoning:o The midpoint of each triangle side is a superconvergent point for the tangential derivative (i.e., the derivative along that side) Ts value is of course the same for the 2 adjacent triangles since piecewise linear trial functions match along the side.o Take a patch of triangles, recover the midside derivatives and do a least square (or energy) fit for the Cartesian nodal derivatives. The patch should not contain too many triangles since that causes the usual polynomial ill-conditioning, but not too few. One compromise is the Irons patch: the set of triangles surrounding a node. (Nodes is the finite-element name for what you call vertices)Ts is a special case of the so-called superconvergent patchrecovery in finite element methods. I remember the results wereerratic. In mesh regions where each node is surrounded by 4 or more triangles the recovery worked fine, but it ran into trouble at boundaries, as most recovery methods do. So after that I just use Wilson weighted averaging, wch is easier to program.If you want to go following keywords (not all at once, of course) finite element superconvergent patch recovery Barlow points Zienkiewicz Zhang === Looking for examples of bad data analysis>...the concern is how many cancers are induced by the>radiation used to acquire the mammogram? The current official>estimate is sometng like 1 in 10,000 (don't quote me) women>who are screened yearly starting at age 40 will develop a cancer>from the x-rays used in the screening proceedure. However, ts>estimate is based upon the so called linear no-threshold model>wch states assumes that if at a radiation dose of X, Y people>out of a thousand develop cancer, then at a dose of 0.001X,>0.001Y people out of a thousand will develop cancer. Sounds somewhat strange. Isn't it natural to assume the> probability to develop cancer due to a certain doze doesn't depend> on the amount of previously absorbed radiation (assuming the later> didn't induce cancer already)?No.>If so, it would mean that given> the probability to develop cancer due to doze d is P(d), one has> P(kd) = 1 - (1 - P(d))^k. Of course, if P(d) is small enough ts> is approximately linear. Is that the motivation for the linear> no-threshold model?No.>However, you said the extrapolated data is> for extremely gh d where P(d) might be not that small?>Data are for atomic-bomb survivors, nuclear accidents, and otherunplanned experiments involving gh doses.See the following for informed insight:Elaine RonNCRP REPORT NO. 136 ON THE SCIENTIFIC BASES FOR LINEARITY IN THEDOSE-RESPONSE RELATIONSP FOR IONIZING RADIATION, Arthur C. Upton etal.Neither of these complex issues are suitable for inclusion in a Ôhowto fit you data to... for a === a symmetric matrix positive definite>> Given the Hessian, H, of a function f: R^n -> R, if H is not pd, and we>> would like to do sometng like H* = H + cI, to make H* pd, is there a way>> to get the information needed from the LAPACK Cholesky routines>> (specifically, dpptrf) to make an estimate of what c needs to be> Hm. You could look at the first negative pivot, and add that much to the> diagonal. That doesn't guarrantee you that on re-factoring you won't get> a negative pivot later, but you could repeat ts process.To the OP: is it important that the correction be a multiple of theidentity? Because, if not, you can stop at each negative pivot andadd enough to it, and it alone, to make it positive. Ts would be adiagonal correction === matrix positive definite >> Given the Hessian, H, of a function f: R^n -> R, if H is not pd, and we>> would like to do sometng like H* = H + cI, to make H* pd, is there a way>> to get the information needed from the LAPACK Cholesky routines>> (specifically, dpptrf) to make an estimate of what c needs to beMy suggestion:>Hm. You could look at the first negative pivot, and add that much to the>diagonal. That doesn't guarrantee you that on re-factoring you won't get>a negative pivot later, but you could repeat ts process.can be found in author = {T.A. Manteuffel},title = {An incomplete factorization technique for positive definite linear systems},journal = {Math. Comp.},volume = {34},year = {1980},pages = {473--497},abstract = {Extension of Meyerink/vdVorst to H-matrices; sfted ICCG: if $A=D-B$ (diagonal) then $A(alpha)=D-{1over 1+alpha}B$; for $alphageqalpha_n>0$ all pivots will be positive; find $alpha_n$ by trial anderror.},keywords = {incomplete factorization, positive definite matrices, H-matrices}}and in the context of incompletely factorisations it works pretty well.Ts suggestion:> To the OP: is it important that the correction be a multiple of the> identity? Because, if not, you can stop at each negative pivot and> add enough to it, and it alone, to make it positive. Ts would be a> diagonal correction to the original matrix.is known as Kershaw's method (also Kershaw's trick), seeauthor = {D.S. Kershaw},title = {The Incomplete Cholesky-Conjugate Gradient Method for the Iterative Solution of Systems of Linear Equations},journal = {J. Comp. Phys.},volume = {26},year = {1978},pages = {43--65},keywords = {conjugate gradient method, incomplete factorization, Kershaw trick},abstract = {Experimental investigation, negative pivot repair by $m_{kk}=sum_{jk}|m_{kj}|$.},folder = {IF2}}and works nowhere near as well, in my experience.Again, both of these are in the context of incomplete factorisations,where you're approximating anyway, so any additional modifications canbe justified with the right amount of hand waving.I wonder what the OP's context is and what he hopes to accomplish byfactoring the modified matrix.V.-- mail === finding and complex numbers ?? dear tutors.I just create a small C program to reproduce the newton method of root finding on equation.. (attached at the end of ts post)But i have 2 basic questions:NOTE: im just evaluating very simple equations, like:y^4+3y^3-2y+3=0ory^2-15=0so notng fancy still, but with ts sample equations i have the following questions:1.- What will be a goot method to guess the initial value of the x0 in the newton algorithm ?2.- how can i know if there its an imaginary root ? (complex number).Here its my code if you want to review it or if sometng helps others, ---// Newton method for root finding demon@demon.com.mx #include #include #include int original[10]; int derivate[10]; int maxexp; double calculate_eq(int roots, int *equation, double guess){ int x; double result; result=0; for(x=0;x<(roots-1);x++) { if(equation[x]) result=result+(equation[x]* pow(guess,roots-(x+1)));// printf(valor en iteracion %i = %.6f = %i * %.6f^%in,x,result, equation[x],guess,roots-(x+1)); } result=result+equation[roots-1]; return result;} void newton(int roots, int *equation, int *derivate){ // buscando raiz double guessing, result, orig, derv, carrier; int x; guessing=1; carrier=guessing; //componer las ecuaciones orig=calculate_eq(roots,equation,guessing); derv=calculate_eq(roots-1,derivate,guessing); // printf(%.6f,orig); // printf(%.6f,derv); for(x=0;x0 && equation[x] > 0) printf(+); if(x != (roots-1)) { if(equation[x]) if(roots-(x+1) > 1) if(equation[x]>1) printf(%iy^%i,equation[x],roots-(x+1)); else printf(y^%i,roots-(x+1)); else if(equation[x]>1) printf(%iy,equation[x]); else if(equation[x]) printf(%iy,equation[x]); else printf(y); } else{ if(equation[x]) printf(%i,equation[x]); } } printf(n);}int main(){ maxexp=3; original[0]=1; original[1]=-2; original[2]=15; derivate_eq(original); printresults(maxexp,original); printresults(maxexp-1,derivate); === Fourier integrals both for your friendly help!I want to add Bailey-Swarztrauber, their papers on fractional FFTcan be found at http://www.nersc.gov/~dhb/ The advantage of that method is: one can specify the wavenumberrange and number of points independently.Gordon Inverarity provides a gher dimensional generalization He provides a Mathematica solution athttp://library.wolfram.com/infocenter/MathSource/4284/Two remarks: mine may be oscillating (mildly) and unfortunatelyi do not know before where to rotate. The other is Simpson: i'played' with it - but by departing from the Ôcenter' errorscould become so uggly that i will not do. May be the extensionssuggested by Richard or in the lectures of Julian === functionOriginator: twomack@cark.greenend.org.uk ([193.201.200.170])[I'm illustrating with some code written in python; it's pretty self-explanatory,but it might be worth mentioning that array indexes start from 0]The solution to the differential equation x''=sin x, startingcondition x=x0, x'=0, is periodic; the period depends on x0, and I'd like totry to estimate it.So I use a standard Runge-Kutta algorithmdef rkfunc(t, xx): V = xx.copy() for i in range(xx.shape[0]/2): V[i*2] = xx[i*2+1] V[i*2+1] = math.sin(xx[i*2]) return Vt=0; xx = [2.5, 0]; h=0.001wle (t < 60.0) k1 = rkfunc(t, xx) k2 = rkfunc(t+h/2, xx + h/2 * k1) k3 = rkfunc(t+h/2, xx + h/2 * k2) k4 = rkfunc(t+h, xx + h*k3) t = t+h old2xx = oldxx; oldxx = xx xx = xx + (h/6)*(k1 + 2*k2 + 2*k3 + k4)and keep a story of the last three values of xx seen (old2xx, oldxx,xx). If I ever see that oldx is the smallest of the three, we've justpassed a minimum; I fit a parabola to the last three values, find theposition of its minimum, and use ts as the estimate of when thefunction t the minimum: M = array([0,0,1,1,1,1,4,2,1], shape=(3,3), type=Float64) V = array(story) paracoeff = la.solve_linear_equations(M,V)# and obtain t_min by observing its minimum tmin = -paracoeff[1] / (2*paracoeff[0]) vtmin = paracoeff[0]*tmin**2 + paracoeff[1]*tmin + paracoeff[2] test = t - h*tminOften, ts gives really good answers: the 1st, 2nd, 3rd ... zero-crossings areto eleven decimals N, 2N, 3N.But sometimes (in the x0=2.5 situation that I've given in the code,for example), the algorithm seems to Ôskip', and I get zero crossingsat N, 2N, 3N+2h, 4N+2h, 5N+2h, 6N+4h ...Can anyone enlighten me as to what's going on and how to fix it? Obviously Ican subtract multiples of 2h from the terms until they fall in line, but that'sputting a sticking-plaster on a === problem finding periodicity of a function [I'm illustrating with some code written in python; it's pretty self-explanatory,> but it might be worth mentioning that array indexes start from 0] The solution to the differential equation x'' = sin x, starting ^^^^^^^^^^^> condition x=x0, x'=0, is periodic; the period depends on x0, and I'd like to> try to estimate it.> Is ts what you mean, or do you mean x + sin x = 0 ?Makes a big difference. In any case, the solution of the equation can bedone as quadratures (the result is an elliptic function) via theintegrating factor x' . That is, x x' + a x' sin x = 0 d/dt ( x'^2 / 2 - a cos x ) = 0 x'^2 /2 - a cos x = B = constantsolve for x'; the result is a separable 1st-order ode, wchyou integrate. You can express the period as an integral.I VERY STRONGLY suggest a course in ode's. Maple ain't allthere is to know, and neither is Python (e.g. there is Forth ;-) . -- Julian V. NobleProfessor Emeritus of ^^^^^^^^^^^^^^^^^^http://galileo.phys.virginia.edu/~jvn/ Science knows only one commandment: contribute to science. -- === Copyright Violations> ::: Everytng said here by James has been snipped ::: <>Now... here's what I have to say..................I've noticed people with webpages extensively quoting from posts I'vemade and am now giving public notice that there is no right to copyextensively, that is, to go beyond fair use, from my writings withoutmy express written permission.People putting up webpages exceeding fair usage are in violation ofinternational copyright law and may face prosecution to the fullextent of those laws.And remember those laws are *international* as I suspect some of youin countries other than the United States may feel safe.However, I am reminding you that you are to obey the law wherever youmay be, or you may find out just how great the reach of those lawsare.Why not just act in accordance with them without stressing me to haveto go after specific people?And... that's === question> If ipiv(i)=j it means that row i is interchanged with row j, so if> you pivot rows k and l, you should interchange ipiv(k) and ipiv(l), AFAIK.Yes, but.iirc (2,3,...) means: first exchange 1 and 2, then 2 and 3. In otherwords, you can not immediately read from ts vector whre a rowultimately goes.V.-- email: lastname at cs utk eduhomepage: cs utk edu tilde === not the appropriate place to ask such questions, but>I was searcng the web about documents wch cover the following>toppics: - multiplicative order in general> - divisors of mersenne numbersUnfortunately I couldn't find anytng except for some smallerIs ts news group the right one for number theoretical questions in>general or are there some wch are more suitable for ts toppic?Any and all help is appreciated :o) sci.math.num-analysis is about numerical algorithmsinvolving ßoating point data (hence working over thereal numbers of complex numbers). A sample topic isthe numerical solution of a partial differential equation. There are some overlaps between number theory and numerical analysis. For example, one might wantan accurate approximation of the Riemann zeta function.But most number theory is about integer arithmetic. Number theory lacks its own newsgroup. Most questions belong in sci.msth.Research questions may go to sci.math.research (moderated).Computer algebra questions (e.g., factorizationof polynomials) may go in sci.math.symbolic. For much information on Mersenne numbers, seethe GIMPS (Great Internet Mersenne Prime Search) pagesat http://www.mersenne.org .-- Adams served two terms as Vice President and one as President, but lostreelection. Later s son became President despite losing the popular vote.That son lost s reelection attempt badly. Now story is repeating itself.pmontgom@cwi.nl Microsoft Research and CWI Home: San === if ts is not the appropriate place to ask such questions, but>I was searcng the web about documents wch cover the following>toppics: - multiplicative order in general> - divisors of mersenne numbersUnfortunately I couldn't find anytng except for some smallerIs ts news group the right one for number theoretical questions in>general or are there some wch are more suitable for ts toppic?Any and all help is appreciated :o) sci.math.num-analysis is about numerical algorithms> involving ßoating point data (hence working over the> real numbers of complex numbers). A sample topic is> the numerical solution of a partial differential equation. There are some overlaps between number theory > and numerical analysis. For example, one might want> an accurate approximation of the Riemann zeta function.> But most number theory is about integer arithmetic. Number theory lacks its own newsgroup. > Most questions belong in sci.msth.> Research questions may go to sci.math.research (moderated).> Computer algebra questions (e.g., factorization> of polynomials) may go in sci.math.symbolic. For much information on Mersenne numbers, see> the GIMPS (Great Internet Mersenne Prime Search) pages> at http://www.mersenne.org . Peter, for that information :o)I already looked at www.mersenne.org, but they don't seem to have anydocuments about the theory bend mersenne numbers. I tnk thatmersenne numbers are much more than just a well of huge prime numbers.You can easily show that for all odd numbers there are infinitemersenne numbers wch are a multiple of ts numbers and some otherinteresting tngs.kind === need?...I have a system of two nonlinear trascendent algebraic equations.One of the unknowns is implicit. What kind of === Re: What kind of algorithm I need?... >I have a system of two nonlinear trascendent algebraic equations. >One of the unknowns is implicit. >What kind of algorithms are there to solve ts problem? >GrrrbauI don't know what an implicit unknown is. I assume you have two nonlinear equations in two unknowns and cannot rewrite it directly as x=f1(x,y) y=f2(x,y)but ts does not matter here. the damped Newtons method should do the job.look forhttp://plato.la.asu.edu/topics/problems/zero.htmlhthpeter== === =Subject: Solution to ts SDE by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id + b*X)*dt + (c + dX)*dWwhere a,b,c and d are constants, and dW is a Wiener processI can solve for X for the cases 1)a=c=0 or 2) b=d=0. So Im guessingthe solution may be a combination of the solutions to 1 and 2. If youcan help me, please email me. === of linear equations that represents loadsat each of the supports for a beam. Calculating the moments resultsgives a singular matrix:0 -4 -42 -604 0 -38 -5642 38 0 -1860 56 18 0The vector for ts matrix is the known moments from the loadingconditions. If anyone has any ideas on resolving ts I'd be verygrateful. I'm not looking for anytng more precise than === trying to solve a a set of linear equations that represents loads> at each of the supports for a beam. Calculating the moments results> gives a singular matrix: 0 -4 -42 -60> 4 0 -38 -56> 42 38 0 -18> 60 56 18 0 The vector for ts matrix is the known moments from the loading> conditions. If anyone has any ideas on resolving ts I'd be very> grateful. I'm not looking for anytng more precise than +/-20%. Simon CornerI am not sure what you want to resolve, and whether you gave all the data.The exact pseudoinverse of your matrix A is B=(1/5142)* 0 2 21 30 -2 0 19 28 -21 -19 0 9 -30 -28 -9 0(A is actually big time singular, its rank is 2 (out of 4)).What B does is: applied to a right-hand side b, the vector B*b solves theleast-squares problem A*x ~ b, and if there are infinitely manysolutions, it selects the one of the smallest length.If that's what you want, use it. If not, supply more information.Accuracy, no matter what percentage, is not guaranteed because it dependson the angle between the column space of A and the right-hand side; itscosine can be anywhere === Matrix> I am trying to solve a a set of linear equations that represents loads> at each of the supports for a beam. Calculating the moments results> gives a singular matrix: 0 -4 -42 -60> 4 0 -38 -56> 42 38 0 -18> 60 56 18 0 The vector for ts matrix is the known moments from the loading> conditions. If anyone has any ideas on resolving ts I'd be very> grateful. I'm not looking for anytng more precise than +/-20%. Simon Corner,your question confused me quite a bit: What are you really trying tocalculate? Moments, suspension forces? What are your unknowns, what isthe right hand side?Could it be, that you are trying to solve a statically undetermined(maybe not the right translation) system? In that case you would alsohave to calculate the bending of the beam.How many === =?ISO-8859-1?Q?MERSENNE_=B4S_PRIMES__NUMBERS?=Que tal, amigos de edumat, los invitamos a formar parte de la b.9csquedadel mayor n.9cmero primo.Con la ayuda de su computadora, usted podra sumarse a uno de losmayores proyectos de computaci.97n distribuida.GIMPS (Great Internet Mersenne Prime Search) fue creado en 1997.Actualmente, esta supercomputadora virtual, formada por m.87s de 70 milm.87quinas distribuidas por todo el mundo, y de la que puede formarparte usted con la suya, tiene el poder de m.87s de 238 de las m.87spoderosas supercomputadoras Cray T932, y hasta el momento ya haencontrado los 6 mayores n.9cmeros primos de Mersenne(www.mersenne.org).Si le interesa esta propuesta, dir.92jase ahttp://www.mersenneforum.org/forumdisplay.php?s=&forumid=38. All.92encontrar.87 toda la informaci.97n referente a este support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1OHt7w00423; by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1OHnei32236 by legacy.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: availability of ghway design programs Terramodel and MX Road === following books:Numerical Recipes in CNumerical Recipes in FortranNumerical Recipes in Fortran90(see www.library.cornell.edu/nr/bookcpdf/c4-5.pdf)there is a procedure for computing roots and coefficients of biquadrature (gaujac). In ts procedure, derivative of bipolynomial is calculated in the following way: ((n(alpha+beta-(2n+alpha+beta)x)Pn(x)+2(n+alpha)(n+beta)Pn-1(x ))Pn'(x)=--- (2n+alpha+beta)(1-x^2)where Pn(x) is the n-th bi polynomial with alphaand beta parameters.Could someone give me a theoretical background for such an equation?I checked the found that way of calculating the === derivative In the following books: Numerical Recipes in C> Numerical Recipes in Fortran> Numerical Recipes in Fortran90 (see www.library.cornell.edu/nr/bookcpdf/c4-5.pdf) there is a procedure for computing roots and coefficients of bi> quadrature (gaujac). In ts procedure, derivative of bi> polynomial is calculated in the following way: ((n(alpha+beta-(2n+alpha+beta)x)Pn(x)+2(n+alpha)(n+beta)Pn-1(x ))> Pn'(x)=---> (2n+alpha+beta)(1-x^2) where Pn(x) is the n-th bi polynomial with alpha> and beta parameters. Could someone give me a theoretical background for such an equation?> I but I haven't found that way of calculating the derivative. for all standardorthogonal polynomials are given.-- Julian V. NobleProfessor Emeritus of ^^^^^^^^^^^^^^^^^^http://galileo.phys.virginia.edu/~jvn/ God is not willing to do everytng and thereby take away our free will and that share of glory that rightfully belongs to us. -- === sin(x)^2 + cos(x)^2Was ts discovered by Euler, or by some famous Greek?Jeff Stout> Can anyone analyze the 1 for me === me ?Reminds me of a certain physicist who went crackers in s old age andkept trying to publish papers in wch he claimed that the ratios of thepreponderance of 0's and 1's. (It was about 35 years ago.)As I remarked to a friend at that time, the effect would have beeneven clearer in base 2.-- Julian V. NobleProfessor Emeritus of ^^^^^^^^^^^^^^^^^^http://galileo.phys.virginia.edu/~jvn/ God is not willing to do everytng and thereby take away our free will and that share of glory that rightfully belongs to us. -- === anyone analyze the 1 for me ?Sure. It's the only number that is the (division) inverse of itself: 1^{-1} = 1 It would seem that you don't consider -1 to be a number! ;-) Both +1 and -1 are their own multiplicative inverses, of course. I knew I forgot one. Just couldn't remember what it was ;-)-- Julian V. NobleProfessor Emeritus of ^^^^^^^^^^^^^^^^^^http://galileo.phys.virginia.edu/~jvn/ God is not willing to do everytng and thereby take away our free will and that share of glory that rightfully belongs to us. -- === educationWhat happens with Geometry later in a Mathematics education, after (longafter?) the study of the remedial lower division kind taken just beforestandard trigonometry? Does any coursework reexamine it for purposes of reviewplus further development? I have the feeling that most mathematical activitiesrely on analytic geometry and trigonometry if any sort of geometry is used. What would be the title of a course wch examines Geometry at and above thatlower division remedial course? G C-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.tnkspot.net/k12math/newsgroup charter: === Geometry later in educationWhen I was in college I had a class called A Survey of Geometry wch didsome brief review of plane geometry and then very quickly went intonon-Euclidian topics such as geometry on different solids like spheres andsaddles (please forgive, but I can't for the life of me remember thecorrect mathematical term). If I remember right it was a junior levelcourse and had several prerequisites that seemed to have notng to do withgeometry.Amy> What happens with Geometry later in a Mathematics education, after (long> after?) the study of the remedial lower division kind taken just before> standard trigonometry? Does any coursework reexamine it for purposes ofreview> plus further development? I have the feeling that most mathematicalactivities> rely on analytic geometry and trigonometry if any sort of geometry isused.> What would be the title of a course wch examines Geometry at and abovethat> lower division remedial course? G C-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.tnkspot.net/k12math/newsgroup charter: === Geometry later in education> What happens with Geometry later in a Mathematics education, after (long> after?) the study of the remedial lower division kind taken just before> standard trigonometry? Does any coursework reexamine it for purposes of review> plus further development? I have the feeling that most mathematical activities> rely on analytic geometry and trigonometry if any sort of geometry is used. > What would be the title of a course wch examines Geometry at and above that> lower division remedial course? Most mathematicians never see axiomatic Euclidean geometry again aftergh school. Analytic geometry and vector spaces are ubiquitous. Theaxiomatic *approach* to math they'll see again, generally in the mostdetail in group theory, number theory, and set theory classes.-- Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karpluslife member (LAB, Adventure Cycling, American Youth Hostels)Effective Cycling Instructor #218-ck (lapsed)Professor of Biomolecular Engineering, University of California, Santa CruzUndergraduate and Graduate Director, BioinformaticsAffiliations for identification only.-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.tnkspot.net/k12math/newsgroup charter: === integrated mathHow about Math books from SingaporeMath.com? The New ElementaryMathematics series (7-10th grade) is integrated.(I'm not advertising for them or anytng). > I tnk the concept of teacng geometry and algebra in parallel so that> the ideas from one branch can be used in the other and thereby a connection> and emphasis between the two branches can be established, is an excellent> goal. However, the last time I looked (about 5 years ago...) I did not see any> integrated math series from the usual textbook publisng companies that I> would have felt worthwle. -- > Sheila King> http://www.tnkspot.net/sheila/> http://www.k12groups.org/> http://www.mathxy.com-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.tnkspot.net/k12math/newsgroup charter: === number combinations 2.I need to compose a list of all possible three number sets using thenumbers 1 through 4. can anyone help me compile a list? All help willbe greatly appriciated(spelling) . -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.tnkspot.net/k12math/newsgroup charter: === number combinations 2.> I need to compose a list of all possible three number sets using the> numbers 1 through 4. can anyone help me compile a list? All help will> be greatly appriciated(spelling) . Can you use numbers more than once?If you can, start with:111111121113111411211122....4444(ie, increment the last digit as many times as you can, then let it roll over to the next digit, just like a clock or odometer)If you can only use each number once, simply eliminate everytng from the above list with duplicate digits. So you'd have (again, smallest to largest)123412431324etc-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.tnkspot.net/k12math/newsgroup charter: === equationswhat are the steps to finding out for example y=2b+5 what are yousuppose to do? Please help thankyou-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.tnkspot.net/k12math/newsgroup charter: === equations> what are the steps to finding out for example y=2b+5 what are you> suppose to do? Please help thankyouFinding what out? I can see quite a few tngs you could be doing here:1) Solving for y2) Solving for b3) Finding the slope4) Grapng the lineetcWhen you ask a math question, the first tng to do is to precisely define the problem. Once you know what it's asking, THEN you can look for the answer.-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.tnkspot.net/k12math/newsgroup charter: === HELP~!!!!!!!!!!!!!!!!!!!!~My math teacher keeps confusing me. Ok, we are on chapter 10 section5(finding the area, apothem, and perimeter of regular polygons andcircle). I know how to find the area of regular polygons, but I don'tknow how to find the apothem of them. For example, if you have atriangle and the line to the dot in the middle of triangle is 8in.,how do I find the apothem of that triangle? Signed, Confused-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.tnkspot.net/k12math/newsgroup charter: === HELP~!!!!!!!!!!!!!!!!!!!!~> ... For example, if you have a triangle and the line to the dot> in the middle of triangle is 8in., how do I find the apothem of> that triangle?If the line to the dot in the middle of triangle is the line from avertex to the center of the equilateral triangle, you should be ableto find the apothem by using the properties of a 30-60-90 degreetriangle.Dom Rosa-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.tnkspot.net/k12math/newsgroup charter: === HELP~!!!!!!!!!!!!!!!!!!!!~> My math teacher keeps confusing me. Ok, we are on chapter 10 section> 5(finding the area, apothem, and perimeter of regular polygons and> circle). I know how to find the area of regular polygons, but I don't> know how to find the apothem of them. For example, if you have a> triangle and the line to the dot in the middle of triangle is 8in.,> how do I find the apothem of that triangle?> Signed,> Confused>A polygon of n sides has 180(n-2) degrees. If you want to work with atriangle, you know there is 180 degrees. If you draw the medians, you get a60 degree angle. When the apothem is drawn, it will bisect the angle sobasically, you have a 30-60-90 triangle. In ts case, the apothem is 8 in. Moran-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.tnkspot.net/k12math/newsgroup charter: === the equation to answer the problemIn ts particular problem, I am given only the answer the product),and I am to figure out how they arrived at product. What is theequation to figure ts out???, Stacy-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.tnkspot.net/k12math/newsgroup charter: === What is the equation to answer the problem> In ts particular problem, I am given only the answer the product),> and I am to figure out how they arrived at product. What is the> equation to figure ts out???> er...not knowing anytng about the problem, that's kind of hard to answer :-)However, product means multiply.-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.tnkspot.net/k12math/newsgroup charter: === the fearsome variable!Seems to me that when early grades use problems such as open box or star or circle or any other symbol + 3 = 5 and expect the youngster to write 2 in the symbol, that's introducing the concept of a variable.Don> I thought ts was a curious and telling comment.> <<<> We might as well be arguing over whether or not we should use x for> a variable. ..and don't even try to tell me that when first> introduced to even ts _English_ letter to denote a variable, some> are not confused for a wle. I have a good friend who teaches first and second grade, and she> assures me that introducing letter names for numbers in simple> equations doesn't faze her students. So I wonder what sort of spiny shell a teacher might be creating in> the minds of students around the introduction of a letter as a> variable in middle school that could cause confusion, and other> problems reported by some teachers. Students are perfectly at home> naming cats, dogs, stuffed animals, game characters, cars, trucks,> trains, and various other macnes -- so one would not expect them to> even tnk it odd to name (even temporarily) some numbers, or> formulas, or functions when the discussion turns to math. These> students learned before going to school how pronouns (substitute names> that may be without specific object reference) work and how they can> be used in sentences, and variables are just the pronouns of math. > How is the teacher whose students react negatively hedging the natural> naming propensity about with fearsome extra verbiage, and why? (Sorry, I wouldn't normally write anytng related to education that> used the word fear or any variant of it - I must have been infected by> the frequent exposure to it in one responder's missives.)> -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.tnkspot.net/k12math/newsgroup charter: === Learning center to catch up in math/reading?Our fourth grade son was recently assessed at a Sylvan LearningCenter. We had m assessed because he is falling bend at school. The special education services he has received have allowed m tofall drastically bend in the area of math. The total math score wasa grade level of 3.2, and our son is currently in s 6th month offourth grade. We also suspected a possible comprehension problem withreading. The reading assessment showed only 70% comprehension, allother reading scores were average. We are considering spending agreat deal of time and money to hopefully have our son catch up tograde level by the time he begins fifth grade in September. We wouldlike to hear about good/bad experiences regarding Sylvan from peoplein similar situations? in advance.-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.tnkspot.net/k12math/newsgroup charter: === Solving the stated problem not the problem you tnk up> Recently, several questions in ts group made me tnking of our> math education in general. Seems to me we put too much emphasis on using daily language to> describe math problems and in the process lost the beauty of> precise of math. As a result, the students sometimes are trying> to solve a problem they tnk are asked. basng the examine designers about item 8 of ts test. see item 8. http://www.mathforum.com/epigone/mathed-news/clarcorpleeTs problem was discussed by Fernando Q. Gouvea in page 6 of the Dec.America.Students had to calculate the perimeter, to the nearest centimeter, ofa pictured regular octagon with the following given data:1. The radius is 4.6 cm (distance from the center to a vertex).2. The apothem is 4cm (perpendicular distance from the center to aside).The choices were 41cm, 36cm, 27cm, and 18cm.As Gouvea pointed out, no octagon with the given measurements exists:If the radius is 4.6, then the apothem would be 4.6(cos22.5) = 4.2.Students who used the Pythagorean theorem would obtain the length ofhalf a side to be 2.2715633, yielding a perimeter that would round to36 cm.Students who used trigonometry, would obtain the length of half a sideto be 4(tan22.5) = 1.6568542, yielding a perimeter that would round to27 cm.How did the test preparers come up with the numbers 4.6 and 4? Gouveasees the following two possibilities:1. For a Hexagon, a radius of 4.6 yields an apothem of 4.6(cos30) =3.9837169, wch rounds to 4.0 cm. So the authors did thecalculations for a hexagon and then changed to an octagon withoutchanging the numbers.2. A more intriguing one is ts: If you compute 4.6(cos22.5) withthe calculator set to radians you get -4.01. Ignore that minus sign,and tadah: ...It just occurred to me that there may be a trd possibility: The testpreparers may have calculated an apothem of 4 cm by carelessly using acentral angle of 60 degrees for an Octagon.posted at:http://mathforum.org/epigone/math-teach/plemzitaiThe first from The Hartford Courant deals with a scoring debacle thathas occurred with the Connecticut Mastery Tests. The second from theSeattle Times is a grim expose of how scoring mills operate.Dom Rosa-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.tnkspot.net/k12math/newsgroup charter: === Graph Isomorpsm Sorry, I meant to say directed graphs instead of connected graphs.> For small graphs, say on the order of 3 vertices, how does one go> about finding isomorpsms if you were to try to enumerate all> possible cases? 3 vertices? To enumerate all cases, you can write out the adjacency matrix (wch will not be symmetric if you are looking at directed graphs.) Index the columns with the vertex labels 1, 2, 3 (or however many vertices you need. Write the rows downward in the same order as above. Then, replacing 1,2,3 on the top with any permutation of (1,2,3) will get you the adjacency matrix of an isomorpc graph. To check if two directed graphs are isomorpc by brute force, fix the adjacency of one of the graphs and permute the other one until all permuations have been checked (or until an identical matrix is found.) Even for brute force, ts is quite inefficient... a much better method uses a kind of backtrack search. Say you have two digraphs you want to test for isomorpsm. Take v1 of G1 and look at in-degree and out-degree, and then out of all vertices in G2 with the same in-degree and out-degree, try mapping v1 to them (individually). Then continue the process for the neighbours of v1, and so on. Of course, a good recursive algorithm for ts would be harder to write than the first brute-force method, but it would perform much better on large graphs. Using the degrees of the vertices is one of many invariants (i.e. properties that are invariant under isomorpsm). The book on Combinatorial Generation by Kreher and Stinson covers many more invariants for use in ts problem and describe backtracking algorithms in detail. The textbook has a homepage here: http://www.math.mtu.edu/~kreher/cages.html where they also offer C-code implementations of their algorithms (but you'd probably have to consult the textbooks to know precisely === bergv@math.uiuc.edu (Maarten Bergvelt)>>clipped.....> 3 vertices?> To enumerate all cases, you can write out the adjacency matrix (wch > will not be symmetric if you are looking at directed graphs.) Index the > columns with the vertex labels 1, 2, 3 (or however many vertices you need. > Write the rows downward in the same order as above.> Then, replacing 1,2,3 on the top with any permutation of (1,2,3) will > get you the adjacency matrix of an isomorpc graph.> To check if two directed graphs are isomorpc by brute force, fix the > adjacency of one of the graphs and permute the other one until all > permuations have been checked (or until an identical matrix is found.) Even for brute force, ts is quite inefficient... a much better method > uses a kind of backtrack search. Say you have two digraphs you want to > test for isomorpsm. Take v1 of G1 and look at in-degree and out-degree, > and then out of all vertices in G2 with the same in-degree and out-degree, > try mapping v1 to them (individually). Then continue the process for the > neighbours of v1, and so on.> Of course, a good recursive algorithm for ts would be harder to write > than the first brute-force method, but it would perform much better on > large graphs. Using the degrees of the vertices is one of many invariants (i.e. > properties that are invariant under isomorpsm). The book on > Combinatorial Generation by Kreher and Stinson covers many more invariants > for use in ts problem and describe backtracking algorithms in detail.> The textbook has a homepage here: > http://www.math.mtu.edu/~kreher/cages.html> where they also offer C-code implementations of their algorithms (but > you'd probably have to consult the textbooks to know precisely what the > programs do.)That is helpful, I assume the rows (i) indicate from wle thecolumns (j) indicate to in the directed graph sense. So from what Iunderstand, are you suggesting that you try all column permutations?But ts doesn't take into account self-loops such as vertex 1 tovertex 1. For instance if I have the matrix0 1 10 0 0 0 0 0can permute to:1 0 10 0 00 0 0wch is not isomorpc since there is a 1 in (i,j)=(1,1). === Graph Isomorpsm undecided-unixOriginator: bergv@math.uiuc.edu (Maarten Bergvelt)> That is helpful, I assume the rows (i) indicate from wle the> columns (j) indicate to in the directed graph sense. So from what I The Ôform' and Ôto' for column/row can be chosen arbitrarily without affecting correctness, but there's probably some convention for wch is commonly wch.> understand, are you suggesting that you try all column permutations?> But ts doesn't take into account self-loops such as vertex 1 to> vertex 1. For instance if I have the matrix I said you try all column permutations, but you still have to write the rows in the same order the columns appear as well. (i.e. apply the same permutation to both column and row.)> 0 1 1> 0 0 0 > 0 0 0 can permute to: 1 0 1> 0 0 0> 0 0 0Let's index the rows and columns: 1 2 31 0 1 12 0 0 03 0 0 0If you are permuting 1 with 2, then 2 1 32 0 0 01 1 0 13 0 0 0> wch is not isomorpc since there is a 1 in (i,j)=(1,1). for> the book suggestion, I'll go look it up. Ts kind of permutation always maintains the value of (i,j). So the (1,1) entry in the first matrix is 0, and in the second matrix, the (2,2) position has now been replaced with the (1,1) entry, wch is === bergv@math.uiuc.edu (Maarten Bergvelt)First Call for Papers 15th Australasian Workshop on Combinatorial Algorithms Ballina Beach Resort Ballina, NSW, Australia http://linus.levels.unisa.edu.au/~pmanyem/awocaSCOPE -----The Fifteenth Australasian Workshop on Combinatorial Algorithms (AWOCAfour-hour drive south from Brisbane. The workshop will follow the styleof its predecessors: problem-oriented papers and an emphasis on informaldiscussions. TOPICS------Authors are invited to submit papers that present original andunpublished research in the following areas: Algorithms and Data Structures Complexity Theory Graph Theory and Combinatorics Cryptography Algorithms on Strings Graph Algorithms Graph Drawing Computational Algebra and Geometry Computational Biology Communications Networks Probabilistic and Randomised Algorithms New Paradigms of Computation Ts is not an exclusive list. Papers related to other theoreticalaspects of Computer Science will also be considered. Papers wch stateand discuss open problems and work-in-progress witn the general themeof algorithms and computational complexity are especially encouraged. SUBMISSION GUIDELINES---------Electronic submissions are solicited. Papers should be submitted on orIn extreme cases, contributions may be submitted by sending a hard copyto: Attention: Ms B. Munday NICTA Bay 15, Australia Technology Park Eveleigh, NSW 1430, Australia. be considered.Simultaneous submission to other conferences with published proceedings is not allowed. PUBLICATION-----------Submitted papers will be refereed and the accepted papers will appear inthe proceedings. Authors of accepted papers must guarantee that theirpaper will be presented at the conference. After the workshop, selectedpapers will be published in a special issue of a journal (to bedecided). IMPORTANT DATES---For further information, seehttp://linus.levels.unisa.edu.au/~pmanyem/awoca or please contact Peter Eades (peter@it.usyd.edu.au) or Seokhee Hong === undecided-unixOriginator: bergv@math.uiuc.edu (Maarten Bergvelt)Here are ts week's titles in the mathematics arXiv, available at: http://front.math.ucdavis.edu/ http://front.math.ucdavis.edu/submissionsTs week in the mathematics arXiv may be freely redistributedwith attribution and without modification.Titles in the mathematics arXiv (9 Feb - 13 Feb)AC: Commutative Algebra-----------math.AC/0402180 Holger Brenner: The rationality of the lbert-Kunz multiplicity in graded dimension twomath.AC/0402179 Holger Brenner: Tight closure and plus closure in dimension twoq-bio.PE/0402015 Bernd Sturmfels, Seth Sullivant: Toric ideals of phylogenetic invariantsmath.AC/0402106 Erika Giorgi: On the irreducible components of the form ring and an application to intersection cyclesAG: Algebraic Geometry----------math.AG/0402196 Johannes Nicaise: Motivic generating series for toric surface singularitiesmath.AG/0402184 Dennis Gaitsgory: On de Jong's conjecturemath.AG/0402158 Grigoriy Blekherman: Volumes of Nonnegative Polynomials, Sums of Squares and Powers of Linear Formsmath.AG/0402154 Oleg N. Popov: The Cox Ring of a Del Pezzo Surface Has Rational Singularitiesmath.AG/0402146 Sam Payne: Fujita's very ampleness conjecture for singular toric varietiesmath.AG/0402143 Ulrich Goertz, Thomas J. Haines: The Jordan-Hoelder series for nearby cycles on some Smura varieties and affine ßag varietiesmath.AG/0402101 Johannes Nicaise: Arcs and resolution of singularitiesmath.AG/0402099 ros Sato: Wild hypersurface bundles over toric varietiesmath.AG/0402098 F. Guillen, V. Navarro, P. Pascual, A. Roig: Moduli Spaces and Formal OperadsAP: Analysis of PDEs--------math.AP/0402203 Ilia Kamotski, Michael Ruzhansky: Regularity properties, representation of solutions and spectral asymptotics of systems with multiplicitiesmath.AP/0402193 b Sterbenz: Global Regularity for General Non-Linear Wave Equations I. (6+1) and gher Dimensionsmath.AP/0402192 b Sterbenz: Angular Regularity and Strichartz Estimates for the Wave Equationmath.AP/0402191 b Sterbenz: Global Regularity and Scattering for General Non-Linear Wave Equations II. (4+1) Dimensional Yang--Mills Equations in the Lorentz Gaugemath.AP/0402170 Jean-Francois Bony, Remi Carles, Dietrich Haefner, Laurent Michel: Scattering theory for the Schrodinger equation with repulsive potentialmath.AP/0402138 D. Del Santo, M. Prizzi: Backward uniqueness for parabolic operators with non-Lipscitz coefficientsmath.AP/0402130 Terence Tao: Global well-posedness and scattering for the gher-dimensional energy-critical non-linear Schrodinger equation for radial datamath.AP/0402129 Jim Colliander, Mark Keel, Gigliola Stafßiani, deo Takaoka, Terry Tao: Global well-posedness and scattering for the energy-critical nonlinear Schrodinger equation in R^3CA: Classical Analysis and ODEs-------math.CA/0402209 Stephen Semmes: Elements of harmonic analysismath.CA/0402175 Palle E. T. Jorgensen: Iterated function systems, representations, and lbert spaceCO: Combinatorics-----math.CO/0402208 Ron M. Adin, Ira M. Gessel, Yuval Roichman: Signed Mahoniansmath.CO/0402206 Jeremy , Victor Reiner: Cyclotomic and simplicial matroidsmath.CO/0402186 M.D. Atkinson, M.M. Murphy, N. Ruskuc: Pattern avoidance classes and subpermutationsmath.CO/0402148 M. Beck, J. A. De Loera, M. Develin, J. Pfeiße, R. P. Stanley: Coefficients and Roots of Ehrhart Polynomialsmath.CO/0402132 Michael Krivelevich, Simon Litsyn, Alexander Vardy: A Lower Bound on the Density of Sphere Packings via Graph Theorymath.CO/0402127 Michel Lassalle, Michael Schlosser: Inversion of the Pieri formula for Macdonald polynomialsmath.CO/0402125 A. K. Kwasniewski: Poisson, Dobinski, Rota and coherent statesmath.CO/0402113 Eric M. Rains: BC_n-symmetric abelian functionsmath.CO/0402092 M. Bradley: Two Identities for Multiple Harmonic $q$-Seriesmath.CO/0402091 M. Bradley: Partition Identities for the Multiple Zeta Functionmath.CO/0402088 Leonid Gurvits: Combinatorics dden in hyperbolic polynomials and related topicsCV: Complex Variables---------math.CV/0402161 Jim Agler, C. McCarthy: Norm preserving extensions of holomorpc functions from subvarieties of the bidiskmath.CV/0402104 Robert Berman: Holomorpc Morse inequalities on manifolds with boundarymath.CV/0402096 S. Benelkourc, B. Jennane, A. Zeria: Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscatesDG: Differential Geometry-math.DG/0402207 Leonardo Biliotti: Exponential map of a weak Riemannian lbert manifoldmath.DG/0402201 Robert L. Bryant: SO(n)-invariant special Lagrangian submanifolds of C^{n+1} with fixed locimath.DG/0402198 Michael T. Anderson: Some results on the structure of conformally compact Einstein metricsmath.DG/0402195 Igor Zelenko: Fundamental form and Cartan's tensor of (2,5)-distibutions coincidemath.DG/0402194 Natasa Sesum: Limiting behaviour of the Ricci ßowmath.DG/0402182 Niky Kamran, Terry Robart: An infinite-dimensional manifold structure for analytic Lie pseudogroups of infinite typemath.DG/0402173 F. T. Farrell, P. Ontaneda: Exotic structures and the limitations of certain analytic methods in geometrymath.DG/0402171 Igor Zelenko: Variational Approach to Differential Invariants of Rank 2 Vector Distributionsmath.DG/0402153 Anna Pratoussevitch: Traces in Complex Hyperbolic Triangle Groupsmath.DG/0402142 Mario Listing: Scalar curvature rigidity of almost Hermitian spin manifolds wch are asymptotically complex hyperbolicmath.DG/0402141 J'anos Koll'ar: Einstein metrics on connected sums of $S^2times S^3$math.DG/0402124 Fuquan Fang: Positive Quaternionic Kaehler manifolds and symmetry rank: IImath.DG/0402122 Takuro Mochzuki: A characterization of semisimple local system by tame pure imaginary pluri-harmonic metricmath.DG/0402114 Marcella Palese: Backlund loop algebras for compact and non-compact nonlinear spin models in $(2+1)$ dimensionsmath.DG/0402100 Thomas Branson, Andreas Cap, Michael Eastwood, Rod Gover: Prolongations of Geometric Overdetermined Systemsmath.DG/0402097 Alexander I. Bobenko, Christian Mercat, Yuri B. Suris: Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green's functionDS: Dynamical Systems---------math.DS/0402165 Elon Lindenstrauss: Rigidity of multiparameter actionsmath.DS/0402147 W. Milnor: On Latt`es Mapsmath.DS/0402144 J.-R. Chazottes, L. Ramirez, E. Ugalde: Finite type approximations of bs measures on sofic subsftsmath.DS/0402133 D. Fisher, D. Morris, K. Whyte: Non-ergodic actions, cocycles and superrigiditymath.DS/0402115 R. L. Adler, B. Kitchens, M. Martens, C. Pugh, M. Shub, C. Tresser: Convex Dynamics and ApplicationsFA: Functional Analysis-----------math.FA/0402202 I. Gasparis, E. Odell, B. Wahl: Weakly null sequences in the Banach space C(K)math.FA/0402160 Joram Lindenstrauss, Preiss: On Fr'echet differentiability of Lipsctz maps between Banach spacesmath.FA/0402150 Miguel Carrion-Alvarez: Variations on a theme of Gelfand and Naimarkmath.FA/0402123 K.Storozhuk: Asymptotical behavior of subspaces under action of asymptotically finite-dimensional semigroups of operatorsGM: General Mathematics-----------math.GM/0402103 William M. Goldman: An exposition of results of FrickeGR: Group Theory----math.GR/0402205 Murray Elder: L_delta groups are almost convex and have a sub-cubic Dehn functionmath.GR/0402187 J.-F. Mestre: Relevements d'extensions de groupe de galois $L3(2)$math.GR/0402116 Franc{c}ois Digne: Pr'esentations duales des groupes de tresses de type affine $tilde A$GT: Geometric Topology----------math.GT/0402197 Howard Masur, Anton Zorich: Multiple Saddle Connections on Flat Surfaces and Principal Boundary of the Moduli Spaces of Quadratic Differentialsmath.GT/0402185 Svetlana Tyurina, Alexander Varchenko: Finite order invariants for $(n,2)$-torus knots and the curve $Y^2=X^3+X^2$math.GT/0402167 Hao Wu: Tight Contact Small Seifert Spaces with $e_0leq-3$math.GT/0402166 Craig A. Jensen, Nathalie Wahl: Automorpsms of free groups with boundariesmath.GT/0402162 William J. Schellhorn: Filamentations for Virtual Linksmath.GT/0402155 Nafaa Chbili: The Casson-Walker-Lescop invariant of periodic three-manifoldsmath.GT/0402131 b A. Rasmussen: Khovanov homology and the slice genusmath.GT/0402121 Michel Boileau, Scheng Wang: Degree one maps between small 3-manifolds and Heegaard genusmath.GT/0402120 Ruifeng Qiu, Scheng Wang: Small knots and large handle additionsmath.GT/0402119 Haibao Duan, Scheng Wang: Non-zero degree maps between $2n$-manifoldsHO: story and Overviewmath.HO/0402204 Gavriel Segre: On the mathematical structure of Tonal HarmonyKT: K-Theory and Homology-math.KT/0402190 Amalendu Krishna: On K_2 of 1-dimensional local ringsMG: Metric Geometry-------math.MG/0402087 Jun Murakami, Akira Usjima: A volume formula for hyperbolic tetrahedra in terms of edge lengthsMP: Mathematical Physicsphysics/0402051 Gerald V. Dunne, Christopher S. Gauter: Simple Soluble Molecular Ionization Modelnlin.SI/0402014 H. Aratyn, K. Bering: Compatible Poisson Structures of Toda Type Discrete erarchymath-ph/0402031 Rossen I. Ivanov: On the Dressing Method for the Generalised Zakharov--Shabat Systemmath-ph/0402030 Xavier Gr`acia, Miguel C. Mu~noz-Lecanda, Narciso Rom'an-Roy: On some aspects of the geometry of differential equations in physicsmath-ph/0402028 Houman Owhadi: Averaging versus Chaos in Turbulent Transport?hep-th/0402043 A.O.Barvinsky, D.V.Nesterov: Nonperturbative heat kernel and nonlocal effective actiongr-qc/0402051 Lars Andersson, Henk van Elst, Claes Uggla: Past asymptotic dynamics in vacuum G2 cosmologymath-ph/0402029 Joshua Feinberg: Fredholm's Minors of Arbitrary Order: Their Representation as a Determinant of Resolvents and an Explicit Formula for Their Functional Derivativemath-ph/0402027 Giuseppe Ruzzi: Restricted Haag duality in locally covariant quantum field theoriesmath-ph/0402026 Timo Korvola, Antti Kupiainen, Jari Taskinen: Anomalous scaling for 3d Cahn-lliard frontsmath-ph/0402025 JM Robbins, M Zyskin: Classification of unit-vector fields in convex polyhedra with tangent boundary conditionsmath-ph/0402024 Hakan Andreasson, Simone Calogero, Reinhard Illner: On Blowup for Gain-Term-Only classical and relativistic Boltzmann equationsmath-ph/0402023 Fabian Brau: On the decrease of the number of bound states with the increase of the angular momentummath-ph/0402022 Fabian Brau, Francesco Calogero: Lower limit in semiclassical form for the number of bound states in a central potentialmath-ph/0402021 Tuncay Aktosun: Inverse scattering on the line with incomplete scattering datamath-ph/0402020 Tuncay Aktosun, Vassilis G. Papanicolaou, Vassilis Zisis: Inverse scattering on the line for a generalized nonlinear Schroedinger equationcond-mat/0402238 Daniel Dantchev, Michael Krech: The critical Casimir force and its ßuctuations in lattice spin models: exact and Monte Carlo resultsmath-ph/0402019 Tuncay Aktosun, Ricardo Weder: Inverse Spectral-Scattering Problem with Two Sets of Discrete Spectra for the Radial Schroedinger Equationmath-ph/0402018 Johan Groenqvist, Thomas Guhr, Heiner Kohler: The k-Point Random Matrix Kernels Obtained from One-Point Supermatrix Modelsmath-ph/0402017 M. Lorente: Raising and lowering operators and their factorization for generalized orthogonal polynomials of hypergeometric type on homogeneous and non-homogeneous latticemath-ph/0402016 Abu Bakr Mehmood, S. Umer Abbas, Ghulam Shabbir: Closed Form Approximation Solutions for the Restricted Circular Three Body Problemmath-ph/0402015 V.Shramchenko: Real doubles of Hurwitz Frobenius manifoldsmath-ph/0402014 V.Shramchenko: Integrable systems related to elliptic branched coveringsmath-ph/0111017 K. Vaninsky: The Atiyah-tcn Bracket for the Cubic Nonlinear Schrodinger Equation. I. General Potentialshep-th/0402057 D.Bashkirov, G.Sardanashvily: Covariant Hamiltonian field theory. Path integral quantizationhep-th/0402006 Bozdar Z. Iliev: Lagrangian quantum field theory in momentum picture. I. Free scalar fieldscond-mat/0305505 M. B. Hastings: Lieb-Schultz-Mattis in gher Dimensionscond-mat/0402142 Leo P. Kadanoff, Marko Kleine Berkenbusch: Trace for the Loewner Equation with Singular Forcingphysics/0401098 L.P. Horwitz, N. Katz, O. Oron: Could the classical relativistic electron be a strange attractor?math-ph/0402013 Nikolai Filonov, Frederic Klopp: Absolute continuity of the spectrum of a Schrodinger operator with a potential wch is periodic in some directions and decays in othersgr-qc/0402032 Kouji Nakamura: General framework of gher order gauge invariant perturbation theoryNA: Numerical Analysis----------math.NA/0402178 Maxim Dvornikov: Spectral properties of numerical differentiationmath.NA/0402164 Yunkai Zhou: Eigenvalue Computation from the Optimization Perspective: On bi-son, IIGD, RQI, and Newton UpdatesNT: Number Theory-----math.NT/0402168 Krzysztof Maslanka: Effective method of computing Li's coefficients and their unexpected propertiesmath.NT/0402163 Gabor Wiese: Dihedral Galois representations and Katz modular formsmath.NT/0402135 Kenic Kawagoe, Masato Wakayama, Yosnori Yamasaki: q-Analogues of the Riemann zeta, the Dirichlet L-functions and a crystal zeta functionmath.NT/0402128 L. J. P. Kilford: A generalisation of a congruence due to Vantieghem only holding for primesmath.NT/0402111 A. J. Scholl: On some l-adic representations of Gal(Qbar/Q) attached to noncongruence subgroupsmath.NT/0402095 Roberto G. Ferretti: Positivity of Heights of Semistable VarietiesOA: Operator Algebras---------math.OA/0402181 Marius Junge, Zhong-Jin Ruan, Sherman: A classification for 2-isometries of noncommutative Lp-spacesmath.OA/0402177 Thomas Eckl: A note on invariants of ßows induced by Abelian differentials on Riemann surfacesquant-ph/0402056 J.A. Holbrook, D.W. Kribs, R. Laßamme: Noiseless subsystems and the structure of the commutant in quantum error correctionmath.OA/0402126 Stephen Allen, Pask, Aidan Sims: A dual graph construction for gher-rank graphs, and $K$-theory for finite 2-graphsmath.OA/0402109 Marius Stefan: An estimate of free entropy and applicationsmath.OA/0402108 Marius Stefan: Indecomposability of free group factors over nonprime subfactors and abelian subalgebrasmath.OA/0402107 Marius Stefan: Infinite multiplicity of abelian subalgebras in free group subfactorsmath.OA/0402105 J.A. Holbrook, D.W. Kribs, R. Laßamme, D. Poulin: Noiseless subsystems for collective rotation channels in quantum information theorymath.OA/0402094 Huaxin Lin, royuki Osaka: The Rokhlin property and the tracial topological rankPR: Probability Theory----------math.PR/0402183 Anatolii A. Puhalskii: Stochastic processes in random graphsmath.PR/0402174 Marzio Cassandro, Enza Orlandi, Pierre Picco, Maria Eulalia Vares: One-dimensional random field Kac's model: localization of the phasesmath.PR/0402169 Remco van der Hofstad, Frank Redig: Maximal clusters in non-critical percolation and related modelsmath.PR/0402156 G.T. Tetzlaff: Criticality in unbounded-types brancng processesmath.PR/0402137 Emilio De Santis, Fabio Spizzicno: Change point models and conditionally pure birth processes; an inequality on the stochastic intensitymath.PR/0402136 Emilio De Santis, Mauro Piccioni: Perfect simulation for unilateral fieldsQA: Quantum Algebra-------math.QA/0402200 Christian Blohmann: Perturbative Symmetries on Noncommutative Spacesmath.QA/0402199 Christian Blohmann: Realization of q-deformed spacetime as star product by a Drinfeld twistmath.QA/0402159 Shlomo Gelaki: Basic quasi-Hopf algebras of dimension n^3math.QA/0402152 Jun-ic Okuda, Yosro Takeyama: On relations for the $q$-multiple zeta valuesmath.QA/0402151 Kornel Szlachanyi: The double algebraic view of finite quantum groupoidsmath.QA/0402118 Marcelo Aguiar, Nicolas Andruskiewitsch: Representations of matched pairs of groupoids and applications to weak Hopf algebrasmath.QA/0402117 James E. McClure, Jeffrey H. Smith: Operads and cosimplicial objects: an introductionmath.QA/0402112 A. Gerasimov, S. Kharchev, D. Lebedev: Representation theory and quantum integrabilitymath.QA/0402102 Bullock, Walter Lo Faro: The Kauffman bracket skein module of a twist knot exteriormath.QA/0402093 M. Bradley: Multiple $q$-Zeta ValuesRA: Rings and Algebras----------math.RA/0402188 Shouchuan Zhang, Yao-Zhong Zhang: Structures and Representations of Generalized Path Algebrasmath.RA/0402176 Peter Jorgensen: Tate cohomology over fairly general ringsmath.RA/0402145 Eli Aljadeff, Christian Kassel: Norm formulas for finite groups and induction from elementary abelian subgroupsRT: Representation Theory-math.RT/0402157 J.M. Landsberg, L. Manivel: The sextonions and $E_{7frac 12}$math.RT/0402140 Dmitri I. Panyushev: Normalizers of ad-nilpotent idealsmath.RT/0402139 Pablo Ramacher: Pseudodifferential operators on prehomogeneous vector spacesmath.RT/0402134 S.Eswara Rao: Toroidal Z-Algebrasmath.RT/0402089 Ian M. Musson: Lie Superalgebras, Clifford Algebras, Induced Modules and Nilpotent OrbitsSG: Symplectic Geometry-----------math.SG/0402189 Fan Ding, Yunfeng Jiang, Jianzhong Pan: The Chen-Ruan Cohomology of Almost Contact Orbifoldsmath.SG/0402149 Yong-Geun Oh: Length minimizing property, Conley-Zehnder index and $C^1$-perturbations of Hamiltonian functionsSP: Spectral Theory-------math.SP/0402172 E. B. Davies: Semi-classical Analysis and Pseudospectramath.SP/0402110 Barry Simon: The sharp form of the strong Szego theoremmath.SP/0402090 Marianne Akian, Ravindra Bapat, Stephane Gaubert: Generic Asymptotics of Eigenvalues and Min-Plus Algebra-- / Greg Kuperberg (UC Davis) / / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/ / * All the math that's fit to === bergv@math.uiuc.edu (Maarten Bergvelt),The following result about the rank of the optimal solution of asemidefinite program is well known:For a semidefinite program with m constraints wch fulfills strongduality, the rank r of the optimal solution X of the primal problemmin s.t. = b_i for i=1,...,m X >= 0satisfies 0.5 r (r+1) <= m.My question is: Does ts result also hold if we can only guarantee astrictly interior point for the dual problem, but not for the primalproblem? Under wch conditions does ts result still hold? for any help, jens === undecided-unixEpigone-thread: tweldprarprilOriginator: bergv@math.uiuc.edu (Maarten Bergvelt)I tnk so.It suffices to show Ext_T^j(X,Y)=Ext_A^j(X,Y) for any two X,Y in T.Fix X and consider both sides as delta-functors in Y. The firstclearly is universal.To see that the second is universal, one has to show that for each Yin T there is an injective module I wch is torsion and has Y as a submodule.There clearly is an injective I with 0 -> Y -> I exact. Since the ringis commutative,I contains the torsion submodule I_tors and clearly Y lies in it.Injectivity is equivalent to full divisibility (Cohn, Algebra vol3)and ts showsthat I_tors again is injective. So the right hand side delta functor is erasable, hence universal === undecided-unixOriginator: baez@math-cl-n03.math.ucr.edu ( Baez)Originator: bergv@math.uiuc.edu (Maarten Bergvelt)>James Propp came up with an interesting idea related to the Golden Object:>consider what happens when we evaluate the generating function for binary >trees at -1. On the one hand we get an alternating sum of Catalan numbers:|T|(-1) = -1 + 1 - 2 + 5 - 14 + 42 + ...On the other hand, we can use the formula|T| = (1 - sqrt(1 - 4x))/2to get|T|(1) = (1 - sqrt(5))/2 Ts was a typo: I meant|T|(-1) = (1 - sqrt(5))/2>Of course, it's possible we >should use the other sign of the square root, and get|T|(1) = (1 + sqrt(5))/2Here too I meant to write |T|(-1).To make ts post a bit less boring, let me note that startingwith a Golden Object - an object G in a distributive categoryequipped with an isomorpsm G^2 = G + 1we can construct isomorpsmsG^3 = G (G+1) = G^2 + 1 = 2G + 1where 2G is short for the coproduct G+G, and similarlyG^4 = G (2G+1) = 2G^2 + G = 3G + 2G^5 = G (3G+2) = 3G^2 + 2G = 5G + 3and so on, leading to a categorified version of the usualrelation between Fibonacci numbers and the golden number.Also, wle I don't know a good concrete *example* of suchan object G in a distributive category, one can get an objectG with an isomorpsm G^2 = G + 1 in the category of representations of quantum SU(2) with a suitable value of q. Ts is not a distributive category since the tensor product is not a product in the categorical sense. But, it still is a symmetric 2-rig: a symmetric monoidal category with colimits, where the tensor product distributes over colimits. === undecided-unixOriginator: bergv@math.uiuc.edu (Maarten Bergvelt)Q2: we knew continuous map preserve topology properties, including>close curve index(or winding number). If W is a continuous map from>R^n -> C, if I have a curve Gamma and a point O in C plane and knew>the winding number of Gamma to O, if I want to make the statement that>the pre-image of Gamma and O in R^n should have the same winding>relationsp, what other conditions I need to assume? do I need to>ask W^-1 is also a continuous map?The concept of winding number is not well-defined in R^n for n>2 even if W^-1(Gamma) is a curve, wch it may not be. Even if n=2,one needs to assume W^-1 is also continuous.Q3: Fredholm integrals are compact operators, can I say Fredholm>integrals are continuous maps from R^n to their ranges(R or C)? why?>is compactness stricter than continuity? or they are equivallent?Integral operators act on functions, not R^n, and typically producefunctions (not elements of R or C). Compactness is a generalization of finite dimensionality. In particular,if T:X -> Y (where X, Y are Banach spaces) and ||T - T_n|| -> 0 whereT_n are continuous linear operators with finite dimensional ranges, thenT is compact. All compact operators are continuous.Dan-- Dan Luecking Department of Mathematical SciencesUniversity of Arkansas Fayetteville, Arkansas 72701To reply by email, change Look-In-Sig to === bergv@math.uiuc.edu (Maarten Bergvelt)> Q1. can I say a complex function W(r)=u(r)+j*v(r) is a continuous map> from R^n to C iif u(r) and v(r) are both continuous function over> R^n?Yes. It's a good exercise to prove ts using the delta-epsilondefinition of continuity.Note that mathematicians usually use the letter i instead of jhere. > Q2: we knew continuous map preserve topology properties, including> close curve index(or winding number). If W is a continuous map from> R^n -> C, if I have a curve Gamma and a point O in C plane and knew> the winding number of Gamma to O, if I want to make the statement that> the pre-image of Gamma and O in R^n should have the same winding> relationsp, what other conditions I need to assume? do I need to> ask W^-1 is also a continuous map?Ts is interesting, but we need to do a little more work to formulatea precise question here.To start, W^-1 is generally not a map at all, as a point in C may havemany preimages in R^n, so it doesn't really make sense to ask if W^-1is continuous. Still, you can talk about the inverse image of a set Sin C, wch is the set of all points x in R^n with W(x) in S.Now the inverse images of Gamma and 0 could in general be reallynasty, making it hard to discuss winding relationsps at all. Soyou probably want to assume that Gamma is a smooth curve and W is ageneric smooth map. Then you know that the inverse image of Gammais an n-1 dimensional submanifold of R^n, call it X, and the inverseimage of 0 is an n-2 dimensional submanifold, call it Y. Also X and Yare oriented, wch is good. Now you still need to decide what youmean by winding number here when n>2. You probably (also when n=2)want to additionally assume that W is proper, i.e. that the inverseimage of a compact set is compact, so that X and Y are compact. Iffurthermore Y is connected, then there is a reasonable notion ofwinding number provided by Alexander duality (but given that youasked your first question, I tnk you need to learn a lot moretopology before you can really understand what Alexander duality is--- ts is explained towards the back of most algebraic topologybooks). If Y has k connected components, then you get k differentwinding numbers.When n=2, I tnk that the sum of these k winding numbers equals mtimes the winding number of Gamma around 0, where m is the degree ofthe restriction of W mapping from X to === undecided-unixOriginator: bergv@math.uiuc.edu (Maarten Bergvelt)> I am an engineering student, however, I am working on some theoretical> problems wch involve topology and complex analysis. I have couple of> questions concerning some basic concepts that I did not find definite> answer from the references I borrowed. I am wondering if anyone can> help me to answer those quesitons or direct me to the right references> that address these questions: Q1. can I say a complex function W(r)=u(r)+j*v(r) is a continuous map> from R^n to C iif u(r) and v(r) are both continuous function over> R^n?Yes.> Q2: we knew continuous map preserve topology properties, including> close curve index(or winding number).No, it doesn't. Take, for instance, f:C --> C defined by f(z) = z^2and take g(t) = exp(t) (with t in [0, 2pi]). Then the winding numberof g with respect to 0 is 1, but the index number of f o g with respectto 0 is 2.> If W is a continuous map from> R^n -> C, if I have a curve Gamma and a point O in C plane and knew> the winding number of Gamma to O, if I want to make the statement that> the pre-image of Gamma and O in R^n should have the same winding> relationsp, what other conditions I need to assume? do I need to> ask W^-1 is also a continuous map?If W and W^-1 are both continuous, then yes, the winding number ispreserved.> is compactness stricter than continuity? or they are equivallent?Compacteness is stricter === === 799Originator: bergv@math.uiuc.edu (Maarten Bergvelt)Subject: naive questions on topology> Q1. can I say a complex function W(r)=u(r)+j*v(r) is a continuous>map from R^n to C iif u(r) and v(r) are both continuous function over>R^n?Yes.If u and v are continuous and as + and * are continuous,the composition w = u + j*v is continuous.Conversely when |w(r) - w(r0)| is small,|w(r) - w(r0)| = |u(r)-u(r0) + j(v(r)-v(r0)| = (u(r) - u(r0))^2 + (v(r) - v(r0))^2shows (u(r) - u(r0))^2 and (v(r) - v(r0))^2are small and hence |u(r) - u(r0)| and |v(r) - v(r0)|are also small. >Q2: we knew continuous map preserve topology properties,Ts may not be so. For example, the continuous image of a space withisolated points may not have === iso-8859-1-unixOriginator: bergv@math.uiuc.edu (Maarten Bergvelt)> why is integration harder than differentiation?Differentiation is a breaking up procedure, but integration puts itall together. A quote from Sam Rayburn: Any donkey can kick down abarn, but it takes a good carpenter to build one. [Absolutely nooffence meant for the first activity though.]A differential equation links/relates the two. Performing integrationinto an analytical form is as difficult as the complexity of tsrelationsp. y''+ y' = 1 may be easily solved, but log(y''+ y')^sin(x y') = 1 is much more difficult to solve than simply writing outsuch a differential relationsp.I too had often wondered why there is no proper mathematicaldefinition of degree of hardness of solvability of differentialequations, accenting on relational complexity. All we have iscategorization into ordinary, partial, non-linear etc. The arrival ofnumerical macne solvers like the Runge-Kutta or implicit functionshas rendered such an enquiry perhaps needless... once and for all ?