mm-231 === Subject: Parallel lines 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. you in advance,-Ash-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === Parallel Lines QuestionAshley,Euclid 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-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === Parallel lines questionHow about let us 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 whether they agree or if i'm k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === 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 advance,>> -Ash-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === Parallel lines questionGreat timing... I'm 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 advance,> -Ash-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === Parallel lines question>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 advance,>-AshCan 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 writing. You have a good point.G C-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === 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 for equality andcongruence.-- 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).-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === Math 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 self-confidence.Best wishes, Mark V.-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === === Math enrichment for a six-year oldSubject: Re: Math enrichment for a six-year oldAuthor: Mark V. >(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.>(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 student anydifferently?Haim-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === Math enrichment for a six-year-oldHaim,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 himself whether he WANTS to do that.)Mark-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === 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 Shakespeare and enjoy it.mattsmom-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === Math 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 preferable to unschool.-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === Math enrichment 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.-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === Math enrichment for a six-year oldAbout 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 functions is one example.G C-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === Math 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 be filled with classes.mattsmom-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === Math 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.-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === Math enrichment for a six-year-old> Kevin, 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'sAll 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.-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === Math Enrichment for a six-year-oldWell, 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 Library.-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === Math Enrichment for a 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.-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === Math enrichment for a six-year-oldWhen we 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 mathinstruction.Hope this helps.mattsmom-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === Potter and the Mathematics of Doom jamsportlandwww.jamsportland.comThe Art of Mental Calculation for K- 4th Grade Students-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === === Math enrichment 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 First, of all, I want it on the record that I did not pay Karl to> 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 (preferred email address)-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === Freeware Scientific 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 or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.thinkspot.net/k12math/newsgroup charter: === Paper published by Algebraic and Geometric TopologyThe following paper has been published:Algebraic and Geometric TopologyURL:http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3- 27.abs.htmlTitle:Generalized orbifold Euler characteristics of symmetric orbifolds and covering spaces Author(s):Hirotaka Tamanoi Abstract:Let G be a finite group and let M be a G-manifold. We introduce theconcept of generalized orbifold invariants of M/G associa to anarbitrary group Gamma, an arbitrary Gamma-set, and an arbitrarycovering space of a connec manifold Sigma whose fundamental groupis Gamma. Our orbifold invariants have a natural and simple geometricorigin in the context of locally constant G-equivariant maps fromG-principal bundles over covering spaces of Sigma to the G-manifoldM. We calculate generating functions of orbifold Euler characteristicof symmetric products of orbifolds associa to arbitrary surfacegroups (orientable or non-orientable, compact or non-compact), in bothan exponential form and in an infinite product form. Geometrically,each factor of this infinite product corresponds to an isomorphismclass of a connec covering space of a manifold Sigma. The essentialingredient for the calculation is a structure theorem of thecentralizer of homomorphisms into wreath products described in termsof automorphism groups of Gamma-equivariant G-principal bundles overfinite Gamma-sets. As corollaries, we obtain many identities incombinatorial group theory. As a byproduct, we prove a simple formulawhich calculates the number of conjugacy classes of subgroups of givenindex in any group. Our investigation is motiva by orbifoldconformal field theory.Secondary: 57S17, 57D15, 20E22, 37F20, 05A15Keywords:Automorphism group, centralizer, combinatorial group theory, coveringspace, equivariant principal bundle, free group, Gamma-sets,generating function, Klein bottle genus, (non)orientable surfacegroup, orbifold Euler characteristic, symmetric products, twissector, wreath productAuthor(s) address(es):Department of Mathematics, University of California Santa Cruz, CA 95064, USA Email: === period?We receive a lot of fixed-record length files fromall kinds of sources, and and I am looking for away to determine the record length. In some casesthe records are termina by several combinationsof ; those are pretty easy. The hard files,however, come from IBM mainframes, in EBCDIC, with no line(or record) terminator. The first thing I do is convertthose to ASCII (with the Unix dd conv=ascii) command).The files contain personnel information, such as SSN,employee's name, phone number, and in many cases thereare several fixed consecutive blanks which were reservedbut not used. I figure that if I divide the charactersin 3 main classes: alphabetic, numeric and blanks it wouldbe much mor eefficient than trying an exhaustive bruteforce approach.I one took a digital design class in which the bigrecent discovery was spectral techniques to minimizethe chip's real state use. I guess I am looking for analgorithm like those, but much simpler, since we aredealing with one-dimension only.Any suggestion or comments are welcome and apprecia.TIA,-Ramon F. === compact groupLet $G$ be a locally compact group, and let $wG$ be its weakly almostperiodic compactification. The kernel $K(wG)$ of $wG$ is then a groupisomorphic to the almost periodic compactification of $G$.There are (non-compact) groups such that $wG = K(wG) cup G$ (the unionis necessarily disjoint). For such groups, the closed ideal $wGsetminus G$ of $wG$ then has an identity.Question: Let $G$ be a locally compact group (not compact) such that $wGsetminus G$ has an identity. Does that necessarily mean that $wGsetminus G = K(wG)$?Any pertinent hints are apprecia.Volker Runde.begin:vcard n:Runde;Volkertel;fax:+1 780 492 6826tel;home:+1 780 480 1181tel;work:+1 780 492 /org:University of Alberta;Mathematical and Statistical Sciencesadr:;;CAB 632;Edmonton;Alberta;T6G 2G1;Canadaversion:2.1email;internet:vrunde@ math.ualberta.catitle:Associate Professorx-mozilla-cpt:;1856fn:Volker === dense in L^2Originator: baez@math-cl-n01.math.ucr.edu (John Baez)What's an easy low-brow way to prove that polynomialfunctions times the Gaussian exp(-x^2) are dense in L^2(R)?I know a general-purpose L^2 Stone-Weierstrass theoremthat does the job, but it requires knowing the Fouriertransform is unitary, which takes a while to prove.I also know the eigenfunctions of the harmonic oscillatorHamiltonian are an orthonormal basis of L^2(R), and thisalso does the job, but again it relies on some stuff thattakes work to prove.For a class I plan to teach, I'd like a quick proofthat doesn't use much machinery, if one === in L^2> What's an easy low-brow way to prove that polynomial> functions times the Gaussian exp(-x^2) are dense in L^2(R)?Suppose that f is in L^2(R) and is orthogonal to all such products. Expanding exp(zx) in a power series and using Fubini's theorem, check thatint_R f(x) exp(zx) exp(-x^2) dx = 0,first for all complex z of sufficiently small modulus, then for all complex z by analytic continuation. In particular,int_R f(x) exp(-itx) exp(-x^2) dx = 0,for all real t. As f(x)exp(-x^2) is clearly an element of L^1(R),this last display means that the Fourier transform of f vanishes. Thus f === Gaussian are dense in L^2> What's an easy low-brow way to prove that polynomial> functions times the Gaussian exp(-x^2) are dense in L^2(R)?See chapter 1 of Igusa's (Springer Ergebnisse, 70's) bookTheta Functions for a low-brow proof. I suspect that thisbook will also have other things you may find useful === Gaussian are dense in L^2 > What's an easy low-brow way to prove that polynomial functions times > the Gaussian exp(-x^2) are dense in L^2(R)? > I know a general-purpose L^2 Stone-Weierstrass theorem that does > the job, but it requires knowing the Fourier transform is unitary, > which takes a while to prove. > I also know the eigenfunctions of the harmonic oscillator Hamiltonian > are an orthonormal basis of L^2(R), and this also does the job, but > again it relies on some stuff that takes work to prove. > For a class I plan to teach, I'd like a quick proof that doesn't use > much machinery, if one exists. >For each compact interval I, polynomials times exp(-x^2) are dense inthe continuous functions on I with the sup norm; you can appeal toStone-Weierstrass, or its special case for polynomials. And continuousfunctions with compact support are dense in L^2(R).The above does not use Fourier transform being === Polynomials times a Gaussian are dense in L^2> What's an easy low-brow way to prove that polynomial functions times> the Gaussian exp(-x^2) are dense in L^2(R)?I know a general-purpose L^2 Stone-Weierstrass theorem that does> the job, but it requires knowing the Fourier transform is unitary,> which takes a while to prove.I also know the eigenfunctions of the harmonic oscillator Hamiltonian> are an orthonormal basis of L^2(R), and this also does the job, but> again it relies on some stuff that takes work to prove.For a class I plan to teach, I'd like a quick proof that doesn't use> much machinery, if one exists.>For each compact interval I, polynomials times exp(-x^2) are dense in>the continuous functions on I with the sup norm; you can appeal to>Stone-Weierstrass, or its special case for polynomials. And continuous>functions with compact support are dense in L^2(R).>The above does not use Fourier transform being unitary.>Is this low-brow enough?Possibly a little too low-brow, since it doesn't _quite_ give aproof. Say f is in L^2. You can choose g continuous withcompact support so ||f-g||_2 < epsilon. And now you showthat one can approximate g _on_ the support of g by afunction of the required type, but you actually have toapproximate g in L^2(R).>Nemo************************David C. === in L^2>What's an easy low-brow way to prove that polynomial functions times>the Gaussian exp(-x^2) are dense in L^2(R)?>I know a general-purpose L^2 Stone-Weierstrass theorem that does>the job, but it requires knowing the Fourier transform is unitary,>which takes a while to prove.>I also know the eigenfunctions of the harmonic oscillator Hamiltonian> are an orthonormal basis of L^2(R), and this also does the job, but>again it relies on some stuff that takes work to prove.>For a class I plan to teach, I'd like a quick proof that doesn't use>much machinery, if one exists.>>For each compact interval I, polynomials times exp(-x^2) are dense in>>the continuous functions on I with the sup norm; you can appeal to>>Stone-Weierstrass, or its special case for polynomials. And continuous>>functions with compact support are dense in L^2(R).>>The above does not use Fourier transform being unitary.>>Is this low-brow enough?> Possibly a little too low-brow, since it doesn't _quite_ give a> proof. Say f is in L^2. You can choose g continuous with> compact support so ||f-g||_2 < epsilon. And now you show> that one can approximate g _on_ the support of g by a> function of the required type, but you actually have to> approximate g in L^2(R).>>Nemo> ************************> David C. === _Probability_Theory_I have pos this to amazon.com, and readers of this forumshould also find it to be of interest. Review of Edwin T. Jaynes' book, _Probability Theory: The Logic of Science_ (Five stars (out of five))To pure mathematicians, probability theory is measure theory inspaces of measure 1. To the extent to which you remain a puremathematician, this book will be incomprehensible to you.To frequentist statisticians, probability theory is the study ofrelative frequencies or of proportions of a population; those areprobabilities.To Bayesian statisticians, probability theory is the study ofdegrees of belief. Bayesians may assign probability 1/2 to theproposition that there was life on Mars a billion years ago;frequentists will not do that because they cannot say that therewas life on Mars a billion years ago in precisely half of allcases -- there are no such cases.To _subjective_ Bayesians, probability theory is about subjectivedegrees of belief. A subjective degree of belief is merely howsure you happen to be.Noninformative _objective_ Bayesians assign noninformativeprobability distributions when they deal with uncertainpropositions or uncertain quantities, and replace them withinformative distributions only when they update them because ofdata. Data, in this sense, consists of the outcomes of randomexperiments.Informative _objective_ Bayesians -- a rare species -- ask whatdegree of belief in an uncertain proposition is logicallynecessita by whatever information one has, and they don'tnecessarily require that information to consist of outcomes ofrandom experiments.Jaynes is an informative objective Bayesian. This book is hisdefense of that position and his account of how it is to be used.Pure mathematicians will not find that this book resembles thatbranch of pure mathematics that they call probability theory.Jaynes rails against those he disagrees with at great length.Often he is right. But often he simply misunderstands them. Forexample, writing in the 1990s, he said that pure mathematiciansreject the use of Dirac's delta function and its derivatives, andrela topics. That is nonsense; the delta function has longbeen considered highly respectable, and required material in thegraduate curriculum. Unfortunately Jaynes's misunderstandings maycause some others to misunderstand him when he is right.Statisticians are more informed than pure mathematicians andwill disagree with Jaynes for better reasons. _Some_statisticians will agree with him.This book many flaws, made all the more annoying by the factthat we need to overlook them in order to understand === Goldbach-Idea - probably dumb!Just thought I'd throw in my 2 cents on Goldbach. Not my area, but thethread made me think of the following.If you were to generate a set of false primes by insisting that inany interval they are distribu amongst the odd numbers with thesame frequency as the real ones (we can forget about 2),What is the probability that such a set has the Goldbach property(i.e. any even number can be expressed as the sum of two falseprimes)? Is it non-zero? What would the limit of this probability beif one only had to start looking after an arbitrarily large n?My really stupid idea was that if (subject to some reasonableverification method for sets of false primes being sta) theprobability was non-zero, then the Goldbach conjecture could be trueby accident.If however the probability does tend to zero, then surely theconjecture is equally fascinating!I'm no number theorist, but would love to === shift theoremsHi all.Let $L$ be a second-order strongly elliptic operator over a region $Omega$.Consider the problem of finding the solution $u$ of the problem $Lu=f$in $Omega$ (with $u=0$ on $partialOmega$), where $f$ is areal-valued function defined over $Omega$.Roughly speaking, the usual shift theorem for such problems says thatif $fin H^r(Omega)$, then $uin H^{r+2}(Omega)$, with$$|u|_{H^{r+2}(Omega)} le C |f|_{H^r(Omega)}.$$Can anybody point me to information about how the shift constant $C$depends on the coefficients of the differential operator $L$? Forexample, suppose that appropriate norms of these coefficients arebounded; is there Werschulz (8-{)} Metaphors be with you. -- bumper stickerGCS/M (GAT): d? -p+ c++ l u+(-) e--- m* s n+ h f g+ w+ t++ r- y? Internet: agw@cs.columbia.eduWWWATTnet: Columbia U. (212) 939-7060, Fordham U. (212) 636-6325