mm- === Subject: Re: High Dimensonal Data Acquisition, Computing and Visualization >Do you know any interesting web links about a problem in the topic? A.L. === Subject: Re: Error bound in Cubic Spline cases for non uniform meshes >I need some references for error estimation of cubic spline S with >the given function f where f is in C^4. Cubic spline may be of >spline Type-1 or Type-2 or Periodic spline where as the meshes >are non-uniform. Any suggestion regarding the query is welcome. >Thanx. >akbchand this is contained already in the Ahlberg-Nilson-Walsh book theory and applications of spline functionsin chapter 2. uniformity of the mesh is not required. hth peter === Subject: Re: Solving a very large non-sparse linear system >Do anyone know on how to solve a linear system Ax=b >with dimension of A equal to 100000x100000 or even larger? >I am glad to hear your comments on this. >Larry you write this under the title nonsparse. If it is really dense, then for me the main question will be how to access these data? the matrix will necessarily reside on a backing store, and then the the number of accesses to this backing store will determine the total time much more than any other fact. (if you consider an iterative solver, then typically any single step will require to access every matrix element at least once per step, this sounds no good for me. and if you consider a direct solver, then a block wise access will be necessary to make this efŽcient. but even if everything would reside in main memory: 10^15/3 žoat ops will require parallel processing to be completed in reasonable time. but maybe and hopefully by nonsparse you meant not completely dense. typically, an iterative scheme plus a good preconditioner must be applied. but to give useful advice, one needs to know much more about your matrix than large and nonsparse. so tell us: where does it come from? hth peter === Subject: Determinant of a large sparse matrix X-AUTHid: lawhiu What is the best way to compute the determinant of a large, sparse asymmetric matrix? The Žrst thing that comes into my mind is to perform a sparse LU decomposition, but would that be inefŽcient? If I adopt instead the incomplete LU decomposition, will the approximation introduced lead to signiŽcant error in the determinant? The context is this. I have a variable n by n matrix X that I want to optimize with small rank of X. So X is usually singular. According to http://www.cds.caltech.edu/~maryam/acc03_Žnal.pdf this can be achieved by the heuristic log(det(X + delta I)), where delta is a small number that seems to require some care to select. That¹s why I may need to compute the determinant of a large, sparse matrix. P.S. My email address is lawhiu at cse dot msu dot edu. === Subject: Maximization Formula help. Hi All, I was wondering if anyone could help with this formula. Sorry for my ignorance with your syntax. I have a list of lengths (rows). I need to maximize those lengths with units of 96 & 72 while leaving no length of less that 12 units in any row. I need to then put the waste back into the formula and the maximize the next length (row). The rows also need to be staggered with no rows have a seam within 24 units of the seams in the row above or below. Any help would be greatly appreciated. I am writting this in VBA. Paul An example list would include length such as: Not to scale: Random seam pattern 300 = | x | x | x | x | x | 300 = | x | x | x | x | x | 300 = | x | x | x | x | x | === Subject: Re: Estimating min/max eigenvalue of a matrix > Formally (double meaning) perhaps, but in reality this is tautological > ^^^^^^^^^^^^ > nonsense. > ^^^^^^^^ > > Perhaps you would care to clarify that rather offensive remark? > > If you found it offensive then you¹ve a problem that I can¹t help you with > but here¹s a heads up: http://www.mhsanctuary.com/narcissistic/dsm.htm. Not > to worry, it¹s a condition endemic among academics and well parodied in > Huxley¹s ŒPoint Counter Point¹, which you ought to read for some layman > insight into the condition. > -- > E&OE > Ciao, > Gerry T. > Stick to topics of the newsgroup instead of trying to justify your > abrasiveness. > -- Lou Pecora This isn¹t Abu Ghraib, pte. Trick or treat: http://www.frankmagazine.ca/ -- E&OE Ciao, Gerry T. ______ The enemy is anybody who¹s going to get you killed, no matter which side he¹s on. -- Yossarian, in Catch 22. === Subject: Re: Estimating min/max eigenvalue of a matrix > > Let t = the trace of the matrix = the sum of its eigenvalues, and u = > the trace of the square of the matrix = the sum of squares of all its > elements = the sum of squares of its eigenvalues. Then m = t/n is the > average eigenvalue, s = sqrt(n*u - t^2)/n is the standard deviation of > the eigenvalues, and the eigenvalues are bounded by m +- s*sqrt(n-1). > > Since there is nothing to prevent the lower bound from being negative, > in practice the upper bound tends to be more useful than the lower. > The trace of the square of the matrix is O(n^2). As I said above, > I don¹t think it is possible to Žnd even rough bounds for general > pos.-def. matrices more cheaply than this. Both the Schmidt norm > and Tiktopoulos¹ norm are O(n^2). The latter tends to be a better > estimate of the largest eigenvalue, however. What you think about the use of Brauer ovals of Cassini? Some that is not a detonator for other question outside the topic (sig!)). === Subject: Psedo inverse by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4P52ol02584; does anybody know how can I Žnd the matrix Pseudo inverse or SVD C code? === Subject: Re: Psedo inverse >does anybody know how can I Žnd the matrix Pseudo inverse or SVD C >code? http://www.netlib.org/clapack or http://www.netlib.org/c/meschach hth peter === Subject: Re: Bland and Altman Plot by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4P52om02570; >Can you and does it make sense to compute a correlation coefŽcient >for data you plot in a Bland Altman plot? On the plots you get a mean >difference score and a CI, but I do not see a correlation coefŽcient. Can you specify your question more clearly. Muthu === Subject: Applications of the Distance from a Point to a Curve by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4PDf2t17285; I would like to know if anyone knows about (real) situations where it is necessary to compute the euclidean distance from a point to a curve/surface (or even the point where it is reached), you may not assume that the point is very close to the curve (unlike the usual Žtting problems). I have special interest in applications to phisics. Dimas === Subject: Re: Applications of the Distance from a Point to a Curve >I would like to know if anyone knows about (real) situations where it >is >necessary to compute the euclidean distance from a point to a >curve/surface (or even the point where it is reached), you may not >assume that the point is very close to the curve (unlike the usual >Žtting problems). >I have special interest in applications to phisics. >Dimas what about gravitational force induced by a planet in connection with a landing manoeuvre ? hth peter === Subject: Re: Applications of the Distance from a Point to a Curve >I would like to know if anyone knows about (real) situations where it >is >necessary to compute the euclidean distance from a point to a >curve/surface (or even the point where it is reached), you may not >assume that the point is very close to the curve (unlike the usual >Žtting problems). > >I have special interest in applications to phisics. > > >Dimas > > what about gravitational force induced by a planet in connection with a > landing manoeuvre ? > hth > peter Unless the planet has a very peculiar shape it is only the distance to the center that is of interest. Isaac Newton spent a lot of time proving that. A physics application perhaps more useful than landing on the planet Cowpatia might be Žnding the electrostatic potential arising from a given charge distribution on some arbitrary surface, using the principle of superposition. -- Julian V. Noble Professor Emeritus of Physics jvn@lessspamformother.virginia.edu ^^^^^^^^^^^^^^^^^^ http://galileo.phys.virginia.edu/~jvn/ For there was never yet philosopher that could endure the toothache patiently. -- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1. === Subject: Re: Minimization on Positive SemideŽnite Problem by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4PLOYE01453; >A positive semideŽnite matrix isn¹t enough to ensure that a minimum >exists; take a look at that second 1-D example again. >If you want to be safe, then instead of solving Ax = b you could solve >(A + epsilon * I) * x = b, where epsilon is small enough that it won¹t add >much error to your solution but large enough that it won¹t be drowned out >by žoating-point error. That will give you a positive deŽnite matrix, >and if your original problem had a hyperplane of minimizers then this new >problem should pick out the point on that hyperplane closest to 0. I did a small test and it turns out quite good except that the elements of the inverse get magniŽed with reciprocal of epsilon. I wonder if the perturbation performed through individual epsilon rather than homogeneous value would help in the accuracy in anyway. >Screwing with your matrix that way will obviously introduce some error, >but you could get some feel for the magnitude of that error by solving: >(A + epsilon * I) * x1 = b >(A + epsilon * I) * x2 = b + epsilon * x1 >(A + epsilon * I) * x3 = b + epsilon * x2 >etc, and you could set an error tolerance for this as well as for the >error in your iterative solver. I am quite unsure of how this error-measuring mechanism works. Could please point me some references? === Subject: Re: Breaking up pseudoinverse >For the problem > >A^t.A.x = A^t.b > >is there any generally valid way to break A^t.A up into a >smaller/sparser matrix - what I mean is, can you put an error bound on >the damage you¹re doing to x for altering A^t.A in some way. > >Eg. by zeroing small elements or even simply chopping it into, say 4x4 >blocks, and solving each block separately (essentially zeroing >elements that don¹t Žt the block structure)? > yes clearly, apply the perturbation theorem for the solution of linear > systems, which is knwon in norms or in componentwise representation. > (look up Golub & van Loan or other standard textbooks on numerical linear > algebra for that) its logical conclusion, I¹ve been wondering about using a diagonal matrix approximation to A^t.A, then inverting the diagonal matrix. Some numerical experiments with randomly generated matrices show that it¹s not a good approximation in those cases, however for other matrices (that perform a convolution) it does seem to be adequate. I don¹t have a good feel for what A^t.A should look like though - obviously symmetric, but what about the diagonal thing? === Subject: Re: difference? > what is difference between gauss-jordan and matrix? > are they used with the same technique? > i dont see any difference My guess is that you have not received any replies because, forgive me, your question makes no sense. Gauss - Jordan is a technique for manipulating matrices. Maybe you could give an example of what is bothering you? -- Paul Sperry Columbia, SC (USA) === Subject: Re: need urgent help on this question.... someone kind enough, pls kindly help/guide me to solve the inverse laplace of ln(1+1/s) using taylor series s at inŽnte expansion/power series of ln(1+t). any guidance is greatly appreciated. answer is t(1-e^-t) using inversion complex contour. > Hi there, > I need urgent help on the following question. > 1. Expand ln(1 + 1/s) in a power series about s = inŽnite. I tried using > taylor but really have great problem in evaluating please help... > URGENT. === Subject: Re: need urgent help on this question.... > someone kind enough, > pls kindly help/guide me > to solve the inverse laplace of ln(1+1/s) using taylor series > s at inŽnte > expansion/power series of ln(1+t). > any guidance is greatly appreciated. > answer is t(1-e^-t) using inversion complex contour. > Hi there, > I need urgent help on the following question. > 1. Expand ln(1 + 1/s) in a power series about s = inŽnite. I tried using > taylor but really have great problem in evaluating please help... > URGENT. The given answer is incorrect. ln(1 + x) = x - x^2/2 + x^3/3 - x^4/4 + ... ln(1 + 1/s) = 1/s - 1/(2*s^2) + 1/(3*s^3) - 1/(4*s^4) + ... Which has inverse Laplace 1 = (1/2!)*t + (1/3!)*t^2 - (1/4!)*t^3 + .... = (1/t)*(t - (1/2!)*t^2 + (1/3!)*t^3 - (1/4!)*t^4 + ...) = (-1/t)*(-1 + (1 - t + (1/2!)*t^2 - (1/3!)*t^3 + (1/4!)*t^4 + ...)) = (-1/t)*(-1 + exp(-t)) = (1 - exp(-t))/t. I¹ll leave it to you to keep track of the interval of convergence. -- Paul Sperry Columbia, SC (USA) === Subject: residual correlated with dependent variable doing a review problem and would appreciate any help suppose you have the model Yi = B1 + B2Xi + ui and the estimator of the residual, ui_hat = a1 + a2Yi + ei how can u Žnd the estimator of a2? i.e. a2_hat? === Subject: matlab can somebody help me graph f(x,y) = 3/2xy for 0<=x<=2, 0<=y<=2, x+y<=2 in matlab -- Hi... - cOS long Œat¹ myvrsix(d0t)com === Subject: Re: i am struck so any help or hint or suggestion is kindly appreciated... > Hi there, > Perhaps you can help me with the following, please do it fast as my deadline > is tomorrow. > 1. Inverse Laplace transforms can often be evaluated in several different > ways. Find > L?1{ln(1 + 1/s)} using each of the following approaches. > (a) Expand ln(1 + 1/s) in a power series about s = inŽnite, and take > inverse Laplace > transforms term by term. (This is allowable provided the series is uniformly > convergent, which it is for |s| > 1.) > Please help me with this asap. > Yours sincerely, > Lim Yoong Ping === Subject: Re: > i am struck so any help or hint or suggestion is kindly appreciated... > Hi there, > Perhaps you can help me with the following, please do it fast as my > deadline > is tomorrow. > 1. Inverse Laplace transforms can often be evaluated in several different > ways. Find > L?1{ln(1 + 1/s)} using each of the following approaches. > (a) Expand ln(1 + 1/s) in a power series about s = inŽnite, and take > inverse Laplace > transforms term by term. (This is allowable provided the series is > uniformly > convergent, which it is for |s| > 1.) > Please help me with this asap. > Yours sincerely, > Lim Yoong Ping [...] > The given answer is incorrect. > ln(1 + x) = x - x^2/2 + x^3/3 - x^4/4 + ... > ln(1 + 1/s) = 1/s - 1/(2*s^2) + 1/(3*s^3) - 1/(4*s^4) + ... > Which has inverse Laplace > 1 = (1/2!)*t + (1/3!)*t^2 - (1/4!)*t^3 + .... = > (1/t)*(t - (1/2!)*t^2 + (1/3!)*t^3 - (1/4!)*t^4 + ...) = > (-1/t)*(-1 + (1 - t + (1/2!)*t^2 - (1/3!)*t^3 + (1/4!)*t^4 + ...)) = > (-1/t)*(-1 + exp(-t)) = (1 - exp(-t))/t. > I¹ll leave it to you to keep track of the interval of convergence. -- Paul Sperry Columbia, SC (USA) === Subject: laplacian problem Hi everyone, I¹m a CS student and I am currently doing a maths course. I¹ve got a question concerning a homework question. Now I know some people get annoyed if you ask a question about homework but I have spent hours trying to answer one sum, have looked on INet and still cannot get the answer right. Question is Find nabla(triangle)(^2)f where f=xy/z put otherwise Žnd the laplacian of f where f =xy/z That is equal to div (grad f) = d(^2)f d(^2)f d(^2)f ------- +------- + ------- dx dx dx The problem i have is when you Žnd the multiple (2 times)partial derivative of fy/z you get 0 right??? I know the answer is 2xy/z(^3) but i¹m interested in how you achieve this. Thx for any help, Nick === Subject: Re: laplacian problem > Hi everyone, > I¹m a CS student and I am currently doing a maths course. I¹ve got a > question concerning a homework question. Now I know some people get annoyed > if you ask a question about homework but I have spent hours trying to answer > one sum, have looked on INet and still cannot get the answer right. > Question is > Find nabla(triangle)(^2)f where f=xy/z > put otherwise Žnd the laplacian of f where f =xy/z > That is equal to div (grad f) = d(^2)f d(^2)f d(^2)f > ------- +------- + ------- > dx dx > dx This is wrong. d(^2)f d(^2)f d(^2)f del^2 f = ------- + ------- ------ (dx)^2 (dy)^2 ((dz)^2 > The problem i have is when you Žnd the multiple (2 times)partial derivative > of fy/z you get 0 right??? > I know the answer is 2xy/z(^3) > but i¹m interested in how you achieve this. > Thx for any help, > Nick -- -- Geo. Michael Henry No! Bad dog! I said sit! anonymous === Subject: surface integral Can anybody write the integral for this? DobbleIntegral((1+z)*sqrt(1+x^2+y^2),dS) where S is (r*cos(),r*sin(),), 0<<2*Pi, 0 Can anybody write the integral for this? > DobbleIntegral((1+z)*sqrt(1+x^2+y^2),dS) > where > S is (r*cos(),r*sin(),), 0<<2*Pi, 0 Are there analog of hopŽan property for groups in the case of modules > over an arbitrary ring? Yes, you can still deŽne the idea (that all surjective endomorphisms are bijective). It¹s easy to see that any noetherian module is HopŽan: if f:M -> M is surjective with nontrivial kernel, the kernels of f, fof, fofof, ... are a strictly increasing sequence of submodules. I don¹t think too much else is known, constructing more examples: ------- MR1757444 (2001b:16056) Liu, Zhongkui(PRC-NWTE) A note on HopŽan modules. Comm. Algebra 28 (2000), no. 6, 3031--3040. 16W60 (16D80) For a ring $R$, a module $M$ is said to be HopŽan if every surjective endomorphism is an isomorphism, a property well known to be possessed by a Noetherian module. The following theorem is the heart of this note: Let $(S, leq)$ be a strictly totally ordered monoid which is Žnitely generated and such that $0 leq s$ for every $s in S$. Then for a left $R$-module $M$ the following are equivalent: $(1)$ $M$ is a HopŽan $R$-module; $(2)$ $[[M ^{S, leq}]]$ is a left $[[R^{S, leq}]]$-module. The $[[R^{S, leq}]]$ and $[[M^{S, leq}]]$ are appropriately deŽned generalized power series ring and module, respectively. This theorem produces more examples of HopŽan modules. ------ William C. Waterhouse Penn State === Subject: Re: On the partial sums of reciprocals of primes >In Prime Numbers: A computational Perspective by Crandall and > Pomerance >you will Žnd this fact on page 32. Their Theorem 1.4.2 (Mertens) > states >that: >As x -> inf, prod_{p le x}(1 - 1/p) ~ e^{-gamma}/ln(x) >where gamma is the Euler constant. Taking the logarithm of this > relation, we >have, >sum_{p le x}{1/p} = ln(ln(x)) + B + o(1), >for the Mertens constant B deŽned as >B = gamma + sum_{p}{(ln(1 - 1/p) + 1/p) > A lot of people say that the proof of >sum_{p le x}{1/p} = ln(ln(x)) + B + o(1), > can be found in Hardy and Wright. I know that. Please note that > I said > S(x)< lnlnx + B + 1/(logx)^2. > The point is the explicit formula for the upper bound. I think this > inequality is very stronger than the asymptotic formula. If S(x) = log log x + B + o(1) then for some C it¹s also true that S(x) < log log x + C. That sufŽces to make it true that S(x) < log log x + constant + 1/(log x)^2; the last term is irrelevant, isn¹t it? If the result you were interested in said, say, that log log x + B < S(x) < log log x + B + 1/(log x)^2 (same B on both sides) then that would indeed be much stronger than the result proved in H&W. -- Gareth McCaughan sig under construc === Subject: Re: Integral on SO(3) Epigone-thread: fenfremterd >Let f be the map from SO(3) to R, deŽned by f(M) = m_{1,3} where M is >the matrix (m_{i,j}) (1<= i,j <= 3). >I¹d like to know int_{SO(3)} g(M) dM, where dM is the Haar measure on >SO(3), and g(M) = 1 if a < f(M) <= b, 0 otherwise. >In other words, I¹m looking for the distribution of the random >variable f(M) (with respect to the Haar measure on SO(3)). >I think there is a geometric way to simplify the calculation, but I >didn¹t Žnd it. As a general rule, any calculation on SO(3) should be perfomred on SU(2) which is a simply connected double cover. Google found 5000000 pages desciribng this map, and I will not add another one. I will say that SU(2) is the unit sphere in the quaternions, and thus calculations there are much easier. HTH David === Subject: Functional analysis Hi everyone. I¹ve got a little problem which I need help with from all you functional analysts out there. Let D be the open unit disk, L1(D) the Banach space of measurable functions on D and A1(D) the Banach space of analytic functions on D which are also in L1(D). There is a projection S from L1(D) onto A1(D) (via the Bergman kernel, but that¹s not important...). Also for every e >0 there is a projection Te from A1(D) into L1(D) such that the norm of (Te - I) on A1(D) is e (where I is the identity). So, my question is - can you show that there is a sufŽciently small e so that the composition (Te o S) is also a projection? Dan Goodman === Subject: Re: Functional analysis > Hi everyone. I¹ve got a little problem which I need help > with from all you functional analysts out there. > Let D be the open unit disk, L1(D) the Banach space of measurable functions > on D and A1(D) the Banach space of analytic functions on D which are also in > L1(D). > There is a pr Subject: Fourier Analysis and Cross Correlation I am trying to complete my thesis and have run into a bit of a snag and am desperately hoping to Žnd some assistance. My thesis involves a cross correlation and Fourier analysis of river stage, discharge and precipitation data. I, however, am not able to Žgure out how to run the data in excel. I have run it on a linex system but was not able to generate the decent graphs and am now trying run the data on excel but cannot Žgure out how to do it. Any assistance is greatly appreciated. === Subject: Re: Fourier Analysis and Cross Correlation > I am trying to complete my thesis and have run into a bit of a snag and am > desperately hoping to Žnd some assistance. My thesis involves a cross > correlation and Fourier analysis of river stage, discharge and precipitation > data. I, however, am not able to Žgure out how to run the data in excel. > I have run it on a linex system but was not able to generate the decent > graphs and am now trying run the data on excel but cannot Žgure out how to > do it. Any assistance is greatly appreciated. There are some very good (free) graphical packages that run under Linux (which is what I assume you mean). See if you can Žnd a Linux guru. Or you can Google for it. Alternatively, if you have run the cross-correlation under Linux you will have Žles that tabulate the results, no? If so, import those Žles into Excel and then use their graphing facilities (which, IMO, are not as good as you will Žnd for Linux--but maybe they are both adequate and easier for a beginner to use). -- Julian V. Noble Professor Emeritus of Physics jvn@lessspamformother.virginia.edu ^^^^^^^^^^^^^^^^^^ http://galileo.phys.virginia.edu/~jvn/ For there was never yet philosopher that could endure the toothache patiently. -- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1. === Subject: Re: InŽnetely many Irreducible Polynomials? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4O2eDq19672; >How would i show there are inŽnetely many irreducible polynomials in >a Želd, even if it is a Žnite Želd? In fact there ARE inf. many (but obviously with increasing degrees). Take F_2 (Žn. Želd with just 0 and 1 as elements), for every degree there is at least one. See: Lidl/Niederreiter Intro to Fin. Fields MJV === Subject: inverse of sum of two squares and singular matrices with form of kronecker product by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4O2eFC19814; Hey, In fact, I have a question also conceren inverse of two matrices, if x denote that the kronecker product of two matrices , I interest to Žnd the inverse or generalized inverse of the following expression : (T1xT1+T2xT2)^-2 where T1 and T2 are sojection S from L1(D) onto A1(D) (via the > Bergman kernel, but that¹s not important...). Also for every > e >0 there is a projection Te from A1(D) into L1(D) such > that the norm of (Te - I) on A1(D) is e (where I is the identity). Stop right here. If the (norm) distance of two commuting idempotents P and Q on a Banach space is less than 1 then they are equal. (If they are not expected to commute, they will still be similar.) Indeed, Pythagorean formula for idempotents says (whether P and Q commute or not, and it can be checked by automatic writing) (P+Q-I)^2 + (P-Q)^2 = I (think of (P+Q-I)^2 as cos(A)^2 and (P-Q)^2 as sin(A)^2) So, if norm (P-Q)^2 < 1 then P+Q-I is invertible, and (more automatic writing) P*Q-Q*P = (P-Q) * (P+Q-I) leads to the above conclusion. (If they are not expected to commute, they will still be similar.) So, it seems to me that the question below disappears, unless I am missing something. Remark: Similarity and interpolation between projectors, Acta Scientiarum mathematicarum (Szeged) 39(1977), 341-351, from which I copied the above idea of proof. It may (must?) have been known before that. > So, my question is - can you show that there is a > sufŽciently small e so that the composition (Te o S) is also a projection? > Dan Goodman === Subject: Re: Log (Log (x)) >Can anything be said about expressing log(log(x*y)) as function of >log(log(x)) and log(log(y))? >>Yes: it can¹t be done. log(log(x*y)) is _not_ a function of >>log(log(x)) and log(log(y)). (Or more simply put, log(s+t) >>is not a function of log(s) and log(t).) >Since log(s+t)=log(exp(log s))+exp(log t)) , I must be >missing something here. Aargh. Yeah, that¹s probably it, you must be missing something... Sorry, don¹t know what I was thinking. >Similarly, log(log(x*y)) appears to me to be >log(log(exp(exp(log(log(x)))*exp(exp(log(log(y)))))) >(modulo possible mismatching of parentheses). >Lee Rudolph ************************ David C. Ullrich === Subject: Re: 3^n - 2^n and relatives Am 24.05.04 01:00 schrieb Gareth McCaughan: >> I got an unequality, which i¹m unable to analytically to disprove. >> Does anyone know useful related material? >> The inequality, formulating a restriction, arose whily studying >> the question of a certain, primitive assumed loop in the collatz >> (3x+1)-problem. Here it goes: >> 3^L - 1 >> powceil2( 3^L ) <= 2^L* ------- >> 2^L - 1 >> where powceil2 ( x) = smallest power 2^S >= x > According to theorems mentioned at the start of chapter 3 > of Baker¹s Transcendental Number Theory, the following is > true. (...) > So ... if I haven¹t botched the above in any way (which > I probably have), you only need to check a few hundred > thousand values of L. You¹d do that in practice by calculating > log 3 / log 2 with sufŽcient accuracy (plain ol¹ double > precision should be Žne) and looking to see how close to > an integer you can get by multiplying it by an integer > up to a few hundred thousand in size. Gareth et.al - to continue the discussion I invite to join into personal communication respecting the current repost of the newsgroup- charter. I think I have a deŽnite disproof of the above inequality, but also it could be faulty. Gottfried Helms === Subject: Associate/Mathematician Wanted! - Unique opportunity for right person! Associate/Mathematician Wanted! - Unique opportunity for right person! Need very bright and resourceful mathematician to help develop pari-mutuel trading system. QualiŽcations Required for Job: PhD in mathematics, specializing in any of the following project applicable Želds: Mathematical Logic, Set Theory, Probability, Stochastic Analysis, Statistics, Games, Operations Research. Strong programming skills and knowledge of ai methodologies A+! Remuneration will be in form of equity participation. www.paritrader.com Please reply to: ajanalytics@ureach.com === Subject: Re: Maple vs Mathematica / community > recently I answer a question about the Mathematica syntax > and in this posting I mention Richad Fatemans MockMma > I got the posting back with the comment that Richard Fateman > is not a friend of this news-group and I should send my > answer directly to the original poster ... That does seem a rather odd reply. As Dr. Fateman notes, several people at WRI know him and have good relationships with him. He has also posted on the newsgroup several times. Bhuvanesh, Wolfram Research. -- Disclaimer: Any opinions expressed herein are my own and not necessarily those of Wolfram Research. === Subject: Re: What happened for 140 years? I would like to add one more reason (No.9) to the list provided before. Dr.M.Basti Summary of some reasons that for 140 years the research on solving polynomials was untouched with differential equations methods. 1-The notion of non-radical solutions of polynomials of degree n>4. 2- Not having solutions of higher order linear differential equations. 3-No attempt in solving polynomials with Riccati differential equation (i.e. Žrst order). 4- The Žrst stone in solving the quadratic equation was not the second order linear differential equations. 5-No idea of the notion of class polynomials. 6-The language of deriving differential equations was complex, lengthy and involved a lot of maneuvering, particularly with differential operators. 7. No example was provided (even one) outside of the classical solved equations : The quadratic, cubic and quartic. No.8-Possibly they mostly felt the polynomial solutions are primarily within the domain of algebra!, particularly with the advancements of Galois theory. 9- I have Žrst considered solving Riccati differential equations through higher order linear differential equations. Polynomial connections similarly expanded (1985) without my knowledge of the papers by Harley and Cockle. While Harley associates polynomials of the type y^n-n*y +(n-1)*x =0 to higher order linear differential equations (LODE), Initially I assigned Riccati differential equations to LODE, and thus the new techniques of solving Riccati differential equations initiated. Harley never mentioned Riccati differential equations associated with polynomials, which is the cornerstone of nature¹s structure of differential equations and polynomials, so far I have several classes of polynomials with Riccati differential equations. === Subject: Re: On Papers of Cockle (1860) and Harley (1862) As I had mentioned before it took me several months to search for a topic of research for my lifetime research in math. I Žnally summarized that in solving exact solutions of Riccati differential equations. I did about a year of extensive research, but unfortunately I noticed that my methods were not much different that others in broad sense and in fact I was adding each day more equations to my Žle. I sensed that a new technique was required totally different than those of conventional methods. Thus I decided to possibly leave this research and comeback to my previous research on Qualitative theory of ODE. I know that embarking upon fundamentally new areas requires some understanding of philosophy. I never liked philosophy, and if you visit Cambridge (U.K) you see that they have philosophical societies associated with math.. Anyway, I happened to read some of the Russel books to have some understanding about basic philosophy. Good thing about Universities such as Cambridge is that you learn in the environment something about the past masters done research there. It happened that I noticed the concept of Universals in one of the Russell.89s books and this was key to create new methods of Riccati and polynomials. Later on I will explain in my lecture notes how precisely I interpreted the statements of Aristotle.89s universals word by word and translated it into solving Riccati differential equations (later on polynomials) with linear higher order differential equations. I wonder how J.Cockle understood about the subject. I have recently requested a copy from the inter-library loans. Possibly I will let you know about my understanding when I receive the copies. Today more than 20 years after those initial studies of Riccati, I am able to solve very complicated Riccati as well as polynomials with differential equations methods. Thus a new science has been borne. Dr.Mehran Basti === Subject: MAPLE: different algorithms for ADD and SUM computations. I have a remark regarding the MAPLE functions ADD and SUM. It looks like the algorithms for computations of the ADD and SUM are different. The example below shows that although the inert forms of both functions are the same (1+B), the Žnal results are different. I belive that the reason for this is that both functions calculate the 0^0 differently. What I need is to have SUM to behave as ADD. Is there a setting in MAPLE which will force SUM behave as ADD, when it comes to computation of 0^0 ? Michael ---------------------------------------------------- > func:=B^k; k func := B > add_func:=add(func,k=0..1); > sum_func:=sum(func,¹k¹=0..1); B := 0 > add_func:=add(func,k=0..1); > sum_func:=sum(func,¹k¹=0..1); === Subject: Re: MAPLE: different algorithms for ADD and SUM computations. 0^k evaluates to 0, so your difference depends on whether B or k is given a value Žrst... > subs(k=0,subs(B=0,B^k)); 0 > subs(B=0,subs(k=0,B^k)); 1 and that¹s a difference between add and sum. Sometimes you can play with quotes Œ Œ to delay evaluation... > B := 0; B := 0 > sum(B^k,k=0..1); 0 > sum(ŒB^k¹,k=0..1); 1 === Subject: Associate/Mathematician Wanted! - Unique opportunity for right person! Content-Length: 573 Originator: rusin@vesuvius Associate/Mathematician Wanted! - Unique opportunity for right person! Need very bright and resourceful mathematician to help develop pari-mutuel trading system. QualiŽcations Required for Job: PhD in mathematics, specializing in any of the following project applicable Želds: Mathematical Logic, Set Theory, Probability, Stochastic Analysis, Statistics, Games, Operations Research. Strong programming skills and knowledge of ai methodologies A+! Remuneration will be in form of equity participation. www.paritrader.com Please reply to: ajanalytics@ureach.com === Subject: Analysis and Applications - Vol. 2, No. 3 Content-Length: 682 Originator: rusin@vesuvius Analysis and Applications View table-of-contents and abstracts at http://www.worldscinet.com/aa.html Contents: An Analogue Of CowlingPrice¹s Theorem And Hardy¹s Theorem For The Generalized Fourier Transform Associated With The Sphericquares and singular matrices. Also it will be useful if I could know the result of this expression is also kronecker product or not cause I need to get a kronecker product from that for my work. Younis Fathy My mail is: younis.fathy@mail.uni-oldenburg.de === Subject: Re: Dual Functional by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4O2eEQ19754; >I encountered the following statement and would appreciate if someone >could shed some light to my question below. >The ordinary dual functional of a minimization problem satisŽes the >following conditions: >(i) L(y) = inf_x { f(x) + y¹g(x)} , if y_i >= 0 forall i >(ii) L(y) = -inŽnity , otherwise >in which f(x) is the original objective function to minimize, g(x) <= >0 is valid condition for constraint, and y is the Lagrange multiplier. >I presume f(x) and g(x) are convex functions and x is unrestricted in >the original problem. My question is why condition (ii) always leads >to -inŽnity? >this allows you to formulate duality by an unconstrained min_max_characterization Right. And I can understand why y>=0 being feasible dual associated to the constraint g(x)<=0. >like the primal is set to +inŽnity outside the feasible domain. ^^^^^ >this is a standard approach in convex duality theory. When you say set, do you mean it is just a matter of telling a reader that the Žnal value is unknown and can lead to anything which may include a stationary point with an objective value being less than inŽnity but deŽnitely not a solution, or, it really tends to one extreme (inŽnity) ? If the latter is true then I would like some explanation. If I consider only one y_j < 0 with the rest y_i >= 0 for all i != j. Let v = {y_i | for all i != j}, then z(y) = inf_x {[f(x) + v¹g(x)] + [y_j * g_j(x)]} is convex + concave. Now I got stuck. I am not able to fully convince myself that z(y) leads to -inŽnity. >for detailed analysis, see Rockafellars book on convex analysis I¹ve looked into it once. But I was soon drown in the theorems, propositions, corollaries, lemmas, etc that took their turns non-stop though. I might need it again when I am more comfortable with it. === Subject: Re: Dual Functional >>I encountered the following statement and would appreciate if >someone >>could shed some light to my question below. >> >>The ordinary dual functional of a minimization problem satisŽes >the >>following conditions: >> >>(i) L(y) = inf_x { f(x) + y¹g(x)} , if y_i >= 0 forall i >>(ii) L(y) = -inŽnity , otherwise >> >>in which f(x) is the original objective function to minimizal Mean Operator C. Chettaoui, Y. Othmani and K. Trimeche On Isometric Immersions Of A Riemannian Space With Little Regularity Sorin Mardare InŽnite Valued Solutions Of Non-Uniformly Elliptic Problems Dominique Blanchard And Olivier Guibe Homogenization Of Two Heat Conductors With An Interfacial Contact Resistance Patrizia Donato And Sara Monsurro For more information, go to http://www.worldscinet.com/aa.html === Subject: Higher-Dimensional Categories: an illustrated guidebook Originator: baez@math-cl-n03.math.ucr.edu (John Baez) Content-Length: 1609 Originator: rusin@vesuvius Aaron Lauda and Eugenia Cheng have just written a book on n-categories: Higher-Dimensional Categories: an illustrated guidebook http://www.dpmms.cam.ac.uk/~elgc2/guidebook If you¹re curious about the deŽnitions of weak n-category that people are studying these days, this book is *the* place to get started. It¹s a friendly, lucidly written introduction which not only presents these deŽnitions, but explains what makes them tick and compares the philosophies behind different approaches. It aims at transmitting the key intuitions rather than crushing the reader with rigor. It has a sense of humor - something more math books could use. And best of all, it¹s packed with pictures! After a general introduction, it treats: Penon¹s deŽnition of weak omega-categories in terms of magmas Batanin and Leinster¹s deŽnitions in terms of globular operads The Baez/Dolan--Hermida/Power/Makkai--Leinster--Cheng opetopic deŽnitions Simpson and Tamsamani¹s deŽnitions in terms of multisimplicial sets Street¹s deŽnition in terms of simplicial sets Joyal¹s deŽnition in terms of cellular sets Trimble and May¹s deŽnitions in terms of iterated weakened enrichment At the end it has a bunch of charts comparing these deŽnitions in various ways. It even has a fold-out model of the 3d associahedron! The current draft of the book has been prepared just in time for the and Applications: http://www.ima.umn.edu/categories/ The authors welcome comments and will use them to prepare the Žnal version. === Subject: Re: Functional analysis Epigone-thread: skongrendjix Content-Length: 581 Originator: rusin@vesuvius >Let T be an (afŽne) surjective isometry of an afŽne real Hilbert space >If H is Žnite-dimensional, it is easy to prove that either >- T has a Žxed point, or >- |T^n(x)| converges to inŽnity for all xin H. >Does this result hold is H is inŽnite-dimensional? Probably you mean a Žxed point not equal to the zero element in H. If so, the answer for inŽnite dimensional Banach spaces (including Hilbert spaces) is ŒNo¹. Simply consider the left shift operator on l^p for any p in 1 <= p < inŽnity. === Subject: Re: Functional analysis Content-Length: 695 Originator: rusin@vesuvius >Let T be an (afŽne) surjective isometry of an afŽne real Hilbert > space >If H is Žnite-dimensional, it is easy to prove that either >- T has a Žxed point, or >- |T^n(x)| converges to inŽnity for all xin H. >Does this result hold is H is inŽnite-dimensional? > Probably you mean a Žxed point not equal to the zero element in H. Probably not, otherwise my question would be stupid. An afŽne isometry is not supposed to Žx zero! However, my question has been solved in the negative for several months by Romain Tessera. -- Yves === Subject: Re: Functional analysis Content-Length: 797 Originator: rusin@vesuvius > Stop right here. If the (norm) distance of two commuting idempotents P > and Q on a Banach space is less than 1 then they are equal. --- Unfortunately, the two projections I¹m interested in won¹t commute. Basically, my problem is that given a projection S onto the closed subspace A1 of the Banach space L1, if we peturb the subspace by a projection Te which is arbitrarily close to the identity on A1 (but need not be close to the identity away from A1, so it may not be the case that the norm of S-Te is less than 1), can we still Žnd a projection onto Te(A1)? My Žrst thought was that Te*S might work, but now I¹m not sure. > (If they are not expected to commute, e, g(x) ><= >>0 is valid condition for constraint, and y is the Lagrange >multiplier. >> >>I presume f(x) and g(x) are convex functions and x is unrestricted >in >>the original problem. My question is why condition (ii) always >leads >>to -inŽnity? >>this allows you to formulate duality by an unconstrained >min_max_characterization >Right. And I can understand why y>=0 being feasible dual associated >to the constraint g(x)<=0. >>like the primal is set to +inŽnity outside the feasible domain. (deŽned) > ^^^^^ >>this is a standard approach in convex duality theory. >When you say set, do you mean it is just a matter of telling a >reader that the Žnal value is unknown and can lead to anything which >may include a stationary point with an objective value being less than >inŽnity but deŽnitely not a solution, or, it really tends to one >extreme (inŽnity) ? >If the latter is true then I would like some explanation. If I >consider only one y_j < 0 with the rest y_i >= 0 for all i != j. Let v >= {y_i | for all i != j}, then >z(y) = inf_x {[f(x) + v¹g(x)] + [y_j * g_j(x)]} >is convex + concave. Now I got stuck. I am not able to fully convince >myself that z(y) leads to -inŽnity. assume that exists a x with g_j(x)>0, (a nonredundant constraint) and if you allow one y_j<0 then it can be arbitrarily large (negative0 and hence the inf value will be -inŽnity >>for detailed analysis, see Rockafellars book on convex analysis >I¹ve looked into it once. But I was soon drown in the theorems, >propositions, corollaries, lemmas, etc that took their turns non-stop >though. I might need it again when I am more comfortable with it. a much shorter and good readable condensation is in avriels book nonlinear programming ,analysis and methods. chapter 5. hth peter === Subject: Re: Dual Functional by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4PLOZv01486; >>like the primal is set to +inŽnity outside the feasible domain. > (deŽned) > ^^^^^ >>this is a standard approach in convex duality theory. >When you say set, do you mean it is just a matter of telling a >reader that the Žnal value is unknown and can lead to anything which >may include a stationary point with an objective value being less than >inŽnity but deŽnitely not a solution, or, it really tends to one >extreme (inŽnity) ? >If the latter is true then I would like some explanation. If I >consider only one y_j < 0 with the rest y_i >= 0 for all i !they will still be similar.) --- Is there anything that can be said when they don¹t commute? === Subject: Formula for area of hyperbolic tetrahedra Content-Length: 171 Originator: rusin@vesuvius Has anyone worked out a formula for the area of hyperbolic tetrahedra, i.e., a three-dimensional version of the Gauss-Bonnet formula for the area of hyperbolic triangles? = j. Let v >= {y_i | for all i != j}, then >z(y) = inf_x {[f(x) + v¹g(x)] + [y_j * g_j(x)]} >is convex + concave. Now I got stuck. I am not able to fully convince >myself that z(y) leads to -inŽnity. > assume that exists a x with g_j(x)>0, > (a nonredundant constraint) > and if you allow one y_j<0 then it can be > arbitrarily large (negative0 and hence the inf value will be > -inŽnity Ah! That is cheating ;) I know what my problem was now. In my head, I kept obeying g_j(x)<=0 without thinking further that the naive unconstraint minimization wouldn¹t care much if the constraint is redundacy) > a much shorter and good readable condensation is in avriels book > nonlinear programming ,analysis and methods. chapter 5. I¹ll get this book. -cheers! === Subject: Re: Free ( or a tleast cheap ) 3D surface plot for Windows ( possibly CAD related ) > My data to plot inherently comes in values equally spaced on an x,y > grid. The z values will values will have a WIDE dynamic range. I am > looking at experimental data, not analytic functions. > My input arrays will be from 10x1000 to 500x15000, or any permutation in > between. > I¹ve tried Scilab ( in which I do my pre-processing ) and gnuplot. > Both have two problems: > 1. they are not native Windows apps > 2. their primary orientation is towards analytic functions > Is there something from the Computer Aided Drafting realm that I should > investigate? OpenDX is very powerful, but will take a while to learn. Gib === Subject: Re: Free ( or a tleast cheap ) 3D surface plot for Windows ( possibly CAD related ) >> My data to plot inherently comes in values equally spaced on an x,y >> grid. The z values will values will have a WIDE dynamic range. I am >> looking at experimental data, not analytic functions. >> My input arrays will be from 10x1000 to 500x15000, or any permutation >> in between. >> I¹ve tried Scilab ( in which I do my pre-processing ) and gnuplot. >> Both have two problems: >> 1. they are not *native* Windows apps ^^^^^^ |- emphasis added in reply >> 2. their primary orientation is towards analytic functions >> Is there something from the Computer Aided Drafting realm that I >> should investigate? > OpenDX is very powerful, but will take a while to learn. > Gib *BUT* , item 15 of FAQ states *NO* native Windows version ;{ I suspect some of my dissatisfaction with Scilab and gnuplot result from unspeciŽed/unknown/??? assumptions/mindsets. E.G. -- In some cultures head nod means Œyes¹, in others Œno¹. Also, the read me for Windows did not inspire conŽdence. Also, no manual nor speciŽcations available without downloading whole package. I may be paranoid, BUT would I be assaulted by tons of unwelcome advertising for just inquiring if OpenDX would be what I¹m looking for? PS. TNSTAFL == There¹s No Such Thing As a Free Lunch I¹m not sure I want to pay the price of free lunch. { apologies to R.A.H. ) === Subject: Re: Žnite-elemet time-domain method for exterior problems > Hello all > However, for what I propose to do (modelling srbitrary 3D acoustic systems) > I now have doubts about whether BEM is the fastest way to do it. > I am considering an iterative time-domain method in which I split the space > into elements and progressively update parameters for each element (sort of > like a cellular automaton). This way I can exploit locality, making each > step very simple and quick for O(n^4) with a low constant (n being the > resolution in each axis, typically proportional to the number of time steps > required). > A problem arises when my sound Želd is in contact with the outside world, > an inŽnite space that would require inŽnite numbers of elements. It will > not sufŽce to simply present a power-absorbing characteristic impedance > here, since curved sound waves propagate in a more complex manner and this > needs to be accounted for if the model is to be of any use (basically, the > impedance of air drops below the characteristic impedance over distances > much shorter than a wavelength). > I have stumbled across the term perfectly matched layer, but I¹m not sure > how this works, whether it exhibits the property I have just described or > whether it is fast enough. > It seems to me that if I arrange for my space to be a sphere, then the > surface has enough symmetry that I can treat every surface element the same > way. Since the whole system is linear and time-invariant, I can determine > the time-domain response of the entire surface to an excitation at a single > element and then simply convolve over time and surface-position to simulate > the external sound Želd. > However, these convolutions would take longer than the whole of the rest of > the simulation if done in the time domain O(n^6). So I want to invent a > cellular automaton-like rule for the surface elements which will yield the > required response function (and obviously also be linear, time-invariant > and space-invariant over the surface of a sphere). > Has anyone else tried this? Am I on the right track? Or is there another, > even faster approach? There is a large body of work in those problems, although a lot of it (dealing with underwater shock, acoustics and EM) is classiŽed since 1970. For a fairly recent overview of the state of the art in unbounded problems, including papers on the major approaches (BEM, inŽnite elements, PML, DAA, scaled FEM, spectral methods, etc) see 27-31 July 1997 (Fluid Mechanics and Its Applications, V. 49) T. L. Geers, ed., Kluwer Academic Publishers December, 1998 ISBN: 0792352661 Hardcover This title can be found by seaching in http://www.addall.com/ About 90% of the work has been in the frequency domain, in both acoustics and electromagnetics, because of the military importance of the signature detection, shadowing and targetting inverse problems. Oil companies have also worked extensively in the topic of resource detection in soil media from data mining of impulsive records, but of course little of that is published. === Subject: logarithmic regression It¹s about logarithmic regression: i can¹t understand what is the meaning of : log(y) = a + b*log(x). (why log(x)??). and there is the relation y = 10^a * x^b. Ok I can¹t understand where does this relation come from. And then how can i implement this regression? Do i need only to pass under logarithmic function the points, and then simply apply the linear regression? Alberto. === Subject: Re: logarithmic regression >It¹s about logarithmic regression: i can¹t understand what is the meaning of >log(y) = a + b*log(x). (why log(x)??). >and there is the relation >y = 10^a * x^b. log(y) = a*log(10) + b*log(x), aa=a*log(10) just another constant >Ok I can¹t understand where does this relation come from. >And then how can i implement this regression? >Do i need only to pass under logarithmic function the points, and then >simply apply the linear regression? yes, what else? simple linear regression . but don¹t forget: compared to the original nonlinear model, this introduces bias and you might get quite different coefŽcients. hth peter === Subject: Re: logarithmic regression > It¹s about logarithmic regression: i can¹t understand what is the meaning of > log(y) = a + b*log(x). (why log(x)??). > and there is the relation > y = 10^a * x^b. > Ok I can¹t understand where does this relation come from. > And then how can i implement this regression? > Do i need only to pass under logarithmic function the points, and then > simply apply the linear regression? > Alberto. Many data are described by exponential growth/decay. Thus, a radioactive sample diminishes in strength according to N(t) = N(0) * exp( -b*t ) and the intensity of x-rays passing through a thickness x of matter decreases according to I(x) = I(0) * exp( - x/L ) . Other quantities, such as population, typically grow exponentially in time. If one suspects the data are described by such an exponential law, then the taking the logarithm of both sides transforms to a linear relation, y = y(0) * exp( -b*x ) ln(y) = z = a - b * x where a = ln[ y(0) ] , and the new dependent variable is z = ln(y). There are also quantities whose natural relationship is a power law. For example, gravitational force decreases as the inverse second power of distance (Newton¹s Law of Universal Gravitation), and electrostatic force decreases in the same way (Coulomb¹s Law). Suppose we want to check Coulomb¹s Law: we would measure the force between two Žxed charges as we vary the distance and would get a table with two columns, F and D. We believe F = (Q*q) * D^{-2} If we take the logarithm of both sides (base 10, base e --it doesn¹t matter) we get log(F) = Z = A + B * log(D) = A + B * u again a linear relation, with u = log(D), z = log(F), and A = log(Q*q). Although we _expect_ B = -2, the point of the measurement is to check this, so we make a straight-line (linear regression) Žt and determine the best value of B. We will probably Žnd it to be -2 pm epsilon where epsilon is a small quantity that represents how well we can determine B from these data. If we want a better value we need more data or a different kind of experiment, or both. I might add that there was recently some excitement in the physics community because people measuring B for gravity were Žnding something slightly different from -2, which they attributed to a hitherto-unknown Žfth force of nature. You should Google for this topic to see how it turned out. (That¹s your HW assignment, to pay for my explanation.) Finally, you should read an introduction to the ideas of mathematical statistics and probability theory which will explain where these ideas come from and why we do such things as regression Žts. Oddly enough, the Cartoon Guide to Statistics, by Larry Gonick and Woollcott Smith, is a Žrst-rate introduction for the beginner. -- Julian V. Noble Professor Emeritus of Physics jvn@lessspamformother.virginia.edu ^^^^^^^^^^^^^^^^^^ http://galileo.phys.virginia.edu/~jvn/ For there was never yet philosopher that could endure the toothache patiently. -- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1. === Subject: High Dimensonal Data Acquisition, Computing and Visualization Do you know any interesting web links about a problem in the topic? M === Subject: Re: High Dimensonal Data Acquisition, Computing and Visualization > Do you know any interesting web links about a problem in the topic? You might try the journal Computing in Science and Engineering, jointly published by the IEEE Computer Society and the American Institute of Physics. The have had special issues devoted to both these topics in the past two years. -- Julian V. Noble Professor Emeritus of Physics jvn@lessspamformother.virginia.edu ^^^^^^^^^^^^^^^^^^ http://galileo.phys.virginia.edu/~jvn/ For there was never yet philosopher that could endure the toothache patiently. -- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1. === Subject: Re: High Dimensonal Data Acquisition, Computing and Visualization >Do you know any interesting web links about a problem in the topic? A.L. === Subject: Re: Error bound in Cubic Spline cases for non uniform meshes >I need some references for error estimation of cubic spline S with >the given function f where f is in C^4. Cubic spline may be of >spline Type-1 or Type-2 or Periodic spline where as the meshes >are non-uniform. Any suggestion regarding the query is welcome. >Thanx. >akbchand this is contained already in the Ahlberg-Nilson-Walsh book theory and applications of spline functionsin chapter 2. uniformity of the mesh is not required. hth peter === Subject: Re: Solving a very large non-sparse linear system >Do anyone know on how to solve a linear system Ax=b >with dimension of A equal to 100000x100000 or even larger? >I am glad to hear your comments on this. >Larry you write this under the title nonsparse. If it is really dense, then for me the main question will be how to access these data? the matrix will necessarily reside on a backing store, and then the the number of accesses to this backing store will determine the total time much more than any other fact. (if you consider an iterative solver, then typically any single step will require to access every matrix element at least once per step, this sounds no good for me. and if you consider a direct solver, then a block wise access will be necessary to make this efŽcient. but even if everything would reside in main memory: 10^15/3 žoat ops will require parallel processing to be completed in reasonable time. but maybe and hopefully by nonsparse you meant not completely dense. typically, an iterative scheme plus a good preconditioner must be applied. but to give useful advice, one needs to know much more about your matrix than large and nonsparse. so tell us: where does it come from? hth peter === Subject: Determinant of a large sparse matrix X-AUTHid: lawhiu What is the best way to compute the determinant of a large, sparse asymmetric matrix? The Žrst thing that comes into my mind is to perform a sparse LU decomposition, but would that be inefŽcient? If I adopt instead the incomplete LU decomposition, will the approximation introduced lead to signiŽcant error in the determinant? The context is this. I have a variable n by n matrix X that I want to optimize with small rank of X. So X is usually singular. According to http://www.cds.caltech.edu/~maryam/acc03_Žnal.pdf this can be achieved by the heuristic log(det(X + delta I)), where delta is a small number that seems to require some care to select. That¹s why I may need to compute the determinant of a large, sparse matrix. P.S. My email address is lawhiu at cse dot msu dot edu. === Subject: Maximization Formula help. Hi All, I was wondering if anyone could help with this formula. Sorry for my ignorance with your syntax. I have a list of lengths (rows). I need to maximize those lengths with units of 96 & 72 while leaving no length of less that 12 units in any row. I need to then put the waste back into the formula and the maximize the next length (row). The rows also need to be staggered with no rows have a seam within 24 units of the seams in the row above or below. Any help would be greatly appreciated. I am writting this in VBA. Paul An example list would include length such as: Not to scale: Random seam pattern 300 = | x | x | x | x | x | 300 = | x | x | x | x | x | 300 = | x | x | x | x | x | === Subject: Re: Estimating min/max eigenvalue of a matrix > Formally (double meaning) perhaps, but in reality this is tautological > ^^^^^^^^^^^^ > nonsense. > ^^^^^^^^ > > Perhaps you would care to clarify that rather offensive remark? > > If you found it offensive then you¹ve a problem that I can¹t help you with > but here¹s a heads up: http://www.mhsanctuary.com/narcissistic/dsm.htm. Not > to worry, it¹s a condition endemic among academics and well parodied in > Huxley¹s ŒPoint Counter Point¹, which you ought to read for some layman > insight into the condition. > -- > E&OE > Ciao, > Gerry T. > Stick to topics of the newsgroup instead of trying to justify your > abrasiveness. > -- Lou Pecora This isn¹t Abu Ghraib, pte. Trick or treat: http://www.frankmagazine.ca/ -- E&OE Ciao, Gerry T. ______ The enemy is anybody who¹s going to get you killed, no matter which side he¹s on. -- Yossarian, in Catch 22. === Subject: Re: Estimating min/max eigenvalue of a matrix > > Let t = the trace of the matrix = the sum of its eigenvalues, and u = > the trace of the square of the matrix = the sum of squares of all its > elements = the sum of squares of its eigenvalues. Then m = t/n is the > average eigenvalue, s = sqrt(n*u - t^2)/n is the standard deviation of > the eigenvalues, and the eigenvalues are bounded by m +- s*sqrt(n-1). > > Since there is nothing to prevent the lower bound from being negative, > in practice the upper bound tends to be more useful than the lower. > The trace of the square of the matrix is O(n^2). As I said above, > I don¹t think it is possible to Žnd even rough bounds for general > pos.-def. matrices more cheaply than this. Both the Schmidt norm > and Tiktopoulos¹ norm are O(n^2). The latter tends to be a better > estimate of the largest eigenvalue, however. What you think about the use of Brauer ovals of Cassini? Some that is not a detonator for other question outside the topic (sig!)). === Subject: Psedo inverse by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4P52ol02584; does anybody know how can I Žnd the matrix Pseudo inverse or SVD C code? === Subject: Re: Psedo inverse >does anybody know how can I Žnd the matrix Pseudo inverse or SVD C >code? http://www.netlib.org/clapack or http://www.netlib.org/c/meschach hth peter === Subject: Re: Bland and Altman Plot by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4P52om02570; >Can you and does it make sense to compute a correlation coefŽcient >for data you plot in a Bland Altman plot? On the plots you get a mean >difference score and a CI, but I do not see a correlation coefŽcient. Can you specify your question more clearly. Muthu === Subject: Applications of the Distance from a Point to a Curve by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4PDf2t17285; I would like to know if anyone knows about (real) situations where it is necessary to compute the euclidean distance from a point to a curve/surface (or even the point where it is reached), you may not assume that the point is very close to the curve (unlike the usual Žtting problems). I have special interest in applications to phisics. Dimas === Subject: Re: Applications of the Distance from a Point to a Curve >I would like to know if anyone knows about (real) situations where it >is >necessary to compute the euclidean distance from a point to a >curve/surface (or even the point where it is reached), you may not >assume that the point is very close to the curve (unlike the usual >Žtting problems). >I have special interest in applications to phisics. >Dimas what about gravitational force induced by a planet in connection with a landing manoeuvre ? hth peter === Subject: Re: Applications of the Distance from a Point to a Curve >I would like to know if anyone knows about (real) situations where it >is >necessary to compute the euclidean distance from a point to a >curve/surface (or even the point where it is reached), you may not >assume that the point is very close to the curve (unlike the usual >Žtting problems). > >I have special interest in applications to phisics. > > >Dimas > > what about gravitational force induced by a planet in connection with a > landing manoeuvre ? > hth > peter Unless the planet has a very peculiar shape it is only the distance to the center that is of interest. Isaac Newton spent a lot of time proving that. A physics application perhaps more useful than landing on the planet Cowpatia might be Žnding the electrostatic potential arising from a given charge distribution on some arbitrary surface, using the principle of superposition. -- Julian V. Noble Professor Emeritus of Physics jvn@lessspamformother.virginia.edu ^^^^^^^^^^^^^^^^^^ http://galileo.phys.virginia.edu/~jvn/ For there was never yet philosopher that could endure the toothache patiently. -- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1. === Subject: Re: Minimization on Positive SemideŽnite Problem by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i4PLOYE01453; >A positive semideŽnite matrix isn¹t enough to ensure that a minimum >exists; take a look at that second 1-D example again. >If you want to be safe, then instead of solving Ax = b you could solve >(A + epsilon * I) * x = b, where epsilon is small enough that it won¹t add >much error to your solution but large enough that it won¹t be drowned out >by žoating-point error. That will give you a positive deŽnite matrix, >and if your original problem had a hyperplane of minimizers then this new >problem should pick out the point on that hyperplane closest to 0. I did a small test and it turns out quite good except that the elements of the inverse get magniŽed with reciprocal of epsilon. I wonder if the perturbation performed through individual epsilon rather than homogeneous value would help in the accuracy in anyway. >Screwing with your matrix that way will obviously introduce some error, >but you could get some feel for the magnitude of that error by solving: >(A + epsilon * I) * x1 = b >(A + epsilon * I) * x2 = b + epsilon * x1 >(A + epsilon * I) * x3 = b + epsilon * x2 >etc, and you could set an error tolerance for this as well as for the >error in your iterative solver. I am quite unsure of how this error-measuring mechanism works. Could please point me some references? === Subject: Re: Breaking up pseudoinverse >For the problem > >A^t.A.x = A^t.b > >is there any generally valid way to break A^t.A up into a >smaller/sparser matrix - what I mean is, can you put an error bound on >the damage you¹re doing to x for altering A^t.A in some way. > >Eg. by zeroing small elements or even simply chopping it into, say 4x4 >blocks, and solving each block separately (essentially zeroing >elements that don¹t Žt the block structure)? > yes clearly, apply the perturbation theorem for the solution of linear > systems, which is knwon in norms or in componentwise representation. > (look up Golub & van Loan or other standard textbooks on numerical linear > algebra for that) its logical conclusion, I¹ve been wondering about using a diagonal matrix approximation to A^t.A, then inverting the diagonal matrix. Some numerical experiments with randomly generated matrices show that it¹s not a good approximation in those cases, however for other matrices (that perform a convolution) it does seem to be adequate. I don¹t have a good feel for what A^t.A should look like though - obviously symmetric, but what about the diagonal thing? === Subject: Re: difference? > what is difference between gauss-jordan and matrix? > are they used with the same technique? > i dont see any difference My guess is that you have not received any replies because, forgive me, your question makes no sense. Gauss - Jordan is a technique for manipulating matrices. Maybe you could give an example of what is bothering you? -- Paul Sperry Columbia, SC (USA) === Subject: Re: need urgent help on this question.... someone kind enough, pls kindly help/guide me to solve the inverse laplace of ln(1+1/s) using taylor series s at inŽnte expansion/power series of ln(1+t). any guidance is greatly appreciated. answer is t(1-e^-t) using inversion complex contour. > Hi there, > I need urgent help on the following question. > 1. Expand ln(1 + 1/s) in a power series about s = inŽnite. I tried using > taylor but really have great problem in evaluating please help... > URGENT. === Subject: Re: need urgent help on this question.... > someone kind enough, > pls kindly help/guide me > to solve the inverse laplace of ln(1+1/s) using taylor series > s at inŽnte > expansion/power series of ln(1+t). > any guidance is greatly appreciated. > answer is t(1-e^-t) using inversion complex contour. > Hi there, > I need urgent help on the following question. > 1. Expand ln(1 + 1/s) in a power series about s = inŽnite. I tried using > taylor but really have great problem in evaluating please help... > URGENT. The given answer is incorrect. ln(1 + x) = x - x^2/2 + x^3/3 - x^4/4 + ... ln(1 + 1/s) = 1/s - 1/(2*s^2) + 1/(3*s^3) - 1/(4*s^4) + ... Which has inverse Laplace 1 = (1/2!)*t + (1/3!)*t^2 - (1/4!)*t^3 + .... = (1/t)*(t - (1/2!)*t^2 + (1/3!)*t^3 - (1/4!)*t^4 + ...) = (-1/t)*(-1 + (1 - t + (1/2!)*t^2 - (1/3!)*t^3 + (1/4!)*t^4 + ...)) = (-1/t)*(-1 + exp(-t)) = (1 - exp(-t))/t. I¹ll leave it to you to keep track of the interval of convergence. -- Paul Sperry Columbia, SC (USA) === Subject: residual correlated with dependent variable doing a review problem and would appreciate any help suppose you have the model Yi = B1 + B2Xi + ui and the estimator of the residual, ui_hat = a1 + a2Yi + ei how can u Žnd the estimator of a2? i.e. a2_hat? === Subject: matlab can somebody help me graph f(x,y) = 3/2xy for 0<=x<=2, 0<=y<=2, x+y<=2 in matlab -- Hi... - cOS long Œat¹ myvrsix(d0t)com === Subject: Re: i am struck so any help or hint or suggestion is kindly appreciated... > Hi there, > Perhaps you can help me with the following, please do it fast as my deadline > is tomorrow. > 1. Inverse Laplace transforms can often be evaluated in several different > ways. Find > L?1{ln(1 + 1/s)} using each of the following approaches. > (a) Expand ln(1 + 1/s) in a power series about s = inŽnite, and take > inverse Laplace > transforms term by term. (This is allowable provided the series is uniformly > convergent, which it is for |s| > 1.) > Please help me with this asap. > Yours sincerely, > Lim Yoong Ping === Subject: Re: > i am struck so any help or hint or suggestion is kindly appreciated... > Hi there, > Perhaps you can help me with the following, please do it fast as my > deadline > is tomorrow. > 1. Inverse Laplace transforms can often be evaluated in several different > ways. Find > L?1{ln(1 + 1/s)} using each of the following approaches. > (a) Expand ln(1 + 1/s) in a power series about s = inŽnite, and take > inverse Laplace > transforms term by term. (This is allowable provided the series is > uniformly > convergent, which it is for |s| > 1.) > Please help me with this asap. > Yours sincerely, > Lim Yoong Ping [...] > The given answer is incorrect. > ln(1 + x) = x - x^2/2 + x^3/3 - x^4/4 + ... > ln(1 + 1/s) = 1/s - 1/(2*s^2) + 1/(3*s^3) - 1/(4*s^4) + ... > Which has inverse Laplace > 1 = (1/2!)*t + (1/3!)*t^2 - (1/4!)*t^3 + .... = > (1/t)*(t - (1/2!)*t^2 + (1/3!)*t^3 - (1/4!)*t^4 + ...) = > (-1/t)*(-1 + (1 - t + (1/2!)*t^2 - (1/3!)*t^3 + (1/4!)*t^4 + ...)) = > (-1/t)*(-1 + exp(-t)) = (1 - exp(-t))/t. > I¹ll leave it to you to keep track of the interval of convergence. -- Paul Sperry Columbia, SC (USA) === Subject: laplacian problem Hi everyone, I¹m a CS student and I am currently doing a maths course. I¹ve got a question concerning a homework question. Now I know some people get annoyed if you ask a question about homework but I have spent hours trying to answer one sum, have looked on INet and still cannot get the answer right. Question is Find nabla(triangle)(^2)f where f=xy/z put otherwise Žnd the laplacian of f where f =xy/z That is equal to div (grad f) = d(^2)f d(^2)f d(^2)f ------- +------- + ------- dx dx dx The problem i have is when you Žnd the multiple (2 times)partial derivative of fy/z you get 0 right??? I know the answer is 2xy/z(^3) but i¹m interested in how you achieve this. Thx for any help, Nick === Subject: Re: laplacian problem > Hi everyone, > I¹m a CS student and I am currently doing a maths course. I¹ve got a > question concerning a homework question. Now I know some people get annoyed > if you ask a question about homework but I have spent hours trying to answer > one sum, have looked on INet and still cannot get the answer right. > Question is > Find nabla(triangle)(^2)f where f=xy/z > put otherwise Žnd the laplacian of f where f =xy/z > That is equal to div (grad f) = d(^2)f d(^2)f d(^2)f > ------- +------- + ------- > dx dx > dx This is wrong. d(^2)f d(^2)f d(^2)f del^2 f = ------- + ------- ------ (dx)^2 (dy)^2 ((dz)^2 > The problem i have is when you Žnd the multiple (2 times)partial derivative > of fy/z you get 0 right??? > I know the answer is 2xy/z(^3) > but i¹m interested in how you achieve this. > Thx for any help, > Nick -- -- Geo. Michael Henry No! Bad dog! I said sit! anonymous === Subject: surface integral Can anybody write the integral for this? DobbleIntegral((1+z)*sqrt(1+x^2+y^2),dS) where S is (r*cos(),r*sin(),), 0<<2*Pi, 0 Can anybody write the integral for this? > DobbleIntegral((1+z)*sqrt(1+x^2+y^2),dS) > where > S is (r*cos(),r*sin(),), 0<<2*Pi, 0 Are there analog of hopŽan property for groups in the case of modules > over an arbitrary ring? Yes, you can still deŽne the idea (that all surjective endomorphisms are bijective). It¹s easy to see that any noetherian module is HopŽan: if f:M -> M is surjective with nontrivial kernel, the kernels of f, fof, fofof, ... are a strictly increasing sequence of submodules. I don¹t think too much else is known, constructing more examples: ------- MR1757444 (2001b:16056) Liu, Zhongkui(PRC-NWTE) A note on HopŽan modules. Comm. Algebra 28 (2000), no. 6, 3031--3040. 16W60 (16D80) For a ring $R$, a module $M$ is said to be HopŽan if every surjective endomorphism is an isomorphism, a property well known to be possessed by a Noetherian module. The following theorem is the heart of this note: Let $(S, leq)$ be a strictly totally ordered monoid which is Žnitely generated and such that $0 leq s$ for every $s in S$. Then for a left $R$-module $M$ the following are equivalent: $(1)$ $M$ is a HopŽan $R$-module; $(2)$ $[[M ^{S, leq}]]$ is a left $[[R^{S, leq}]]$-module. The $[[R^{S, leq}]]$ and $[[M^{S, leq}]]$ are appropriately deŽned generalized power series ring and module, respectively. This theorem produces more examples of HopŽan modules. ------ William C. Waterhouse Penn State === Subject: Re: On the partial sums of reciprocals of primes >In Prime Numbers: A computational Perspective by Crandall and > Pomerance >you will Žnd this fact on page 32. Their Theorem 1.4.2 (Mertens) > states >that: >As x -> inf, prod_{p le x}(1 - 1/p) ~ e^{-gamma}/ln(x) >where gamma is the Euler constant. Taking the logarithm of this > relation, we >have, >sum_{p le x}{1/p} = ln(ln(x)) + B + o(1), >for the Mertens constant B deŽned as >B = gamma + sum_{p}{(ln(1 - 1/p) + 1/p) > A lot of people say that the proof of >sum_{p le x}{1/p} = ln(ln(x)) + B + o(1), > can be found in Hardy and Wright. I know that. Please note that > I said > S(x)< lnlnx + B + 1/(logx)^2. > The point is the explicit formula for the upper bound. I think this > inequality is very stronger than the asymptotic formula. If S(x) = log log x + B + o(1) then for some C it¹s also true that S(x) < log log x + C. That sufŽces to make it true that S(x) < log log x + constant + 1/(log x)^2; the last term is irrelevant, isn¹t it? If the result you were interested in said, say, that log log x + B < S(x) < log log x + B + 1/(log x)^2 (same B on both sides) then that would indeed be much stronger than the result proved in H&W. -- Gareth McCaughan sig under construc === Subject: Re: Integral on SO(3) Epigone-thread: fenfremterd >Let f be the map from SO(3) to R, deŽned by f(M) = m_{1,3} where M is >the matrix (m_{i,j}) (1<= i,j <= 3). >I¹d like to know int_{SO(3)} g(M) dM, where dM is the Haar measure on >SO(3), and g(M) = 1 if a < f(M) <= b, 0 otherwise. >In other words, I¹m looking for the distribution of the random >variable f(M) (with respect to the Haar measure on SO(3)). >I think there is a geometric way to simplify the calculation, but I >didn¹t Žnd it. As a general rule, any calculation on SO(3) should be perfomred on SU(2) which is a simply connected double cover. Google found 5000000 pages desciribng this map, and I will not add another one. I will say that SU(2) is the unit sphere in the quaternions, and thus calculations there are much easier. HTH David === Subject: Functional analysis Hi everyone. I¹ve got a little problem which I need help with from all you functional analysts out there. Let D be the open unit disk, L1(D) the Banach space of measurable functions on D and A1(D) the Banach space of analytic functions on D which are also in L1(D). There is a projection S from L1(D) onto A1(D) (via the Bergman kernel, but that¹s not important...). Also for every e >0 there is a projection Te from A1(D) into L1(D) such that the norm of (Te - I) on A1(D) is e (where I is the identity). So, my question is - can you show that there is a sufŽciently small e so that the composition (Te o S) is also a projection? Dan Goodman === Subject: Re: Functional analysis > Hi everyone. I¹ve got a little problem which I need help > with from all you functional analysts out there. > Let D be the open unit disk, L1(D) the Banach space of measurable functions > on D and A1(D) the Banach space of analytic functions on D which are also in > L1(D). > There is a projection S from L1(D) onto A1(D) (via the > Bergman kernel, but that¹s not important...). Also for every > e >0 there is a projection Te from A1(D) into L1(D) such > that the norm of (Te - I) on A1(D) is e (where I is the identity). Stop right here. If the (norm) distance of two commuting idempotents P and Q on a Banach space is less than 1 then they are equal. (If they are not expected to commute, they will still be similar.) Indeed, Pythagorean formula for idempotents says (whether P and Q commute or not, and it can be checked by automatic writing) (P+Q-I)^2 + (P-Q)^2 = I (think of (P+Q-I)^2 as cos(A)^2 and (P-Q)^2 as sin(A)^2) So, if norm (P-Q)^2 < 1 then P+Q-I is invertible, and (more automatic writing) P*Q-Q*P = (P-Q) * (P+Q-I) leads to the above conclusion. (If they are not expected to commute, they will still be similar.) So, it seems to me that the question below disappears, unless I am missing something. Remark: Similarity and interpolation between projectors, Acta Scientiarum mathematicarum (Szeged) 39(1977), 341-351, from which I copied the above idea of proof. It may (must?) have been known before that. > So, my question is - can you show that there is a > sufŽciently small e so that the composition (Te o S) is also a projection? > Dan Goodman === Subject: Re: Log (Log (x)) >Can anything be said about expressing log(log(x*y)) as function of >log(log(x)) and log(log(y))? >>Yes: it can¹t be done. log(log(x*y)) is _not_ a function of >>log(log(x)) and log(log(y)). (Or more simply put, log(s+t) >>is not a function of log(s) and log(t).) >Since log(s+t)=log(exp(log s))+exp(log t)) , I must be >missing something here. Aargh. Yeah, that¹s probably it, you must be missing something... Sorry, don¹t know what I was thinking. >Similarly, log(log(x*y)) appears to me to be >log(log(exp(exp(log(log(x)))*exp(exp(log(log(y)))))) >(modulo possible mismatching of parentheses). >Lee Rudolph ************************ David C. Ullrich === Subject: Re: 3^n - 2^n and relatives Am 24.05.04 01:00 schrieb Gareth McCaughan: >> I got an unequality, which i¹m unable to analytically to disprove. >> Does anyone know useful related material? >> The inequality, formulating a restriction, arose whily studying >> the question of a certain, primitive assumed loop in the collatz >> (3x+1)-problem. Here it goes: >> 3^L - 1 >> powceil2( 3^L ) <= 2^L* ------- >> 2^L - 1 >> where powceil2 ( x) = smallest power 2^S >= x > According to theorems mentioned at the start of chapter 3 > of Baker¹s Transcendental Number Theory, the following is > true. (...) > So ... if I haven¹t botched the above in any way (which > I probably have), you only need to check a few hundred > thousand values of L. You¹d do that in practice by calculating > log 3 / log 2 with sufŽcient accuracy (plain ol¹ double > precision should be Žne) and looking to see how close to > an integer you can get by multiplying it by an integer > up to a few hundred thousand in size. Gareth et.al - to continue the discussion I invite to join into personal communication respecting the current repost of the newsgroup- charter. I think I have a deŽnite disproof of the above inequality, but also it could be faulty. Gottfried Helms === Subject: Associate/Mathematician Wanted! - Unique opportunity for right person! Associate/Mathematician Wanted! - Unique opportunity for right person! Need very bright and resourceful mathematician to help develop pari-mutuel trading system. QualiŽcations Required for Job: PhD in mathematics, specializing in any of the following project applicable Želds: Mathematical Logic, Set Theory, Probability, Stochastic Analysis, Statistics, Games, Operations Research. Strong programming skills and knowledge of ai methodologies A+! Remuneration will be in form of equity participation. www.paritrader.com Please reply to: ajanalytics@ureach.com === Subject: Re: Maple vs Mathematica / community > recently I answer a question about the Mathematica syntax > and in this posting I mention Richad Fatemans MockMma > I got the posting back with the comment that Richard Fateman > is not a friend of this news-group and I should send my > answer directly to the original poster ... That does seem a rather odd reply. As Dr. Fateman notes, several people at WRI know him and have good relationships with him. He has also posted on the newsgroup several times. Bhuvanesh, Wolfram Research. -- Disclaimer: Any opinions expressed herein are my own and not necessarily those of Wolfram Research. === Subject: Re: What happened for 140 years? I would like to add one more reason (No.9) to the list provided before. Dr.M.Basti Summary of some reasons that for 140 years the research on solving polynomials was untouched with differential equations methods. 1-The notion of non-radical solutions of polynomials of degree n>4. 2- Not having solutions of higher order linear differential equations. 3-No attempt in solving polynomials with Riccati differential equation (i.e. Žrst order). 4- The Žrst stone in solving the quadratic equation was not the second order linear differential equations. 5-No idea of the notion of class polynomials. 6-The language of deriving differential equations was complex, lengthy and involved a lot of maneuvering, particularly with differential operators. 7. No example was provided (even one) outside of the classical solved equations : The quadratic, cubic and quartic. No.8-Possibly they mostly felt the polynomial solutions are primarily within the domain of algebra!, particularly with the advancements of Galois theory. 9- I have Žrst considered solving Riccati differential equations through higher order linear differential equations. Polynomial connections similarly expanded (1985) without my knowledge of the papers by Harley and Cockle. While Harley associates polynomials of the type y^n-n*y +(n-1)*x =0 to higher order linear differential equations (LODE), Initially I assigned Riccati differential equations to LODE, and thus the new techniques of solving Riccati differential equations initiated. Harley never mentioned Riccati differential equations associated with polynomials, which is the cornerstone of nature¹s structure of differential equations and polynomials, so far I have several classes of polynomials with Riccati differential equations. === Subject: Re: On Papers of Cockle (1860) and Harley (1862) As I had mentioned before it took me several months to search for a topic of research for my lifetime research in math. I Žnally summarized that in solving exact solutions of Riccati differential equations. I did about a year of extensive research, but unfortunately I noticed that my methods were not much different that others in broad sense and in fact I was adding each day more equations to my Žle. I sensed that a new technique was required totally different than those of conventional methods. Thus I decided to possibly leave this research and comeback to my previous research on Qualitative theory of ODE. I know that embarking upon fundamentally new areas requires some understanding of philosophy. I never liked philosophy, and if you visit Cambridge (U.K) you see that they have philosophical societies associated with math.. Anyway, I happened to read some of the Russel books to have some understanding about basic philosophy. Good thing about Universities such as Cambridge is that you learn in the environment something about the past masters done research there. It happened that I noticed the concept of Universals in one of the Russell.89s books and this was key to create new methods of Riccati and polynomials. Later on I will explain in my lecture notes how precisely I interpreted the statements of Aristotle.89s universals word by word and translated it into solving Riccati differential equations (later on polynomials) with linear higher order differential equations. I wonder how J.Cockle understood about the subject. I have recently requested a copy from the inter-library loans. Possibly I will let you know about my understanding when I receive the copies. Today more than 20 years after those initial studies of Riccati, I am able to solve very complicated Riccati as well as polynomials with differential equations methods. Thus a new science has been borne. Dr.Mehran Basti === Subject: MAPLE: different algorithms for ADD and SUM computations. I have a remark regarding the MAPLE functions ADD and SUM. It looks like the algorithms for computations of the ADD and SUM are different. The example below shows that although the inert forms of both functions are the same (1+B), the Žnal results are different. I belive that the reason for this is that both functions calculate the 0^0 differently. What I need is to have SUM to behave as ADD. Is there a setting in MAPLE which will force SUM behave as ADD, when it comes to computation of 0^0 ? Michael ---------------------------------------------------- > func:=B^k; k func := B > add_func:=add(func,k=0..1); > sum_func:=sum(func,¹k¹=0..1); B := 0 > add_func:=add(func,k=0..1); > sum_func:=sum(func,¹k¹=0..1); === Subject: Re: MAPLE: different algorithms for ADD and SUM computations. 0^k evaluates to 0, so your difference depends on whether B or k is given a value Žrst... > subs(k=0,subs(B=0,B^k)); 0 > subs(B=0,subs(k=0,B^k)); 1 and that¹s a difference between add and sum. Sometimes you can play with quotes Œ Œ to delay evaluation... > B := 0; B := 0 > sum(B^k,k=0..1); 0 > sum(ŒB^k¹,k=0..1); 1 === Subject: Associate/Mathematician Wanted! - Unique opportunity for right person! Content-Length: 573 Originator: rusin@vesuvius Associate/Mathematician Wanted! - Unique opportunity for right person! Need very bright and resourceful mathematician to help develop pari-mutuel trading system. QualiŽcations Required for Job: PhD in mathematics, specializing in any of the following project applicable Želds: Mathematical Logic, Set Theory, Probability, Stochastic Analysis, Statistics, Games, Operations Research. Strong programming skills and knowledge of ai methodologies A+! Remuneration will be in form of equity participation. www.paritrader.com Please reply to: ajanalytics@ureach.com === Subject: Analysis and Applications - Vol. 2, No. 3 Content-Length: 682 Originator: rusin@vesuvius Analysis and Applications View table-of-contents and abstracts at http://www.worldscinet.com/aa.html Contents: An Analogue Of CowlingPrice¹s Theorem And Hardy¹s Theorem For The Generalized Fourier Transform Associated With The Spherical Mean Operator C. Chettaoui, Y. Othmani and K. Trimeche On Isometric Immersions Of A Riemannian Space With Little Regularity Sorin Mardare InŽnite Valued Solutions Of Non-Uniformly Elliptic Problems Dominique Blanchard And Olivier Guibe Homogenization Of Two Heat Conductors With An Interfacial Contact Resistance Patrizia Donato And Sara Monsurro For more information, go to http://www.worldscinet.com/aa.html === Subject: Higher-Dimensional Categories: an illustrated guidebook Originator: baez@math-cl-n03.math.ucr.edu (John Baez) Content-Length: 1609 Originator: rusin@vesuvius Aaron Lauda and Eugenia Cheng have just written a book on n-categories: Higher-Dimensional Categories: an illustrated guidebook http://www.dpmms.cam.ac.uk/~elgc2/guidebook If you¹re curious about the deŽnitions of weak n-category that people are studying these days, this book is *the* place to get started. It¹s a friendly, lucidly written introduction which not only presents these deŽnitions, but explains what makes them tick and compares the philosophies behind different approaches. It aims at transmitting the key intuitions rather than crushing the reader with rigor. It has a sense of humor - something more math books could use. And best of all, it¹s packed with pictures! After a general introduction, it treats: Penon¹s deŽnition of weak omega-categories in terms of magmas Batanin and Leinster¹s deŽnitions in terms of globular operads The Baez/Dolan--Hermida/Power/Makkai--Leinster--Cheng opetopic deŽnitions Simpson and Tamsamani¹s deŽnitions in terms of multisimplicial sets Street¹s deŽnition in terms of simplicial sets Joyal¹s deŽnition in terms of cellular sets Trimble and May¹s deŽnitions in terms of iterated weakened enrichment At the end it has a bunch of charts comparing these deŽnitions in various ways. It even has a fold-out model of the 3d associahedron! The current draft of the book has been prepared just in time for the and Applications: http://www.ima.umn.edu/categories/ The authors welcome comments and will use them to prepare the Žnal version. === Subject: Re: Functional analysis Epigone-thread: skongrendjix Content-Length: 581 Originator: rusin@vesuvius >Let T be an (afŽne) surjective isometry of an afŽne real Hilbert space >If H is Žnite-dimensional, it is easy to prove that either >- T has a Žxed point, or >- |T^n(x)| converges to inŽnity for all xin H. >Does this result hold is H is inŽnite-dimensional? Probably you mean a Žxed point not equal to the zero element in H. If so, the answer for inŽnite dimensional Banach spaces (including Hilbert spaces) is ŒNo¹. Simply consider the left shift operator on l^p for any p in 1 <= p < inŽnity. === Subject: Re: Functional analysis Content-Length: 695 Originator: rusin@vesuvius >Let T be an (afŽne) surjective isometry of an afŽne real Hilbert > space >If H is Žnite-dimensional, it is easy to prove that either >- T has a Žxed point, or >- |T^n(x)| converges to inŽnity for all xin H. >Does this result hold is H is inŽnite-dimensional? > Probably you mean a Žxed point not equal to the zero element in H. Probably not, otherwise my question would be stupid. An afŽne isometry is not supposed to Žx zero! However, my question has been solved in the negative for several months by Romain Tessera. -- Yves === Subject: Re: Functional analysis Content-Length: 797 Originator: rusin@vesuvius > Stop right here. If the (norm) distance of two commuting idempotents P > and Q on a Banach space is less than 1 then they are equal. --- Unfortunately, the two projections I¹m interested in won¹t commute. Basically, my problem is that given a projection S onto the closed subspace A1 of the Banach space L1, if we peturb the subspace by a projection Te which is arbitrarily close to the identity on A1 (but need not be close to the identity away from A1, so it may not be the case that the norm of S-Te is less than 1), can we still Žnd a projection onto Te(A1)? My Žrst thought was that Te*S might work, but now I¹m not sure. > (If they are not expected to commute, they will still be similar.) --- Is there anything that can be said when they don¹t commute? === Subject: Formula for area of hyperbolic tetrahedra Content-Length: 171 Originator: rusin@vesuvius Has anyone worked out a formula for the area of hyperbolic tetrahedra, i.e., a three-dimensional version of the Gauss-Bonnet formula for the area of hyperbolic triangles?