mm-318 === Subject: : Re: Covering number by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i2C1JWx04056;i think you are looking for something like Ininuga. Go tohttp://www2.arnes.si/~ubolti/ininuga.nina>A Covering Design c(v,k,t,m,b) is a pair (V,B), where V is a set of v>elements (called points)>and B is a collection of b k-subsets of V (called blocks) such that >every m-subset of V>intersects at least one member of B in at least t points.>It is required that v >= k >= t and v >= m >= t.>Given some parameters of a design and some subsets,>how can you determine fast the number of Covered/Uncoverd blocks>within these parameters?>I'm looking for an algorithm or pseudocode to count the number>of blocks that are covered/uncovered, with a method other than bruteforce.>Some examples of partially covered designs below:>Example #1,>Parameters: v=12 k=5 t=4 m=6 b=9> 1 3 5 6 12> 1 4 7 9 12> 1 8 9 10 11> 2 3 4 7 8> 2 3 8 9 12> 2 4 5 6 9> 2 7 10 11 12> 3 4 5 10 11> 5 6 7 8 10>The above design covers 770 blocks of the possible 924.>It leaves uncovered 154 blocks.>My question again is as of methods/algorithms to count the number of>covered/uncoverd blocks as fast as possible.>Example #2,>Parameters: v=18 k=7 t=4 m=7 b=11> 1 2 3 4 9 14 18> 1 2 3 4 11 12 13> 1 2 5 6 7 8 9> 1 2 8 15 16 17 18> 1 7 8 10 11 15 16> 2 10 12 13 14 15 16> 3 4 5 6 7 14 17> 3 4 5 6 8 10 18> 3 4 9 10 15 16 17> 5 6 7 11 12 13 18> 8 9 11 12 13 14 17>Covered blocks=31,692 of the possible 31,824.>Uncovred blocks=132>Example #3,>Parameters: v=15 k=6 t=3 m=4 b=14> 1 2 3 4 5 6> 1 2 3 7 10 13> 1 3 6 8 12 13> 1 4 7 8 10 14> 1 5 9 12 13 14> 1 6 7 9 11 13> 2 4 10 11 12 15> 2 5 6 7 12 14> 2 8 9 11 14 15> 3 4 7 9 12 15> 3 5 10 11 14 15> 4 5 7 8 11 13> 4 6 9 13 14 15> 5 6 8 9 10 15>Covered blocks=1363 of the possible 1365.>Uncovred blocks=2>Your ideas or comments are welcome. === Subject: : Modelling dynamics of linked membersI'm looking at a problem of modelling a mechanical system that can be approximated by a number (say 20) of rigid members (rods), linked to each other endwise to form a chain. Imagine that the chain of members is strung between two fixed points, and that the members are acted on by varying external forces + gravity + damping.I am (currently) ignoring rotational effects, so a member is treated as having mass but no moment of inertia about its axis. In other words, I'm assuming that the state of the system is defined completely by the positions and velocities of the junctions. A junction is like a universal joint, fully flexible in all directions, i.e. zero (or small) bending stiffness. Members and joints are considered to be very stiff in tension/compression.What is the best approach to modelling this system? So far I have been looking at the static case, and solving for the steady-state shape with constant external loading. I have adopted the quick and dirty approach of treating the members like springs, with a stiffness that I increase as the equilibrium is approached, and with artificially high damping. This allows me to use the ODE solvers in Matlab without much sweat.I may want to solve for the true dynamics, and it seems to me that this could be quite a hard problem, since in some sense it is very stiff. Is this a well-known problem?Gib === Subject: : Re: Modelling dynamics of linked members> I'm looking at a problem of modelling a mechanical system that can be > approximated by a number (say 20) of rigid members (rods), linked to > each other endwise to form a chain. Imagine that the chain of members > is strung between two fixed points, and that the members are acted on by > varying external forces + gravity + damping.> I am (currently) ignoring rotational effects, so a member is treated as > having mass but no moment of inertia about its axis. In other words, > I'm assuming that the state of the system is defined completely by the > positions and velocities of the junctions. A junction is like a > universal joint, fully flexible in all directions, i.e. zero (or small) > bending stiffness. Members and joints are considered to be very stiff > in tension/compression.> What is the best approach to modelling this system? So far I have been > looking at the static case, and solving for the steady-state shape with > constant external loading. I have adopted the quick and dirty approach > of treating the members like springs, with a stiffness that I increase > as the equilibrium is approached, and with artificially high damping. > This allows me to use the ODE solvers in Matlab without much sweat.> I may want to solve for the true dynamics, and it seems to me that this > could be quite a hard problem, since in some sense it is very stiff. Is > this a well-known problem?> GibGo to http://www3.addall.com/ type mechanism linkage Formore hits, go to Used Books there. Then visit your university library. === Subject: : Re: Modelling dynamics of linked members> Go to http://www3.addall.com/ type mechanism linkage For> more hits, go to Used Books there. Then visit your university library.I think what makes the problem I'm interested in rather different from the problems treated in robotics and mechanical machinery etc. is that all rotational degrees of freedom exist at the junctions. Effectively my linked chain of members is more like a string with attached weights than a mechanical linkage system. I can't think of a mechanical system with this character. I'm guessing that it might have chaotic behaviour.I'm looking for suggestions on appropriate mathematical simulation methods.Gib === Subject: : Re: Modelling dynamics of linked members> I'm looking at a problem of modelling a mechanical system that can be > approximated by a number (say 20) of rigid members (rods), linked to > each other endwise to form a chain. > I may want to solve for the true dynamics, and it seems to me that this > could be quite a hard problem, since in some sense it is very stiff. Is > this a well-known problem?You'd consult books about robotics, e.g., J. Angeles, Fundamentals of Robotic Mechanical Systems, Springer, New York 1997.This is all about your systems..sArnold Neumaier === Subject: : Beam FEA help...I was wondering if anyone can help.I am doing a research project and need to do FEA of a beam.It was suggested to me that FEMLAB would be my best route, but ata minimum price tag of $1295.00, its a little out of my league.I need the following criteria met:1) Nonlinear analyses for large deflections, the software will have toiteratively compute the deflections, changing the node locations andstiffness matrix for each time step.2) Ability to do transient dynamic simulations, where forces change asfunction of time3) Ability to program forces as a function of time.4) Use beam or frame elements for 2D analyses. Each node should have 3degree of freedom (two translations and one rotation) at each nodeThe most important criteria right now is 4.I was looking at the freeware on this site:http://www.engr.usask.ca/~macphed/finite/fe_resources/ node12.htmlbut there is a lot to look through.I was hoping, either someone can point me to a freeware program or let meknowof some software that is a lot lower priced.Thanks for your help.If you reply to me directly, remove the 'z' from jfrancoz@diet... === Subject: : Re: Beam FEA help...> I was wondering if anyone can help.> I am doing a research project and need to do FEA of a beam.> It was suggested to me that FEMLAB would be my best route, but at> a minimum price tag of $1295.00, its a little out of my league.> I need the following criteria met:> 1) Nonlinear analyses for large deflections, the software will have to> iteratively compute the deflections, changing the node locations and> stiffness matrix for each time step.> 2) Ability to do transient dynamic simulations, where forces change as> function of time> 3) Ability to program forces as a function of time.> 4) Use beam or frame elements for 2D analyses. Each node should have 3> degree of freedom (two translations and one rotation) at each node> The most important criteria right now is 4.> I was looking at the freeware on this site:> http://www.engr.usask.ca/~macphed/finite/fe_resources/node12. html> but there is a lot to look through.> I was hoping, either someone can point me to a freeware program or let me> know> of some software that is a lot lower priced.> Thanks for your help.> If you reply to me directly, remove the 'z' from jfrancoz@diet...There is another FEM resources web site:http://www.engr.usask.ca/%7Emacphed/finite/fe_resources/ fe_resources.htmlthough I suspect many of the links are the same.In FEM, as in most things in life, there is a time-money tradeoff. If you dont want to spend money, it will cost you time.So it depends: how valuable is your time? If you are an aerospaceengineer at ,say, Lockheed-Martin, your labor cost is $140 to$200/hr. Ergo if a $1500 black-box package saves you one week of timeyou are way ahead. If you are an RA or postdoc at a school, your costis $20-40 /hr so doing your own code could be cost effective if youare fast with your fingers, and you start from a grey box tool likeMatlab.In between there is the option of downloading free low-level publicdomain software. This often has to be modified and then you have todebug it. The plus side is that you may gain a better understanding ofthe method by suffering through a debug process. === Subject: : .8bs.b9.b2.82274H.81fi.8f[RegisteredTrademark][@w[ CapitalCCedilla].86n.b9BAfi[DownQuestion]s[ CapitalYAcute][CapitalYAcute], z|[Micro]o.b2{.b3ouO[CapitalIGrave][Hyphen].be|.b3http:// www.msi-team.com/wchen === Subject: : New source of the math!!!I have come across a source providing good support to thoseexperiencing difficulties with doing homework in maths includingarithmetic(http://www.bymath.com/studyguide/ari/form1 .htm), geometry(http://www.bymath.com/studyguide/geo/pro/pro1/pro1. htm), algebra(http://www.bymath.com/studyguide/alg/pro/pro1/pro1.htm ), functionsand graphics (http://www.bymath.com/studyguide/fun/pro/pro.htm),principles of analysis(http://www.bymath.com/studyguide/ana/pro/pro1/pro1. htm),all this is also supported by a good deal of examples andillustrations. === Subject: : differencing two audio samplesHi:I've gave considerable thought into this but my math sucks.. I havetwo sound samples, each 10-20 seconds of sound at something like 44Khz- I need to subtract one from another, so I end up with a prototypeand a difference, where the difference should be very small, andoverall uses less storage. Supposed both samples are extremelysimilar to our human ears, and the beat of the samples are at quitegood matches. This problem is very similar to the joint-stereocompression used by mp3/ogg/etc, if any different at all, but I'mstuck at differencing, here are the things I've tried:1. I could do subtraction in time domain, but the result is more likea union of the two samples rather then difference - the problem is thephase is all different even though they sounded the same to our ear. The difference computed this way is useless to me because they are toobig.2. I tried it in DCT domain - the problem is when I subtract thecoefficients of the two time series in this domain, the resultdifference is still too large - afterall, I'm not dealing with'stereo' difference, but rather similarity between samples that couldbe off sync at multiple places with different - say - backgroundaccompanyment.Are there existing good algorithms for dealing with this? E.g.combine time-warping with DCT?Bill === Subject: : Re: differencing two audio samplesTry comp.dsp .> Hi:> I've gave considerable thought into this but my math sucks.. I have> two sound samples, each 10-20 seconds of sound at something like 44Khz> - I need to subtract one from another, so I end up with a prototype> and a difference, where the difference should be very small, and> overall uses less storage. Supposed both samples are extremely> similar to our human ears, and the beat of the samples are at quite> good matches. This problem is very similar to the joint-stereo> compression used by mp3/ogg/etc, if any different at all, but I'm> stuck at differencing, here are the things I've tried:> 1. I could do subtraction in time domain, but the result is more like> a union of the two samples rather then difference - the problem is the> phase is all different even though they sounded the same to our ear. > The difference computed this way is useless to me because they are too> big.> 2. I tried it in DCT domain - the problem is when I subtract the> coefficients of the two time series in this domain, the result> difference is still too large - afterall, I'm not dealing with> 'stereo' difference, but rather similarity between samples that could> be off sync at multiple places with different - say - background> accompanyment.> Are there existing good algorithms for dealing with this? E.g.> combine time-warping with DCT?> Bill === Subject: : Re: Is there any book on system of ordinary differential equations? > Can somebody introduce me some good books on how to solve system of >ordinary differential equations by finite difference? I only have >Numerical Analysis by Richard L. Burden and J. Douglas Faires. But >this book doesn't tell much about system of ODEs. >Peng1) the book: Hairer-Norsett-Wanner Solving ordinary differential equations I Hairer-Wanner Solving ordinary differential equations II (stiff systems) has a lot of inforamtion, well written, even with software but may be a bit advanced2) Eriksson-Estep-Hansbo-Johnson: computational differential equations studentlitteratur Lund, ISBN 91-44-49311-8 intended for beginners, used succesfully in student courses, contains the essentials3) Asher & Petzold: computer methods for ordinary differential equations and similar to 2)hthpeter === Subject: : Re: eigenvalues of large, symmetric, matrces with many zeroes. >I need to compute the eigenvalues of a symmetric, >real, nXn matrix A where: >1. each row of A is a vector of zeroes with >at most 4 ones >2. n is very large, of order 10^5 or 10^6. >Any suggestions are greatly appreciated. >Federico Echenique I assume you want only some ... (the largest , the smallest, some in someinterval .... , otherwise you must proceed completely different ...)http://www.ime.unicamp.br/~chico/arpack++http:// www.caam.rice.edu/software/ARPACKhthpeter === Subject: : Re: Hexagonal and pentagonal finite element> You are right: misprint, it should say face 235.> I beleieve you are using wrong natural coordinates and wrong> shape functions for the pyramid.OK. I can see the paper error: Wrong natural coordinates.The element I call Pyra5 has been used for over 25 years in 3D FEMsolid codes, initially in the nuclear reactor codes of the mid 1970s. It is in Ansys and (I believe) Abaqus. I use it occasionally (as atransition element) and works OK.Generally too stiff for stress analysis (Pyra13 and Pyra14 are betterbutmore expensive) but fine in acoustics, thermal and hydro. Passes thepatch test when mixed with Hexa8 and Tetr4, which is important sincethat's the whole purpose of transition elements. One has to be a bitcareful with gradient recovery at the apex , because of the 0/0ratios. Extrapolating from Gauss points does the trick.So when I saw a post claiming this element does not exist, I lookedat the calendar, but it is not April 1 yet. === Subject: : Re: Hexagonal and pentagonal finite element> Is this the paper you are talking about ?> http://www.cs.brown.edu/publications/techreports/reports/CS-00 -03.html> As you seems to be very aware of those kind of problems, did you ever > heard of a method for hexagonal/pentagonal shape functions ? I only find > x^2y^2 shape function (and it is only in 2D)> Thanksp://www.cs.brown.edu/publications/techreports/reports/CS -00-03.html> As you seems to be very aware of those kind of problems, did you ever > heard of a method for hexagonal/pentagonal shape functions ? I only find > x^2y^2 shape function (and it is only in 2D)> ThanksLet me try to answer this one. Conforming shape functions forpolyhedra elements of arbitrary shape can be constructed in severalways. One method I am aware of, although I have not used it myself,is the extended FEM of Belytschko and Moran. This was callednatural FEM in earlier publications. Extended means that theelement can be enriched with discontinuities on arbitrary planes forproblems like cracks. Some references: IJNME, v48, pp1475ff, 2000and IJNME, v 43, pp839ff, 1998.My friend and colleague Sergio Idelhsohn has used a variation of thatidea for Lagrangian hydrodynamic simulations, in which a new 3D meshis constructed on the fly at each timestep using Voronoi methods. Such mesh typically consists of a high proportion of tetrahedra, say90%, but the rest can be polyhedra of arbitrary number of faces:5, 6, .. etc . This happens because his generator tries to avoidslivers when there are fragmentation, spills or recombinations. There was a recent publication of this method in EC, but I dont havethe reference handy. === Subject: : Re: R_E_A_D T_H_I_SIn sci.math.num-analysis, susan.thornburn@stees.nhs.uk:> Could someone please send me some info about the thread above this one.> Any pictures would be helpful too.> Thanks.> Susan.thornburn@stees.nhs.ukThe thread above this one is highly meaningless, since Usenet isan inherently dynamic medium. Also, it depends on how yournewsreader presents information; most modern newsreaders sortthe subjects alphabetically (which means the previous thread, asof today on my newsreader, has to do with Runge-Kutta). However,many also take threading into account; subject changes may notnecessarily factor if they're not a root. (This post [a reply],for instance, is branching off your post, with your post as root.)As for pictures -- it is possible to send pictures but it'sdiscouraged in most newsgroups (there are however Usenetnewsgroups dedicated to pictures -- the most well-knownexample arguably being alt.binaries.pictures.erotica, butalt.binaries.pictures has other stuff too). In thisnewsgroup the pictures I would hope to see are graphplots,which might be better done by simply sending lists of numbersinto a program such as gnuplot, or a spreadsheet such asExcel, Gnumeric, or OpenOffice Calc.For example, the list of x,y coordinates-10 100-8 64-6 36-4 16-2 40 02 44 166 368 6410 100is an approximation of the parabola y = x^2, and should be easilyplottable.It should also be easier to feed to various statistical analysis packages.-- #191, ewill3@earthlink.netIt's still legal to go .sigless. === Subject: : Re: constrained brownian motion>> You are missing a lot, including your specification.>> When x reaches a boundary, just changing the sign of>> s will not keep it from exceeding the boundary, so I>> do not know exactly what is your model. Also, simulation>> must be done much more carefully than usual; I had a>> somewhat different problem 30 years ago.>ok, i was not clear enough to describe what i am doing. I first only changed>the sign and clamped the X value to the boundary. Then i computed the>missing random walk, i.e, knowing that the collision occurs between t_i>rather at x=0.6-eps, eps being here the difference between the value>computed with s=1 and the boundary.This is very unclear. Also, when it comes to crossings,simulation of a brownian motion at anything other thanan ultrafine set of points is not adequate; this is thesimulation problem I handled for maxima and minima.One could either use a process which is constrained tokeep away from the boundary; this would be locally aBessel process of order 3. Or one could continuallyreflect the process (not the trend) until it leaves theboundary. This reflection procedure will still have anuncountable number of collisions. Either method mightwell be treatable, the second probably easier. The behavior of a brownian motion with constant drift between two points, given the values at the points, does not depend on the drift.It is still not clear what your problem is.-- This address is for information only. I do not claim that these viewsare those of the Statistics Department or of Purdue University.Herman Rubin, Department of Statistics, Purdue Universityhrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: : Fourier transform of exp(-at+bt^2) functionThe Fourier transform of a function exp(-at) is a/(a^2+(2*PI*f)^2)and the Fourier transform of a function exp(-bt^2) is p*exp(-qf^2) where pand q are known fucntions of b.Now, what is the Fourier transform of a function exp(-at-bt^2)?Can it be represented by an analytical expression?Can it be approxamated by c1/(a^2+(2*PI*f)^2) +c2*exp(-q1*f^2)?Thanks a lot.Haifang Li === Subject: : Re: Fourier transform of exp(-at+bt^2) function> The Fourier transform of a function exp(-at) is a/(a^2+(2*PI*f)^2)> and the Fourier transform of a function exp(-bt^2) is p*exp(-qf^2) wherep> and q are known fucntions of b.> Now, what is the Fourier transform of a function exp(-at-bt^2)?Do your own homework.> Can it be represented by an analytical expression?Yes> Can it be approxamated by c1/(a^2+(2*PI*f)^2) +c2*exp(-q1*f^2)?No.Hint: complete the square.> Thanks a lot.You're welcome,Gerry T. === Subject: : Re: all the roots of a nonlinear equationDistribution: inet >(sorry for posting to sci.math!) > I realize this is probably a dumb question but I've looked at numerical >analysis books and racked my brain but I must be missing something obvious. >I have equations which are sums of sines and cosines an example being: >2 Cos[x] - Cos[2 x] - Sin[x] + 3 Sin[2 x] - Sin[3 x] = 0 >I need to find all the roots of this equation in the range -pi to pi. >The algorithms I've looked at only give me one root and if based on a >bisection technique dont always give an answer in the range I mentioned >above. >I tried using NSolve in mathematica for this above equation and it does'nt >return any roots (there are 4 when looking at the plot). >I am interested in an algorithm that I could implement in C / Fortran for >later use that would give me all the roots of the above types of functions >automatically. >Is this possible? Or have I missed something very obvious? > Thanks, as you later detected, mathematica can do that (for this simple system).there are harder cases, but as long asthe dimension of the system doesn't becomelarge (in this context maybe 10 to 20...) thenhttp://www.netlib.org/toms/681 (INTBIS) does the job.hthpeter === Subject: : Re: Linear (Nonlinear) Curve Fitting with Constraints >Hello to all, >I have a set of data points that I wish to fit to a model of the >following form under two situations. >y(x)=Summation_Over_i ( Ai / (1 + 10^(x-Bi) ) >In situation 1 the Bi parameters are known. Thus, I will be only >fitting for the Ai. >In situation 2 both Ai and Bi must be fitted for. >In both cases there is the additional constraint: >Summation_Over_i (Ai) = 1 >I know that in the first case I need a general linear least square >curve fitting program with constraints and a non linear one also with >constraints in the second case. >Is there any specific FORTRAN code available on the net that will >allow me to do what I want? Thanks for your help >Zacenlsip does that job and also nlscon (since you provide only an equation as constraint). are there no constraints on Bi ???also dqed is a choice (which uses the partially separable structure of theproblem) but this might get trouble with bad fits (since it employs nolinesearch/trust region test) because of the simplicity of that constraint you could also eliminate formally, then express the remaining Ai as functions of the Bi via linear least squares and so reduce the problem to an unconstrained one with respect to the Bi and feed this into elsunc.you find these here:http://plato.la.asu.edu/topics/problems/ nlolsq.htmlhthpeter === Subject: : How to do feedback analysis on signals by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i2D0Fj820225;How to do feedback analysis on signals === Subject: : Re: laplace transformHea, I encouter the following problem. could you please give some suggestions.Thanks in advance. Suppose x1,x2...xm are i.i.d. exponential non-positive random variable with expected value 1/alpha; Suppose y1,y2...yn are i.i.d. exponential non-positive random variable with expected value 1/beta; Define L=x1+x2+...xm; and Q = y1+y2+...+yn; Then, L,Q are Erlang distributed random variable. Let f_Z(z) and F_Z(z) are pdf and cdf of the random varialbe Z. Now, I hope to find the Laplace transform of the following expression. C(t)=f_L(t)*F_Q(a-t) where a is positive fixed quantity.-------------ZHANG Yanhttp://www.ntu.edu.sg/home5/pg01308021/ === Subject: : skewed dualities: functions and setsR.Goldberg Methods of Real Analysis 2nd Edition p11:f^-1(X union Y) = f^-1(X) union f^-1(Y)f^-1(X intersect Y) = f^-1(X) intersect f^-1(Y)f(X union Y) = f(X) union f(Y)However the relationf(X intersect Y) ?= f(X) intersect f(Y)is Conspicuosly absent from this listConspicuosly absent from this list R.Goldberg Methods of Real Analysis 2nd Edition p11:> f^-1(X union Y) = f^-1(X) union f^-1(Y)> f^-1(X intersect Y) = f^-1(X) intersect f^-1(Y)> f(X union Y) = f(X) union f(Y)> However the relation> f(X intersect Y) ?= f(X) intersect f(Y)> is Conspicuosly absent from this list Any insights?It should be included and it should read f(A / B) subset f(A) / f(B)which quickly follows from for all A,B A subset B ==> f(A) subset f(B)Exercise: show the same for arbitrary union and intersections. === Subject: : Re: skewed dualities: functions and sets> R.Goldberg Methods of Real Analysis 2nd Edition p11:> f^-1(X union Y) = f^-1(X) union f^-1(Y)> f^-1(X intersect Y) = f^-1(X) intersect f^-1(Y)> f(X union Y) = f(X) union f(Y)> However the relation> f(X intersect Y) ?= f(X) intersect f(Y)> is Conspicuosly absent from this list My intuition says that following is true:> Two integers x,y are coprime iff x does not divide y & y does not divide x> Could someone give me a rigorous proof of this if my claim is true or> a counterexample if it is not true.Counterexample: 4 and 6.Two integers are coprime if there does not exists an integer x>1 thatdivides both.-- Jon Haugsand Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@ifi.uio.no http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 85 24 92 === Subject: : Re: Hypocritical Fortune Tellers (What are the odds?)Zaphod Beeblebrox the 1st i presume..... oh wait... his name is in his emailaddress colin dickson...... anyway....> i asked you to provide> links to the other forums that you supposedly post at. i got nothing.> www.christianity.com is one of the biggest christian forums on theinternet.> if God told you to post on religious forums, then you would think thatthis> site would be one of your first stops.step off christianity.com... it isnt that good... crosswalk.com has morecontent.... everyone who belives in the Lord Jesus Christ will be saved and theres nouse fighting over petty little crap.... if you accept Jesus in your lifeand heard and are in his word to better understand him he will work throughyou... and hes called _everyone_ to do something acording to his talent ....hes not gonna make a hammer and tell it to wipe noses...... zaphod i thinkyou should leave the poor guy alone... daryl you should stop doing things inthe name of the lord and demanding payment...if you believe in him he isyour suffency.... remember the money changers in ther temple?....its jus notkosher :)stev-o === Subject: : Re: Closure property question === Subject: : Closure property question >Consider a collection of sets F. Define a set C_n of the form > oo oo >C_n = U / (B_njk), where B_njk belongs to F for all j and k. > j=1 k=1 >{C_n} is the collection of all such sets. Question: Is {C_n} closed >under countable unions?Why should it be? For example let every B_njk be the set of rationalsThen C_n = Q for all n and also their union /{ C_n | n in N } = Qwhich is neither closed nor open in the reals. >I believe it is since the resulting iterated unions can be >expressed as one union, and the resulting set is some C_k.Huh?? The countable union of countable unions is a countable union. >{C_n} would not necessarily be closed under countable intersections? Inthe example above, /{ C_n | n in N } = Q which isn't closed nor open.---- === Subject: : Re: Closure property question> Then C_n = Q for all n and also their union /{ C_n | n in N } = Q> which is neither closed nor open in the reals.He didn't say closed but closed under (ie: stable under) and from whatI remember all his assertions seemed true to me. === Subject: : Differentiability of Besov functions?Content-Length: 747Originator: rusin@vesuviusLet B_alpha^{p, q} be the standard Besov spaceon the unit circle (if your favorite notation for Besovspaces is different the comments below shouldclarify whether my p,q are the same as yoursor swapped...)Let X_q = B_1^{infinity, q}. Then X_infinityis the Zygmund class, which contains nowhere-differentiable functions. On theother hand it's easy to see that if f is inX_2 then f is absolutely continuous, hencedifferentiable almost everywhere (f in X_2implies f' in B_0^{infinity, 2}, which iscontained in B_0^{2, 2} = L^2(T).)So one might conjecture that if f is inX_q, 2 < q < infinity, then f must bedifferentiable on a set of Hausdorff dimension at least 2/q. Is this well-known?************************ === Subject: : Re: Matrix norms.>> Is it possible to define an unbounded norm on matrices?> Perhaps that I am missing something here, but the only vector> that I know in which you can define a bounded norm is {0}.Hm, in the (german) terminology I know, a linear map f:V->U, where V and Uare normed spaces, is said to be bounded iff there is a c>0 such that||f(x)|| <= c ||x||The smallest such c then is the norm of f. It is bounded iff it iscontinuous.BTW, what is the english noun for continuous?-- I'm on warm milk and laxativesCherry-flavored antacidsreverse my forename for mail! - saibot === Subject: : Re: Design QuestionHi Ron : I can't help thinking you are going to have problems determiningwhat exactly in phase means. Is there any way you can expand your problemdescription with some sort of mathematical definition of how similar thesecomponents need to be before you stop ignoring them?Best of Luck - Mike> I need to extract and amplify only the components of two signals that are> exactly in phase. Out of phase signals of any amplitude are noise. I am> thinking this is sort of the opposite of common mode rejection. Should Ibe> pursuing an electronic (hardware) solution or a math (DSP) solution? Is> there some way to utilize the phase part of an FFT?> Ron H. === Subject: : what are holomorphic/mero sectionsDear NG,I wanted to know how would one define meromorphic and holomorphic sections (of say a vector bundle of rank n). Thanks in advance.Sincerely,Jose Capco