mm-230 === Subject: Partial differentiation This question has got me baffled. Do you guys have any idea?Where are the stationary points/critical points in h(x,y) = cos (x +y)?When you partially differentiate with respect to x and then by y youget -sin(x+y) which you can't really set to zero to find thestationary points. So how do you find the points?Also if z=f(x-y) how do you show that (dz/dx)(with y constant) === help/advice you can give-- tmlSubject: Re: Partial differentiation> Where are the stationary points/critical points in h(x,y) = cos (x + > When you partially differentiate with respect to x and then by y you> get -sin(x+y) which you can't really set to zero to find the> stationary points. So how do you find the points?Why can't you solve sin(x+y)=0? That seems to me to be just x+y=pi*kfor integer k, which is a set of parallel lines.> Also if z=f(x-y) how do you show that (dz/dx)(with y constant) +> (dz/dy)(with x constant) is zero?Take the two derivatives using the chain rule, and add them.-- Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karpluslife member (LAB, Adventure Cycling, American Youth Hostels)Effective Cycling Instructor #218-ck (lapsed)Professor of Computer Engineering, University of California, Santa CruzUndergraduate and Graduate Director, BioinformaticsAffiliations for identification only.-- === tmlSubject: Re: Partial differentiationI see how you get x+y=pi*k by taking the inverse sin to both sides butI don't see how that helps you to find the stationary pointsh=cos(x+y)? If they are both parallel sides, then the can't be anystationary points can there?> Where are the stationary points/critical points in h(x,y) = cos (x +> y)?> When you partially differentiate with respect to x and then by y you> get -sin(x+y) which you can't really set to zero to find the> stationary points. So how do you find the points? Why can't you solve sin(x+y)=0? That seems to me to be just x+y=pi*k> for integer k, which is a set of parallel lines. Also if z=f(x-y) how do you show that (dz/dx)(with y constant) +> (dz/dy)(with x constant) is zero? Take the two derivatives using the chain rule, and add them. > -- > Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karplus> life member (LAB, Adventure Cycling, American Youth Hostels)> Effective Cycling Instructor #218-ck (lapsed)> Professor of Computer Engineering, University of California, Santa Cruz> Undergraduate and Graduate Director, Bioinformatics> === Affiliations for identification only.-- tmlSubject: Good algebra book?Can anyone recommend a good high school algebra book? It should covermatrices and complex numbers but it's ok === if it doesn't.-- tmlSubject: RE: graphing toolsRE: question below. If it is for mac os x, we have a graph building tool where students can build graphs and experiment with them from scratch, it is very interactive and provides immediate feedback. We can help your k-12 schools with software and support, is so desired. The cost is very minimal. We can even make custom applications for your students (at no cost) according to your curriculum and individual needs.Lance Blandmailto:lbland@vvi.comVVI888-VVI-PLOThttp:// === www.vvi.comSubject: graphing toolsDoes anyone know any good mathematical graphing tools for computers Ican download?-- === tmlSubject: Re: GREI totally agree with your views on probability and statistics, not only forfuture teachers but also for just about everyone else. Probability andstatistics, or stochastics as the combined subjects are often called inEurope, are the most useful math for everyday life after arithmetic.Lynn>.. I have a few questions about the GRE. The school I've all but choose requires both the general and subjecttest. When during my undergrad should I take the general test? I have one full> year of classes left plus one semester of student teaching after that.So> I'm looking at entering grad school Fall 2005. I don't think there is> anything else left in my undergrad that will help with the general test,I> don't have any English classes left and I've taken all the math anynon-math> major would have.> Take the general test whenever you feel like it---it's not much> different from an SAT---that is it basically tests high-school-levelskills.> That's probably the reason that the mode on the GRE math test is 800> (the highest possible score). Some stats that were recently given to me:> Verbal: fairly normal bell curve, with high-end fat tail and> mean: 477> median: 480> mode: 460 (on best fit curve)> Quant: rising end to curve, with> mean: 579> median: 590> mode: 800 (yes, by far the most common score!)> For all Engineering majors,> Quant: curve looks like rising exponential> mean: 716> median: 740> mode: 800> If you want to get into grad school as a math major, you should have> GRE scores at least as good as most engineering students (740 or above).> Can anyone recommend books to review for the math subject test? I thinkI'll> be ok on the general test, but the math subject worries me a bit.However, I> have the three highest level classes yet to go in my undergrad.> I have no idea what they put on the Math Subject test these days.> Long, long ago, when I took it, I remember thinking that it didn't> test any math that I had learned since my freshman year, just a little> calculus and linear algebra. It may well be different now, though a> little web browsing reveals that recent exams have been about 50%> calculus, 25% algebra, linear algebra, and abstract algebra, and 25%> additional topics studied areas.> (http://www.gre.org/subdesc.html#math)> The other seems to come in large part from simple probability and> combinatorics---so if you've never had that, take it!> Pure mathematicians often turn up their noses at probability and> statistics, but they are far more useful for teachers than most of> what is taught in graduate math programs. In fact, if I were hiring a> math teacher, I'd much rather see a masters in statistics than in pure> math.> I'd also like to see more college and college-bound students taking an> elementary probability and statistics course in place of the> badly-misnamed precalculus course that so many take as their final> exposure to math.> -- > Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karplus> life member (LAB, Adventure Cycling, American Youth Hostels)> Effective Cycling Instructor #218-ck (lapsed)> Professor of Computer Engineering, University of California, Santa Cruz> Undergraduate and Graduate Director, Bioinformatics> === Affiliations for identification only.-- tmlSubject: Re: GRE I'm going to schedule the test for spring break. So mid March.John>.. I have a few questions about the GRE. The school I've all but choose requires both the general and subjecttest. When during my undergrad should I take the general test? I have one full> year of classes left plus one semester of student teaching after that.So> I'm looking at entering grad school Fall 2005. I don't think there is> anything else left in my undergrad that will help with the general test,I> don't have any English classes left and I've taken all the math anynon-math> major would have. Can anyone recommend books to review for the math subject test? I thinkI'll> be ok on the general test, but the math subject worries me a bit.However, I> have the three highest level classes yet to go in my undergrad.> As it happens, I just took the general test recently.> I first took it during my senior year of college, and I wish I'd taken> it earlier; there was really nothing I learned my last year that helped> at all. For the subject test, sure, you want to wait as long as> possible, but you might as well take the general test now. Plus, if you> screw up (I did - wasn't feeling well that day and got under 700 on two> sections) you have time to retake it :-)> The math section is IMO quite easy; both times that I took the test I> ended up with 800. (Computer science major here BTW) You basically> need to be able to do basic algebra, geometry, and statistics, and be> able to read a graph. (Of course, being able to multiply quickly and> accurately by hand helps :-)) When you register for the test they'll> send you a free CD with two sample tests on it, so you can get an idea> of what === you'll be seeing.-- tmlSubject: studenti was need help in calculas.the topic are integration. Are you solve my problem === you solve my problem via enternetPose your problem and somebody will try, I'm sure.-- charlie dickThe right to be left alone -- the most comprehensive of rights, and the right most valued by a free people. - Justice Louis Brandeis, === Olmstead v. U.S. (1928).-- tmlSubject: Re: studentPlease post the problem and what you have done so far.Also try http://www.sosmath.com for math help.> i was need help in calculas.> the topic are integration. Are you solve my problem === via enternet-- tmlSubject: Re: Grad schoolI am a high school counselor in Michigan and I always refer my students to ThePrinceton Review. I am sure that in addition to printed sources available atyour local bookstore, they have an online version available also. The Princeton Review is known for ranking schools in specific subject areas,however, there is also a book out there that I always refer my student to. Itis titled,Index of Majors and Graduate Degrees. It indexes different degreesby subject area and majors. By using these two sources together, you should be able to narrow down whooffers your desired degree and how they rank against other === Salisbury LeVasseurMEd, LPC, NCC/NCSC-- tmlSubject: Re: Grad schoolsusilev2@aol.comnojunk responds:>I am a high school counselor in Michigan and I always refer my students to>The>Princeton Review. I am sure that in addition to printed sources available at>your local bookstore, they have an online version available also. >The Princeton Review is known for ranking schools in specific subject areas,>however, there is also a book out there that I always refer my student to.>It>is titled,Index of Majors and Graduate Degrees. It indexes different>degrees>by subject area and majors. >By using these two sources together, you should be able to narrow down who>offers your desired degree and how they rank against other Salisbury LeVasseur>MEd, LPC, NCC/NCSCWhat is in the school that would make it better or worse for studyingMathematics than in another school? The subject is difficult to study; thestudent must do his own learning, since the instructor can NOT do the learningfor the student. Are the differences based on the oratory skill of faculty;choice of instructional textbooks in use; how well the faculty know theirsubject; course grading policy; the way the courses are sequenced and eachcourse assembledcourse design?This subject started and is still Grad school, so some componant of researchmay be relevant here which can also be treated in any responses to === myadditional inquiry.G C -- tmlSubject: Re: Grad school> What is in the school that would make it better or worse for studying> Mathematics than in another school? The subject is difficult to study; the> student must do his own learning, since the instructor can NOT do the learning> for the student. Are the differences based on the oratory skill of faculty;> choice of instructional textbooks in use; how well the faculty know their> subject; course grading policy; the way the courses are sequenced and each> course assembledcourse design?The faculty generally choose the subject material for the first yearof grad school and present it to the students. The choice ofmaterial, the clarity of the presentation, the enthusiasm of thefaculty for teaching, the textbooks chosen, the intelligence of thefellow students, the amount and difficulty of work assigned, ... allaffect the grad school experience. You want to go to a departmentwhere you are in the top 1/4 (so that faculty will want to payattention to you) but are not the very best (so that classes aren'tbeing dumbed down for your classmates).> This subject started and is still Grad school, so some component of research> may be relevant here which can also be treated in any responses to my> additional inquiry.For a PhD student, the research component should dominate the choiceof grad school. You want to go to a school where you can study asubject of interest to you with someone who is passionately interestedin the material. If you've no idea what you want to do for research,it is best to go to a large department that covers many subjects, sothat you have some choices, and be prepared to change departments ifit turns out that the subject you end up loving isn't well coveredthere. If you know what you want to study, find out who the expertsare in that field and where they are teaching.For a Masters in math, I'm less certain of the best way to choose agrad school---many do not have a high regard for master's students, soone could get short-changed even at schools that do a very good jobfor their PhD students. A school's reputation depends more on thefaculty and the PhD students than the Master's students, so you can'trely just on reputation, and a Master's student is not going to bespecializing so much that a close fit in research interests mattersmuch.-- Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karpluslife member (LAB, Adventure Cycling, American Youth Hostels)Effective Cycling Instructor #218-ck (lapsed)Professor of Computer Engineering, University of California, Santa CruzUndergraduate and Graduate Director, BioinformaticsAffiliations for identification only.-- === tmlSubject: Re: Grad school For a Masters in math, I'm less certain of the best way to choose a> grad school---many do not have a high regard for master's students, so> one could get short-changed even at schools that do a very good job> for their PhD students. A school's reputation depends more on the> faculty and the PhD students than the Master's students, so you can't> rely just on reputation, and a Master's student is not going to be> specializing so much that a close fit in research interests matters> much.> I myself have only a BA in mathematics, and have been recentlyexploring obtaining a master's in mathematics. (Full-time collegeteaching usually requires a subject master's, minimum. Still plan toget a master's in education as well, though - the people in k12 whohave the power of hiring and firing usually want that, not a subjectmaster's. [There are plenty of online master's degrees in education,but none that I know of in math.] Quick note: Only after havinggraduated from a state-approved teacher education program can onequalify for state certification reciprocity, which covers the vastmajority of states. If one obtains certification via an alternativeroute, then one's state certificate is good only for that state. Somehave learned the hard way that if one wants to move to another state,then tough luck.)The student advisor of a math dept that is trying to become recognizedas a top-notch Ph.D program directly told me to stay away from theprogram: The professors there won't think you're serious if all youwant is a Master's.So I'm sort of in the same boat. To obtain what you've talked about -the personal attention by the professors, etc. - I wondered aboutchecking out schools where the Master's program is the terminalprogram. Don't know whether it's a good idea. If it is, anyone havesome thoughts on good, reputable === schools in this regard?Paul-- tmlSubject: Re: Grad school> I myself have only a BA in mathematics, and have been recently> exploring obtaining a master's in mathematics. (Full-time college> teaching usually requires a subject master's, minimum. Still plan to> get a master's in education as well, though - the people in k12 who> have the power of hiring and firing usually want that, not a subject> master's. [There are plenty of online master's degrees in education,> but none that I know of in math.]Are you aware of this program? It is an online/video-based MA inTeaching Math. The good news is that of the required courses only 2 arein education and the rest can be math courses.http://www.uidaho.edu/eo/newhtml/progmath.htmRich-- === tmlSubject: Re: Grad school>The student advisor of a math dept that is trying to become recognized>as a top-notch Ph.D program directly told me to stay away from the>program: The professors there won't think you're serious if all you>want is a Master's.>So I'm sort of in the same boat. To obtain what you've talked about ->the personal attention by the professors, etc. - I wondered about>checking out schools where the Master's program is the terminal>program. Don't know whether it's a good idea. If it is, anyone have>some thoughts on good, reputable schools in this regard?>PaulThe state college or state universities such as California State UniversityCityName have bachelor and master programs, but no PhD programs (or is this notcompletely correct?); The University schools, such as University ofCalifornia at CityName have bachelor, master, and PhD programs. Maybe thisforms a possible but somewhat incomplete way of judging which sort ofinstitutions would want masters students or PhD students --- This in relationto what Paul Tanner has stated. G === C-- tmlSubject: stats/probabilityI am doing an individual project on oreos... however my instructorsays it needs to be a probability project, not just statistical... andI am drawing a blank on how to relate it to probability.the original project I am thinking about is are we really gettingdouble stuff? comparing the weight of the double stuff oreo fillingto regular oreo filling.how do i turn it into something with === stats/probability> I am doing an individual project on oreos... however my instructor> says it needs to be a probability project, not just statistical... and> I am drawing a blank on how to relate it to probability. the original project I am thinking about is are we really getting> double stuff? comparing the weight of the double stuff oreo filling> to regular oreo filling. how do i turn it into something with probability?You could look at the probability that a double-stuff oreo filling isgreater than 1.9 times the weight of a regular oreo. For that, youare not looking at summary statistics of the double-stuff oreos, butthe distribution of weights.You could also look at the probability of a cookie in the bag being abroken cookie.-- Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karpluslife member (LAB, Adventure Cycling, American Youth Hostels)Effective Cycling Instructor #218-ck (lapsed)Professor of Computer Engineering, University of California, Santa CruzUndergraduate and Graduate Director, BioinformaticsAffiliations for identification only.-- === tmlSubject: Re: stats/probabilityYou already have it, just compute the probability of getting a certainweight. In fact we did a similar problem in my stats class.We made a stem and leaf diagram to display their weights. Ours was candybars, but Oreos would work fine. Then use the h(x) function to computeprobabilities.You could also use this a quality control feature, randomly select cookiesoff the production line and weigh them to see if they conform to the factorystandards. Using something called a OC function.Good Luck.> I am doing an individual project on oreos... however my instructor> says it needs to be a probability project, not just statistical... and> I am drawing a blank on how to relate it to probability.> the original project I am thinking about is are we really getting> double stuff? comparing the weight of the double stuff oreo filling> to regular oreo filling.> how do i turn it === tmlSubject: combosif u have 10 numbers how many combonations === of 6 are there.and if there is a formula?-- tmlSubject: Re: combos<>http://lzkiss.netfirms.com/cgi-bin/igperl/igp.pl?dir= === probabilities&name=counting-- tmlSubject: Re: combos<59afvv0ji3h8lne9f256k3q0fit3gl52je@4ax.com>:> if u have 10 numbers how many combonations of 6 are there.> and if there is a formula?Here is a way to think about it.You have 10 numbers to select from. You will choose six for a combinationout of the 10.of your six numbers?10.You have 10 choices for the first number.Once that number has been selected, how many ways can you choose the secondnumber of your six? 9.You have nine numbers left to choose from after you have selected thefirst.How many ways can you choose the third number after the first two have beenchosen?8.There are 8 numbers left to choose from.And so on.Therefore, the number of ways to choose six number from the 10 is:10 * 9 * 8 * 7 * 6 * 5However, many of these combinations are equivalent, since in combinationsthe order in which the numbers are arranged is not important. (Inpermutations it is.) So you need to divide by the number of ways that youcan rearrange the six numbers that you have selected, which is 6! or6*5*4*3*2*1Thus, your final answer is(10 * 9 * 8 * 7 * 6 * 5)/6!or10!/(4!6!)since10!/4! = (10*9*8* ... *3*2*1)/(4*3*2*1) = 10 * 9 * 8 * 7 * 6 * 5and10!/(4!6!) = (10*9*8*7)/(4*3*2*1) = 10*3*7 = 210-- Sheila Kinghttp://www.thinkspot.net/sheila/http://www.k12groups.org/- === - tmlSubject: Re: combosYou have to use factorials....I'm not sure though if it's 10! or 6!.> if u have 10 numbers how many combonations of 6 are there.> and if there is a formula?-- === tmlSubject: Re: combos> You have to use factorials....I'm not sure though if it's 10! or 6!. >> if u have 10 numbers how many combonations of 6 are there.>> and if there is a formula?The term combinations is a confusing one, which has little intuitivemeaning for most people.When I teach combinatorics, I find it easier to talk about choosingout of a set. If you want to choose 6 distinct elements from a set of10 elements, then there are ( 10 ) ( ) , ( 6 )read as 10 choose 6, ways to do it. The formula for n choose k is n! --------- k! (n-k)!I only have my students memorize two formulas fromcombinatorics---this one and that the number of ways to permute ndistict elements is n! .I show them the tricks that are used to prove other formulas, so thatthey can rederive the formulas almost as fast as they can dredge themout of memory, and with greater chance of getting the right formula.-- Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karpluslife member (LAB, Adventure Cycling, American Youth Hostels)Effective Cycling Instructor #218-ck (lapsed)Professor of Computer Engineering, University of California, Santa CruzUndergraduate and Graduate Director, BioinformaticsAffiliations for identification only.-- === tmlSubject: matrices, linear algebra, working the processIs it fair in studying Matrices and systems of linear equations from a collegealgebra textbook to resort to a graphing calculator to do the various problemswhich use 3 x 3 or higher matrices? When I follow the examples in the book, Iunderstand well about the matrix operations; WHEN I WORK THEM BY HAND, if theyare more than 2 x 2, they take a long time to do, and most of the work containserrors that are difficult to find. When I redo them, AGAIN BY HAND, I makeother errors which are just as difficult to find. Still, I understand theexample problems, and some of the exercises I handle successfully. When I usea graphing calculator on which I have been learning how to enter matrixelements and invoke inverse-matrix and reduced row echelon functions, thecorrect results are obtained. Is it common today in college courses of algebra involving the learning ofsystems of linear equations or matrices to let the students use a graphingcalculator during classtime on quizes and tests? If a matrix is 3 x 3 orlarger, it can become very time-consuming and as described, many errors canoccur doing manual problem solving. How is all this managed in instruction? Certainly, understanding the concepts and showing that one can perform themanual skill is important; just that for only the simplest problems time is notheavily taxed. For other problems, so much time is needed for an === assessmentproblem.G C-- tmlSubject: why does lb. mean === poundsthats what i'm trying to find out!!!!!!!!-- tmlSubject: Re: why does lb. mean pounds> thats what i'm trying to find out!!!!!!!!A quick Google on lb pounds led me to:http://www.unc.edu/~rowlett/units/dictP.htmlYou'll find === your answer there.MT-- tmlSubject: Transition to College MathI am looking for Online help with my math. If you know of a websitethat offers free tutoring please e-mail me. THANKS-- === tmlSubject: Problem with equationsI would like to know hot to solve equations like:cos(h1*a1)-cos(h1*a2)+..+cos(h1*an)=0cos(h2*a1)-cos(h2*a2 )+..+cos(h2*an)=0..cos(hn*a1)-cos(hn*a2)+..+cos(hn*an)=0my === questionOk maybe someone can help me. My math teacher wants us to write apaper on whether or not a line is parallel to itself and why. Thusfar, i'm pretty sure that a line is not parallel to itself becauseevery definition i have found specifically says two or more lines. Can anyone tell me whether they agree === QuestionEuclid originally defined parallel lines as lines which, whenproduced (or extended) indefinitely, never intersect. So you can askyourself whether a line ever intersects itself or not.One of the nice things about words is that they allow you be creative,and sound like you're really saying something when actually whatyou've said is completely unclear. Of course the desire to be clear iswhat led the movement in mathematics to using other symbols whosemeaning can be made more precise; but as long as your teacher has theaudacity to give you a writing assignment in a math class, you mightas well have fun with it - in other words, you can probably chooseeither answer and find a way to verbally justify it.(Euclid: Euclid's Elements Volume One, tranla by Thomas Heath, goodchance its at your local library.)Mark-- === pick 4 points out of the space, A,B,C and D.We know that (A,B) pair forms a line and (C,D) pair forms another line.Somehow magically A and B are both on (C,D) line. (because we pick thepoints in the first place, we can make them so).Sometimes people define parallel lines as co-planer lines that never meet.You now have (A,B) line and (C,D) line meet more than once -- and maybeinfinitely times. Can you show that every point on (A,B) is also on (C,D)?Sometimes people define parallel lines as lines with constant distancebetween them. The distance between any point on (A,B) and line (C,D) iszero. (if you can show that every point on (A,B) is also on (C,D)?> Ok maybe someone can help me. My math teacher wants us to write a> paper on whether or not a line is parallel to itself and why. Thus> far, i'm pretty sure that a line is not parallel to itself because> every definition i have found specifically says two or more lines.> Can anyone tell me === Parallel lines questionI've always defined parallel lines (in part) as lines in the sameplane which share no common points, or never intersect. Using thisdefinition, I'd say that a line is not parallel to itself since all ofits points are common.SFS>> Ok maybe someone can help me. My math teacher wants us to write a>> paper on whether or not a line is parallel to itself and why. Thus>> far, i'm pretty sure that a line is not parallel to itself because>> every definition i have found specifically says two or more lines.>> Can anyone tell me whether they agree or if i'm wrong?>>>-- === modern geometry, and we had to define a line. This isthe 300 level in college definition which was a mind opening idea. Thinkabout the symmetries involved in a line, what makes lines parallel? How canyou prove this? Trying thinking in 3D, commonly known as 3 space. If youhave a line in space is it parallel to itself?Good Luck,John> Ok maybe someone can help me. My math teacher wants us to write a> paper on whether or not a line is parallel to itself and why. Thus> far, i'm pretty sure that a line is not parallel to itself because> every definition i have found specifically says two or more lines.> Can anyone tell me whether they agree or if i'm === someone can help me. My math teacher wants us to write a>paper on whether or not a line is parallel to itself and why. Thus>far, i'm pretty sure that a line is not parallel to itself because>every definition i have found specifically says two or more lines. >Can anyone tell me whether they agree or if i'm wrong? Can you document that? If you can, then express what you have to say. Referto the documentation if you believe it to be necessary. It seems to me that iftwo lines in a plane have the same slope, then they are parallel. But a linebeing parallel to itself? Your teacher may be trying to provoke thoughtfulcritical reasoning from you to express this in === parallel to itself? Your teacher may be trying to provoke thoughtful>critical reasoning from you to express this in writing.Likely that is so, as there are reflexive properties === enrichment for a six-year oldYour six-year-old son is a natural whiz at mathematics and you'rewondering how to teach him?I don't get it. Try:(1): Show him where the library is, and agree to take him there.(2): Show him a bookstore with a good science/math section, and agreeto take him there.(3): Find out if there are other kids in your community who are gifat and interes in mathematics, see if you can bring them together.You could consider hiring a math tutor just to chat with them; I thinkthat would be far more valuable than creating a formal program ofinstruction for them. Just let them experience the pleasure andvalidation that comes from community with others who share similarinterests - and then get out of their way! Or rather: be glad thatthey can enjoy learning and that you have been able to provide anenvironment for them where this is possible, without feeling that youhave to control or direct them. (As children there will of coursestill be other areas in their lives where this is appropriate.)(4): If your son's abilities are limi to mathematics and if you donot wish to homeschool (or to unschool), then you might want tosuggest to his public school teachers (and the principal) that theymake a small adjustment: during the time in which the others kids arelearning math, let your son use the school library. Or, if your son isinteres and the other students are receptive to this (and if yourson is capable of being sympathetic to the lesser abilities ofothers), then let your son assist the teacher in teaching math to theother children.A fundamental tension in childrearing is the tension between allowingchildren to enjoy the present and preparing them to be successful inthe future. I would suggest that your son has less reason to beanxious about the future than most, and that you can afford to letyour son make more of his own choices than most -- choices which willnaturally focus most on enjoying the present. My suggestion to you is: don't worry about it! Or to put it more positively: learn toidentify with your son's natural === === enrichment for a six-year oldSubject: Re: Math enrichment for a six-year oldAuthor: Mark V. (2): Show him a bookstore with a good science/math section, and agree>to take him there.>(3): Find out if there are other kids in your community who are >gif>(4): If your son's abilities are limi to mathematics and if you do>not wish to homeschool (or to unschool), then you might want to>suggest to his public school teachers...Mark V., I appreciate your sentiment, but please try the following littlethought experiment. In your essay, replace every reference to mathwith a reference to music and see how it parses. I suspect that no parent whose child is seriously gif andinteres in music would ever consider any one of your suggestions. On the contrary, the main mission of musically gif families is toget their progeny into serious music schools like Juilliard, Manes,etc. No doubt, the students will do some hanging out with other studentsof a similar bent, but that will happen in a context of a program ofstudy that is very serious, indeed. Why would you advise a mathematically talen === for a six-year-old,My suggestions #1 - #3 were not really intended as suggestions, but asquestions: if this kid truly understands that some infinities arebigger than others (i.e. can follow Cantor's arguments), then hewould seem quite ready to decide for himself what his interests wereand where to focus his efforts. Naturally the adults in his life canstill be helpful to him in the sense of informing him about possibleresources, finding like-minded people, etc. But I decided to respondto test whether something else was going on: did this kid's motherfeel the need to turn this kid into the next Einstein. Was she in factrespectful of his interests, looking for ways to be helpful, or wasshe trying to program his life for him? Should this kid be readingEnder's Game, to get a glimpse of how controlling and manipulativeadults can be?Changing Math to Music doesn't really change things as far as I cansee; at some point you will want to inform them of the existence ofplaces like Juilliard, perhaps take a field trip to visit the placeearly on to see if the kid is interes; but that doesn't mean youhave to decide for them You are going to be a musician and you aregoing to spend x hours a day practicing in order to accomplish that,and that's just the way it is. (Of course one of the things that canbe learned by meeting the kids at Juilliard is that it DOES takepractice and discipline to get there. But you can still allow yourchild to decide for === Math enrichment for a six-year oldMark V's suggestions #1,2, 3 are good. In fact that's what we didwith Matt. He had an unerring eye for the math sections of bookstores.We were also lucky to have a Math Circle that he could join.But I beg to differ with suggestion 4. We've encountered teachers whodid not believe in tracking. This applied to math and science;theytracked in humanities without realizing it, like the teacher who'dgiven my older son her own teaching resources on a topic he'd gotteninteres in, or another who let him write a 19- page paper when theassignment called for a minimum of 4 because he'd gotten fascina bythe topic. But to return to Matt, besides that egalitarian teacher,we encountered inexperienced teachers who were too overwelmed toenrich, or they demanded that he do all the required work before hecould do more challenging problems. This meant he spent too much timedoing busywork to be able to tackle the enrichment problems we sentalong, but had too much time left after completing the busywork. Mattput his geometry skills building origami paper airplanes that gotflown in class and him in trouble. He fiercely resen doing mathhe'd learned several grades earlier, and he had little respect forteachers whose math knowledge was inferior to his (that was in 5th and6th grade). The tears we experienced was because the math was tooeasy, not because it was too hard. We had a near-delinquent on ourhands until he was allowed to do calculus in 7th grade on his own. Don't get me star on the gif child as assistant. Each child hasthe right to learn to the highest of his or her potential. No childshould be an unpaid teacher, though this is what is probably going tohappen in one course for which Matt knows 2/3 of the materials for theyear (the price to pay for auditing a math course in college). Mattused to say: I know so much more math than my classmates, so manythings are obvious to me that it does not occur to me that I shouldexplain them. The tension I saw with Matt in grade school was not about enjoying thepresent vs. preparing for the future. Each child's present isdifferent. Matt's happened to be 4-5 grades above that of hisclassmates. But he was forced to live in his own past by the school'sidea of what the present curriculum should be (and to be fair, whatthe majority of his classmates were). It's the equivalent of beinggiven a diet of primers when one can read === enrichment for a six-year-oldI myself have had the experience of being an unpaid tutor; these daysits called cooperative learning.I can sympathize, but do not really see your comments as a response tomy suggestion. If you will reread my comments, you will notice the keyphrase if your son is interes.On the other hand I do feel now that it would be just as well towithdraw my suggestion #4. Few activities that are compulsory areeither fun or good for you; it would be far === for a six-year old> Your six-year-old son is a natural whiz at mathematics and you're> wondering how to teach him?> I don't get it. Try:> (1): Show him where the library is, and agree to take him there.> (2): Show him a bookstore with a good science/math section, and agree> to take him there....One of the problems with aiding a mathematically gif child isfinding materials at the appropriate level---too easy and they areboring, too hard and they are frustrating. Most small publiclibraries have no text books and precious little other math materialsuitable for a mathematically gif child. Same for bookstores.School libraries often have textbooks, but they are often poorlychosen for gif students, being heavy on repetitive drill ofmaterial that the gif students master quickly. There are a fewbookstores that may have suitable material (Math'n'stuff in Seattle,for example), but it is buried among piles of remedial math books, sojust showing a child the store is hardly going to be helpful.Unlike choosing a fiction book, you can't just browse a page or two tosee if it is at the right level and well enough written. There are alot of rather crummy math books out there (since most people who buythe books are incapable of distinguishing a good one from a bad one),so critical advice from knowledgeable people is very valuable.The other advice (finding a community of math-interes children ofsufficiently similar skill levels) is also very difficult toaccomplish. This may be why the distance-learning programs like EPGYhave become such a big deal---they provide a community of sorts forthe kids who otherwise are quite isola in their interests.Having a child skip useless math instruction in school may helpalleviate boredom, but does not further his or her math learning.Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karpluslife member (LAB, Adventure Cycling, American Youth Hostels)Effective Cycling Instructor #218-ck (lapsed)Professor of Computer Engineering, University of California, Santa CruzUndergraduate and Graduate Director, BioinformaticsAffiliations for identification only.-- === MATT, with Mattsmom,Did Matt skip (omit) College Algebra-PreCalculus-Elementary Functions PlusTrigonometry, the typically recommended course of study for before studyingFreshman first year Calculus? This is the course in which polynomialfunctions are studied in a rigorous manner including finding zeros, graphing,the remainder and factor theorems, finding roots of polynomial functions, thefundamental theorem of algebra...... usually topics beyond what one studies inthe Intermediate Level of Algebra.The College Algebra course is typically more difficult since it typically goesbeyond intermediate. At Intermediate level, you usually deal with quadraticfuncitons and conic section equations; in college Algebra you deal withpolynomial functions and again at least review of conic sections. Further, Matt may like computer programming. Some of Algebra study can includebasic problems examined by a custom-made-by-student program. Estimating zerosof polynomial === enrichment for a six-year oldDid Matt skip (omit) College Algebra-PreCalculus-Elementary FunctionsPlusTrigonometry, the typically recommended course of study for beforestudyingFreshman first year Calculus?Answer:Matt's 7th grade teacher recommended a precalculus textbook (Stewart,Redlin & Watson) which Matt finished in one semester; so he then gavehim a copy of Finney/Demana/Waits/Kennedy which Matt finished half-waythrough 8th grade.Matt loves pure math rather than applied math. He began to readAbelson & Susssman in the summer of 7th grade. He covered the first 3chapters but has so far not found computer science captivating enoughto finish the last two chapters (he did learn Scheme, though).He studied Multivariable Calculus and Linear Algebra at the HarvardExtension School last year. There are some applied math coursesavailable through the Extension School, but he does not want to takethem, so we're trying to make it possible for him to audit a mathproof course. All are willing for him to do so; it's theincompatible schedules that are difficult, and the Mass legislature'sinsistence that all slots on high school schedules === enrichment for a six-year-oldFirst let me apologize to everyone for accidentally starting a newthread on this topic. But I'm glad that it crea enough interest fora couple people (so far) to take the trouble to respond.Kevin, I've lived the last 25 years in Santa Cruz, San Diego, andBerkeley, and it's true that I've been spoiled by having easy accessboth to good science/math bookstores and to good libraries (Universityand Community Colleges). For many people this could indeed be aproblem; but since mxlptlx mentioned that her son understands someinfinities are bigger than otheand he can make up and doFibonacci sequences in his head, I assumed that apparently mxlptlxhad local access to math/science books.As you say, critical advice from knowledgable people is veryhelpful; in fact that is just as true when looking for good fictionto read.But rather than recommending particular resources (and without havingprior knowledge of me, how could my suggestions in this area betrus?), I chose to focus on the idea that, in addition to beingable to learn mathematics by reading a math book someone has given himto read, mxlptlx's son just might be able to learn how to do some ofthe other things that the rest of us at some point need to learn howto do, like recognize a well-written book when you see one.Poincare mentioned that the talent needed for creative mathematicalwork was an aesthetic sense, and Einstein said similar things. Itseems to me that one of the ways you might begin to develop this senseis by learning on your own, by trial-and-error, how to tell goodwriting - and - good mathematics, from bad writing - and - badmathematics.For the sake of clarity, I am talking here about balance; of course itsaves time to make use of Amazon Reviews and the opinions of peopleyou know; I just don't think you have to do it all for him.Now, speaking of recommendations: books on the history of mathematicsoften don't have much mathematics in them, but they can be quiteuseful for discovering interesting topic areas.Another book that may be interesting in this sense may be Garrity'sAll the Mathematics You Missed which is only fairly well written,but I think is unique in that it provides an overview of theUndergraduate Math Curriculum.Another interesting one might be a practice exam book for the MathGRE - showing what kind of problems you WILL be able to solve once youlearn the rela theory.-- === I've lived the last 25 years in Santa Cruz, San Diego, and> Berkeley, and it's true that I've been spoiled by having easy access> both to good science/math bookstores and to good libraries (University> and Community Colleges). Santa Cruz has very little in the way of library or bookstore sourcesfor math and science material appropriate for an advanced 6-year-old.I know, because I've spent a lot of time looking here. He's not readyfor college-level math yet, and there isn't much locally availablebetween the remedial elementary school level and college level. I'vehad to rely heavily on purchases made over the web.> For many people this could indeed be a> problem; but since mxlptlx mentioned that her son understands some> infinities are bigger than otheand he can make up and do> Fibonacci sequences in his head, I assumed that apparently mxlptlx> had local access to math/science books.As you say, critical advice from knowledgable people is very> helpful; in fact that is just as true when looking for good fiction> to read.> But rather than recommending particular resources (and without having> prior knowledge of me, how could my suggestions in this area be> trus?), I chose to focus on the idea that, in addition to being> able to learn mathematics by reading a math book someone has given him> to read, mxlptlx's son just might be able to learn how to do some of> the other things that the rest of us at some point need to learn how> to do, like recognize a well-written book when you see one.> Poincare mentioned that the talent needed for creative mathematical> work was an aesthetic sense, and Einstein said similar things. It> seems to me that one of the ways you might begin to develop this sense> is by learning on your own, by trial-and-error, how to tell good> writing - and - good mathematics, from bad writing - and - bad> mathematics.When the books are not available locally to look at, this becomes verydifficult. In any case, my son relies heavily on adultrecommendations for books to read---he'll pick up a series or authorhe is familiar with and he'll browse through very short books, but forlonger books he does not browse the library shelves, but relies onadults (librarians or parents) to suggest books to him. For math books, the appropriate level for him would be books thatinclude material he does not already know but for which he has thenecessary foundations to learn them quickly. This is a very difficultthing for him to judge. It is hard for ANYONE to determine whethersomething they do not know yet is of the appropriate difficulty. Thisis quite a different matter than choosing a fiction book to read,where a small sample of the book usually suffices to show whether ornot you have the necessary vocabulary and reading skills to be able toread the book (whether or not you enjoy the book make take a muchlarger sample).As a university professor, I find it difficult to choose the besttexts for a course where I already KNOW the material. Expecting a6-year-old to be able to find the right books to study from in alibrary that more often than not won't HAVE the right books is simplyridiculous.> For the sake of clarity, I am talking here about balance; of course it> saves time to make use of Amazon Reviews and the opinions of people> you know; I just don't think you have to do it all for him.Now, speaking of recommendations: books on the history of mathematics> often don't have much mathematics in them, but they can be quite> useful for discovering interesting topic areas.> Another book that may be interesting in this sense may be Garrity's> All the Mathematics You Missed which is only fairly well written,> but I think is unique in that it provides an overview of the> Undergraduate Math Curriculum.> Another interesting one might be a practice exam book for the Math> GRE - showing what kind of problems you WILL be able to solve once you> learn the rela theory.These suggestions may be appropriate in a few years---a 6-year-oldwho is just beginning to play with Fibonacci sequences and differentnotions of infinities is not ready for the GRE.-- Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karpluslife member (LAB, Adventure Cycling, American Youth Hostels)Effective Cycling Instructor #218-ck (lapsed)Professor of Computer Engineering, University of California, Santa CruzUndergraduate and Graduate Director, BioinformaticsAffiliations for identification only.-- === Kevin, I happen to think that the math section at Logos indowntown Santa Cruz is pretty good, although of course not as good asCody's in Berkeley.It was my understanding that this particular six-year-old was able tofollow Cantor's arguments about different infinities, in which case hewould seem to be ready to start making use of, for example, the UCSCScience === six-year-old> Well, Kevin, I happen to think that the math section at Logos in> downtown Santa Cruz is pretty good, although of course not as good as> Cody's in Berkeley.Logos has a random collection of used books---and almost no math booksfor 6-year-olds (not even bright 6-year-olds).> It was my understanding that this particular six-year-old was able to> follow Cantor's arguments about different infinities, in which case he> would seem to be ready to start making use of, for example, the UCSC> Science Library.Nope, that doesn't follow at all. A child may be able to understand Cantor diagonalization whencarefully explained without being able to read college-level mathbooks. I went through Cantor diagonalization with my seven-year-oldson once last spring when we were talking about infinities, and hepretty much understood it (though I suspect he did not grasp itcompletely and could not reproduce the argument 6 months later). Thisdoesn't mean he is ready for a college freshman text like Rosen'sDiscrete Math and Its Applications, nor even for high-schoolalgebra, and he certainly isn't ready to start browsing the graduatetexts in the UCSC Science Library!Even college students can have difficulty finding a book that explainsthings clearly enough in the UCSC Science Library---or any librarywhich has a lot of graduate-level math texts. When I have gonelooking for a book to explain some statistics or in some other fieldwhere I have not had much training, it often takes me hours goingthrough dozens of books to find something comprehensible that coversthe subjects I'm looking for.The challenge with bright young children is to find them material thatpiques their interest and challenges them without overwhelming them.Mark's suggestions seem appropriate for a bright teenager, but not forbright 6- and 7-year-olds.-- Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karpluslife member (LAB, Adventure Cycling, American Youth Hostels)Effective Cycling Instructor #218-ck (lapsed)Professor of Computer Engineering, University of California, Santa CruzUndergraduate and Graduate Director, BioinformaticsAffiliations for identification only.-- === found out that Matt liked math, we trea math as a form ofentertainment and had no particular program in mind.Below are some books he used. I cannot remember exactly when, butdefinitely all before 3rd grade was over. Sometimes he read the booksby himself, sometimes he asked that someone pose him a problem, oftenover dinner. --Charles Barry Townsend, World's Toughest Puzzles (NY: Sterling,1990).______________________, World's Most Baffling Puzzles (NY; Sterling,1991)--William Simon, Mathematical Magic, (NY: Dover, 1964).--Arthur Benjamin & Michael Brant Schermer, Mathemagics: How to LookLike a Genius Without Really Trying (Los Angeles: Lowell House, 1993).--C. Lukas & E. Tarjan, Mathematical Games (NY: Barnes & Noble, 1963)Pentagram, Puzzlegrams (NY: Simon & Schuster, 1989)--David Colton, Mensa Presents The Covert Challenge (NY: Barnes &Noble, 1999).--Carolyn Skitt, Mensa Presents Mind Games for Kids 9NY: Barnes &Noble, 1997)--Robert Allen, Mensa Presents Mind Mazes for Kids (NY: Barnes &Noble, 1995).He also read many many logic puzzles books which fell apart and whichI discarded.Other resources for younger kids are lis at:http://www.hoagiesgif.org/math.htmWe have not made use of these as we discovered that site after Mattbegan precalculus and thus embarked on a structured program of === Harry Potter and the Mathematics of Doom jamsportlandwww.jamsportland.comThe Art of Mental Calculation === === for a 6 year oldSubject: Re: Math enrichment for a 6 year oldAuthor: remove .de twice to email Subject: Re: Math enrichment for a 6 year old> Author: remove .de twice to email lob me this softball. But, Karl, as long as your asking...[snip of a lot of good stuff]I'm glad I asked. Of course I agree that the basic problem here is thecompulsory, sole-source provider, government-opera school system, whichhas so much built-in inflexibility that a lot of issues that ought to befeasible become difficult.I have a lot of interest, as a home educator of mathematics, in knowing whena child is really done with one level, and ready to go on to the next. Theradically different approach I see to various mathematical topics in booksfrom China, say, as contras with books from the Uni States makes meever eager to reconsider topics that we supposedly have already covered, butfrom another point of view. In that sense I am probably not accelerating asrapidly as is strictly possible for my son, but I hope I am laying a goodfoundation for him so that he doesn't hit the wall later. He has just begunthe accelera Algebra I & II course in the University of MinnesotaTalen Youth Mathematics Program. My biggest regret about this so far isthat my son drew the section (one out of five sections) with NO girls. Thetheory is that to achieve a critical mass of girls in sections that havegirls at all, it's best to have close to a 50:50 ratio of boys and girls ina section with girls, and enrollment imbalance is such that one section endsup with no girls at all, and my son was placed in that section. I reallywan him to meet more girls who have an interest in math. I think by nextyear (geometry and precalculus in one year) all the sections will havegirls, after attrition, and then my son can get that social benefit that Ican't provide him at home with his two younger brothers and infant youngersister.Best wishes on finding what fits for your son.-- Karl M. Bunday Christ has set us free. Galatians 5:1Learn in Freedom (TM) http://learninfreedom.org/kmbunday AT earthlink DOT net === CalculatorI have a freeware calculator that I have spent about 3 months updatingand now have up on my website for download. I need user feedback onusability, any bugs, sugges features, etc. It does expression evaluation, programable with VB Scriptinglanguage, does 2D (polar, cartesian and parametric) and 3D plotting,Matrix manipulation, complex number math, 1-variable equation solving,simultaneous equation solving (2x2 to 8x8), Integer math to 32000places, 2D statistics with graphing and regression curve fitting, unitconversions and geometric area/volume calculations. For documentingcalculations, it has a built in RTF editor, image editor, formulaillustrator and schematic capture module. The design star out as an aid for engineers but has evolved moreinto a learning tool for math students. The link is: www.dazyweblabs.com/dazycalc/index.htmlAny feedback would be === -Could someone tell me, in general terms, since I'm not really amathematician --if: 1) one were to be thinking about a specific topic, and 2) then within seconds, randomly pick up a 1500 page book thataddresses 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 pagethat the topic existsI know you probably don't have all the data to make any kind ofanalysis, but what are the ballpark chances of something like thathappening? Or, what are the minimum chances === 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 doubtless worrying about whether you are afflicted withprecognition or some other form of ESP. Don't---the probabilityis not so small it could never happen in the lifetime of theuniverse. So I would be willing to bet you do not possess anunsuspected psychic talent. or any other, for that matter. (Iwould also bet you find that my blanket dismissal of psychicabilities annoying... ;-)Same with dreams, where you dream about someone you haven't seenfor a while and unexpectedly meet them the next day. I once estim-ated it should happen to everyone every few years, by sheer chance.To do your book problem, first ask How many distinct topics arethere in my head? Then ask yourself how many books you own, whetheryou have ever read this book before, whether it is new or used,whether the topic is perhaps a popular one. In the latter case,and if the book were from a library, it would not be surprisingthat it fell open to the right place, since that is easier thanopening it to another place. But in any event, the frequency ofunlikely events occurring by sheer chance surprises most peoplewho trouble to work out the probabilities.You also have to rule out unconscious or subliminal clues, evensmells. Our non-visual senses are much sharper than we realize,and we depend on them much more than we think we do.-- Julian V. NobleProfessor Emeritus of ^^^^^^^^^^^^^^^^^^http://galileo.phys.virginia.edu/~jvn/ Science knows only one commandment: contribute to science. -- === Bertolt Brecht, Galileo.Subject: Calculate numerical value of special integralI'd like to calculate the value of the integralint_0^infty x^{n-2}exp(-x^n) dxSimple methods like Runge-Kutta and similar failed. Can anyone give me some hints on how to solve such a problem?Christian-- Christian Meisl www.amft.tu-graz.ac.at Inst. f. Apparatebau, Mech. Verfahrenstechnik und Feuerungstechnik---------------- What color is a chameleon on a mirror? ----------------PGP fingerprint: 33C7 9C54 6E92 6861 === E674 3F71 28DC 7A6F F5A3 D5A9Subject: 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.Change variable, y=x^n/2. After some algebra,int_0^infty x^{n-2}exp(-x^n) dx= n/2 int_0^infty y^{1-2/n}exp(-y^2) dyUse an adaptive quadrature code that understands the possibleintegrand singularity at y=0. A very good code (C, Fortran both) isdint() from the JPL Math77 library now sold commercially. This isavailable in C and Fortran. Search Google for Math a la Carte. Youwill be lucky.You truncate the upper limit to some value, say 5, to get the accuracyyou === need.DTBSubject: 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 CantrellSubject: 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, it's not numerical analysis but:If you're familiar with the Gamma function, then your integral is just 1/n * Gamma((n-1)/n)if n is, say, a === positive integer.David CantrellSubject: 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, it's not numerical analysis but:> If you're familiar with the Gamma function, then your integral is just 1/n * Gamma((n-1)/n) if n is, say, a positive integer. David CantrellDoesn't require n a positive integer. All that is necessary for convergenceis Re[(n-1)/n] > 0 . And other values can be defined by analytic continuationvia the reflection formula for gamma functions.-- Julian V. NobleProfessor Emeritus of ^^^^^^^^^^^^^^^^^^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> Simple methods like Runge-Kutta and similar failed. Can anyone give> me some hints on how to solve such a problem? Well, it's not numerical analysis but:> If you're familiar with the Gamma function, then your integral is just 1/n * Gamma((n-1)/n) if n is, say, a positive integer. David Cantrell Doesn't require n a positive integer.True. (I hadn't said it was a requirement.)> All that is necessary for convergence is Re[(n-1)/n] > 0 .Perhaps it should be noted now, since we're speaking more generally, thatthe integral may converge, but not to 1/n * Gamma((n-1)/n). For example,it converges to the negative of that for real n < 0. So, for real n notin [0,1], one might write that the integral is 1/|n| * Gamma((n-1)/n)David> And other values can be defined by> analytic continuation via the === reflection formula for gamma functions.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?Matlab's quad8-Function works fine for me for n>1 (it uses Simpson's Rule). It also depends on how you set your upper integration limit - it might not be necessary to integrate up to 10000 or 1000 - depending on your n 10 might also be sufficient. You can check that by increasing your upper integration limit and === comparing the integration result.NilsSubject: Re: Calculate numerical value of special integral>> int_0^infty x^{n-2}exp(-x^n) dxIs there a special need for a numerical solution? You can also do that analytically:int_0^infty === x^{n-2}exp(-x^n) dx=n=2: 1/2*sqrt(Pi)NilsSubject: What is a Jacobi Line Preconditioner?Could some kind person explain (preferably in simple English) what aJacobi Line Preconditioner is? My reason for asking: I am solvingequations for a mesh structure using conjugate gradient. The meshproduces a large number of equations, but they structure into atridiagonal banded sparse matrix. The diagonal terms are postive andall are greater than zero. There are two upper and two lower bands andthey are symmetric. So its very typical of the equations for a meshand - most important for a PC - very efficent in storage. I use Jacobias a preconditioner, but I've read that there are better ones.However, whenever I look at a new candidate, the preconditioniningmatrix - or rather its inverse - tends to fill in between the bandsand results in too high a storage requirement. I must, in other words,find a preconditioner which does not increase storage. Literature doesnot often talk about the storage requirements of the inverse matrix!I've read references to a Jacobi Line Preconditioner, but I can't seemto find a description anywhere, and I wondered what it was and if itwould work === without increasing my storage.Subject: Re: What is a Jacobi Line Preconditioner?> Could some kind person explain (preferably in simple English) what a> Jacobi Line Preconditioner is? My reason for asking: I am solving> equations for a mesh structure using conjugate gradient. The mesh> produces a large number of equations, but they structure into a> tridiagonal banded sparse matrix. The diagonal terms are postive and> all are greater than zero. There are two upper and two lower bands and> they are symmetric. So its very typical of the equations for a mesh> and - most important for a PC - very efficent in storage. I use Jacobi> as a preconditioner, but I've read that there are better ones.> However, whenever I look at a new candidate, the preconditionining> matrix - or rather its inverse - tends to fill in between the bands> and results in too high a storage requirement. I must, in other words,> find a preconditioner which does not increase storage. Literature does> not often talk about the storage requirements of the inverse matrix! I've read references to a Jacobi Line Preconditioner, but I can't seem> to find a description anywhere, and I wondered what it was and if it> would work without increasing my storage.In 2D problems, dont waste time and energy with iterative solvers.Unless you are talking about tens of === millions ofequations, use a direct solver.Subject: Re: What is a Jacobi Line Preconditioner?> equations for a mesh structure using conjugate gradient. The mesh> produces a large number of equations, but they structure into a> tridiagonal banded sparse matrix. The diagonal terms are postive and> all are greater than zero. There are two upper and two lower bands You're contradicting your self. Your matrix is *penta* diagonal, Ipresume.> I use Jacobi> as a preconditioner, but I've read that there are better ones.In your case you might even use FFT to solve the system directly. Checkout Fishpack.> However, whenever I look at a new candidate, the preconditionining> matrix - or rather its inverse - tends to fill in between the bands> and results in too high a storage requirement.That happens if you perform a full factorisation. Try looking at ILU(Incomplete LU).> find a preconditioner which does not increase storage. Literature does> not often talk about the storage requirements of the inverse matrix!That is because the inverse is usually full. Good thing that you don'tneed the inverse in general.> I've read references to a Jacobi Line Preconditioner, but I can't seem> to find a description anywhere, and I wondered what it was and if it> would work without increasing my storage.I suggest you check out www.netlib.org/templates.Line Jacobi uses the diagonal blocks as the entities that you divide by.However, that is the mathematical formulation and it doesn't mean youexplicitly invert them in practice. You can factor them, which is veryspace efficient, and then you solve systems with them.V.-- === mail me at lastname at cs utk eduSubject: Re: Positive Definate MatricesThe following paper might be of interestK.C. Toh, A note on the calculation of step-lengths in interior-pointmethods for semidefinite programming, Computational Optimization andApplications, 21 (2002), pp. === 301--310.Subject: Re: Positive Definate MatricesYou can also evaluate the eigen values of the given matrix and test if all of them are positive. For a positive definite matrix they have to be positive.I am not sure if this is computationally more efficient but is one of the possible methods.raju Can anyone tell me how to/point me to some literature telling me how> to detrermine wether a given matrix is positive definate> programmatically. Moreover, can I write some code to determine whether a given matrix is> positive definate? The basic definition of positive definate implies the need to test the> given NxN matrix against all non-zero N-dimensional vectors (which> obviously aint gonna happen). Are there any techniques === Mitch.Subject: Re: Positive Definite Matrices You can also evaluate the eigen values of the given matrix and test if> all of them are positive. For a positive definite matrix they have to be> positive. I am not sure if this is computationally more efficient but is one of> the possible methods.Calculating the spectrum takes significantly more time than computingthe Cholesky factorization.Both are O(n^3) for a dense matrix, but witha larger factor in the O for the spectral approach. For narrow band matrices, factorization is O(n) while eigenvalue computationstake O(n^2) work with QR and O(n log n) with divide and conquer.For irregular sparsity patterns the ratio is usually even worse.Thus definiteness should always be checked by attempting Cholesky or LDL^T, not by calculating eigenvalues. For hand calculations, LDL^T is preferable, even for n=2, where it amounts to checking A11>0, A22>(A12+A21)^2/4A11, or equivalently A11>0, 4*A11*A22>(A12+A21)^2, and in the symmetric case, A11>0, === A11*A22>A12^2.Arnold NeumaierArnold NeumaierSubject: Re: Positive Definate Matrices> You can also evaluate the eigen values of the given matrix and test if> all of them are positive. For a positive definite matrix they have to be> positive.... or have positive real part. Some people include symmetry in thedefinition of pd, others don't.V.-- mail me at === lastname at cs utk eduSubject: Re: Positive Definate Matrices > You can also evaluate the eigen values of the given matrix and test if> all of them are positive. For a positive definite matrix they have to be> positive. ... or have positive real part. Some people include symmetry in the> definition of pd, others don't. V.> --> mail me at lastname at cs utk eduPositive-definite requires hermiticity, otherwise it has no meaning.Thus the eigenvalues are always real (and of course, positive).-- Julian V. NobleProfessor Emeritus of ^^^^^^^^^^^^^^^^^^http://galileo.phys.virginia.edu/~jvn/ Science knows only one commandment: contribute to science. -- === Bertolt Brecht, Galileo.Subject: Re: Positive Definate Matrices> Positive-definite requires hermiticity, otherwise it has no meaning.Why on earth? What is so meaningless about x^tAx>0 for all x in R^n?And there is plenty of theory that only needs the x^tAx condition tohold. M-matrices, convergence of iterative methods,convection-diffusion, ...Over to you.V.-- === mail me at lastname at cs utk eduSubject: 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 threadm=bb4e0k%242k17%241%40ns.felk.cvut.cz&rnum=1&prev=/ groups%3Fq%3Dpositive%2Bdefinite%2BHurak%2Bgroup:sci.math.*% 26hl%3Den%26lr%3D%26ie%3DUTF-8%26oe%3DUTF-8%26group%3Dsci.math .*%26selm%3Dbb4e0k%25242k17%25241%2540ns.felk.cvut.cz%26rnum% === dataI'm looking at several questions from a 10,000 person survey in whichrespondents were asked about risk behavior. I want to create a new variablein my dataset that scores respondents1 2 or 3 based on level of riskaversion.The score will be based on a composite of the risk questions. Each of therisk questions asked respondents to rate their level of avoidance of certainrisky behaviors on a scale of 1 to 10. The distribution of responses isextremely different from question to question.How can I tell what the cut off point is for each of the three risk groupsin the new variable? Has anyone worked === with a model like this?Subject: Fast-Bezier curve ; unicity ?Let us denote SUM=SUM_{k=0 to k=3} and for g:[0,1]-->Rput G[g]={(x,y) | x in [0,1] , y=g(x)}.For k in {0,1,2,3} let x_k=k(k+1)/12 P_k=(x_k , y_k) , k in {0,1,2,3} ,where y_k= f(x_k) , f:[0,1]-->R .In order to describe the shape of G[f], I tried to find a curve (C) having parametric equations x=X(t)=t(C) , t in [0,1] , y=Y(t)=SUM c_k(t)*y_k where I imposed the following conditions on Y(t) : a) Y must be continuous on [0,1] with Y(0)=y_0 , Y(1)= y_3 , b) Y must be derivable of any order on (0,1) , c) c_k(t) >= 0 , t in [0,1] , k in {0,1,2,3} d) SUM c_k(t)=1 , SUM c_k(t)*x_k = t , e) if G(f) is convex on [0,1], then (C) must be convex . I found a solution of the problem , namely c_0(t)=(1+3*y)*(1-y)^3 , y:=y(t)=sqrt(t) c_1(t)=6*t*(1-y)^2c_2(t)=4*t*y*(1-y)c_3(t)=t^2 advance, Alex. ( Note: These c_k(t) are notpolynomials in classic sense. However Y(t) is a generalized polynomial in the === powers {1,y,y^2,y^3,y^4 }. )Subject: Re: Looking for examples of bad data analysis> I wonder if the entire Polywater event could be considered as an> example.Polywater, the 5th Force, High-Temperature Superconductivity, and Cold Fusion are all examples of fairly recent observations. Of these, only one of them stood up to close scrutiny. However, I'm not sure what role bad data analysis played in these.$.02 -Ron === ShepardSubject: Re: Looking for examples of bad data analysisThe easiest mistake is in cosmology, when calculating the function>between a thing's distance from us and it's speed. Normally it's taken>that V = D*H where H is a constant (I think it's called the hubble>constant.) is exact, but this violates relativity in that you can get>V = c. Uh-huh. And does anyone want to guess what happens when you get to such distances? (Hint: Hubble's constant has had to be revised as data comes in, since its value is related to the age of the === universe.) :)-- Matthew Funke (mff@hopper.unh.edu)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.N-rays--major dustup about 100 years back, reported in ScientificAmerican many moons ago. A French physicist, Rene Blondlot, wasconvinced that he had discovered a new kind of radiation, which had alot of unusual effects--increasing visual acuity, stabilizing arcdischarges, and several others. Everyone in France replicated theexperiment easily, but nobody outside Europe could do it. Finally,R. W. Wood was dispatched to Blondlot's lab, to try to get to thebottom of the discrepancy. In one experiment, he was ushered into adarkened room in which a hardened steel file (a strong N-ray source)was placed at the entrance slit of an N-ray spectrograph using a solidaluminum prism, with an electric arc at the movable exit slit. Bynoting positions where the arc was more stable than others, an N-rayspectrum could be produced, which showed spectral lines.In the darkness, Wood reached over and surreptitiously_removed_the_prism_ from the spectrograph. An N-ray spectrum wasproduced all the same.When the story came out, everyone outside France abandoned N-raysright away, but Blondlot and some others went to their graves claimingto === believe in them.Subject: Re: Looking for examples of bad data analysis>The easiest mistake is in cosmology, when calculating the function>between a thing's distance from us and it's speed. Normally it's taken>that V = D*H where H is a constant (I think it's called the hubble>constant.) is exact, but this violates relativity in that you can get>V = c.> There are several misconceptions hereIndeed.> 1) That the recessional velocity is a physical velocity constrained to> be less than c 2) That V = D*H holds without modification over cosmological distancesV - D*H DOES hold without modification over cosmological distances, but neither the velocity nor the distance concerned are observables. For low redshifts, various distances are degenerate, so one can apply this equation to other distances and velocities. V = D*H is necessary in a universe which is homogeneous and isotropic and is expanding (or contracting); no dynamics here, just kinematics.> 3) That H is large enough that the recessional velocity approaches c> for less than cosmological distances 4) That cosmologists are not, in fact, aware of all this.Unfortunately, some are not. Even some professional ones. (I'm not aware of anyone who has actually published wrong results based on some misunderstanding here, but some cosmologists who work mainly in other fields do unfortunately teach cosmology courses and === spread misconceptions.)Subject: Re: Looking for examples of bad data analysis> Everyone who deals with radiation biology is aware of the existence> of a threshold. However, the International Atomic Energy Agency (IAEA)> has agreed for the purpose of setting dosage standards for the public,> to use linear extrapolation, because it provides an upper bound to> the number of radiation-induced cancers. (That is, the straight line> plotted from high doses and passing through zero dose lies _above_> the actual curve.)> Has there been any consensus on where this threshold is, and what the> with little biology background?There are many. The most comprehensive library-type book that I've found onthe LET/ threshold / benefit discussion is Radiation Hormesis, T.D.Luckey, 1991, === CRC Pressgreywolf42ubi dubium ibi libertasSubject: Re: Looking for examples of bad data analysis[elided]See the following for informed insight:Elaine RonNCRP REPORT NO. 136 ON THE SCIENTIFIC BASES FOR LINEARITY IN THEDOSE-RESPONSE RELATIONSHIP FOR IONIZING RADIATION, Arthur C. Upton etal.-- Ciao,Gerry T.______It (classical thermodynamics) is the only physical theory ofuniversal contentwhich I am convinced will === never be overthrown, ... A. Einstein.Subject: Why are irrational numbers not countable?If I'm reading right, countable sets are defined as those that can bemapped one to one to the natural numbers. Integers and rationalnumbers are considered countable, even though there are more of them(e.g. 2 integers for every natural number). Sets of countable setsare considered countable. So why not irrational numbers?Consider any number with N decimal digits. The set of those numberscan be mapped to 10^N or fewer natural numbers, and so is countable. This will remain true even if you insert a decimal point somewhere inthe sequence of N digits. The set of all sets of numbers with Ndecimal digits (N from 0 to infinity) must be countable as well. Butthat necessarily includes the (infinite) sequences of decimal digitswhich represent irrational numbers.Or to put it really simplistically, put a 1 in front of any irrationalnumber (to handle cases like 0.0000ABCD... vs 0.ABCD... vs AB.CD...)and remove the decimal point, and you're left with a unique naturalnumber - one which corresponds to no other irrational number. So allirrational numbers map to a subset of the natural numbers, and anysubset of natural numbers can map one to one to the natural numbers -i.e. countable.So why aren't === irrational numbers considered countable?Subject: Re: Why are irrational numbers not countable?> If I'm reading right, countable sets are defined as those that can be> mapped one to one to the natural numbers. Integers and rational> numbers are considered countable, even though there are more of them> (e.g. 2 integers for every natural number). Sets of countable sets> are considered countable. So why not irrational numbers? Consider any number with N decimal digits. The set of those numbers> can be mapped to 10^N or fewer natural numbers, and so is countable. > This will remain true even if you insert a decimal point somewhere in> the sequence of N digits. The set of all sets of numbers with N> decimal digits (N from 0 to infinity) must be countable as well. But> that necessarily includes the (infinite) sequences of decimal digits> which represent irrational numbers.The last sentence of that paragraph is false. For any positiveinteger N, it is true that the set of all N-digit numbers iscountable. (You refer to the set of all sets of N-digitnumbers, but I think that's a mistake.) But infinity is nota positive integer, and the set of all infinite digit stringsis *not* countable.> Or to put it really simplistically, put a 1 in front of any irrational> number (to handle cases like 0.0000ABCD... vs 0.ABCD... vs AB.CD...)> and remove the decimal point, and you're left with a unique natural> number - one which corresponds to no other irrational number. So all> irrational numbers map to a subset of the natural numbers, and any> subset of natural numbers can map one to one to the natural numbers -> i.e. countable. So why aren't irrational numbers considered countable?Your construction only works for terminating decimals.Irrational numbers have non-terminating decimals.So, for instance, the number pi is irrational; its decimalexpansion begins 3.1415926 and continues for ever. Yourconstruction would require it to be mapped to a naturalnumber that begins 131415926 and continues for ever, butthere is no such natural number because every natural numberhas a finite number of digits.(There's another, much less important, error in your construction;the numbers 1234 and 12.34 both appear to be mapped to the integer11234. This error can be fixed. The other cannot.)--