mm-5530 === Subject: R.A. Heinline on Mathematics and those who cannot cope with it. Robert A. Heinlein said: * Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable sub-human who has learned to wear shoes, bathe, and not make messes in the house. o Robert Heinlein This is particularly applicable to Lester Zick, who I am certain wears shoes and does not make messes in his house. Bob Kolker === Subject: Re: R.A. Heinline on Mathematics and those who cannot cope with it. > Robert A. Heinlein said: * Anyone who cannot cope with mathematics is not fully human. At > best he is a tolerable sub-human who has learned to wear shoes, bathe, > and not make messes in the house. > o Robert Heinlein > I seem to recall that this was said by one of Heinlein's fictional characters, not by Heinlein himself. The author is not bound to agree with everything he makes his characters say. -- Chris Henrich http://www.mathinteract.com God just doesn't fit inside a single religion. === Subject: Re: R.A. Heinline on Mathematics and those who cannot cope with it. > Robert A. Heinlein said: > * Anyone who cannot cope with mathematics is not fully human. At > best he is a tolerable sub-human who has learned to wear shoes, bathe, > and not make messes in the house. I seem to recall that this was said by one of Heinlein's fictional > characters, not by Heinlein himself. The author is not bound to agree > with everything he makes his characters say. Yes, but we can assume that a 2,000-year old man did not survive all those years by tolerating uneducated fools. === Subject: Re: R.A. Heinline on Mathematics and those who cannot cope with it. > Robert A. Heinlein said: > * Anyone who cannot cope with mathematics is not fully human. At > best he is a tolerable sub-human who has learned to wear shoes, bathe, > and not make messes in the house. > I seem to recall that this was said by one of Heinlein's fictional > characters, not by Heinlein himself. The author is not bound to agree > with everything he makes his characters say. Yes, but we can assume that a 2,000-year old man did not > survive all those years by tolerating uneducated fools. Wait, are we talking about Heinlein or Mel Brooks? -- If you like high adventure, come with me. If you like the stealth of intrigue, come with me. If you like blood and thunder, come with me. But first listen to a word from our sponsor. -- Adventures by Morse === Subject: Re: R.A. Heinline on Mathematics and those who cannot cope with it. > Wait, are we talking about Heinlein or Mel Brooks? Anything said by a Heinlein character can be attributed to Heinlein. He is the ventriloquist. Bob Kolker === Subject: Re: R.A. Heinline on Mathematics and those who cannot cope with it. > Robert A. Heinlein said: * Anyone who cannot cope with mathematics is not fully human. At > best he is a tolerable sub-human who has learned to wear shoes, bathe, > and not make messes in the house. > o Robert Heinlein This is particularly applicable to Lester Zick, who I am certain wears > shoes and does not make messes in his house. Bob Kolker funny, I always visualize Lester Zick as bag lady. === Subject: Re: R.A. Heinline on Mathematics and those who cannot cope with it. > Robert A. Heinlein said: * Anyone who cannot cope with mathematics is not fully human. At > best he is a tolerable sub-human who has learned to wear shoes, bathe, > and not make messes in the house. > o Robert Heinlein This is particularly applicable to Lester Zick, who I am certain wears > shoes and does not make messes in his house. > Heinlein was an elitist and as such had this simplistic view of linear ranking. He was caressing his own ego. And did You hear the undertone of :...and when not at best, he is an intolarable sub-human...? Times did change for better and nowadays we can look at humans with their brain-diversity in a non-linear and a non frightening way. With friendly greetings Hero === Subject: Re: R.A. Heinline on Mathematics and those who cannot cope with it. > Robert A. Heinlein said: * Anyone who cannot cope with mathematics is not fully human. At > best he is a tolerable sub-human who has learned to wear shoes, bathe, > and not make messes in the house. > o Robert Heinlein > This is particularly applicable to Lester Zick, who I am certain wears > shoes and does not make messes in his house. Bob Kolker Do you have any proof for that? -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: R.A. Heinline on Mathematics and those who cannot cope with it. > Do you have any proof for that? None. I have no way of knowing if Zick wears shoes or makes a mess in his house. I was being (uncharacteristically) charitable. Bob Kolker === Subject: Re: R.A. Heinline on Mathematics and those who cannot cope with it. > Robert A. Heinlein said: * Anyone who cannot cope with mathematics is not fully human. At > best he is a tolerable sub-human who has learned to wear shoes, bathe, > and not make messes in the house. > o Robert Heinlein This is particularly applicable to Lester Zick, who I am certain wears > shoes and does not make messes in his house. Bob Kolker You may be making way too many assumptions about him. === Subject: Re: Linear functionals on C(X) <5703119.1194883877381.JavaMail.jakarta@nitrogen.mathforum.org>, It is know that the continuous linear functionals > on C_c(X), the space of > continuous compactly supported functions on a > compact space X are given by > Borel measures. If X is already compact, the _c is redundant. Inspired by this i have the following question: > Could one give a counterexample when it is > impossible to represent an > element of dual, say C_R(X),the space of continuous > functions(not necessary > having a compact support) as integrating wrt to a > Borel measure ? Perhaps what you mean is the continuous, bounded > real-valued functions on the > reals, with supremum norm? Then you can use the > Hahn-Banach theorem to get > linear functionals P such that lim inf_{x -> infty} > f(x) <= P(f) <= lim sup_{x > -> infty} f(x) for all f. In particular, P(f) = 0 if > f has compact support. > Of course, these can't be constructed explicitly. > They certainly can't be > represented as integrating wrt a measure. you apply Hahn-Banach theorem here to get existence of such a functional > P(f). Let C_b be the set of bounded continuous functions on R, and let C_L = {f in C_b: lim_x->oo f(x) exists}. C_L is a closed subspace of C_b under the supremum norm. For f in C_L, define T(f) = lim_x->oo f(x). Then T is a bounded linear functional on C_L, which by Hahn-Banach extends to a bounded linear functional on C_b. If this extended T were given by a complex Borel measure mu on R, then for every f in C_b with compact support we would have int_R f dmu = 0. By the Riesz Representation theorem, mu = 0. But T is obviously not the zero functional; therefore T is not given by such a measure. === Subject: Re: Linear functionals on C(X) > <5703119.1194883877381.JavaMail.jakarta@nitrogen.mathf > orum.org>, > It is know that the continuous linear > functionals > on C_c(X), the space of > continuous compactly supported functions on a > compact space X are given by > Borel measures. If X is already compact, the _c is redundant. Inspired by this i have the following question: > Could one give a counterexample when it is > impossible to represent an > element of dual, say C_R(X),the space of > continuous > functions(not necessary > having a compact support) as integrating wrt to > a > Borel measure ? Perhaps what you mean is the continuous, bounded > real-valued functions on the > reals, with supremum norm? Then you can use the > Hahn-Banach theorem to get > linear functionals P such that lim inf_{x - infty} > f(x) <= P(f) <= lim sup_{x > -> infty} f(x) for all f. In particular, P(f) = > 0 if > f has compact support. > Of course, these can't be constructed explicitly. > They certainly can't be > represented as integrating wrt a measure. give an idea how exactly > you apply Hahn-Banach theorem here to get existence > of such a functional > P(f). Let C_b be the set of bounded continuous functions on > R, and let C_L = > {f in C_b: lim_x->oo f(x) exists}. C_L is a closed > subspace of C_b > under the supremum norm. For f in C_L, define T(f) = > lim_x->oo f(x). > Then T is a bounded linear functional on C_L, which > by Hahn-Banach > extends to a bounded linear functional on C_b. If > this extended T were > given by a complex Borel measure mu on R, then for > every f in C_b with > compact support we would have int_R f dmu = 0. By the > Riesz > Representation theorem, mu = 0. But T is obviously > not the zero > functional; therefore T is not given by such a > measure. === Subject: Re: Linear functionals on C(X) > <5703119.1194883877381.JavaMail.jakarta@nitrogen.mathforum.org>, > It is know that the continuous linear functionals > on C_c(X), the space of > continuous compactly supported functions on a > compact space X are given by > Borel measures. If X is already compact, the _c is redundant. Inspired by this i have the following question: > Could one give a counterexample when it is > impossible to represent an > element of dual, say C_R(X),the space of continuous > functions(not necessary > having a compact support) as integrating wrt to a > Borel measure ? Perhaps what you mean is the continuous, bounded > real-valued functions on the > reals, with supremum norm? Then you can use the > Hahn-Banach theorem to get > linear functionals P such that lim inf_{x -> infty} > f(x) <= P(f) <= lim sup_{x > -> infty} f(x) for all f. In particular, P(f) = 0 if > f has compact support. > Of course, these can't be constructed explicitly. > They certainly can't be > represented as integrating wrt a measure. exactly > you apply Hahn-Banach theorem here to get existence of such a functional > P(f). Let C_b be the set of bounded continuous functions on R, and let C_L = > {f in C_b: lim_x->oo f(x) exists}. C_L is a closed subspace of C_b > under the supremum norm. For f in C_L, define T(f) = lim_x->oo f(x). > Then T is a bounded linear functional on C_L, which by Hahn-Banach > extends to a bounded linear functional on C_b. If this extended T were > given by a complex Borel measure mu on R, then for every f in C_b with > compact support we would have int_R f dmu = 0. By the Riesz > Representation theorem, mu = 0. But T is obviously not the zero > functional; therefore T is not given by such a measure. I might just add that ||T|| = 1 and T(1) = 1 imply the condition lim inf_{x -> infty} f(x) <= T(f) <= lim sup_{x -> infty} f(x). Note first that if f >= 0, 0 <= ||f|| - f <= ||f|| so ||f|| - T(f) = T(||f|| - f) <= || ||f|| - f || <= ||f|| and thus T(f) >= 0. Thus if f >= g, T(f) >= T(g). Now if r = lim sup_{x -> infty} f(x) and epsilon > 0, there is g in C_L with g >= f and lim_{x -> infty} g(x) < r + epsilon. Therefore T(f) <= T(g) < r + epsilon. Take epsilon -> 0 to get T(f) <= r. Apply to -f to get the bound with the lim inf. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Fermat's Last Theorem <33027786.1194887724334.JavaMail.jakarta@nitrogen.mathforum.org> .................................. > ** > Great counterexample! > Obviously this is much older in Bassam's head than > what he thought at > first...*sigh* Tonio > Pd . What will his explanations be? Perhaps Z has to > be a number that > does NOT end with a 2, or maybe p has to be a prime 3? I have already made my final statement where you certainly can see it What else can you add? Nothing, I guess! B.Karzeddin- ****** Well, first of all *Plonk!*, and then: ** Bassam's First Conjecture: If (X & Y) are two odd coprime positive integers, then the following integer equation doesn't have any solution in all integer numbers Z^P = X^P + Y^P + 2*N*X*Y*Z Where (P) is odd prime number, (Z & N) are positive integers [Nov 10, 3:12 PM] ** Counterexample: Let x = 19, y = 21, z = 52, p = 3, n = 3. > Marcus. [Nov 10, 6:32 PM] ** Bassam's answer And Bassam's Second Conjecture Nice, May be I have forgotten to add that (N is prime to P) necessary condition I will verify and come back with more explanations since it is too old in my head [Nov 11, 5:48 PM] ** Counterexample: Here's a counterexample where n is relatively prime to p: x = 1, y = 7, z = 172, p = 3, n = 2113 quasi [Nov 11, 7:46 PM] ** Basaam's answer and Bassam's Third Conjecture: You can't find a counter example to the following integer equation: Z^P = X^P + Y^P + N*X*Y*Z Where: (X, Y, Z,P) are coprime positive integers pairwise Z is even positive integer P is odd prime number N is non-negative even integer that is divisible by P [Please do notice this time there is no 2 in the last memeber of the right side] *** Counterexample: Right -- I missed that. But later I posted another example which avoids that issue. quasi [Nov 12, 1:27 PM] ** Bassam's answer and, believe it or nut(s), Bassam's Fourth Conjecture: Fast & Unbelievable May be I have forgotten to mention the last condition: that is N is only twice an odd positive integer & gcd(N/2,XYZ)=1 [Nov 12, 3:47 PM]...*Pant, pant!* Bassam, yah havivi, you're right: I've nothing else to add to all the stuff above. Tonio === Subject: Re: $50 for arXiv endorsement > I will pay $50 to the person which will endorse me for arXiv preprint. > http://www.mathematics21.org/algebraic-general-topology.html and want to put them also to arXiv at math.GN category. Sadly nobody > endorses me. I have already contacted five endorses, but without > reply. I have the last choice to offer money. There is a complexity behind this offer however. I do not know whether > arXiv's server will inform me who endorses me, and so may not know to > whom pay the money if several persons will claim that endorsed me. For a solution I will consider two cases: > 1. arXiv will inform me the name of the endorser - in this case I will > pay $50 to that person. > 2. arXiv will not inform me about the name of the endorser - for > possibility of this case I will apply the algorithm below. ** Algorithm ** 1. I offer to send me letters saying I want to consider to endorse > you. 2. If arXiv informs me about the endorser, pay to that person. 3. Otherwise select the first letter I want to consider to endorse > you. and reply to that person, Yes, please endorse me. > then make the decision whether to endorse them (based, dependently on > to gain $50). Afterwards the person in question is expected to endorse > me and to send me the letter OK, I have endorsed you. In reply to > that last letter I will send $50. 4. If this person does not reply as expected, repeat this with the > second, third, etc. person until success. ** Below is the endorsement letter ** (Victor Porton should forward this e-mail to someone who's registered > as > an endorser for the math.GN (General Topology) subject class of > arXiv.) math.GN section of arXiv. To tell us that you would (or would not) > like > to endorse this person, please visit the following URL: http://arxiv.org/auth/endorse.php?x=8KIXKG If that URL does not work for you, please visit http://arxiv.org/auth/endorse.php and enter the following six-digit alphanumeric string: Endorsement Code: 8KIXKG ***** Hey! Aren't you the guy who was offering money to people that'll nominate you for some maths prize (perhaps the Wolf prize)?? Privyiet, moi druk!! I see you've now some slightly more humble aspirations: to be simply endorsed for the ArXiv....well, good luck! Tonio === Subject: Re: #274 Heckman and Tribble are Cantor fools and why Cantor's diagonal is all wet; new textbook: Mathematical Physics (Reals & Counting Numbers/AP-adics Primer) for age 6 years onward (1) The Reals are All Possible Digit Arrangements rightward > (2) The Counting Numbers are All Possible Digit Arrangements leftward > (3) The Cantor Diagonal can *never work* on All Possible Digit Arrangements > (4) To apply the Cantor Diagonal on Reals or Counting Numbers means > the person is daft or stupid. > (5) For the Counting Numbers have to be All Possible Digit Arrangements because > that is what makes them countable !!! How you may ask? Because All > Possible Digit Arrangements orders the Counting Numbers because every digit > gives the existence of a ** predecessor and successor**. Let's apply (1) and (5) to an actual example. What is the predecessor and successor of .333... within the list of All Possible Digit Arrangements? What are the numbers that come before and after 1/3 in your sequence of all possible numbers? === Subject: #278 All Possible Digit Arrangements gives order and pattern to the Reals and makes them Countable; new textbook: Mathematical Physics (Reals & Counting Numbers/AP-adics Primer) for age 6 years onward (1) The Reals are All Possible Digit Arrangements rightward > (2) The Counting Numbers are All Possible Digit Arrangements leftward > (3) The Cantor Diagonal can *never work* on All Possible Digit Arrangements > (4) To apply the Cantor Diagonal on Reals or Counting Numbers means > the person is daft or stupid. > (5) For the Counting Numbers have to be All Possible Digit Arrangements because > that is what makes them countable !!! How you may ask? Because All > Possible Digit Arrangements orders the Counting Numbers because every digit > gives the existence of a ** predecessor and successor**. Let's apply (1) and (5) to an actual example. What is the predecessor > and successor of .333... within the list of All Possible Digit > Arrangements? > What are the numbers that come before and after 1/3 in your sequence > of all possible numbers? I take it you mean the Real number 0.33333..... and I believe I addressed that in a previous post. The predecessor of the Real Number 0.3333....... is 0.33333......3333332 and the successor is 0.333.......33333334 Because the Reals and the Counting Numbers are All Possible Digit Arrangements directly makes them Countable since digits are ORDERED so the ordering of digits forces the Arrangements into a Countable Pattern where you simply add one higher digit or subtract one lower digit from the EndView. All very simple that a bright 6 year old should fly through this with his/her parents guidance. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: f(x)=exp(-1/x^2)) >I wonder if someone hint me how to find (d^n/dx)f(x). By far the easiest way is to notice that the k-th derivative of f(x) is of the form P_k(1/x)*f(x). These polynomials can be generated recursively by P_0(x) = 1, P_{k+1}(1/x) = (2/x^3)*P_k(1/x) + d(P_k(1/x))/dx. So P_1(1/x) = 2/x^3, P_2(1/x) = 4/x^6 - 6/x^4, P_3(1/x) = 8/x^9 - 36/x^7 + 24/x^5, etc. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: radius of convergenceI >I'm finding a radius of convergence functional series sum((1/ >(2n)!)g_n(t)*x^2n ), where g_n(t)=(D^n)a(t) for a(t) : (D^k)a(0) = 0 >for any k =0,1.. So I'm trying use Ratio test but i think it is wrong. >Please get me same hints. If you have a function which is analytic in a region except for isolated singularities, the radius of convergence at any point of a valid power series about that point is the distance to the nearest singularity. If the function, as a function of a real variable, has the properties you indicated, and is not 0, then 0 is an essential singularity, and any kind of power series one gets at 0 will be meaningless. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: 1^2 =3, Discovered Your computer returned zero, because existing computers(C Program) were configured to understand 1^2 = 1, not=3. Hence, 2(1^2 - 1)/2 = 0 1^2 = 3, is a new discovery. As of now, the open-minded approach is the best tool necessary to understand this new truth. When the 'qubit' generation of computers are eventually built, humans will be able to peer beyond the limitations of our present computers, into the inner isosceles structure of the physical units that natural numbers represent. Quantum computers will be able to capture the comprehensive structures of physical units, and when they do, humanity will come to understand why and how Square One Equals Three.-Aiya-Oba. === Subject: Re: #276 Where Cantor went abysmally wrong; new textbook: Mathematical Physics (Reals & Counting Numbers/AP-adics Primer) for age 6 years onward > The number 43333.....333333 has predecessor and successor as > 43333.....3333332 and 433333.....33333334. So why is this difficult for you to grasp Victor? Explain to me what 4333.....3333 means. I can sort of understand ......3333: that number has a 3 on the i-th position counting leftward, for all i. So that's infinite, and I guess with adics you can define the semantics of that sort of numbers. But I don't see what the dots in 4333....3333 stand for. Are there an infinite number of digits on the dots? Where is the 4? Is that the i-th digit from the right for some value of i? Victor. -- Victor Eijkhout -- eijkhout at tacc utexas edu === Subject: Looking for Fluid Mechanics Book The book is Fluid Mechanics Fundamentals and applications by Yunus === Subject: Finding the width of a convex polytope along a given dimension hello, does anyone know of algorithms (efficient or otherwise) for finding the width of a convex polyhedron/polytope along an axis (ie the maximum distance between projections of the extreme points onto an axis) ? it seems to me the problem is somehow related to extreme points and cheers === Subject: Re: Finding the width of a convex polytope along a given dimension > hello, does anyone know of algorithms (efficient or otherwise) for > finding the width of a convex polyhedron/polytope along an axis (ie the > maximum distance between projections of the extreme points onto an axis) > ? it seems to me the problem is somehow related to extreme points and > cheers Is what you are looking for not simply max{} - min{} where the p_i are the vertices and v is the direction of the axis? hagman === Subject: Re: eigen values,history and applications to problems in all engine3ering fields and applied maths > I wish to know ,who had first given the idea of eigen values. Is > there a list of problems which are solved by egien values like those > in vibrations.Let us collect a comprehensive list of such problems n.m.damle > See e.g. . and -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Positive convex cones: a unique decomposition result? > Hello all My question is set in convex geometry and should be fairly standard, > but I have so far been unsuccessful in finding an answer. We have a positive polyhedral convex cone C in R^m and a finite set of > extreme directions {e_i} of C. For each extreme direction e_i, chose > an arbitrary parallel vector c_i and call this a standard ray. Then > any element in x in C can be written as a conic combination of {c_i}, > i.e. x = k_1 c_1 + ... + k_n c_n for some non-negative real numbers k_i. So far, this should be > standard. Now to my question: is the above decomposition unique? I.e. are there > no l_i distinct from k_i such that x = l_1 c_1 + ... + l_n c_n No, it's not unique. Consider e.g. the cone {[x,y,z] in R^3: z >= 0, |x|+|y| <= z}. This has four extreme directions: you can take the c_i to be [1,0,1], [-1,0,1], [0,1,1] and [0,-1,1]. Then c_1 + c_2 = c_3 + c_4. To have uniqueness, you need a cross-section of the cone to be a simplex. > ? My feeling is that unique decomposition should hold. However I have > been unable to find any results in the literature confirming this. My > attempts at coming up with a proof have also failed. The standard > approach to prove unique decomposition in linear algebra relies on > basis vectors being linearly independent, but we do not necessarily > have that our standard rays are linearly independent. We only have the > weaker, conical independence. I hope that my formulation of the problem makes sense. Any > suggestions, pointers to relevant literature or other usenet groups > would be greatly appreciated. Michael. PS: Another formulation of the problem is in the context of Petri net > theory, which I myself am more familiar with: standard rays correspond > to minimal semiflows. We know that any semiflow can be written as a > positive linear combination of minimal semiflows. The question then > becomes whether this positive linear combination is unique. > -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: [] hello > .... > Guys, your scholarly discussion scared away the chinese person asking > for help.... That's exactly why I tried to follow up the wrangling with a sensible answer, and also mailed it to the OP. Let's hope it repairs the damage. Ken Pledger. === Subject: Re: [] hello > Though, I must admit that when you're complaining about my offensive > language, it's a little hard to figure out what you mean by dearth of > vulgarities. If I wasn't convinced that you're a total wordsmith guy > with a wicked command of English, I might even guess that dearth is > not the word you wanted. But what are the odds? Look up dearth. Means scarcity. He mocks you > for lack of imaginative invective. You think I didn't know what dearth meant? Seems odd to complain that someone is being vulgar and then to use the phrase dearth of vulgarity. It's possible that your interpretation is correct, but I find it pretty unlikely. -- Scientists have calculated that the chance of anything so patently absurd actually existing are millions to one. But magicians have calculated that million-to-one chances crop up nine times out of ten. -- Terry Pratchett on Intelligent Design. Or something. === Subject: make 100 by using 1, 7, 7, 7, 7 So, using only +, -, x, /, and parentheses, and ONLY these numbers: 1, 7, 7, 7, 7 how can you make 100? Is there more than one solution? === Subject: Re: make 100 by using 1, 7, 7, 7, 7 dangerousgame95@gmail.com a .8ecrit : > So, using only +, -, x, /, and parentheses, and ONLY these numbers: 1, 7, 7, 7, 7 how can you make 100? Is there more than one solution? > Another one (return two '7') : 1 L7 L7 === Subject: Re: make 100 by using 1, 7, 7, 7, 7 Raymond Manzoni a .8ecrit : > dangerousgame95@gmail.com a .8ecrit : > So, using only +, -, x, /, and parentheses, and ONLY these numbers: > 1, 7, 7, 7, 7 > how can you make 100? Is there more than one solution? Another one (return two '7') : > 1 L7 L7 Sorry Silver (probably your idea...) === Subject: Re: make 100 by using 1, 7, 7, 7, 7 > So, using only +, -, x, /, and parentheses, and ONLY these numbers: 1, 7, 7, 7, 7 how can you make 100? Is there more than one solution? 1 () () And I didn't even use any 7s. ;-) === Subject: Re: make 100 by using 1, 7, 7, 7, 7 > So, using only +, -, x, /, and parentheses, and ONLY these numbers: > 1, 7, 7, 7, 7 > how can you make 100? Is there more than one solution? > 1 () () > And I didn't even use any 7s. ;-) We can use your idea with the 7's. 1 <> <> === Subject: Re: make 100 by using 1, 7, 7, 7, 7 dangerousgame95@gmail.com a .8ecrit : > So, using only +, -, x, /, and parentheses, and ONLY these numbers: 1, 7, 7, 7, 7 how can you make 100? Is there more than one solution? > 177 - 77 No? === Subject: Re: make 100 by using 1, 7, 7, 7, 7 > So, using only +, -, x, /, and parentheses, and ONLY these numbers: 1, 7, 7, 7, 7 how can you make 100? Is there more than one solution? > 177-77 comes to mind the most quickly to me (but is concatenation legal here? => implies that you mean digits instead of numbers). === Subject: Re: make 100 by using 1, 7, 7, 7, 7 > So, using only +, -, x, /, and parentheses, and ONLY these numbers: 1, 7, 7, 7, 7 how can you make 100? Is there more than one solution? > How about this: 7+7+7+7+7+7+7+7+7+7+7+7+7+7+1+1. I think, that's the only solution. -- Thomas Nordhaus === Subject: Re: make 100 by using 1, 7, 7, 7, 7 > So, using only +, -, x, /, and parentheses, and ONLY these numbers: > > 1, 7, 7, 7, 7 > > how can you make 100? Is there more than one solution? > How about this: 7+7+7+7+7+7+7+7+7+7+7+7+7+7+1+1. I think, that's the >only solution. The implication of showing four 7's is that you are supposed to use only those four 7's, and hence, only one 1. If you allow any number of 7's and 1's, it should be obvious that there are infinitely many solutions. quasi === Subject: Re: make 100 by using 1, 7, 7, 7, 7 So, using only +, -, x, /, and parentheses, and ONLY these numbers: 1, 7, 7, 7, 7 how can you make 100? Is there more than one solution? > How about this: 7+7+7+7+7+7+7+7+7+7+7+7+7+7+1+1. I think, that's the > only solution. The implication of showing four 7's is that you are supposed to use > only those four 7's, and hence, only one 1. Well, right you are - I guess, coming home pub, looking through the newsgroup and making funny replies doesn't work all the time. -- Thomas Nordhaus === Subject: Re: make 100 by using 1, 7, 7, 7, 7 > > So, using only +, -, x, /, and parentheses, and ONLY these numbers: > 1, 7, 7, 7, 7 > how can you make 100? Is there more than one solution? > How about this: 7+7+7+7+7+7+7+7+7+7+7+7+7+7+1+1. I think, that's the > only solution. > > The implication of showing four 7's is that you are supposed to use > only those four 7's, and hence, only one 1. Well, right you are - I guess, coming home pub, looking through the >newsgroup and making funny replies doesn't work all the time. Sorry, I should have realized your reply was being deliberately silly. quasi === Subject: Re: make 100 by using 1, 7, 7, 7, 7 So, using only +, -, x, /, and parentheses, and ONLY these numbers: > 1, 7, 7, 7, 7 > how can you make 100? Is there more than one solution? How about this: 7+7+7+7+7+7+7+7+7+7+7+7+7+7+1+1. I think, that's the >only solution. The implication of showing four 7's is that you are supposed to use > only those four 7's, and hence, only one 1. If you allow any number of 7's and 1's, it should be obvious that > there are infinitely many solutions. quasi So anyone know of any arrangement? === Subject: Re: make 100 by using 1, 7, 7, 7, 7 So, using only +, -, x, /, and parentheses, and ONLY these numbers: 1, 7, 7, 7, 7 how can you make 100? Is there more than one solution? How about this: 7+7+7+7+7+7+7+7+7+7+7+7+7+7+1+1. I think, that's the > only solution. -- > Thomas Nordhaus 7*7 + 7*7 + 1 + 1 Maybe there is a restriction to use each symbol once and to only use the given numbers only once? === Subject: Re: infinitely differentiable functions > >Let S be the set of C^infinity functions f from R to R such that f and >all derivatives of f are everywhere positive. >Then S is closed under addition, multiplication, composition, and of >course, differentiation. Also, S is closed under scalar multiplication >by positive constants. >Questions: >(1) Is S closed under all positive powers? In other words, if f is in >S, and a is a positive constant, must exp(a*ln(f)) be in S? >(2) If f is in S, must there exist a positive integer n such that the >limit as x approaches infinity of f^n(x)/e^x approaches infinity. Note >-- for this question, f^n means the n'th power of f. >(3) If there is a positive integer n such that f^(n) is in S, must f >be in S? Note -- for this question, f^(n) means the n'th iterate of >f. > > (4) If f is in S, must 1/f(-x) be in S? > > quasi >No, take f(x) = e^(e^x). Then f is in S, but 1/f(x) = 1/e^(e^(x)), >which is bounded on R. Hence 1/f(-x) is bounded on R, which no >function in S is. > > Ok, but then I think (4) can be revised to avoid that type of example: > > (4) [revised] If f is in S, and if range(f) = (0,infinity), must > 1/f(-x) be in S? > > quasi No, let f(x) = (e^x - 1)/x. Then range(f) = (0,infinity), but 1/f(-x) >= -x/(e^(-x) - 1), which grows like x near +oo. But every function in >S grows faster than any power of x at oo. Nice explanation. Your many counterexamples have helped strengthen my somewhat limited > intuition about these functions. > quasi You're welcome. Also note that if f(x) = int_(0,1) t^p e^(xt), p > -1, then the change of variables t = s/|x| shows f(x) is like 1/|x|^(1+p) at -oo. So decay at -oo can be pretty slow for these functions. === Subject: Re: infinitely differentiable functions > Here's a related problem ... Can a (possibly infinite) sum of functions of the form a*e^(b*x), > where the b-values are positive and pairwise distinct, and the > a-values are nonzero reals, be identically equal to 0? Remarks: The case of finite sums is easy -- the answer is no. For the case of infinite sums, I'm pretty sure the answer is also > no, and my feeling is it should be easy. Following the same strategy > which worked for finite sums, I tried taking derivatives (infinitely > many times), and then I plugged in x = 0. My intuition was that this > would lead to a contradiction, and maybe it does, but if so, I don't > see it. Assume sum_(n=1,oo) |a_n| < oo and sum(n=1,oo) a_n*e^(b_nx) = 0 all x in R. Define a real finite Borel measure mu on [0,oo) by mu = sum_n a_n*delta(b_n). Then int_[0,oo) e^(tx) dmu(t) = 0 for all x. But finite linear combinations of exponentials e^(xt), x < 0, which form an algebra, are dense in C_0, the continuous functions on [0,oo) that -> 0 at oo; this is a Stone-Weierstrass type argument. So int_[0,oo) f(t) dmu(t) = 0 for all f in C_0, and Riesz Representation theorem gives mu = 0, ie, all a_n = 0. Don't know what to do without the assumption sum_(n=1,oo) |a_n| < oo. === Subject: Re: infinitely differentiable functions , Here's a related problem ... Can a (possibly infinite) sum of functions of the form a*e^(b*x), > where the b-values are positive and pairwise distinct, and the > a-values are nonzero reals, be identically equal to 0? Remarks: The case of finite sums is easy -- the answer is no. For the case of infinite sums, I'm pretty sure the answer is also > no, and my feeling is it should be easy. Following the same strategy > which worked for finite sums, I tried taking derivatives (infinitely > many times), and then I plugged in x = 0. My intuition was that this > would lead to a contradiction, and maybe it does, but if so, I don't > see it. Assume sum_(n=1,oo) |a_n| < oo and sum(n=1,oo) a_n*e^(b_nx) = 0 all x > in R. Define a real finite Borel measure mu on [0,oo) by mu = sum_n > a_n*delta(b_n). Then int_[0,oo) e^(tx) dmu(t) = 0 for all x. But > finite linear combinations of exponentials e^(xt), x < 0, which form > an algebra, are dense in C_0, the continuous functions on [0,oo) that > -> 0 at oo; this is a Stone-Weierstrass type argument. So int_[0,oo) > f(t) dmu(t) = 0 for all f in C_0, and Riesz Representation theorem > gives mu = 0, ie, all a_n = 0. Don't know what to do without the assumption sum_(n=1,oo) |a_n| < oo. Intesting fact: Again suppose sum_(n=1,oo) |a_n| < oo. For x < 0, define f(x) = sum(n=1,oo) a_n*e^(b_nx), where the b's are positive. Suppose f(x_m) = 0 for a sequence of negative x_m's such that sum 1/|x_m| = oo. (For example, x_m = -m). Then f(x) = 0 for all x < 0. Proof: Extend f to an analytic function F on Re(z) < 0 by setting F(z) = sum(n=1,oo) a_n*e^(b_n*z). Then F is a bounded analytic function in Re(z) < 0. Now consider F o g, where g is a linear fractional transformation of the unit disc onto Re(z) < 0. Then F o g is a bounded analytic function in the disc that vanishes along a sequence w_m in the disc such that sum (1-|w_m|) = oo. That violates the Blaschke condition, so F o g is identically 0, which implies F is identically 0. Hence f(x) = 0 for all x < 0. === Subject: integer zeros and critical numbers for a polynomial My questions is this. Let f(x) be a polynomial over the integers with degree at least 1 with only simple integer zeros. Can the critical numbers ever be integers? To put it another way. If f(x) is a polynomial over the integers and both f(x) and its derivative both factor into linear factors over the integers, must f have a zero of multiplicity greater than 1? This is just a question I had after years of creating comprehensive graphing questions for calculus tests. Greg === Subject: Re: integer zeros and critical numbers for a polynomial My questions is this. Let f(x) be a polynomial over the integers > with degree at least 1 with only simple integer zeros. Can the > critical numbers ever be integers? To put it another way. If f(x) is a polynomial over the integers > and both f(x) and its derivative both factor into linear factors over > the integers, must f have a zero of multiplicity greater than 1? This is just a question I had after years of creating comprehensive > graphing questions for calculus tests. See my prior posts for many links to related literature --Bill Dubuque === Subject: Re: integer zeros and critical numbers for a polynomial My questions is this. Let f(x) be a polynomial over the integers > with degree at least 1 with only simple integer zeros. Can the > critical numbers ever be integers? To put it another way. If f(x) is a polynomial over the integers > and both f(x) and its derivative both factor into linear factors over > the integers, must f have a zero of multiplicity greater than 1? This is just a question I had after years of creating comprehensive > graphing questions for calculus tests. > Greg No, consider x(x-2). === Subject: Re: integer zeros and critical numbers for a polynomial On Nov 12, 4:17 pm, The World Wide Wade with degree at least 1 with only simple integer zeros. Can the > critical numbers ever be integers? To put it another way. If f(x) is a polynomial over the integers > and both f(x) and its derivative both factor into linear factors over > the integers, must f have a zero of multiplicity greater than 1? This is just a question I had after years of creating comprehensive > graphing questions for calculus tests. > Greg No, consider x(x-2). Ok. What about degree 3 or higher? -- Greg === Subject: Re: integer zeros and critical numbers for a polynomial > On Nov 12, 4:17 pm, The World Wide Wade with degree at least 1 with only simple integer zeros. Can the > critical numbers ever be integers? To put it another way. If f(x) is a polynomial over the integers > and both f(x) and its derivative both factor into linear factors over > the integers, must f have a zero of multiplicity greater than 1? This is just a question I had after years of creating comprehensive > graphing questions for calculus tests. > Greg No, consider x(x-2). Ok. What about degree 3 or higher? Nice Cubic Polynomials for Curve Sketching Tom Bruggeman; Tom Gush Mathematics Magazine Vol. 53, No. 4 (Sep., 1980), pp. 233-234 Polynomials All of Whose Derivatives Have Integer Roots C. E. Carroll The American Mathematical Monthly > Vol. 96, No. 2 (Feb., 1989), pp. 129-130 Nice Cubic Polynomials, Pythagorean Triples, and the Law of Cosines Jim Buddenhagen; Charles Ford; Mike May Mathematics Magazine > Vol. 65, No. 4 (Oct., 1992), pp. 244-249 -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: integer zeros and critical numbers for a polynomial On Nov 12, 4:17 pm, The World Wide Wade with degree at least 1 with only simple integer zeros. Can the > critical numbers ever be integers? To put it another way. If f(x) is a polynomial over the integers > and both f(x) and its derivative both factor into linear factors over > the integers, must f have a zero of multiplicity greater than 1? This is just a question I had after years of creating comprehensive > graphing questions for calculus tests. > Greg No, consider x(x-2). Ok. What about degree 3 or higher? Nice Cubic Polynomials for Curve Sketching > Tom Bruggeman; Tom Gush > Mathematics Magazine Vol. 53, No. 4 (Sep., 1980), pp. 233-234 Polynomials All of Whose Derivatives Have Integer Roots > C. E. Carroll > The American Mathematical Monthly > Vol. 96, No. 2 (Feb., 1989), pp. > 129-130 Nice Cubic Polynomials, Pythagorean Triples, and the Law of Cosines > Jim Buddenhagen; Charles Ford; Mike May > Mathematics Magazine > Vol. 65, No. 4 (Oct., 1992), pp. 244-249 Another reference is MR1752251 (2001c:11035) Buchholz, Ralph H.; MacDougall, James A. (5-NEWC) When Newton met Diophantus: a study of rational-derived polynomials and their extension to quadratic fields. (English summary) J. Number Theory 81 (2000), no. 2, 210[CapitalEth]233. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: integer zeros and critical numbers for a polynomial My questions is this. Let f(x) be a polynomial over the integers > with degree at least 1 with only simple integer zeros. Can the > critical numbers ever be integers? To put it another way. If f(x) is a polynomial over the integers > and both f(x) and its derivative both factor into linear factors over > the integers, must f have a zero of multiplicity greater than 1? This is just a question I had after years of creating comprehensive > graphing questions for calculus tests. > Greg No, consider x(x-2). Ok. What about degree 3 or higher? -- Greg Consider X*(X+9)*(X-15): Zeroes at -9, 0, 15 critical at -5, 9 hagman === Subject: Re: integer zeros and critical numbers for a polynomial My questions is this. Let f(x) be a polynomial over the integers > with degree at least 1 with only simple integer zeros. Can the > critical numbers ever be integers? To put it another way. If f(x) is a polynomial over the integers > and both f(x) and its derivative both factor into linear factors over > the integers, must f have a zero of multiplicity greater than 1? This is just a question I had after years of creating comprehensive > graphing questions for calculus tests. > Greg No, consider x(x-2). Ok. What about degree 3 or higher? -- Greg > Consider X*(X+9)*(X-15): > Zeroes at -9, 0, 15 > critical at -5, 9 hagman > -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: integer zeros and critical numbers for a polynomial > On Nov 12, 4:17 pm, The World Wide Wade My questions is this. Let f(x) be a polynomial over the integers > with degree at least 1 with only simple integer zeros. Can the > critical numbers ever be integers? To put it another way. If f(x) is a polynomial over the integers > and both f(x) and its derivative both factor into linear factors over > the integers, must f have a zero of multiplicity greater than 1? This is just a question I had after years of creating comprehensive > graphing questions for calculus tests. > Greg No, consider x(x-2). Ok. What about degree 3 or higher? -- Greg > Consider X*(X+9)*(X-15): > Zeroes at -9, 0, 15 > critical at -5, 9 hagman And f''(x)= 6*x - 12 has an integer root, too. === Subject: Re: A help with a proof on measure theory <30885032.1194805490690.JavaMail.jakarta@nitrogen.mathforum.org>, > I'd like an opinion about my proof, please. Let (X, M u) be a measure space, where X is a set, M > is a sigma-algebra on X and u is a measure defined > on M. Let f_n be a sequence of functions defined on > X and with values on [0, oo] such that lim f_n = f. > Suppose that lim Integral f_n du = Integral f du < > oo (the integrals taken over X). Show that, for > every set E of M, Integral_E f_n du = Integral_E f > du, where Integral_E means integral over E. Also, > show that this conclusion may fail if we have lim > Integral f_n du = Integral f du = oo My proof: The given conditions imply that, for sufficiently > large n, f_n is integrable over X and, therefore, > over each set E of M. So, we can assume, WLOG, that > such conditions hold for every n. Since e f_n is a > sequence in L+ (the space of measurable functions on > X with values on [0, oo]), for every n we have 0 <= > Int_E f_n du <= Int f_n du. Since the sequence (Int > f_n du) is bounded (for it converges on R), these > inequalities show (Int_E f_n du) is bounded and, > therefore, contains convergent subsequences Let > (Int_E f_n_k du) be one of such subsequences. Then, > (f_n_k) is subsequence of (f_n), so that lim f_n_k = > f. According to the properties of the integral, for > every k we have Int_E f_n_k du + Int_E' f_n_k du = Int f_n_k du , > where E' is the complement of E with respect to X. Since (Int f_n_k du) is a subsequence of (Int f_n > du), which converges to Int f du, the same is true > of (Int f_n_k du). And since Int_E f_n_k du > converges, the equality above shows that Int_E' > f_n_k du converges and that lim Int_E f_n_k du + lim Int_E' f_n_k du = lim Int > f_n_k du = Int fdu. In addition, since Int fdu = > Int_E f du + Int_E' fdu, it follows that lim Int_E f_n_k du + lim Int_E' f_n_k du = Int_E f > du + Int_E' fdu (1) Since (f_n_k) is a sequence in L+ whose limit is f > (on X and on every set of M), it follows from > Fatou's Lemma that lim Int_E f_n_k du >= Int_E f du (2) and, similarly, > lim Int_E' f_n_k du >= Int_E' f du (3). In order for (1), (2) and (3) to be simultaneously > satisfied, (2) and (3) must be equalities, which > implies that lim Int_E f_n_k du = Int_E f du (4). Since (Int_E > f_n du) is bounded and (4) holds for all of its > convergent subsequences, it follows (Int_E f_n du) > itself is convergent and lim Int_E f_n du = Int_E f du, valid for every > measurableset E. This proves the theorem. > To see this conclusion may fail if lim Int f_n du = > Int f du = oo, we can take X = (0, oo), M = Lebesgue > sigma-algebra on X , u = Lebesgue measure and f_n(x) = 1/(nx) if x is in (0, 1] > f_n(x) = 1 if x is in (1, oo) n=1,2,3.... .... Then, f_n conveges to the function f given by f(x) = 0, if x is in (0, 1] > f(x) = 1 if x is in > (1, oo). Let E = (0, 1]. Then, recalling in this > case the Lebesgue and Riemann integrals yields the > same value, we have, > for every n, Int_E f_n du = Int (0 to 1)1/nx dx = > 1/n ln(x) (0 to 1) = oo, Int_(1,oo) f_n du = 1 * oo > = oo and, therefore, Int f_n du = oo. It follow that lim Int_E f_n du = oo and lim Int f_n > n du = oo. On the other hand, Int_E f du = 0, Int_(0, 1) f du = > oo and Int f du =oo > So, we have lim Int f_n du = Int f du = oo, but > lim_E f_n du = oo > 0 = Int_E f du. This shows the > condtion Int f du < oo cannot be dropped. > Amanda if you know some L^p theory another way would be: f_n -> f a.e. (wrt u) and | f_n| -> | f | (L^1(mu)) norm and so f_n -> f in L^1 which gives us the desired result. Right: Using Fatou's Lemma, int lim inf (|f| + |f_n| - |f-f_n|) = int 2|f| <= lim inf int (|f| + |f_n| - |f-f_n|) = 2 int |f| - lim sup int |f-f_n|, which gives int |f-f_n| -> 0. Sneaky ... === Subject: Basic probability Hello folks, let us assume that X and Y are normal-distributed random variables. What can we say about the random variable Z = X + Y? Is Z also normal-distributed? How can I check that? Do I simply add the probability density functions of X and Y? Also, what happens if we consider X*Y or X/Y? Ed === Subject: Re: Basic probability > Hello folks, let us assume that X and Y are normal-distributed random variables. What can we say about the random variable Z = X + Y? > Is Z also normal-distributed? You need to know something about how X and Y are related. It's true if X and Y are independent, or more generally if X and Y are jointly normal. > How can I check that? > Do I simply add the probability density functions of X and Y? No. If X and Y have a joint density f(x,y), the density of Z is int_{-infty}^infty f(x,z-x) dx > Also, what happens if we consider X*Y or X/Y? Almost certainly not normal. If X and Y have mean 0 and are independent, X/Y has a Cauchy distribution while the density of X Y is given by a modified Bessel function of the second kind. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Basic probability > Hello folks, let us assume that X and Y are normal-distributed random variables. What can we say about the random variable Z = X + Y? > Is Z also normal-distributed? If X and Y are independent, Z is normal. The mean of Z is the sum of the means of X and Y, and the variance is the sum of the variances of X and Y. > How can I check that? Do you mean prove it? This Wiki page sketches out some proofs. http://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables > Do I simply add the probability density functions of X and Y? No. You need the theory of functions of random variables. > Also, what happens if we consider X*Y or X/Y? More complicated things. See this page for answers to all three of your questions: http://en.wikipedia.org/wiki/Normal_distribution (under Properties, items 2 and 3). (Product and ratio distributions given for the case where X and Y are zero-mean only). - Randy === Subject: Re: Basic probability > Hello folks, let us assume that X and Y are normal-distributed random variables. What can we say about the random variable Z = X + Y? In general: nothing. > Is Z also normal-distributed? However, if X,Y are independent, then Z is normal with expected value E(Z)=E(X)+E(Y) and variance V(Z)=V(X)+V(Y). How can I check that? > Do I simply add the probability density functions of X and Y? No, you need to calculate a double integral. Also, what happens if we consider X*Y or X/Y? These are definitely not normally distributed. Ed === Subject: Re: Confirmation of Shannon's Mistake about Perfect Secrecy of One-time-pad > Just for a moment, forget the OTP problem. Pretend that you've never > even heard of the OTP. Please study the following question. > I have two coins, one red and one blue. Both are marked 0 on one > side and 1 on the other, and both are fair (i.e. they both come up > 0 with probability 1/2, and 1 with probability 1/2). I toss both > coins. The red coin comes up with the number R. The blue coin comes up > with the number B. I calculate C = R Xor B, and I find that C = 0. > Given that C = 0, what is the probability that R = 0? > Do you think that this question has a well-defined single numerical > answer? > (Please try to give a direct reply to the actual question that I have > asked. If possible, please begin your reply with the word yes or the > word no.)- - > - - > yes Great. I think that's the first of your replies that I've completely > understood! Now let me change the problem slightly. Instead of the red coin being > fair, it will be biased. I'll state the new problem in full: I have two coins, one red and one blue. Both are marked 0 on one > side and 1 on the other. The red coin is biased and comes up 0 > with probability p and 1 with probability 1-p. The blue coin is > fair: it comes up 0 with probability 1/2, and 1 with probability > 1/2. I toss both coins. The red coin comes up with the number R. The > blue coin comes up with the number B. I calculate C = R Xor B, and I > find that C = 0. Given that C = 0, what is the probability that R = > 0? Do you think that this new problem has a well-defined single answer > (in terms of p)? If so, what do you think the answer is?- - - - you just no see the contrastion. take the question easy. > If c fixed, k and P are dependant, > but you just use the probablity c not fixed, that is a mixture, and > changged the probablity characteristic,including the value. > can you prove probabilities when c not fixed, the same as > probabilities when c not fixed. I have no idea what you are talking about. I asked you two simple questions: 1. Do you think that this new problem has a well-defined single answer > (in terms of p)? 2. If so, what do you think the answer is? Please just answer the questions. Please answer yes or no to > question 1. If you answered yes to question 1 then please also > answer question 2.- - - - I just deny and say I have no idea what you are talking about. > i do know how you discuss with me so long when you have no idea what i > am talking about It occurred to me later that perhaps this is a misunderstanding based on language difficulties. When I say I have no idea what you are talking about, I mean that your answer has nothing to do with the question I asked. You launched straight into this stuff about you just use the probablity c not fixed, that is a mixture, and changged the probablity characteristic,including the value, but in fact I haven't used any probabilities to do anything at all. I haven't done *any* calculations, and, indeed, I haven't even said whether *I* think the problem has a well-defined answer. What I have done is describe a procedure that you or I could easily carry out. (Just imagine that you physically have the coins, and you physically do what I described.) Then I asked you whether you think the stated probability can be calculated. If you don't think it can then just say no. === Subject: Re: Randomness of digits within pi <30695245.1194621803816.JavaMail.jakarta@nitrogen.mathforum.org> <87hcjrwv4x.fsf@nonospaz.fatphil.org> On Nov 12, 11:39 am, Phil Carmody Benton's law? Benford's law. === Subject: Re: shortest route from point to point on discontinuous line > Hi all, I've been braking my head over this, and i'm just stuck. I'm writing > an algorithm that applies limits to a collection of objects and then > tries to find the largest subset of that collection that fits all > these limits. I think the term you want to use is constraint or bound, not limit, which mathematically is something completely different. > Now one of these limits can be simplified to this situation. Every point in my collection has an X and a Y value. One limit states > that the limit for the average value for Y is dependent on the average > value for X through this function: LimitY(x) = > if x <= 36 then .5 > if x > 36 and <= 39 then .45 > if x > 39 and <= 42 then .41 > if x > 42 and <= 45 then .38 > if x > 45 and <= 48 then .36 > if x > 48 then .33 so this limit is a line that looks something like this: ______ > _____ > _____ > ______ > _______ > To find out if the average for a certain collection, a point (x, y), > is above the line is easy, put the average for x into the function and > check if the average y is bigger than LimitY(x). If the average is above the line, i have to delete some elements in > the collection to bring the average under the line. But i want to > delete as few objects as possible to bring it under the line. So take > these examples: for points (X,Y) > * (48.1 , .35) in this case i would delete objectswith a large X to > bring down the average X to under 48. That would make the limit .36 > and i would be ok. * (47,9 , .37) in this case i would delete objects with a large Y to > bring down the average Y to .36. * 48.1 , .37) in this case i woudl delete object with a large X and > a large Y, because the nearest point below the line is just to the > left and below. Now i would like to find some generic algotithm that, for any point > (x,y), when above the limit, tells me if i should bring the average X > down, bring the average Y down, or maybe both. So i need to find the shortest 'route' to a point below the line for > a given point (X, Y) Does anyone have any experience with this type of problem? It's easy > to spot when you draw it out, but i'm finding it difficult to program > it with line segements and all. If I understand you correctly, given a point P = (x,y) with y > LimitY(x), you want to find the closest point to P on the polygonal curve consisting of the graph of LimitY plus the vertical jump segments. If there are just a few segments, as in your example, the simplest thing to do is identify the possible candidates and find the closest one. Say LimitY(x) = y_i for x_{i-1} < x <= x_i, i=1..n, where x_0 = -infinity and x_n = infinity, and y_{i+1} < y_i for each i as in your example. Then the possible candidates are (x, y_j) if x_{j-1} < x < x_j, and (x_j,y) if y_{j+1} < y < y_j, and all (x_i, y_i). -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: shortest route from point to point on discontinuous line Hi all, I've been braking my head over this, and i'm just stuck. I'm writing > an algorithm that applies limits to a collection of objects and then > tries to find the largest subset of that collection that fits all > these limits. I think the term you want to use is constraint or bound, not > limit, which mathematically is something completely different. Now one of these limits can be simplified to this situation. Every point in my collection has an X and a Y value. One limit states > that the limit for the average value for Y is dependent on the average > value for X through this function: LimitY(x) = > if x <= 36 then .5 > if x > 36 and <= 39 then .45 > if x > 39 and <= 42 then .41 > if x > 42 and <= 45 then .38 > if x > 45 and <= 48 then .36 > if x > 48 then .33 so this limit is a line that looks something like this: ______ > _____ > _____ > ______ > _______ > To find out if the average for a certain collection, a point (x, y), > is above the line is easy, put the average for x into the function and > check if the average y is bigger than LimitY(x). If the average is above the line, i have to delete some elements in > the collection to bring the average under the line. But i want to > delete as few objects as possible to bring it under the line. So take > these examples: for points (X,Y) > * (48.1 , .35) in this case i would delete objectswith a large X to > bring down the average X to under 48. That would make the limit .36 > and i would be ok. * (47,9 , .37) in this case i would delete objects with a large Y to > bring down the average Y to .36. * 48.1 , .37) in this case i woudl delete object with a large X and > a large Y, because the nearest point below the line is just to the > left and below. Now i would like to find some generic algotithm that, for any point > (x,y), when above the limit, tells me if i should bring the average X > down, bring the average Y down, or maybe both. So i need to find the shortest 'route' to a point below the line for > a given point (X, Y) Does anyone have any experience with this type of problem? It's easy > to spot when you draw it out, but i'm finding it difficult to program > it with line segements and all. If I understand you correctly, given a point P = (x,y) with y > LimitY(x), > you want to find the closest point to P on the polygonal curve > consisting of the graph of LimitY plus the vertical jump segments. > If there are just a few segments, as in your example, the simplest thing > to do is identify the possible candidates and find the closest one. > Say LimitY(x) = y_i for x_{i-1} < x <= x_i, i=1..n, where x_0 = -infinity > and x_n = infinity, and y_{i+1} < y_i for each i as in your example. > Then the possible candidates are (x, y_j) if x_{j-1} < x < x_j, > and (x_j,y) if y_{j+1} < y < y_j, and all (x_i, y_i). Let me amend that: if y_{j+1} < y <= y_j and x_{k-1} < x <= x_k, the candidates are (x_j, y), (x_i, y_i) for j < i < k, and (x, y_k). -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: 1-2=3; Root( -49 ) = -7; Bus Traveling, Linearity. > 1-2=3; Root( -25 ) = -5; Bus Traveling, Linearity. Let's setup the following Rule: (Call this Bus Traveling Rule since I worked this out when I was > traveling on a bus). Bus Traveling Rule: We can go either to the left or to the right or zero, but never > approach zero from left or right. > So: We can only go from 0 to -infinity or to infinity. Then (Corollary?): x+y = x-y = y-x = y+x. > i.e: 1-2 = 2-1, etc equals either -3 or +3. etc. a times (b+c) = a times (b-c) = a times (c-b) = a times (b+c). But: a/(b-c) not equal a/(b+c), because we only can go to either right > or left from 0. Thus you think that x=y does not imply z/x = z/y ? hagman === Subject: Re: 1-2=3; Root( -49 ) = -7; Bus Traveling, Linearity. > 1-2=3; Root( -25 ) = -5; Bus Traveling, Linearity. > Now combine this rule with discovery made by poet philosopher Ayia-Oba that 1^2 = 3, and we have a system more powerful than Archimedes Plutonium's AP-Adics. === Subject: Polynomials in Two Variables Taking Distinct Integer Values at Lattice-Points Some time ago quasi and others were discussing this kind of problem. I've come across the paper, John S Lew, Polynomials in two variables taking distinct integer values at lattice-points, Amer Math Monthly 88 (1981) 344-346. It may be of interest. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Four Color Theorem A planar graph does not need five mutually adjacent vertices to be 5 chroma. Should the above statement be called a hypothesis , a conjecture, or something else? ---Bill J === Subject: Re: Four Color Theorem > A planar graph does not need five mutually adjacent vertices to be 5 > chroma. Should the above statement be called a hypothesis , a conjecture, > or something else? ---Bill J The statement can be rewriten to There exists a planar 5-chromatic graph without five mutually adjacent vertices. However, such a planar graph does not exist (neither a 5-chromatic one nor one containing a K_5). === Subject: Re: Four Color Theorem A planar graph does not need five mutually adjacent vertices to be 5 > chroma. Should the above statement be called a hypothesis , a conjecture, > or something else? ---Bill J The statement can be rewriten to > There exists a planar 5-chromatic graph without five mutually adjacent > vertices. > However, such a planar graph does not exist (neither a 5-chromatic one > nor one containing a K_5). I knew that! But you cannot say which of the two statements are true without first proving the four color theorem! I have looked up the definitions of hypothesis and conjecture and neither word seems appropriate. But what else would you call such a statement? Bill J === Subject: Re: Invalid Assumptions and the search for Truth On Nov 8, 6:57 am, Timothy Golden BandTechnology.com > I am fairly sure that we should discuss assumption to progress on this > thread. > Assumptions are like faith. > I am caught with assumptions at the base even as I attempt to delete > them. For instance I assume that reality exists regardless of my > belief system. > In other words its laws are independent of my own (or anyone's) > beliefs of it. > I further assume that reality is consistent. In other words there is a > stability or consistency that will not change which yields the results > of our experience. While the resultant system has dynamics those > dynamics are the result of a persistent basis. This is a direct mapping to axiomatic thinking and it is the stable > properties that I (we) are in search of. A series of axioms will > suffice. These axioms will have to be fairly productive to yield the > dynamics which we observe. As such there may be multiple mappings of > them (parallel models) and so even if they are found the rigorous > decomposition might still be ongoing. Especially if general > dimensional behavior enters into such a theory then the extensions of > it will not necessarily reach any final conclusion but instead like > modern science yield an ever branching tree of knowledge with > frustrating complexity for the human. formalisation has been an extremely productive enterprise it has helped make mathematical results much more repeatable assisting prediction and education alike this is why i have tried to stress that constructivists need not be against formal methods despite that stand from many intuitionists when we establish as well we can a static semantics for manipulating a given set of symbols the symbols gain a power of control semantics gives them meaning and the more static the more repeatable but give an axiomatic system to a four year old and see how they do it is likely there will be many failures of the child to bisimulate your expectations for the system we are obsessive beings we hypostatise and reify we think we are headed somewhere fixed when that is not something we have justified when semantics change temporal and causal relationships change things that were static in one model may become dynamic in a latter take the metric of space the extra decimal places of experimental precision completely reinterpreted when the semantic shifts occur even though we had very good precision distance measures an effect that eluded the decimal places the curvature of spacetime turned out a major effect in the universe what is the local truth? what is global? what part of our ontology is temporal? these are properties of symbols we cannot determine with certainty no phenomena completely validates just as software validation is never complete instead we get reliability tolerances it has functioned as expected in the past for future expectations what is assumed? assumption is the a priori hubris of kant a posteriori synthesis is knowledge our pasts our memories now is the test of our knowledge the information used to validate our models (our future guesses) every step we learn what works well the building blocks aren't assumptions the building blocks are symbols > Without the presumption of existence of a full model I would only > appropriate a small investment. > With such a presumption we can then assess the current position and > attempt to identify what is wrong with it i.e. what are the current > invalid assumptions? The paradigm of the invalid assumption is > somewhat the skeptical position and it is those weak links in the > chain that need mending. Wether this can be a differential transition > is dubious at this point. So the freedom of the constructivist > paradigm is felt. Further some assumptions may be decomposable or > derivable just as some branches of mathematics are decompositions of > previous versions. One would hope though that such a step would yield > some profitable consequences. I can accept a direct accusation of hubris though I would liken it > back to the human position and upon divorcing oneself from that > position to think largely enough then this problem is no longer an > emotional problem; it is an informational problem which for the human > induces emotions. We all are subject to egoism and without challenging > it continually one will neither think largely enough nor accurately > enough. The twists there can be untwisted by others. An uncensored > medium such as this is a fine platform for the errant human. expectations of system fault are regularly justified validating the validators has never ended -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: =?utf-8?q?--_Da_pobijem_Je=C5=A1e,_i_gotova_stvar!_--?= Rise up and shine, white sons and daughters Rise up and shine, you gotta fight to part those waters When we swim in the light, all will be okay The black, yellow and brown man will wash away. -------------------- Ovu pesmicu pevala mi je mama kad sam bio mali, dok me je uspavljivala. Bio sam u.8au.8akan u dekicu sa likom na.8aeg velikog vojskovo©¢e Dragoljuba Mihailovi.8da, sa zidnog postera su me gledale tople, nebesko-plave o.8bi na.8aeg velikog Firera, a tata je u drugoj sobi isprobavao .8aubaru sa kokardom, koja mu je ostala od tatinog tate (tj. mog dede). Sa tom .8aubarom na glavi tata je ostvario velike pobede na vukovarskom rati.8atu. Doneo nam je te '91. godine mnogo suvenira: ogrlicu sa odrezanim prsti.8dima usta.8ake dece, maramicu u kojoj su bile o.8bi nekoliko usta.8akih bojovnika uhva.8denih kako u civilu oru zemlju ispred ku.8de (a sigurno su zakopavali oru.93je), kao i neke .8budne crvenkasto- bele kuglice, koje je nehajno bacio na sto. Dok su one odskakivale po stolu, kao da su .93ive, tata se .8aeretski nasmejao i rekao mi: Nenade, tatin mudonjo, ako ostane.8a bez ovoga, vi.8ae nisi mu.8ako! I tako, sad sam veliki. Roditelji su mi usadili ogromnu ljubav prema svom narodu i jo.8a ve.8du mr.93nju prema drugim narodima. Jo.8a uvek mi u glavi odzvanjaju re.8bi moga tate, koji mi je neprestano ponavljao: Sine Nenade, upamti da su za sva zla ovog sveta krivi .91idovi, a posle njih Hrvati i poturice. I zato, Nenade, tatin sokole, kada bude.8a stasao u pravog, arijevskog mu.8akarca, kad god bude.8a imao priliku, treba da ostvari.8a zadatak koji su nam u amanet ostavili Sveti Sava, Dimitrije Ljoti.8d, Dra.93a Mihailovi.8d i drugi veliki sinovi srpskog milosti; odrasli, .93ene deca... Ma sve pod kamu ili pred cev, jer ima bre da im se napijemo krvi za sve .8ato su nam vekovima radili! Eto, to sam ukratko ja, Nenad Mili.8bevi.8d, na juznetu znan kao Raul Endymion, ina.8be dete sa beogradskog asfalta. Ukoliko imate neka dodatna pitanja, ili .93elite da sa mnom podelite va.8aa razmi.8aljanja o endimion@myrealbox.com ili kontaktirajte me na moj ICQ: 208030128 === Subject: Prime number (proof) Hi all! This is the question: Show that for any n there are at least n consecutive natural numbers none of which is prime. [HINT: Consider (n + 1)! + 2, (n + 1)! + 3, ... , (n + 1)! + (n + 1) ] So, how do I proceed? Should I go for induction or try a direct proof? I can obviously prove that (n + 1)! + (n + 1) is NOT prime, since: (n + 1)! + (n + 1) = n!(n + 1) + (n + 1) = (n + 1)(n! + 1), what makes it NOT prime. But how do I prove it for all n numbers in the sequence? Any hints/enlightments will be welcome. === Subject: Re: Prime number (proof) > Hi all! This is the question: Show that for any n there are at least n consecutive natural numbers > none of which is prime. [HINT: Consider (n + 1)! + 2, (n + 1)! + 3, ... , (n + 1)! + (n + 1) ] That's not a _hint_, that's an answer. Phil -- -- Microsoft voice recognition live demonstration === Subject: Re: Prime number (proof) > Hi all! This is the question: Show that for any n there are at least n consecutive natural numbers > none of which is prime. [HINT: Consider (n + 1)! + 2, (n + 1)! + 3, ... , (n + 1)! + (n + 1) ] So, how do I proceed? Should I go for induction or try a direct proof? I can obviously prove that (n + 1)! + (n + 1) is NOT prime, since: (n + 1)! + (n + 1) = n!(n + 1) + (n + 1) = (n + 1)(n! + 1), what makes > it NOT prime. But how do I prove it for all n numbers in the sequence? > Any hints/enlightments will be welcome. Hint: What do you suppose (n+1)! + j is divisible by, if j <= n+1? -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Prime number (proof) On 12 nov, 20:18, Robert Israel none of which is prime. [HINT: Consider (n + 1)! + 2, (n + 1)! + 3, ... , (n + 1)! + (n + 1) ] So, how do I proceed? Should I go for induction or try a direct proof? I can obviously prove that (n + 1)! + (n + 1) is NOT prime, since: (n + 1)! + (n + 1) = n!(n + 1) + (n + 1) = (n + 1)(n! + 1), what makes > it NOT prime. But how do I prove it for all n numbers in the sequence? > Any hints/enlightments will be welcome. Hint: What do you suppose (n+1)! + j is divisible by, if j <= n+1? > -- > Robert Israel isr...@math.MyUniversitysInitials.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada j? Since: (n + 1)! = 1 * 2 * ... * j * ... * (n + 1), so i can say that: (n+1)! + j = j((n+1)!/j + 1) Does this solve the whole problem? === Subject: Re: Prime number (proof) > j? Since: > (n + 1)! = 1 * 2 * ... * j * ... * (n + 1), so i can say that: > (n+1)! + j = j((n+1)!/j + 1) > Does this solve the whole problem? By your result, (n + 1)! + 2 is divisible by 2. So, it's not prime. Likewise (n + 1)! + 3 is divisible by 3. Hence, it's also not prime. etc. === Subject: Re: Algebra with splitting field and mod 103. > Hello sir~ K is the splitting field of {7x^3 - 4x^2 + 2x - 5} over Z_103. Find the [K : Z_103] ------------------------------------------------ > I think... > 7x^3 - 4x^2 + 2x - 5 = (x-1)(7x^2 + 3x + 5) I think...it's impossible problem. What's form of extension field of Z_103 ? There is no solution such that 7x^2 + 3x + 5 = 0 (mod 103) Because, 7x^2 + 3x + 5 = 7x^2 + 518x + 5 = 7(x^2 + 74 + 1369 - 1368) + 5 = 7(x + 37)^2 - 9578 so, 7(x + 37)^2 = 9578 = 102 = 308 (mod 103) Let X = x + 37. so, 7X^2 = 308 (mod 103) so, X^2 = 44 (mod 103) Legendre (44 / 103) = ((2^2)*11 / 103) = (11 / 103) = (-1)^{((11-1)/2)*((103-1)/2)}(103 / 11) = - (4 / 11) = -1 so, no solution. Anyway, Let a, b are zeros such that 7x^2 + 3x + 5 = 0. (Namely, a, b = [-3 + i*(sqrt(131))] / 14) and Z_103(a,b)...What's form ? a, b are complex number. If c in Z_103, c+a or c+b is impossible calculation. If... Let E be a extension of Z_103. Let a, b in E. so, Z_103 <= Z_103(a) <= Z_103(a,b) [Z_103(a) : Z_103] = 2. How do you show that [Z_103(a,b) : Z_103(a)] = 1 or [Z_103(a,b) : Z_103] = 2 ? === Subject: Re: Algebra with splitting field and mod 103. I think... > 7x^3 - 4x^2 + 2x - 5 = (x-1)(7x^2 + 3x + 5) I think...it's impossible problem. > What's form of extension field of Z_103 ? There is no solution such that > 7x^2 + 3x + 5 = 0 (mod 103) > Because, > 7x^2 + 3x + 5 = 7x^2 + 518x + 5 > = 7(x^2 + 74 + 1369 - 1368) + 5 > = 7(x + 37)^2 - 9578 so, 7(x + 37)^2 = 9578 = 102 = 308 (mod 103) Let X = x + 37. > so, 7X^2 = 308 (mod 103) > so, X^2 = 44 (mod 103) Legendre (44 / 103) = ((2^2)*11 / 103) > = (11 / 103) = (-1)^{((11-1)/2)*((103-1)/2)}(103 / 11) > = - (4 / 11) = -1 so, no solution. So the polynomial is irreducible, something that Bill has already shown you. Anyway, > Let a, b are zeros such that 7x^2 + 3x + 5 = 0. > (Namely, a, b = [-3 + i*(sqrt(131))] / 14) > and > Z_103(a,b)...What's form ? > a, b are complex number. > If c in Z_103, c+a or c+b is impossible calculation. If... > Let E be a extension of Z_103. Let E be a extension of Z_103 in which 7x^2 + 3x + 5 has roots a,b > Let a, b in E. > so, Z_103 <= Z_103(a) <= Z_103(a,b) > [Z_103(a) : Z_103] = 2. How do you show that [Z_103(a,b) : Z_103(a)] = 1 > or [Z_103(a,b) : Z_103] = 2 ? For this listen to Hagen. The point is the polynomial (7x^2 + 3x + 5) has one linear factor in Z_103(a)[X]the other one must also be there, because (7x^2 + 3x + 5)/(x-a) belongs to Z_103(a)[X]. So, b belongs to Z_103(a). Muhammad === Subject: dimension of the space of linear maps V->V, V a vector space It seems like the space of linear maps V->V where V is a vector space of dimension n, has dimension n^2. Can somebody help me to prove or disprove this claim? === Subject: Re: dimension of the space of linear maps V->V, V a vector space > It seems like the space of linear maps V->V where V is a vector space > of dimension n, has dimension n^2. > Can somebody help me to prove or disprove this claim? > Given a basis for V, what is a nice basis for the space of linear maps? === Subject: Derivative of Riemanian exponential map Hi All, I need help with the derivative of the Riemannian exponential map. If b(x) = exp_x (v) = x^(1/2) * exp(x^(-1/2) * v * x^(-1/2)) * x^(1/2), How would I go about calculating db(x)/dx, where x in Sym(+,n) and v is a constant symmetric matrix, i.e., v in Sym(n). I'm working on formulation of a gradient descent problem on the Riemannian manifold of symmetric positive definite Sym(+,n) matrices, and this is the last term, that I need to know to calculate the gradient with respect to x in Sym(+,n) Any help/pointers are appreciated. TIA Ambrish -- Ambrish Tyagi Computer Vision Laboratory Dept. of Computer Science and Engineering Ohio State University === Subject: Unusual Problem I have 12 persons numbered from 1 to 12 that I am making a work plan for. They are going to work together two and two for a week. I am trying to accomplish this under the following conditions: 1) A person should try to work with all the other persons. 2) All persons should work the same number of weeks before the plan restarts. 4) The plan should be well distributed. Something like: (1,2)(1,3)(1,4)(1,5)(1,6)(1,7)(1,8)(1,9)(1,10)(1,11) (1,12) should be avoided since person 1 will get tired. First I found that there are 66 combinations of unique pairs (since identical pairs are not allowed and reversed order should not be counted either): (1,2)(1,3)(1,4)(1,5)(1,6)(1,7)(1,8)(1,9)(1,10)(1,11) (1,12) (2,3)(2,4)(2,5)(2,6)(2,7)(2,8)(2,9)(2,10)(2,11) (2,12) (3,4)(3,5)(3,6)(3,7)(3,8)(3,9)(3,10)(3,11) (3,12) (4,5)(4,6)(4,7)(4,8)(4,9)(4,10)(4,11) (4,12) (5,6)(5,7)(5,8)(5,9)(5,10)(5,11) (5,12) (6,7)(6,8)(6,9)(6,10)(6,11) (6,12) (7,8)(7,9)(7,10)(7,11) (7,12) (8,9)(8,10)(8,11) (8,12) (9,10)(9,11) (9,12) (10,11)(10,12) (11,12) I seems that the worst problem is 4). I have tried to group the paris in 6 rows containing 11 pairs but after filling out the first two rows the problems get impossible to keep track of. Are there any kind of techniques to distributes this kind of data based on the condition that there should be a few weeks before a person should work again and that the same pair is not allowed before all have been working together? === Subject: Re: Unusual Problem > I have 12 persons numbered from 1 to 12 that I am making a work plan for. > They are going to work together two and two for a week. I am trying to > accomplish this under the following conditions: 1) A person should try to work with all the other persons. 2) All persons should work the same number of weeks before the plan > restarts. 4) The plan should be well distributed. Something like: > (1,2)(1,3)(1,4)(1,5)(1,6)(1,7)(1,8)(1,9)(1,10)(1,11) (1,12) > should be avoided since person 1 will get tired. First I found that there are 66 combinations of unique pairs (since > identical pairs are not allowed and reversed order should not be counted > either): (1,2)(1,3)(1,4)(1,5)(1,6)(1,7)(1,8)(1,9)(1,10)(1,11) (1,12) > (2,3)(2,4)(2,5)(2,6)(2,7)(2,8)(2,9)(2,10)(2,11) (2,12) > (3,4)(3,5)(3,6)(3,7)(3,8)(3,9)(3,10)(3,11) (3,12) > (4,5)(4,6)(4,7)(4,8)(4,9)(4,10)(4,11) (4,12) > (5,6)(5,7)(5,8)(5,9)(5,10)(5,11) (5,12) > (6,7)(6,8)(6,9)(6,10)(6,11) (6,12) > (7,8)(7,9)(7,10)(7,11) (7,12) > (8,9)(8,10)(8,11) (8,12) > (9,10)(9,11) (9,12) > (10,11)(10,12) > (11,12) I seems that the worst problem is 4). I have tried to group the paris in 6 > rows containing 11 pairs but after filling out the first two rows the > problems get impossible to keep track of. Are there any kind of techniques to distributes this kind of data based on > the condition that there should be a few weeks before a person should work > again and that the same pair is not allowed before all have been working > together? Sure, here's a Python program. The key here is converting the combinations to bit patterns and then using Hamming Distance (number of changed bit positions) to ensure workers are spread out. A Hamming Distance of 4 means no one works two weeks in a row. There's a fallback to accept Hamming Distance of 2 (a single worker works 2 weeks in a row). The random shuffling helps break up patterns such as person 0 working every other week for 21 weeks. You'll still see individuals working every other week 4 or 5 times (track how often a person works by scanning down an individual column of the last part of the output). With random shuffling, the fallback never occured. # Python import gmpy import random def ooloop6(a, n, perm=True, repl=True): if (not repl) and (n>len(a)): return r0 = range(n) r1 = r0[1:] if perm and repl: # ok v = ','.join(['c%s' % i for i in r0]) f = ' '.join(['for c%s in a' % i for i in r0]) e = ''.join([p = [''.join((,v,)) ,f,]]) exec e return p if (not perm) and repl: # ok v = ','.join(['c%s' % i for i in r0]) f = ' '.join(['for c%s in a' % i for i in r0]) i = ' and '.join(['(c%s>=c%s)' % (j,j-1) for j in r1]) e = ''.join([p = [''.join((,v,)) ,f, if ,i,]]) exec e return p if perm and (not repl): # ok v = ','.join(['c%s' % i for i in r0]) f = ' '.join(['for c%s in a' % i for i in r0]) i = ' and '.join([' and '.join(['(c%s!=c%s)' % (j,k) for k in range(j)]) for j in r1]) e = ''.join([p = [''.join((,v,)) ,f, if ,i,]]) exec e return p if (not perm) and (not repl): # ok v = ','.join(['c%s' % i for i in r0]) f = ' '.join(['for c%s in a' % i for i in r0]) i = ' and '.join(['(c%s>c%s)' % (j,j-1) for j in r1]) e = ''.join([p = [''.join((,v,)) ,f, if ,i,]]) exec e return p # # use 'a' for person 0, 'b' for person 1, etc. # person 0 will be mapped to bit 0, person 1 to # bit 1, etc. # p = ooloop6('abcdefghijkl',2,False,False) print 'Combinations without Replacement: %6d' % (len(p)) print print 'The Combinations without Replacement' for i in p: print i, print # # convert to bit patterns # pbin = [(2**(ord(i[0])-97) + (2**(ord(i[1])-97))) for i in p] print The combinations as 12-bit numbers each with 2 bits set for i in pbin: print '%6d %s' % (i,gmpy.digits(i,2).zfill(12)) random.shuffle(pbin) print print The 12-bit numbers randomly shuffled for i in pbin: print '%6d %s' % (i,gmpy.digits(i,2).zfill(12)) # # initialize the schedule with the first pattern # schedule = [pbin.pop(0)] found_all_4 = False forwards = True while (not found_all_4) and (pbin): all = len(pbin) start = 0 got_it = False while (not got_it) and (start != all): # # if both workers different, Hamming Distance is 4 # hd = gmpy.hamdist(pbin[start],schedule[-1]) if hd == 4: got_it = True else: if forwards: start += 1 else: start -= 1 if got_it: schedule.append(pbin.pop(start)) else: found_all_4 = True found_all_2 = False while (not found_all_2) and (pbin): all = len(pbin) start = 0 got_it = False while (not got_it) and (start < all): # # settle for Hamming Distance of 2 if we run out of 4s # hd = gmpy.hamdist(pbin[start],schedule[-1]) if hd == 2: got_it = True else: start += 1 if got_it: schedule.append(pbin.pop(start)) else: found_all_2 = True print print The schedule calculated by Hamming Distance for i in schedule: print '%6d %s' % (i,gmpy.digits(i,2).zfill(12)) ## Combinations without Replacement: 66 ## ## The Combinations without Replacement ## ab ac ad ae af ag ah ai aj ak al ## bc bd be bf bg bh bi bj bk bl cd ## ce cf cg ch ci cj ck cl de df dg ## dh di dj dk dl ef eg eh ei ej ek ## el fg fh fi fj fk fl gh gi gj gk ## gl hi hj hk hl ij ik il jk jl kl ## ## The combinations as 12-bit numbers each with 2 bits set ## 3 000000000011 ## 5 000000000101 ## 9 000000001001 ## 17 000000010001 ## 33 000000100001 ## 65 000001000001 ## 129 000010000001 ## 257 000100000001 ## 513 001000000001 ## 1025 010000000001 ## 2049 100000000001 ## 6 000000000110 ## 10 000000001010 ## 18 000000010010 ## 34 000000100010 ## 66 000001000010 ## 130 000010000010 ## 258 000100000010 ## 514 001000000010 ## 1026 010000000010 ## 2050 100000000010 ## 12 000000001100 ## 20 000000010100 ## 36 000000100100 ## 68 000001000100 ## 132 000010000100 ## 260 000100000100 ## 516 001000000100 ## 1028 010000000100 ## 2052 100000000100 ## 24 000000011000 ## 40 000000101000 ## 72 000001001000 ## 136 000010001000 ## 264 000100001000 ## 520 001000001000 ## 1032 010000001000 ## 2056 100000001000 ## 48 000000110000 ## 80 000001010000 ## 144 000010010000 ## 272 000100010000 ## 528 001000010000 ## 1040 010000010000 ## 2064 100000010000 ## 96 000001100000 ## 160 000010100000 ## 288 000100100000 ## 544 001000100000 ## 1056 010000100000 ## 2080 100000100000 ## 192 000011000000 ## 320 000101000000 ## 576 001001000000 ## 1088 010001000000 ## 2112 100001000000 ## 384 000110000000 ## 640 001010000000 ## 1152 010010000000 ## 2176 100010000000 ## 768 001100000000 ## 1280 010100000000 ## 2304 100100000000 ## 1536 011000000000 ## 2560 101000000000 ## 3072 110000000000 ## ## The 12-bit numbers randomly shuffled ## 513 001000000001 ## 68 000001000100 ## 20 000000010100 ## 1040 010000010000 ## 10 000000001010 ## 1032 010000001000 ## 1152 010010000000 ## 129 000010000001 ## 2112 100001000000 ## 640 001010000000 ## 6 000000000110 ## 130 000010000010 ## 9 000000001001 ## 18 000000010010 ## 36 000000100100 ## 192 000011000000 ## 384 000110000000 ## 1025 010000000001 ## 12 000000001100 ## 257 000100000001 ## 2052 100000000100 ## 48 000000110000 ## 768 001100000000 ## 34 000000100010 ## 2560 101000000000 ## 258 000100000010 ## 514 001000000010 ## 1056 010000100000 ## 1088 010001000000 ## 2049 100000000001 ## 1280 010100000000 ## 516 001000000100 ## 132 000010000100 ## 72 000001001000 ## 2304 100100000000 ## 544 001000100000 ## 264 000100001000 ## 260 000100000100 ## 144 000010010000 ## 2176 100010000000 ## 1536 011000000000 ## 96 000001100000 ## 272 000100010000 ## 1026 010000000010 ## 33 000000100001 ## 3 000000000011 ## 5 000000000101 ## 80 000001010000 ## 2056 100000001000 ## 288 000100100000 ## 40 000000101000 ## 2064 100000010000 ## 3072 110000000000 ## 160 000010100000 ## 320 000101000000 ## 66 000001000010 ## 528 001000010000 ## 65 000001000001 ## 520 001000001000 ## 17 000000010001 ## 2080 100000100000 ## 24 000000011000 ## 576 001001000000 ## 2050 100000000010 ## 136 000010001000 ## 1028 010000000100 ## ## The schedule calculated by Hamming Distance ## 513 001000000001 ## 68 000001000100 ## 1040 010000010000 ## 10 000000001010 ## 20 000000010100 ## 1032 010000001000 ## 129 000010000001 ## 2112 100001000000 ## 1152 010010000000 ## 6 000000000110 ## 640 001010000000 ## 9 000000001001 ## 130 000010000010 ## 36 000000100100 ## 18 000000010010 ## 192 000011000000 ## 1025 010000000001 ## 384 000110000000 ## 12 000000001100 ## 257 000100000001 ## 2052 100000000100 ## 48 000000110000 ## 768 001100000000 ## 34 000000100010 ## 2560 101000000000 ## 258 000100000010 ## 1056 010000100000 ## 514 001000000010 ## 1088 010001000000 ## 2049 100000000001 ## 1280 010100000000 ## 516 001000000100 ## 72 000001001000 ## 132 000010000100 ## 2304 100100000000 ## 544 001000100000 ## 264 000100001000 ## 144 000010010000 ## 260 000100000100 ## 2176 100010000000 ## 1536 011000000000 ## 96 000001100000 ## 272 000100010000 ## 1026 010000000010 ## 33 000000100001 ## 80 000001010000 ## 3 000000000011 ## 2056 100000001000 ## 5 000000000101 ## 288 000100100000 ## 2064 100000010000 ## 40 000000101000 ## 3072 110000000000 ## 160 000010100000 ## 320 000101000000 ## 528 001000010000 ## 66 000001000010 ## 520 001000001000 ## 65 000001000001 ## 2080 100000100000 ## 17 000000010001 ## 576 001001000000 ## 24 000000011000 ## 2050 100000000010 ## 136 000010001000 ## 1028 010000000100 === Subject: Re: Unusual Problem > I have 12 persons numbered from 1 to 12 that I am making a work plan for. > They are going to work together two and two for a week. I am trying to > accomplish this under the following conditions: 1) A person should try to work with all the other persons. 2) All persons should work the same number of weeks before the plan > restarts. 4) The plan should be well distributed. Something like: > (1,2)(1,3)(1,4)(1,5)(1,6)(1,7)(1,8)(1,9)(1,10)(1,11) (1,12) > should be avoided since person 1 will get tired. First I found that there are 66 combinations of unique pairs (since > identical pairs are not allowed and reversed order should not be counted > either): (1,2)(1,3)(1,4)(1,5)(1,6)(1,7)(1,8)(1,9)(1,10)(1,11) (1,12) > (2,3)(2,4)(2,5)(2,6)(2,7)(2,8)(2,9)(2,10)(2,11) (2,12) > (3,4)(3,5)(3,6)(3,7)(3,8)(3,9)(3,10)(3,11) (3,12) > (4,5)(4,6)(4,7)(4,8)(4,9)(4,10)(4,11) (4,12) > (5,6)(5,7)(5,8)(5,9)(5,10)(5,11) (5,12) > (6,7)(6,8)(6,9)(6,10)(6,11) (6,12) > (7,8)(7,9)(7,10)(7,11) (7,12) > (8,9)(8,10)(8,11) (8,12) > (9,10)(9,11) (9,12) > (10,11)(10,12) > (11,12) I seems that the worst problem is 4). I have tried to group the paris in 6 > rows containing 11 pairs but after filling out the first two rows the > problems get impossible to keep track of. Are there any kind of techniques to distributes this kind of data based on > the condition that there should be a few weeks before a person should work > again and that the same pair is not allowed before all have been working > together? You can formulate your problem as in integer programming problem (depending on the form of the objective). For n people, there are N = n(n-1)/2 slots and each person must work (n-1) of them; in your case, n = 12 and N = 66. Define binary variables x(i,j,k) = 1 if pair (i,j) works in slot k, for i <> j and k = 1,...,N, and x(i,j,k) = 0 otherwise. Define binary variable y(i,k) = 1 if person i works in slots k and k+1, y(i,k) = 0 otherwise. Say we want to minimize the total number of adjacent slots worked by all people. A formulation is as follows: minimize Z = sum(y(i,k): i=1..n, k=i..N), subject to the following constraints: (1) each pair works a single slot, so sum(x(i,j,k): k = 1..N) = 1 for all pairs i <> j; (2) each slot is worked by a single pair, so sum(x(i,j,k): i <> j) = 1 for all k; (3) if pair (i,j) works slot k then so does pair (j,i), so x(i,j,k) = x(j,i,k) for all i <> j and all k; (4) y(i,k) is forced to be 1 if worker i works both slots k and k+1, so y(i,k) >= sum(x(i,j,k) + x(i,j,k+1); j=1..n, j <> i) - 1. Note that when some x(i,j,k) = 1 and some x(i,j',k+1) = 1 we have y(i,k) >= 1, hence y(i,k) = 1 because y is binary. When x(i,j,k) = 0 for all j or x(i,j',k+1) = 0 for all j', then y(i,k) >= 0 or y(i,k) >= -1, hence y(i,k) >=0; minimizing y will force y = 0 in this case. Thus, Z will properly represent the total number of adjacent slots for all the workers. Of course, this formulation may be rather large. In your example it has 66^2 binary variables x, 12*66 binary variables y, 66 constraints of type (1), 66 of type (2), 66^2 of type (3) and 12*66 of type (4). Good IP codes may be able to handle the problem, however. Also, it might be worth exploring the possibility of smaller formulations. Also, of course, you may not agree with the objective given above. R.G. Vickson === Subject: Re: Unusual Problem >I have 12 persons numbered from 1 to 12 that I am making a work plan for. >They are going to work together two and two for a week. I am trying to >accomplish this under the following conditions: 1) A person should try to work with all the other persons. 2) All persons should work the same number of weeks before the plan >restarts. 4) The plan should be well distributed. Something like: >(1,2)(1,3)(1,4)(1,5)(1,6)(1,7)(1,8)(1,9)(1,10)(1,11) (1,12) >should be avoided since person 1 will get tired. First I found that there are 66 combinations of unique pairs (since >identical pairs are not allowed and reversed order should not be counted >either): (1,2)(1,3)(1,4)(1,5)(1,6)(1,7)(1,8)(1,9)(1,10)(1,11) (1,12) >(2,3)(2,4)(2,5)(2,6)(2,7)(2,8)(2,9)(2,10)(2,11) (2,12) >(3,4)(3,5)(3,6)(3,7)(3,8)(3,9)(3,10)(3,11) (3,12) >(4,5)(4,6)(4,7)(4,8)(4,9)(4,10)(4,11) (4,12) >(5,6)(5,7)(5,8)(5,9)(5,10)(5,11) (5,12) >(6,7)(6,8)(6,9)(6,10)(6,11) (6,12) >(7,8)(7,9)(7,10)(7,11) (7,12) >(8,9)(8,10)(8,11) (8,12) >(9,10)(9,11) (9,12) >(10,11)(10,12) >(11,12) I seems that the worst problem is 4). I have tried to group the paris in 6 >rows containing 11 pairs but after filling out the first two rows the >problems get impossible to keep track of. Are there any kind of techniques to distributes this kind of data based on >the condition that there should be a few weeks before a person should work >again and that the same pair is not allowed before all have been working >together? Check out round-robin tournaments: Rob Johnson take out the trash before replying === Subject: random walk on Z^2 Hi all, suppose that X_n is a random walk on Z^2 (probability p=1/4 to go up/ down/right/left) starting from 0. Let Q(a,b) be a point in Z^2. It is well known that the random walk will reach that point with probability 1. I am looking for E(T_A) where T_A = inf(n > 0 : X_n = A) ? (that is: how long will this random walk take to reach that point) How would you do that ? Fedor === Subject: Re: random walk on Z^2 >Hi all, > suppose that X_n is a random walk on Z^2 (probability p=1/4 to go up/ >down/right/left) starting from 0. Let Q(a,b) be a point in Z^2. It is >well known that the random walk will reach that point with probability >1. I am looking for E(T_A) where T_A = inf(n > 0 : X_n = A) ? (that >is: how long will this random walk take to reach that point) How would >you do that ? Even in one dimension it is infinite. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Another of what appears every now & again;stupid questions! > If Hydrogen can lift 1.1 Kilograms per M=meter 3^ cubed, how much > hydrogen would it take to lift 110,000 lbs.; specifically to a > launchable height for an Earth orbit? What would the size of something > be? Mathematically, the answer is 45360 m^3, which is > the volume of a sphere of radius 22.12... metres, > or diameter 145 feet. The biggest airships of the > 1930s had over 4 times that volume, and a useful > lift of 250,000 lbs, more than twice your target. Physically, however, the diminishing density of > air with height means that this not a practicable > method of reaching orbital altitude. Not orbital altitude, of course, but it might be a feasible way of reaching an altitude high enough that air resistance is not as much of a problem when you do launch your rocket. Some of the competitors for the Ansari X-prize launched from balloons. That 1.1 kg/m^3 is at standard temperature and pressure (i.e. essentially at sea level). If you want to lift to a higher altitude, the mass of the hydrogen won't change, but you need a larger envelope to put it in (so that as you ascend, the hydrogen can expand and keep the same pressure as the surrounding air). -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Another of what appears every now & again;stupid questions! > If Hydrogen can lift 1.1 Kilograms per M=meter 3^ cubed, how much > hydrogen would it take to lift 110,000 lbs.; specifically to a > launchable height for an Earth orbit? What would the size of something > be? Watching Mars rising so it peaked my interests! > Keith Putting something in orbit is not so much about acheiving a particular height, but of a particular velocity, namely orbital velocity. http://en.wikipedia.org/wiki/Orbital_speed === Subject: Re: Another of what appears every now & again;stupid questions! If Hydrogen can lift 1.1 Kilograms per M=meter 3^ cubed, how much > hydrogen would it take to lift 110,000 lbs.; specifically to a > launchable height for an Earth orbit? What would the size of something > be? Watching Mars rising so it peaked my interests! Putting something in orbit is not so much about acheiving a particular > height, but of a particular velocity, namely orbital velocity. http://en.wikipedia.org/wiki/Orbital_speed Yes, but at a sufficient height, that orbital velocity relative to the earth is zero (geostationary orbit). Still, as this is well above the atmosphere, it does not help the original poster. -- === Subject: Re: Determinant proof I have three matrices (in Matlab notation): a = rand(4,4); b = rand(3,3); p = rand(3,4); I'm trying to find an analytical proof of the following statement: det(eye(size(b))+b*p*inv(a)*p') = det(eye(size(a))+inv(a)*p'*b*p) > http://en.wikipedia.org/wiki/Sylvester%27s_determinant_theorem I.e. det(I+AB) = det(I+BA). I don't ever recall seeing this called > Sylvester's determinant theorem. Does anyone know a reference? The proof is easy: in the polynomial ring Z[Aij,Bij] simply > > What is det(A) when A is non-square? Append 0's to A,B till square. It doesn't change det of I+AB, I+BA. > >Perhaps he meant: >(1) prove it for square matrices, as above. >(2) If A and B are non-square, say A is m x n and B is n x m with >m > n, then append m-n columns of 0's to A and m-n rows of 0's to B, >and note that > [ B ] >det(I + AB) = det(I + [A 0] [ 0 ]) > [I 0 ] [ B ] >det(I + BA) = det([0 I ] + [ 0 ] [A 0]) Expanding on this a bit, suppose A is mxn and B is nxm where n >= m. >Using this result, it is pretty simple to show that n-m > det(Ix - BA) = x det(Ix - AB) That is, the characteristic polynomial of BA is x^{n-m} times that of >AB. I assume this is well known, does it have a name? I have tried >looking a bit, but I have not found anything yet. I have found a reference for the m = n case. In The Characteristic Polynomial of a Product, Ralph Howard calls this the most notorious of all qualifying exam questions: Wikipedia mentions the full result, but only sketches a proof of the m = n case: Rob Johnson take out the trash before replying === Subject: Re: Epistemology 501: Angular Mechanics > So if distance r doesn't interact with linear momentum p=mv to produce > rotation something else must interact with linear p=mv to produce > rotation and distance r instead.And the something else is transverse > acceleration, moron. Why certainly. If velocity changes there must be >an acceleration. I don't believe I've ever said >anything different. Nobody would. You may have even >heard the term centripetal acceleration around here >somewhere. Are you making the claim that I, or somebody else, >claimed there's no such thing as centripetal >acceleration or that acceleration is not involved >in rotational motion? If you care to amplify that we're all going to >learn that once again you have completely misunderstood >some elementary point somebody was making. But >feel free to make a fool of yourself in a new way. >Complete with the profanity if that helps. Well you see, imbecile, the classical formulation for angular momentum L=r x p you defend doesn't acknowledge transverse acceleration. ~v~~ === Subject: Re: Epistemology 501: Angular Mechanics > Distance doesn't interact with velocity, asshole. > of which p is compounded produces distance in turn and not vice versa. Velocity and distance have a relationship. Distance >(or rather position) is the integral of velocity, >velocity is the derivative of position. You can get >one from the other. Not sure why you say that the >integral relationship is a more privileged one than >the derivative relationship. In both cases you derive >one from the other. Even swearing won't change that. Only because acceleration produces velocity and velocity produces distance, asshole. ~v~~ === Subject: Re: Epistemology 501: Angular Mechanics > I asked whether distance affects p in terms > of velocity. I can't make sense of that except as asking whether >velocity changes with distance, and the answer is >the one I gave: sometimes yes. Then hie thee to a nunnery. Or a convent might be more appropriate. ~v~~ === Subject: Re: Epistemology 501: Angular Mechanics > I didn't enquire of your stupid little majesty whether gravitational > force varies with distance. Yes, and that's too vague a question to answer. So >I gave an example where it does. In other cases it >might not. Gravitational force might not vary with distance? Hmmm. Even your stupidity would appear to know no boundaries. Have you ever considered becoming a Jesuit? I understand they need a few good ignorati. ~v~~ === Subject: Re: Epistemology 501: Angular Mechanics >Too vague a question. Distance obviously affects >velocity in the case of a planet moving around >a sun, as moving farther away from the sun requires >a climb upward in potential energy and thus a loss >of kinetic energy. > So when I ask a question you consider too vague you just claim you > answered it anyway? You're a ing cretin. Nope. Never claimed to have answered that specific >question, since I just saw it for the first time. >Stop lying, Lester. I know, Randy. You just claimed to have answered all my questions. I would assume all would have included that question. But maybe the predicate all is a word which you don't understand without examples. ~v~~ === Subject: Re: Epistemology 501: Angular Mechanics >Zick probably does not know that bxb is the zero vector. Actually it >isn't that he doesn't know it, it is that he cannot comprehend it. Zick >is unable to cope with mathematics which, according to Robert Heinlein, >shows that he is a subhuman. So are you suggesting that the functional dependence of L=r x p in the classical formulation of angular momentum doesn't depend on rp and is subhuman, Bobby? ~v~~ === Subject: Re: Epistemology 501: Angular Mechanics > Are you seriously suggesting the product rp has no bearing on the > calculation of the cross product L=r x p? Are you really brain dead? It has a bearing. Good. Then we're done here. ~v~~ === Subject: Re: Epistemology 501: Angular Mechanics >But it's easy enough to show exactly what L is for >motion in a straight line. Shall we calculate it? >Or shall we prove your intellectual dishonesty >again by having you refuse to look at the actual >calculation of L, and instead just declaim what >you think it is because you don't understand what >that little x means? > Who needs to look at the actual calculation of L when we have the > classical formulation for angular momentum L=r x p which would be true > for all calculations of L? Because without seeing what r x p is, you feel free >to make all sorts of untrue statements about r x p. >You don't KNOW what the classical formulation is >because you refuse to look at what actual values >of r x p are for different situations. The point is the classical formulation for angular momentum L=r x p is supposed to be true for all situations. ~v~~ === Subject: Re: Epistemology 501: Angular Mechanics <6j5hj3darsr619i328eralk0di4sm4b3gg@4ax.com > So you mean you're unable to deduce the properties of the functional > dependence of angular momentum on r from classical formulation of > angular momentum L=r x p? One is forced to wonder why bother with > formulas at all? You can deduce the functional dependence of angular momentum on r. But it helps to know what r x p denotes. Assuming that it entails the product rp would lead to errors, as you've so ably illustrated. PD === Subject: Re: Epistemology 501: Angular Mechanics > So you mean you're unable to deduce the properties of the functional > dependence of angular momentum on r from classical formulation of > angular momentum L=r x p? One is forced to wonder why bother with > formulas at all? You can deduce the functional dependence of angular momentum on r. Good. Then we're done here. > But >it helps to know what r x p denotes. You don't know what r x p denotes? Jesus, you're an idiot. > Assuming that it entails the >product rp would lead to errors, as you've so ably illustrated. It doesn't entail the product rp? Go back to philosophy, Paul. ~v~~ === Subject: Re: Epistemology 501: Angular Mechanics <1t9aj39aaq9i02bv1o7mo0bij9r4949okf@4ax.com> It was instead something that you just made up. Or perhaps you'd like >to deny saying it? BORING!!! I certainly deny saying I deduced it from L=r x p. > Ah, fine. Apparently you just made it up, then. Wonderful, we just needed some clarity on that. PD === Subject: Re: Epistemology 501: Angular Mechanics >It was instead something that you just made up. Or perhaps you'd like >to deny saying it? > BORING!!! I certainly deny saying I deduced it from L=r x p. Ah, fine. Apparently you just made it up, then. Wonderful, we just >needed some clarity on that. Take my word for it, you need some clarity on a lot more than that. ~v~~ === Subject: Re: Epistemology 501: Angular Mechanics <1t9aj39aaq9i02bv1o7mo0bij9r4949okf@4ax.com> entirely possible that for a body in straight line motion and r=00, >the value of L can be 3. The value of L is certainly not driven to >infinity when r goes to infinity. Good. Then we can dismiss the classical formulation of angular > momentum as L=r x p as incorrect out of hand on your say so. Not so. You just don't know what the notation r x p denotes. You apparently think it entails the product rp. it does not. The fact that you misled yourself on the denotation is not anyone's problem but yours. PD === Subject: Re: Epistemology 501: Angular Mechanics >entirely possible that for a body in straight line motion and r=00, >the value of L can be 3. The value of L is certainly not driven to >infinity when r goes to infinity. > Good. Then we can dismiss the classical formulation of angular > momentum as L=r x p as incorrect out of hand on your say so. Not so. You just don't know what the notation r x p denotes. Good. That qualifies me to talk to you. > You >apparently think it entails the product rp. it does not. L=r x p does not entail the product rp? Paul, you're an imbecile. ~v~~ === Subject: Re: Epistemology 501: Angular Mechanics <1t9aj39aaq9i02bv1o7mo0bij9r4949okf@4ax.com> Well you see, Lester, it was YOU that deduced from the classical >formulation L=rxp a statement that L decreases as tranverse >acceleration increases. > You're an idiot, Paul. I deduced nothing of the kind from the > classical definition of angular momentum L=r x p. >Well, since you said it, you either deduced it or you made it up. In >your case it could be either. > momentum L=r x p I deduced the functional dependence of L on r ranges > from L=0 at r=0 to infinite at r=00 for motion in a straight line. No, it does NOT. As I've told you repeatedly, what you've just said is >true if and only if the x is to be taken as an algebraic times. >However, in the definition of L=r x p, x is NOT an algebraic >times, it is a vector cross product operation. Your complete lack of >understanding of what it means doesn't change that. Well see, Paul, the root of your stupidity lies in your arrogant > pronouncement of arbiter dicta as if you knew whereof you spake. Are you seriously suggesting L=r x p does not entail the product rp? > That's correct. There's more to it than the product rp. It appears that after hollering in your ear for days about this, it's beginning to dawn on you. I'm not surprised it has taken this long for you to get a glimmer. L=r x p is a *vector* equation, with r, p, and L each being vectors. Vectors can either be represented by sets of components (3 components in the classical formulation), or by a magnitude and two directional angles. In the coordinate representation, where the vector Y is represented as [Y1, Y2, Y3], this classical formulation is written as L = [L1, L2, L3] = [(r2*p3 - r3*p2), (r3*p1 - r1*p3), (r1*p2 - r2*p1)] which as you can see is quite a bit different than rp. Alternatively, from the 2 directional angles belonging to r and p each, you can deduce a angle between r and p, which we'll call theta. Realizing that L is a *vector*, the *magnitude* of the L vector is given by |L| = r*p*sin(theta) which as you can see is quite a bit different than L=rp. Furthermore, you're not done, as you've not specified the 2 directional angles of L, which you still need to do as L is a vector. Those are determined from the directional angles of r and p by a tedious but straightforward process which is generally captured by the right-hand rule mnemonic. All of this apparently comes as a surprise to you, as you've apparently come to several conclusions based on the assumption that L = r x p entails the product rp, despite repeated warnings otherwise. That's because you, Lester, are an uneducated pretender, a buffoon who prefers prose to comprehension. PD PD === Subject: Re: Epistemology 501: Angular Mechanics [Blah, blah, blah] yadayada whatever BORING ~v~~ === Subject: Re: Epistemology 501: Angular Mechanics > Well see, Paul, the root of your stupidity lies in your arrogant > pronouncement of arbiter dicta as if you knew whereof you spake. > Are you seriously suggesting L=r x p does not entail the product rp? That's correct. There's more to it than the product rp Did I say there wasn't more to it? The point is that the product rp is there. In other words L=r x p does entail the product rp. ~v~~ === Subject: Re: Epistemology 501: Angular Mechanics <004hj3tt9m7v984uli6eci75i03j8hjeul@4ax.com> Now consider r x p L = r x p = (a + bt) x mb > = (a x m) + (b x b)t > = a x m, a constant. That should be m*(axb). Arggh. Yes, of course. Typed too fast. > I am sure this is just a type. Since a and b are > constant, so ia axb. L = (a x mb) + (b x mb)t = a x mb, which is constant. > Zick probably does not know that bxb is the zero vector. He can't comprehend that the cross-product is different from the ordinary product, and refuses to look at any actual cross-product calculations as that would disturb his world view. They're not relevant, you see. But pronouncements without having ever learned the properties of cross-products, why those are relevant. - Randy === Subject: Re: Epistemology 501: Angular Mechanics >He can't comprehend that the cross-product is different >from the ordinary product, and refuses to look at any >actual cross-product calculations as that would >disturb his world view. They're not relevant, you >see. But pronouncements without having ever learned >the properties of cross-products, why those are >relevant. Are you seriously suggesting the cross product L=r x p doesn't depend on the product rp? ~v~~ === Subject: Re: The Virgin Birth of Points > > So if you unionize an infinite number of points, would the converse > operation be decertification of the union and wouldn't that constitute > division by zero? You are making no sense here. Division is the inverse operation to >multiplication. So? I'm just asking what the inverse operation of the unionization of points is. ~v~~ === Subject: Re: The Virgin Birth of Points So? I'm just asking what the inverse operation of the unionization of > points is. There is none. THe set operations do not form a group. But, of course, you knew that. The set operations constitute a lattice. Bob Kolker === Subject: Re: The Virgin Birth of Points > I think you need to learn some measure theory. This is a question about > mathematics, by the way, not philosophy. Zick is totally incapable of understanding either mathematics or >physics. Well I know enough of mathematics to have convinced you there is no real number line. ~v~~ === Subject: Re: The Virgin Birth of Points > Well I know enough of mathematics to have convinced you there is no > real number line. So what. The theory of real numbers can and is developed without any geometric content of all. Any geometrical associations with real numbers are merely aids to intuition, not logical necessity. In the nineteenth century a purely analytic foundations for the theory of real and complex variables was developed. Geometry was purged as a logical necessity. Of course, geometry can be very helpful for the right-brain operations associated with discovering new theorems to prove or new mathematical systems. Bob Kolker === Subject: Re: The Virgin Birth of Points > > > Then I'm curious about this unionizing of points people talk about. You are curious about sets (and no wonder, you know nothing about them). Yes but on the plus side I suffer fools gladly. ~v~~ === Subject: Re: The Virgin Birth of Points > > > No. Your main purpose in this thread is the same as in any other of > your threads. And that is the intentional obfuscation of established > mathematical concepts. > > >He is unable to do otherwise. He cannot comprehend standard mathematical >concepts. Zick cannot cope with mathematics. Robert Heinlein had some >interesting things to say about people like Zick. Yes, I know, Bobby. He said they don't suffer heresies like modern mathematics, relativity, and quantum theory gladly. ~v~~ === Subject: Re: The Virgin Birth of Points > The Virgin Birth of Points > ~v~~ > The Jesuit heresy maintains points have zero length but are not of > zero length and if you don't believe that you haven't examined the > argument closely enough. >Clearly points don't have zero length, they have a positive infinitesimal >length for which zero is just the closest real approximation. > Erm, no. Points (or rather singletons) have zero length. >I agree. Also, like I said in the other post, points can only exist as >boundaries of higher dimensional regions. Lines, surfaces, solids etc >can exist as regions in their own world and as boundaries in higher >dimensions. When they are in the role of a boundary they are not part >of any regions (of higher dimension). >We can't observe life of a point as a region in its own dimensional >space. > Except the main purpose of this thread is less to discuss the zero > length of points than the heresy of maintaining self contradictory > predicates, as in has zero length and is not of zero length. >No. Your main purpose in this thread is the same as in any other of >your threads. And that is the intentional obfuscation of established >mathematical concepts. I know, Igor. I'm just making an exception in this particular case. ~v~~ === Subject: Re: The Virgin Birth of Points > > Except the main purpose of this thread is less to discuss the zero > length of points than the heresy of maintaining self contradictory > predicates, as in has zero length and is not of zero length. Points do not have a length (0 or not). Some -sets- of points have >-measure-. In particular a set consisting of a single point has measure 0. You have manage to confuse an object with a set whose element is that >object. Hey it's not my problem, Bobby. I'm not the one who claims points have zero length but are not of zero length.Modern mathematics is a heresy. ~v~~ === Subject: Re: The Virgin Birth of Points > Hey it's not my problem, Bobby. I'm not the one who claims points have > zero length but are not of zero length.Modern mathematics is a heresy. Neither does any one else. You have created a straw man here. Measure is associated with certain -sets of points-, not the points themselves. Bob Kolker === Subject: Re: The Virgin Birth of Points > Hey it's not my problem, Bobby. I'm not the one who claims points have > zero length but are not of zero length.Modern mathematics is a heresy. Ultra-heretic Zick accusing others of his own sin? It is to laugh! === Subject: Re: The Virgin Birth of Points > Lines (which are sets of points) sometimes have a non-zero set >intersection which consists of a single point. And apparently sometimes they don't? So what is it exactly you're sometimes saying, Bobby? >Once again you do not distinguish between objects and the sets of which >the objects are elements. Another evidence that you cannot cope with >mathematics. But I can certainly cope with the likes of homo habilis. ~v~~ === Subject: Re: The Virgin Birth of Points > And apparently sometimes they don't? So what is it exactly you're > sometimes saying, Bobby? Parellel lines have no points of intersection. Next question? Bob Kolker === Subject: Re: The Virgin Birth of Points > > > I think you need to learn some measure theory. This is a question about > mathematics, by the way, not philosophy. Zick is totally incapable of understanding either mathematics or >physics. Robert Heinlein had some clever things to say about people who >cannot cope with mathematics. Heinlein said they are subhuman but >capable of wearing shoes and keeping clean. Unlike yourself, Bobby. ~v~~ === Subject: Re: The Virgin Birth of Points <96ggj3pohi18rck2tl0gackdfsn95eeg9q@4ax.com> <2j2hj3dvclqnq3qomcnm15bt4d31c984gh@4ax.com unionizing of points Wasn't that a mafia racket that started on the waterfront? Oh, hang on, that was the unionizing of punts. Never mind. === Subject: Re: The Virgin Birth of Points > unionizing of points Wasn't that a mafia racket that started on the waterfront? Oh, hang on, >that was the unionizing of punts. Never mind. Actually in the New Yorker, the unionizing of puns. ~v~~ === Subject: Re: The Virgin Birth of Points The Virgin Birth of Points > ~v~~ The Jesuit heresy [snip crap] Idiot. The Giant Flying Spaghetti Monster heresy. The thinner thighs in 30 days heresy. 1) Who gives one what god thinks? 2) Who gives one what borign idiot Lester Zick spews? Uncle Al sold his soul to Satan in a simple contract: Post mortality, for all of remaining eternity, Uncle Al will torture meat puppets like you, Zick, in a choice part of Hell. Uncle Al gets ice in his drink and a fine party into eternity. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/lajos.htm#a2 === Subject: Re: The Virgin Birth of Points > > The Virgin Birth of Points > ~v~~ > > The Jesuit heresy >[snip crap] Idiot. The Giant Flying Spaghetti Monster heresy. The thinner thighs >in 30 days heresy. 1) Who gives one what god thinks? Plainly Jesuits do. > 2) Who gives one what borign idiot Lester Zick spews? borign? I'm not familiar with that word. Pray tell, is it a learned borrowing from the Jesuit? >Uncle Al sold his soul to Satan in a simple contract: Post mortality, >for all of remaining eternity, Uncle Al will torture meat puppets like >you, Zick, in a choice part of Hell. Uncle Al gets ice in his drink >and a fine party into eternity. Al, look, you're just raving now. Obviously you have even less coherence than usual. Tell you what, why not take a couple steps back from the subject and go yourself. ~v~~ === Subject: Re: The Virgin Birth of Points > Uncle Al sold his soul to Satan in a simple contract: Post mortality, > for all of remaining eternity, Uncle Al will torture meat puppets like > you, Zick, in a choice part of Hell. Uncle Al gets ice in his drink > and a fine party into eternity. And a hummer from his Lady. Bob Kolker > === Subject: Re: The Virgin Birth of Points intersection which consists of a single point. Once again you do not distinguish between objects and the sets of which > the objects are elements. Another evidence that you cannot cope with > mathematics. Bob Kolker A line is not a set of points because sets are indifferent to order. However, if you care to order points we still do not have a minimal definition of a line. === Subject: Re: The Virgin Birth of Points > A line is not a set of points because sets are indifferent to order. > However, if you care to order points we still do not have a minimal > definition of a line. Consider E2, the set of number pairs (x,y) x,y real taken as points. Along with the pythagorian metric and the obvious definition of lines (sets of (x,y) which satisfy a*x + b*y = c for some constants a,b,c) you get a structure that satistfies Hilberts postulates for plane geometric space. Since the axioms are categorica, all instances of Euclidea plane geometry (as axiomatized by Hilbert) are isometric. So a model where lines consist of points yields an instance of the geometry. Since the line can be parametrized by a single variable it can be easily ordered. Where did you get you degree? I need to know, so I won't send my kids there. Bob Kolker > === Subject: Re: The Virgin Birth of Points the objects are elements. Another evidence that you cannot cope with > mathematics. > A line is not a set of points because sets are indifferent to order. > However, if you care to order points we still do not have a minimal > definition of a line. I've been thinking about the links to Euclid's and Hilbert's axioms presented in some of the other geometry threads: http://en.wikipedia.org/wiki/Hilbert%27s_axioms These last few posts are posing the question, is a point an _element_ of a line, or is a point a _subset_ of a line? The correct answer is neither. For let us review Hilbert's axioms again: The undefined primitives are: point, line, plane. There are three primitive relations: Betweenness, a ternary relation linking points; Containment, three binary relations, one linking points and lines, one linking points and planes, and one linking lines and planes; Congruence, two binary relations, one linking line segments and one linking angles. So we see that line is an undefined _primitive_, and that there is a _primitive_ to be known as containment, so that a line may be said to contain points. Notice that the primitive contain has _nothing_ to do with the membership primitive of a set theory such as ZFC. Why? Because this is a geometric theory that is not even written in the _language_ of ZFC. So both a point is an element of a line and a point is a subset of a line are incorrect. The other question concerns what the intersection of two lines is. Well, first we must define intersection -- in terms of our _primitives_, of course -- before we can answer. And the only answer we can possibly give is in terms of _containment_: the intersection of two lines a,b is a point A such that a contains A and b contains A as well, provided that such a point exists. We can't call it a position, since position is not a _primitive_ of our theory. So now all we have to do is prove that if such a point exists, it must be unique. But this follows directly from Axiom I.1. For if there were two points of intersection A,B, then I.1 tells us that two points determine a line, so that AB = a and AB = b as well, therefore a = b. So if a,b are distinct and intersect, then they intersect in a unique point of intersection. === Subject: Re: The Virgin Birth of Points > Once again you do not distinguish between objects and the sets of which > the objects are elements. Another evidence that you cannot cope with > mathematics. > A line is not a set of points because sets are indifferent to order. > However, if you care to order points we still do not have a minimal > definition of a line. I've been thinking about the links to Euclid's and Hilbert's >axioms presented in some of the other geometry threads: Guesswork gives me a headache. Please spare us undemonstrated assumptions of truth. ~v~~ === Subject: Re: The Virgin Birth of Points Once again you do not distinguish between objects and the sets of which > the objects are elements. Another evidence that you cannot cope with > mathematics. > A line is not a set of points because sets are indifferent to order. > However, if you care to order points we still do not have a minimal > definition of a line. I've been thinking about the links to Euclid's and Hilbert's > axioms presented in some of the other geometry threads: http://en.wikipedia.org/wiki/Hilbert%27s_axioms These last few posts are posing the question, is a > point an _element_ of a line, or is a point a > _subset_ of a line? The correct answer is neither. For let us review > Hilbert's axioms again: The undefined primitives are: point, line, plane. > There are three primitive relations: Betweenness, a ternary relation linking points; > Containment, three binary relations, one linking > points and lines, one linking points and planes, > and one linking lines and planes; > Congruence, two binary relations, one linking line > segments and one linking angles. So we see that line is an undefined _primitive_, > and that there is a _primitive_ to be known as > containment, so that a line may be said to > contain points. Notice that the primitive contain has _nothing_ > to do with the membership primitive of a set > theory such as ZFC. Why? Because this is a > geometric theory that is not even written in > the _language_ of ZFC. So both a point is an element of a line and a > point is a subset of a line are incorrect. The other question concerns what the intersection > of two lines is. Well, first we must define > intersection -- in terms of our _primitives_, > of course -- before we can answer. And the only > answer we can possibly give is in terms of > _containment_: the intersection of two lines a,b > is a point A such that a contains A and b contains > A as well, provided that such a point exists. We > can't call it a position, since position is > not a _primitive_ of our theory. So now all we have to do is prove that if such a > point exists, it must be unique. But this follows > directly from Axiom I.1. For if there were two > points of intersection A,B, then I.1 tells us that > two points determine a line, so that AB = a and > AB = b as well, therefore a = b. So if a,b are > distinct and intersect, then they intersect in a > unique point of intersection. A position may well not be a primitive, but the intersections of lines construct positions, not points. Primitives are incommensurables. Points, lines, planes, etc are incommensurables which do not contain the properties of one within the other. Their 'synthesis' is not a synthesis of properties or objects, but of the frameworks that establish objects and properties (see Kant). === Subject: Re: The Virgin Birth of Points point exists, it must be unique. But this follows > directly from Axiom I.1. For if there were two > points of intersection A,B, then I.1 tells us that > two points determine a line, so that AB = a and > AB = b as well, therefore a = b. So if a,b are > distinct and intersect, then they intersect in a > unique point of intersection. > A position may well not be a primitive, but the intersections of lines > construct positions, not points. Primitives are incommensurables. > Points, lines, planes, etc are incommensurables which do not contain > the properties of one within the other. Their 'synthesis' is not a > synthesis of properties or objects, but of the frameworks that > establish objects and properties (see Kant). I'm sorry, but I must concur with Mr. Kolker here. The intersection of two lines is a point. One way to see what's going on here is to consider the standard model of Hilbert, namely R^3. (Kolker uses the notation E2 for 2D Hilbert, but let us consider the third dimension now as well.) Now we can determine what this model happens to map the primitives to. As Kolker has said, line is mapped to the set of ordered triples (x,y,z) satisfying a linear relation. But what about point? Is point mapped to a triple itself (an element of a line), or is it a singleton whose sole element is an ordered triple (a subset of a line)? This is, of course, closely related to what the primitive containment is mapped to. It could be membership (point = ordered triple) or inclusion (point = singleton of ordered triple). I believe that mapping containment to membership will be awkward. Let us recall what Hilbert Containment, three binary relations, one linking points and lines, one linking points and planes, and one linking lines and planes. So we see that lines contain points, planes contain points, and planes contain lines. And here lies the problem. If we let containment be mapped to membership, then planes would have both points and lines as distinct elements. And even if we only allowed planes to have lines as elements, which lines would be the elements of the plane anyway. For the plane z = 0, for example, are x = constant the elements of the plane, or y = constant, or all of them? So it makes much more sense to map containment to inclusion. Thus points are singletons and subsets of the lines and planes that happen to contain them. And therefore the intersection of two lines is the set intersection -- which is exactly the point. Of course, what about the ordered triples -- the elements of points, lines -- themselves? We may call them positions, if we want. So the single element of a point is a position, and the elements of a line are positions. And so answering the OP's question, positions don't have a measure, but points do -- at least, in the standard model R^3 of Hilbert, where subsets in R^3 have a Lebesgue measure. Of course, this is all only in the standard model of Hilbert. In other models, point, line, may be mapped to something other than sets, so we can't always refer to the element of a point as a position, because point may be mapped to something that doesn't have an element. To see what I mean, let us take a page from Han de Brujin's book and come up with a new model of some subset of Hilbert's axioms. (If you don't know who HdB is, it's not that important for this example.) Consider Hilbert's Axioms of Incidence only -- the ones labeled I.1 to I.7. Notice one can't prove from these axioms alone that more than finitely many points exist. Indeed, we observe I.7: I.7: Upon every straight line there exist at least two points, in every plane at least three points not lying in the same straight line, and in space there exist at least four points not lying in a plane. Apparently, by I.1 through I.7, we can't even prove the existence of more than _four_ points, and indeed, we can construct a model of I.1 through I.7 in which only four points exist. Now in this model, we will map our four points to natural numbers -- in particular, the natural numbers 1, 2, 4, and 8. (Those familiar with HdB should know by now where I am heading with this.) Lines contain exactly two points -- mapped to the sum of the two points that lie on them. Planes contain exactly three points -- mapped to the sum of the three points that lie on them. So we have: 1. point (1) 2. point (2) 3. line (1+2) 4. point (4) 5. line (1+4) 6. line (2+4) 7. plane (1+2+4) 8. point (8) 9. line (1+8) 10. line (2+8) 11. plane (1+2+8) 12. line (4+8) 13. plane (1+4+8) 14. plane (2+4+8) 15. space (1+2+4+8) In this model, containment is mapped to a more complicated matter -- one can try bitwise AND (or OR) to come up with the relation onto which containment is mapped. The important part is that this points don't have positions at all. === Subject: Re: The Virgin Birth of Points Once again you do not distinguish between objects and the sets of which > the objects are elements. Another evidence that you cannot cope with > mathematics. > A line is not a set of points because sets are indifferent to order. > However, if you care to order points we still do not have a minimal > definition of a line. I've been thinking about the links to Euclid's and Hilbert's > axioms presented in some of the other geometry threads: http://en.wikipedia.org/wiki/Hilbert%27s_axioms These last few posts are posing the question, is a > point an _element_ of a line, or is a point a > _subset_ of a line? The correct answer is neither. For let us review > Hilbert's axioms again: The undefined primitives are: point, line, plane. > There are three primitive relations: Betweenness, a ternary relation linking points; > Containment, three binary relations, one linking > points and lines, one linking points and planes, > and one linking lines and planes; > Congruence, two binary relations, one linking line > segments and one linking angles. So we see that line is an undefined _primitive_, > and that there is a _primitive_ to be known as > containment, so that a line may be said to > contain points. Notice that the primitive contain has _nothing_ > to do with the membership primitive of a set > theory such as ZFC. Why? Because this is a > geometric theory that is not even written in > the _language_ of ZFC. So both a point is an element of a line and a > point is a subset of a line are incorrect. The other question concerns what the intersection > of two lines is. Well, first we must define > intersection -- in terms of our _primitives_, > of course -- before we can answer. And the only > answer we can possibly give is in terms of > _containment_: the intersection of two lines a,b > is a point A such that a contains A and b contains > A as well, provided that such a point exists. We > can't call it a position, since position is > not a _primitive_ of our theory. So now all we have to do is prove that if such a > point exists, it must be unique. But this follows > directly from Axiom I.1. For if there were two > points of intersection A,B, then I.1 tells us that > two points determine a line, so that AB = a and > AB = b as well, therefore a = b. So if a,b are > distinct and intersect, then they intersect in a > unique point of intersection. === Subject: Re: The Virgin Birth of Points The Virgin Birth of Points > ~v~~ > The Jesuit heresy maintains points have zero length but are not of > zero length and if you don't believe that you haven't examined the > argument closely enough. > ~v~~ Points have zero length when construed as lying in a spatial >framework. However, points have no length because points are not >objects that arise in a spatial framework. Positions, not points, >arise in the spatial framework, and positions are always >constructions. >I conclude that the question about points cannot be a logical inquiry >or someone here would have been able to sort it out... Logically or illogically? ~v~~ The intersections of lines are positions, not points. There is no precedent for creating a new metaphysical entity from the arbitrary arrangement of lines. I would have thought it obvious. But plainly I was mistaken. === Subject: Re: The Virgin Birth of Points > The Virgin Birth of Points > ~v~~ > The Jesuit heresy maintains points have zero length but are not of > zero length and if you don't believe that you haven't examined the > argument closely enough. > ~v~~ >Points have zero length when construed as lying in a spatial >framework. However, points have no length because points are not >objects that arise in a spatial framework. Positions, not points, >arise in the spatial framework, and positions are always >constructions. >I conclude that the question about points cannot be a logical inquiry >or someone here would have been able to sort it out... > Logically or illogically? > ~v~~ The intersections of lines are positions, not points. There is no >precedent for creating a new metaphysical entity from the arbitrary >arrangement of lines. I would have thought it obvious. But plainly I >was mistaken. Plainly you were, are, and will be for if a point is a metaphysical entity then so I suggest are lines which I should have thought was obvious. ~v~~ === Subject: Re: The Virgin Birth of Points The intersections of lines are positions, not points. There is no > precedent for creating a new metaphysical entity from the arbitrary > arrangement of lines. I would have thought it obvious. But plainly I > was mistaken. Take a pair of linear equations in two variables each of which define a line. If the equations are not linearly dependent they determine a unique solution (x,y) which is --- aha!---- the point of intersection. In a Euclidean Plane lines when they intersect at all, have a unique point of intersection. And mathematical objects are not metaphysical entities. They are brain farts the blow about in our skulls. Bob Kolker > === Subject: Re: The Virgin Birth of Points The Virgin Birth of Points > ~v~~ > The Jesuit heresy maintains points have zero length but are not of > zero length and if you don't believe that you haven't examined the > argument closely enough. > In Euclidean space a set which has exactly one pont as a member has > measure zero. But you can take the union of an uncountable set of such > singleton sets and get a set with non-zero measure. > What measure will give a non-zero number/value? > Lebesgue measure will do so, not for all possible uncountable sets, but > for some. For example, the Lebesgue measure of an interval [a,b] is its > length, b-a. > -- > Dave Seaman > Oral Arguments in Mumia Abu-Jamal Case heard May 17 > U.S. Court of Appeals, Third Circuit > positions are constructions, and it is not appropriate to analyse a > construction in spatial terms. I think you need to learn some measure theory. This is a question about > mathematics, by the way, not philosophy. -- > Dave Seaman > Oral Arguments in Mumia Abu-Jamal Case heard May 17 > U.S. Court of Appeals, Third Circuit > - Hide quoted text - - Show quoted text - I think you need to distinguish between a position and a point before wildly conflating them in both a philosophical and mathematical confusion. === Subject: Re: The Virgin Birth of Points > What measure will give a non-zero number/value? > Lebesgue measure will do so, not for all possible uncountable sets, but > for some. For example, the Lebesgue measure of an interval [a,b] is its > length, b-a. > An interval [a,b] is composed of positions, not points. But even > positions are constructions, and it is not appropriate to analyse a > construction in spatial terms. > I think you need to learn some measure theory. This is a question about > mathematics, by the way, not philosophy. > - Show quoted text - > I think you need to distinguish between a position and a point before > wildly conflating them in both a philosophical and mathematical > confusion. In my statement that you quoted, I used neither of the terms position or point. I mentioned only Lebesgue measure, uncountable sets, and intervals. Exactly what is your, er, point? Why do I need to distinguish between terms that I didn't use? Neither of those is a precise mathematical term, by the way. The meaning depends on context, but to me a point is a member of some abstract space (possibly a vector space, or a topological space, or a metric space, or a measure space, or a Banach space, or whatever). A position, on the other hand, suggests a point that is given in some coordinate system. That doesn't always apply. Lots of times we talk about points in situations where there are no coordinates in sight. I consider position to be too limited a term for that reason. -- Dave Seaman Oral Arguments in Mumia Abu-Jamal Case heard May 17 U.S. Court of Appeals, Third Circuit === Subject: Re: The Virgin Birth of Points Clearly points don't have zero length, they have a positive infinitesimal > length for which zero is just the closest real approximation. You don't need to resort to non-standard analysis. Within the realm of >standard real numbers, the matter is settle using measure (either Borel >or Lebesque) I wouldn't call the calculus non standard analysis. Calculus does not require infinitesimals or NSA. I believe that was Leibniz's method, but we mostly follow Newton's development which only requires a theory of limits. - Randy === Subject: Re: The Virgin Birth of Points > Clearly points don't have zero length, they have a positive infinitesimal > length for which zero is just the closest real approximation. >You don't need to resort to non-standard analysis. Within the realm of >standard real numbers, the matter is settle using measure (either Borel >or Lebesque) > I wouldn't call the calculus non standard analysis. Calculus does not require infinitesimals or NSA. I >believe that was Leibniz's method, but we mostly >follow Newton's development which only requires >a theory of limits. So differentials are points? ~v~~ === Subject: Re: The Virgin Birth of Points > > > I think you need to learn some measure theory. This is a question about > mathematics, by the way, not philosophy. > Zick is totally incapable of understanding either mathematics or > physics. Robert Heinlein had some clever things to say about people who > cannot cope with mathematics. Heinlein said they are subhuman but > capable of wearing shoes and keeping clean. Zick is in my killfile. He is not the person I was responding to. -- Dave Seaman Oral Arguments in Mumia Abu-Jamal Case heard May 17 U.S. Court of Appeals, Third Circuit === Subject: Re: The Virgin Birth of Points > I think you need to learn some measure theory. This is a question about > mathematics, by the way, not philosophy. > Zick is totally incapable of understanding either mathematics or > physics. Robert Heinlein had some clever things to say about people who > cannot cope with mathematics. Heinlein said they are subhuman but > capable of wearing shoes and keeping clean. Zick is in my killfile. He is not the person I was responding to. Aw c'mon, Davey. Everybody knows a Seaman just loves semen. === Subject: Re: The Virgin Birth of Points . > The Virgin Birth of Points > ~v~~ The Jesuit heresy maintains points have zero length but are not of > zero length and if you don't believe that you haven't examined the > argument closely enough. . > In Euclidean space a set which has exactly one pont as a member has > measure zero. But you can take the union of an uncountable set of such > singleton sets and get a set with non-zero measure. . > What measure will give a non-zero number/value? . > The Lebesgue measure of the interval [0,1] is 1. The > Lebesgue measure of every finite and countable subset > of that interval is 0. The Lebesgue measure of the Cantor > set, which is uncountable, is also 0. Is that what you were asking? . Now we have all we need to point to the important fact: It is measuring sets of points, not measuring points or measuring many points. A point A is different from the set { A }. And a hint to the standard topology of the real number line, which according to Kuratowski, gives to every two points the intervall in between, a set of points. With friendly greetings Hero === Subject: Re: The Virgin Birth of Points . > Is that what you were asking? . Now we have all we need to point to the important fact: > It is measuring sets of points, not measuring points or measuring many > points. > A point A is different from the set { A }. And a hint to the standard topology of the real number line, which > according to Kuratowski, gives to every two points the intervall in > between, a set of points. > same hour too, i didn't know about. But anyhow, the truth can be expressed twice, without getting twisted. With friendly greetings Hero === Subject: polarized family l.s., in a family of polarized varieties, why is the Kodaira-Spencer class of a tangent vector cupped with the polarization zero? In fact, why are these classes (that map to zero under 'cupping with polarization') precisely the classes that infinitesimally preserve the polarization? And what does this mean precisely? (this is what Griffiths remarks on p816 of 'periods of integrals on algebraic manifolds II', but forme it's not really obvious). I hope somebody can explain this for me, or give a readable refference. cheers, i.s. === Subject: Discontinuity What is the exact definition for second kind discontinuity? Any help would be appreciated Gustav === Subject: Re: Discontinuity On Nov 12, 5:15 pm, Francisco de Le?n-Sotelo y Esteban What is the exact definition for second kind discontinuity? Any help would be appreciated > Gustav No left or right limits.smn === Subject: spectral radius and M-matrix Hi all, I hope somebody can help on this. Suppose you have an M-matrix A of the following form: A = s I - B (where s is a constant which is larger than the spectral radius of B, I is the identity matrix, and B is a positive matrix) My question is: suppose you know that A is an M-matrix and that the spectral radius of B is smaller than 1. Could you then state that you can always express A as A = I - B? Note that the constant s is now fixed to 1. Nikos === Subject: Re: Implementable Set Theory and Consistency of ZFC > But I wouldn't say: E x A y - (y in x) , because I want to develop my > set theory without mathematical logic as a prerequisite, and certainly > without needing a specification axiom, like in: x = { y : - (y in x) } . x = { y : - (y in x) } What set do you think that is? Do you think that any of the axioms of modern set theory allow one to define a set that way? (Note that the term x, the thing being defined, appears on the right hand side of the definition!) I am sure I don't know what set that is supposed to define. I suppose you meant to define the empty set by specification. This can certainly be done: x = { y : ~(y = y) } But that does not mean that the axiom of empty set needs a specification axiom. -- Jesse F. Hughes You're ketchup, so I'll put you on meatloaf! -- Quincy P. Hughes, age five, tries his hand at insults === Subject: Re: Implementable Set Theory and Consistency of ZFC <871wb37dqf.fsf@phiwumbda.org> <87ejf35ph9.fsf@phiwumbda.org> <87ejf2i7ha.fsf@phiwumbda.org> <87tznykob0.fsf@phiwumbda.org> <87tznxj8ip.fsf@phiwumbda.org> <87sl3g9ukh.fsf@phiwumbda.org> <9b56f$47381372$82a1e228$356@news2.tudelft.nl> <42ba2$47382e31$82a1e228$13052@news1.tudelft.nl> snippet below. You are right Chas, in that the axiom of Extensionality > is simply a theorem in my IST model (: in one of your previous posters). > I've also found that the axiom of Pairing is overhead when compared with > the axiom of Singleton, as is presented in 'Finitist Set Theory'. Here > comes the snippet, which hopefully clarifies the idea of constructivist > application of axioms. There are only _three_ now: empty set, Singleton, > Union. Together with the (axiom 0) statement that they do all the work. program Hughes; > [snipped] So in your implementation, the universe is a large but finite string of bits. Thus the empty set is a string of all zeros. But how do you construct the powerset P(S) from a set S? Or does your system not include a powerset axiom? === Subject: Re: Implementable Set Theory and Consistency of ZFC Here's what I would say. I would say: the empty set is a set with a > particular property, namely that it has no elements. In other words, > when I say the empty set exists, what I mean is that there is a set > with no elements. Would you say something different? No. But I wouldn't say: E x A y - (y in x) , because I want to develop my > set theory without mathematical logic as a prerequisite, and certainly > without needing a specification axiom, like in: x = { y : - (y in x) } . http://en.wikipedia.org/wiki/Axiom_of_empty_set Han de Bruijn Then you want to proclaim a set that has no definition? === Subject: Re: Implementable Set Theory and Consistency of ZFC >A circle is a set, right? Now simply _show_ me the powerset of a circle. >Any picture would be nice. First show me an actual circle. >Work in progress. Then don't come back til you finish it. >http://hdebruijn.soo.dto.tudelft.nl/jaar2007/cirkel.jpg >Han de Bruijn As the points in your circle are not all at the same distance from any > given point, what you have shown is not a circle, at least not by any > mathematically valid definition of circle. After infinitely many hours of number crunching, my computer has come up > with an infinitely much better result, which is proudly presented here: If your computer had crunched anything for infinitely many hours, the universe would have come to an end. http://hdebruijn.soo.dto.tudelft.nl/jaar2007/ideaal.jpg Looks more like the map of the empty set than a circle. Han de Bruijn === Subject: Re: Implementable Set Theory and Consistency of ZFC > Output, as expected: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 > 24 25 26 27 28 29 30 31 Er, what do you think you have shown? What is the relevance of this code and its output? -- Jesse F. Hughes What do you tremble your *soul* before it for? he cried. You don't learn algebra with your blessed soul. Can't you look at it with your clear simple wits? -- D.H. Lawrence, /Sons And Lovers/ === Subject: Re: Implementable Set Theory and Consistency of ZFC > Sure, but even if we allow {} as a primitive, we have to add an axiom > defining it. That is, we would need (Ay)~(y in {}). > I am not sure, but I think this fact is eluding Han. No. We _don't_ have to add an axiom defining it, but I think this fact > is eluding Jesse. Yes, indeed, it *is* eluding me. -- Civilizations have risen and crumbled as my people fight your people, and still it remains the same old battle. I come from a line that mostly walks alone, fighting for the truth against people [like you], but my people always win. -- James S. Harris === Subject: Re: Implementable Set Theory and Consistency of ZFC >I don't mind saying that there is an axiom that the empty set exists. >That's fine. I ask once again: Do you think the string {} expresses this axiom? >That is, is this one of your axioms: Axiom of empty set: {} Or, as is somehow more sensible, is your axiom: Axiom of empty set: (Ex)(Ay)~(y in x) or something similar. If so, then the axiom is just as Ullrich said. >It alleges that there is some x satisfying a particular property. >Why would (Ex)(Ay)~(y in x) be somehow more sensible than just {} ? >We just _have_ {} . Then we _make_ { {} } e.g. with a singleton axiom, >then we _make_ { {} , { {} } } with a pairing axiom. Constructively. >What's so difficult about this? It's not logicism, admittedly .. Why does {} any more claim the existence of an empty set than does > [ ] or ( ), or even ? ? It is the notation everybody has agreed upon That agreement only counts once the existence of an empty set has already been established. My question was how does the use of {} do any more to establish that existence than does any other symbol? You did not answer that question. http://en.wikipedia.org/wiki/Empty_set But in e.g. Delphi Pascal, the empty set is noted as [] , indeed. Any suggestion that the notation is identical to the idea is childish. > You cannot even talk about a chair, if you insist that a chair is just > a string: chair. Then why does HdB insist that {} and there exists an empty set denoted by {} mean the same thing. === Subject: Re: Implementable Set Theory and Consistency of ZFC On Nov 12, 12:48 am, Han de Bruijn That is, is this one of your axioms: >Axiom of empty set: {} >Or, as is somehow more sensible, is your axiom: >Axiom of empty set: (Ex)(Ay)~(y in x) >I suppose one could write {} is a set, or even >{} in V, i.e., that the empty class is a set, >an element of the class of all sets. This is how >we write 0 (or 1) is a natural number in PA. >Of course, this would probably require making >{} and V primitives, rather than the single >primitive of ZFC. >Sure, but even if we allow {} as a primitive, we have to add an axiom >defining it. That is, we would need (Ay)~(y in {}). >I am not sure, but I think this fact is eluding Han. >No. We _don't_ have to add an axiom defining it, but I think this fact >is eluding Jesse. So what you're saying is Everybody knows what {} means. So there is > reason to actually say what it means; because everybody who's anybody > knows already. Well, if everybody knows, then you must know. So what does {} or > the empty set or whatever you claim to be an equivalent expression / > mean/? Nothing? If, as you say, it has no meaning, why bother to include it? === Subject: Re: Implementable Set Theory and Consistency of ZFC >That is, is this one of your axioms: Axiom of empty set: {} Or, as is somehow more sensible, is your axiom: Axiom of empty set: (Ex)(Ay)~(y in x) >I suppose one could write {} is a set, or even >{} in V, i.e., that the empty class is a set, >an element of the class of all sets. This is how >we write 0 (or 1) is a natural number in PA. >Of course, this would probably require making >{} and V primitives, rather than the single >primitive of ZFC. Sure, but even if we allow {} as a primitive, we have to add an axiom > defining it. That is, we would need (Ay)~(y in {}). I am not sure, but I think this fact is eluding Han. No. We _don't_ have to add an axiom defining it, but I think this fact > is eluding Jesse. Han de Bruijn So it defines itself? Then it must be a definition, not a set. === Subject: Re: The Road to Poverty >Helping clueless auk coffeeboys one at a time---nightbat Lame as ever, fro0tbat. Gay as ever, Jew-Boi Deco! Your Pal, HJ === Subject: Re: Difficulty with a Spiral Equation ! > Hello; 1) I've 6 analytically derived data points (T*=0.09) What does (T*=0.09) mean in this context? and the 1st > derivative at the last point: > i x y > 1 -0.236435 0.937134 > 2 -0.232600 0.951276 > 3 -0.233333 0.926882 > 4 -0.242256 0.955982 > 5 -0.228409 0.974592 > 6 -0.211085 0.949008 & (dy/dx)= - 7.2113388 2) My analytical model postulates that the above 6 points lies on a > smooth CLOCKWISE spiral (with no intersecting turns) joining pnt# 1, > pnt# 2, ..., pnt# 6 in the same order. Does your model not tell you anything about the functional form of the > spiral then? There will be infinitely many ways to do this, so it's > hard to know where to start, and hard to see how absolutely any > spiral that fits could make sense in the context of whatever it is > you're doing. Looking at the points, it doesn't seem to me as if you're going to get > a nice spiral -- it'll be squashed and mis-shapen. Is that what > you're expecting? Couple of other things: as far as I can see, your example below *does* > have intersecting turns after it leaves point 6. Or do you only > require the spiral to be non-intersecting on its path from point 1 to > point 6? Do you have any requirements on the number of turns between > points? For example, your curve below makes approximately a quarter- > turn between points 1 and 2, but it could in theory make 10 and a > quarter, or a hundred and a quarter, or whatever... And finally, > presumably you don't care where the centre is? Is that right? 3) I've tried a number of possible spiral formulations with no > success. The most promising attempt was to represent the spiral by > the equation: > r = (a + b.th + c.th^2 + d.th^3).Exp(m.th) > with its centre O at (f,g) > r is the distance from the centre O to pnt # i, i=1, 6 > th is the angle measured clockwise from the vector: O(f,g) ----> pnt > # 1 > (obviously, th for pnt # 1 is zero) 4) So we have 7 unknowns: a, b, c, d, m, f, g > and 7 conditions: 6 points i=1, 6 and the slope (dy/dx) at the > last point i=6 5) I couldn't analytically solve the problem !!! The best I could > get: > a = 0.0094859 > b = - 0.0020353 > c = 0.0002411 > d = - 0.0000122 > m = 0.2693740 > f = - 0.2281596 > g = 0.9417704 > The spiral looks good but has one critical problem! It refuses to > pass through pnt # 5 !!!! It passes through pnt 1, 2, 3, 4, 6 and > satisfies the slope condition at pnt 6. 6) The beauty of the above spiral formula (item 3.) is its flexibility > in accommodating 5-pnt, 4-pnt, and 3-pnt spirals: > > for a 5-point + {(dy/dx) at i=5}, one drops the d term; > > for a 4-point + {(dy/dx) at i=4}, then one drops the c & d > terms ; and > > for a (min) 3-point + {(dy/dx) at i=3}, one drops the terms b, > c & d 7) There might be a robust analytical way to derive the 7 unknowns a, > b, c, ..., etc., and/or a better spiral representation, or even having > pnt # 1 as its centre !! Monir- Hide quoted text - - Show quoted text - matt271829; > i) T*=0.09 means absolutely nothing to the reader! It's my ref to > the sample 6-point data set. ii) My model does not predict the form of the spiral. iii) If you plot the 6 points: > i x y > 1 -0.236 435 0.937 134 > 2 -0.232 600 0.951 276 > 3 -0.233 333 0.926 882 > 4 -0.242 256 0.955 982 > 5 -0.228 409 0.974 592 > 6 -0.211 085 0.949 008 & (dy/dx)= - 7.211 338 8 iv) and the proposed spiral for th=0.0 to (max) 9.5345 : > r = (a + b.th + c.th^2 + d.th^3).Exp(m.th) ,with its centre at > O(f,g) and: > a = 0.009 485 9 > b = - 0.002 035 3 > c = 0.000 241 1 > d = - 0.000 012 2 > m = 0.269 374 0 > f = - 0.228 159 6 > g = 0.941 770 4 you'll find it's a nice smooth spiral that completely misses pnt # > 5 !!!! v) The spiral terminates at the last point of the set, i.e.; pnt # 6 > in this 6-pnt sample. vi) The number of turns should be the absolute MINIMUM. The values of > th associated with the attempted solution (item iv) and listed below > take care of the no. of turns: > pnt i th (clockwise from vector O(f,g) ---->pnt 1 > 1 0.0 > 2 1.644 445 2 > 3 5.557 427 8 > 4 7.583 320 5 > 5 8.357 023 5 > 6 9.534 502 2 vii) The centre of spiral O(f,g) is expected to be close to point # > 1(or right on it should a different spiral form allow that!!). > Further, as a good initial guess the centre O(f,g) could be the > assumed at the centre of the 3-pnt circle pnts 1, 2, 3: > f = g(y1-y2)/(x2-x1) + {-(y1+y2)(y1-y2)+(x2-x1)(x2+x1)} / (2(x2-x1)) > f = g(y2-y3)/(x3-x2) + {-(y2+y3)(y2-y3)+(x3-x2)(x3+x2)} / (2(x3-x2)) > For the sample case, the initial guess could be: > f = -0.213457 > g = 0.938493 > I think this is made easier by stretching in the x-direction, looking for a spiral to fit the stretched points, and then squashing it back. If you stretch by a factor of, say, 2-ish, pick a likely point for the centre, and plot r against theta for the six given points, then you can get some r-theta plots that look nice and smooth and amenable to curve-fitting. Trying things at random, I got this rather sexy example: http://img46.imageshack.us/img46/3307/spiraltp7.gif Unfortunately the equation is horrible: totally arbitrary, with lots of nasty big numbers that resulted from solving a simultaneous equation. === Subject: Re: Difficulty with a Spiral Equation ! Hello; 1) I've 6 analytically derived data points (T*=0.09) What does (T*=0.09) mean in this context? and the 1st > derivative at the last point: > i x y > 1 -0.236435 0.937134 > 2 -0.232600 0.951276 > 3 -0.233333 0.926882 > 4 -0.242256 0.955982 > 5 -0.228409 0.974592 > 6 -0.211085 0.949008 & (dy/dx)= - 7.2113388 2) My analytical model postulates that the above 6 points lies on a > smooth CLOCKWISE spiral (with no intersecting turns) joining pnt# 1, > pnt# 2, ..., pnt# 6 in the same order. Does your model not tell you anything about the functional form of the > spiral then? There will be infinitely many ways to do this, so it's > hard to know where to start, and hard to see how absolutely any > spiral that fits could make sense in the context of whatever it is > you're doing. Looking at the points, it doesn't seem to me as if you're going to get > a nice spiral -- it'll be squashed and mis-shapen. Is that what > you're expecting? Couple of other things: as far as I can see, your example below *does* > have intersecting turns after it leaves point 6. Or do you only > require the spiral to be non-intersecting on its path from point 1 to > point 6? Do you have any requirements on the number of turns between > points? For example, your curve below makes approximately a quarter- > turn between points 1 and 2, but it could in theory make 10 and a > quarter, or a hundred and a quarter, or whatever... And finally, > presumably you don't care where the centre is? Is that right? 3) I've tried a number of possible spiral formulations with no > success. The most promising attempt was to represent the spiral by > the equation: > r = (a + b.th + c.th^2 + d.th^3).Exp(m.th) > with its centre O at (f,g) > r is the distance from the centre O to pnt # i, i=1, 6 > th is the angle measured clockwise from the vector: O(f,g) ----> pnt > # 1 > (obviously, th for pnt # 1 is zero) 4) So we have 7 unknowns: a, b, c, d, m, f, g > and 7 conditions: 6 points i=1, 6 and the slope (dy/dx) at the > last point i=6 5) I couldn't analytically solve the problem !!! The best I could > get: > a = 0.0094859 > b = - 0.0020353 > c = 0.0002411 > d = - 0.0000122 > m = 0.2693740 > f = - 0.2281596 > g = 0.9417704 > The spiral looks good but has one critical problem! It refuses to > pass through pnt # 5 !!!! It passes through pnt 1, 2, 3, 4, 6 and > satisfies the slope condition at pnt 6. 6) The beauty of the above spiral formula (item 3.) is its flexibility > in accommodating 5-pnt, 4-pnt, and 3-pnt spirals: > > for a 5-point + {(dy/dx) at i=5}, one drops the d term; > > for a 4-point + {(dy/dx) at i=4}, then one drops the c & d > terms ; and > > for a (min) 3-point + {(dy/dx) at i=3}, one drops the terms b, > c & d 7) There might be a robust analytical way to derive the 7 unknowns a, > b, c, ..., etc., and/or a better spiral representation, or even having > pnt # 1 as its centre !! Monir- Hide quoted text - - Show quoted text - matt271829; > i) T*=0.09 means absolutely nothing to the reader! It's my ref to > the sample 6-point data set. ii) My model does not predict the form of the spiral. iii) If you plot the 6 points: > i x y > 1 -0.236 435 0.937 134 > 2 -0.232 600 0.951 276 > 3 -0.233 333 0.926 882 > 4 -0.242 256 0.955 982 > 5 -0.228 409 0.974 592 > 6 -0.211 085 0.949 008 & (dy/dx)= - 7.211 338 8 iv) and the proposed spiral for th=0.0 to (max) 9.5345 : > r = (a + b.th + c.th^2 + d.th^3).Exp(m.th) ,with its centre at > O(f,g) and: > a = 0.009 485 9 > b = - 0.002 035 3 > c = 0.000 241 1 > d = - 0.000 012 2 > m = 0.269 374 0 > f = - 0.228 159 6 > g = 0.941 770 4 you'll find it's a nice smooth spiral that completely misses pnt # > 5 !!!! v) The spiral terminates at the last point of the set, i.e.; pnt # 6 > in this 6-pnt sample. vi) The number of turns should be the absolute MINIMUM. The values of > th associated with the attempted solution (item iv) and listed below > take care of the no. of turns: > pnt i th (clockwise from vector O(f,g) ---->pnt 1 > 1 0.0 > 2 1.644 445 2 > 3 5.557 427 8 > 4 7.583 320 5 > 5 8.357 023 5 > 6 9.534 502 2 vii) The centre of spiral O(f,g) is expected to be close to point # > 1(or right on it should a different spiral form allow that!!). > Further, as a good initial guess the centre O(f,g) could be the > assumed at the centre of the 3-pnt circle pnts 1, 2, 3: > f = g(y1-y2)/(x2-x1) + {-(y1+y2)(y1-y2)+(x2-x1)(x2+x1)} / (2(x2-x1)) > f = g(y2-y3)/(x3-x2) + {-(y2+y3)(y2-y3)+(x3-x2)(x3+x2)} / (2(x3-x2)) > For the sample case, the initial guess could be: > f = -0.213457 > g = 0.938493 I think this is made easier by stretching in the x-direction, looking > for a spiral to fit the stretched points, and then squashing it back. > If you stretch by a factor of, say, 2-ish, pick a likely point for the > centre, and plot r against theta for the six given points, then you > can get some r-theta plots that look nice and smooth and amenable to > curve-fitting. Trying things at random, I got this rather sexy example: http://img46.imageshack.us/img46/3307/spiraltp7.gif Unfortunately the equation is horrible: totally arbitrary, with lots > of nasty big numbers that resulted from solving a simultaneous > equation. I meant to also post the pleasing unsquashed spiral -- the same as the first one except stretched by a factor of s in the x-direction. It's at http://img160.imageshack.us/img160/1102/spiral2bc8.gif I think I might draw some snails now... === Subject: Re: #268 Elliptic and Hyperbolic Geometry Coordinate System; new textbook: Nntp-Posting-Host: hera.cwi.nl ... > He had to *define* it in all particulars. > > Alright, I agree with you, although initially I was repulsed by the > notion. Yes, so there is *no* natural method. > So here I define the Elliptic and Hyperbolic Coordinate System. > > And so it is not natural. You *define* it. > > I am not defining what it means to be natural. I am defining what > the Coordinate System is. And the use that accrues will tell me and > others that it is true. If no use, then it is false. If alot of use, > then it is true. So truthness depends on the use? That is not mathematics. So what is 2 + 2? Is it 4? That has a lot of uses. Is it 0? That also has a lot of uses. So which of the two is true? > The outer surface is Elliptic Geometry of one Hemisphere. This > hemisphere is marked with the numbers 1 to ...99999 and their > radix fractional portion such as say 1r1 or 1r78 the fractional > unit is the finite portion rightwards. > > How is that hemisphere marked that way? Suppose we take the northern > hemisphere of the earth. How is Berlin marked? How is New York marked? > How is Mexico marked? How do you mark the points on a hemisphere in a > linear fashion? > > Well I think I have to start first with a point which I call the North > Pole and designate as 0 = 2(pi) and then I call its antipode point > (pi) = 0 you stated that you did not want favourites points and directions. Now you are doing exactly the same. So why did you discard my method that shows that the points on a circle can have an arithmetic defined on them them makes them a field? > And then I pick any one of those longitude lines and call it the > imaginary Greenwich longitude and this hemisphere has the numbers 1 to > 9999...9999 as Integers but also with radix point as Fractions between > Integers, and different from Reals in that these fractions can only be > finite strings so they have alot of holes . Sorry, you do not define a hemisphere, but only half of a circle. So what you are trying to define is arithmetic on a circle, not on a sphere. > Now if the hemisphere of the 1 to 999....99999 were chosen that > coincides with Berlin, New York, Mexico the Berlin would be about 45 > degrees down from the Greenwich longitude which corresponds to the > AP-adic of 25000....00000 and about 15 degrees on the Equator line which > is the AP-adic of approx 08000....000000 Eh? You did *not* define numbers on the Equator line. > So the coordinates for Berlin would be 25000....00000, and > 08000.....000000 where the first is the longitude reckoning and the > second the latitude reckoning. So, again, you have a preferred starting point and a preferred direction, numbering along the equator. And you accuse *me* of using a preferred starting point and a preferred direction when I defined an arithmetic field on the circle. Let me define it again, now on the circle through the north pole, Greenwich and the south pole. We can assign to all those points real numbers from 0 to (but no including) 360, going all the way from the north pole, through Greenwich, to the south pole, and the other side back to the north pole. Given two points A and B, I define addition and multiplication: A '+' B = A + B mod 360 A '*' B = (A * B) / 360 mod 360 where '+' and '*' are the new addition and multiplication, and + and * are the standard addition and multiplication. It is easy to show that '+' and '*' create an arithmetic field on that circle. > The second hemisphere is imaginary beginning with (pi) as South Pole > then pi +1, pi + 2 on up to pi + ...999999 and then 2pi as North Pole. > > The x-axis in Elliptic geometry is the Greenwich longitude and the > y-axis is the Equator. > > do not mark points on a hemisphere with a single number, you do so on > a half circle. So much for precision of your definitions. > > Well if you wanted the antipode coordinates of Berlin they would be in > the imaginary hemisphere of pi and 2pi Do you not know the difference between a hemisphere and a half circle? > All the numbers in Elliptic Geometry are positive numbers and the > operations as defined in this textbook. > > According to your definition just above, the actual numbers you use are > pairs of numbers. > > Well Descartes ends up with using pairs of numbers for the plane in > Euclidean geometry where he uses a X axis and Y-axis. I use > the Greenwich longitude as one axis and the Equator as the second > axis. Yes, but when you try to define them you use the term hemisphere, which is inapproriate. Descartes did *not* define the numbers on a half-plane, but on a line. > So that is how I garner both Elliptic and Hyperbolic geometry all > nested into > one single model. That is why I have to put all the numbers from 1 > to ....99999 > into a hemisphere so as to make the Sphere both geometries. Getting from worse to worse. You do not put them in a hemisphere, you put them on a half-circle. -- 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: #269 discovery rights of a new math concept; new textbook: Mathematical Physics (Microscopic & Telescopic Nntp-Posting-Host: hera.cwi.nl > Where is the frontview? That was being discussed, the front view. > > The logic there was that whoever discovered Frontview of Counting > Numbers would have discussed or discovered what the world's largest > Counting Number was, and perhaps even discovered the world's largest > prime number. They would not have discovered frontview and stopped > on a button. In 1993 I discovered All Possible Digit Arrangements > and that ....999999 was the world's largest integer. In 2007 I > discovered Frontview and endview for Reals. > > So you discovered the frontview in 2007. Tony Orlow discovered it one > or two years earlier. > > No. When you discover something you have to give it a name and you > have to show that you consciously are aware of something new. And you have > to put it to some use or application, otherwise you have nothing new. Why? When Fermat discovered his last theorem (actually a conjecture), he did not give it a name. Or do you think that Pythagoras called his triples the Pythagorean triples? And I do not think that Fermat's last Theorem has any application, so according to you it is nothing new. > So Tony never did any one of those things. He did not give ....999999 > as 99999......99999 a name. He did not call that a name. He did > not spend time discussing 999....9999 and why it was new from > ....99999. And most important of all, he did not put it to use or > application. He did spend a lot of time discussing it. He did put it to use or application (some of his very own uses and applications), and so did you. > But most important I put it to use. Noone before has ever had the > audacity > to say that there exists a world's largest prime number and write that > number out: > > 99999......999998888899998888999888997 > > and write out some sampling of the world's ten largest prime numbers. Have you *ever* given a proof that that number is prime? As your arithmetic is still ill-defined, I have no idea how to verify it. > So the priority of discovery of math like physics should have some > verifiable tests, but the most important of these tests is whether > someone consciously knows they have discovered something new and has > put it to use and application. Pray show the verifiable tests that show that the above number is prime. But before you do that, *define* definition on such numbers. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Why is the Quartic Formula Much Longer than the Cubic Formula? Why is the Quartic Formula Much Longer than the Cubic Formula? === Subject: A proof on weak continuity needed Let X be a Banach space and T a bounded linear operator on X. Prove that T is a continuous map from X to X, when X is equipped with the weak topology (for both the domain and the range), i.e. the topology induced by the space of continuous linear functionals on X. I have a proof that makes use of T** on X** being an extension of T. I want to know a direct proof. Anyone? === Subject: Re: Computer Networking solution manual Does anyone have computer networking: a top down approach 4th edition solution manual? === Subject: #277 ; Frontview does not mean last digit; new textbook: Mathematical Physics (Reals & Counting Numbers/AP-adics Primer) for age 6 years onward <47327EE3.3050104@hotmail.com> Do you ever think about what you are saying, you nutter. Above you are > saying a number has a first digit but logically impossible to have a digit at > infinity. > That's correct. An infinite sequence of digits has no last digit. > Otherwise it wouldn't be infinite, would it? > I did speak of first digit or second digit but I was careful in never saying > last digit. And this is probably why David [ad hominen crap snipped] > Frontview does not mean last digit. Or first digit either? In your example 1000...000788...889, what do you call the > 1 digit, and what do you call the 9 digit? If you numbered > all the digits from right to left, starting with the 9 digit in > position 1, what position is the 7 in, and what position is > the 1 digit in? > David, the first question you should have is whether that number is one of all possible digit arrangements. And if it is, then it exists and it belongs in mathematics. That is the most important question. Frontview and Endview are merely a help and a tool. Just because a number is infinity in its digits does not mean that it zooms off the planet Earth and zooms out into outerspace where its frontview can never be seen. This is the impression you make with your philosophical wishy washyness. That if a number is infinite that somehow forbades or makes hidden from view. Everyone is comfortable with the frontview of a number like 190! even though it is a huge number in our standards of numbers for it is the number of Coulomb Interactions inside a plutonium atom to hold it together. Comfortable because we know there are zeroes digits that occupy the number out to infinity. So why should you be comfortable with zeroes digits out to infinity but uneasy and restless and nervous if there were 7s digits out to infinity? So this a matter of personal taste but not mathematics on your part. So the number for example of 77 is really the AP-adic of 00000......000000077 and we have its Frontview of 0 and its Endview of 7. 1000...000788...889 to answer your question above has a Frontview of 1 and Endview of 9. So why does that bother you, yet for 77 it is of no concern. Both are numbers because both are one possible digit arrangement in the scheme of All Possible Digit Arrangements. As for the question of the 7 digit in that above number, well I do not have to be precise in telling you what Place Value that the 7 occurs because I am not concerned just as I am not concerned about what digit occupies the middle of infinity place value in this number: ............22212019181716151413121110987654321 I can tell you that its Frontview digit is a 9 and its Endview digit is a 1, but I cannot tell you what its middle of infinity place value digit is. But none of that harms any of the concepts gained. The trouble with David is that he never has realized that I am not > some ordinary person doing math and never realized that I am > gifted in doing math, and so, for David everything I do in > math, for him, it has to be all wrong, when in fact, most everything > that David does in math [ad hominen crap snipped] > he thinks the Internet could never be graced with a unrecognized > genius of the subject. [ad hominen crap snipped] > but I really do not know why the most difficult thing is to spot a > genius early on. But for most people in a subject, one > of their most difficult psychological tests in life is to spot a genius in their midsts. Yes, I do have difficulty with that. > I've ignored all your ad hominen attacks on me, because I am a > nicer person than you. You may feel you are nice but when you falsify the history of math by saying that Cantor used the concept of All Possible Digit Arrangements, then you provoke anger and you deserve a tongue lashing. P.S. I forgot a number in a few posts back, I think it was #275. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies