mm-719 === Subject: Re: Length of sequence of consecutive primes starting at 2 (snip) > integer less than or equal to N, because N! is divisible by those > integers, thus (N! + 1) mod P = 1 (where P is an integer less than or > equal to N). So, the first integer (er, I forgot to mention--although 1 > does divide N! + 1, clearly... I'm excluding 1) that divides N!+1 must > be greater than N, and it must be a prime itself (otherwise, it itself > must be divisible by another prime smaller than itself, then that > smaller prime would be dividing N!+1, and we see that the previous > number is no longer the first integer that divides N!+1... need I go > further?). > I'd prefer you start over. While usenet postings often seem like > conversation, it is not necessary to leave in er, I forgot to mention. > The message can be edited before sending so that all your logic is in order > and intact. I'd prefer that you not be too picky about minor details. It's a literary style, for Pete's sake! By putting it in parenthetical notes, I'm stressing (although it's an odd way of using the word stress) that it's a minor point, a triviality that wouldn't be mentioned at all except for my verbose tendency (as you can clearly see....) Besides, I've always hated how some mathematicians rework their logic (or the sequence in which they conduct their proof) over and over until the initial motivation is as cleverly hidden as the proof seems. Besides, clearly, our posts illustrate who is a better editor--I have cleverly snipped out all irrelevant details, while keeping the conversation in context, whereas you have kept all the previous posts which has absolutely nothing to do with what you are talking about. So, who you criticizin'? (This, once again, is also a literary style, as you can see from my fastidious use of apostrophe to mark the elision.) I protest that my proof is still clearly in order and intact, as far as logic goes (and if you can't keep what I had been talking about in the proof while you read that parenthetical note, well...I guess you forgot what I was saying about my logic, eh?). === Subject: Re: Explicit definition of f(n) Can anyone tell me what is the explicit definition of the following > function? f: N --> N f(0)=0 > f(n)=2*f(n-1)+n f(0) = 0 = 1 - 1 > f(1) = 2*0+1 = 1 = 2 - 1 > f(2) = 2*1 + 1 = 3 = 4 - 1 > ^ > n=2 > so you should have gotten f(2) = 4, etc. My new glasses are being ground as we speak. > DWC > f(3) = 2*3 + 1 = 7 = 8 - 1 > f(4) = 2*7 + 1 = 15 = 16 - 1 > .... > f(n) = 2^n - 1 === Subject: Re: Random Walk On a Unit Hypersphere (Repost) > I have a hyperspace (n-dimensional sphere) and a starting point > P (x_1,x_2,...x_n) where (x_1)^2+(x_2)^2+...+(x_n)^2 = 1. Mathematicians call this an (n-1)-dimensional sphere. > I have also a good uniform random number generator U [-1,1]. > How can I randomly walk on (not in ) the sphere using U. > What do you think about the below algorithm ? Does it really > walk randomly (uniform) ? (Think about the poles) Step 1 is the main problem: it generates a random vector in a box. > 1. Generate N dimensional V vector using U[-1,1] > 2. P = P + V/STEPSIZE > 3. Normalize P (such that P = P / |P| ) |P|=Length of a vector P. > 4. .... ( Do something ) > 5. Goto step 1 The key to generating your uniform random walk is being able to pick points uniformly at random on the surface of a hypersphere. This is easy to do incorrectly. Four correct ways to do it are given by Dave Seaman at http://www.math.niu.edu/~rusin/known-math/96/sph.rand The best method involves generating normally distributed random numbers, so you'll want to find out how to do this as well. Here's one site that tells you: http://www.taygeta.com/random/gaussian.html Okay, let's suppose you've now got the random-points-on-a-hypersphere subroutine written and tested. Nice work. That was the hard part. Here's the rest: If we assume that we want to travel a distance THETA (measured as a spherical arc, not as a straight line) at each step, then it is useful to precompute CDIST = cos(THETA) and SDIST = sin(THETA). Here is some pseudocode that takes a unit vector and returns its updated value (after a single random step): unitVector update(unitVector u) { do { v = new uniform random unitVector; c = u.v; s2 = 1. - c^2; } while (s2 > 0.01); // 0.01 is somewhat arbitrary v = SDIST*v + (CDIST*sqrt(s2) - SDIST*c)*u; return v/sqrt(v.v); // v.v should be s2, but enforce unit length } This code selects a unit vector randomly and uniformly from all those whose distance (along the sphere) from u is THETA. Note that in this implementation THETA can be any real number: e.g., if THETA = 3*pi, then the code correctly returns -u. -- Jim Ferry at U of Illinois, Urbana-Champaign, (no, I don't educate) 2 email me l o o k up 1 row === Subject: Re: Random Walk On a Unit Hypersphere (Repost) > unitVector update(unitVector u) { > do { > v = new uniform random unitVector; > c = u.v; > s2 = 1. - c^2; > } while (s2 > 0.01); // 0.01 is somewhat arbitrary > v = SDIST*v + (CDIST*sqrt(s2) - SDIST*c)*u; > return v/sqrt(v.v); // v.v should be s2, but enforce unit length Oops: that should read while (s2 < 0.01); === Subject: Re: Random Walk On a Unit Hypersphere (Repost) > I have a hyperspace (n-dimensional sphere) and a starting point > P (x_1,x_2,...x_n) where (x_1)^2+(x_2)^2+...+(x_n)^2 = 1. > Mathematicians call this an (n-1)-dimensional sphere. i would have thought it was n-dimensional sphere.. was this a typo or am i just understanding it wrongly? === Subject: Re: Random Walk On a Unit Hypersphere (Repost) > I have a hyperspace (n-dimensional sphere) and a starting point > P (x_1,x_2,...x_n) where (x_1)^2+(x_2)^2+...+(x_n)^2 = 1. > > Mathematicians call this an (n-1)-dimensional sphere. > > i would have thought it was n-dimensional sphere.. was this a typo or > am i just understanding it wrongly? In 3-space, the sphere {(x,y,z) : x^2+y^2+z^2=1} is a surface. It is 2-dimensional. So it is called the 2-sphere. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Function difference question Consider an interval (a,b], and a function F which is increasing and right-continuous everywhere. If {(a_j, b_j]} is a countable partition of (a,b], then it can be shown that: F(b) - F(a) = Sum (j=1 --> oo) [F(b_j) - F(a_j)] How does one prove that this holds for (-oo, b] and (a, oo), using the respective limits for F at -oo and +oo? All help and suggestions appreciated. === Subject: Re: A.D. History (was Re: Agnostic MorituriMax) > Interesting. Looks like the Jewish calendar is the oldest. Is It? No. The Jewish calendar counts the number of years since the Creation, calculated by adding up the ages of people in the Bible (taking careful note of the 'begats'!). This was done in the 2nd Century A.D. by Rabbi R. Yose. It doesn't quite accord with Bishop Ussher's christian chronology derived using a similar methodology. Personally, I admire Ussher for his startling precision in the face of such dubious data: the world, he tells us, was created on October 23, 4004 B.C. At noon. [You can probably guess what I think of both chronologies] The Mayan calendar (for example) is much older. It evolved along with the Mayan civilisation, but the long count seems to have been put in place around 2300 years ago. That's about 800 years older than the Christian calendar, and 500 years older than the Jewish calendar. The Egyptian calendar is even older, seemingly having begun (the calendar itself, not its 'year 0') in 4326 BC. === Subject: Re: Agnostic MorituriMax > You mean you were able to think of something truly original and > unique? What was it? I believe it was a great privilege to hear from > God directly. Your point eludes me. I don't regard the concept of God as something original and unique. On the contrary, I regard it as a rather obvious invention to explain away phenomena that were otherwise impossible for ancient peoples to comprehend. On top of this basic human need to find answers has been constructed a series of vast ideological edifices (religions) that persist long after the justification for them (ignorance) has been vanquished. The argument that mankind can never create anything original and unique seems to me entirely spurious. What the hell do you mean? Shakespeare's plays weren't original and unique because he used words that other people had used and included characters who were quite clearly recognisable as human beings? Mozart's music wasn't original and unique because he used the same notes as other composers? The more I think about your assertion, the more ludicrous it becomes. > Wouldn't you give anything to know with absolute > certainty that God exists, that He is your true father, and that you > can get help from him? Er, no. I have made alternative arrangements. > That was it, I though, I will > go completely blind. I asked God for help, and went to see an > ophthalmologist. He examined me and did all kinds of tests. You don't > have the condition you are telling me about, he said. It is true that > something has damaged one of your eyes, but you will not go blind. Was > that a miracle? I don't know. I can still see. I am pleased for you that you still have your sight. You will not be surprised, however, that I would prefer to seek a non-supernatural explanation for your recovery. === Subject: Re: Agnostic MorituriMax > You mean you were able to think of something truly original and > unique? What was it? I believe it was a great privilege to hear from > God directly. > Your point eludes me. I don't regard the concept of God as something > original and unique. On the contrary, I regard it as a rather obvious > invention to explain away phenomena that were otherwise impossible for > ancient peoples to comprehend. On top of this basic human need to find > answers has been constructed a series of vast ideological edifices > (religions) that persist long after the justification for them > (ignorance) has been vanquished. > The argument that mankind can never create anything original and unique > seems to me entirely spurious. What the hell do you mean? Shakespeare's > plays weren't original and unique because he used words that other > people had used and included characters who were quite clearly > recognisable as human beings? Mozart's music wasn't original and > unique because he used the same notes as other composers? The more I > think about your assertion, the more ludicrous it becomes. > Wouldn't you give anything to know with absolute > certainty that God exists, that He is your true father, and that you > can get help from him? > Er, no. I have made alternative arrangements. > That was it, I though, I will > go completely blind. I asked God for help, and went to see an > ophthalmologist. He examined me and did all kinds of tests. You don't > have the condition you are telling me about, he said. It is true that > something has damaged one of your eyes, but you will not go blind. Was > that a miracle? I don't know. I can still see. > I am pleased for you that you still have your sight. You will not be > surprised, however, that I would prefer to seek a non-supernatural > explanation for your recovery. I envy you. You seem to know everything, and I feel I know nothing. Get a laugh. Somebody sent me the e.9amail below. Peter === Subject: Donkey One day a farmer's donkey fell down into a well. The animal cried >piteously for hours as the farmer tried to figure out what to do. >Finally, he decided the animal was old, and the well needed to be >covered up anyway; it just wasn't worth it to retrieve the donkey. >He invited all his neighbors to come over and help him. They all >grabbed a shovel and began to shovel dirt into the well. At first, the >donkey realized what was happening and cried horribly. Then, to >everyone's amazement he quieted down. >A few shovel loads later, the farmer finally looked down the well. He was astonished at what he saw. With each shovel of dirt that hit his back, the donkey was doing something amazing. He would shake it off and take a step up. >As the farmer's neighbors continued to shovel dirt on top of the >animal, he would shake it off and take a step up. Pretty soon, everyone was amazed as the donkey stepped up over the edge of the well and happily trotted off! >Life is going to shovel dirt on you, all kinds of dirt. The trick to >getting out of the well is to shake it off and take a step up. Each of >our troubles is a steppingstone. We can get out of the deepest wells >just by not stopping, never giving up! Shake it off and take a step up. >Remember the five simple rules to be happy: >1. Free your heart from hatred - Forgive. >2. Free your mind from worries - Most never happen. >3. Live simply and appreciate what you have. >4. Give more. >5. Expect less >NOW -------- >Enough of that crap . . >The donkey later came back and bit the out of the farmer who had >tried to bury him. The gash from the bite got infected, and the farmer >eventually died in agony from septic shock. >MORAL FROM TODAY'S LESSON: >When you do something wrong and try to cover your ass, it always comes back to bite you. === Subject: Re: Consultancy available ($1K - $5K US) for work on dim 2 partial orders and (ordered) DAGs Your conjecture may be wrong. It seems you assume that the following holds: [Claim] Let G be the comparability graph of a poset P of dimension 2. Then the transitive reduction (i.e., removal of all arcs u->v such that there is a vertex w such that u->w and w->v are arcs) gives a DAG such that any two vertices in it share either all or none of their immediate descendants. Coversely, the transitive closure of a DAG having the mentioned property given a poset of dimensionality 2. I have a counterexample to the first part: Chain 1: 5->1->4->2->3 Chain 2: 2->1->5->3->4 Then in the transitive reduction of the poset (which is, BTW, the comparability graph of the poset itself in this case) vertices 1 and 2 have 3 as a common immediate descendant, but 4 is an immediate descendant of 1 and not of 3. Stas http://www.busygin.dp.ua > !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! > Note: this is a corrected version of the posting > to sci.math.research (which had several confusing > typos in the statement of the relevant directed > edge definition in the two DAG examples. It also > Dushnik and Miller, the relevant section of Trotter, > and the relevant link at Eric Weisstein's > MathWorld. Cumulative Inquiry apologizes for > the typos in the previous posting to sci.math.research. > !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! > Cumulative Inquiry, Inc, a start-up R&D firm > chartered in Huntsville AL (USA), will pay a > consultancy fee of $1K - $5K in order to obtain > a proof of the following conjecture. > *********************************************** > Let o(dag) be any ordered DAG (directed acyclic > graph) in which any two vertices share either all > or none of their immediate descendants (note: > pre-ordered trees and forests pre-ordered in the > obvious way belong to the none leg of this > condition.) > There is then a 1:1 mapping from the set O(DAG) > of all such o(dag)'s INTO (not ONTO) the set of > all dim 2 partial orders (including the trivial > dim 1 partial order or chain.) > See two examples below for a tree and non-tree case > Also, see this link at Eric Weisstein's MathWorld: > http://mathworld.wolfram.com/PosetDimension.html > Miller referenced in this link. > *********************************************** > Fiduciary references for prompt payment of > consultancy fees by Cumulative Inquiry Inc > can be obtained from William F. Mann at > wfmann@alum.mit.edu. > If a proof is obtained and the author thinks > it worth submitting to any peer-reviewed journal, > then rights will belong to the author of the proof. > Further, if the fee is not exhausted by proof of > the above, then the remainder can be applied to > a precise definition of those ordered DAGs which > are not in O(DAG) but are uniquely representable > by dim 2 partial orders (this is tricky and not > immediately obvious.) > Finally, if the proof is so trivial as not to be > worth even the min of the fee, could someone please > sketch out its beginning in a posting to this group ? > If the consultancy is desired, please respond to David > Halitsky > dhalitsky@cumulativeinquiry.com > David Halitsky > 2928 Stewart Campbell Pte > Thompsons Station TN 37179 USA > Cell: 256-426-6243 (USA) > on this matter. > ************************************************* > Example of a tree case > partial order 1: 0 1 2 3 4 5 6 > partial order 2: 0 6 1 5 2 4 3 > vertices: (0,0), (1,6), (2,1), (3,5), (4,2), (5,4), (6,3) > edges: ((xi,yi),(xj,yj)) such that xi < xj and yi < yj > and no (xk,yk) such that (xi < xk < xj) and > (yi < yk < yj) > Adjacency matrix giving just the DIRECTED edges for first (tree) > example (trivial edges from vertex (xi,yi) to itself are omitted) > (0,0) (1,6) (2,1) (3,5) (4,2) (5,4) (6,3) > (0,0) 1 1 > (1,6) > (2,1) 1 1 > (3,5) > (4,2) 1 1 > (5,4) > (6,3) > ******************************************************** > Example of a non-tree case > partial order 1: 0 1 2 3 4 5 6 > partial order 2: 0 2 1 6 5 4 3 > vertices: (0,0), (1,2), (2,1), (3,6), (4,5), (5,4), (3,3) > edges: ((xi,yi),(xj,yj)) such that xi < xj and yi < yj > and no (xk,yk) such that (xi < xk < xj) and > (yi < yk < yj) > Adjacency matrix giving directed edges for second (non-tree) example > (trivial edges from vertex (xi,yi) to itself are omitted) > (0,0) (1,2) (2,1) (3,6) (4,5) (5,4) (3,3) > (0,0) 1 1 > (1,2) 1 1 1 1 > (2,1) 1 1 1 1 > (3,6) > (4,5) > (5,4) > (6,3) === Subject: A fall-back rewording of the consultancy fee conjecture (to avoid Stas' example, if necessary) > I have a counterexample to the first part: > Chain 1: 5->1->4->2->3 > Chain 2: 2->1->5->3->4 > Then in the transitive reduction of the poset (which is, BTW, > the comparability graph of the poset itself in this case) > vertices 1 and 2 have 3 as a common immediate descendant, but 4 > is an immediate descendant of 1 and not of 3. If this is in fact a counterexample (see my previous reply to Stas), then it can be avoided by a minor fallback modification to the original conjecture as follows (note that added text is CAPTIALIZED): *************************************************************** Let o(dag) be any ordered DAG (directed acyclic graph) ON JUST N+1 VERTICES in which any two vertices share either all or none of their immediate descendants (note: pre-ordered trees and forests pre-ordered in the obvious way belong to the none leg of this condition.) There is then a 1:1 mapping from the set O(DAG)v(N+1) of all such o(dag)'s ON JUST N+1 VERTICES into (not onto) the set of all dim 2 posets for which one chain is 0,1,...,n (including the trivial dim 2 poset in which both chains are 0,1,...,n) ********************************************************* This will avoid Stas' case and still leave the force of the conjecture intact, i.e. to establish that ordered trees, ordered forests, and certain other orderable DAGs have unique representations as dim 2 posets. I have asked/encouraged Stas to accept an honorarium for forcing this issue, if he will not accept the consultancy itself. === Subject: Re: Consultancy available ($1K - $5K US) for work on dim 2 partial orders and (ordered) DAGs > I have a counterexample to the first part: > Chain 1: 5->1->4->2->3 > Chain 2: 2->1->5->3->4 > Then in the transitive reduction of the poset (which is, BTW, > the comparability graph of the poset itself in this case) > vertices 1 and 2 have 3 as a common immediate descendant, but 4 > is an immediate descendant of 1 and not of 3. David replies: If one graphs Stas's example as as set of vertices (xi,yi) (i <= 1 <= 5) with the chain 1 values as xi and the chain 2 values as yi, one still gets an ordered tree of the form: [ [ [ d]] e] A B C where A = (1,1), B = (2,3), C = (3,4), d = (4,5), e = (5,2) Being an ordered tree, this o(dag) is in O(DAG), so I'm not sure this is a counter-example. But if I'm right that it's NOT a counterexample, then the conjecture obviously needs more precise rewording to avoid this confusion. If it IS a counterexample, then the conjecture needs REWORKING (not mere rewording) to ensure that the basic idea can be proven, namely that ordered trees, ordered forests, and certain other orderable DAGs can be represented as dim 2 posets (whatever these certain other DAGs may turn out to be.) Obviously, I hopoe Stas can be persuaded to take on the consultancy, though I'm sure he'll refuse! David Halitsky === Subject: Re: Consultancy available ($1K - $5K US) for work on dim 2 partial orders and (ordered) DAGs >*************************************************************** >Let o(dag) be any ordered DAG (directed acyclic graph) what's dag and why does o depend on it ? Isn't it better to just use G=(V,E) as usual or similar instead of o(dag) ? >ON JUST N+1 VERTICES in which any two vertices share >either all or none of their immediate descendants (note: >pre-ordered trees and forests pre-ordered in the >obvious way belong to the none leg of this >condition.) >There is then a 1:1 mapping from the set O(DAG)v(N+1) >of all such o(dag)'s ON JUST N+1 VERTICES into >(not onto) the set of all dim 2 posets for which >one chain is 0,1,...,n (including the trivial >dim 2 poset in which both chains are 0,1,...,n) why not use N instead of N+1 and omit 0 ? Shouldn't you specify the vertex-set explicitely ? Else there is an infinite number of ODAGS of given order... With into you mean injection (injective mapping) ? So, just that the cardinality of dim 2 posets over a finite set V of vertices is larger than the cardinality of ODAGS over V ? === Subject: Re: Rational bases??? That exists? X-CompuServe-Customer: Yes X-Coriate: interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: George Cox X-Punge: Micro$oft X-Sanguinate: The MVS Guy X-Terminate: SPA(GIS) X-Tinguish: Mark Griffith X-Treme: C&C,DWS at 12:29 PM, jen@fct.unl.pt (Joaquim Nogueira) said: >It is well known that people write numbers in base 10, and that >computers use base 2 (or a power of 2). Some do. There were cultures that used a base other than 10, there were early computers that used a base of 10 and there has been research on computers that use a base of 3. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org === Subject: Re: Solve: cos(th) / th = c ? > I would suggest > th = (.52909974123020777467+.82571186544402814274*c)/ > (.33683514045080888358+.86250543486276334096*c+ > .52594955842441307971*c^2+.13323045571783326803*c^3 > -0.025429795485872842078*c^4) > The maximum of |cos(th)/th - c| for 0 0.0000018. > That's excellent. Did you determine the coefficients using Chebyshev > polynomials, perhaps with a utility available in Maple? I used Maple. Actually a slightly better choice is (0.52761771283563303569 + 0.82821936722283528882*c)/(0.33589139744120485496 + (0.86317229702550238657 + (0.52740947484215323768 + (0.13340502130298481752 - 0.025399472189974922406*c)*c)*c)*c) where the maximum of |cos(th)/th - c| for 0 I'm guessing that your objective was to minimize the maximum > of |cos(th)/th - c| for 0 coefficients are really accurate to anything like the number of significant > digits shown. I say this because, unless Mathematica's numerics are > terrible, the absolute values of the seven extrema of |cos(th)/th - c| > on [0,1] are clearly not quite equal. For this one, the absolute values of the extrema should be approximately equal. Not exactly so, because minimax doesn't let you directly minimize that maximum: in approximating th(c) by th_app(c) it lets you minimize the maximum of w(c) |th(c) - th_app(c)| for some weighting function w. So what I did today was take w(c) = g'(th_1(c)) where g(x) = cos(x)/x and th_1(c) is an approximation of th(c). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: abelian.. hello....doctor~ 1. If every proper subgroup of group G is cyclic group then G is abelian. um.....i think......it's false. counterexample, S_3 = {p0, p1, p2, u1, u2, u3} because, every proper subgroup S_3 is , =, , , . so, all is cyclic. but S_3 is not abelian. right ?? --------------------------------- 2. If every proper subgroup of group G is abelian group then G is abelian. um....i think....false~ conterexample, D_4 = {p0, p1, p2, p3, u1, u2, d1, d2} because, every proper subgroup D_4 is {p0},{p0,u1},{p0,u2},{p0,p2},{p0,d1},{p0,d2} {p0,p2,u1,u2},{p0,p1,p2,p3},{p0,p2,d1,d2} so, all is abelian. but D_4 is not abelian. right ?? thank you for your advice. thank you very much. === Subject: Re: abelian.. >hello....doctor~ >If every proper subgroup of group G is cyclic group >then G is abelian. >um.....i think......it's false. >counterexample, S_3 = {p0, p1, p2, u1, u2, u3} >because, every proper subgroup S_3 is >, =, , , . >so, all is cyclic. >but S_3 is not abelian. >right ?? >--------------------------------- >If every proper subgroup of group G is abelian group >then G is abelian. >um....i think....false~ >conterexample, D_4 = {p0, p1, p2, p3, u1, u2, d1, d2} >because, every proper subgroup D_4 is >{p0},{p0,u1},{p0,u2},{p0,p2},{p0,d1},{p0,d2} >{p0,p2,u1,u2},{p0,p1,p2,p3},{p0,p2,d1,d2} >so, all is abelian. >but D_4 is not abelian. >right ?? Very good! Here is a harder one. If every nontrivial proper subgroup of group G has order p for same prime p then G is abelian? Derek Holt. === Subject: Re: abelian.. >hello....doctor~ >If every proper subgroup of group G is cyclic group >then G is abelian. >um.....i think......it's false. >counterexample, S_3 = {p0, p1, p2, u1, u2, u3} >because, every proper subgroup S_3 is >, =, , , . >so, all is cyclic. >but S_3 is not abelian. >right ?? >--------------------------------- >If every proper subgroup of group G is abelian group >then G is abelian. >um....i think....false~ >conterexample, D_4 = {p0, p1, p2, p3, u1, u2, d1, d2} >because, every proper subgroup D_4 is >{p0},{p0,u1},{p0,u2},{p0,p2},{p0,d1},{p0,d2} >{p0,p2,u1,u2},{p0,p1,p2,p3},{p0,p2,d1,d2} >so, all is abelian. >but D_4 is not abelian. >right ?? You could use S_3 as a counter-example for both problems. Brian >thank you for your advice. >thank you very much. === Subject: Re: Elementary Proofs 2 > Hello again. I was going to type one more attempt at a theorem before I > sent the last one. > > Theorem: Let c be a positive integer that is not a prime number. Show there > is some positive integer b such that > b|c and b <= sqrt(c). >Your theorem is trivially true, just take b=1. THen b is a positive >integer, b divides c, and b <= sqrt(c). >So I assume that you really meant to say: >Theorem: Let c be a positive integer that is not a prime number. Show >there is some positive integer b >= 2 such that b|c and b <= sqrt(c). > Just one tiny nitpick...c also has to be >= 2. > -- Erick That is nitpicky, but you're right! Actually, if I were going to assign this as an exercise, I'd say Let c be a positive composite number, which has the advantage of being an affirmative statement while excluding both primes and units. Is that better? J === ... > With those (sketches of) definitions, it is clear that Google's > threading system is ... bust. It is worse. Google sometimes even breaks threads where the subject stays the same, and connects threads many years apart. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === ... > Google's threading system is well-known to be bust. > > I created no new thread. > It has nothing to do with Google. The new Subject title yields a new > thread. The New subj was: Old Subj just gives enough information > to connect the old thread with the new one and ... oh I see. What do you think is the purpose of the References line? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: JSH: Groupthink [3] requires uf=1 > > [2] reduces to mf^2 = 5 > > which reduces [1] to 65 > > There is no algebraic conflict between these two requirements. > > Hope this helps to reduce the dialog to fundamental issues, such as > whether > > u*f=1 violates JSH's stipulation that: > > f is a non unit, non zero algebraic integer coprime > > to 3 and x, and u a non unit, non zero algebraic integer > > coprime to f. I thought there was another stipulation: m is a (rational) integer. > This would be a contradiction even in his object ring. > > Right. I was surprised that Andrzej Kolowski dropped his entire line of > objection after I made this post. He had started to imply he would show a > conflict in the ring of algebraic integers but never finished and, I > thought, implied a simple algebraic error in substituting uf=1 in one place > and sqrt(5) in another. > > My There is no algebraic conflict between these two requirements. i.e. > uf=1 and mf^2 = 5, was in reference to this simple algebraic substitution > complaint. > > But what we really have is uf=1 and mf^2=5, with m an integer and JSH's: > f is a non unit, non zero algebraic integer coprime > to 3 and x, and u a non unit, non zero algebraic integer > coprime to f. > > So JSH now must argue that f=sqrt(5/m) and u=1/sqrt(5/m) do not conflict > with his algebraic-integer stipulations. > > I was confident Kolowski would pick back up on this. > > Keith, > what Harris said was that if > [*] A*x^3 + B*x^2*y + C*x*y^2 + y^3 > (A, B, and C fixed) can be factored in the form > [**] (a1*x + y)*(a2*x + y)*(a3*x + y), > where a1, a2, a3 and y are algebraic integers, then > [***] A*x^3 + B*x^2 + C*x + 1 > can be factored as > (a1*x + 1)*(a2*x + 1)*(a3*x + 1), > i.e., with the *same values* a1, a2, and a3 as for the > factorization of the first polynomial. > This needs to be proved. Harris is thinking of x as a polynomial > must have > A = a1*a2*a3, > B = a1*a2 + a1*a3 + a2*a3, > C = a1 + a2 + a3. > This is true regardless of what y is. The solutions to these > three equations are (up to permutations) equal to the negative > inverse of the roots of [***]. > Now suppose that you can somehow prove that for a given > value of y that exactly two of a1, a2, and a3 are divisible > by some algebraic integer, say, g. Then you can say the > same if you replace y by 1. > Here is how this applies to the Harris polynomial, P(m): > (m^3*f^6 - 3*m^2*f^4 + 3*m*f^2)*x^3 - 3*(-1 + m*f^2)x*u^2*f^2 + u^3*f^3: > Replace u*f by y. Note that the coefficient of the x term in > this polynomial is 0, so it too can be regarded as multiplied by > u*f. This now fits the description of the polynomial [*] given > above. Harris says that when this is factored in the form > [**], two of a1, a2, and a3 are divisible by f. He chooses > f = sqrt(5), u = 1. Thus he concludes that in the factorization > of 65 x^3 - 60*x + 5*sqrt(5), two of a1, a2, and a3 are divisible > by f. Therefore by the remark above, the same is true with the > factorization of 65 x^3 - 12*x + 1. > This seems right to me and actually is a fairly astute observation. > I am sure Harris was hugely relieved to find that it worked out. > Of course it does him no real good at all because his belief that > he can prove that if P(m) is factored as above, then two of a1, a2 > and a3 are divisible by f, is wrong. His conclusion regarding > 65 x^3 - 12 x + 1 is incorrect also, as can be seen in at least three > ways. But he thinks that he has managed to elude the consequences > of his error in calculation in APF. >And that's what's wrong with sci.math posters, an inability to be >reasonable about mathematics when they're emotionally involved. What crap. Here I explained in detail for someone else why your reasoning for your fix is correct. I gave you credit for being astute. Has anyone else here *ever* done that? You had no comments on the math or on the 'astute' part. Then I went on to say why you were wrong anyway about the bigger picture, which is the absolute God's truth, and you instantly begin whining about my emotional involvement. Clearly the only thing you can accept is total agreement with everything you say. >Note that the poster got emotionally involved with the notion that I >was perpetrating this huge fraud, and made a lot of posts with that >claim. I never said 'huge' fraud. I said your error was a pimple on a pimple. I did say that in my opinion, for you to not mention the error to the editor or to withdraw your paper or submit a revision, constitutes fraud. I still think that is correct. The idea behind your fix as I have explained it here is not even hinted at in the present draft of the paper. It is not completely trivial. It will require a nontrivial revision of the paper. It is not being honest to cover up what you know to be a blatant mistake and hope the reviewers will miss it and accept the paper. >I pointed out that my typos were minor errors and *explained* to him >in a reply to one of his posts accusing me of fraud. >He simply kept accusing, but now apparently is willing to accept >*basic* algebra, but decides to continue to make swipes at me, as if >such a simple concept is something I only recently found out, when I >explained it to him!!! I was willing to accept and even explain the logic of your fix. As explained above: you now have a known error in the paper you have submitted to a journal. That particular error can be fixed but in my view the fix is not obvious, i.e., it takes a nontrivial revision of your paper to get around it. You know this and you are concealing this information from the editor. On top of which, there are other deeper errors which invalidate your conclusions and which your fix doesn't fix at all. See below for an example, if you still have an interest in the math. >Next notice that he promptly goes back to what he really wants to do: >just call me wrong. You ARE wrong. I very definitely do not want to leave the impression with anyone that simply because you found a way to compensate partially for a computational error, you have resolved the other more important problems with the paper! You most definitely have not. >My assessment of posters like Andrzej Kolowski is that they don't care >about the math itself, but about what a math result can do for them, >or how it will make them feel. I addressed the math above, *justifying* what you claimed! You have not said one word about the math, just your amateur psychologizing about my motivations and emotions. >It's one of the reasons sci.math is basically useless for math >discussions where there's disagreement, as posters who *decide* that >you are wrong don't care what the math says. I DO care what the math says. Part of what you said was right, and that is what this post was mostly about, answering someone else's question. Part of what you have in APF is totally wrong. I am not going to leave anyone with the impression that you have in any way fixed the major errors. >They will argue with you day and night for months, repeating errors, >denying those errors when they're pointed out, and all the time >claiming it's you who are doing it. to >convince Ioannis Argyros, the chief editor at that journal, to back >off a paper with a correct proof The APF main result and the application of it to 65 x^3 - 12 x + 1 are incorrect. See below. >(despite minor typos as noted), is >just a slight taste to you all of the hell that comes in when people's >FEELINGS trump mathematics. >See http://rattler.camero n.edu/swjpam/vol2-03.htmlor what if's. Agreed. It's real. >Posters from sci.math actually went out and badly affected the math >world by using social pressure against a solid proof. No! The proof is just plain wrong. Any time an error in published research is noted, the right and proper thing to do is to notify the editor. What the editor does with that information is up to him. >The damage from that *social* intervention is something that will play >out over time, but it clearly shows that mathematicians can just be >social creatures as well. >Now maybe Ioannis Argyros lied to me when he sent me an email saying >my paper had passed formal peer review, You resort to accusations of lying far too often. I think he simply screwed up through carelessness. However he should have given you an honest explanation of what happened later, and perhaps he omitted that. >but that doesn't change the >reality that mathematicians are here clearly SOCIAL CREATURES and the >supposed sanctity of mathematical proof and adequate demonstration has >been broken. All humans are social creatures; it would be stupid to deny that. As for sanctity of proof: you think your proof is correct. I absolutely disagree. Am I supposed to betray my own beliefs and just bow down to whatever you say? You have for some time stopped answering the mathematical objections, preferring evidently to harp endlessly on social factors. >You are lost without a rudder. >If there were objectivity here then there wouldn't be all these >arguments about my work, as then I could just explain it out in detail >and that would be it. There IS objectivity. I have explained here why you were right about your fix. You are ignoring that and seeing just what you want to see. 'Objectivity' to you must mean total obeisance to anything YOU say. Like, Nazis under Hitler were 'objective'. >But the trouble is that no matter how much I explain there are posters >like Andrzej Kolowski who have an emotional investment in just saying >I'm wrong, no matter what the mathematics says. If you are right I will say so, as I have done here. If you are wrong, again I will say so. That seems like straight- forward rational behavior to me. >They don't care about mathematics more than their own personal >feelings, which is why sci.math is useless for serious researchers. >There are too many emotional and loud people willing to just argue >with you because they don't like you who don't give a damn what the >math says. Here's what your math in APF says: that if 65 x^3 - 12 x + 1 is factored in the form (a_1 x + 1)*(a_2 x + 1)*(a_3 x + 1), where a_1, a_2, and a_3 are algebraic integers, then two of a_1, a_2, and a_3 are divisible in the algebraic integers by f = sqrt(5). Suppose this is true. Say without loss of generality that a_1 is divisible by sqrt(5). That is, a_1 = b * sqrt(5), where b is an algebraic integer. You can then show with a little algebra that 5 b^6 - 144 b^4 + 312 b^2 - 169 = 0. The polynomial on the left is a non-monic, irreducible, primitive cubic in the variable b^2. Because it is non- monic, b^2 cannot be an algebraic integer, and therefore b itself cannot be an algebraic integer. Thus a_1 cannot be divisible by f = sqrt(5). This directly contradicts your assertion in APF. That is, the conclusion of APF is contradicted by a straightforward algebraic number theory argument. The math in APF therefore cannot be correct. Tell me what is wrong with this argument. Andrzej >James Harris === Subject: Re: JSH: Groupthink > Posters from sci.math actually went out and badly affected the math > world by using social pressure against a solid proof. My logic is undeniable., V.I.K.I -- I, RobotMy proof is irrefutable., JSH -- sci.math resident crank. > James Whak-A-Mole Harris -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: Re: JSH: Groupthink ... stuff deleted ... > This seems right to me and actually is a fairly astute observation. >I am sure Harris was hugely relieved to find that it worked out. >Of course it does him no real good at all because his belief that >he can prove that if P(m) is factored as above, then two of a1, a2 >and a3 are divisible by f, is wrong. His conclusion regarding >65 x^3 - 12 x + 1 is incorrect also, as can be seen in at least three >ways. But he thinks that he has managed to elude the consequences >of his error in calculation in APF. > And that's what's wrong with sci.math posters, an inability to be > reasonable about mathematics when they're emotionally involved. Is that really what's wrong? I'm guessing that you're the most ego-invested one of the lot, and your sense of pride prevents you from acknowledging that you are mistaken. > Note that the poster got emotionally involved with the notion that I > was perpetrating this huge fraud, and made a lot of posts with that > claim. Aren't you the person who initiated the fraud discussion? > I pointed out that my typos were minor errors and *explained* to him > in a reply to one of his posts accusing me of fraud. That was a single point. You continue to avoid addressing the point that I've brought up. > He simply kept accusing, but now apparently is willing to accept > *basic* algebra, but decides to continue to make swipes at me, as if > such a simple concept is something I only recently found out, when I > explained it to him!!! > Next notice that he promptly goes back to what he really wants to do: > just call me wrong. > My assessment of posters like Andrzej Kolowski is that they don't care > about the math itself, but about what a math result can do for them, > or how it will make them feel. > It's one of the reasons sci.math is basically useless for math > discussions where there's disagreement, as posters who *decide* that > you are wrong don't care what the math says. > They will argue with you day and night for months, repeating errors, > denying those errors when they're pointed out, and all the time > claiming it's you who are doing it. Perhaps you're referring to my argument that shows your Primary Argument to reach an incorrect conclusion? You have stated without so much as a word of explanation that I am using some assumption regarding the existence of rings (other than the ring of algebraic integers) that satisfy some property regarding their units. Here again is yet another request to have my errors pointed out. I have, on several occasions, requested that you point out which of the following steps is invalid: The correctness of all of the following statements is sufficient to prove that none of the a_i is coprime to 5 in the ring of algebraic integers, so for your argument to stand, at least one of the following must fail. Here is an outline of the argument; if you could just point out which of the statements relies on the alleged false assumption then you will assist in furthering this discussion. In this outline, a refers to one of the coefficients that you claim must be coprime to 5. P(x) refers to the polynomial x^3 - 12 x^2 + 65, and I note that P(-a) = 0, for each of your values a. 1. The following formulas are true: q(x)r(x) = (64 x + 128)P(x) + 5 r(x)s(x) = (32 x + 72)P(x) + x, where q,r,s are defined as follows: q(x) = 8 x^2 - 76 x - 185 r(x) = 8 x^2 - 4 x - 45 s(x) = 4 x^2 - 37 x - 104 2. Since -a is a root of P(x), one has the following factorizations: q(-a)r(-a) = 5 r(-a)s(-a) = -a 3. The minimal polynomial of r(-a) is given as: MP_r = x3 - 969 x2 + 315 x + 5 which is irreducible over Q. This shows that r(-a) is an algebraic integer, but not a unit in that ring. shared by a and 5 is not a unit in the ring of algebraic integers. 5. a and 5 are not coprime in the ring of algebraic integers, since they share a non-unit factor in common. I claim that not one of these individual points depends on the assumption you say I've made. The above constitutes the entire argument and all commentary should be restricted to the argument, so let's see where your objection lies. > convince Ioannis Argyros, the chief editor at that journal, to back > off a paper with a correct proof (despite minor typos as noted), is > just a slight taste to you all of the hell that comes in when people's > FEELINGS trump mathematics. I will take the blame for convincing Argyros that the paper contained a false conclusion. In fact, it was the above argument that did the convincing. I made that step because your paper *does* contain the error I've outlined above. I did not request the paper be withdrawn, but I have no control over other people's actions. By maintaining that your paper is correct, you should at least make a tiny effort to indicate which of my above statements is incorrect. > See http://rattler.cameron.edu/swjpam/vol2-03.html > Check that link people. The discussion here isn't over abstractions > or what if's. > Posters from sci.math actually went out and badly affected the math > world by using social pressure against a solid proof. Again with that unsupported claim. Back it up, why don't you? > The damage from that *social* intervention is something that will play > out over time, but it clearly shows that mathematicians can just be > social creatures as well. > Now maybe Ioannis Argyros lied to me when he sent me an email saying > my paper had passed formal peer review, but that doesn't change the > reality that mathematicians are here clearly SOCIAL CREATURES and the > supposed sanctity of mathematical proof and adequate demonstration has > been broken. > You are lost without a rudder. > If there were objectivity here then there wouldn't be all these > arguments about my work, as then I could just explain it out in detail > and that would be it. But I've repeatedly given you the opportunity to explain this one little thing, and you've assiduously avoided just that opportunity. Then, as you say, that would be it. > But the trouble is that no matter how much I explain there are posters > like Andrzej Kolowski who have an emotional investment in just saying > I'm wrong, no matter what the mathematics says. explicit detail, enough to be verified or refuted on its on grounds, and simple enough that it only takes some arithmetic to verity (or refute). You don't seem to pick up on that: if I'm incorrect, I've given you enough evidence to prove it. If you truly believe in the correctness of your position, you would seize the chance. > They don't care about mathematics more than their own personal > feelings, which is why sci.math is useless for serious researchers. Well, there is sci.math.research. The present topic is not really at the research level. > There are too many emotional and loud people willing to just argue > with you because they don't like you who don't give a damn what the > math says. If it's all about what the math says, in your parlance, then scroll on up and do the math. If it's all about your precious ego, then I suppose you'll continue along your predetermined path. > James Harris Dale === Subject: Re: JSH: Groupthink Fixing bad formatting: in item 3 below, the carets indicating exponents were omitted. I've included them here, without further indication: ... stuff deleted ... > Here is an outline of the argument; if you could just point out which > of the statements relies on the alleged false assumption then you will > assist in furthering this discussion. > In this outline, a refers to one of the coefficients that you claim > must be coprime to 5. P(x) refers to the polynomial x^3 - 12 x^2 + 65, > and I note that P(-a) = 0, for each of your values a. > 1. The following formulas are true: > q(x)r(x) = (64 x + 128)P(x) + 5 > r(x)s(x) = (32 x + 72)P(x) + x, > where q,r,s are defined as follows: > q(x) = 8 x^2 - 76 x - 185 > r(x) = 8 x^2 - 4 x - 45 > s(x) = 4 x^2 - 37 x - 104 > 2. Since -a is a root of P(x), one has the following > factorizations: > q(-a)r(-a) = 5 > r(-a)s(-a) = -a > 3. The minimal polynomial of r(-a) is given as: > MP_r = x^3 - 969 x^2 + 315 x + 5 > which is irreducible over Q. This shows that > r(-a) is an algebraic integer, but not a unit > in that ring. > shared by a and 5 is not a unit in the ring > of algebraic integers. > 5. a and 5 are not coprime in the ring of algebraic > integers, since they share a non-unit factor in > common. > I claim that not one of these individual points depends on the > assumption you say I've made. The above constitutes the entire > argument and all commentary should be restricted to the argument, > so let's see where your objection lies. ... the rest deleted ... > Dale Dale. === Subject: A matrix question What can one say about a mxn (mWhat can one say about a mxn (mare linearly independent, other than that it is of rank m? Any reference? This is equivalent to saying that every nonzero vector in the null space of A has at least m+1 nonzero entries. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: A matrix question > What can one say about a mxn (m are linearly independent, other than that it is of rank m? Any reference? One can say its rectangular (m I know about it. I somehow do not like anything with a period. Even if > the period is *huge*. Maybe it's stupid of me, but I'd rather have > something that will definately not have, is garuanteed not to have, a > period. The digits of Sqrt(Prime) will never have a period. > While the base-n digits may not have a period, the >coefficients of its continued fraction representation >do. > See: http://mathworld.wolfram.com/PeriodicContinuedFraction.html >[Sorry, I don't know how ln associates in that expression :)] The only one that would make much sense is ln(X) sqrt(X). Otherwise why not just say O(ln X)? > Given a (large?) sequence of digits of Sqrt(X), I believe >that it would be possible to compute a periodic continued >fraction representation and, ultimately, X. Indeed. For example, from .704699910719625109102098274228621088823310518282 you get a continued fraction that starts [0;1,2,2,1,1,2,2,1,22,1,2,2,1,1,2,2,1,22,1,2,2,1,1,2,2,1,22,... ] If you guess that the 1,2,2,1,1,2,2,1,22 should repeat forever, you can solve x = [0; 1,2,2,1,1,2,2,1,22+x] = (2384+105*x)/(3383+149*x) to get x = -11 (+/-) sqrt(137), and thus these digits are from sqrt(137). But for a large X, if you need anywhere near ln(X) sqrt(X) terms of the continued fraction it's not going to be so easy. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Generating fractional part of Sqrt(Prime) >I know about it. I somehow do not like anything with a period. Even if >the period is *huge*. Maybe it's stupid of me, but I'd rather have >something that will definately not have, is garuanteed not to have, a >period. The digits of Sqrt(Prime) will never have a period. > While the base-n digits may not have a period, the >coefficients of its continued fraction representation >do. > See: http://mathworld.wolfram.com/PeriodicContinuedFraction.html >[Sorry, I don't know how ln associates in that expression :)] > The only one that would make much sense is ln(X) sqrt(X). Otherwise > why not just say O(ln X)? > Given a (large?) sequence of digits of Sqrt(X), I believe >that it would be possible to compute a periodic continued >fraction representation and, ultimately, X. > Indeed. For example, from > .704699910719625109102098274228621088823310518282 > you get a continued fraction that starts > [0;1,2,2,1,1,2,2,1,22,1,2,2,1,1,2,2,1,22,1,2,2,1,1,2,2,1,22,... ] > If you guess that the 1,2,2,1,1,2,2,1,22 should repeat forever, you > can solve > x = [0; 1,2,2,1,1,2,2,1,22+x] = (2384+105*x)/(3383+149*x) > to get x = -11 (+/-) sqrt(137), and thus these digits are from sqrt(137). > But for a large X, if you need anywhere near ln(X) sqrt(X) terms of > the continued fraction it's not going to be so easy. But will this continued fraction be of any use, if I would start using (and therefore an adversary would only be able to obtain) digits from a certain point, say e.g. the 2^16-th digit (in whatever base), in other words if I would just discard the first 2^16-1 digits of the fraction? Would X still be obtainable from that using today's math/computers? My primes would be like 1024 (but preferrable more -> 2048, 4096, ...) bits, by the way. Cryptographically sized primes (RSA/PGP/...). M. === Subject: Re: number theory question In sci.math, John : > A friend of mine mentioned that if you take the sequence of integer > squares modulo 20 then you definitely see a recurring pattern: > 1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,... > || > VV > 1,4,9,16,5,16,9,4,1,0,1,4,9,16,5,... > It is obvious that this sequence will just keep repeating, but how would > you prove that? It seems like the same will hold true even if 20 is > replaced by any integer n, but again I don't quite see how to prove that > the above sequence has to repeat. > -John Well, for starters, if m = n (modulo 20), then m^k = n^k (modulo 20) for any integer k > 0, so an arbitrary polynomial f(x) with integer coefficients is going to repeat every 20 no matter what. Since it turns out 10^2 = 100 = 0 (mod 20), and (x + 10)^2 = x^2 + 20 + 100 = x^2 (mod 20), it will repeat every 10 as well. -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: Uncountable number of isolated points on [0,1] part: >What's the prove that there can't be a subset of [0,1] >countaining more than countable number of isolated points? >Can't prove that :( I presume that an isolated point is a point which belongs to some finite interval [a,b] such that the point is > a and < b and no other member of the subset is in that interval. The statement is obviously true. How might we prove that? Two steps. First thing we need to prove that for all the isolated points we have, we can modify the intervals in which they lie that we use to show they are isolated so that *none of these intervals are overlapping*. Obviously, we can do this for any *two* intervals, otherwise the points wouldn't be isolated. If we can prove *that*, of course, the rest of the proof is trivial... each of these now non-overlapping intervals contains at least one *rational number*. John Savard http://home.ecn.ab.ca/~jsavard/index.html === Subject: Re: Uncountable number of isolated points on [0,1] >part: >What's the prove that there can't be a subset of [0,1] >countaining more than countable number of isolated points? >Can't prove that :( >I presume that an isolated point is a point which belongs to some finite >interval >[a,b] such that the point is > a and < b and no other member of the subset is in >that interval. >Two steps. >First thing we need to prove that for all the isolated points we have, we can >modify the intervals in which they lie that we use to show they are isolated so >that *none of these intervals are overlapping*. >Obviously, we can do this for any *two* intervals, otherwise the points wouldn't >be isolated. >If we can prove *that*, of course, the rest of the proof is trivial... each of >these now non-overlapping intervals contains at least one *rational number*. Even easier: make all the intervals have rational endpoints, and you don't need to worry about disjointness. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: 'Uncountable' doesn't exist I choose not to accept Cantor's delusions, and I should seek help? > Sorry, 'delusions' was too strong a word. I was making a reference to > Cantor's mental illness, since the previous poster accused me of being > crazy! > I am a Platonist, and I can't imagine that all the universe of > mathematics doesn't exist somehow in the mind of God. I simply can't > accept that an uncountable could ever exist in that sense. > So which is it? Is your objection that uncountable sets don't exist, as > you say here, or is it that uncountable sets can be manipulated like > other sets, as you say above? If your objection is the former, then for > you the latter objection does not even arise; so why mention it? If your > objection is the latter, then why do you keep talking about AC, when > there are lots of things that can be done to manipulate uncountable sets > in ZF? I would prefer not to describe uncountable collections as 'sets'. The power-set axiom, applied to infinite sets, is very problematic, as the power-set isn't well defined. In ZF, we can do only a few things with uncountable sets. We can take their power-set, and we can make subsets based on an explicit formula. By giving a well-ordering for every set, AC greatly increases what we can do. I say, and I know that some mathematicians agree, that none of the added theorems we can prove is useful or meaningful. They can not be constructed using the logic we know; to say that they exist implies some sort of 'meta-logic' that is (presumably) beyond Man's power to know. This idea is rather disconcerting, for it implies that all our philosophy is in vain, that we can never comprehend the first cause (including necessarily the totality of all logic) that must govern all that can exist. Since all of our intellectual activity can only be understood as part of that infinite endeavor, the existence of any such 'meta-logic' would make us, in effect, worthless. Andrew Usher === Subject: Re: 'Uncountable' doesn't exist > would make us, in effect, worthless. Speak for yourself. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: 'Uncountable' doesn't exist I choose not to accept Cantor's delusions, and I should seek help? > Sorry, 'delusions' was too strong a word. I was making a reference to > Cantor's mental illness, since the previous poster accused me of being > crazy! > I am a Platonist, and I can't imagine that all the universe of > mathematics doesn't exist somehow in the mind of God. I simply can't > accept that an uncountable could ever exist in that sense. > So which is it? Is your objection that uncountable sets don't exist, as > you say here, or is it that uncountable sets can be manipulated like > other sets, as you say above? If your objection is the former, then for > you the latter objection does not even arise; so why mention it? If your > objection is the latter, then why do you keep talking about AC, when > there are lots of things that can be done to manipulate uncountable sets > in ZF? I would prefer not to describe uncountable collections as 'sets'. The power-set axiom, applied to infinite sets, is very problematic, as the power-set isn't well defined. In ZF, we can do only a few things with uncountable sets. We can take their power-set, and we can make subsets based on an explicit formula. By giving a well-ordering for every set, AC greatly increases what we can do. I say, and I know that some mathematicians agree, that none of the added theorems we can prove is useful or meaningful. They can not be constructed using the logic we know; to say that they exist implies some sort of 'meta-logic' that is (presumably) beyond Man's power to know. This idea is rather disconcerting, for it implies that all our philosophy is in vain, that we can never comprehend the first cause (including necessarily the totality of all logic) that must govern all that can exist. Since all of our intellectual activity can only be understood as part of that infinite endeavor, the existence of any such 'meta-logic' would make us, in effect, worthless. Andrew Usher === Subject: Re: 'Uncountable' doesn't exist > I am a Platonist, and I can't imagine that all the universe of > mathematics doesn't exist somehow in the mind of God. I simply can't > accept that an uncountable could ever exist in that sense. > So which is it? Is your objection that uncountable sets don't exist, as > you say here, or is it that uncountable sets can be manipulated like > other sets, as you say above? If your objection is the former, then for > you the latter objection does not even arise; so why mention it? If your > objection is the latter, then why do you keep talking about AC, when > there are lots of things that can be done to manipulate uncountable sets > in ZF? > I would prefer not to describe uncountable collections as 'sets'. The > power-set axiom, applied to infinite sets, is very problematic, as the > power-set isn't well defined. The power set axiom tells us all we need to know about the power set. > In ZF, we can do only a few things with uncountable sets. We can take > their power-set, and we can make subsets based on an explicit formula. > By giving a well-ordering for every set, AC greatly increases what we > can do. I say, and I know that some mathematicians agree, that none of > the added theorems we can prove is useful or meaningful. They can not > be constructed using the logic we know; to say that they exist implies > some sort of 'meta-logic' that is (presumably) beyond Man's power to > know. There are lots of things we can do without AC. We can construct the ordinals and prove that they are well ordered. We can construct the integers, the rationals, the reals, and the complex numbers. We can talk about algebraic structures such as groups, rings, and fields. We can define topologies. ZF is the basis of a very rich field of mathematics. > This idea is rather disconcerting, for it implies that all our > philosophy is in vain, that we can never comprehend the first cause > (including necessarily the totality of all logic) that must govern all > that can exist. Since all of our intellectual activity can only be > understood as part of that infinite endeavor, the existence of any > such 'meta-logic' would make us, in effect, worthless. Adopting AC does not mean using a different kind of logic. It's still the very same logic that we use in ZF, but with one extra tool in the toolbox. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: 'Uncountable' doesn't exist > This is true, as you have an explicit formula. But proofs by induction > are certainly infinite (in fact, we need an axiom to allow them), > because they expand to an infinite number of steps. > > Depends on your starting point. Induction is an axiom if you are working > in PA, but it's a theorem if you are working in ZF. Hence your statement > (we need an axiom) is not necessarily true. > OK. Nonetheless it does compass a countable infinity of steps. > It's only one step in a proof. The hypotheses of AC are satisfied, and > therefore a certain set exists. ? What does AC have to do with induction? > But the axiom of choice _doesn't_ give an explicit formula. Hence it > is necessary to perform each choice individually. > You have that backwards. If it were possible to perform each choice > individually, we would not need the axiom of choice. AC was devised > precisely to cover those cases where we cannot perform each step > individually. > So basically it says 'We can't really do this, but we'll assume we > can.'? > That's not what we assume; AC says nothing at all about what we can do. > The axiom merely says that a certain set exists, not that we can > construct it. Is it your belief, then, that AC does not truly describe part of a mathematical system, but simply enlarges the universe of what we term sets? > Remember that the axiom of choice is really about what we choose to > call proper sets. If I say that uncountable collections aren't sets, > I'm not claiming they could not 'exist' in any sense. At least one > does, the set of all countable ordinals, but the construction if very > artificial. > The set aleph_1 of all countable ordinals does not require AC for its > construction. > No, but it the absence of AC, it doesn't correspond to any other > mathematical objects, which is why I called it 'very artificial'. > I have no idea what you are talking about. The set aleph_1 (or omega_1) > is an object of ZF. Its construction involves at least two applications > of the power set axiom, as I recall. It is no more artificial than > anything else in ZF. AC is completely irrelevant here. You're confusing cardinals and ordinals, I think. Aleph_1 is an unordered set, while omega_1 is totally ordered. I repeat, in the absence of AC, there is no object described by omega_1, the countable ordinals. Andrew Usher === Subject: Re: 'Uncountable' doesn't exist > It's only one step in a proof. The hypotheses of AC are satisfied, and > therefore a certain set exists. > What does AC have to do with induction? Huh? > So basically it says 'We can't really do this, but we'll assume we > can.'? > That's not what we assume; AC says nothing at all about what we can do. > The axiom merely says that a certain set exists, not that we can > construct it. > Is it your belief, then, that AC does not truly describe part of a > mathematical system, but simply enlarges the universe of what we term > sets? Sets are mathematical objects. Set theory is a mathematical system. > The set aleph_1 of all countable ordinals does not require AC for its > construction. > No, but it the absence of AC, it doesn't correspond to any other > mathematical objects, which is why I called it 'very artificial'. > I have no idea what you are talking about. The set aleph_1 (or omega_1) > is an object of ZF. Its construction involves at least two applications > of the power set axiom, as I recall. It is no more artificial than > anything else in ZF. AC is completely irrelevant here. > You're confusing cardinals and ordinals, I think. Aleph_1 is an > unordered set, while omega_1 is totally ordered. I repeat, in the > absence of AC, there is no object described by omega_1, the countable > ordinals. A cardinal is an initial ordinal. Aleph_1 and omega_1 are the same set. We use one name when we want to think of it as a cardinal, and the other when we want to think of it as an ordinal. I added (or omega_1) because the usual construction is to define the ordinal omega_1 first, and then define the cardinal aleph_1 to be the same set. You may repeat your assertion if you like, but it is still wrong. The definition of omega_1 does not depend on AC. You seem to have some very strange notions about what AC says. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: 'Uncountable' doesn't exist Discussion, linux) > > It's only one step in a proof. The hypotheses of AC are satisfied, and > therefore a certain set exists. > ? > What does AC have to do with induction? > Huh? Huh yourself. The paragraph right above this was: > Depends on your starting point. Induction is an axiom if you > are working in PA, but it's a theorem if you are working in ZF. > Hence your statement (we need an axiom) is not necessarily > true. The immediate discussion was about whether induction was one step or not, and I think you somehow took it back to AC. -- Analysis/ editors have evaluated the paper, they accepted it for publication and they have the copyright of its contents - and thus they are responsible for its correctness,' she [said]. === Subject: Re: 'Uncountable' doesn't exist > > It's only one step in a proof. The hypotheses of AC are satisfied, and > therefore a certain set exists. > ? > What does AC have to do with induction? > Huh? > Huh yourself. The paragraph right above this was: > Depends on your starting point. Induction is an axiom if you > are working in PA, but it's a theorem if you are working in ZF. > Hence your statement (we need an axiom) is not necessarily > true. > The immediate discussion was about whether induction was one step or > not, and I think you somehow took it back to AC. Ok, I see. To answer the question, what induction has to do with AC is that: (1) Andrew claims induction involves infinitely many steps in a proof, (2) Andrew claims AC involves infinitely many steps in a proof, (3) Andrew is wrong on both counts. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: 'Uncountable' doesn't exist > Virgil says... > What is IZF set theory? >Excessively red queenish. > I don't agree. I think it's pretty cool. IZF shows that much of the mathematical > content of set theory can be maintained in a constructive > theory with much more modest ontological commitments. In IZF, there are > basically only integers and formulas --- higher-order sets only exist > to the extent that they can be defined via a formula. I was thinking about that the other day, in terms of set theory, about numbers not being sets. That's the notion that if you just have containers and nothing but containers to put in them, that it's just empty. Then in terms of mathematics you could consider numbers that are not sets. Functions have to do with type theory and lambda calculus. Then again I'm still working on designing a programming language and programming tools, and I've come to the notion that containers should be on a par with, say, objects in the design and development of programs, because so many things are containers. The file is a container, the file on a web server is a container, the list is a container, the string is a container, the database is a container, its query result is a container. The graph is the generic container. I think all other containers are specialized graphs. All the containers can be serialized as sequences. Then, the computer program can be visualized of a flowgraph of sorts where as each statement, or rather expression or command, is evaluated then the edge or path to the next statement is followed determined by the flow-of-control operators so that the program resembles a flowchart. My idea here is that I've been working on generative programming tools to define program in terms of patterns simple enough to be usable and concise yet complex enough to utilize the operating system environments' capabilities. The patterns on different systems for the same conceptual operation reflect different types, different orders of the statements to be called, and otherwise various variations on the theme of the state machine. For example, where the idea is to write a program once and generate code for UNIX, Windows, Macintosh, and Java using C, C++, Objective C, C#, and Java, using the standard libraries of UNIX, C, Win32/COM, Macintosh Carbon and Cocoa, C++ and the STL, web forms and middleware, and Java, to open a file on UNIX, you might use the statement of calling a function, the imperative command: FILE* f = open(/usr/home/x/data.txt); or int fd = fopen(/usr/home/x/data.txt, O_RDONLY, 0777); That might contain errors. In Java, you might use an InputStream. InputStream is = new RandomAccesFile(/usr/home/x/data.txt).getInputStream(); Again, that might not be the actual correct syntax and semantics, not compiling. Then to test to see if the file is readable, you need compare f against NULL, (void*)0, or 0, and as well check ferror, and perhaps perror, or otherwise handle errors and exceptions, external events such as signals to be handled by your program. In Java, the file opening method on some file-type object is enclosed within a try/catch block to as well be informed and able to catch the exception and handle it gracefully, else the program might fall off the flowgraph into the undefined behavior of all that the state machine could be within the operating environment. Anyways there's a file, or source for reading, in terms of sources, sinks, and filters, which may be a socket or pipe or memory block or file view or other source that as an object has a related function, a method to read the data into the scope of the program and then process it. This is a diversion from high level set-theoretical treatments of infinite sets, the point is that it takes a lot to translate allocate necessary resources, find a file, open it, read and process its contents, close the file and deallocate the resources. Then there's the entire concept of what a program is, and all the object-level semantics that make JavaBeans and COM objects, and how to handle a signal versus an exception versus an error code return value among the hundreds of possible values on each system. It's like considering how to make a simple graphical application, a terminal front-end, and a web page with JavaScript validation by pressing one button. The idea is software reuse to the extreme. Anyways that's quite off-topic here in a discussion about the transfinite cardinals, but it is some consideration IZF. I searched sci.math for mention of IZF and some related terms, and got to read some notes from 1991 that IZF is the product of Beeson and Dummett and is quite constructivist in its aims and direction, and as well a nice post by Keith Ramsay about constructivism. I can not at this point readily define constructivism. Now then, the FOM post says that in IZF it is not inconsistent to have a function between the natural integers and the reals. That's exactly what was written. I simply agree. Virgil, your intellect is evidenced by your mathematical knowledge, usage, and analytical skills, and your vocabulary, and to some extent your literary references. It's sometimes difficult to be objective about oneself, but not impossible. In fact, you know yourself more thoroughly than anyone else does. I'm intellectually pleased to see another theory that gets some acceptance that allows a mapping from the naturals to and from the reals, although at the same time I hoped to make more developments before the dam breaks. I don't know the development of that line of argument, as soon as I heard of it I sent it to sci.math. It may well work correctly with a variety of my logical and mathematical statements. (Insert standard spiel of my motivations for writing to and recollections of sci.math.) Ezra: A single touch..., to ignite her. naturals and reals? Is not a function's range most often its image or co-image? Does McCarty's statement as mentioned by Frank on the draconianly moderated FOM list imply the consistency of bijections between the naturals and reals? Ross Finlayson === Subject: Euclid and textbooks Euclid's Elements was used to teach elementary mathematics for over 20 centuries. It is hard to imagine any sort of textbook in modern times being used for even one. Why is this? Was it in irrational reverence for the ancients that caused no one to attempt a better exposition than Euclid? Or, is it something wrong with modern teaching? The results derived using basic calculus have not changed since Euler, but all freshman calculus texts are new; this doesn't seem to make sense. Also, I have a question (out of ignorance, not stupidity). When Euclid's proofs are referred to as 'synthetic' rather than 'analytic', what exactly do those terms mean? I know that analysis refers to the calculus, but that doesn't seem to help. Andrew Usher === Subject: Re: Euclid and textbooks > Euclid's Elements was used to teach elementary mathematics for over 20 > centuries. It is hard to imagine any sort of textbook in modern times > being used for even one. > Why is this? Was it in irrational reverence for the ancients that > caused no one to attempt a better exposition than Euclid? Or, is it > something wrong with modern teaching? The results derived using basic > calculus have not changed since Euler, but all freshman calculus texts > are new; this doesn't seem to make sense. > Also, I have a question (out of ignorance, not stupidity). When > Euclid's proofs are referred to as 'synthetic' rather than 'analytic', > what exactly do those terms mean? I know that analysis refers to the > calculus, but that doesn't seem to help. > Andrew Usher Analytic geometry is the Cartesian geometry of Rene de Carte, based on coordinate systems and algebraic properties corresponding to geometric objects, like a x + b y = c being a linear equation corresponding to the line in geometry. It attempts to derive and prove geometric relationships by means of algebraic analysis. Synthetic geometry is pure in the sense that it has minimal dependence on algebraic analysis, and does not rely on coordinate systems nor the analysis of algebra to derive or prove geometric relationships, but does it strictly with geometric properties. === Subject: Euclid and textbooks Euclid's Elements was used to teach elementary mathematics for over 20 centuries. It is hard to imagine any sort of textbook in modern times being used for even one. Why is this? Was it in irrational reverence for the ancients that caused no one to attempt a better exposition than Euclid? Or, is it something wrong with modern teaching? The results derived using basic calculus have not changed since Euler, but all freshman calculus texts are new; this doesn't seem to make sense. Also, I have a question (out of ignorance, not stupidity). When Euclid's proofs are referred to as 'synthetic' rather than 'analytic', what exactly do those terms mean? I know that analysis refers to the calculus, but that doesn't seem to help. Andrew Usher === Subject: p-value and asset managment 1/Is it possible to know asset managment and not to know what the p-value is? I was asked it during an interview and did not know 2/ What is the most important to work in asset managment -(math part)? === Subject: Re: ~ Proof techniques for surjectivity. <6ZxLc.2321$rCl.392@news04.bloor.is.net.cable.rogers.com> used. Yes, x + 1 ranges over R as long as x is allow free range. > That's a non-sequitur. You need take in account that within the > definion of f, the x in x + 1 is limited to x < 0. > Ah! Yes, I did take the range into account. I suppose that each case > does not have to range over the R, but that the combination of the two of > them must. That way any member of the codomain can be mapped to. For > instance, if a function consists of two parts (I'm not sure the correct > term) with one part ranging over R+ and the other over R-, then the function > really maps to all of R combined. Is that correct reasoning? As I recall, for you have not included the proof which is the topic of discussion, as I recall, no mention of limiting x to x < 0 was made. Yes it may have been in your mind, but it wasn't as I recall not on paper. Please keep adequate context for more accurate replies, we don't remember your problems and proofs as vividly as do you who have been working on them. > Stick with it. > I sure will. I plan to start studying every day now and so will progress > much more rapidly. Don't bit off more than you can chew. Math is infinite, mind finite. === Subject: Re: The real effect of centrifugal force > Centrifugal force is mistakenly thought to cause a body to fly out of > its circular path; along a tangent direction which represents the > inertial path of Newtons inertial motion. > > Actually centrifugal force, when not restrained as by a string or > sling causes a body to flee the center of an evolute radially; along > involutes that remain directed radially away from the center: > > Like the sling used by David to slay Goliath; when it is released the > missile travels radially away from the center to points equidistant > along the tangent.; at potentially lethal speeds depending on the > speed it has when it is released from the evolute. > > There is nothing fictitious about centrifugal force! > > not true. you're mistaking the difference between letting go of the > other end of the string and the string vanishing completely. if u let > go of the string sure it'll fly off radially. > No. It'll fly off tangentially. > Or, to be perfectly accurate, it'll fly off _both_ radially and > tangentially. > If you stay with the rotating frame, looking down your arm at the > sling stone, it'll appear to fly off radially. At least until > Coriolis rears its head and the stone veers off anti-spinward. > (Ever played catch on a merry-go-round? Amazingly difficult.) > If you work from an inertial frame, standing behind the slinger > and watching the stone, it'll appear to fly off tangentially. > Both descriptions work and both descriptions wind up with the stone > impacting Goliath's head at the same relative speed. Not true. If you do it this way your stone drops limply to the ground about halfway to Goliath's head. It is probably better to stay with the inertial frame in this case, because the sling projectile's velocity after release is not simply the linear speed during rotation. During the release interval, most of the speed is generated by moving the center of rotation a large distance forward. Therefore your rotating frame becomes much more complicated than your inertial frame. The assumption that the sling whirls around a stationary center and the speed is the radius times the angular velocity is as bogus as the assumed spherical horse in the old joke. MOST of the sling effect comes from moving the center (all of it if you include the fact that small linear motions of the center cause the whirling). That should change your picture of the direction the mass travels as well. === Subject: Re: argh > i just posted a message and I don't think it went through. STUPID > INTERNET EXPLORER! :) didnt save :( Then don't use internet explorer! :-) -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Parallel computation possibilities? Hello again, Some earlier discussions on the RSA factoring challenge got me thinking. A popular video game for a console might end up in 4 million homes. Optimistically assume that, on average, there'd be 10 hours wasted computational time for each copy (e.g. while paused, waiting for vertical blank etc.) So a single game could provide 40 million CPU hours that could be used for *something*. Of course, the thing that makes this interesting (?!) is that there's no communication available between the machines - all they can do is work independently and notify the user if they obtain a useful result (e.g. You won a prize! Phone us with these factors for a £1000 reward). Given the unusual assumptions, can anyone think of potential applications? Factoring was the original motivation but a quadratic sieve appears to require that you collate all the work in a final step to solve a big linear system. Alternatively, randomly choosing factors would take way too long. Interested to hear your ideas, G.A. === Subject: Re: Parallel computation possibilities? > Given the unusual assumptions, can anyone think of potential > applications? You could collate lots of statics about human reaction times, as a simple extension of the existing application. BTW, some of those game consoles are now talking to each other. -- CodeCutter - good, fast and cheap; pick two. === Subject: Re: Expressing e as a function of Pi > possibly related web pages. For example, you can learn that > This comes from number theory and is connected with the fact > that the imaginary quadratic number field with discriminant > -163 has class number 1. There is no discriminant beyond -163 > with class number 1 I'm astonded. Is there no positive upper limit? === Subject: Re: Expressing e as a function of Pi In sci.math, David Bandel e as a function of Pi, in the sense of shortest formula, > i.e.: least number of functions and/or digits needed, > surely is this: > e = 640320^(3/Pi/Sqrt(163)) (approx., not truly equal) > = 2.7182818284590450+ > while > e = 2.7182818284590452+ > so this simple formula gives nearly 17 correct digits, > with an error of only 2 ulps in the 17th place. > Of course, Sqrt is the square root function. > how about: e ~ (-1)^(1 / [i*PI]) > this gives e correct to about infinity places Yes, but how does one compute (-1)^(irrational pure imaginary number)? ;-) That doesn't look like a very easy computational problem, somehow. (Not that 640320^(3/Pi/Sqrt(163)) looks all that easy, either.) -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: Expressing e as a function of Pi > In sci.math, David Bandel > One of the best ways of approximately expressing > e as a function of Pi, in the sense of shortest formula, > i.e.: least number of functions and/or digits needed, > surely is this: > > > e = 640320^(3/Pi/Sqrt(163)) (approx., not truly equal) > > = 2.7182818284590450+ > > while > > e = 2.7182818284590452+ > > so this simple formula gives nearly 17 correct digits, > with an error of only 2 ulps in the 17th place. > Of course, Sqrt is the square root function. > > how about: e ~ (-1)^(1 / [i*PI]) > this gives e correct to about infinity places > Yes, but how does one compute (-1)^(irrational pure imaginary number)? ;-) > That doesn't look like a very easy computational problem, somehow. > (Not that 640320^(3/Pi/Sqrt(163)) looks all that easy, either.) i believe the brilliant and eminent math professor: james harris recently published a paper on the exponentation of complex numbers. i believe it had some critical errors pointed out by some middle school math flunkies.. but he's still shooting for peer review! go james harris you homosexual pedophile you! we all love you! === Subject: Re: Graphs, planarity and dual graphs > All sources that I've seen say that the dual of a plane graph (or > plane map, depending on how you define the embedding of a graph in the > plane) is obviously a plane graph, with no further proof, but I > don't really see that as instantly as the authors seem to do. > Consider any face of the graph in isolation. You can deform it > (imagine it is drawn on a rubber sheet and you are stretching the > rubber in some way) until the face becomes a regular polygon. Connect A self dual graph is * / / * How do you propose to stretch the face of that graph into a regular polygon? > the centre of the face to the midpoints of the edges by straight > lines. Now undo that deformation (i.e. let go of the rubber sheet) and > you will have drawn the part of the dual graph that lies inside that > face. Repeat this for each of the interior faces. The exterior face > doesn't quite work exactly the same way, but you can still do > something similar. Then (provided you make sure that all the ends of > the lines you have drawn meet up) you get the whole dual graph. What is the dual of a chain? The empty graph? The empty graph and * / / * are self dual graphs. Are there any others? === Subject: Re: Graphs, planarity and dual graphs 3QLpj-NoP*NzsIC,boYU]bQ]H'y<#4ga3$21: > The empty graph and > * > / > / > * > are self dual graphs. Are there any others? Yes, lots. Any pyramid, for instance. -- David Eppstein Computer Science Dept., Univ. of California, Irvine http://www.ics.uci.edu/~eppstein/ === Subject: Re: Question about PhD math programs reworded titles are supposed to represent are not. May I recommend students take legal action upon this deception in advertising. === Subject: Re: Question about PhD math programs reworded > Courses with the titles are offered, but the courses which those > titles are supposed to represent are not. > May I recommend students take legal action upon this deception in > advertising. Right on!!! I support that! I further support students who graduate high school who can't read and write to sue their school system. If enough students do this the school systems might realize that it might be cheaper to educate these youth then to keep paying out hugh court settlements. === Subject: Re: Roger Bacon on math > Neglect of mathematics works injury to all knowledge, since he who > ignorant of it cannot know the other sciences or the things of this > world. And what is worse, is that men who are thus ignorant are > unable to perceive their own ignorance and so do not seek a remedy. > ---Roger Bacon Ignorance is bliss, it's the fashionable fad these days, from education to news media to what Bush is doing in the White Lie House and to what politicians city, state and national, are doing to US. === He describes the basic ideas of the proof, his recent contribution with Steve Smith on the quasithin portion of the classification, and gives a historical summary of events to date. Interestingly, he says the quasithin classification was noticed to have a serious gap about 10 years ago, and he and Smith took seven years to close the gap; this part of the classification was written up last year, and will be published soon. Aschbacher: I have described the Classification as a theorem, and at this time I believe that to be true. === Subject: Re: Pandoras Box Wormholes: It seems wormholes imply a closed Universe (at least along one axis). Picture the 2D version, where there's a flat 2D plane. A wormhole is like a pipe leading from one point to another through the 3rd dimension. A 2D object can move along the surface of the pipe from one end to the other. However note the sides of the tube are finite width, and can be quite narrow. If half way down the pipe, you move in a perpendicular direction, you'll quickly reach your starting point, as you go around the pipe. In a sufficiently narrow tube, you should be able to see yourself in the distance. Or see multiple copies of yourself in a line out to infinity, similar to the view when you have two mirrors facing each other. This implies there are no apparent edges of a wormhole as you go through it, since you can always see in all directions regardless of the topology. Puzzles: What if the wormhole is narrow enough that you can't fit through it? Will you bump into the side of yourself trying to squeeze in? What if the wormhole's coordinate system is defined such that the pipe is of zero length, i.e. you more or less teleport from one location to another? What happens then if you run into a wormhole you can't fit through? Will you cut yourself in two? Describe the 3D Universe equivalents of the above. Recursive Universe: Consider a recursive or fractal Universe, which is defined in terms of itself! Assume a 2D Universe defined as a 3x3 grid of squares. The middle square is defined as a copy of the Universe, i.e. the middle square can also be divided into a 3x3 grid of squares, where its middle square is another copy of the Universe, and so on forever. Similarly the outer boundary of the Universe leads into a larger version of the whole thing, where the current Universe is just the middle square of a larger 3x3 grid. With a sufficiently powerful telescope, or if the Universe is small enough, you can see a small version of yourself in the inner Universe, and smaller and smaller versions of yourself deeper in. Similarly there's larger and larger versions of yourself in outer Universes. Puzzles: What happens when you move into the middle square or the next smaller Universe? Note that implies each copy of yourself moves into the next smaller Universe on all levels. If you repeat the process, won't you become larger and larger relative to the rest of the Universe? Could you crush yourself like stepping on an ant? ;) Are you increasing the total mass in the Universe? If there's gravity in these Universes, will you be subject to infinite gravity from the infinitely large versions of yourself in outer Universes? Note if the coordinate system were defined such that you don't get any smaller as you move into an inner Universe, that's the equivalent to the whole thing being one simple Universe on a torus surface. O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O * Walter D. Cruiser1 Pullen :) ! Astara@msn.com * O Get lost in my Labyrinth Web page: http://www.astrolog.org/labyrnth.htm O *O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O* === Subject: Re: Pandoras Box >Wormholes: It seems wormholes imply a closed Universe (at least along one >axis). Err, no they don't. They imply a closed loop within the universe, but in your 2D example the universe itself is open. >Puzzles: What if the wormhole is narrow enough that you can't fit through >it? Will you bump into the side of yourself trying to squeeze in? That's what I'd think, yes -- if you entered feet first, your right shoulder would contact your left shoulder and you'd go no further. >What if >the wormhole's coordinate system is defined such that the pipe is of zero >length, i.e. you more or less teleport from one location to another? What >happens then if you run into a wormhole you can't fit through? I think the length of the wormhole makes no difference; you'll still get stuck, with part of your body on one side of the hole and part on the other. It's just that your entire body will be in flat space, minus one line (or plane, in 3D), so it won't be obvious why you're stuck. I don't see why this would cut you in two, though. (I'm considering only geometry here, not gravity...) === Subject: Re: Pandoras Box > Wormholes: It seems wormholes imply a closed Universe (at least along > one axis). Picture the 2D version, where there's a flat 2D plane. A > wormhole is like a pipe leading from one point to another through the > 3rd dimension. A 2D object can move along the surface of the pipe from > one end to the other. However note the sides of the tube are finite > width, and can be quite narrow. If half way down the pipe, you move in > a perpendicular direction, you'll quickly reach your starting point, > as you go around the pipe. In a sufficiently narrow tube, you should > be able to see yourself in the distance. Or see multiple copies of > yourself in a line out to infinity, similar to the view when you have > two mirrors facing each other. This implies there are no apparent > edges of a wormhole as you go through it, since you can always see > in all directions regardless of the topology. > Puzzles: What if the wormhole is narrow enough that you can't fit > through it? Will you bump into the side of yourself trying to squeeze > in? What if the wormhole's coordinate system is defined such that the > pipe is of zero length, i.e. you more or less teleport from one > location to another? What happens then if you run into a wormhole you > can't fit through? Will you cut yourself in two? Describe the 3D > Universe equivalents of the above. This is becoming part of my argument for warps and bend/wormholes in space being defined as a kind of object. A 3D equivalent of this could be my Pandoras Box that contains a wormhole inside it which would of course make it smaller than the box, at least the entrance. Then whichever way you try to push the box into the wormhole or try to pull it away from the wormhole, part of it will be going into the wormhole, and will either cause the box to become disjointed, or the box will try and stretch like an elastic object. You tell us what you think Walter... Another question is if the box did become seperated into two pieces would they still remain connected? > Recursive Universe: Consider a recursive or fractal Universe, which is > defined in terms of itself! Assume a 2D Universe defined as a 3x3 grid > of squares. The middle square is defined as a copy of the Universe, > i.e. the middle square can also be divided into a 3x3 grid of squares, > where its middle square is another copy of the Universe, and so on > forever. Similarly the outer boundary of the Universe leads into a > larger version of the whole thing, where the current Universe is just > the middle square of a larger 3x3 grid. With a sufficiently powerful > telescope, or if the Universe is small enough, you can see a small > version of yourself in the inner Universe, and smaller and smaller > versions of yourself deeper in. Similarly there's larger and larger > versions of yourself in outer Universes. > Puzzles: What happens when you move into the middle square or the next > smaller Universe? Note that implies each copy of yourself moves into > the next smaller Universe on all levels. If you repeat the process, > won't you become larger and larger relative to the rest of the > Universe? Could you crush yourself like stepping on an ant? ;) Are you > increasing the total mass in the Universe? If there's gravity in these > Universes, will you be subject to infinite gravity from the infinitely > large versions of yourself in outer Universes? Note if the coordinate > system were defined such that you don't get any smaller as you move > into an inner Universe, that's the equivalent to the whole thing being > one simple Universe on a torus surface. This recursive universe you are describing seems very similar to a kind of Cantor dust... see Sierpinski carpet... You are asking many different questions here. won't you become larger and larger relative to the rest of the Universe? If you move from a larger square to a smaller square, you would become larger yes. Could you crush yourself like stepping on an ant? Well the box you step into might be too small for you, which might mean you couldn't step into it at all depending on your deffinitions. If you have two twins one 2 units big and the other 1 unit big, and they both start out next to each other in a 3 unit square and a 2 unit square, then they can both take one step, into the 2 unit square and into the 1 unit square, then niether of them will be able to take a step anymore, so neither of them will arive on the same square and be able to crush the other one. If both of the twins were smaller or started out in larger squares they could both take more steps, but they would always take the same number of steps if they were twins in the way it is defined. Are you increasing the total mass in the Universe? The total mass of the universe is a cardinal number already it seems. If there's gravity in these Universes, will you be subject to infinite gravity from the infinitely large versions of yourself in outer Universes? Your relationship to your other twins doesn't change at all. The interesting thing about the Sierpinski carpet and Sierpinski triangle to me is not this game you are playing.. but the Chaos Game.. The Chaos Game involves randomly selecting points within the Sierpinski triangle or carpet, which will create a perfect order that is the fractile you have constructed. If you have any order to the points you pick, any pattern, you will not create the fractile.. it is only when chaos is involved that the order emerges... see: http://en.wikipedia.org/wiki/Sierpinski_triangle for more info, including source code... > O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O > *O*O * Walter D. Cruiser1 Pullen :) ! > Astara@msn.com * O Get lost in my Labyrinth Web page: > http://www.astrolog.org/labyrnth.htm O > *O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O*O* > O*O* === Subject: Re: Pandoras Box > Imagine a Box in R3, that is closed and orientable. Inside of the volume > of this box, but not on the surface of the box, there is a portal or > wormhole that stretches R3 through the 4th dimension to the outside of the > box, but also not on the surface of the box at this end either. You can > see a diagram of this box in our dimension here: > http://nodalpoints.net/pandorasbox.php Let's generalize some things: Let U be the universe (e.g. R3), including definitions of adjacency and connectedness. Let S be the surface (e.g. a box). Let two points x and y (neither belonging to S) be related if they can be connected by at least one continuous curve within U that does not intersect S, and unrelated otherwise. Note that relatedness and unrelatedness are both reflexive, and relatedness is also symmetric and transitive. (There are alternate definitions of connectedness for which this is not the case, but those are outside the scope of this discussion.) S divides U into one or more subsets, as follows: 1) Pick an arbitrary point p1 in (U - S) 2) Let S1 be the set of all points related to p1 3) If S + S1 = U, stop 4) Pick an arbitrary point p2 in (U - S - S1) 5) Let S2 be the set of all points related to p2 6) If S + S1 + S2 = U, stop [etc.] The usual sense of inside and outside depends on this process leading to exactly two sets. (Furthermore, one set has finite volume and is inside, the other has infinite volume and is outside). The problem is, by creating the wormhole - even if it has zero dimensions and zero added points, e.g. (0,0,0) and (0,0,1) are considered adjacent while all other adjacencies follow the usual rules for R3 - you've changed U, and S ceases to have an inside and outside. This is not surprising if you consider other cases of changing U (to something that's easier to grasp intuitively) while keeping S the same. For instance, a circle embedded in R2 has an inside and outside; the same circle embedded in R3 does not. I'm not sure whether the addition of the wormhole also causes the box to become non-orientable. I believe you're correct to suspect that changing U while keeping S the same *can* change the orientability of S. === Subject: Re: Pandoras Box > Imagine a Box in R3, that is closed and orientable. Inside of the > volume of this box, but not on the surface of the box, there is a > portal or wormhole that stretches R3 through the 4th dimension to the > outside of the box, but also not on the surface of the box at this > end either. You can see a diagram of this box in our dimension here: > http://nodalpoints.net/pandorasbox.php > Let's generalize some things: > Let U be the universe (e.g. R3), including definitions of adjacency > and connectedness. > Let S be the surface (e.g. a box). > Let two points x and y (neither belonging to S) be related if they can > be connected by at least one continuous curve within U that does not > intersect S, and unrelated otherwise. > Note that relatedness and unrelatedness are both reflexive, and > relatedness is also symmetric and transitive. (There are alternate > definitions of connectedness for which this is not the case, but > those are outside the scope of this discussion.) > S divides U into one or more subsets, as follows: > 1) Pick an arbitrary point p1 in (U - S) > 2) Let S1 be the set of all points related to p1 > 3) If S + S1 = U, stop > 4) Pick an arbitrary point p2 in (U - S - S1) > 5) Let S2 be the set of all points related to p2 > 6) If S + S1 + S2 = U, stop > [etc.] > The usual sense of inside and outside depends on this process > leading to exactly two sets. (Furthermore, one set has finite > volume and is inside, the other has infinite volume and is > outside). > The problem is, by creating the wormhole - even if it has zero > dimensions and zero added points, e.g. (0,0,0) and (0,0,1) are > considered adjacent while all other adjacencies follow the usual > rules for R3 - you've changed U, and S ceases to have an inside > and outside. This is not surprising if you consider other cases > of changing U (to something that's easier to grasp intuitively) > while keeping S the same. For instance, a circle embedded in R2 > has an inside and outside; the same circle embedded in R3 does not. > I'm not sure whether the addition of the wormhole also causes the > box to become non-orientable. I believe you're correct to suspect > that changing U while keeping S the same *can* change the > orientability of S. Okay I understand what you say, and it is extremely helpfull... after reading you though I suspect I am not being clear enough in a formal sense.. I like your idea of remapping points to create a wormhole which would allow us to simply work in R^3, but in my idea I transformed R^3 to have a Handle which extended into R^4. Now that we agree it may change the properties of the box as such... does this mean we can no longer consider R^3 to be R^3 after a transformation in the sense that after you transform a cube into a sphere it is no longer a polygon, and as such loses certain properties like its ability to have sides? a complete nutter or kook that wasn't worth the time for a reply. :D :D :D === Subject: Re: Pandoras Box > Imagine a Box in R3, that is closed and orientable. Inside of the > volume of this box, but not on the surface of the box, there is a > portal or wormhole that stretches R3 through the 4th dimension to the > outside of the box, but also not on the surface of the box at this end > either. You can see a diagram of this box in our dimension here: > http://nodalpoints.net/pandorasbox.php > Let's generalize some things: > Let U be the universe (e.g. R3), including definitions of adjacency and > connectedness. > Let S be the surface (e.g. a box). > Let two points x and y (neither belonging to S) be related if they can > be connected by at least one continuous curve within U that does not > intersect S, and unrelated otherwise. > Note that relatedness and unrelatedness are both reflexive, and > relatedness is also symmetric and transitive. (There are alternate > definitions of connectedness for which this is not the case, but those > are outside the scope of this discussion.) > S divides U into one or more subsets, as follows: > 1) Pick an arbitrary point p1 in (U - S) 2) Let S1 be the set of all > points related to p1 3) If S + S1 = U, stop > 4) Pick an arbitrary point p2 in (U - S - S1) 5) Let S2 be the set of > all points related to p2 6) If S + S1 + S2 = U, stop > [etc.] > The usual sense of inside and outside depends on this process > leading to exactly two sets. (Furthermore, one set has finite volume > and is inside, the other has infinite volume and is outside). > The problem is, by creating the wormhole - even if it has zero > dimensions and zero added points, e.g. (0,0,0) and (0,0,1) are > considered adjacent while all other adjacencies follow the usual rules > for R3 - you've changed U, and S ceases to have an inside and > outside. This is not surprising if you consider other cases of > changing U (to something that's easier to grasp intuitively) while > keeping S the same. For instance, a circle embedded in R2 has an inside > and outside; the same circle embedded in R3 does not. > I'm not sure whether the addition of the wormhole also causes the box to > become non-orientable. I believe you're correct to suspect that > changing U while keeping S the same *can* change the orientability of S. > Okay I understand what you say, and it is extremely helpfull... after > reading you though I suspect I am not being clear enough in a formal > sense.. I like your idea of remapping points to create a wormhole which > would allow us to simply work in R^3, but in my idea I transformed R^3 to > have a Handle which extended into R^4. Now that we agree it may change > the properties of the box as such... does this mean we can no longer > consider R^3 to be R^3 after a transformation in the sense that after you > transform a cube into a sphere it is no longer a polygon, and as such > for a response.. I was beginning to think I came off like a complete > nutter or kook that wasn't worth the time for a reply. :D :D :D The difference between a cube and a sphere is geometric, not topological. (Informal explanation of topology: Imagine that the cube is made of rubber; you can stretch it all you want, but you can't poke holes or glue pieces together.) doughnut =/= sphere Changing the universe of discourse from R^3 to (R^3 plus a wormhole) does induce a topological change, yes - not in the box directly, but in the points within it (and therefore in how the box relates to those points). === Subject: Re: Pandoras Box > > Imagine a Box in R3, that is closed and orientable. Inside of the > volume of this box, but not on the surface of the box, there is a > portal or wormhole that stretches R3 through the 4th dimension to > the outside of the box, but also not on the surface of the box at > this end either. You can see a diagram of this box in our > dimension here: http://nodalpoints.net/pandorasbox.php > > Let's generalize some things: > > Let U be the universe (e.g. R3), including definitions of adjacency > and connectedness. > > Let S be the surface (e.g. a box). > > Let two points x and y (neither belonging to S) be related if they > can be connected by at least one continuous curve within U that does > not intersect S, and unrelated otherwise. > > Note that relatedness and unrelatedness are both reflexive, and > relatedness is also symmetric and transitive. (There are alternate > definitions of connectedness for which this is not the case, but > those are outside the scope of this discussion.) > > S divides U into one or more subsets, as follows: > > 1) Pick an arbitrary point p1 in (U - S) 2) Let S1 be the set of > all points related to p1 3) If S + S1 = U, stop > 4) Pick an arbitrary point p2 in (U - S - S1) 5) Let S2 be the set > of all points related to p2 6) If S + S1 + S2 = U, stop > [etc.] > > The usual sense of inside and outside depends on this process > leading to exactly two sets. (Furthermore, one set has finite > volume and is inside, the other has infinite volume and is > outside). > > The problem is, by creating the wormhole - even if it has zero > dimensions and zero added points, e.g. (0,0,0) and (0,0,1) are > considered adjacent while all other adjacencies follow the usual > rules for R3 - you've changed U, and S ceases to have an inside > and outside. This is not surprising if you consider other cases of > changing U (to something that's easier to grasp intuitively) while > keeping S the same. For instance, a circle embedded in R2 has an > inside and outside; the same circle embedded in R3 does not. > > I'm not sure whether the addition of the wormhole also causes the > box to become non-orientable. I believe you're correct to suspect > that changing U while keeping S the same *can* change the > orientability of S. > > > > Okay I understand what you say, and it is extremely helpfull... after > reading you though I suspect I am not being clear enough in a formal > sense.. I like your idea of remapping points to create a wormhole > which would allow us to simply work in R^3, but in my idea I > transformed R^3 to have a Handle which extended into R^4. Now that > we agree it may change the properties of the box as such... does this > mean we can no longer consider R^3 to be R^3 after a transformation > in the sense that after you transform a cube into a sphere it is no > longer a polygon, and as such loses certain properties like its > beginning to think I came off like a complete nutter or kook that > wasn't worth the time for a reply. :D :D :D > The difference between a cube and a sphere is geometric, not > topological. (Informal explanation of topology: Imagine that > the cube is made of rubber; you can stretch it all you want, > but you can't poke holes or glue pieces together.) > doughnut =/= sphere > Changing the universe of discourse from R^3 to (R^3 plus a wormhole) > does induce a topological change, yes - not in the box directly, but > in the points within it (and therefore in how the box relates to > those points). Great, so now that we know how the box relates to points within it changes some of the properties of the box and that space itself continues to exist with all of its properties, we can ask a few more questions. Is it possible to make a smooth translation of the box to either let it go through the portal and come out the other side or to take it off of the portal entranace? The way I see it is if you pull it in any direction, only part of one of the sides will enter the portal. If you continue pulling this would either cause the box to react elastictly and stretcha round the portal, or cause it to become disjointed and have part of the box travel inside of the portal while the other part of the box detatched itself from the portal entrance. If this last scenario happened, would the two pieces of the box continue to remain connected even though they existed in different parts of space? I think we will have to define formally what we mean by an object when we talk about the box, first of all. Then perhaps we can begin asking questions like, can portals exibit some of the properties of objects? If a portal can redifine properties of an object, and if a box resting around a portal can't be moved away from it... It would seem that it may very well be possible to construct a new model of what an object is simply by 'changing the universe of discourse' Even if not, the conclusion that we have now come to where the space around objects can redefine an objects propertie by changing the way the object relates to the points around it leads me to suspect it is becoming harder to distinguish the box from R^3... and doesn't it seem natural to suspect it may become harder to distinguish R^3 from the box? Certain phenomina in physics are already linked specificly to space/time. Gravity is usually thought simply to be a propertie of similar transformations to the ones we have been making in R^3. In physics a wormhole like we have created may be entirely possible as well. So if our discussion were to hold true in a physical theory of reality, then not only would the mass of an object displace the space around it to create gravity, but space could reorder fundemental properties of an object it surounded. Perhaps if we could then come to a conclusion of how a kind of object could be made from remappings or transformations of space, we might have a mathmatical model that could express a unifide field theory? This might seem to be getting a little out of hand, but if you are going to tie electromagnetism, gravity, and strong/weak nuclear forces together.... a good way to do it would to make them all properties of space, the way gravity is often thought to be. === Subject: Help needed: numbers that are equal up to two digits. Hi This problem has kept me busy, but could not find anything. I hope any of you can push me in the right direction. Consider two 10-digit numbers (A and B) and delete in each number 2 digits. What remains is A' and B' both 8 digit numbers. What I am interested in is the situation when A<>B but A'=B'. Of course this depends on which digits you delete. When this is possible for two numbers A and B, define this as A differs 2 digits from B. Example: A=5361728317 and A'=53*1728*17 B=8531742817 and B'=*5317*2817 (the * indicates a deleted digit) Then A'=B' My basic question is Given a number A, which or how many numbers B are there that differ 2 digits from A? I would like to generalize the numbers to words over a finite alphabet. And also generalize the distance 2 to a fixed but arbitrary distance. This question came up when some people at my work suggested that some subsequence of a series of digits would be representative enough of the original number. Mark === Subject: Higher dimensional complex analysis? What about a complex analysis for numbers with more than just two components? The most basic problem seems to be to be able to define these numbers as objects having unique inverses. I somewhat have the imagination that this leads to group theory, ie. to a factoring of groups... Is there a generalization of complex analysis to higher dimensions (n>2)? How is this achieved? Mark === Subject: Re: Higher dimensional complex analysis? > What about a complex analysis for numbers with more than just two > components? The most basic problem seems to be to be able to define > these numbers as objects having unique inverses. I somewhat have the > imagination that this leads to group theory, ie. to a factoring of > groups... > Is there a generalization of complex analysis to higher dimensions > (n>2)? How is this achieved? > Mark Yes there is, and it goes beyond the Cayley-Dickson algebras - Real & Complex, together with the Quaternions & Octonions mentioned in Alex Hunsley's reply. I provide a somewhat convoluted explanation in http://library.Wolfram.com/infocenter/Mathsource/4894 I am hoping to produce a shortened version acceptable to sci.math.research. Meanwhile, in brief, omitting many details:- Start with Moufang loops, which are Cayley multiplication tables with a unit and a division property, i.e. for all ELEMENTS a there is a right ar and left al inverse such that a*ar=al*a=1. Groups and Octonions are the only Moufang loops. They can be used as multiplication tables for (generalized) unsigned vectors A={a,b,..}. Negation and subtraction are not defined for them, so they do not yet describe algebras. Roots of unity s^r=1 provide generalized signs s^j (j What about a complex analysis for numbers with more than just two > components? The most basic problem seems to be to be able to define > these numbers as objects having unique inverses. I somewhat have the > imagination that this leads to group theory, ie. to a factoring of > groups... > Is there a generalization of complex analysis to higher dimensions > (n>2)? How is this achieved? > Mark Yes, and it goes past the Cayley Dickson algebras (quaternions and Octonions) mentioned in another reply. I have provided a rather convoluted explanation in http://Library.Wolfram.com/infocenter/MathSource/4894 and I am trying to produced a compressed version that will be acceptable to sci.math.research. Very briefly, all groups and the Octonions are Moufang Loops - m*m Cayley multiplication tables with a division property, but with negation and subtraction undefined. Generalised signs are r'th roots of unity (and not just the square root - and fourth root i) that collapse a Moufang loop to Hoop algebras (my term). These are (m/r)*(m/r) signed Cayley tables such as Quaternion Pauli-sigma Clifford2 Davenport Dihedral D3 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 5 6 2 -1 4 3 2 1 4i -3i 2 1 4 3 2 -1 4 -3 2 1 6 5 4 3 3 -4 -1 2 3 -4i 1 2i 3 -4 -1 2 3 4 -1 -2 3 4 5 6 1 2 4 3 -2 -1 4 3i -2i 1 4 -3 -2 1 4 -3 -2 1 4 3 2 1 6 5 5 6 1 2 3 4 6 5 4 3 2 1 (I am particularly interested in those with ternary symmetry, such as These act as division algebras for generalized vectors A={a,b,..}; there are right and left multiplicative inverses for every combination of hoop and vector (of suitable length). Only Moufang-loop-derived tables provide inverses Ainv with (A*B)*Ainv=B. Moufang loops, and the hoops, conserve their determinants, Det[A]*Det[B]=Det[A*B] (Frobenius conservation). The factors of these determinants are conserved, and are denominators in the expressions for the inverses. Example factorisations- Quaternion (a^2+b^2+c^2+d^2)^2. Pauli-sigma (a^2-b^2-c^2-d^2)^2 Clifford2 (a^2-b^2-c^2-d^2)^2 Davenport === Subject: Re: Higher dimensional complex analysis? > What about a complex analysis for numbers with more than just two > components? The most basic problem seems to be to be able to define > these numbers as objects having unique inverses. I somewhat have the > imagination that this leads to group theory, ie. to a factoring of > groups... > Is there a generalization of complex analysis to higher dimensions > (n>2)? How is this achieved? > Mark I'm a little rusty on the details, but you're thinking of quaternions (4D) and octonions (8D). These higher complex spaces lose qualities like nonassociativity (i.e. the octonions). http://math.ucr.edu/home/baez/Octonions/octonions.html alex === Subject: Re: Higher dimensional complex analysis? > What about a complex analysis for numbers with more than just two > components? The most basic problem seems to be to be able to define > these numbers as objects having unique inverses. I somewhat have the > imagination that this leads to group theory, ie. to a factoring of > groups... > > Is there a generalization of complex analysis to higher dimensions > (n>2)? How is this achieved? > > Mark > I'm a little rusty on the details, but you're thinking of quaternions (4D) > and > octonions (8D). Even quaternions fail to work for analysis as well as the complexes. Definition of analytic function becomes several inequivalent items. You might say, for example, that f is (right-) differentiable at a point a if limit(h->0) (f(a+h)-f(a))*h^{-1} exists, limit in the sense of quaternions. Well, even the function f(z) = z^2 fails to be differentiable by that definition. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Integral of product of 3 MeijerG Functions > I have a ? on evaluating an integral containing 3 MeijerG functions. > The integral looks like: > Integrate(0,Infinity) t^a*MeijerG(t)*MeijerG(t)*MeijerG(t), > where a = constant, and the 3 MeijerG's are appopriately defined. > I do not find much on it while searching on the net or integral tables. > Though Wolfram site, gives the answer for the integral, it does not > specify the conditions under which it is applicable. > Hence, any suggestions/references in this regard? Perhaps that Victor Adamchik or Kerry Gaskell (they implemented definite integration using MeijerG functions for mathematica/reduce) could help you (I'm not sure that you'll find the 3 terms integral in these papers...) : Adamchik's 1990 paper 'The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system' and other papers : http://www-2.cs.cmu.edu/~adamchik/papers.html Gaskell's paper for an overview and references : http://www.zib.de/Symbolik/reduce/moredocs/defint.pdf The classical (1936) paper of Meijer 'Neue Integraldarstellungen aus der Theorie der Whittakerschen und Hankelschen Funktionen' is here : http://134.76.163.65/servlet/digbib?template=view.html&id=37341&startpage=47 3&endpage=493&image-path=http://134.76.176.141/cgi-bin/letgifsfly.cgi&image- s ubpath=/1579&image-subpath=1579&pagenumber=473&imageset-id=1579 Hoping it helped, Raymond === Subject: A forced three-dimensional ODE system ? Hi all, I am looking for a third-order differential equation that exhibits interesting chaotic properties when an external modulation is applied. I have spent some time on topological methods for analyzing chaotic behavior in three-dimensional systems, including forced nonlinear oscillators with phase space R^2 x S^1 (see home page). I am investigating possible extensions of this approach to higher-dimensional systems and I would like to make some tests on a system that lives in R^3 x S^1, which is the simplest structure above dimension three. I often work with laser models but the simplest multimode one has natural phase space R^4 x S^1 because each new mode comes with least an inversion population variable. Thus if some kind soul could suggest an interesting forced third-order differential equation, I would be very grateful. -- Marc http://www-phlam.univ-lille1.fr/perso/lefranc/ === Subject: Re: A forced three-dimensional ODE system ? >Hi all, >I am looking for a third-order differential equation that exhibits >interesting chaotic properties when an external modulation is applied. >I have spent some time on topological methods for analyzing chaotic >behavior in three-dimensional systems, including forced nonlinear >oscillators with phase space R^2 x S^1 (see home page). >I am investigating possible extensions of this approach to >higher-dimensional systems and I would like to make some tests on a >system that lives in R^3 x S^1, which is the simplest structure above >dimension three. I often work with laser models but the simplest >multimode one has natural phase space R^4 x S^1 because each new mode >comes with least an inversion population variable. >Thus if some kind soul could suggest an interesting forced third-order >differential equation, I would be very grateful. Just an idea, a starting point: x' = y y' = x - x^3 z' = -z This is just Duffings equation made three-dimensional. Notice that the origin now becomes a hyperbolic fixed point with two stable and one unstable eigenvalue. Furthermore there is a (non-transversal) homoclinic orbit which lies in the plane z=0 and is fully part of the stable manifold which is the product of the figure eight with a line. This system in itself is not yet interesting, because the last equation is uncoupled from the others. You could couple the equations by adding a term alpha*z on the RHS of equation 2 (for example). The origin is still hyperbolic with the same eigenvalues and there is still a homoclinic orbit in the plane z=0. Maybe forcing (but where to put it?, what is force in this system?) will split that homoclinic orbit as it does in Duffings equation which could lead to chaotic behavior. Maybe that helps. Thomas === Subject: Re: Raatikainen's Complexity Complex > I will pen a more detailed response in due time. Meanwhile, I think a > preliminary answer can be that the nature is not necessarily > rational the way logicist philosophy has defined it (Incidentally, I > think this is one of the places where logical positivism has failed, a > branch in the school of thought known as positivism) I don't see the relevance. What does logical positivism have to do with Chaitin's claims or Raatikainen's criticism of them? -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Please comment > Hmm..I thought you were talking about mathematics as a human > process (you said Is mathematics all about building > domains...). Agree. My insight into mathematics is meaningless, I may thus mix things (you see, I still imagine a set as a collection of n-tuples of symbols, and an inherence predicate as some or other rewrite rule with a goal symbol being either the name of the set or the object 'truth'). I am sorry about that. > As to the Church-Turing thesis, in very vague > terms, yes it is consistent, as long as one accepts that all > these human constructs/manipulations (visualizations, > informal proofs) are formalizable. I am glad to hear it just one more time. > Did you read my remarks regarding the stratification of existence > from self subsistent objects to numbers? What do you think? > Do you mean about objects->concepts->sets->numbers? Yes. > It's an interesting classification. It's rubbish, right? > Some might say that you could combine concepts and sets > (that is concepts are pretty much the same thing as sets) [...] You mean, ontologically, sets and concepts are the same objects? > and you might even get away with having only one object (the empty set). I see. What would be the power set of the empty set, please? > Sounds like you'd get a lot from set theory to help expand > your ideas. Set theory is _everything_. I know. Andrew P.S. As for the typology of existence I would add: 1. There are objects, objects are self subsistent. 2. There are sets, sets are qualities of objects. 3. There are numbers, numbers are qualities of sets (whereas truth functions are _pure_ thoughts). (If I bore you, please discontinue posting.) === Subject: Disproofs of Riemann Hypothesis http://www.i-b-r.org/docs/JiangRiemann.pdf Also resolves Goldbach, twin primes and (x^2+1)-primes. === Subject: Re: Disproofs of Riemann Hypothesis > http://www.i-b-r.org/docs/JiangRiemann.pdf > Also resolves Goldbach, twin primes and (x^2+1)-primes. Does it find the Yeti and Bigfoot too? === Subject: Re: Disproofs of Riemann Hypothesis > http://www.i-b-r.org/docs/JiangRiemann.pdf A quotation from this paper: 2. Disproofs of Riemann's Hypothesis Theorem 1. zeta(s) has no zeros in the critical strip, that is zeta(s) =/= 0, where 0 <= sigma <= 1. The journal that gave us a special issue on Fermat's Last Theorem (vol 15 no.3, 1998). MR1677713 (2000b:11026) Athanasopoulos, S. N.(GR-PATR-TP) An algebraic proof of Fermat's last theorem. (English. English summary) Algebras Groups Geom. 15 (1998), no. 3, 247--288. 11D41 This is an attempt to prove Fermat's last theorem using techniques known only in Fermat's days. Assuming the existence of a nontrivial solution to $x^p+y^p=z^p$ the author shows the existence of certain integers $lambda,.8c[Micro],phi,m$ which satisfy a complicated polynomial relation, number (7.1.9). The author then proceeds to assume that $lambda=.8c[Micro]$. It is no wonder that one subsequently obtains a contradiction. Apparently we were not allowed to take $lambda=.8c[Micro]$. Instead, the author concludes that Fermat's last theorem is proved. Reviewed by F. Beukers Next Item MR1677709 (2000b:11027) Obi, C. Fermat's last theorem. (English. English summary) Algebras Groups Geom. 15 (1998), no. 3, 289--298. 11D41 This is a naive attempt to prove Fermat's last theorem using techniques known only in Fermat's days. This time the error is on the bottom of page 292. The author states that some numbers, having nothing to do with the equation, satisfy $a_0^2+b_0^2=c_0^2$. The conclusion is that a certain case of Fermat's last theorem is solved. Using this technique one could prove any conjecture made in mathematics. Reviewed by F. Beukers Previous Item Next Item MR1677705 Trell, E.(S-LINKH) Isotopic proof and re-proof of Fermat's last theorem verifying Beal's conjecture. (English. English summary) Algebras Groups Geom. 15 (1998), no. 3, 299--318. 11D41 {There will be no review of this item.} Previous Item Next Item MR1677701 Jiang, C. X. On the Fermat-Santilli isotheorem. (English. English summary) Algebras Groups Geom. 15 (1998), no. 3, 319--349. 11D41 {There will be no review of this item.} Previous Item MR1677697 (2000c:11214) Jiang, C. X. Foundations of Santilli's isonumber theory. I. Isonumber theory of the first kind. (English. English summary) Algebras Groups Geom. 15 (1998), no. 3, 351--393. 11Z05 (11A99 11N05 11P32) Summary: In this paper we study, apparently for the first time, the foundations of Santilli's isonumber theory of the first kind, characterized by the axiom-preserving isotopic lifting of the multiplicative unit, product and elements. In particular, we study the isogroup, the isodivisibility, the unique isofactorization theorem, the isoprime number theorem, the isocongruences, etc. We introduce a new branch of number theory: Santilli's additive isoprime theory. By using the arithmetic function $J_n(omega)$ we prove about fifty theorems including the iso-Goldbach theorem, isoprime twins theorem, Yu's mathematical problem, $k$-tuples of isoprimes,break $(hat{p}+hat{a})^{hat2}+hat{a}, hat{p}+hat{b}, hat{p}+hat2$ and $hat{p}+hat6$, etc. We introduce a generalization of Euler's proof of the existence of infinitely many primes. We study as an application that the limit for the periodic table of the stable elements is uranium with an atomic number of 92. Previous Item -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Disproofs of Riemann Hypothesis > The journal that gave us a special issue on > Fermat's Last Theorem (vol 15 no.3, 1998). If they're that bad, how do they make it to 21 volumes? Is this a vanity journal? V. -- email: lastname at cs utk edu homepage: cs utk edu tilde lastname === Subject: Re: Disproofs of Riemann Hypothesis >The journal that gave us a special issue on >Fermat's Last Theorem (vol 15 no.3, 1998). > If they're that bad, how do they make it to 21 volumes? Is this a vanity > journal? In the early days of its existence, believe it or not, it was a respectable journal. Then its publisher, Santilli, became more active in the management of the journal and it went downhill. I believe the remaining legitimate editors resigned when the Fermat volume was issued. What is more surprising is that Santilli manages to continue to have Kluwer and Elsevier publish his voluminous monographs. (In earlier times he also got published by Springer.) === Subject: Re: Disproofs of Riemann Hypothesis > The journal that gave us a special issue on > Fermat's Last Theorem (vol 15 no.3, 1998). >If they're that bad, how do they make it to 21 volumes? Is this a vanity >journal? /s/va/insa Read http://www.i-b-r.org/ itself. Lee Rudolph === Subject: Re: Disproofs of Riemann Hypothesis > The journal that gave us a special issue on > Fermat's Last Theorem (vol 15 no.3, 1998). >If they're that bad, how do they make it to 21 volumes? Is this a vanity >journal? >/s/va/insa I'm probably the only one who has no idea what you mean by that... >Read http://www.i-b-r.org/ itself. The

Scientific Works

alone makes the site well worth a visit. >Lee Rudolph ************************ David C. Ullrich === Subject: Re: Disproofs of Riemann Hypothesis > http://www.i-b-r.org/docs/JiangRiemann.pdf > Also resolves Goldbach, twin primes and (x^2+1)-primes. Hi all, As some of you may have noted from previous posts, I'm almost a complete mathematical 'nitwit', so what I post here may be completely wrong :) I looked in the document, and the first thing I see is that Jiang uses the definition of zeta valid (convergent) for sigma>= 1 (equation 1) for his proof (starting at equation 4). Because he wants to disproof RH, I think he should use another definition of zeta, valid (convergent) for 0 20.} 20 degrees solution for angle CDF using Euler Lemma, where principle of concurrecy is used,so this method is not entirely into trig BF&I. bc=50;dtr=Pi/180; lower case for lengths; mb=N[bc*Sin[70*dtr]/Sin[50*dtr]] mc=N[bc*Sin[60*dtr]/Sin[50*dtr]] me=N[mc*Sin[10*dtr]/Sin[40*dtr]] md=N[mb*Sin[20*dtr]/Sin[30*dtr]] lhs=(mb*me)/(mb+me)*Sin[50*dtr]; rhs=md/(Cot[30*dtr]+Cot[x*dtr]); FindRoot[lhs==rhs,{x,19,25}] Hindsight observations: DE and BC produced meet at F. CMEF is a cyclic quadrilateral. CD is tangent to circle through A,D and E, making triangles CDE and CAD similar. > ) If three concurrent lines are cut by a transversal making intercepts > ) ) approximately MB=62,ME=15,MD=38 to the nearest millimeter. Someone may > ) please improve on trig accuracy. Solution by FindRoot in Mathematica > ) gives CDE = 22.83 degrees confirmed also by actual protractor > ) measurement. > Obviously millimeter accuracy is pretty bad, because the actual answer is a > nice round 20 degrees as proven elsethread (in a purely geometric fashion). > SaSW, Willem === Subject: Re: real area venn diagram > We're having a conversation in my office about doing a venn diagram > for 3 sets (hence 3 overlapping shapes) such that the areas of the > shapes, and of the overlaps of the shapes, is propertional to the > number of items in those areas. > Now, it's pretty obvious that if you use circles of different sizes > and shapes, this 'area accurate' diagram cannot be done for numbers of > set greater than 2. > I think it's the same case for rectangles too. So what shapes could we > use? > Do it on a sphere. There you can get exactly the 8 distinct > ratios you need. Can you describe a little more what you mean? Are you saying use three spheres? Or put shapes on a single sphere? alex === Subject: Re: real area venn diagram > We're having a conversation in my office about doing a venn diagram > for 3 sets (hence 3 overlapping shapes) such that the areas of the > shapes, and of the overlaps of the shapes, is propertional to the > number of items in those areas. > Now, it's pretty obvious that if you use circles of different sizes > and shapes, this 'area accurate' diagram cannot be done for numbers > of set greater than 2. > I think it's the same case for rectangles too. So what shapes could > we use? > Do it on a sphere. There you can get exactly the 8 distinct > ratios you need. > Can you describe a little more what you mean? Are you saying use three > spheres? Or put shapes on a single sphere? Sorry. You have one sphere. 3 circles on the sphere, intersecting when appropriate, creating at most 8 areas on the sphere. On one side of a circle is positive, other side is negative. Move and resize the circles appropriately to get the areas for the 8 parts in corresponding ratio to the number of elements in each subset. For 1 and 2 circles (i.e. 1 and 2 properties, it seems obvious to me that you can get any ratio of 2 and 4 subsets that you want. For 3, it seems likely but I can't quickly come up with a way of calculating the coordinates of the circles from the 8 values. -- Mitch Harris (remove q to reply) === Subject: set definition of n-tuple What is the standard way to represent an ordered n-tuple as a set, such that the equality of any two tuples also holds for the corresponding sets? alex === Subject: Re: set definition of n-tuple Originator: anno4000@lublin.zrz.tu-berlin.de (Anno Siegel) > What is the standard way to represent an ordered n-tuple as a set, such that > the equality of any two tuples also holds for the corresponding sets? Define a pair: (x_1, x_2) = { {x_1}, {x_1, x_2}} It is easy to see that (x_1, x_2) = (y_1, y_2) exactly if x_1 = y_1 and x_2 = y_2. (Distinguish the cases x_1 = x_2 and x_1 != x_2 for the proof.) Then define an n-tuple for n >= 2 as (x_1, ..., x_n) = ((x_1, ..., x_n-1), x_n) If follows recursively that (x_1, ..., x_n) = (y_1, ..., y_m) exactly if n = m and x_i = y_i for all i. Anno === Subject: Re: set definition of n-tuple > What is the standard way to represent an ordered n-tuple as a set, such that > the equality of any two tuples also holds for the corresponding sets? I am not sure that there is a standard way. This way is often used: The ordered 0-tuple () and the ordered 1-tuple (x) are not very interesting. The ordered 2-tuple (ordered pair) (x,y) is defined to be the set {{x},{x,y}}. This is often called a Kuratowski pair. The ordered n-tuple (for n>2) (x_1,x_2,...,x_n) is defined to be the ordered pair (x_1,(x_2,...,x_n)). Thus (x_1,x_2,...,x_n) is a set with at most n elements. Note that, for example, (x,x)={{x},{x,x}}={{x},{x}}={{x}}. -- Clive Tooth http://www.clivetooth.dk === Subject: Re: set definition of n-tuple >What is the standard way to represent an ordered n-tuple as a set, such > that >the equality of any two tuples also holds for the corresponding sets? > I am not sure that there is a standard way. This way is often used: > The ordered 0-tuple () and the ordered 1-tuple (x) are not very interesting. > The ordered 2-tuple (ordered pair) (x,y) is defined to be the set > {{x},{x,y}}. > This is often called a Kuratowski pair. > The ordered n-tuple (for n>2) (x_1,x_2,...,x_n) is defined to be the ordered > pair (x_1,(x_2,...,x_n)). Surely your definition here loses the information about the order of x_2, x_3 etc., as they are all just shoved in a set! > Thus (x_1,x_2,...,x_n) is a set with at most n elements. > Note that, for example, (x,x)={{x},{x,x}}={{x},{x}}={{x}}. === Subject: Re: set definition of n-tuple > What is the standard way to represent an ordered n-tuple as a set, such > that > the equality of any two tuples also holds for the corresponding sets? > I am not sure that there is a standard way. This way is often used: > The ordered 0-tuple () and the ordered 1-tuple (x) are not very > interesting. > The ordered 2-tuple (ordered pair) (x,y) is defined to be the set > {{x},{x,y}}. > This is often called a Kuratowski pair. > The ordered n-tuple (for n>2) (x_1,x_2,...,x_n) is defined to be the > ordered > pair (x_1,(x_2,...,x_n)). > Surely your definition here loses the information about the order of > x_2, x_3 etc., as they are all just shoved in a set! bad. === Subject: Re: set definition of n-tuple > What is the standard way to represent an ordered > n-tuple as a set, such that > the equality of any two tuples also holds for the > corresponding sets? > alex You can use this abbreviation: (a_1,...,a_n) = { { a_j | 1<=j<=k } | 1<=k<=n } So you get (a_1,...,a_n) = { {a_1}, {a_1,a_2}, ..., {a_1,a_2,...a_n} } eg: (a,b,c,d) = { {a}, {a,b}, {a,b,c}, {a,b,c,d } } (a,b,c) = { {a}, {a,b}, {a,b,c} } (a,b) = { {a}, {a,b} } Dirk Vdm === Subject: Re: set definition of n-tuple > What is the standard way to represent an ordered > n-tuple as a set, such that > the equality of any two tuples also holds for the > corresponding sets? > alex > You can use this abbreviation: > (a_1,...,a_n) = { { a_j | 1<=j<=k } | 1<=k<=n } > So you get > (a_1,...,a_n) > = { {a_1}, {a_1,a_2}, ..., {a_1,a_2,...a_n} } > eg: > (a,b,c,d) = { {a}, {a,b}, {a,b,c}, {a,b,c,d } } > (a,b,c) = { {a}, {a,b}, {a,b,c} } > (a,b) = { {a}, {a,b} } > Dirk Vdm So... (1,2,2) = { {1}, {1,2}, {1,2,2} } = { {1}, {1,2} } = { {1}, {1,1}, {1,1,2} } = (1,1,2) Not acceptible! -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: set definition of n-tuple >What is the standard way to represent an ordered >n-tuple as a set, such that >the equality of any two tuples also holds for the >corresponding sets? >alex >You can use this abbreviation: > (a_1,...,a_n) = { { a_j | 1<=j<=k } | 1<=k<=n } >So you get > (a_1,...,a_n) > = { {a_1}, {a_1,a_2}, ..., {a_1,a_2,...a_n} } >eg: > (a,b,c,d) = { {a}, {a,b}, {a,b,c}, {a,b,c,d } } > (a,b,c) = { {a}, {a,b}, {a,b,c} } > (a,b) = { {a}, {a,b} } >Dirk Vdm > So... (1,2,2) = { {1}, {1,2}, {1,2,2} } = { {1}, {1,2} } > = { {1}, {1,1}, {1,1,2} } = (1,1,2) > Not acceptible! Does the recursive definition seem ok to you? e.g. (a, b, c) = { a, {a, {b, {b, c} } } or, the other notation I mentioned, which avoids arbitrary depth recursive sets: (a, b, c) = { {0, {a, E - {a} }}, {1, {b, E - {b} }}, {2, {c, E - {c} } } (where E = universal set) does that seem bulletproof? alex === Subject: Re: set definition of n-tuple > Does the recursive definition seem ok to you? > e.g. > (a, b, c) = { a, {a, {b, {b, c} } } You seem to have } missing. I assume that you mean to define (a,b) as {a,{a,b}} and (a,b,c) as {a,{a,{b,{b,c}}}}. But this does not work as the ordered triple (a,b,c) is now indistinguishable from the ordered pair (a,(b,c)). -- Clive Tooth http://www.clivetooth.dk === Subject: Re: set definition of n-tuple > Does the recursive definition seem ok to you? > e.g. > (a, b, c) = { a, {a, {b, {b, c} } } > You seem to have } missing. > I assume that you mean to define (a,b) as {a,{a,b}} and (a,b,c) as > {a,{a,{b,{b,c}}}}. But this does not work as the ordered triple (a,b,c) is > now indistinguishable from the ordered pair (a,(b,c)). Ha, but {a,{a,{b,{b,c}}}} would be *defined* as (a,(b,c)) so that would be no problem :-) Dirk Vdm === Subject: Re: set definition of n-tuple > What is the standard way to represent an ordered > n-tuple as a set, such that > the equality of any two tuples also holds for the > corresponding sets? alex > You can use this abbreviation: > (a_1,...,a_n) = { { a_j | 1<=j<=k } | 1<=k<=n } > So you get > (a_1,...,a_n) > = { {a_1}, {a_1,a_2}, ..., {a_1,a_2,...a_n} } > eg: > (a,b,c,d) = { {a}, {a,b}, {a,b,c}, {a,b,c,d } } > (a,b,c) = { {a}, {a,b}, {a,b,c} } > (a,b) = { {a}, {a,b} } > Dirk Vdm > So... (1,2,2) = { {1}, {1,2}, {1,2,2} } = { {1}, {1,2} } > = { {1}, {1,1}, {1,1,2} } = (1,1,2) > Not acceptible! Indeed not :-) Meanwhile I have made a correction such that (1,2,2) = { 1, {2, {2}}} (1,1,2) = { 1, {1, {2}}} Dirk Vdm === Subject: Re: set definition of n-tuple >What is the standard way to represent an ordered >n-tuple as a set, such that >the equality of any two tuples also holds for the >corresponding sets? >alex > You can use this abbreviation: > (a_1,...,a_n) = { { a_j | 1<=j<=k } | 1<=k<=n } > So you get > (a_1,...,a_n) > = { {a_1}, {a_1,a_2}, ..., {a_1,a_2,...a_n} } > eg: > (a,b,c,d) = { {a}, {a,b}, {a,b,c}, {a,b,c,d } } > (a,b,c) = { {a}, {a,b}, {a,b,c} } > (a,b) = { {a}, {a,b} } > Dirk Vdm Hi Dirk I've seen this definition somewhere before. It does meet my criteria above. However, another erquirement I have just realised I have is that you can construct the original tuple from the set! The above scheme fails in the following case: (a,a) = { {a}, {a, a} } = { {a}, {a} } = { {a} } so the set { {a} } could be representing a tuple of any dimension that is composed of all a's! I need to be able to get from the set back to the exact tuple... alex === Subject: Re: set definition of n-tuple >What is the standard way to represent an ordered >n-tuple as a set, such that >the equality of any two tuples also holds for the >corresponding sets? alex > You can use this abbreviation: > (a_1,...,a_n) = { { a_j | 1<=j<=k } | 1<=k<=n } > So you get > (a_1,...,a_n) > = { {a_1}, {a_1,a_2}, ..., {a_1,a_2,...a_n} } > eg: > (a,b,c,d) = { {a}, {a,b}, {a,b,c}, {a,b,c,d } } > (a,b,c) = { {a}, {a,b}, {a,b,c} } > (a,b) = { {a}, {a,b} } > Dirk Vdm > Hi Dirk > I've seen this definition somewhere before. It does meet my criteria above. > However, another erquirement I have just realised I have is that you can > construct the original tuple from the set! The above scheme fails in the > following case: > (a,a) = { {a}, {a, a} } = { {a}, {a} } = { {a} } > so the set { {a} } could be representing a tuple of any dimension that is > composed of all a's! I need to be able to get from the set back to the exact > tuple... Oops, yes, you are right - I went too fast. Since writing down a closed form is not so easy in this case, I'll work the other way around: (a,b) = { {a}, {a,{b}} } (a,b,c) = { {a}, {a,{b}}, {a,{b,{c}}} } (a,b,c,d) = { {a}, {a,{b}}, {a,{b,{c}}}, {a,{b,{c,{d}}}} } (a_1,...,a_n) = { {a_1}, {a_1,{a_2}}, ..., {a_1,{a_2,{...{a_n}...}} } Now for the challenge: (a_1,...,a_n) = Dirk Vdm > alex === Subject: Re: set definition of n-tuple <10g1vh4kgtj9ec6@corp.supernews.com> <410103fa$1@usenet01.boi.hp.com Since writing down a closed form is not so easy > in this case, I'll work the other way around: > (a,b) = { {a}, {a,{b}} } > (a,b,c) = { {a}, {a,{b}}, {a,{b,{c}}} } > (a,b,c,d) = { {a}, {a,{b}}, {a,{b,{c}}}, {a,{b,{c,{d}}}} } > (a_1,...,a_n) > = { {a_1}, {a_1,{a_2}}, ..., {a_1,{a_2,{...{a_n}...}} } > Now for the challenge: > (a_1,...,a_n) = A n-tuple (a1,.. a_n) is a map f from n = { k integer | 0 <= k <= n-1 } to the elements with f(k - 1) = a_k === Subject: Re: set definition of n-tuple >A n-tuple (a1,.. a_n) is a map f from n = { k integer | 0 <= k <= n-1 } >to the elements with f(k - 1) = a_k What is a map? I presume it is a function. What is a function? A standard formalization of the notion defines a function as a binary relation with appropriate restrictions. What is a binary relation? The same standard formalization defines it to be a subset of a certain set of ordered pairs. What are ordered pairs? Ah, well, now... Lee Rudolph === Subject: Re: set definition of n-tuple Originator: anno4000@lublin.zrz.tu-berlin.de (Anno Siegel) >A n-tuple (a1,.. a_n) is a map f from n = { k integer | 0 <= k <= n-1 } >to the elements with f(k - 1) = a_k > What is a map? I presume it is a function. What is a function? > A standard formalization of the notion defines a function as a > binary relation with appropriate restrictions. What is a binary > relation? The same standard formalization defines it to be a subset > of a certain set of ordered pairs. What are ordered pairs? > Ah, well, now... Yes. But you *can* define pairs (without defining general n-tuples), then go on to relations, then functions, and then define an n-tuple over a set M as a function t: { 1, ..., n } ---> M. That has the advantage that an n-tuple is a set of n elements (not a matter of course, as the thread has shown). I seem to remember that was one of the OP's requirements. Anno === Subject: Re: set definition of n-tuple What is the standard way to represent an ordered >n-tuple as a set, such that >the equality of any two tuples also holds for the >corresponding sets? alex > You can use this abbreviation: > (a_1,...,a_n) = { { a_j | 1<=j<=k } | 1<=k<=n } > So you get > (a_1,...,a_n) > = { {a_1}, {a_1,a_2}, ..., {a_1,a_2,...a_n} } > eg: > (a,b,c,d) = { {a}, {a,b}, {a,b,c}, {a,b,c,d } } > (a,b,c) = { {a}, {a,b}, {a,b,c} } > (a,b) = { {a}, {a,b} } Dirk Vdm > Hi Dirk > I've seen this definition somewhere before. It does meet my criteria above. > However, another erquirement I have just realised I have is that you can > construct the original tuple from the set! The above scheme fails in the > following case: > (a,a) = { {a}, {a, a} } = { {a}, {a} } = { {a} } > so the set { {a} } could be representing a tuple of any dimension that is > composed of all a's! I need to be able to get from the set back to the exact > tuple... > Oops, yes, you are right - I went too fast. > Since writing down a closed form is not so easy > in this case, I'll work the other way around: > (a,b) = { {a}, {a,{b}} } > (a,b,c) = { {a}, {a,{b}}, {a,{b,{c}}} } > (a,b,c,d) = { {a}, {a,{b}}, {a,{b,{c}}}, {a,{b,{c,{d}}}} } > (a_1,...,a_n) > = { {a_1}, {a_1,{a_2}}, ..., {a_1,{a_2,{...{a_n}...}} } > Now for the challenge: > (a_1,...,a_n) = But ({{a}},{a},a) ={ {{{a}}}, {{{a}},{{a}}}, {{{a}},{{a},{a}}} } ={{{{a}}}} =({{a}},{a}) So we do not know if we have a 3-tuple or a 2-tuple. -- Clive Tooth http://www.clivetooth.dk === Subject: Re: set definition of n-tuple >What is the standard way to represent an ordered >n-tuple as a set, such that >the equality of any two tuples also holds for the >corresponding sets? alex > You can use this abbreviation: > (a_1,...,a_n) = { { a_j | 1<=j<=k } | 1<=k<=n } > So you get > (a_1,...,a_n) > = { {a_1}, {a_1,a_2}, ..., {a_1,a_2,...a_n} } > eg: > (a,b,c,d) = { {a}, {a,b}, {a,b,c}, {a,b,c,d } } > (a,b,c) = { {a}, {a,b}, {a,b,c} } > (a,b) = { {a}, {a,b} } Dirk Vdm Hi Dirk > I've seen this definition somewhere before. It does meet my criteria > above. > However, another erquirement I have just realised I have is that you can > construct the original tuple from the set! The above scheme fails in the > following case: (a,a) = { {a}, {a, a} } = { {a}, {a} } = { {a} } so the set { {a} } could be representing a tuple of any dimension that > is > composed of all a's! I need to be able to get from the set back to the > exact > tuple... > Oops, yes, you are right - I went too fast. > Since writing down a closed form is not so easy > in this case, I'll work the other way around: > (a,b) = { {a}, {a,{b}} } > (a,b,c) = { {a}, {a,{b}}, {a,{b,{c}}} } > (a,b,c,d) = { {a}, {a,{b}}, {a,{b,{c}}}, {a,{b,{c,{d}}}} } > (a_1,...,a_n) > = { {a_1}, {a_1,{a_2}}, ..., {a_1,{a_2,{...{a_n}...}} } > Now for the challenge: > (a_1,...,a_n) = ={ > {{{a}}}, > {{{a}},{{a}}}, > {{{a}},{{a},{a}}} > } > ={{{{a}}}} > =({{a}},{a}) > So we do not know if we have a 3-tuple or a 2-tuple. Now, *that* is devious! Aarg ;-) Dirk Vdm === Subject: Re: set definition of n-tuple > What is the standard way to represent an ordered > n-tuple as a set, such that > the equality of any two tuples also holds for the > corresponding sets? > alex > You can use this abbreviation: > (a_1,...,a_n) = { { a_j | 1<=j<=k } | 1<=k<=n } > So you get > (a_1,...,a_n) > = { {a_1}, {a_1,a_2}, ..., {a_1,a_2,...a_n} } > eg: > (a,b,c,d) = { {a}, {a,b}, {a,b,c}, {a,b,c,d } } > (a,b,c) = { {a}, {a,b}, {a,b,c} } > (a,b) = { {a}, {a,b} } > Dirk Vdm > Hi Dirk > I've seen this definition somewhere before. It does meet my criteria > above. However, another erquirement I have just realised I have is that > you can construct the original tuple from the set! The above scheme > fails in the following case: > (a,a) = { {a}, {a, a} } = { {a}, {a} } = { {a} } > so the set { {a} } could be representing a tuple of any dimension that > is composed of all a's! I need to be able to get from the set back to > the exact tuple... ok, here's an idea to do this: we encode the indexes of the tuple items along with the value for each item at that index. for example, to represent (a, b, c): { {0, {a, a-1}}, {1, {b, b-1}}, {2, {c, c-1}} } ^ index=0 ^ index=1 ^index=2 Note that I'm writing {a, a-1} to be guaranted a two-value set that wouldn't = 0 it would collapse to a singleton.) Is there a better way to create a guaranteed two value set to store one value? Of course, I'm sure there's a nefarious recursive set definition as well, but it sounds evil. Here's my first try for recursive definition of (a, b, c): { a, {a, {b, {b, c} } } alex === Subject: Re: set definition of n-tuple What is the standard way to represent an ordered > n-tuple as a set, such that > the equality of any two tuples also holds for the > corresponding sets? alex You can use this abbreviation: > (a_1,...,a_n) = { { a_j | 1<=j<=k } | 1<=k<=n } > So you get > (a_1,...,a_n) > = { {a_1}, {a_1,a_2}, ..., {a_1,a_2,...a_n} } > eg: > (a,b,c,d) = { {a}, {a,b}, {a,b,c}, {a,b,c,d } } > (a,b,c) = { {a}, {a,b}, {a,b,c} } > (a,b) = { {a}, {a,b} } Dirk Vdm > Hi Dirk > I've seen this definition somewhere before. It does meet my criteria > above. However, another erquirement I have just realised I have is that > you can construct the original tuple from the set! The above scheme > fails in the following case: > (a,a) = { {a}, {a, a} } = { {a}, {a} } = { {a} } > so the set { {a} } could be representing a tuple of any dimension that > is composed of all a's! I need to be able to get from the set back to > the exact tuple... > ok, here's an idea to do this: we encode the indexes of the tuple items along > with the value for each item at that index. > for example, to represent (a, b, c): > { {0, {a, a-1}}, {1, {b, b-1}}, {2, {c, c-1}} } > ^ index=0 ^ index=1 ^index=2 > Note that I'm writing {a, a-1} to be guaranted a two-value set that wouldn't > = 0 it would collapse to a singleton.) Is there a better way to create a > guaranteed two value set to store one value? > Of course, I'm sure there's a nefarious recursive set definition as well, but > it sounds evil. > Here's my first try for recursive definition of (a, b, c): > { a, {a, {b, {b, c} } } But why not simply (a,b,c) = { a, {b, {c} } } such that the outermost element is the first component of the tuple and the innermost element is the last component? Dirk Vdm === Subject: Re: set definition of n-tuple >What is the standard way to represent an ordered >n-tuple as a set, such that >the equality of any two tuples also holds for the >corresponding sets? > >[...] But why not simply > (a,b,c) = { a, {b, {c} } } >such that the outermost element is the first >component of the tuple and the innermost >element is the last component? Are you suggesting {a, {b}} for an ordered pair? Then if b = {c}, we cannot distinguish (a,b,c) from (a,b). Also, I believe the definition of (a,b) = {a, {b}} will not work without the axiom of foundation (or regularity), which is generally unnecessary for mathematics. Sorry, I am trouble coming up with an example right now, and I must go. I leave it as an exercise. :-) -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: set definition of n-tuple > Are you suggesting {a, {b}} for an ordered pair? Then if b = {c}, > we cannot distinguish (a,b,c) from (a,b). Also, I believe the > definition of (a,b) = {a, {b}} will not work without the axiom of > foundation (or regularity), which is generally unnecessary for > mathematics. Sorry, I am trouble coming up with an example right now, > and I must go. I leave it as an exercise. :-) With that definition of an ordered pair the set {{x},{y}} can mean ({x},y) or ({y},x). -- Clive Tooth http://www.clivetooth.dk === Subject: Re: set definition of n-tuple >What is the standard way to represent an ordered >n-tuple as a set, such that >the equality of any two tuples also holds for the >corresponding sets? alex You can use this abbreviation: > (a_1,...,a_n) = { { a_j | 1<=j<=k } | 1<=k<=n } >So you get > (a_1,...,a_n) > = { {a_1}, {a_1,a_2}, ..., {a_1,a_2,...a_n} } >eg: > (a,b,c,d) = { {a}, {a,b}, {a,b,c}, {a,b,c,d } } > (a,b,c) = { {a}, {a,b}, {a,b,c} } > (a,b) = { {a}, {a,b} } Dirk Vdm >Hi Dirk >I've seen this definition somewhere before. It does meet my criteria >above. However, another erquirement I have just realised I have is that >you can construct the original tuple from the set! The above scheme >fails in the following case: >(a,a) = { {a}, {a, a} } = { {a}, {a} } = { {a} } >so the set { {a} } could be representing a tuple of any dimension that >is composed of all a's! I need to be able to get from the set back to >the exact tuple... >ok, here's an idea to do this: we encode the indexes of the tuple items along >with the value for each item at that index. >for example, to represent (a, b, c): > { {0, {a, a-1}}, {1, {b, b-1}}, {2, {c, c-1}} } > ^ index=0 ^ index=1 ^index=2 >Note that I'm writing {a, a-1} to be guaranted a two-value set that wouldn't >= 0 it would collapse to a singleton.) Is there a better way to create a >guaranteed two value set to store one value? >Of course, I'm sure there's a nefarious recursive set definition as well, but >it sounds evil. >Here's my first try for recursive definition of (a, b, c): >{ a, {a, {b, {b, c} } } > But why not simply > (a,b,c) = { a, {b, {c} } } > such that the outermost element is the first > component of the tuple and the innermost > element is the last component? > Dirk Vdm Because I presumed that mathematically notation speaking, counting the amount of items in a set is an 'easier' thing to do than saying 'is this item X a set or just an atomic, primitive other item?' Or, to put it another way, why is the standard pair definition this: (a, b) --> { {a}, {a, b} } ... and not this? (a, b) --> { a, {b} } It seems to me that the thing imposing the ordering in the represntation { {a}, {a, b} } is the counts of the inner sets. the one with count 1 is the first item, the one with the count 2 minus the first set is the second item, etc. === Subject: Re: set definition of n-tuple What is the standard way to represent an ordered >n-tuple as a set, such that >the equality of any two tuples also holds for the >corresponding sets? You can use this abbreviation: > (a_1,...,a_n) = { { a_j | 1<=j<=k } | 1<=k<=n } >So you get > (a_1,...,a_n) > = { {a_1}, {a_1,a_2}, ..., {a_1,a_2,...a_n} } >eg: > (a,b,c,d) = { {a}, {a,b}, {a,b,c}, {a,b,c,d } } > (a,b,c) = { {a}, {a,b}, {a,b,c} } > (a,b) = { {a}, {a,b} } I've seen this definition somewhere before. It does meet my criteria >above. However, another erquirement I have just realised I have is that >you can construct the original tuple from the set! The above scheme >fails in the following case: (a,a) = { {a}, {a, a} } = { {a}, {a} } = { {a} } so the set { {a} } could be representing a tuple of any dimension that >is composed of all a's! I need to be able to get from the set back to >the exact tuple... ok, here's an idea to do this: we encode the indexes of the tuple items along >with the value for each item at that index. for example, to represent (a, b, c): { {0, {a, a-1}}, {1, {b, b-1}}, {2, {c, c-1}} } > ^ index=0 ^ index=1 ^index=2 Note that I'm writing {a, a-1} to be guaranted a two-value set that wouldn't when a >= 0 it would collapse to a singleton.) Is there a better way to create a >guaranteed two value set to store one value? Of course, I'm sure there's a nefarious recursive set definition as well, but >it sounds evil. >Here's my first try for recursive definition of (a, b, c): { a, {a, {b, {b, c} } } > But why not simply > (a,b,c) = { a, {b, {c} } } > such that the outermost element is the first > component of the tuple and the innermost > element is the last component? > Because I presumed that mathematically notation speaking, counting the amount > of items in a set is an 'easier' thing to do than saying 'is this item X a set > or just an atomic, primitive other item?' > Or, to put it another way, why is the standard pair definition this: > (a, b) --> { {a}, {a, b} } > ... and not this? > (a, b) --> { a, {b} } > It seems to me that the thing imposing the ordering in the represntation > { {a}, {a, b} } > is the counts of the inner sets. the one with count 1 is the first item, the > one with the count 2 minus the first set is the second item, etc. Defining (a,b) as {a,{b}} means that the ordered pair ({x},y) is the same as the ordered pair ({y},x). -- Clive Tooth http://www.clivetooth.dk === Subject: Re: set definition of n-tuple > What is the standard way to represent an ordered > n-tuple as a set, such that > the equality of any two tuples also holds for the > corresponding sets? alex > You can use this abbreviation: > (a_1,...,a_n) = { { a_j | 1<=j<=k } | 1<=k<=n } > So you get > (a_1,...,a_n) > = { {a_1}, {a_1,a_2}, ..., {a_1,a_2,...a_n} } > eg: > (a,b,c,d) = { {a}, {a,b}, {a,b,c}, {a,b,c,d } } > (a,b,c) = { {a}, {a,b}, {a,b,c} } > (a,b) = { {a}, {a,b} } > Dirk Vdm > Hi Dirk > I've seen this definition somewhere before. It does meet my criteria > above. However, another erquirement I have just realised I have is > that you can construct the original tuple from the set! The above > scheme fails in the following case: > (a,a) = { {a}, {a, a} } = { {a}, {a} } = { {a} } > so the set { {a} } could be representing a tuple of any dimension that > is composed of all a's! I need to be able to get from the set back to > the exact tuple... > ok, here's an idea to do this: we encode the indexes of the tuple items > along with the value for each item at that index. > for example, to represent (a, b, c): > { {0, {a, a-1}}, {1, {b, b-1}}, {2, {c, c-1}} } > ^ index=0 ^ index=1 ^index=2 > Note that I'm writing {a, a-1} to be guaranted a two-value set that > example, when a = 0 it would collapse to a singleton.) Is there a better > way to create a guaranteed two value set to store one value? > Of course, I'm sure there's a nefarious recursive set definition as > well, but it sounds evil. > Here's my first try for recursive definition of (a, b, c): > { a, {a, {b, {b, c} } } > alex answering myself... I didn't realise it, but my attempt at recursive definition appears to be identical to the recursive Kuratowski pair method that the last danish pastry just mentioned! I think it might be a pain for writing about tuple-set manipulations as there's not much notation I know of for denoting the nth sub set of a set etc. (or if there is, it would be confusing?) Maybe I could do something with my above definition: > for example, to represent (a, b, c): > { {0, {a, a-1}}, {1, {b, b-1}}, {2, {c, c-1}} } > ^ index=0 ^ index=1 ^index=2 however, the problem with this is that writing 'a-1' etc. presumes that a is a number. I perhaps need a notation to mean 'some thing that is not equal to a? this sounds laborious/stupid though.... hmm.. I suppose I could write { E - a } when I want to mean 'not a'. (where E is the 'universe' or universal set} alex === Subject: Re: mathematicians and longhand arithmetic >All this reminds me of a *terrible* introduction to a book I read. Some >completely clueless idiot was trying to make analogies in subjects he hadn't a >scooby about. He was talking about how people manage with what's available, and >one of the his analogies to that was basically this: > Einstein formulated general relativity and there are only ten digits! >Talk about missing the point on soooo many levels. >The other analogies were also laughable. One was that Da Vinci painted >beautiful paintings when there are only three prime colours. DUH! And the >final nail in the coffin for this fool was saying that Beethoven composed >beautiful music when there are only 12 notes in the scale.. AARGH! Kill this >moron! > I have a sudden compulsive need to read that introduction. What's > the title of the book? I'll go back and find it. It was the introduction to a little book someone got me, it was relating to work happyness or something cheesy like that. I'll try and find it, it must be at home somewhere! alex === Subject: Newton's First In addition to defining a body's inertia as that which opposes any external force exerted on, and/or by it, with an equal and opposite force; the initial position, orientation and velocity - uniform speed and unchanging direction - of Newton's First Law of motion also serves as an inertial reference _motion_; an initial _inertial_ velocity [vi] from which all subsequent positions, orientations and velocities can easily be seen to proceed. === Subject: What is a four vector? Just one of Unky Al's favorites is a four vector. What the heck is it? === Subject: Re: What is a four vector? > Just one of Unky Al's favorites is a four vector. What the heck is it? It's a word you don't understand following a number too high for you to count. -E === Subject: Re: What is a four vector? Four-Vector http://mathworld.wolfram.com/Four-Vector.html Position Four-Vector http://mathworld.wolfram.com/PositionFour-Vector.html More === Subject: Re: What is a four vector? >Just one of Unky Al's favorites is a four vector. What the heck is it? The easy version has imaginary time, s = (x, y, z, ict) So that the magnitude is s dot s = x^2 + y^2 + z^2 - c^2 t^2 Under Lorentz transformations it is this magnitude that is preserved, compared to Galilean transformations preserving the length with no time dependence, r^2 = x^2 + y^2 + z^2 In Newtonian mechanics there's no meaningful vector operations that include time as a component, so it would just be an ordered quadruplet. -- Very well, he replied, I allow you cow's dung in place of human excrement; bake your bread on that. -- Ezekiel 4:15 === Subject: Re: What is a four vector? > Just one of Unky Al's favorites is a four vector. What the heck is it? A four-dimensional vector, with one time-like and three space-like components, which transforms under Lorentz transformations like the position four-vector. Examples: position and time: x^mu = (ct, x, y, z) energy and momentum: p^mu = (E/c, p_x, p_y, p_z) charge and current density: j^mu = (c rho, j_x, j_y, j_z) electrodynamics potentials: A^mu = (phi, A_x, A_y, A_z) derivatives: del_mu = (1/c del/del t, nabla) Etc. Note: ^ and _ implies super- and subscripts, respectively; this distinguishes contra- from covariant four vectors. And has nothing to do with to the power of, BTW. === Subject: Re: What is a four vector? > Just one of Unky Al's favorites is a four vector. What the heck is it? > A four-dimensional vector, with one time-like and three space-like > components, which transforms under Lorentz transformations like the > position four-vector. > Examples: > position and time: x^mu = (ct, x, y, z) > energy and momentum: p^mu = (E/c, p_x, p_y, p_z) > charge and current density: j^mu = (c rho, j_x, j_y, j_z) > electrodynamics potentials: A^mu = (phi, A_x, A_y, A_z) > derivatives: del_mu = (1/c del/del t, nabla) > Etc. > Note: ^ and _ implies super- and subscripts, respectively; this > distinguishes contra- from covariant four vectors. And has nothing to > do with to the power of, BTW. You'll soon find out that Shead's four-vector is actually ( f, w, g, a ) Dirk Vdm === Subject: Re: What is a four vector? > You'll soon find out that Shead's four-vector is actually > ( f, w, g, a ) > Dirk Vdm ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Actually, I think he prefers [4]. Ari === Subject: Re: Optimal Strategy in playing Stockmarket via VonNeumann Gametheory; Crossover technique; 30 free shares BMY today > Portfolio of PAF as of 22JUL04: > BCE 8,000 20.30 $162,400.00 > BLS 10 25.57 $2,55.70 > BMY 6870 23.45 $161,101.50 > SBC 7,200 23.59 $169,848.00 > 22,080 sum of units of share-wealth > realestate land 3APR03 of 3 lots $19,000. > science-art of pictures,porcelain etc starting JAN03 for $15,805. > realestate land 30JUL03 another lot $11,500. > Last Crossover was 25May04 and today is 22July04. So that was > about a 2 month pause in action. Today was good news that AT&T > is halting its practice of invading baby-bell territory and SBC price > jumped to around 23.55 and luckily BMY price is down to 23.24. > Back on 25May04 BMY was priced at 24.98 and > SBC was priced at 23.85 so one can see the pretty crossover > of today, although it is not as big of its approx 30 cents. But > who cares, free shares are nothing to snicker at. Today I netted 30 > free shares of BMY by selling 2,000 > SBC at approx 23.55 and buying 2,020 BMY at approx 23.24. But then > things got sticky in that BMY started to jump ahead fast and > so had to sell 4710 SBC at 23.59 and bought 4720 BMY at 23.45. So I > have 30 free shares of BMY plus a tiny cash extra (before taxes). > What is the greatest single thing that I like about the Crossover > Technique? Not only is it the VonNeumann gametheory Optimal Strategy > for Playing the StockMarket with its proof that this technique uses the > maximum forms of currency-- i.e.-- 2 forms of currency in money currency > and in stock itself as currency because on 25May the portfolio had only > 22,050 shares and today it owns 22,080 shares. But the greatest feature > of Crossover Technique is its purification, or cleansing, or getting > clean > in finances by a pure crystal clear nice warm bathe. Never before in all > my financial life have I experienced such pretty, and idyllic and lovely > and thrilling and exhilarating cleaning up. Other investors with their > silly > techniques never have a chance of getting clean. > I mean, if I had just sat on the portfolio above of say for the past > several years > the total number of shares would be less than 21,500. But I did not sit > still, > instead I watched for crossovers and applied the technique and those > 21,500 > has grown to 22,080. And I expect it to grow to 23,000 then 24,000 etc > etc. > Every time I partake in a crossover is as if all the past and previous > crossovers were justified and this is what I call a Cleaning up and a > Purification, > perhaps even an anointing of oil as in biblical times only of course > after one is > cleansed and purified by clean pure water to apply a vaseline or > chapstick > oinment to the hands and feet. > Crossover is great, is beautiful is electric and is cleansing pure. Why > do I feel > this way? Because after every Crossover I feel I have been thoroughly > washed > of all past sins and inequities and wrongheadedness. No other technique > to the > stockmarket allows for this cleansing pure bath washing of all past > errors. > Archimedes Plutonium > www.archimedesplutonium.com > www.iw.net/~a_plutonium > whole entire Universe is just one big atom where dots > of the electron-dot-cloud are galaxies same old idiot. same old garbage. same misconceptions about game theory. buzz off you annoying pest. learn the difference between opinion and mathematical proof. stupid fag. === Subject: sci.idiots you can expect here archimedes plutonium - a moron who has no idea about game theory but posts about it like he's the modern day von neumman. he thinks proving a game that is not chess has a draw OS proves that chess has a draw OS. (absurd i know but so is this idiot's every word) donald shead - not only is he full of hot air on all topics related to physics.. he has no idea about calculus so thinks everything is a simple multiplication or division process.. even gravity fields that vary over time and space.. and he thinks force is a fundamental quantity and claims to be a bridge designer (he designed lego bridges for 3 year olds and again, this was only for one purpose: to get closer to the fresh young meat he loves so much.. this man has repeatedly sent me emails admitting to being a homosexual pedophile) james harris - this jerk got an incorrect paper published in a journal. his claim to fame is that his paper passed peer review sure. i agree to this. his peers are the only idiots that could possibly let that error-ridden paper pass any sort of review. he's an idiot so his peers would be idiots no? now fully equipped with the knowledge of his mistakes, he has moved to trying to get it published in ANOTHER journal. hooray for how stupid this fraud-attempting idiot is. i suggest these 3 men be deported to iraq as they are adversely affecting the average-IQ of whatever country they currently inhabit. === Subject: Re: sci.idiots you can expect here !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw [Rubbish...] > i suggest these 3 men be deported to iraq as they are adversely > affecting the average-IQ of whatever country they currently inhabit. Judging from your posting, they had a severe impact on you at least. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Probability by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6NC55b02354; What can we say about the distribution of a random variable X if E(1/X) = 1/E(X) === Subject: Re: Probability > What can we say about the distribution of a random variable X if > E(1/X) = 1/E(X) Assuming X is positive, we may conclude X is constant (= deterministic). I used Cauchy-Schwarz for that. For negative X, same conclusion. What about signed X? Or complex X? === Subject: Re: Probability > What can we say about the distribution of a random variable X if > E(1/X) = 1/E(X) >Assuming X is positive, we may conclude X is constant >(= deterministic). I used Cauchy-Schwarz for that. >For negative X, same conclusion. >What about signed X? Or complex X? Don't know exactly what _can_ be said, but we certainly can't conclude that X is constant. (For example P(X=1) = P(X=-1) = 1/2.) ************************ David C. Ullrich === Subject: Re: Probability > What can we say about the distribution of a random variable X if > E(1/X) = 1/E(X) ... >What about signed X? Or complex X? >Don't know exactly what _can_ be said, but we >certainly can't conclude that X is constant. >(For example P(X=1) = P(X=-1) = 1/2.) There's no nontrivial solution with a two-valued X. Hint: factor (p a + (1-p) b) (p/a + (1-p)/b) - 1. But you could take e.g. P(X=-1) = 1/36 P(X=1) = 7/12 P(X=2) = 7/18 Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Probability > What can we say about the distribution of a random variable X if > > E(1/X) = 1/E(X) > >Assuming X is positive, we may conclude X is constant >(= deterministic). I used Cauchy-Schwarz for that. >For negative X, same conclusion. >What about signed X? Or complex X? > Don't know exactly what _can_ be said, but we > certainly can't conclude that X is constant. > (For example P(X=1) = P(X=-1) = 1/2.) ??what?? Then E(X)=0, right? So certainly not E(1/X)=1/E(X). [unless you are in one of those weird 1/0 threads here...] === Subject: Re: Probability What can we say about the distribution of a random variable X if > > E(1/X) = 1/E(X) > >Assuming X is positive, we may conclude X is constant >(= deterministic). I used Cauchy-Schwarz for that. For negative X, same conclusion. What about signed X? Or complex X? > Don't know exactly what _can_ be said, but we > certainly can't conclude that X is constant. > (For example P(X=1) = P(X=-1) = 1/2.) >??what?? Then E(X)=0, right? So certainly not E(1/X)=1/E(X). Um. >[unless you are in one of those weird 1/0 threads here...] Drat - actually I was going to suggest that as a way out, but you beat me to it. Now I'm really stuck. ************************ David C. Ullrich === Subject: Re: Probability > What can we say about the distribution of a random variable X if > > E(1/X) = 1/E(X) > > >Assuming X is positive, we may conclude X is constant (= deterministic). > I used Cauchy-Schwarz for that. >For negative X, same conclusion. >What about signed X? Or complex X? > Don't know exactly what _can_ be said, but we certainly can't conclude > that X is constant. (For example P(X=1) = P(X=-1) = 1/2.) I don't see how you could say that. If P(X=1)=P(X=-1)=0.5 then E(X)= 0.5*-1 + 0.5*1 = 0 and 1/E(X) is not defined in the reals. However, E(1/X) = -0.5/1 + 0.5/1 = 0. Clearly E(1/X) is defined in the reals. > ************************ > David C. Ullrich -- Lance Lamboy Go F*ck Yourself ~ Dick Cheney === Subject: Re: Probability > What can we say about the distribution of a random variable X if > > E(1/X) = 1/E(X) > > >Assuming X is positive, we may conclude X is constant (= deterministic). > I used Cauchy-Schwarz for that. >For negative X, same conclusion. >What about signed X? Or complex X? > Don't know exactly what _can_ be said, but we certainly can't conclude > that X is constant. (For example P(X=1) = P(X=-1) = 1/2.) >I don't see how you could say that. Actually neither do I. But I did, somehow... >If P(X=1)=P(X=-1)=0.5 then E(X)= >0.5*-1 + 0.5*1 = 0 and 1/E(X) is not defined in the reals. However, >E(1/X) = -0.5/1 + 0.5/1 = 0. Clearly E(1/X) is defined in the reals. > ************************ > David C. Ullrich ************************ David C. Ullrich === Subject: Re: What is Forced (accelerated) Motion? > Acceleration is defined as a change in velocity. > No, > That is the mathematical definition. > In the real world, > aceleration is a change of speed (velocity) and nothing to do with vectors. > Speed=velocity. I can't decide if you just make these things up and then post them to be provocative, or if you are really this ignorant of the basics of motion. It requires only a little effort and thought to learn the basics of motion. Why are you even posting here if you don't know anything about motion or math. Vectors are necessary to describe things in 3D. The things you are saying could only be true if everything was restricted to a line. Orbits and trajectories are in 2D, so one needs vectors. I am now going to try to stop wasting my time on this drivel. Van === Subject: Re: Knowledge: Formal and informal > Somewhere along the line the intellectual giants of our time took a > wrong turn and lost their way; as well as what the intellectual giants > of yesteryear tried so hard to teach them. > And you're qualified to state that, how? Please state any degrees you have that > qualify you to judge people who have degrees of their own. > It doesn't take a degree for anybody to realize that mechanics is the > science of Force and motion, and question Einstein's theories of > warped space-time; they are just plain nonsensical, Morry. Einstein's theories are not nonsense. The premises are quite reasonable, like mass curves space and moves along geodesics of that space. Its hard to find other, let alone better explanations. GR is a great example of a construct of the mind, which logically follows from very reasonable assumptions. It also has the advantage of excellent agreement with observations (outside the quantum domain). IMO, there are more conceptual problems with QM than with any classical theory like SR or GR. But QM also has excellent agreement with experiment, which is how these things are judged in physics. Van === Subject: Newton's Second With his Second Law of Motion, Newton united force and weight through acceleration; showing that the ratio of a net external force [f], exerted on and/or by any body or mass of matter, when divided by the acceleration [a] that it causes is a constant [f/a]: Then he showed that weight is that particular force exerted by terra firma; that prevents free fall; where bodies resting on Earth's surface are restrained from an acceleration [g] of about 32'/sec^2; that while the weight of a body varies at various locations, it varies in proportion to the acceleration at which it will free fall; so that at any particular location the ratio of it's weight [w] to the acceleration [g] at which it will free fall is a constant [w/g]. These two constants are equal, and can be expressed mathematicaly as: f/a = w/g By mathematical transposition, we get: f = (w/g)a, and w = (f/a)g. Force = weight/acceleration of free fall times the acceleration: Weight = force/acceleration times the acceleration of free fall. === Subject: Re: Newton's Second > With his Second Law of Motion, Newton united force and weight through > acceleration.... Newton's Second Law http://scienceworld.wolfram.com/physics/NewtonsSecondLaw.html A force F acting on a body gives it an acceleration a which is in the direction of the force and has magnitude inversely proportional to the mass m of the body: F = ma F = ma is a differential equation and therein lies its power. Some examples of application: Case 1. The force is constant: Assuming that the mass remains constant, we have constant acceleration: F/m = dv/dt = a = constant direct integration (with respect to t) gives formulas such as: v - v_o = at v = at + v_o A second integration results in: x - xo = 1/2 at^2 + v_ot x = 1/2 at^2 + v_ot + x_o eliminating t, we get 2a(x-x_o) = v^2 - v_o^2 Case 2. The force is a function of position only: F(x) = m d^2x/dt^2 In a great many instances the force that an object experiences depends only on the object's position with respect to other bodies, for example, electrostatic and gravitational forces. The equation can be rewritten as: F(x) = mv dv/dx = m/2 d(v^2)/dx = dT/dx where the quantity T = 1/2 mv^2 The integral [ F(x)dx ] + constant = T oh it gets good from here.... the upshot for a freely falling body under the influence of gravity is: v = dx/dt = ± sqrt (v_o^2 - 2gx), where g is the acceleration due to gravity. Gosh I didn't even cover the equations for the force of a spring! Case 3. The force as a function of velocity only: F(v) = m dv/dt It often happens that the force on a object depends only on the velocity of the object. This is, for example, true of viscous A single integration t = integral [ m dv/F(v)] = t(v) results in: v = v(t), and a second integration gives: x = integral [ v(t) dt] = x(t) You see, Shead, if you know a bit of calculus and the circumstances of some rectilinear motion problem, you can figure out a hell of a lot of physics just from the relationship between Force and acceleration that Isaac Newton figured out: F = m d^2x/dt^2 = ma === Subject: Re: Newton's Second > With his Second Law of Motion, Newton united force and weight through > acceleration; showing that the ratio of a net external force [f], > exerted on and/or by any body or mass of matter, when divided by the > acceleration [a] that it causes is a constant [f/a]: Hey Dumb Donny Head, you STILL don't know the difference among active, passive, inertial, and gravitational masses. You are a boring psychotic ing imbecile. http://www.apa.org/journals/psp/psp7761121.html http://insti.physics.sunysb.edu/~siegel/quack.html -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: Finding equation ... > David Cantrell continued.. That is, taking any monotonic odd bounded function , adding half range it so it is positive and then integrating it.So,in the power series all are even powers except one x term. === Subject: Re: Finding equation >The function to be found is of the form R->R:f(x), defined for all x >belonging to R, and continuous. Moreover for all x, f'(x)>0 and >f''(x)>0. The trick is, it must NOT be of the form f(x)=C*a^x, for >which C,a are rela numbers and a>0. This question appeared on another forum, and yet no one has been able >to give an equation. It is quite clear that a polynomial will not do, >your time! > How about f(x) = 1 + e^x > As I mentioned in sci.math.num-analysis when the same question was asked, > using the idea of a hyperbola does work. For example: > f(x) = x + Sqrt(x^2 + 1). > David Cantrell You may be implying a class of functions, I am not sure, choosing an even function y'' and >0, adding a constant to integrand y' to keep it also >0.Right? I got one more in this fashion in f(x)= 2 x + x*ArcTan(x) -Log(1+x^2)/2. Another one is x+Log(Cosh(x)) but OP may not be happy with as it is an Exp related function.I wonder what is characteristic of the general term in a monotonically increasing power series. === Subject: Re: Newton's Second {snip] Not this stuff again! === Subject: Inter-expressibility among periodic functions Is it possible to express any periodic function in terms of another periodic function of the same period? === Subject: Re: Inter-expressibility among periodic functions They're all expressible in terms of sines and cosines, therefore... > Is it possible to express any periodic function in terms of another > periodic function of the same period? === Subject: Re: Inter-expressibility among periodic functions >Is it possible to express any periodic function in terms of another >periodic function of the same period? If f is periodic then g = 2*f has the same period, and it's not too hard to express f in terms of g. Maybe you had some other conditions in mind? ************************ David C. Ullrich === Subject: Re: Is this number rational or irrational? > (And why should I care?) > sqrt(24)*sqrt(54) > Now, if I process the above in this order > sqrt(4*6)*sqrt(9*6); > 2sqrt(6)*3sqrt(6); > I get a product of rationals times irrationals-- > which surely (?) is irrational. Theorem 1: Let A and B be two real numbers. If A is rational (A<>0) and B is irrational, then their product A*B is irrational. Proof: First of all, the set Q of rational numbers is closed under the operation of division, that is, if X and Y are rational numbers, then X/Y is also rational. So, let A*B be rational. Then, A*B/A is the division of two rational numbers so it is also rational. But A*B/A = B , which is irrational. Since we have a contradiction from the premise that A*B is rational, we conclude that it is actually an irrational number. Theorem 2: If C and D are two irrational numbers, then their product may be rational. Proof: Just choose C to be any irrational number, and D=1/C, which is also irrational (we could apply theorem 1 to show this. If D would be rational, then 1/D=C would be also rational). However, it is obvious to see that C*D=1, a rational number... === Subject: Re: Is this number rational or irrational? > (And why should I care?) > sqrt(24)*sqrt(54) > Now, if I process the above in this order > sqrt(4*6)*sqrt(9*6); > 2sqrt(6)*3sqrt(6); > I get a product of rationals times irrationals-- > which surely (?) is irrational. > Theorem 1: Let A and B be two real numbers. If A is rational (A<>0) > and B is irrational, then their product A*B is irrational. > Proof: First of all, the set Q of rational numbers is closed under the > operation of division, that is, if X and Y are rational numbers, then > X/Y is also rational. > So, let A*B be rational. Then, A*B/A is the division of two rational > numbers so it is also rational. But A*B/A = B , which is irrational. > Since we have a contradiction from the premise that A*B is rational, > we conclude that it is actually an irrational number. > Theorem 2: If C and D are two irrational numbers, then their product > may be rational. > Proof: Just choose C to be any irrational number, and D=1/C, which is > also irrational (we could apply theorem 1 to show this. If D would be > rational, then 1/D=C would be also rational). However, it is obvious > to see that C*D=1, a rational number... George === Subject: Re: Is this number rational or irrational? > I get a product of rationals times irrationals-- > which surely (?) is irrational. > But if I rearrange the factors, I can get > 2*3*sqrt(6)*sqrt(6) = 2*3*6 = 36; the product of two irationals can be rational. you have two irationals here each is the sqrt(6) which when multiplied surely make a rational #. operations in math are mostly binary. the rules of a rational times an irational being irational is a binary operation which means it refers to one irational quantity multiplied by one rational quantity. > Now, it appears to me that the rationality or > irrationality of a number should not depend > on my noticing how its factors can be commuted > and/or associated. So, is the number > sqrt(24)*sqrt(54) rational or irrational? rational as any idiot could see > (Again, why should I care?) > George with an attitude like that, you definitely should NOT care. you're missing the point of mathematics completely when you ask questions like that. please by all means, continue not caring and try to steer clear of an area of study where your limited understandings cause elementary mistakes. === Subject: Re: Is this number rational or irrational? irrationality of a number should not depend > on my noticing how its factors can be commuted > and/or associated. So, is the number > sqrt(24)*sqrt(54) rational or irrational? > rational as any idiot could see And how does 522nd reply to a trivial question makes you look like? === Subject: Re: Is this number rational or irrational? > (And why should I care?) > sqrt(24)*sqrt(54) > Now, if I process the above in this order > sqrt(4*6)*sqrt(9*6); > 2sqrt(6)*3sqrt(6); > I get a product of rationals times irrationals-- > which surely (?) is irrational. No it isn't. Why do you say surely? And why do you question yourself on that point? You should just admit to yourself that you are making a guess and that you prove that your guess was wrong below. > But if I rearrange the factors, I can get > 2*3*sqrt(6)*sqrt(6) = 2*3*6 = 36; > which surely is rational. > Now, it appears to me that the rationality or > irrationality of a number should not depend > on my noticing how its factors can be commuted > and/or associated. True. Not only that but the quantity itself does not change if the factors are commuted or associated differently. The real numbers are commutative and associative under multiplication (and addition for that matter). > So, is the number > sqrt(24)*sqrt(54) rational or irrational? You've already answered your own question. You proved that the number in question is 36. And yes, 36 is rational. What's your point? I can only fathom that you do not know that the real numbers are commutative and associative under multiplication, in which case you may want to find a mathematical source that proves these properties or at least postulates them. Note, it is obvious that 2*3 = 3*2 (it all boils down to counting). But it's not obvious that pi*e = e*pi (quiz: what does pi*e even mean?). > (Again, why should I care?) (how should I know, you asked the question.) Leonard > George === Subject: Re: Is this number rational or irrational? > (And why should I care?) > sqrt(24)*sqrt(54) > Now, if I process the above in this order > sqrt(4*6)*sqrt(9*6); > 2sqrt(6)*3sqrt(6); > I get a product of rationals times irrationals-- > which surely (?) is irrational. I think you are assuming that the fact that a rational number times an irrational number being irrational (easily proved) extends when two irrational numbers are used. It doesn't as your example shows. === Subject: Point Counting on Elliptic Curves can anyone recommend useful papers that are available on the web for point counting on elliptic curves. Curves over a prime filed (GF(p)) or a binary field (GF((2^m)) are both good. to those that others think have merit. Also, if you know code (like MIRACL has some), that would be helpful too. X-mailer: xrn 9.02 === Subject: Re: Elmentary Proofs Mail-To-News-Contact: abuse@dizum.com >I was wondering if any of you would be so kind to make sure they are correct >and to suggest improvements to my style. >2.3.4 Show that the product of a non-zero rational number and an irrational >number is irrational. An alternative approach that I haven't seen anybody else suggest yet is: First, prove the following lemma: The product of two rational numbers is a rational number. (Use the definition of rational number and the closure of Z under multiplication to prove this.) Armed with this, multiply a non-zero rational a times an irrational b. Now, you get to do proof by contradiction. Assume that the product c is rational. Since a is non-zero, what does it have? What can you do with this and with c? What is the result when you do that? Since they're both rational, what does the lemma tell you about b? -- Michael F. Stemper #include Time flies like an arrow. Fruit flies like a banana. === Subject: Re: Elmentary Proofs > Let the first number be a/b in reduced form (where (a,b)=1), and call the > irrational number c. Assume their product is rational. Put it in reduced > form as x/y, where x and y are integers and (x,y)=1. > x/y = (ac)/b. Cross multiply and we get xb = ayc. xb is an integer, as is > ay. > So divide both sides by ay and we get xb/ay = c, where xb and ay are > integers. > So c is rational--which is a contradiction. So their product is irrational. > Insisting on reduced form is not needed, and thus confusing. Yes, I was thinking of a similar problem I did years ago. I forgot to omit the references to reduced form. Also, when sending, I immediately realized I'd also forgotten to add which you can do because they are non-zero after divide both sides by ay. Oh, well. Michael X-mailer: xrn 9.02 === Subject: Re: Fractions v natural numbers in The Road to Reality Mail-To-News-Contact: abuse@dizum.com >The magic is that the idea of a fraction actually works despite the fact >that we do not really directly experience things in the physical world that >are exactly quantified by fractions - pieces of pie leading only to >approximations. (This is quite unlike the case of natural numbers, such as >1, 2, 3, which do precisely quantify numerous entities of our direct >experience.) >I must say that, for me, fractions and natural numbers are on the same >footing: we do not directly experience things in the physical world that are >exactly quantified by them. Well, I have exactly one son. My bedroom has exactly two closets. My whiteboard has exactly three erasers in its chalk tray.My office has exactly four guest chairs in it. Although I've never seen a natural number in the wild, I've certainly encountered plenty of things that are exactly quantified by them. -- Michael F. Stemper #include === Subject: Re: Fractions v natural numbers in The Road to Reality >The magic is that the idea of a fraction actually works despite the >fact that we do not really directly experience things in the physical >world that are exactly quantified by fractions - pieces of pie leading >only to approximations. (This is quite unlike the case of natural >numbers, such as 1, 2, 3, which do precisely quantify numerous entities >of our direct experience.) >I must say that, for me, fractions and natural numbers are on the same >footing: we do not directly experience things in the physical world that >are exactly quantified by them. > Well, I have exactly one son. My bedroom has exactly two closets. My > whiteboard has exactly three erasers in its chalk tray.My office has > exactly four guest chairs in it. > Although I've never seen a natural number in the wild, I've certainly > encountered plenty of things that are exactly quantified by them. The same can be said for fractions, and so I disagree with the quotation above. For example, if one of your guest chairs is black and the others are green, then exactly 3/4 of your guest chairs are green. DWC === Subject: Re: Captain Bat strikes again (Was Re: Riley Michelle Johnson > The following (courtesy of Waxy.org) is sort of an unofficial FAQ > explaining the psychotic nonsense posted to Usenet by Shawn Daryl > Kabatoff AKA Dar, AKA Probababbilities. And now AKA marcia and > me. > WARNING: Read below before even thinking about responding to this > twit. > http://www.waxy.org/archive/2002/05/21/dar_kaba.shtml#000643 Can you do us a favour and not keep posting the contents of the web page? The URL would be enough! alex === Subject: Re: Are all changes in velocity acceleration > I'd have to say yes; even though a force is not exerted on the > observer, but on what is being observed, the acceleration observed > required an accelerating force: > Average acceleration is (defined as) the rate of change in velocity [a > = (vt-vi)/t], and instantaneous acceleration is defined as something > else. Idiot. a = dv/dt. v = ds/dt. === Subject: Re: Are all changes in velocity acceleration charset=iso-8859-1 Yes... or deceleration. === Subject: Re: Are all changes in velocity acceleration Yes. Christ, learn some calculus. === Subject: Re: Are all changes in velocity acceleration In sci.math, Eric Gisse : > Yes. > Christ, learn some calculus. An interesting command, since Jesus pre-dated Newton and Leibnitz by over a millennium.... :-) -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: Are all changes in velocity acceleration > In sci.math, Eric Gisse > : >Yes. >Christ, learn some calculus. > An interesting command, since Jesus pre-dated Newton and Leibnitz > by over a millennium.... > :-) Calculate the miracle volume if a point-source saviour, moving at 5 furlongs a second, encounters a incontinent crippled man who is executing brownian motion ..... [5 marks] etc. etc. === Subject: Re: Are all changes in velocity acceleration charset=iso-8859-1 > Calculate the miracle volume if a point-source saviour, moving at 5 furlongs a > second, encounters a incontinent crippled man who is executing brownian motion > ..... > [5 marks] > etc. etc. Ah dam man, warn me next time.. I got pepsi everywhere.... ROFL.. === Subject: Re: Are all changes in velocity acceleration <10g2b2m81sf8e54@corp.supernews.com In sci.math, Eric Gisse > : > Yes. Christ, learn some calculus. > > > An interesting command, since Jesus pre-dated Newton and Leibnitz > by over a millennium.... > > :-) > Calculate the miracle volume if a point-source saviour, moving at 5 furlongs a > second, encounters a incontinent crippled man who is executing brownian motion > ..... > [5 marks] I'm sorry but I find it somewhat sacreligious to assume a spherical Jesus. :P === Subject: Re: Are all changes in velocity acceleration charset=iso-8859-1 > Calculate the miracle volume if a point-source saviour, moving at 5 furlongs a > second, encounters a incontinent crippled man who is executing brownian motion > ..... > [5 marks] > I'm sorry but I find it somewhat sacreligious to assume a spherical Jesus. Ahh, no problem then.. Jesus is a point source, his *area of effect volume* is spherical. === Subject: Re: Are all changes in velocity acceleration > I'd have to say yes; even though a force is not exerted on the > observer, but on what is being observed, the acceleration observed > required an accelerating force: > Average acceleration is (defined as) the rate of change in velocity [a > = (vt-vi)/t], and instantaneous acceleration is defined as something > else. > Average acceleration is proposed to be the result of an external > force, and is found to be proportional to the force exerted on and/or > by a body's mass, and/or inertia. > So in any event any observed acceleration is either a forced > acceleration of the observer, and/or a forced acceleration of whatever > is observed. > That should pretty well explain the rest of Newton' s first law; that > a body continues to remain at rest, or move in a straight line with > constant velocity. so first you say yes.. then you deviate to a completely unrelated topic about observers. why did you send me an email bragging about your rights to being a homosexual pedophile? === Subject: Re: JSH: Revocation of a Ph.D > I'm not sure about these constructions. In Pittsburgh, there is an > (abominable) alternative pronounced and spelled yinz. Is youins a > misspelling of yinz here? Or yinz a slur of youins? > Now that you mention it, youins doesn't sound as awful to my ears as > yinz. Maybe some Texas relatives were Arkansans in disguise. Carnegie-Mellon has classes that have been created by students and are taught by students. One of these is Pittsburgheze for the non-Yinzers, which is a linguistic study of the Pittsburgh dialect for non-natives. === Subject: Re: JSH: Revocation of a Ph.D <87briibzmj.fsf@phiwumbda.org> <87acxrkaeo.fsf@phiwumbda.org> Discussion, linux) > I'm not sure about these constructions. In Pittsburgh, there is an > (abominable) alternative pronounced and spelled yinz. Is youins a > misspelling of yinz here? Or yinz a slur of youins? > Now that you mention it, youins doesn't sound as awful to my ears as > yinz. Maybe some Texas relatives were Arkansans in disguise. > Carnegie-Mellon has classes that have been created by students and are > taught by students. One of these is Pittsburgheze for the non-Yinzers, > which is a linguistic study of the Pittsburgh dialect for non-natives. Not coincidentally, CMU is my alma mater. -- It has been shown that no man can sit down to write without a very profound design. Thus to authors in general trouble is spared. A novelist, for example, need have no care of his moral. It is there -- that is to say, it is somewhere -- and the moral and the critics can take care of themselves. --E.A. Poe === Subject: [no subject] by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6NDpxp14945; about Mandelbrot's book re. coastline section & Hausauff fractal dimension from Mandelbrots fractal geometry of nature in the Hausdorff fractal dimension section: ... these lengths are raised to the power D=1, the Euclidean dimension of a straight line... do one dimensional lines have to be straight? surely an arc from a circle or a line with a sharp corner are one dimensional lines? ...The surface measure of a closed polygon's interior is similarly calculated by paving it with squares, and adding the squares' sides raised to the power D=2, the Eucleidean dimension of a plane... when he says the squares' sides, surely he means *two* *sides* at right angles to each other of each square, not all four sides? or is this a new way of getting the area of a square? (that's a serious question -- having just started to try and learn maths it seems some things are completely turned on their heads from what little i know about maths, but sometimes not -- it just sounds like it, and it's nearly impossible to know which things are being turned on their heads, and which things are just loosely described) ...When, on the other hand, the wrong power is used, the result gives no specific information: the area of every closed polygon is zero, and the length of its interior is infinite... what does that mean? surely it depends *which* wrong power is used -- surely there's a lot of wrong powers (whatever that actually means) that will result in, i can imagine wrong, but, non-zero results? any explenation/description of that sentance would be great. ...Let us proceed likewise for a polygonal approximation of a coastline made up of small intervals of length E. If their lengths are raised to the power D, we obtain a quantity we may call tentatively an approximate measure in the dimension D. Since according to Richardson the number of sides is N=F*E^-D, said approximate measure takes the value F*E^D*E^-D = F. how and why does it get from N=F*E^-D to F*E^D*E^-D = F ? also with F*E^D*E^-D = F that's just saying F = F right? with E^D*E^-D resulting in nothing? (or E^D*E^-D resulting in 1 which when used in multiplication means nothing? yes i think it results in one.) yes i'd guess it does mean F = F because it then says: ...Thus, the approximate measure in the dimension D is independent of E. so you can just ignore all that and use F as is, whatever F is. i still don't know what F is. can anyone tell me what F is please? *someone* must know what F is?! :) just having an answer to what F is would be great. === Subject: A model for the natural numbers Originator: anno4000@lublin.zrz.tu-berlin.de (Anno Siegel) The current thread about n-tuples (set definition of n-tuple) reminds me of a question I have been meaning to ask here for some time. There are many ways of constructing the natural numbers starting from (naive) set theory. One goes like this: Define 0 = {} (the empty set) If n is a natural number, define its successor n+1 = union( n, { n } ) That results in a system where each natural number is the set of all strictly smaller natural numbers. It seems pretty clear that it fulfills the Peano axioms. It is somewhat unusual to include 0 in the naturals, but that is by no means a new idea. In fact, indexing from 0 instead of 1 has advantages in some situations, as programmers know. Starting the naturals at 0 is an invitation to do that where appropriate. What I like about his particular system is that it gives a convenient notation for the set { 0, ..., n-1 }, which is simply n. So i IN n is a way to let an index run from 0 to n-1. Other ways of writing that are always a little cumbersome. My question: Has this system been proposed before? Does it have a name? Is it in use in some circles? Anno === Subject: Re: A model for the natural numbers > The current thread about n-tuples (set definition of n-tuple) reminds > me of a question I have been meaning to ask here for some time. > There are many ways of constructing the natural numbers starting from > (naive) set theory. One goes like this: > Define 0 = {} (the empty set) > If n is a natural number, define its successor n+1 = union( n, { n } ) > That results in a system where each natural number is the set of all strictly > smaller natural numbers. It seems pretty clear that it fulfills the Peano > axioms. > It is somewhat unusual to include 0 in the naturals, In my experience, it is unusual not to see 0 included in the natural numbers. But it makes no difference. > but that is by no > means a new idea. In fact, indexing from 0 instead of 1 has advantages > in some situations, as programmers know. Starting the naturals at 0 > is an invitation to do that where appropriate. > What I like about his particular system is that it gives a convenient > notation for the set { 0, ..., n-1 }, which is simply n. So i IN n > is a way to let an index run from 0 to n-1. Other ways of writing that > are always a little cumbersome. > My question: Has this system been proposed before? Yes. But apparently you have independently invented it (probably about a century too late), and that is commendable. > Does it have a name? I don't know. > Is it in use in some circles? Yes. It is quite standard. It appears in many introductory books on set theory. See _Elements of Set Theory_ by Herbert Enderton for example (which will probably also tell you who is credited with the invention). -Leonard > Anno === Subject: Re: A model for the natural numbers Originator: anno4000@lublin.zrz.tu-berlin.de (Anno Siegel) > Define 0 = {} (the empty set) > If n is a natural number, define its successor n+1 = union( n, { n } ) > [...] > It is somewhat unusual to include 0 in the naturals, > In my experience, it is unusual not to see 0 included in the natural > numbers. Ah, so things have moved on. I was taught (Germany, 60's) that we really ought to include zero, but tradition has it.... > But it makes no difference. No big deal, no. [...] > My question: Has this system been proposed before? > Yes. But apparently you have independently invented it (probably > about a > century too late), and that is commendable. I wouldn't say I invented it. I don't remember having heard about it explicitly, but apparently it has been around for a while. These things transmit through osmosis. It's also pretty obvious. > Does it have a name? > I don't know. We have learned in this thread it's the v. Neumann system, which I knew only by name. Live and learn... Anno === Subject: Re: A model for the natural numbers > Define 0 = {} (the empty set) > If n is a natural number, define its successor n+1 = union( n, { n } ) > So i IN n is a way to let an index run from 0 to n-1. > Does it have a name? Yawn v. Neumann -- Rainer Rosenthal, r.rosenthal@web.de _____________________ | _ | | | (_) | Given A, P and a circle. Find B, C on the | | A P | circle with P on BC and area(ABC)=maximum. | |__________|___(Ingmar Rubin in de.sci.mathematik) ________| === Subject: Re: A model for the natural numbers > The current thread about n-tuples (set definition of n-tuple) reminds > me of a question I have been meaning to ask here for some time. > There are many ways of constructing the natural numbers starting from > (naive) set theory. One goes like this: > Define 0 = {} (the empty set) > If n is a natural number, define its successor n+1 = union( n, { n } ) > That results in a system where each natural number is the set of all strictly > smaller natural numbers. It seems pretty clear that it fulfills the Peano > axioms. > It is somewhat unusual to include 0 in the naturals, but that is by no > means a new idea. In fact, indexing from 0 instead of 1 has advantages > in some situations, as programmers know. Starting the naturals at 0 > is an invitation to do that where appropriate. > What I like about his particular system is that it gives a convenient > notation for the set { 0, ..., n-1 }, which is simply n. So i IN n > is a way to let an index run from 0 to n-1. Other ways of writing that > are always a little cumbersome. > My question: Has this system been proposed before? Does it have a name? > Is it in use in some circles? http://www.wordiq.com/definition/Ordinal_number === Subject: Re: A model for the natural numbers Originator: anno4000@lublin.zrz.tu-berlin.de (Anno Siegel) [...] > Define 0 = {} (the empty set) > If n is a natural number, define its successor n+1 = union( n, { n } ) [...] > My question: Has this system been proposed before? Does it have a name? > Is it in use in some circles? > http://www.wordiq.com/definition/Ordinal_number So could I write things like sum( i, i IN n) = n(n+1)/2 without raising eyebrows? Anno === Subject: Re: A model for the natural numbers sum( i, i IN n) = n(n+1)/2 > without raising eyebrows? Yes, the idea is standard particularly in treatments of ordinal numbers in set theory, but no, if you write a sum like that in a different context without explanation, it'll still raise eyebrows. -- Kevin === Subject: Re: A model for the natural numbers Originator: anno4000@lublin.zrz.tu-berlin.de (Anno Siegel) [following up to myself] [...] > So could I write things like > sum( i, i IN n) = n(n+1)/2 > without raising eyebrows? Ugh. That ought to raise eyebrows, not because of the way it's written, but because it's wrong. The sum goes to n-1, so it's sum( i, i IN n) = n(n-1)/2 Sorry about that. Anno === Subject: Re: A model for the natural numbers because it's wrong. The sum goes to n-1, so it's > sum( i, i IN n) = n(n-1)/2 Chuckle... Well, in case it wasn't clear from my previous post, even in a corrected form, it'll still raise eyebrows. -- Kevin === Subject: Re: A model for the natural numbers Originator: anno4000@lublin.zrz.tu-berlin.de (Anno Siegel) > Ugh. That ought to raise eyebrows, not because of the way it's written, but > because it's wrong. The sum goes to n-1, so it's > sum( i, i IN n) = n(n-1)/2 > Chuckle... Well, in case it wasn't clear from my previous post, even > in a corrected form, it'll still raise eyebrows. Pity, really. It's a handy idiom, lacking nothing but -- idiomacity. Anno === Subject: Re: _Re: VOTE on whether 1/oo = 0 > Should the question be whether the limit of 1/x as x approaches > infinity equals 0? > And if not 0, what other value could it be. > The other value could be undefined. The question is not even > well-posed until the specific number system is declared. The symbol oo > is not a number in the reals, for example, so 1/oo is undefined in the > reals. > It's true that lim x->oo (1/x) is defined on the reals, but it is not > the same as 1/oo, unless you stipulate a special meaning to the symbol > 1/oo. Although there are exceptions, generally speaking, a limit and > an evaluation are two different things. > Patrick I wonder how far one can expect Clarke's Three Laws to apply, in whether it is possible or impossible to define zero and infinity as numbers, in that formula, 1/oo = 0, and remain logically consistent and without contradictions, with the definition(s) of all numbers, in whatever system? Science fiction author Arthur C. Clarke formulated the following three laws: http://en.wikipedia.org/wiki/Clarke%27s_Three_Laws (1) When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong. (2) The only way of discovering the limits of the possible is to venture a little way past them into the impossible. (3) Any sufficiently advanced technology is indistinguishable from magic. If (0) is the only number that is neither positive nor negative and all the others are non-zero numbers, (1/oo) is the number either zero and/or non-zero. That's its magical definition, in a sense, (1/oo) is the a number that is zero and non-zero. Clearly a contradiction of supernatural proportions! g! The constant e, for example can be defined as the limit of ( 1 + (1/x) )^x as x approaches infinity. That's ( 1 + (1/oo) )^oo isn't it. If (1/oo) is treated as the number 0, then it simplifies to (1 + 0)^oo, which looks like the number 1 multiplied by itself. Hmm? Euler's number as 1 rather than e, 2.7182818284590... Maybe the number (1) should have some identity definitions and non-identity definitions to accommodate such bifurcations of logic. i as the square root of (-1)? It does sort of ring a bell and blow out the candle as the book is closed and the absolute nature of the number one's definition of its > Identity <<< get its arithmetical EXORCISM! LOL! If (1/oo) is treated as a non-zero number, then the sum of the infinite series of 1/k! as k approaches infinity, converges to the value of e, 2.7182818284590... Maybe e is like the holy water used in mathematical magic! g! When a number can be both zero and non-zero, I confess, it looks like magic is at work! Perhaps a Magical Number System should expand on such spooky numbers! Sort of like the forbidden operation of taking the root of a negative value did for, what were they once called, Imaginary Numbers? Call 'em Exorcised Numbers of the forbidden operations with Infinity and it's reciprocal. If the ghost in the machine have a mathematical exorcism! Plunge into the abyss and sprinkle generously with magic inCOUNTations. g! eeeeeeeiiiiiii.... I hear the spirit screaming! peace is of the pi, d8>D === Subject: Re: _Re: VOTE on whether 1/oo = 0 <... The constant e, for example can be defined as the limit of ( 1 + (1/x) )^x > as x approaches infinity. > That's ( 1 + (1/oo) )^oo isn't it. Only if you ignore the definition of limit. === Subject: 130 free shares BMY Re: Optimal Strategy in playing Stockmarket via VonNeumann Gametheory; Crossover technique > Portfolio of PAF as of 22JUL04: > BCE 8,000 20.30 $162,400.00 > BLS 10 25.57 $2,55.70 > BMY 6870 23.45 $161,101.50 > SBC 7,200 23.59 $169,848.00 > 22,080 sum of units of share-wealth > realestate land 3APR03 of 3 lots $19,000. > science-art of pictures,porcelain etc starting JAN03 for $15,805. > realestate land 30JUL03 another lot $11,500. > Last Crossover was 25May04 and today is 22July04. So that was > about a 2 month pause in action. Today was good news that AT&T > is halting its practice of invading baby-bell territory and SBC price > jumped to around 23.55 and luckily BMY price is down to 23.24. > Back on 25May04 BMY was priced at 24.98 and > SBC was priced at 23.85 so one can see the pretty crossover Portfolio of PAF as of 23JUL04: BCE 8,000 20.25 $162,000.00 BLS 10 26.24 $262.40 BMY 12,000 23.60 $283,200.00 SBC 2,200 24.33 $53,526.00 22,210 sum of units of share-wealth realestate land 3APR03 of 3 lots $19,000. science-art of pictures,porcelain etc starting JAN03 for $15,805. realestate land 30JUL03 another lot $11,500. What a difference a day makes. I should have waited yesterday for today because the spread on SBC to BMY was about 75 cents today whereas yesterday it was at best 30 cents. So today, because of the Crossover, I sold 5000 SBC at approx 24.33 and bought 5130 BMY at 23.60. I tried to get the BMY at 23.49 but it is wise in these cases to go with the flow rather than end up with zip of BMY. Who knows, perhaps on Monday BMY will be at 24.50 and SBC drop back to 23.50. The Crossover is beautiful and pretty because it allows me to Wash and Clean and Purify the portfolio everytime it happens. Sort of like a cleaning of the slate, a cleaning of the tabla rosa, and sort of like new beginnings, and sort of like the end of a school year and the start of a new school year where it is a fresh new beginning. So, from yesterday the share wealth of the portfolio was 22,080 and today with the 130 free shares of BMY plus cash extras (before taxes of course) the share wealth total is now 22,210. I sort of wish I picked up another free 10 shares of BMY to make it a cool, cool 22,220. But we should never get so piggish greedy. Archimedes Plutonium www.archimedesplutonium.com www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: Why my post did not appear here during 24 hours? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6NF6YP26540; >Couldn't anyone help me with explanation? IMHO, because the brilliant henchmen of the moderator need some time to solve your problems. They want to the delay, I think. >Also, any idea on account of what should I do to increase >the chances that my posts appear at the first stroke? You may try http://at.yorku.ca Ady. >Best wishes, >Vladimir Bondarenko === Subject: Re: solvable numbers by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6NF6Y426528; > Let S be the set of all numbers that are solvable, that is, >they are the roots of solvable polynomials with integer coefficients. > Let T be the set of all numbers obtained by performing on the >integers(in any order, and with parentheses anywhere they make >sense)finite sequences of multiplying, dividing, adding, subtracting, >and extracting nth roots.(for natural numbers n) > (a) Obviously T is a field. I think it's called a root field. > (b) S is a subset of T. > (c) S is closed under the operation of extracting nth roots. > Q1. Is S a field? (If S is a field then it is a root field, > and then clearly S equals T.) I believe that subsequent taking of the n_th root produces only solvable Galois groups. (the galois group of x^n-a should be commutative) === Subject: Re: [no subject] by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6NF6YW26533; >about Mandelbrot's book re. coastline section & Hausauff fractal >dimension >from Mandelbrots fractal geometry of nature in the Hausdorff fractal >dimension section: >... these lengths are raised to the power D=1, the Euclidean dimension >of a straight line... >do one dimensional lines have to be straight? surely an arc from a >circle or a line with a sharp corner are one dimensional lines? No, but straight lines are one dimensional- he's using the easiest example >...The surface measure of a closed polygon's interior is similarly >calculated by paving it with squares, and adding the squares' sides >raised to the power D=2, the Eucleidean dimension of a plane... >when he says the squares' sides, surely he means *two* *sides* at >right angles to each other of each square, not all four sides? or is >this a new way of getting the area of a square? (that's a serious >question -- having just started to try and learn maths it seems some >things are completely turned on their heads from what little i know >about maths, but sometimes not -- it just sounds like it, and it's >nearly impossible to know which things are being turned on their heads, >and which things are just loosely described) No, he means the length of *one* side of each sub-square: its area is s^2. He can say the squares' sides since they are all of the same length. >...When, on the other hand, the wrong power is used, the result gives >no specific information: the area of every closed polygon is zero, and >the length of its interior is infinite... >what does that mean? surely it depends *which* wrong power is used -- >surely there's a lot of wrong powers (whatever that actually means) that >will result in, i can imagine wrong, but, non-zero results? any >explenation/description of that sentance would be great. No, it doesn't. If you use any power less than d (the dimension of the set) the sequence of area approximations will go to 0. If you use any power larger than d, it does not converge. Only one d will give a non-zero number. >...Let us proceed likewise for a polygonal approximation of a coastline >made up of small intervals of length E. If their lengths are raised to >the power D, we obtain a quantity we may call tentatively an >approximate measure in the dimension D. Since according to Richardson >the number of sides is N=F*E^-D, said approximate measure takes the >value F*E^D*E^-D = F. >how and why does it get from N=F*E^-D to F*E^D*E^-D = F ? E^D is the length of each segment. If there are N segments then the total length is F= N*E. Since N= F*E^(-D), F= (F*E^(-D))*E^(D)= F(E^(-D)*E^D)= F. >also with F*E^D*E^-D = F that's just saying F = F right? with E^D*E^-D >resulting in nothing? (or E^D*E^-D resulting in 1 which when used in >multiplication means nothing? yes i think it results in one.) Yes, E^(-D) is the same as 1/E^D so E^(-D)*E^D= 1. >yes i'd guess it does mean F = F because it then says: >...Thus, the approximate measure in the dimension D is independent of >E. >so you can just ignore all that and use F as is, whatever F is. i still >don't know what F is. can anyone tell me what F is please? *someone* >must know what F is?! :) F is the total length (or area, or volume, depending on the problem). The point is that there exist just one D for which that works and that, even if it is a fraction, is the dimension of the set. >just having an answer to what F is would be great. === Subject: Lagrange Inversion Formula ? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6NF6Yi26563; a time not very long ago someone mentioned the LAGRANGE INVERSION FORMULA, but I could not find the posting. Could you point me to a reference, in the internet or a text book, that features it ? DR === Subject: Re: Lagrange Inversion Formula ? > a time not very long ago someone mentioned the LAGRANGE INVERSION FORMULA, but I could not find the posting. Could you point me to a reference, in the internet or a text book, that features it ? > DR Try Google. If you really meant the Lagrange inversion formula: Google Lagrange inversion: 16700 hits Or, if it was the Lagrange interpolation formula: Google Lagrange interpolation: 38700 hits. There's probably some explanation for you. On the other hand, both expressions yielded a few hits when looking in Google (advanced groups search in sci.math for each of those expressions in subject, sorted by date): Dale === Subject: Re: Lagrange Inversion Formula ? > a time not very long ago someone mentioned the LAGRANGE INVERSION FORMULA, but I could not find the posting. Could you point me to a reference, in the internet or a text book, that features it ? > DR Is this the one about inverting a power series? Here is a reference: Ch. 5, Ex. 33, N. Asmar, Applied Complex Analysis with Partial Differential Equations, Prentice Hall, 2002. === Subject: Re: Point Counting on Elliptic Curves by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6NF6Y826523; >can anyone recommend useful papers that are available on the web for point >counting on elliptic curves. This links to specific point counting method: However, pages 3,4 contain a list of who's who in point counting. To the best of my knowledge this list is more or less up to date. HTH D.L. === Subject: Preparation for Graduate School This fall I will be entering graduate school to begin working on an MA in Mathematics and am seeking your advice as I prepare. I anticipate my studies to be rather broad including both pure and applied topics (though perhaps with an emphasis in pure). I received my undergraduate degree in Math and CS two years ago and, for the past two years, have been working as a web developer. While from a number of perspectives my work has been rewarding, it had not helped to keep my mathematical senses keen. If you were to recommend 3 key concepts to review before returning to school, what would they be? Another way to pose the question would be: what should be the key concepts from a broad (both pure and applied) undergraduate math education? I am looking for concepts that are neither too broad as to be unmanageable (proofs) nor too narrow to not be generally useful (Runge-Kutta Method). A few thoughts I have had are mathematical induction, Fundamental Theorem of Calculus, and Least Squares Approximation. Any thoughts? === Subject: Re: Preparation for Graduate School > This fall I will be entering graduate school to begin working on an MA > in Mathematics and am seeking your advice as I prepare. I anticipate > my studies to be rather broad including both pure and applied topics > (though perhaps with an emphasis in pure). > I received my undergraduate degree in Math and CS two years ago and, > for the past two years, have been working as a web developer. While > from a number of perspectives my work has been rewarding, it had not > helped to keep my mathematical senses keen. > If you were to recommend 3 key concepts to review before returning to > school, what would they be? Another way to pose the question would > be: what should be the key concepts from a broad (both pure and > applied) undergraduate math education? > I am looking for concepts that are neither too broad as to be > unmanageable (proofs) nor too narrow to not be generally useful > (Runge-Kutta Method). A few thoughts I have had are mathematical > induction, Fundamental Theorem of Calculus, and Least Squares > Approximation. > Any thoughts? Make an appointment to talk to the Graduate Student Advisor in the department were you are going to study and talk to him/her about it. If you already know what courses you are going to be taking and what the textbooks are then go ahead and get them and start reading .. === Subject: Re: Preparation for Graduate School > This fall I will be entering graduate school to begin working on an MA > in Mathematics and am seeking your advice as I prepare. I anticipate > my studies to be rather broad including both pure and applied topics > (though perhaps with an emphasis in pure). > I received my undergraduate degree in Math and CS two years ago and, > for the past two years, have been working as a web developer. While > from a number of perspectives my work has been rewarding, it had not > helped to keep my mathematical senses keen. > If you were to recommend 3 key concepts to review before returning to > school, what would they be? Another way to pose the question would > be: what should be the key concepts from a broad (both pure and > applied) undergraduate math education? > I am looking for concepts that are neither too broad as to be > unmanageable (proofs) nor too narrow to not be generally useful > (Runge-Kutta Method). A few thoughts I have had are mathematical > induction, Fundamental Theorem of Calculus, and Least Squares > Approximation. > Any thoughts? > Make an appointment to talk to the Graduate Student Advisor in the > department were you are going to study and talk to him/her about it. If you > already know what courses you are going to be taking and what the textbooks > are then go ahead and get them and start reading .. Your preparation time could be better spent by broadening your vistas and opening up your brain to new things. Once in school, it will be time to focus on specific topics. There will be plenty of help on specific topics, but you are ultimately responsible for acquiring your own breadth of knowledge... One of the keys is to always look for the more general perspective. Mathematics is about general patterns, not specific techniques... Here's what I have to say about what you've suggested for yourself: - Mathematical Induction: Great, but the realm of integers is only a special case. Learn to prove things by induction for other mathematical objects that are recursively defined WITHOUT reducing this first to a problem about integers. Trees are one good example (a tree is either one external node or an internal node with a finite number of offsprings that are trees themselves) if you know that something is true of trivial trees [consisting of just one external node] and can show that it's true of a tree whenever it's true of its subtrees, then you've shown it's true of all trees... Investigate J.H. Conway's Surrreal Numbers which are constructed purely recursively (the aim is more to learn about structural induction than about the numbers themselves, although they ARE interesting). - Fundamental Theorem of Calculus: Fine, but this kind of Calculus is by definition something that should be taken for granted in graduated studies, like long division... Practice a bit if you have to, but that's it. If you want to work out difficult definite integrals like the pros do, get familiar with contour integrals and Cauchy's residue theorem and learn how powerful this simple idea can be. - Least Square Approximations: Again, look for the general pattern that makes this an interesting solution to a practical problem... HINT: You're projecting a point of a larger space onto a simpler one, just like the projection of a point onto a plane is simply the closest point of the plane. There's not much difference between minimizing a [multi-dimensional] Euclidean distance and minimizing a sum of squares, is there? Projections are easy whereas minimizations can be tough, so we are happy when we can reduce the latter to the former... Allow me to plug my own site, which has already helped a number of people along the road you're about to travel: www.numericana.com. Let me know if it helps you... Gerard P. Michon, Ph.D. g.michon@att.net www.numericana.com === Subject: Post-sci-math s.a.v.e 0n Meds. Va.l.ium < XAN@x + V|@gRa > /So:m:a boundary=--8371262958009061523 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i6NFbcH09199; by support2.mathforum.org (8.12.10/8.12.10/The Math Forum, $Revision: 1.6 secondary) with SMTP id i6NFakX2032692; X-Originating-IP: 144.118.94.13 ---------------------------------------------------------------------

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this comm unication in error, you may purge your email address from our database. ----8371262958009061523-- === Subject: Re: Hawking's talk in Dublin GR 17 On the spot report! I want to bring up a very important point. Hawking has never traveled to a black hole; none of the experts on black holes have. They merely speculate upon their nature from the comforts of their own offices. I don't think their theories are very relevant and are just intellectual games. Craig === Subject: Re: Hawking's talk in Dublin GR 17 On the spot report! > I want to bring up a very important point. Hawking has never traveled > to a black hole; none of the experts on black holes have. They > merely speculate upon their nature from the comforts of their own > offices. I don't think their theories are very relevant and are just > intellectual games. The universe does not care what you think... thank the heavens. === Subject: Re: Hawking's talk in Dublin GR 17 On the spot report! In sci.math, Uncle Al <40FFE233.DC10AF82@hate.spam.net>: > Part 1 of several > First of all no one really understood what Hawking was saying in detail. > This included Kip Thorne, John Preskill, Matt Visser and many others. > The jury is still out. Hawking's paper with the details will be out in a > month. > There are several levels of objections: > Add Jacko Sarfatti to head the list of no one really > understood what Hawking was saying. I can't say I do. :-) Of course I'm not sure I really wanted to count the hairs on a black hole anyway. (I certainly don't want to add mine thereto! :-) ) [.sigsnip] -- #191, ewill3@earthlink.net -- insert random event horizon here It's still legal to go .sigless. === Subject: Re: Hawking's talk in Dublin GR 17 On the spot report! > Part 1 of several > First of all no one really understood what Hawking was saying in detail. > This included Kip Thorne, John Preskill, Matt Visser and many others. I would be just happy if I could believe that Hawking actually knew what he was talking about in general. === Subject: Re: Hawking's talk in Dublin GR 17 On the spot report! charset=iso-8859-1 > Part 1 of several > First of all no one really understood what Hawking was saying in detail. > This included Kip Thorne, John Preskill, Matt Visser and many others. > I would be just happy if I could believe that Hawking actually knew > what he was talking about in general. Hmm, and you base this state of depression on having located errors where in his work? Please pray enlighten us.. === Subject: Re: Hawking's talk in Dublin GR 17 On the spot report! Part 1 of several First of all no one really understood what Hawking was saying in detail. > This included Kip Thorne, John Preskill, Matt Visser and many others. > I would be just happy if I could believe that Hawking actually knew > what he was talking about in general. > Hmm, and you base this state of depression on having located errors where in his > work? Please pray enlighten us.. First of all, I am not in a state of depression. As far as I am concerned the situation ranks up there with astrology, and I dont argue with fortune tellers. But I will say that It is sad that mathematics is being used to buttress (or cover-up) such blatant acts of brazen stupidity and cowardice, but then it is well known that enshrined morons respond well to intimidation by fancy mathematics name calling... As long as I don't hear: And therefore God exists I am ok, and will relegate your research to the level of witchcraft . I know that your ice castle is coming down soon. EOD. === Subject: Re: Hawking's talk in Dublin GR 17 On the spot report! Hello crackpot. === Subject: Re: Hawking's talk in Dublin GR 17 On the spot report! charset=iso-8859-1 : > First of all no one really understood what Hawking was saying in detail. : > This included Kip Thorne, John Preskill, Matt Visser and many others. : > I would be just happy if I could believe that Hawking actually knew : > what he was talking about in general. : : Hmm, and you base this state of depression on having located errors where in his : work? Please pray enlighten us.. *Chortle* Hey, for anyone who isn't familiar with....... http://www.mchawking.com/ I'd have to say that my favorite tune here is the Creationists. === Subject: div~. hello....doctor~ 1-(1/2)+(1/3)+(1/4)-(1/5)+(1/6)+(1/7)-(1/8)+(1/9)+.... show that this diverge. ----------------------------- i think...... i used regrouping. {1-(1/2)+(1/3)} + {(1/4)-(1/5)+(1/6)} + {(1/7)-(1/8)+(1/9)} +.... so, 00 sigma [(1/k) - (1/(k+1)) + (1/(k+2))] k=1 >= sigma [(1/(k+2))] = diverge (by integral test) so, regrouping sequence diverge. i know that sigma a_n converge => every regrouping converge. n=1~00 thus, since regrouping diverge, given initial sequence diverge. um......that's right ?? thank you for your advice. thank you very much. === Subject: Re: div~. > hello....doctor~ > 1-(1/2)+(1/3)+(1/4)-(1/5)+(1/6)+(1/7)-(1/8)+(1/9)+.... > show that this diverge. > ----------------------------- > i think...... > i used regrouping. > {1-(1/2)+(1/3)} + {(1/4)-(1/5)+(1/6)} + {(1/7)-(1/8)+(1/9)} +.... > so, > 00 > sigma [(1/k) - (1/(k+1)) + (1/(k+2))] > k=1 That's not right - it should be sum(k=0,oo) [1/(3k+1) - 1/(3k+2) + 1/(3k+3)]. >= sigma [(1/(k+2))] = diverge (by integral test) > so, regrouping sequence diverge. But this idea will work just fine. === Subject: Re: div~. > hello....doctor~ > 1-(1/2)+(1/3)+(1/4)-(1/5)+(1/6)+(1/7)-(1/8)+(1/9)+.... > show that this diverge. > ----------------------------- > i think...... > i used regrouping. > {1-(1/2)+(1/3)} + {(1/4)-(1/5)+(1/6)} + {(1/7)-(1/8)+(1/9)} +.... > so, > 00 > sigma [(1/k) - (1/(k+1)) + (1/(k+2))] > k=1 >= sigma [(1/(k+2))] = diverge (by integral test) > so, regrouping sequence diverge. > i know that > sigma a_n converge => every regrouping converge. > n=1~00 > thus, > since regrouping diverge, given initial sequence diverge. > um......that's right ?? > thank you for your advice. > thank you very much. Looks to me like the series can be represented by the formula (-1)^n(1/(n+1)) Now apply the Alternating Series Test. Dave === Subject: Re: div~. > hello....doctor~ > 1-(1/2)+(1/3)+(1/4)-(1/5)+(1/6)+(1/7)-(1/8)+(1/9)+.... > show that this diverge. > ----------------------------- > i think...... > i used regrouping. > {1-(1/2)+(1/3)} + {(1/4)-(1/5)+(1/6)} + {(1/7)-(1/8)+(1/9)} +.... > so, > 00 > sigma [(1/k) - (1/(k+1)) + (1/(k+2))] > k=1 >= sigma [(1/(k+2))] = diverge (by integral test) > so, regrouping sequence diverge. > i know that > sigma a_n converge => every regrouping converge. > n=1~00 > thus, > since regrouping diverge, given initial sequence diverge. > um......that's right ?? > thank you for your advice. > thank you very much. > Looks to me like the series can be represented by the formula > (-1)^n(1/(n+1)) > Now apply the Alternating Series Test. > Dave It's not an alternating series. === > Why are you posting this off-topic garbage here? Why are you creating a whole new thread about it? > > Interesting. For some reason, you also created a new thread by > prepending a Re: Argh! That was a bit daft :-( > > Based on the stuff I'm seeing on my TTY, you (Robin) didn't > prepend Re: . > You'e right --- I wasn't being so daft after all :-) This started in comp.text.tex, and I won't bore you with all the details. Is it true that every time I reply it starts a new thread? Has everyone here just been so courteous that noone has pointed out that I am doing something wrong in the way that I post? I include the origin of this discussion. > ... > P.S.: You shouldn't start a new thread for an old question. >As for starting a new thread with my font question, are you saying >that I should do a search and find a thread on fonts >from the past and post to it? >As for Google, its strange you should mention it. I was about >to do as you suggested (I used to use trn long before Google >which I liked, so I am using it now. >This is the first complaint I have gotten on either of these points. > Only because your readers in sci.math (myself included) have been > unwontedly courteous to you. Is this true? > In fact, as you are using it (I > at the headers of this post, and you'll see that the only > which I am replying--there is no Message-ID for Harry Schmidt's > post), and thus you are, indeed, starting a new thread with > every one of your posts. It doesn't appear that I am starting a new thread with every post to me. I don't see the relevance of the headers--you clearly know a lot more about this than I. It appears you saying that I should > (At least, with the definition of > thread I sketched in a recent post to sci.math with threads === > prepended to the Subject: header, which if filled out would be the > correct and complete definition of thread; cf. David Ullrich's > recent related posts in sci.math, which refer explicitly to the > relevant RFCs.) > Followups set to sci.math, where this is at least marginally > on-topic (given the possibility of a formal mathematical discussion > of what the thread data-type is or should be, and the high > improbability--though I wouldn't put it past some of the denizens > of c.t.t.--of managing to make such a discussion relevant to TeX). > Lee Rudolph I don't quite understand the last paragraph, but if there is something wrong with the way I post, please let me know. === >Is it true that every time I reply it starts a new thread? they don't always when you reply (as I mentioned in the part of this post I cut...by the way, if you were replying to my comp.text.tex post, then something is *very* strange, since === inherited from it). There's probably a good reason for that Google is calling the service you're using beta, you know. Lee Rudolph === Subject: Re: heavy trolling was almost, but not quite, entirely unlike tea: > Besides which, I've often done dozens of Google searches a day and never > got charged. > Google is *going to start* charging. They ain't > started yet. And once they do, caveat empty. Then fill it! > I think you people are putting me on. > You'll see. > But not like mittens. > I like mittens, except when I need dexterity. > Mittens are warmer than gloves. I prefer ear muffs. They are sexy. -- TimC -- http://astronomy.swin.edu.au/staff/tconnors/ cpu time/usefulness ratio too high -- core dumped. === Subject: Re: heavy trolling was almost, but not quite, entirely unlike tea: > I mean, some of the best trolling I've ever seen consisted > in nothing more than the bland assertion of a patent absurdity. > > Even on Usenet I find it difficult to believe that anyone would be so > gullible as to respond to a bland assertion of a patent absudity. If > the assertion is patently absurd, only a true naif would respond. > Either that, or an egocentric person with a need to show himself or > herself as somehow intellectually superior to the poster of the patent > absurdity. Seems unlikely, though. There are 4 ways to makes a bland assertion. And since 4 is prime, I am obviously correct. > its search engine? You'll get some small number of free searches per > day (I think 10), and after that you have to use Microsoft's Hotmail > Billing System to pay some sort of micro-payment per search. > Unbelievable! The bastards! -- TimC -- http://astronomy.swin.edu.au/staff/tconnors/ I bet the human brain is a kludge. -- Marvin Minsky === Subject: Re: heavy trolling <40FEE8C7.1030805@netscape.net> <6uqtf09sj1mohc0bjdeap34ipv2700b4jr@4ax.com> day (I think 10), and after that you have to use Microsoft's Hotmail > Billing System to pay some sort of micro-payment per search. > Unbelievable! > The worst part is that they don't even tell you this up front. It's in > the terms of use agreement that nobody reads. They are going to bill > your ISP without even telling you, and then your ISP will pass the bill > along to you. > Besides which, I've often done dozens of Google searches a day and never > got charged. > I think you people are putting me on. > But not like mittens. Dork. --Jeremy -- Jeremy Impson jdimpson can be contacted at acm dot org http://impson.tzo.com/~jdimpson === Subject: Re: For all p q distinct odd primes exists n, (2^n)p+q prime > Can this conjecture > For all p q distinct odd primes exists n, (2^n)p+q prime > be proven, disproven, or related to a standard problem? > I think it's false, and not too hard to show that it's false. > It's known that there are odd integers k such that 2^n + k > is never prime, and k can be taken to be prime. Also, it seems that for p=16985143930825825363784428577 (the smallest prime Brier number in [1]), we have: for all n, (2^n)p+1 and (2^n)p-1 are composite. While this does not yet disprove the conjecture, it is a strong hint that the conjecture is false, and can be proven so with a counterexample. Now who will exhibit one? Fran.8dois Grieu [1] Yves Gallot: A search for some small Brier numbers === Subject: Re: For all p q distinct odd primes exists n, (2^n)p+q prime wrotd: >While this does not yet disprove the conjecture, it is >a strong hint that the conjecture is false, and can be >proven so with a counterexample. >Now who will exhibit one? It can be shown that 78557*2^n + 1 is divisible by one of 3, 5, 7, 13, 19, 37, and 73, (depending on the value of n mod 36). So I guess you can find an explicit counterexample by finding primes p, q with p = 78557 (mod m) and q = 1 (mod m) for m = 3*5*7*13*19*37*73. None come immediately to mind... but, of course they exist. === Subject: Re: For all p q distinct odd primes exists n, (2^n)p+q prime > wrotd: >While this does not yet disprove the conjecture, it is >a strong hint that the conjecture is false, and can be >proven so with a counterexample. >Now who will exhibit one? >It can be shown that 78557*2^n + 1 is divisible by one of 3, 5, 7, 13, >19, 37, and 73, (depending on the value of n mod 36). So I guess you >can find an explicit counterexample by finding primes p, q with p = >78557 (mod m) and q = 1 (mod m) for m = 3*5*7*13*19*37*73. None come >immediately to mind... but, of course they exist. p = 1401087257 and q = 700504351 will do, I believe. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Recantation on inettia > I've been saying right along that the first law of physics defines > inertial motion; but not inertia: > I just got a new physics textbook that quotes Newton's first law as: > If no outside force acts on a body, it will continue at rest or will > continue to move .... For some reason it finally struck me that _this > is_ the definition of inertia! > Up to now, what I've been defining as inertia is not the _concept_ of > inertia, but it's value, or quantity; which is expressed in units of > mass. > A body's inertia is what opposes any force with an equal and opposite > force, and the value, or quantity of a body's inertia is a ratio of > the force that it exerts, and/or is exerted on it, divided by the > acceleration that it causes. This ratio of force divided by > acceleration is a measure of the amount of mass - or quantity of > matter - in the body; which is the _quantity_ of inertia. > Force and weight are related to each other as follows: The ratio of > any body's force to its accelertion is equal in magnitude to the ratio > of its weight to the acceleration at which it will free fall. > Mathematically: f/a = w/g > Force = weight/acceleration of free fall times the > acceleration: > Weight = force/acceleration times the acceleration of free > fall: > Mathematically: f = (w/g)a, and w = (f/a)g. > *Plonk* David's right, Sam. *Plonk* is the sound made when somebody is dropped into your killfile. This is the THIRD time you've killfiled Shead this month. What kind of game are you playing here? Killfile the troll ONCE and be done with it. Please. Jake === Subject: Re: Recantation on inettia > I've been saying right along that the first law of physics defines > inertial motion; but not inertia: > I just got a new physics textbook that quotes Newton's first law as: > If no outside force acts on a body, it will continue at rest or will > continue to move .... For some reason it finally struck me that _this > is_ the definition of inertia! > Up to now, what I've been defining as inertia is not the _concept_ of > inertia, but it's value, or quantity; which is expressed in units of > mass. Sort of like a little light bulb going on in your head? Whatever you're drinking, drink more of it! Double-A === Subject: Re: Recantation on inettia > A body's inertia is what opposes any force with an equal and opposite > force, > Confusing Newton's second law with Newton's third law. Par for the course. > Newton's third law is not about inertia resisting an impressed force. > It is the simple observation that when you push on a wall, the > wall pushes back on you. Nah! Walls don't push back. They just resist until the impressed force exceeds their design strength. When you push your hands together, the > force of your right hand on your left matches the force of the > left hand on the right. Newton's third law applies no matter whether > the wall is made of balsa or of lead. It applies no matter whether > you are wearing a steel gauntlet on your right hand or your left. > Newton's second law is the one that gives a quantitative relationship > between force, inertia and motion. Make that the quantitative relationship between Displacement [distance displaced]; Force [and weight], and Time [periods during which force endures and displacement continues]; so that the ratio of impressed force [f], divided by the rate of displacement per second [s/t^2] is the measure of the material matter which the force is acting and/or acted upon: Like the inertia of one unit of matter [mass] is the result of one pound of force acting for one second; causing one foot of displacement per each sucessive second that it is maintained; which is equal to thirty-two pounds of force acting for one second; causing thirty-two feet of displacement for each successive second that it is maintained... > John Briggs === Subject: Re: Fractions v natural numbers in by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6NICLR29369; >Roger Penrose's new book The Road to Reality arrived today. >In the preface (p. xviii) he is talking about fractions. We find: >The magic is that the idea of a fraction actually works despite the fact >that we do not really directly experience things in the physical world that >are exactly quantified by fractions - pieces of pie leading only to >approximations. (This is quite unlike the case of natural numbers, such as >1, 2, 3, which do precisely quantify numerous entities of our direct >experience.) I presume that you mean one can't have 1/2 of a pie because it is impossible to measure EXACTLY 1/2 pie. But if you talk about measurement, it is also impossible to have exactly 1 pie if (say) a whole pie is defined by some physical characteristic (e.g. weight). According to this viewpoint one might argue: Suppose you have a pile of wood; ostensibly a collection of 6 2 x 4's. How many 2 x 4's do you REALLY have? Each of those pieces of lumbar is only an APPROXIMATION to a 2 x 4. How close the approximation is depends on how much care was taken in cutting them. However, with enough granularity of measurement, even 1 inch is an approximation. In the real world, EVERYTHING is an approximation. :-) === Subject: Re: Fractions v natural numbers in >Suppose you have a pile of wood; ostensibly a collection of 6 2 x 4's. >How many 2 x 4's do you REALLY have? Each of those pieces of lumbar >is only an APPROXIMATION to a 2 x 4. How close the approximation is >depends on how much care was taken in cutting them. Maybe not the best example: a (trimmed) 2 x 4 is approximately 1.5 inches by 3.5 inches. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Fractions v natural numbers in >Roger Penrose's new book The Road to Reality arrived today. >In the preface (p. xviii) he is talking about fractions. We find: >The magic is that the idea of a fraction actually works despite the >fact that we do not really directly experience things in the physical >world that are exactly quantified by fractions - pieces of pie leading >only to approximations. (This is quite unlike the case of natural >numbers, such as 1, 2, 3, which do precisely quantify numerous entities >of our direct experience.) > I presume that you mean one can't have 1/2 of a pie because it is > impossible to measure EXACTLY 1/2 pie. But if you talk about > measurement, it is also impossible to have exactly 1 pie if (say) a > whole pie is defined by some physical characteristic (e.g. weight). > According to this viewpoint one might argue: > Suppose you have a pile of wood; ostensibly a collection of 6 2 x 4's. > How many 2 x 4's do you REALLY have? Each of those pieces of lumbar > is only an APPROXIMATION to a 2 x 4. How close the approximation is > depends on how much care was taken in cutting them. A 2 x 4 does not deserve that name once it has been milled. IIRC, it is then, at least in theory, actually a 3/2 x 7/2 instead. DWC === Subject: Re: real area venn diagram by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6NICNX29474; ... >Sorry. You have one sphere. 3 circles on the sphere, >intersecting when appropriate, creating at most 8 areas on >the sphere. On one side of a circle is positive, other >side is negative. Move and resize the circles appropriately >to get the areas for the 8 parts in corresponding ratio to >the number of elements in each subset. >For 1 and 2 circles (i.e. 1 and 2 properties, it seems >obvious to me that you can get any ratio of 2 and 4 subsets >that you want. For 3, it seems likely but I can't quickly >come up with a way of calculating the coordinates of the >circles from the 8 values. Doesn't look likely. Assume all eight areas are nonzero. Prob(A) determines the radius for circle A; likewise B, C. Prob(A intersect B) then determines the distance between the centers of A and B; likewise the other two pairs. The resulting triangle (formed by the centers of the circles) is rigid, and uniquely determines Prob(A intersect B intersect C), which is no good. Put another way, six parameters (diameters of three circles, distances between each pair of centers) completely determine the Venn diagram, but it should have seven degrees of freedom. === Subject: Re: Probability by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6NICNk29457; > What can we say about the distribution of a random variable X if > E(1/X) = 1/E(X) >Assuming X is positive, we may conclude X is constant >(= deterministic). I used Cauchy-Schwarz for that. >For negative X, same conclusion. >What about signed X? Or complex X? Signed X: Prob(X=-1)=1/2. Prob(X=3+sqrt(8))=Prob(X=3-sqrt(8))=1/4. E(X)=E(1/X)=1. === Subject: help needed badly by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6NICMu29402; For n>=1, let An be the number of ways to write n as as ordered sum of positive integers where each summand is at least 2. (For example, A5=3, because here we may represent 5 by 5,by 2+3, and by 3+2.)Find and solve a recurrence relation for An. This sum is #12 of ex10.2 from ralph grimaldi's'discrete and combinatorial maths(4th ed) Would the solution be a(n)=a(n-1)+a(n-2), a1=1,a2=1? === Subject: Re: help needed badly > For n>=1, let An be the number of ways to write n as as ordered sum > of positive integers where each summand is at least 2. (For example, > A5=3, because here we may represent 5 by 5,by 2+3, and by 3+2.) Find > and solve a recurrence relation for An. > This sum is #12 of ex10.2 from ralph grimaldi's discrete and > combinatorial maths (4th ed) Fix your newsreader; your carriage return key doesn't seem to be firing right. > Would the solution be a(n)=a(n-1)+a(n-2), a1=1,a2=1? Of course not! But given that we /know/ that A1 is 0, A2 is 1, A3 is 1, A4 is 2, A5 is 3, and A6 is 5, it's a reasonable guess that perhaps A(n) = A(n-1) + A(n-2); A(1) = 0, A(2) = 1. ^^^^^^^^^ The proof is pretty trivial; just consider what happens if you split a number like this: 2, ... and what happens when you split a number like this: n, ... for n > 2. (Hint: the latter case is the same as 1+m, ... for m = n-1, m >= 2.) HTH, -Arthur === Subject: Re: help needed badly > For n>=1, let An be the number of ways to write n as as ordered sum of > positive integers where each summand is at least 2. (For example, A5=3, > because here we may represent 5 by 5,by 2+3, and by 3+2.)Find and solve > a recurrence relation for An. This sum is #12 of ex10.2 from ralph > grimaldi's'discrete and combinatorial maths(4th ed) > Would the solution be a(n)=a(n-1)+a(n-2), a1=1,a2=1? It does not look right. Consider a(1) = 1 a(2) = 1 a(3) = a(2) + a(1) = 1 + 1 = 2 a(4) = a(3) + a(2) = 2 + 1 = 3 a(5) = a(4) + a(3) = 2 + 3 = 5 -- Lance Lamboy Go F*ck Yourself ~ Dick Cheney === Subject: Re: Modular arithmetic question > Is there some simple method for finding integers n s.t. n^2 = 1 (mod k) > for a given k? Perhaps I should say, 'efficient' too to rule out > testing all values from 0 to k-1. > When I did a little course on number theory at university this question > came up (with some specific value for k, IIRC it was a 4 digit prime). I > didn't get it then (nor did my supervisor) and it has bugged me > occasionally ever since. > I think you want to reword your question. Let n == 1 mod k. Then n^2 == > 1 mod k for all k. I have trivially satisfied your query. Do you mean to > find all integers n with that property? > If a is relatively prime to k then a^{phi(k)}==1 mod k. For example > 2^4==1 mod 5. Let t be such that phi(k)=2*t and n=a^t. Then n^2==1 mod > k. So if you know phi(k), it should be easy to find other values of n. The n's found this way may be only part of the solution set. Take k=105 for example. phi(105)=48, fi(105)/2=12/2=6. Number of solutions of n^2== 1 mod 105 is 48/6 = 8, but a^6 == 1 or 64 mod 105. Four numbers are known 1,64,41,104. The other four solutions have to be found some other way (perhaps by examining a^3 when a^6==1 mod 105). > snip === Subject: Re: Modular arithmetic question > snip > For larger numbers the question becomes how many solutions are there > for the equation x^2-1=0 mod k where k is composite. Ah it's a lot easier than I thought. Phi(k) gives the number of relatively prime residues. Fi(k)/2 gives the number of relatively prime quadratic residues. Phi(k)/(fi(k)/2) gives the number of solutions for each quadratic residue. === Subject: Re: Modular arithmetic question > Its slightly better than that, you only need test half the number (d-1)/2 (d > odd) - see reply. This is simply because the roots 'n' are symmetric about > the mid-point k/2 as per all residues since for any x But for each jp there are two numbers to check jp-1 and jp+1. So in total there are 2(d-1) numbers to check but symetry says 2(d-1)/2 = d-1. Unless I missed something. Hanford === Subject: F(b)- F(a) question Consider an interval (a,b], and a function F which is increasing and right-continuous everywhere. If {(a_j, b_j]} is a countable partition of (a,b], then it can be shown that: F(b) - F(a) = Sum (j=1 --> oo) [F(b_j) - F(a_j)] How does one prove that this holds for (-oo, b] and (a, oo), using the respective limits for F at -oo and +oo? All help and suggestions appreciated. === Subject: Errata in a Nutshell?? > ... > The final questioner asked him what problem he intended to tackle next, > now that he had solved the paradox of the black hole. > ``I don't know,'' Hawking quickly replied, > bringing the house down with laughter. .... With 9 milion copies out.... how about Errata in a Nutshell. === Subject: Algebraic Topology Question: Let S1 and S2 be two simplicial complexes. If S1 and S2 are homotopy equivalent, and Si (i=1,2) is homotopy equivalent to a wedge of ni (i=1,2) spheres, then is it true that n1=n2? (excume me, if the question is stupid!!?, All the best Alireza Abdollahi === Subject: Re: Algebraic Topology Question: > Let S1 and S2 be two simplicial complexes. If S1 and S2 are homotopy > equivalent, and Si (i=1,2) is homotopy equivalent to a wedge of ni > (i=1,2) > spheres, then is it true that n1=n2? Yes, it's true. Otherwise, a wedge of spheres of dimension n_1 would be homotopy equivalent to a wedge of spheres of dimension n_2, with say n_1 < n_2. But this yields a contradiction: the n_2 dimensional homology of the first wedge is 0, whereas the n_2 dimensional homology of the second wedge is a direct sum of as many copies of Z as there are spheres in the second wedge. (I am assuming you mean nonempty wedges.) === Subject: Re: Solution to Random Walk in a Unit HyperSphere > I developed a new (as far as I know) solution for random walk > in a unit hypersphere. I am waiting your opinions. what is a typical application? alex === Yes, I see. I am now using the other posting option of is less flexible. that I am starting a new thread, or that my posts are different than anyone else's. (I don't worry about the headers.) Van >Is it true that every time I reply it starts a new thread? > they don't always when you reply (as I mentioned in the part > of this post I cut...by the way, if you were replying to my > comp.text.tex post, then something is *very* strange, since === > inherited from it). I was replying to your post in the sense that I was thinking of it, but I decided to move to sci.math and find the topic you referred to, and I replied to Robin Chapman's post in this thread. (sci.math is the only newgroup I look at regularly, I almost missed your reply in comp.text.tex). After that I found a thread with your post in it, posted yesterday I think, which would have been a more appropriate thread, but I didn't see it until to late, and didn't want to delete my post and repost to the other thread. > There's probably a good reason for that Google is calling > the service you're using beta, you know. > Lee Rudolph === Yes, I see. I am now using the other posting option of is less flexible. that I am starting a new thread, or that my posts are different than anyone else's. (I don't worry about the headers.) Van >Is it true that every time I reply it starts a new thread? > they don't always when you reply (as I mentioned in the part > of this post I cut...by the way, if you were replying to my > comp.text.tex post, then something is *very* strange, since === > inherited from it). I was replying to your post in the sense that I was thinking of it, but I decided to move to sci.math and find the topic you referred to, and I replied to Robin Chapman's post in this thread. (sci.math is the only newgroup I look at regularly, I almost missed your reply in comp.text.tex). After that I found a thread with your post in it, posted yesterday I think, which would have been a more appropriate thread, but I didn't see it until to late, and didn't want to delete my post and repost to the other thread. > There's probably a good reason for that Google is calling > the service you're using beta, you know. > Lee Rudolph === Subject: Re: Ito integration > I'd be gratful if someone provided me with an example of adopted > function which is not predictible. What is the intuition of > predictible sets? A rather simple one is the Poisson counting process adapted to its natural filtration. Its unpredictability (or, equivalently, the unpredictability of its jumps) is its fundamental feature. Intuitively, a predictable function (or process) is one where it's always possible to tell where the process will be *at* time t given information up to but not including t. Any process with continuous or even caglad (left continuous, right-limited) paths is predictable. Because of the left continuity, you know where the process will be at time t by looking at what point it's heading for at times approaching t from below. On the other hand, for a Poisson counting process, information up to time t will tell you when the process has jumped before time t, but nothing before time t will tell you whether there's going to be a jump *at* time t or not. -- Kevin === Subject: Re: Ito integration > Let your filtration be that generated by a one-dimensional standard > Brownian motion B(t), suitably completed. The process Z(t) defined as > follows is adapted (even progressively measurable) but not predictable: > Z(t) = 1, if B(t) = 0 and there exists e > 0 such that B(s) = 0 for all > s in the interval (t,t+e). > Z(t) = 0 for all other t. I'm not sure but it looks like Z=0 almost surely. ??????????? > The non-predictability of Z follows from the strong Markov property. === Subject: Re: Ito integration > Let your filtration be that generated by a one-dimensional standard > Brownian motion B(t), suitably completed. The process Z(t) defined as > follows is adapted (even progressively measurable) but not predictable: > > > Z(t) = 1, if B(t) = 0 and there exists e > 0 such that B(s) = 0 for all > s in the interval (t,t+e). > > Z(t) = 0 for all other t. > I'm not sure but it looks like Z=0 almost surely. > ??????????? Indeed, P[ Z(t) = 0 ] = 1 for each t > 0, but P[ Z(t) = 1 for uncountably many t > 0 ] = 1. -- A. === Subject: Re: Ito integration > Let your filtration be that generated by a one-dimensional standard > Brownian motion B(t), suitably completed. The process Z(t) defined as > follows is adapted (even progressively measurable) but not predictable: > > > Z(t) = 1, if B(t) = 0 and there exists e > 0 such that B(s) = 0 for all > s in the interval (t,t+e). > > Z(t) = 0 for all other t. > I'm not sure but it looks like Z=0 almost surely. > ??????????? > Indeed, > P[ Z(t) = 0 ] = 1 > for each t > 0, but > P[ Z(t) = 1 for uncountably many t > 0 ] = 1. Actually, I think Kuba was pointing out a small typo in your definition. I think you *meant* to have Z(t)=1 if B(s) is *non*-zero for all s in some interval (t,t+e). -- Kevin === Subject: Re: Ito integration > Let your filtration be that generated by a one-dimensional standard > Brownian motion B(t), suitably completed. The process Z(t) defined as > follows is adapted (even progressively measurable) but not predictable: > > > Z(t) = 1, if B(t) = 0 and there exists e > 0 such that B(s) = 0 for all > s in the interval (t,t+e). > > Z(t) = 0 for all other t. > > I'm not sure but it looks like Z=0 almost surely. > ??????????? > Indeed, > > P[ Z(t) = 0 ] = 1 > > for each t > 0, but > P[ Z(t) = 1 for uncountably many t > 0 ] = 1. > Actually, I think Kuba was pointing out a small typo in your > definition. I think you *meant* to have Z(t)=1 if B(s) is *non*-zero > for all s in some interval (t,t+e). -- A. === Subject: Re: Non math question. What kind of eraser do you use > I am a paper conversator looking into the properties of the I-Z Cleaner > yours? We ours in Japan and have not been able to locate any information > on them in the states. Never used one, never needed one -- Paul Townsend I put it down there, and when I went back to it, there it was GONE! Interchange the alphabetic elements to reply === Subject: Re: [Ancient_Egypt_Group] The rise and fall of civilizations boundary=------------060600060004040905050800 --------------------------------------------------------------------- The xi are the measurable variable of a civilization: major categories are economic, population, resources, mood,i,1,...,n , M an nbyn nonlinear matrix dx(i)/dt=M.x(i) M is the nonlinear matrix of the culture structure. The cycles are very like R.9assler or Lorenz chaotics... In most cases the structure cycle seems to be a year of years : 365 years This number was derived from an analysis of ancient history dates like the fall of Egypt, the fall of Babylon, the fall of Persia. the fall of Rome , etc. It seems to be more influence by solar and climatic variables than actual civilization. Global warming and similar trends do seem to accelerate the process. Over fishing of cod, tuna and anchovies fisheries also seem to indicate that environmental effects of over use of resources are also a heavy indicator. It is said that the African deserts and middle eastern are a result of 'civilization. In my Allometric approach I use: M=N*A^3-K*A^2 where the powers are based on the fish weight type anabolic and catabolic ( building of and breaking down of biological structure: a healthy culture is healthy physically/ biologically: volume to surface area related). The straight solution is a population saturation sigmoid, but bifurcation cycles add the chaotic effects. Older civilizations were less in control of their destinies. As for the challenge and response argument: In 1974-5 the oil /gas lines made it clear our days on oil were numbered. We have failed in 30 years to respond to that challenge. It really isn't a very hard theory and basic versions of this kind of math are used in adjusting interest rates in Federal control of banking in the USA based on economic measured fluctuations. Although the stock market type predictions are the most well documented, other people like the Club of Rome have been following this kind of mathematical production for a long time. A Russian in the 30's named Nicholai D. Kondratieff became famous for his prediction of the great depression with his long waves. with a 58 year average period. His cartoon is very much like Dr. Mandelbrot's in it's form and they both resemble viscoelastic solid models like that of Maxwell and Voight. The treatment of historical cycles of all types as mathematical simulation systems isn't new but it is useful. And it is very much a hard earned modern science. > Civilization is a movement and not a > condition, a voyage and not a harbor. > Arnold Toynbee > Arnold Toynbee's A Study of History is a ten-volume > analysis of the rise and fall of human civilizations. > The British historian Arnold Joseph Toynbee (1889-1975) > includes virtually all civilizations --the Egyptian, > the Sumeric, the Mayan, the Iranian, the Japanese, the > Hellenic, etc. -- in his monumental work. > Toynbee's thesis is that all societies rise and fall > through a process of challenge and response. The > rise of a civilization is a response to a challenge, > a challenge which must be just right: Too little > and the civilization does not rise at all, too great > and the civilization will collapse. Thus when a > civilization is able to respond to challenges, it grows. > When it fails to respond to a challenge, it enters its > period of decline. According to Toynbee, these challenges > may be physical, as when the Minoans conquered the sea; > or social, as when Athens reacted to the Persian onslaught. > This is something I write in my book in the section 5.3 > about catastrophes as catalysts: A catastrophe can be > a challenge for a complex adaptive system to grow in > complexity. If this challenge is too low, adaption > or adaptation are not necessary, if it is too high, > the agent is unable to adapt itself to the new situation > and is destroyed. If it is just right, the challenge > can be a catalyst to reach new levels of complexity. > According to Toynbee's view, if a challenge from the > outside strengthens a civilization, the civilization > can dominate nearby cultures. If a challenge from the > outside weakens a culture, too much, it is eventually > subjugated by nearby civilizations. > Why can the same threat to a society from the outside > either increase or decrease a civilization's integrity ? > A catastrophe or a threat causes bustling activity (similar > to the activity in an ant colony if it is disturbed). > If there is a place, a river or a oasis where conditions > are still acceptable, people will gather at these points, > especially if there is no culture at all, and there is > nothing to lose. In this case a challenge can increase > integration and unity, it can melt a group together. > If there is already a culture, and the challenge is > small, the threat draws off the attention from internal > problems, and helps to stabilize the civilization. > If there is already a culture, the challenge is huge > and there is nowhere to go and no further potential for > improvement, a huge challenge can cause disintegration > of a culture. Again a catastrophe or a threat causes > bustling activity, but there is no place to go, since > people are already at the best place. People will disperse > in all directions, and the culture collapses. > Arnold J. Toynbee (1889-1975) > http://en.wikipedia.org/wiki/Arnold_J._Toynbee > A Study of History > Arnold J. Toynbee > Oxford University Press, 1934-1961 > Yahoo! Groups Sponsor > ADVERTISEMENT > click here > > ------------------------------------------------------------------------ > Yahoo! Groups Links > * To visit your group on the web, go to: > http://groups.yahoo.com/group/Ancient_Egypt_Group/ > > * To unsubscribe from this group, send an email to: > Ancient_Egypt_Group-unsubscribe@yahoogroups.com > > > * Your use of Yahoo! Groups is subject to the Yahoo! Terms of > Service . -- Respectfully, Roger L. Bagula tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : URL : http://home.earthlink.net/~tftn URL : http://victorian.fortunecity.com/carmelita/435/ --------------060600060004040905050800-- === Subject: Re: [Ancient_Egypt_Group] The rise and fall of civilizations What does the fall of a civilization mean to the ordinary people? 1) no jobs 2) no food/ starvation 3) unable to support their family 4) increase in violence and crime 5) many children die young 6) alcohol and drug abuse increase 7) fewer schools and libraries 8) upper class people are killed by riots 9) population levels decrease abruptly 10) religion/ superstition increases and sometimes takes control 11) no running water or electricity in cities 12) no public transportation 13) no trains or airplanes between far cities 14) telephone communication breaks down I could go on like this for a long time, but I think you get the idea. There is a reason that they call it chaos > The xi are the measurable variable of a civilization: > major categories are > economic, population, resources, mood,i,1,...,n , M an nbyn nonlinear > matrix > dx(i)/dt=M.x(i) > M is the nonlinear matrix of the culture structure. > The cycles are very like R.9assler or Lorenz chaotics... > In most cases the structure cycle seems to be a year of years : 365 years > This number was derived from an analysis of ancient history dates like > the fall of Egypt, > the fall of Babylon, > the fall of Persia. > the fall of Rome , etc. > It seems to be more influence by solar and climatic variables than > actual civilization. Global warming and similar trends > do seem to accelerate the process. Over fishing > of cod, tuna and anchovies fisheries also seem to indicate that > environmental effects of over use of resources are also > a heavy indicator. It is said that the African deserts and middle eastern > are a result of 'civilization. > In my Allometric approach I use: > M=N*A^3-K*A^2 > where the powers are based on the fish weight type anabolic and catabolic > ( building of and breaking down of biological structure: > a healthy culture is healthy physically/ biologically: volume to surface > area related). > The straight solution is a population saturation sigmoid, but > bifurcation cycles > add the chaotic effects. > Older civilizations were less in control of their destinies. > As for the challenge and response argument: > In 1974-5 the oil /gas lines made it clear our days on oil were numbered. > We have failed in 30 years to respond to that challenge. > It really isn't a very hard theory and basic versions > of this kind of math are used in adjusting interest rates > in Federal control of banking in the USA based on economic > measured fluctuations. > Although the stock market type predictions are the most well documented, > other people like the Club of Rome have been following this kind of > mathematical production for a long time. > A Russian in the 30's named Nicholai D. Kondratieff became famous for > his prediction of the great depression > with his long waves. with a 58 year average period. His cartoon is > very much like Dr. Mandelbrot's > in it's form and they both resemble viscoelastic solid models like that > of Maxwell and Voight. > The treatment of historical cycles of all types as mathematical > simulation systems isn't new > but it is useful. And it is very much a hard earned modern science. > Civilization is a movement and not a > condition, a voyage and not a harbor. > Arnold Toynbee > Arnold Toynbee's A Study of History is a ten-volume > analysis of the rise and fall of human civilizations. > The British historian Arnold Joseph Toynbee (1889-1975) > includes virtually all civilizations --the Egyptian, > the Sumeric, the Mayan, the Iranian, the Japanese, the > Hellenic, etc. -- in his monumental work. > Toynbee's thesis is that all societies rise and fall > through a process of challenge and response. The > rise of a civilization is a response to a challenge, > a challenge which must be just right: Too little > and the civilization does not rise at all, too great > and the civilization will collapse. Thus when a > civilization is able to respond to challenges, it grows. > When it fails to respond to a challenge, it enters its > period of decline. According to Toynbee, these challenges > may be physical, as when the Minoans conquered the sea; > or social, as when Athens reacted to the Persian onslaught. > This is something I write in my book in the section 5.3 > about catastrophes as catalysts: A catastrophe can be > a challenge for a complex adaptive system to grow in > complexity. If this challenge is too low, adaption > or adaptation are not necessary, if it is too high, > the agent is unable to adapt itself to the new situation > and is destroyed. If it is just right, the challenge > can be a catalyst to reach new levels of complexity. > According to Toynbee's view, if a challenge from the > outside strengthens a civilization, the civilization > can dominate nearby cultures. If a challenge from the > outside weakens a culture, too much, it is eventually > subjugated by nearby civilizations. > Why can the same threat to a society from the outside > either increase or decrease a civilization's integrity ? > A catastrophe or a threat causes bustling activity (similar > to the activity in an ant colony if it is disturbed). > If there is a place, a river or a oasis where conditions > are still acceptable, people will gather at these points, > especially if there is no culture at all, and there is > nothing to lose. In this case a challenge can increase > integration and unity, it can melt a group together. > If there is already a culture, and the challenge is > small, the threat draws off the attention from internal > problems, and helps to stabilize the civilization. > If there is already a culture, the challenge is huge > and there is nowhere to go and no further potential for > improvement, a huge challenge can cause disintegration > of a culture. Again a catastrophe or a threat causes > bustling activity, but there is no place to go, since > people are already at the best place. People will disperse > in all directions, and the culture collapses. > Arnold J. Toynbee (1889-1975) > http://en.wikipedia.org/wiki/Arnold_J._Toynbee > A Study of History > Arnold J. Toynbee > Oxford University Press, 1934-1961 > Yahoo! Groups Sponsor > ADVERTISEMENT > click here > > ------------------------------------------------------------------------ > Yahoo! Groups Links > * To visit your group on the web, go to: > http://groups.yahoo.com/group/Ancient_Egypt_Group/ > * To unsubscribe from this group, send an email to: > Ancient_Egypt_Group-unsubscribe@yahoogroups.com > > > * Your use of Yahoo! Groups is subject to the Yahoo! Terms of > Service . -- Respectfully, Roger L. Bagula tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : URL : http://home.earthlink.net/~tftn URL : http://victorian.fortunecity.com/carmelita/435/ === Subject: Re: [Ancient_Egypt_Group] The rise and fall of civilizations >What does the fall of a civilization mean >to the ordinary people? Wildly off-topic posts in sci.math? Lee Rudolph === Subject: Re: I am playing OT a bit but the subject line reminds me of a conversation I had in work with the boss's daughter. I was actually in the middle of thrashing out the finer points of a database for a point-of-sale system, but how do you explain that to a seven-year-old? The conversation went something like this: She: What are you doing? I: I'm working. She: No you're not, you're playing! The boss seemed to believe his daughter rather than his employee :) -- Paul Townsend I put it down there, and when I went back to it, there it was GONE! Interchange the alphabetic elements to reply === Subject: Re: Imaginary primes? : > Hi all, > I just read something about imaginary numbers factoring prime numbers, > like: > (1+4*i)*(1-4*i) = > 1 - 4*i + 4*i - 16*(i^2) = > 1 - 16*(-1) = > 1 + 16 = 17. > This was new to me. OK, I'm not a mathematician... > I guess maybe all normal primes could somehow be factored like that? > Can they? If so, could one say, primes as we know them do not exist when > you allow for imaginary factors? Or are there numbers that remain prime > nonetheless? > But: are there imaginary primes, numbers of the form (a + b*i), that > can *not* be factored into factors having the form (x + y*i) with x and > y being integer numbers? > Just curious. > M. If you define a complex integer a+ib as even or odd according to whether the real integer a+b is even or odd, you will find: (a) Addition and multiplication of complex integers follow the same odd/even combination rules as for real integers (b) There is only one even prime (1+i) -- Paul Townsend I put it down there, and when I went back to it, there it was GONE! Interchange the alphabetic elements to reply === Subject: Re: Imaginary primes? > I just read something about imaginary numbers factoring prime numbers, > like: > > (1+4*i)*(1-4*i) = > 1 - 4*i + 4*i - 16*(i^2) = > 1 - 16*(-1) = > 1 + 16 = 17. > > This was new to me. OK, I'm not a mathematician... > > I guess maybe all normal primes could somehow be factored like that? > Can they? If so, could one say, primes as we know them do not exist when > you allow for imaginary factors? Or are there numbers that remain prime > nonetheless? > Over the ring of Gaussian (complex) integers, rational primes of the form 4*n + 3 do not factor (remain prime) but all other rational primes factor in the form (a + bi)(a - bi) for rational integers a and b. E.g., 3, 7, 11, 19, 23, etc., do not factor over the gaussian integers, but 2 = (1 + i)( 1 - i), 5 = (2 + i)( 2 - i), 13 = (3 + 2i)(3 - 2i), etc. === Subject: Re: graphs and tours Hi > let's say there is a graph with 6 nodes and each of the nodes have 10 > edges incoming. > Such graphs are exceedingly numerous, so that does not specify > a unique graph. I mean to say that the graph is fully connected. but it appears to me that the fully connected graphs only have undirected edges between the nodes as well as have self loops. I want a graph without self loops that is connected using two directed edges incoming from all the other nodes. (color one edge red the other blue) so, all the nodes should have 5 red edges from the other 5 nodes and 5 blue edges from the same 5 nodes. then all 6 nodes have 10 incoming edges total > so each of the nodes have two edges incoming from each of the other > nodes in the graph. > Are you assuming this to be the case, or is it given? > If given, the problem should say the graph contains two > distinct copies of the complete graph on 6 nodes. yes... this is given. I think two distinct copies of the complete graph is what I mean. maybe one red edges and blued edges the other. > how many possible tours are there? > Don't know. If the graph isn't connected, zero. Also, define > tour, because if tours are denoted by lists of nodes you will > get a different tour count than if denoted by lists of edges. by tour I mean visiting each node exactly once using the available edges exactly once (whether blue or red edge is used first isn't relevant, but they are different edges) I'm not sure if that is what tour is... maybe i mean hamiltonian paths... Sorry for such newbie q's. > How do i figure this out? > First disambiguify problem. i hope that makes it clearer. > -jiw ames === Subject: Re: Automorphisms of complex numbers >What is the cardinality of the automorphism group of the complex >numbers, C? >I say 2^c, where c:=|C| Right, you get full marks for this. I assume that you mean the structure (C; 0, 1, +, x), i.e., the complex field. This field is built up by starting with the rational field as its prime subfield. Then we add c transcendentals. Then we take the algebraic closure. Any permutation of those c transcendentals (the Hamel basis) generates an automorphism of the field. So we get 2^c automorphisms. But only *two* of those automorphisms are continuous, viz. the identity and the complex conjugates. --Herb Enderton === Subject: Proof of group theory (G/Z cyclic implies G abelian) it? Please be rigorous! Let G be a group and Z(G) the center of G. If G/Z(G) is cyclic, prove G is abelian. For simplicity denote Z(G)=Z. By assumption G/Z = for some g in G. Since the cosets of Z in G form a partition of G then we can write G= U Za (U stands for union, a in G). Since Za is an element of G/Z= then Za=Z*g^i for some integer i. Therefore G = U Z*g^i where i is in Z. Let a, b be in G we want to show that ab=ba. If a lies in G then a is an element of Z*g^j for some integer j. Similarly if b is in G then b is in Z*g^k for some integer k. This implies that a=z*g^j where z is in the Z , and b=z'*g^k with z' in Z. But then ab=(z*g^j)(z'*g^k)=z'*g^(j+k)*z'=z'*g^k*z*g^j=ba. (Since z and z' are elements of the center of G). Thus ab=ba for all a,b in G, whence G is abelian. === Subject: Re: Can you find anything wrong with this solution to the Halting Problem? | | >No, you still don't get it. Let me try again. THERE IS NO | >CORRECT ANSWER ON THIS CLASS OF PROBLEMS. Period. Punct. | > . C'est suisse, et tres, tres precis. | > | > http://home.att.net/~olcott/halts.html | > | > That is all well and good, but, then how do you | > explain that no one had been able to correctly | > refute this proof? | > let me say this one last time | > you didnt solve the halting problem | | You didn't bother to read my proof. If you bothered to read | my proof you would not be sore sure that I did not solve the | Halting Problem. Twenty people have been trying to refute it | for five days. So far none has succeeded. Most argue points | that would have nothing to do with a correct refutation. Peter, 'edens morgan...etc' is clever and knowledgeable. His 'meow arf...' exterior hides an intellect which I reckon is the best which has ever hit talk.bizarre. Do you have anything useful to add to this discussion, Peter? Nor do I. -- )>==ss$$%PARR(.bc> Parr === Subject: Re: Can you find anything wrong with this solution to the Halting Problem? | | > | >No, you still don't get it. Let me try again. THERE IS NO | >CORRECT ANSWER ON THIS CLASS OF PROBLEMS. Period. Punct. | . C'est suisse, et tres, tres precis. | | http://home.att.net/~olcott/halts.html | | That is all well and good, but, then how do you | explain that no one had been able to correctly | refute this proof? Kid spoke trollspeak -- )>==ss$$%PARR(.bc> Parr === Subject: Re: Can you find anything wrong with this solution to the Halting Problem? | > If Olcott is right, then he's setting his sights far too low. A method for | > creating software whose output cannot be accessed by other software would be | > valuable in other ways besides disproving the Halting Problem. The MPAA, | > for instance, would be very interested. Perhaps someone should put him in | > touch with them. | > -- | > Matthew Skala | > mskala@ansuz.sooke.bc.ca Embrace and defend. | > http://ansuz.sooke.bc.ca/ | | Why would this be so difficult? | done is that encryption is powerful enough to not require the | additional expense of special hardware. Which returns us nicely to my observation, an observation which you have not bothered to reply to yet, that you have discovered a principle which I have now patented and will maake me famous and rich, stinking rich. -- )>==ss$$%PARR(.bc> Parr === Subject: Re: Can you find anything wrong with this solution to the Halting Problem? | It is obvious <...> | Please make sure that | you read the section on Possible Bases for Refutation | before responding. Apparently, your silly little list does not permit me to live. Because I am not permitted to live, I cannot read your silly little list. Because I cannot read your silly little list, I am ignorant of the contens of that silly little list. If you follow me so far, please nod. I didn't see you nodding. -- )>==ss$$%PARR(.bc> Parr === Subject: Re: Can you find anything wrong with this solution to the Halting Problem? <...> | All the modification of my program was (under the new | set of conditions) totally fruitless. Yep. -- )>==ss$$%PARR(.bc> Parr === Subject: Re: Can you find anything wrong with this solution to the Halting Problem? | | > Seriously, man, forget the Halting Problem. Even if by some miracle you | > manage to convince the entire anal-retentive comp sci establishment, big | > freaking whoop. The absolute most you could get for that would be a Fields | > Medal, which only impresses geeks. You can't sell it for any significant | | I have my reasons. -- )>==ss$$%PARR(.bc> Parr === Subject: Re: Can you find anything wrong with this solution to the Halting Problem? | In any case since this does not address any | of the points that I made it is not any form of valid refutation. any arbitrary... Sheesh! We dont have to put up with this kook, but we will until 30 July. Thereafter, end. -- )>==ss$$%PARR(.bc> Parr === Subject: Re: Can you find anything wrong with this solution to the Halting Problem? | | > And no one has correctly pointed out any mistakes | > of mine. | > Well, that is incorrect Peter. | > It is also a deliberate lie. | > Well that is actionable libel. | > No, it isn't. I was raised by a lawyer, and I've | > been on the net dealing with kooks like you who | > think every true description of their kook behavior | > is libel, for almost 20 years now. Somehow all | > that hot air has not resulted in a single lawsuit. | | yet. At last, we meet someone has bested Kent!!!! But why do so on usenet when the money's in the lawcourts? -- )>==ss$$%PARR(.bc> Parr === Subject: Re: Can you find anything wrong with this solution to the Halting Problem? | | > Well, that is incorrect Peter. | > It is also a deliberate lie. | Well that is actionable libel. | If you can prove me wrong, then | have at it, otherwise just shut up! libel tee hee and tickle my goolies with a feather duster -- )>==ss$$%PARR(.bc> Parr === Subject: Re: Can you find anything wrong with this solution to the Halting Problem? <...> | Which of course would make it not work. It is most difficult to make a | thing that can not possibly exist, work correctly. Allow me to correct your statement, please, Peter. I will write computerese as you do not understand human languages Your statement should read: It is to make a thing that can not possibly exist, work correctly Isn't this where we came in? -- )>==ss$$%PARR(.bc> Parr === Subject: Re: Can you find anything wrong with this solution to the Halting Problem? <...> | It is not really obvious that two functions that compute the same | thing using the same method derive different results merely | because of a difference in their return value. When you see | how this directly effects the processing of the input, then it | becomes more clear. No it's not. Please eludidate. -- )>==ss$$%PARR(.bc> Parr === Subject: Re: Can you find anything wrong with this solution to the Halting Problem? | That is very good, you made an excellent attempt. This is probably | the very best attempt so far. G Frege did a pretty good job too, | but, I think that yours is better. I've never before encountered a troll who is patronising. Gloat Peter, gloat. Your gloating says, to those who think calmy and logically, and incidently, to me too, that you think you are playing a game with us. Or are we playing a game with you? As I've stated before, Peter, you have created a concept which will, because I have copyrighted it, make me famous and rich, stinking rich. You thought you were playing a game. Yes you were playing a game, child. Now tell me whether I am playing games. -- )>==ss$$%PARR(.bc> Parr === Subject: Re: Can you find anything wrong with this solution to the Halting Problem? | http://home.att.net/~olcott/halts.html | | Just walk through an execution trace table and see for yourself. | Use the concrete examples provided on this link. See how the | execution trace table differs when the WillHalt() function is run | using the LoopIfHalts() function as input, on each of these two | different variations of the WillHalt() function. | | (1) bool WillHalt(M, w) | (2) void WillHalt(M, w) | | In the first case WillHalt can not correctly determine if LoopIfHalts() | will halt or not. In the second case LoopIfHalts() won't even compile | because it is a syntax error to check the return value of a function that | has no return value. Sorry Peter, That did not answer my question. -- )>==ss$$%PARR(.bc> Parr === Subject: Re: Can you find anything wrong with this solution to the Halting Problem? | | http://www.netaxs.com/people/nerp/automata/halting2.html | | Try walking through an execution trace of LoopIfHalts() | using these two versions of WillHalt() | (1) bool WillHalt() | (2) void WillHalt() Your programme ended up in an indeterminate state. Could you clarify what happened afterwards, please? Did it halt, or did it loop indefinitely? -- )>==ss$$%PARR(.bc> Parr === Subject: Re: Can you find anything wrong with this solution to the Halting Problem? | > Now suppose you created such a WillHalt program, call it P1. Suppose I then | > take a copy of P1, and make a small change to it so that its output _is_ | > accessible to other programs -- I return the output rather than writing it | > to the screen. Call this new program P2. Obviously P2 is susceptible to | > Turing's proof, so P2 cannot exist. But your claim is that P1 can exist, so | > it must be the case that it's impossible to modify P1 as I've described. | | No you can go ahead and modify it, and all you succeed in doing | is making another modified version that will no longer work correctly. | This does not change the fact that the original unmodified version | will continue to produce correct results every time So you now claim that there are an infinity of programs which your magic program cannot handle. OK I can deal with that. Can you? -- )>==ss$$%PARR(.bc> Parr === Subject: Re: Can you find anything wrong with this solution to the Halting Problem? | | > You are able to correctly refute what I am saying | > without even reading a single word. | > That is exactly so. | > I hate to see a grown kook cry; take it easy. <...> | I usually never resort to giving back the rudeness | that I am dealt, but you are a presumptuous ass. Kent is not an ass. Kent is not a donkey. Kent is not a zebra. Kent is not a pony. Kent is not a horse. In fact, I'm convinced that Kent is not a member of the equine family at all. Now trot off into your stable where you belong, foal. -- )>==ss$$%PARR(.bc> Parr === Subject: Re: Can you find anything wrong with this solution to the Halting Problem? | | edens morgan mair fheal greykitten tomys des anges | > if you succeed at that youll be famous for refuting the church-turing thesis | See if you can find any reason why this would not work. | http://home.att.net/~olcott/halts.html There are 3 fundamental errors. See if you can find one. -- <.bc)&&PARR##sss==<( === Subject: Re: Can you find anything wrong with this solution to the Halting Problem? > > You'd have to ask my quadrapelegic friend Jesse Sanchez-Velasquez. > I'm asking you. Beating up quadraplegics with a nerf bat, or going > on and on about the kookiness of people like Olcott, is kooky in > itself. Wouldn't you agree? That would make you one of the great > kooks on the net. >Simply stating the facts should be enough for a scientific subject. Hi Eray! Please forgive me for using your post as sort of a template to chronicle events making this story of Torkel popping out of nowhere to ask these apparently pointless questions of daddy Kent, add up. Needless to say, the following are all honest opinions based on reasonable inference. Links and detail available on request, corrections appreciated and gladly made. Email works too if you zap zap this and part. 1. Months ago, Dolan gently flamed Torkel in c.a.p. Actually not a flame, more of a correction, easily justified. No doubt immediately groups count, is *slighted*! He slinks away silently, wounded but apparently still breathing and thinking, Ve vill meet again! 2. Somewhat later, Larry (yours truly) offers a short description of Goedel's theorem to a friend in sci.logic of a few sentences which, lo and behold, do not exactly explicate the theorem in precise theoretical detail! This, ostensibly, attracts the attention of Torkel with the to think troll here if they like.) 3. Author of the short Goedel piece (can you guess?): Kent Paul Dolan. 4. Ooooh, the plot thickens! 5. But Larry doesn't know anything about the Torkel-Dolan agenda yet, or who the hell Torkel is. Torkel is planning to embarrass Dolan 6. Torkel posts a nice suggestion to Larry about the piece. But it is a ruse, a trap, designed to embarrass Dolan via Larry. Larry sees the thing that doesn't look right, makes a qualification just in case, but makes a very nice reply to Torkel as contract for friendly dialog. 7. Torkel stupidly springs the trap anyway, snipping the qualification. Why would he do that to Friendly Larry The Nobody? Made no sense. Anyway. Larry unsnips and uses the misfired ruse to flame the hell out of Torkel. Torkel learns fast not to post content to Larry because it turns out Larry is some kind of logic razor guy, able to chop Torkel content into little pieces while hopping on one foot and telling jokes. 8. So instead of that, Torkel, now aka the sarcasm guy, continues with short cryptic questions, like at the top of this post, thus risking nothing, simply to aggravate Larry. 9. Larry still doesn't know about the c.a.p. Dolan-Torkel thing yet, so he and two siblings royally flame Torkel into oblivion, a very old and long-running game in my family in various venues including decades of nightclub performance, and with targets many notches more challenging than Torkel. Just old times for us. 10. Torkel thinks the et al are sock puppets , but can't tell for sure. The computer professor from Sveden hasn't figured out yet that Larry and Milt develop and market an Email / Usenet reader among other things, have 10 servers, 50 nics, bogus headers, etc., often sit side-by-side and alternately use the same boxes, etc., and poor ol' Torkel ain't never gonna figure it out without a court order, even from Sveden, and probably not even then. (Not to mention some other associates playing this round-robin game mostly for the benefit of next years sales forecast, but also the two Johns - Ashcroft and Poindexter, and our proclivity to post political content under different but sometimes rotated nics.) 11. There is controversy with a few other posters, but that is a side issue except for that fact that *not one* heavy hitter in sci.logic joins the debate on Goedel - substantiating that the criticism of the Dolan piece is obviously trivial and way out of context of theoretical accuracy for such an exercise in short-copy, which even then did raise the question in my mind: What's this Torkel, no slouch in mathematical logic, really up to?. Finally, Torkel slinks away again, almost, and remarkably *almost* admitting defeat on the Goedel question as the word apparently creeps into his remarks - can you believe it? Larry et al go to Red Snapper to celebrate their newfound status as mathematical logician wizards! 12. But remember, Larry The Nobody is only the vehicle at hand. The real goal is Dolan, top Usenet gun to beat for every new gunslinger in town, and author of the original c.a.p. *slight*. 13. Enter Peter Olcott, kook. Torkel takes a look at Olcott and - in a greater display of stupidity than I've seen in a long time considering he is some sort of computer professor - posts that Olcott is a Larry sock puppet!!! . Well, what's funny there is that you could inscribe what I know about Turing's Halting Problem on a typical CPU with a magic marker, and what I *care* about it with a large paintbrush. 25+ years of developing software for a living, and no sign of Turing, and very little chance of ever meeting the guy! 14. Torkel soon realizes that embarrassing error, as does everyone else, no doubt including half the planet in Torkel's imagination. Not surprisingly, no more about Olcott sockpuppet for Larry! Torkel slinks away again in embarrassment. Hmmn, what's next from this guy? 15. Dolan flames Ocott, identifies him as a kook, which Larry and one other had done previously as well, beating up on Olcott quite more than Dolan without a peep from Torkel who is certainly reading. 16. Enter Torkel again with the quote at top, now with heartfelt sorrow for poor Peter Olcott and quadraplegics everywhere, asking his cryptic, safe, no-risk, apparently pointless, 3rd grade questions of daddy Kent, infernal star poster of everywhere, immutably silent thru the whole sci.logic affair and even now, obstinately and - damn that Kent! - apparently *purposefully* inaccessible to Torkel's revenge fantasy. 17. For what reason are these questions at the top, you might ask? Beats me! This guy must really love getting slammed! Everyone who's been watching this, and of course zillions of spectators in Torkel's mind, anxiously awaits the next big Torkel-clobberfest. The only question is, how? 18. How, is starting to become obvious. The atom bomb of Usenet - ignore him completely. Thus the perennial loser, Torkel, sits looking at his stupid grade-school questions, now only too obviously snipping at the tails of his intellectual superior, just sitting there at the top of this post, forever lonely, no doubt aware that I and everyone else following this is looking as well, and waiting for good ol' Torkel to slink away once more, muttering, That Dolan dared to ignore *me* the great Torkel again! and, one infers, planning his next revenge... Well, he can always take a shot at me as a consolation prize, except that I have him plonked of course! Unless you again do me the service of quoting the latest developments in this important affair. Being an avid reader of misc.misc, I should notice any of your crossposts to there. Watch: true to form, I predict Torkel will snip all but one or two sentences, and ask one of his no-risk, Larry-safe, Dolan-safe, questions. Oops, there I go infecting the experiment again! >Discussion about kooks is kookish indeed. Uh oh. Then I'm in trouble! Nice talking to you again. Also again, apologize for snipping the rest. I like both you and Dolan. I don't want to get involved in your ongoing Usenet chat, though I've been aware of it. There's a big difference between you and Torkel. Intellectual honesty, good faith, detail offered for scrutiny and criticism, and context to allow people to know where you're coming from. I will always one of the fastest learners! (I liked the elephants in the Usenet sewer quote - just so. That's one thing we can all agree on!) Larry