mm-3109 === > Could someone tell me, in general terms, since I'm not really a > mathematician -- > if: > 1) one were to be thinking about a specific topic, and > 2) then within seconds, randomly pick up a 1500 page book that > addresses this topic only once in a couple paragraphs only, > 3) open it to the exact page > 3) and then, to specifically look at the exact place within the page > that the topic exists > I know you probably don't have all the data to make any kind of > analysis, but what are the ballpark chances of something like that > happening? Or, what are the minimum chances of it happening? I'd suggest ignoring the second #3 (which you probably intended to be #4...), because if there's a particular thing you're looking for and you look at a page that has that on it, then your attention may be drawn to the relevant bit subconsciously. Other than that, as someone else said: if it's covered on only one page and there are 1500 pages then 1/1500 is a pretty good estimate given that one does pick up the book. Since you're more likely to open the book in the middle than right at the start, and since a topic covered only once is probably more likely to be somewhere in the middle than at the start, the correct figure would be a little higher (i.e., less improbable) than that. What's harder to quantify is the probability of picking up an appropriate book at all. The probability is likely to be higher than you'd expect, because if a topic interests you you're likely to have books on related subjects, and vice versa; and because seeing the book might trigger thought about a topic related to its subject matter; and so on. Back-of-envelope calculation: Suppose your life consists of a succession of 1-hour contemplations about various topics, interspersed with periods of complete inertness. Suppose there are four such periods per day. Suppose that the probability of your picking up the nearest book to hand at some point during any one of those contemplations is 1/10. And suppose the probability that this book is marginally relevant (so that it has at most a few directly relevant pages, but does have at least one) is 1/100. And suppose further that this book is always approximately 1500 pages long, and assume for convenience that in fact it always has only one relevant page. Then the probability of the event you describe taking place during any one contemplation is about 1/1500000. Suppose these contemplations happen only during your working life, which we'll take as 30 years in length, and suppose that you contemplate on 5 days each week, for 40 weeks of each year. That's a total of 6000 contemplative days, and therefore 24000 contemplations. Then the probability that the event you describe happens at least once during your lifetime is about 1.6%. Not very likely, but not terribly unlikely, either. Suppose this sort of contemplative existence occurs only in the wealthy Western nations; let's say their total population is half a billion people. And suppose it's only 1/1000 of that population that's privileged to have such a life. And, finally, suppose that this sort of contemplative life is the only way to have the sort of experience we're discussing. Then the number of people now living who have had, or will have, such an experience would be about 1.6% of 500000, or approximately 8000. Those are mostly very conservative estimates. I expect the real number is one or two orders of magnitude greater. -- Gareth McCaughan .sig under construc === Subject: Re: Probability calculation > Hello - > > Could someone tell me, in general terms, since I'm not really a > mathematician -- > if: > > 1) one were to be thinking about a specific topic, and > 2) then within seconds, randomly pick up a 1500 page book that > addresses this topic only once in a couple paragraphs only, > 3) open it to the exact page > 3) and then, to specifically look at the exact place within the page > that the topic exists > > I know you probably don't have all the data to make any kind of > analysis, but what are the ballpark chances of something like that > happening? Or, what are the minimum chances of it happening? > I'd suggest ignoring the second #3 (which you probably > intended to be #4...), because if there's a particular > thing you're looking for and you look at a page that has > that on it, then your attention may be drawn to the > relevant bit subconsciously. > Other than that, as someone else said: if it's covered on > only one page and there are 1500 pages then 1/1500 is a > pretty good estimate given that one does pick up the book. > Since you're more likely to open the book in the middle > than right at the start, and since a topic covered only > once is probably more likely to be somewhere in the middle > than at the start, the correct figure would be a little > higher (i.e., less improbable) than that. > What's harder to quantify is the probability of picking > up an appropriate book at all. The probability is likely > to be higher than you'd expect, because if a topic > interests you you're likely to have books on related > subjects, and vice versa; and because seeing the book > might trigger thought about a topic related to its > subject matter; and so on. > Back-of-envelope calculation: > Suppose your life consists of a succession of 1-hour > contemplations about various topics, interspersed with > periods of complete inertness. Suppose there are > four such periods per day. > Suppose that the probability of your picking up the > nearest book to hand at some point during any one of > those contemplations is 1/10. > And suppose the probability that this book is marginally > relevant (so that it has at most a few directly relevant > pages, but does have at least one) is 1/100. > And suppose further that this book is always approximately > 1500 pages long, and assume for convenience that in fact > it always has only one relevant page. > Then the probability of the event you describe taking place > during any one contemplation is about 1/1500000. > Suppose these contemplations happen only during your > working life, which we'll take as 30 years in length, > and suppose that you contemplate on 5 days each week, > for 40 weeks of each year. That's a total of 6000 > contemplative days, and therefore 24000 contemplations. > Then the probability that the event you describe happens > at least once during your lifetime is about 1.6%. Not > very likely, but not terribly unlikely, either. > Suppose this sort of contemplative existence occurs > only in the wealthy Western nations; let's say their > total population is half a billion people. And suppose > it's only 1/1000 of that population that's privileged > to have such a life. And, finally, suppose that this > sort of contemplative life is the only way to have the > sort of experience we're discussing. Then the number > of people now living who have had, or will have, such > an experience would be about 1.6% of 500000, or > approximately 8000. > Those are mostly very conservative estimates. I expect > the real number is one or two orders of magnitude greater. === Subject: Re: Positive Definate Matrices > Positive-definite requires hermiticity, otherwise it has no meaning. > Thus the eigenvalues are always real (and of course, positive). > I would like to direct you to a recent thread Looked at it. Didn't see any examples. Please post an example of a real, non-symmetric matrix that is positive, i.e. (x,Ax) > 0 for all (real) x. If it is real and symmetric, then it is ipso-facto Hermitian. -- Julian V. Noble Professor Emeritus of Physics jvn@lessspamformother.virginia.edu ^^^^^^^^^^^^^^^^^^ http://galileo.phys.virginia.edu/~jvn/ Science knows only one commandment: contribute to science. -- Bertolt Brecht, Galileo. === Subject: Re: Positive Definate Matrices > Please post an example of a real, non-symmetric matrix that is > positive, i.e. (x,Ax) > 0 for all (real) x. If it is real and > symmetric, then it is ipso-facto Hermitian. ( a 1 ) ( -1 a ) Eigenvalues are a+i and a-i, so a>0 places them in the right half place, so A rotates any vector over less than 90 degrees, so x^t(Ax) > 0. V. -- mail me at lastname at cs utk edu === Subject: Re: Positive Definate Matrices <1g0f9q6.1iws6a71x5upbzN%see.signature@for.address> <3F4F6F22.5F97519F@virginia.edu> <3F5277C6.268C60C0@virginia.edu Looked at it. Didn't see any examples. > Please post an example of a real, non-symmetric matrix that is > positive, i.e. (x,Ax) > 0 for all (real) x. If it is real and > symmetric, then it is ipso-facto Hermitian. (Not to intrude, but this comes up quite a bit in control theory, so I thought I might jump in.) Let A_s is the symmetric part of A, i.e. A_s = 0.5(A+A') and let A_a be the anti- (or skew) symmetric part of A, i.e. A_a = 0.5(A-A') Notice that A_s is symmetric, A_a is skew, and A = A_s + A_a Then, x' A x = x' A' x (transpose of a real number is the same number) = 0.5 x' A x + 0.5 x' A' x = x' A_s x That is, the quadratic form only depends on the symmetric part of the matrix A. Thus, you can choose any symmetric positive definite matrix, add a skew symmetric matrix to it, and you still have a positive definite matrix. To be positive definite, the real parts of the eigenvalues of A_s must be positive. The relationship between the eigenvalues of A and A_s is generally nontrivial, and A can have negative eigenvalues and still be positive definite. Hope that helps, Rick === Subject: Re: Looking for examples of bad data analysis > For instructional purposes I am looking for true examples of bad data > analysis in the published literature. A linear regression which was > not robust and eventually misleading would be such an example. They > can be in any field of science and may involve either real data or > computer simulations, but should be transparent, instructive, and > verifiable. Major historical errors are ideal. > If you happen to know of any illustrative example that mislead many > researchers, please let me know. > norb1@yahoo.com Slide #11 of the Helena, MT smoking ban study, from this site: http://www.no-smoke.org/HelenaPowerPoint.pdf It's all integer data, so pretty easy to read. Multiple linear regression by month, forcing a common slope to all curves for the no-ban data, and then plotting the repeating pattern. No idea how they got a curve for the ban period. I'd love to see what ANYONE has to say on this! === Subject: Excess N notation Hello Im trying to understand the excess N notation. Suppose I can represent 5 digits (base 10) in fraction and 2 digits (base 10) in the exponent and we are talking about excess 49. How can you determine that it is in fact excess 49? I thought the formula for excess N notation was 2^(n-1) - 1 which is clearly not 49 if we substitute 1 for n === Subject: Re: Excess N notation >Im trying to understand the excess N notation. >Suppose I can represent 5 digits (base 10) in fraction and 2 digits (base >10) in the exponent and we are talking about excess 49. >How can you determine that it is in fact excess 49? I thought the formula >for excess >N notation was 2^(n-1) - 1 which is clearly not 49 if we substitute 1 for n Think of it as (10^n)/2 - 1 === Subject: search for the proper method for solving differential equation it's a big problem for me to find the way to solve any of these two given differential equations (second-order and ordinary, with variable coefficients). x^2*y'' - x*(x+2)*y' + (x+2)*y=x^3 and (1-x^2)*y'' - 4*x*y' - (1+x^2)*y(x) = x Solutions are not given, but solutions with mathematics software are possible. Please give me a hint. A. Kratzer === Subject: Re: search for the proper method for solving differential equation > it's a big problem for me to find the way to solve any of these two > given differential equations (second-order and ordinary, with variable > coefficients). > x^2*y'' - x*(x+2)*y' + (x+2)*y=x^3 > and > (1-x^2)*y'' - 4*x*y' - (1+x^2)*y(x) = x > Solutions are not given, but solutions with mathematics software are > possible. > Please give me a hint. > A. Kratzer Have a look at the chapter on Special Functions in the book Mathematical Methods of Physics by Mathews & Walker. Should be in any decent Physics Department's library. Look under the topic singularities and look under Riemann P-symbol. Also, once you can solve the _homogeneous_ equations, you can solve the inhomogeneous equations you have given above by construction the Green's function. (Also known as the method of variation of parameters.) -- Julian V. Noble Professor Emeritus of Physics jvn@lessspamformother.virginia.edu ^^^^^^^^^^^^^^^^^^ http://galileo.phys.virginia.edu/~jvn/ Science knows only one commandment: contribute to science. -- Bertolt Brecht, Galileo. === Subject: Re: Calculate numerical value of special integral > I'd like to calculate the value of the integral > > int_0^infty x^{n-2}exp(-x^n) dx > Some thoughts- > The integral exists - with real values of n - for n > 1. > For real n: > The integral is real-valued for n > 1, yes, but also for n < 0. > David Cantrell As David Cantrell has observed, the problem is solved if there is a way to approximate (1/n)*Gamma(1-1/n) , or Gamma(x) for small (positive ) x . Denote z(p)=zeta(p):=Sum_{k=1 to k=infty}1/k^p , in our case we need p>1 , g=gamma:= Euler's constant . For instance for practical reason, it's possible to use approximation (-1 As David Cantrell has observed, the problem is solved if > there is a way to approximate (1/n)*Gamma(1-1/n) , or > Gamma(x) for small (positive ) x . Denote > z(p)=zeta(p):=Sum_{k=1 to k=infty}1/k^p , in our case we need p>1 , > g=gamma:= Euler's constant . > For instance for practical reason, it's possible to use approximation > (-1 ln(Gamma(x+1))=approx= 0.5* ln(Pi*x/(sin(Pi*x)))- > -arctanh*(x)+A*x- B*x^3/3- C*x^5/5- ... > with > A=1-g=1-0.5772156649015328606...= .4342944819... > B=z(3)-1=Apery-1=.2020569031595943.. > C=z(5)-1=.0369277551433700... . > I consider that above formuala was established by De Morgan , or try Gnu Scientific Library (GSL) http://sources.redhat.com/gsl/ref/gsl-ref_toc.html and for your problem http://sources.redhat.com/gsl/ref/gsl-ref_7.html#SEC119 Maybe that helps. Axel === Subject: Re: Calculate numerical value of special integral > I'd like to calculate the value of the integral > int_0^infty x^{n-2}exp(-x^n) dx > Simple methods like Runge-Kutta and similar failed. Can anyone give me some > hints on how to solve such a problem? Well, for n > 1 Mathematica 5.0 gives In[1]:= Assuming[n > 1, Integrate[x^(n - 2) E^(-x^n), {x, 0, Infinity}]] Out[1]= Gamma[(n - 1)/n]/n while for n < 0, In[2]:= Assuming[n < 0, Integrate[x^(n - 2) E^(-x^n), {x, 0, Infinity}]] Out[2]= Gamma[-(1/n)]/n^2 Paul -- Paul Abbott Phone: +61 8 9380 2734 School of Physics, M013 Fax: +61 8 9380 1014 The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling Highway Crawley WA 6009 mailto:paul@physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul === Subject: Re: Why are irrational numbers not countable? <3F50BE37.FF1DE4E3@virginia.edu> so ALL the reals have cardinality aleph_1 . > [etc.] > 2^aleph_0 is the actual answer. > The hypothesis aleph_1 = 2^aleph_0 is undecidable. In some set > theory models it holds, in others it does not hold. > (Search keyphrase: Continuum Hypothesis). True for you. I thought, however, that this would be a bit abstruse for this group. -- Julian V. Noble Professor Emeritus of Physics jvn@lessspamformother.virginia.edu ^^^^^^^^^^^^^^^^^^ http://galileo.phys.virginia.edu/~jvn/ Science knows only one commandment: contribute to science. -- Bertolt Brecht, Galileo. === Subject: Re: Why are irrational numbers not countable? <3F50BE37.FF1DE4E3@virginia.edu> <3F535F95.633D1098@virginia.edu [...] > so ALL the reals have cardinality aleph_1 . > [etc.] > 2^aleph_0 is the actual answer. > The hypothesis aleph_1 = 2^aleph_0 is undecidable. In some set > theory models it holds, in others it does not hold. > (Search keyphrase: Continuum Hypothesis). > True for you. I thought, however, that this would be a bit > abstruse for this group. > -- If it is true for an advanced set theory amateur (like myself) then it is true for a less advanced reader, too. It may be appreciated differently, though. (And you may be surprised how much detailed knowledge is avidly absorbed by amateurs, advanced or not.) As a factual detail for onlookers, the hypothesis 2^aleph_0 = aleph_1 holds true in the minimal model of set theory, where everything is, in a well-defined way, constructible (transfinitely, of course). The set theory gourmets consider this model boring (too much is decidable there). > Julian V. Noble > Professor Emeritus of Physics > jvn@lessspamformother.virginia.edu > ^^^^^^^^^^^^^^^^^^ > http://galileo.phys.virginia.edu/~jvn/ > Science knows only one commandment: contribute to science. > -- Bertolt Brecht, Galileo. === Subject: need online differential equation solver better than maple Need an online differential equation solver better than maple. Free linux rpm packages would also do. Particularly, I have to see how does the solution of (f'(t))^2=p/f(t)-q with p, q positive real numbers look like. === Subject: Text on Applied Analysis Can anyone recommend a good text for a self-study of applied analysis? I have been trying to understand the respective properties of linear operators in Hilbert, Banche, and normed spaces from Hunter and Nachtergaele's Modern Applied Analysis text but I am finding it too abstract and lacking in practical examples. I am mainly interested in the properties of operators in vector spaces as they apply to statistical (parameter) estimation problems. GW === Subject: Re: Text on Applied Analysis >Can anyone recommend a good text for a self-study of applied analysis? > I have been trying to understand the respective properties of linear >operators in Hilbert, Banche, and normed spaces from Hunter and >Nachtergaele's Modern Applied Analysis text but I am finding it too >abstract and lacking in practical examples. I am mainly interested in >the properties of operators in vector spaces as they apply to >statistical (parameter) estimation problems. I like Optimization by Vector Space Methods for an (engineering-oriented) introduction to these topics (normed, Hilbert, Banach spaces, and so on). If you're completely new to analysis, then it might be good to get ahold of a less-applied introductory book on real analysis, such as Rudin or Royden, to use as a reference. cheers, Rick === Subject: integral computation for non-smooth integrands ? I have to compute 1D integrals in an integro-differential equation, i.e. the integrand can not be expected to be smooth. Apart from own very basic and slow integration routines, I aim at extending my numerical integration by more sophisticated routines. I deal with delay equations, hence the Fourier approach appears not possible. Does anybody deal with a similar problem and might give me a hint for improved and thus faster methods? Axel === Subject: Re: My prime research, focus on a feature REVISED >Well, that's just not true. Christian Bau made a specific claim >saying that my work was something else. That claim has been one he >has made over a period of many months, and it has been refuted many >times--but Christian Bau just keeps coming back making the claim. >Now then he is lying to you and others by continually claiming >something that has been refuted, but possibly you're one of those >people who likes it when others think they can lie to you without >concern about consequences. Of course I don't like people lying to me at all, and I didn't say that I think C.B. is right. The thing is that neither of you is lying on purpose, everybody just states his point of view, may it be right or wrong or somewhere inbetween. Concerning your function/algorithm... Of course C.B. is right when he says that there have been other prime counting functions (not algorithms) and other functions (not algorithms) that know when a number is prime. The problem with all those functions however, is that they're recursive functions, and that's what kills their efficiency. Your's is just another recursive function, with a complexity that won't revolutionize anything. Sorry to say that, but that's how it is. But, nonetheless, your function is interesting. It is a new way of approaching the problem, and might help yourself or others to actually get a better solution to it. >> You could just as well publish your work in scientific journals, but >> there will be no less fighting against your ideas. >> Just my two cent, and I'm sorry for them. >> CMW >Yes, there will be a lot of fighting against the mathematics. >But the neat thing about math is that people can fight against it all >they want, as they can't change mathematical truth. You're right there of course, but the truth is not a truth, until it has been proven right. And as long as there is no proof, mathematicians will not believe the truth. Except if it's their own truth of course. >So in the meantime I get to point out liars like Christian Bau and >wonder at those of you who don't care about people like him who lie to >you. >But then again, there must be a market for liars, as some people care >less about the truth than what they *want* to hear is the truth. >Christian Bau tells those of you who don't want to accept that I could >have found this great math that it's not so great, which is what you >want to hear, so you accept his lies. No, it's not what we want to hear, we just don't believe that a solution that's as good as you claim it to be does exist. So, if you come up with a thorough proof that your method is right, a right complexity calculation, and some runtime tests, which all show that your solution is better than all the others we know so far, we will not go accept your work as great. And that is what these people that fight your idea want to make you do: Make it water proof. >James Harris CMW === Subject: iterative solver I must solve a complex linear system Ax = b; The matrix A is not available but I know the result of the multiplication of A with an arbitrary vector v. I tried CG but do not seem to be convergent. What solver / preconditioner do you recomend ? === Subject: Re: iterative solver > I must solve a complex linear system Ax = b; > I tried CG but do not seem to be convergent. CG will only work if your matrix is Hermitian. If it's complex symmetric CG will also work, but you need to use the x^ty inner product rather than x^*y. If your operator is not symmetric or hermitian, take GMRES. I believe there is also a complex solver in QMRpack. V. -- mail me at lastname at cs utk edu === Subject: Re: iterative solver Originator: elsner@mathematik.tu-chemnitz.de (Ulrich Elsner) According to Victor Eijkhout : >> I must solve a complex linear system Ax = b; >> I tried CG but do not seem to be convergent. >CG will only work if your matrix is Hermitian. If it's complex symmetric >CG will also work, but you need to use the x^ty inner product rather >than x^*y. >If your operator is not symmetric or hermitian, take GMRES. >I believe there is also a complex solver in QMRpack. You snipped the problematic part. The original poster only mentioned access to A*x for arbitrary x. GMRES, QMR et al need multiplication with A^T as well. If that is available,, GMRES, QMR or other Krylov- subspace methods _might_ work well (depending on the nature of the matrix). If only A*x is available, transpose free methods like TFQMR CGS or Bi-CGStab might be worth a look. For an overview and additional references, have a look at http://www.netlib.org/linalg/html_templates/report.html Ulrich Elsner >-- >mail me at lastname at cs utk edu === Subject: Re: iterative solver > You snipped the problematic part. The original poster only mentioned > access to A*x for arbitrary x. GMRES, QMR et al need multiplication > with A^T as well. GMRES doesn't. I was mistaken about QMR. > matrix). If only A*x is available, transpose free methods like TFQMR > CGS or Bi-CGStab might be worth a look. Yep. > For an overview and additional references, have a look at > http://www.netlib.org/linalg/html_templates/report.html V. -- mail me at lastname at cs utk edu === Subject: Optimize BLAS1-like operation Are there any optimized subroutine libraries similar to BLAS to handle vector operations of the following type? double complex a(n), b(n), c(n) do i = 1, n a(n) = a(n) + b(n)*c(n) end do The operations are somewhat similar to the BLAS1 zdotu so I would guess it can be highly optimized for most machines. Either C or Fortran code would be fine. -- James C. West (Jim) jwest@okstate.edu Professor Electrical and Computer Engineering Oklahoma State University === Subject: Re: Optimize BLAS1-like operation > a(n) = a(n) + b(n)*c(n) > end do > The operations are somewhat similar to the BLAS1 zdotu so I would guess > it can be highly optimized for most machines. You would guess, but you would be wrong. The number of loads & stores is too high to do much optimising. Unless you have a Cray, or so. I know of no libraries that provide optimised implementations of this operation. V. -- mail me at lastname at cs utk edu === Subject: Re: Optimize BLAS1-like operation * Jim West: > double complex a(n), b(n), c(n) > do i = 1, n > a(n) = a(n) + b(n)*c(n) > end do > The operations are somewhat similar to the BLAS1 zdotu so I would guess > it can be highly optimized for most machines. What do you highly optimized code expect to accomplish? It will still need to do the read accesses to the memory, the multiply-and-add instruction, and the final write. This is pretty much what your loop above expresses. Everything else depends on your hardware; your optimizing compiler will probably do a better job in figuring ou how to partially unroll your loop than some strangers on the net (be sure to provide it with highly optimized hints by using the += operator in C and array expressions in Fortran 90). -- Rupert === Subject: Re: Optimize BLAS1-like operation > It will still need to do the read accesses to the memory, > the multiply-and-add instruction, and the final write. This > is pretty much what your loop above expresses. The difference between a reference implementation and an optimized one can be enormous. See Atlas (netlib.org/atlas). > Everything > else depends on your hardware; your optimizing compiler will > probably do a better job in figuring ou how to partially > unroll your loop than some strangers on the net Maybe you're right about the strangers on the net, but an optimizing compiler will not nearly be as fast as a well constructed optimized implmentation. Again, see Atlas. V. -- mail me at lastname at cs utk edu === Subject: Re: commutative === >Subject: commutative >I am not a mathematician, so please forgive my trivial question: >Is it possible to define addition in a way that it is not commutative? > a + b <> b + a (<> is supposed to mean unequal) >Does that make sense in any way? > Most introductory texts on Modern Algebra will have examples of > groups in which the group operation (addition) is not commutative. > A common example are the rotations applied to a physical object > like a box with width, length and depth all different. Gordan is correct in citing noncommutative group operations as a possible model, but ordinarily the label of addition and a use of plus sign as symbol will signify the operation is commutative. Noncommutative operations are ordinarily called multiplication and indicated by a product (or composition) notation.