mm-389 === Subject: : (Linear algebra + Differential equation) Help me solve this problem!! Problem : Let x(t) be an infinitely differentiable real vector which satisfies x'(t)=Ax(t) where x'(t) denotes differentiation of x(t). Let A be an 3x3 matrix satisfying < 0 where z is any nonzero vector and < , > denotes inner product. Show that lim = 0 as t->infinite In my opinion , If A= -I , then conclusion is obvious. But general case is not easy to me. Thank you === Subject: : Re: (Linear algebra + Differential equation) Help me solve this problem!! > Problem : Let x(t) be an infinitely differentiable real vector which > satisfies x'(t)=Ax(t) where x'(t) denotes differentiation of x(t). Let > A be an 3x3 matrix satisfying < 0 where z is any nonzero vector > and < , > denotes inner product. Show that lim = 0 as > t->infinite > In my opinion , If A= -I , then conclusion is obvious. But general > case is not easy to me. Thank you First: if you define exp(t*A) = sum(k=0 to infty)[ A^k / k! ] (A is the 3x3 matrix), then you can prove that x(t) = exp(t*A)*x(0) (where x(0) is the initial condition on the ODE) is solution of x'(t) = A*x(t). So, you can write x'(t) = A*exp(t*A)*x(0). Applying norm, you have norm(x'(t)) = norm(A*exp(t*A)*x(0)) <= norm(A)*norm(exp(t*A))*norm(x(0)) As A and x(0) are constants, it's enough to prove that norm(exp(t*A)) tends negative definite, ie that -A is positive definite. This implies that exists a regular matrix P, and a diagonal matrix D such that A = - P*D*(P^-1) and that all the elements in the diagonal of D are estrictly positive. So, you can write exp(t*A) = sum(k=0 to infty)[ (-tP*D*(P^-1))^k / k! ] = sum(k=0 to infty)[ P*(-tD)^k*(P^-1) / k! ] = P* sum(k=0 to infty)[ (-tD)^k / k! ] *(P^-1) = P*exp(-tD)*(P^-1) And: norm(exp(t*A)) <= norm(P)*norm(exp(-tD))*norm(P^-1) So, we just need to show that exp(-tD) ----> 0 (the matrix 0) as t --> infty. It's easy to prove that, if D is diagonal: exp(-tD) is a diagonal matrix, with elements exp(-t*D11), exp(-t*D22) and exp(-t*D33) in the diagonal (D11, D22 & D33 are the elements of the diagonal of D). But, we saw above that D11, D22 & D33 are estrictly positive, so we have exp(-t*Dii) ----> 0 as t---> infty, for i=1,2,3 And this implies exp(-tD) ---> 0, so norm(x'(t)) ----> 0 as t-->infty. If we choose the euclidean norm, we have norm(x) = sqrt(), ie = norm(x'(t))^2 ------->0 as t-->infty. David Gomez dgomez@ing.uchile.cl === Subject: : Re: (Linear algebra + Differential equation) Help me solve this problem!! >Problem : Let x(t) be an infinitely differentiable real vector which >satisfies x'(t)=Ax(t) where x'(t) denotes differentiation of x(t). Let >A be an 3x3 matrix satisfying < 0 where z is any nonzero vector >and < , > denotes inner product. Show that lim = 0 as >t->infinite Hint: All the eigenvalues of A are negative. I think this should get you what you want. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: : Re: (Linear algebra + Differential equation) Help me solve this problem!! >> Problem : Let x(t) be an infinitely differentiable real vector which >> satisfies x'(t)=Ax(t) where x'(t) denotes differentiation of x(t). Let >> A be an 3x3 matrix satisfying < 0 where z is any nonzero vector >> and < , > denotes inner product. Show that lim = 0 as >> t->infinite >negative definite, ie that -A is positive definite. This implies that exists >a regular matrix P, and a diagonal matrix D such that >A = - P*D*(P^-1) No, a positive-definite matrix need not be diagonalizable. The 2x2 matrix [[3,1], [-1,1]] is a counterexample (2 is the only eigenvalue so it can't be diagonalizable unless it's diagonal, which it isn't.) (A _symmetric_ positive-definite matrix must be diagonalizable...) >[...] >David Gomez >dgomez@ing.uchile.cl ************************ David C. Ullrich === Subject: : Re: (Linear algebra + Differential equation) Help me solve this problem!! >>Problem : Let x(t) be an infinitely differentiable real vector which >>satisfies x'(t)=Ax(t) where x'(t) denotes differentiation of x(t). Let >>A be an 3x3 matrix satisfying < 0 where z is any nonzero vector >>and < , > denotes inner product. Show that lim = 0 as >>t->infinite >Hint: All the eigenvalues of A are negative. I think this should >get you what you want. I don't see how the result follows immediately, because A need not be diagonalizable. ************************ David C. Ullrich === Subject: : Re: (Linear algebra + Differential equation) Help me solve this problem!! >Problem : Let x(t) be an infinitely differentiable real vector which >satisfies x'(t)=Ax(t) where x'(t) denotes differentiation of x(t). Let >A be an 3x3 matrix satisfying < 0 where z is any nonzero vector >and < , > denotes inner product. Show that lim = 0 as >t->infinite >In my opinion , If A= -I , then conclusion is obvious. But general >case is not easy to me. Thank you Are you also given that A is symmetric? If so the replies that other people have given will work. If not you could argue as follows: First note that there exists a number c > 0 such that (*) <= -c for all z (proof below). Now let f(t) = . Then f'(t) = 2 = 2 <= -2c = -2c f(t). That is, f' + 2c f <= 0. This shows that g'(t) <= 0, if g(t) = exp(2ct) f(t). So g is non-increasing, so g(t) <= g(0) for t > 0. That is, f(t) <= exp(-2ct) f(0). So f(t) = -> 0 as t -> infinity, and I can think of at least two ways to deduce from this that also tends to 0 (one way is to note that x' satisfies the same differential equation as x does.) Proof of (*): Let S be the set of all z with ||z|| = 1. Then the function is continuous and negative on S; since S is compact it follows that there exists c > 0 with <= -c on S, and this implies (*). ************************ David C. Ullrich === Subject: : Re: (Linear algebra + Differential equation) Help me solve this problem!! >Problem : Let x(t) be an infinitely differentiable real vector which >satisfies x'(t)=Ax(t) where x'(t) denotes differentiation of x(t). Let >A be an 3x3 matrix satisfying < 0 where z is any nonzero vector >and < , > denotes inner product. Show that lim = 0 as >t->infinite >Hint: All the eigenvalues of A are negative. I think this should >>get you what you want. I don't see how the result follows immediately, because A need not >be diagonalizable. Even then, doesn't a basis for the solution space consist of functions of the form t |-> p(t) exp(lam t) v, where p is a polynomial, lam is an eigenvalue, and v is an (eigen?)vector? Maybe not - I am rusty on my diff. eq. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: : Re: (Linear algebra + Differential equation) Help me solve this problem!! >>Problem : Let x(t) be an infinitely differentiable real vector which >>satisfies x'(t)=Ax(t) where x'(t) denotes differentiation of x(t). Let >>A be an 3x3 matrix satisfying < 0 where z is any nonzero vector >>and < , > denotes inner product. Show that lim = 0 as >>t->infinite >Hint: All the eigenvalues of A are negative. I think this should >get you what you want. >>I don't see how the result follows immediately, because A need not >>be diagonalizable. >Even then, doesn't a basis for the solution space consist of functions >of the form t |-> p(t) exp(lam t) v, where p is a polynomial, lam >is an eigenvalue, and v is an (eigen?)vector? Maybe not - I am rusty >on my diff. eq. I'm a little rusty myself, but it seems clear to me that this is not so. If functions of that form formed a basis for the solution space then x(t) would be a linear combination of eigenvectors for every t. But the eigenvectors don't span R^n, while x(0) can be any vector in R^n. That last sentence _could_ be a lie, but I don't think so. Anyway, it could very well be that even if the above is correct it's still true that the fact that all the eigenvalues are negative would immediately imply the desired result if we weren't rusty on this. But I don't see how offhand. (Ah. I bet we need generalized eigenvectors, whatever the heck they are.) ************************ David C. Ullrich === Subject: : Re: (Linear algebra + Differential equation) Help me solve this problem!! >Problem : Let x(t) be an infinitely differentiable real vector which >satisfies x'(t)=Ax(t) where x'(t) denotes differentiation of x(t). Let >A be an 3x3 matrix satisfying < 0 where z is any nonzero vector >and < , > denotes inner product. Show that lim = 0 as >t->infinite >>Hint: All the eigenvalues of A are negative. I think this should >>get you what you want. >I don't see how the result follows immediately, because A need not >be diagonalizable. Who cares? If the space is finite dimensional, < 0 for all vectors z implies <= -c . So the derivative of = 2 <= -2c. This makes <= exp(-2ct) from which the result follows. -- 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: (Linear algebra + Differential equation) Help me solve this problem!! >>Problem : Let x(t) be an infinitely differentiable real vector which >>satisfies x'(t)=Ax(t) where x'(t) denotes differentiation of x(t). Let >>A be an 3x3 matrix satisfying < 0 where z is any nonzero vector >>and < , > denotes inner product. Show that lim = 0 as >>t->infinite >Hint: All the eigenvalues of A are negative. I think this should >get you what you want. >>I don't see how the result follows immediately, because A need not >>be diagonalizable. >Who cares? If the space is finite dimensional, < 0 >for all vectors z implies <= -c . So the >derivative of = 2 <= -2c. >This makes <= exp(-2ct) from which >the result follows. Uh, thanks. If you look closely at the thread you see that I already said that... ************************ David C. Ullrich === Subject: : (n to the 5th power) mod 10 Is there an easy way to prove that, when a is an integer, the equation `(a ^ 5) mod 10 = a mod 10' holds? By the way, it also seems to work for powers of 9, 13, 17, etc. That is: `(a ^ (4b + 1)) mod 10 = a mod 10'. Thanks, Mauro === Subject: : Re: (n to the 5th power) mod 10 By induction. Note that: (a+1)^5 = a^5 + 5a^4 +10a^3 + 10a^2 +5a + 1 = a^5 + 5*(a^4 + 2a^3 + 2a^2 +a) + 1 = a^5 + 1 + 5*(something even) Assume true for a So (a+1)^5 mod 10 = a^5+1+5*(something even) mod 10 = a^5 + 1 mod 10 = a+1 mod 10 Presumably the 4b+1 case is solved in the same manner, using the full binomial expansion expression. > Is there an easy way to prove that, when a is an integer, > the equation `(a ^ 5) mod 10 = a mod 10' holds? > By the way, it also seems to work for powers of 9, 13, 17, > etc. That is: `(a ^ (4b + 1)) mod 10 = a mod 10'. > Thanks, > Mauro electron-dot-cloud are galaxies === Subject: : Re: 100 free SBC; VonNeumann Gametheory how to play StockMarket Another crossover between BLS which was 25.80 today and BMY which was 25.60. These two may have been the best switching partners for the past 6 months, however, I have not been enthusiastic with BLS to make that an active switching cell. And I am surprized to see WYE sort of falling, but whether if falls below $40 remains to be seen. And I have seen VZ now fall over labor union disputes in August for the past several years. I may just be able to capitalize on this sort of periodic weakness. Just as I try to capitalize on the weakness in the dividend pay periods. Archimedes Plutonium, a_plutonium@hotmail.com whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: : Re: [OT] de mortuis nil nisi bene (was: Armand Borel Finally dead) Does that apply to Idi Amin as well, who also died at > the same (official) age? > But outside math you are responsible for what you do and > say. Besides true and false you are confronted with > right and wrong. I wanted to express my belief that talking > badly about people, who cannot defend themselves, is wrong. I reject your polemics and wish you all the best. Rainer Rosenthal > r.rosenthal@web.de I completely agree with you Rainer. Here's an extra quote from a famous author: There is a solution to everything except death and death is the solution to everything! PS: I conjecture that Mr. Hans Aberg has *some* frustration or something... As far as I know, if a person does something not worthwhile (or useless), it'll come out sooner or later (time will tell). Ergo, why be so aggressive about Serre's contribution if sooner or later they'll be forgotten (which I think is highly unlikely!). === Subject: : Abel Prize >> So there is a sorry combination here, a celebrity status that is allowed >> to deflect much needed resources away from topics that would provide for a >> more proper scientific development. Serre is, of course, a great talent, >> but not a Nobel Prize level scientist or anything on that level. >In that case, you'd better notify the Abel Committee immediately. >No doubt they'll revoke Serre's Abel Prize when they find that such a >notable as yourself thinks him unworthy. There was a discussion in this group with a Norwegian mathematician about this at the time the prize was announced. For details, I refer to that thread. In brief, though, it turns out that the statutes of the Abel prize are written differently than that of the Nobel prize, not requiring work of (whatever the formulation now is) exceptional significance, but it is acceptable with generally successful careers, just as in the case of the Fields medal. So the Fields medalists are qualified for the Abel prize though several of them would not qualify for a Nobel Prize in mathematics, which requires higher standards. It looks as though what one may surmise is the control group of the Fields medal now tries to translate into the Abel Prize. The difference is of course, that the Fields medal procedure seems to be secret and thus can be run as a business board room, whereas the Abel Prize procedures are public, under scrutiny of the Norwegian government. So it is going to be interesting to see what happens with this prize the next few years. Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: === Subject: : Re: About Russell's first paradox |Consider the set M to be The set of all sets that do not contain > |themselves as members. Formally: A is an element of M if and only > |if A is not an element of A. In the sense of Cantor, M is a > |well-defined set. No, from Cantor's point of view there are multiplicities that are > not possible to gather together as sets. Frege's system was > the one that foundered on the paradox. (It's been quite awhile since I've studied this stuff in any detail but....) Hmmm. Maybe that was Cantor's personal point of view, but I don't recall anything in his formalism that leads the reader to think that his axiom of comprehension is restricted in any serious way. As well, I don't recall anything in his tacit use of comprehension that would lead one to believe that it was restricted - e.g., Cantor's work was prior to Grungesetze (sp?), so there would be little mathematical motivation for Cantor to restrict comprehension in such a manner. No? (italicized words are such because axioms like extension, comprehension, and the like weren't explicitly pointed to by Cantor, if memory serves.) cdj === Subject: : Re: About Russell's first paradox X-ID: Xuy4xUZLwe8vL33CHhhManR7LUfMO+k2AA7fXNDZdgfQsdyElB60oo Hmmm. Maybe that was Cantor's personal point of view, but I don't > recall anything in his formalism that leads the reader to think that > his axiom of comprehension is restricted in any serious way. Actually, he didn't have an _axiom of comprehension_, since his system was not explicitly formulated as an axiomatic system. > well, I don't recall anything in his tacit use of comprehension that > would lead one to believe that it was restricted - e.g., Cantor's work > was prior to Grundgesetze, so there would be little mathematical > motivation for Cantor to restrict comprehension in such a manner. No? > Actually, Cantor (!) was well aware of the danger of unrestricted comprehension. And there ARE documents where he clearly points out that unrestricted comprehension in the sense of Frege as mathematical blunder. multiplicities that are not possible to gather together as sets. Unfortunately Cantor didn't publish this ideas officially, as far as I know. :-( F. === Subject: : Re: About Russell's first paradox X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Punge: Micro$oft X-Sanguinate: themvsguy@email.com X-Terminate: SPA(GIS) X-Tinguish: Mark Griffith X-Treme: C&C,DWS >By using the empty set, we can show that W has the structure of a set >that contain itself as a member of itself: No. >By definition A, the set of all sets that contain themselves as >members, must have some kind of the above self structural similarity >over scales, by a recursive process. No. >Also by definition A, the set of all sets that do not contain >themselves as members, must not have this property, Again, no. >Through this structural point of view, there is no paradox. Of course there is. >What do you think? I think that you need to sit down and try to work up formal proofs of your claims. You will, of course, fail, but in the process you may learn where you wnet wrong. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Any unsolicited bulk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org === Subject: : Re: About Russell's first paradox |Actually, Cantor (!) was well aware of the danger of unrestricted |comprehension. And there ARE documents where he clearly points out that |unrestricted comprehension in the sense of Frege as mathematical |blunder. | |multiplicities that are not possible to gather together as sets. I remember reading a paper by Kreisel in which he says comprehension, before he appears to have known about Russell's paradox. I'm pretty sure he also had his notion of a distinction between infinite and absolutely infinite (i.e. what we would call proper classes). He was concerned about possible theological opposition from pious folks who thought the infinite was God's own private turf. Keith Ramsay === Subject: : Affine Connection & Gauge Theory Geometry on the space with affine connection is more general than Riemanian geometry. I thought the Grassmann algebla on the former space might be relevant with the gauge transformation. Namely the structure equation on the Grassmann algebla corresponds to the gauge field. For details: http://139.134.5.123/tiddler2/gauge4/gauge.htm === Subject: : algrithm to list all possible partitioning of a network Pardon me for ignorance. Lets say, I want to generate all possible partition on a set networked nodes. Is there an algrithm to do that? I understand that for a general set, this would be related to Bell Number. But in my case, a network relation is defined on the set. Theofore, using a set partition algrithm is very ineffecient because it would generate a lot of invalid partitioning. I assume a fairly large network of 50 nodes but each node is connected to just a few (usually less than 3) other nodes. Does anyone know of an algrithm to list all partitioning of a network? Many thanks in advance for the help. === Subject: : Re: algrithm to list all possible partitioning of a network > Pardon me for ignorance. Lets say, I want to generate all possible partition on a set networked > nodes. Is there an algrithm to do that? I understand that for a > general set, this would be related to Bell Number. But in my case, a > network relation is defined on the set. Theofore, using a set > partition algrithm is very ineffecient because it would generate a lot > of invalid partitioning. I assume a fairly large network of 50 nodes > but each node is connected to just a few (usually less than 3) other > nodes. Does anyone know of an algrithm to list all partitioning of a network? Many thanks in advance for the help. The above sentence I want to generate all possible partition on a set networked nodes. should be I want to generate all possible partition on a set of networked nodes. === Subject: : Re: algrithm to list all possible partitioning of a network >>Pardon me for ignorance. If only the whole world would consider such apologies in advance... and mean them. But then it would lose its force and sincerity. >>Lets say, I want to generate all possible partition on a set networked >>nodes. Is there an algrithm to do that? I understand that for a >>general set, this would be related to Bell Number. The easy problem. A reference is better than explaining it in full here. See: Nijenhuis and Wilf, Combinatorial Algorithms It explains generation of lots of combinatorial algorithms. even gives code, but that is essentially meaningless because it is not structured. So just look at the very cogent explanations. The short explanation is that you can generate the (many) set partitions recursively (all at once as a set) or one at a time or in order, -totally- based on the basic recurrence for set partitions. A set partition on n with m parts is either - one on n-1 with m parts: then place the nth object in one of the m parts. or - one on n-1 with m-1 parts: then add a singleton set with the nth object. This gives the recurrence S(n,m) for the -number- of partitions of a set of size n into m parts: S(n, m) = m S(n-1,m) + S(n-1,m-1) So what, you say, given that this just -counts- the partitions. It doesn't create them. But you can still use it to create the kth such partition (given n and m). Compute recursively or dynamic programmingly S(n-1,m). - If k is less than m times this, then you'll be putting object n into the floor((k/S(n-1,m))th part of the set partition which is computed recursively as the kth set partition of S(n-1, m). - If k is equal or greater, than you'll be creating a singleton set for object n, find the k - m*S(n-1,m) th partition from S(n-1,m-1). This method can work for all kinds of combinatorial structures. See the book to do incremental construction (one after the other). >>But in my case, a >>network relation is defined on the set. Theofore, using a set >>partition algrithm is very ineffecient because it would generate a lot >>of invalid partitioning. I assume a fairly large network of 50 nodes >>but each node is connected to just a few (usually less than 3) other >>nodes. >>Does anyone know of an algrithm to list all partitioning of a network? The above sentence I want to generate all possible partition on a set > networked > nodes. should be I want to generate all possible partition on a set > of networked nodes. So you have a graph rather than a bunch of unrelated objects. The connections in your network (the edges in your graph) imply that some of the nodes are similar and some not. Suppose you have the graph: c / | a--b | | d c and d are some how similar and you want to -not- deal with partitions that treat these two nodes similarly (because you'll just be repeating an action that is essentially the same. e.g. of the 15 set partitions on {a,b,c,d}: 1:{{a},{b},{c},{d}}, 2:{{a,b},{c},{d}}, 3:{{a,c},{b},{d}}, 4:{{a,d},{b},{c}}, 5:{{a},{b,c},{d}}, 6:{{a},{b,d},{c}}, 7:{{a},{b},{c,d}}, 8:{{a,b},{c,d}}, 9:{{a,c},{b,d}}, 10:{{a,d},{b,c}}, 11:{{a,b,c},{d}}, 12:{{a,b,d},{c}}, 13:{{a,c,d},{b}}, 14:{{a},{b,c,d}}, 15:{{a,b,c,d}}, of these 3 and 4, 9 and 10, 11 and 12, are the same. You only want to construct the (uh...counting) 12 -distinct- such vertex partitions (i.e. don't construct 4,9, and 12) (I think this is what you intend; if not, apologies, disregard, clarify) So what you need to do are: 1) figure out which pts are similar 2) construct distinct partitions with respect to the distinct vertices. To do 1, you have to compute the graph isomorphisms and -the- partition.of the vertices as orbits under the isomorphism group. To do this, get the excellent nauty package at http://cs.anu.edu.au/~bdm/nauty/ To do 2, uh...uh...ok I'm sure this is not bad but my fingers are tired from typing (er really my brain froze) so the only appropriate intelligent thing I can say at this point is that almost all graphs -do not- have any symmetries, that is, the graph isomorphism group of most graphs is trivial. Another way of saying this is that you probably won't save yourself much effort (computer time) by eliminating these essentially different partitions. -unless- you know ahead of time that your graph/network is really structured (like a hypercube). Then you'd probably use an ad hoc generation algorithm anyway. So what I'm trying to say is that you shouldn't worry about the symmetries. Even with 50 nodes you'll have quite a few partitions to worry about no matter what. Mitch === Subject: : Re: algrithm to list all possible partitioning of a network ... >>Lets say, I want to generate all possible partition on a set [of networked nodes where a network relation is defined on the set but a set partition algorithm would generate a lot of invalid partitioning and I assume a fairly large network of 50 nodes but each node is connected to just a few (usually less than 3) other nodes.] ... > [Reference to Nijenhuis and Wilf, Combinatorial Algorithms] ... > So you have a graph rather than a bunch of unrelated objects. The > connections in your network (the edges in your graph) imply that some of > the nodes are similar and some not. Suppose you have the graph: > c > / | > a--b | > | > d > c and d are some how similar and you want to -not- deal with partitions > that treat these two nodes similarly (because you'll just be repeating > an action that is essentially the same. e.g. of the 15 set partitions on > {a,b,c,d}: ... > 9:{{a,c},{b,d}}, > 10:{{a,d},{b,c}}, ... > of these 3 and 4, 9 and 10, 11 and 12, are the same. You only want to > construct the [...] 12 -distinct- such vertex partitions (i.e. > don't construct 4,9, and 12) (I think this is what you intend; if not, apologies, disregard, clarify) ... I think similarity is not the issue; rather, each element of a network partition of N is to be a connected subgraph of N. If so, neither 9 nor 10 is valid, since none of {a,c}, {b,d}, {a,d}, and {b,c} are connected graphs. I imagine there is some literature for this problem but don't know what's relevant. Google for all connected subgraphs turns up 80 urls, eg, http://arxiv.org/pdf/math.CO/0211436 which says, We propose a novel algorithm for enumerating and listing all minimal cutsets of a given graph. It is known that this problem is NP-hard and scanning all minimal cutsets is an important issue in many applications, such as evaluating the reliability of networks. -jiw === Subject: : Re: algrithm to list all possible partitioning of a network I will be interested in the answer too. I was trying to find out the number of such partitions. I think, the algorithm will be closely linked with the enumeration scheme. The following 2 examples will probably elucidate the problem. Mr. mathphyer, pl. let me know if this is what you have in mind. Let there be 3 objects: a, b, c Partition# 1: {a,b,c} [1 part in the patition] Partition# 2: {a},{b,c} [2 parts in the partition] Partition# 3: {b},{a,c} [2 parts in the partition] Partition# 4: {c},{a,b} [2 parts in the partition] Partition# 5: {a},{b},{c} [3 parts in the partition] Total # of partitions=5 For 4 objects: a, b, c, d Partition#1: (abcd) = 1 [1 part in partition] Partition# 2-8: [2 parts in each partition] (a)(bcd),(b)(acd),(c)(abd),(d)(abc), (ab)(cd),(ac)(bd),(ad)(bc) = 7 Partition# 9-14: [3 parts in each partition] (a)(b)(cd), (a)(c)(bd),(a)(d)(bc),(c)(d)(ab),(b)(c)(ad),(b)(d)(ac) = 6 Partition# 15: (a)(b)(c)(d) = 1 [4 parts in partition] Total=15 Thanks. -Samik >Pardon me for ignorance. >>Lets say, I want to generate all possible partition on a set networked >>nodes. Is there an algrithm to do that? I understand that for a >>general set, this would be related to Bell Number. But in my case, a >>network relation is defined on the set. Theofore, using a set >>partition algrithm is very ineffecient because it would generate a lot >>of invalid partitioning. I assume a fairly large network of 50 nodes >>but each node is connected to just a few (usually less than 3) other >>nodes. >>Does anyone know of an algrithm to list all partitioning of a network? >>Many thanks in advance for the help. > The above sentence I want to generate all possible partition on a set > networked > nodes. should be I want to generate all possible partition on a set > of networked nodes. -- Samik Raychaudhuri University of Wisconsin, Madison http://samik.freeshell.org/ === Subject: : Re: An equation to match a graph Hi! WOW! That is exactly the function I was looking for. Thanks! How did you figure it out? I checked the graph and derivative and they fit like a charm. I'm now going to try and add a parameter (not sure where to put it yet) to experiment with different levels of distortion (proportion of the steepest derivative to the smallest derivative). Thanks again Dave! Daniel > Hi I need to create an equation (y=f(x)) that has a certain shape and > qualities. I know exactly how I want it to look but can't figure out a > good equation. It should satisfy: * Differentiable (Has first derivative at every point) > * Slopes (think of y=x^2): > slope(0)=0 > slope(x > 0) > 0 (never = 0) > slope(x < 0) < 0 (never = 0) *The derivative near the points where x is an ODD INTEGER should be > far larger (in the absolute value sense of the word) than the > derivative near the points where x is an EVEN INTEGER. The definition > of near is flexible. Something like y = (3.2x - sin(pi*x))^2 appears to satisfy your requirements. === Subject: : Re: Another look on Russell's first paradox > Consider the set M to be The set of all sets that do not contain > themselves as members. Formally: A is an element of M if and only > if A is not an element of A. [...] this must be a contradiction in > the underlying theory. Right. We would have a contradiction in (some variant of) _naive_ set > theory here. Solution: Use some advanced set theory which does not allow for this > contradiction. (For example, ZF(C), TT, NBG, MK, NF, NFU, etc., etc.) > (As you can see, it's not THAT hard to > avoid the mentioned contradiction.) > F. If it's so easy, why are there all of those different solutions that you list (and etc. etc.) above? Mathematicians will never understand the paradoxes until they generalize (make variable) the base of computing. They talk about Godel's semantic (based on soundness) and syntactic (based on w-consistency) proofs of his 1st Incompleteness Theorem. These are simply using the true and the provable wffs as the base. But rather than treating each as a special thing, consider instead using any 2-place relation D(a,b) as the base: N#P(x) means that P(a) iff D(N,a). [ We say that program N solves recursion P(x). ] P(x) means there is an N such that N#P(x). When multiple values of D are being considered, we write N#P(x)[D] and P(x)[D]. [ Verbally, we add w.r.t. D. ] P(I) means P(x) and ~P(x). -E for any expression E means that E is false. Theorem: -~D(x,x) Now consider various values of D: 1. YES(a,b) Turing Machine a with input b halts yes. 2. PR(a,b) Wff a with its one free variable replaced by the symbols for b is provable. 3. SE(a,b) Set a contains element b. 4. TS(a,b) English sentence a with its one pronoun replaced by noun phrase b is true. Then by substitution, we have: 1. -~YES(x,x): The set of Turing Machines that do not halt yes on themselves is not recursively enumerable. 2. -~PR(x,x): The set of the Godel numbers of wffs that are not provable on themselves is not representable. 3. -~SE(x,x): There is no set of those sets that do not contain themselves. 4. -~TS(x,x): There is no sentence that is true iff its pronoun refers to a sentence that is false of itself. (With further logic, this translates into the Liar Paradox. As is, it is the paradox 'It is not true of itself.' is true of itself.) Turing's proof of the Unsolvability of the Halting Problem takes just a little more logic. (See my papers below.) People (e.g. Gregory Chaitin) who say that Turing is to computability as Godel is to Logic don't realize that these are different theorems. The analog in Logic to Turing's result is that whether a wff is decidable (provable or refutable) is not P(I) e. g. define DEC(a) to mean decidable and we have -DEC(I). The analog in computing to the theorem based on w-consistency is the always-halting problem (is not solvable, i.e. recursive.) When you do it that way, then it is easy to see why Russell's Paradox happens: We are simply trying to put an aleph-0 set into 1-1 with an aleph-1 set. It is a one step proof. Avoiding it is not so simple. We are in effect asking for an axiomitization of the recursively enumerable sets w.r.t. relation SE. We can work with the sets that are not representable by YES but not those not representable by SE! Using the above formalism, we can greatly simplify and clarify the various attempts to axiomatize Set Theory (and probably other theories.) For example, the fact that there is an empty set is ~TRUE(x) where D is SE, i.e. ~TRUE(x)[SE]. The fact that there is a universal set is TRUE(x). Relation TRUE is defined by DEF:P(a),P(a)^TRUE(a) where for any wffs w and v, DEF:w,v means w and v are equivalent. Peano's 5 axioms are reduced to simply TRUE(x)[YES], or TRUE(x)[PR] if you want to be old-fashioned. It is fun to formalize the various Set Theories using this formalism - and watch the various authors struggle with the complex, semi-formal notation of Set Theory! (It is also easy to generate Recursion Theory theorems.) Charlie Volkstorf Cambridge, MA http://www.mathpreprints.com/math/Preprint/CharlieVolkstorf/ 20021008.1/1 http://www.arxiv.org/html/cs.lo/0003071 === Subject: : Re: Another look on Russell's first paradox X-ID: Zj2vYwZGge8lO5alT8ZrlLXsC8UN59LGJWPT+IUHjdd3sFWPClPfou Solution: Use some advanced set theory which does not allow for this > contradiction. (For example, ZF(C), TT, NBG, MK, NF, NFU, etc., etc.) > various attempts to axiomatize Set Theory Fine. DID you actually do that for one particular set theory? I'm sure you did, after all: It is fun to formalize the various Set Theories using this formalism - and watch the various authors struggle with the complex, semi-formal notation of Set Theory. F. === Subject: : Re: Another look on Russell's first paradox > Using the above formalism, we can greatly simplify and clarify the > various attempts to axiomatize Set Theory > Fine. DID you actually do that for one particular set theory? I'm sure you did, after all: It is fun to formalize the various Set > Theories using this formalism - and watch the various authors struggle > with the complex, semi-formal notation of Set Theory. > F. Yes. I started with ZFC. It was pretty easy. (Try it yourself. I'll be glad to help if you'd like. It's just another application of my axiomatization of programming on which my Program Synthesis system is based. I already gave you the formal wffs for 2 or 3 ZFC axioms.) Charlie Volkstorf Cambridge, MA === Subject: : Re: Another look on Russell's first paradox X-ID: SPateUZQYe6QEx2z4OGVmkCb5vXSkpxRdb74wTdrbfJi-2Mo-pN96O Using the above formalism, we can greatly simplify and clarify the > various attempts to axiomatize Set Theory Fine. DID you actually do that for one particular set theory? I'm sure you did, after all: It is fun to formalize the various Set > Theories using this formalism - and watch the various authors struggle > with the complex, semi-formal notation of Set Theory. Yes. I started with ZFC. It was pretty easy. Great. Where did you publish your results? Any reference to some Web-Site? F. === Subject: : Re: Any idea for a math career? > I don't think the university finance departments have a monopoly on grade > inflation. I have seen grade inflation in all areas, but particularly > business colleges. I had a room-mate who couldn't even pass a freshman > calculus course, but nearly had a 4.0 gpa in business. > No need to talk down about people who have different talents than you. A member of the family got an Advanced level on the state-wide math test for high-schoolers, but was obviously not going to make the grade in calculus and therefore dropped out. The difference between high-school math and calculus is that you can learn by rote and apply that math *until* you get to calculus and apply its methods to mathematical objects. Then different considerations and modes of learning apply. David Ames === Subject: : any software can help finding patterns from time series? I was just wondering if some software can help finding frequent pattern from time series/data streams, and finding unusual pattern from a new stream based on those frequent patterns. === Subject: : A parking problem Consider a parking lot with N spaces. You arrive at the lot at a random instant and find all N spaces occupied. Assuming that the occupancy times have uniform distribution in (0,1) hours what is the average waiting time and its standard deviation until a space becomes vacant. === Subject: : Re: A parking problem If nobody else is added to the queue, or anyone who is added agrees to let you park first, I think it is 3600 / ( 2*N ) seconds. I have a finite algebra text with business applications and a section on queing theory that I haven't read thoroughly. It's right here and I can look something up.... Yours, Doug Goncz, Replikon Research, Seven Corners, VA The hormones work at different speeds. In a fight-or-flight scenario, glucocorticoids are the ones drawing up blueprints for new aircraft carriers; epinephrine is the one handing out guns. === Subject: : Re: A parking problem Amending previous post: That's assuming the arrivals before your were uniformly distributed in the {0,1} interval as well. But I am still not sure. Yours, Doug Goncz, Replikon Research, Seven Corners, VA The hormones work at different speeds. In a fight-or-flight scenario, glucocorticoids are the ones drawing up blueprints for new aircraft carriers; epinephrine is the one handing out guns. === Subject: : Re: Are all mathematicians music lovers? X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Punge: Micro$oft X-Sanguinate: themvsguy@email.com X-Terminate: SPA(GIS) X-Tinguish: Mark Griffith X-Treme: C&C,DWS In , on > === Subject: : Are all mathematicians music lovers? No. A lot are. >It seems every math/science type I meet is a Bach lover, having >fallen for the propaganda that music expands the mind. No doubt, but is that because of the people you meet or only your perceptions of them? Do you imagine that everyone who loves Music loves Bach? Or even that every Mathematician who loves Music loves Bach? If so, you are seriously in error. >having fallen for the propaganda that music expands the mind. And what gives you this keen insight into their psychology? Perhaps a visit to Snopes would do you some good. >Actually, there are people >who go crazy from repetitive tunes that won't stop inside their >head. And your evidence for that claim is? >I've been able to drive otherwise calm-and-collected math/science types >berserk by saying music is stupid. Telling you that you are an ignorant fool is not the same as going berserk. >Are there any mathematicians today who have the >courage to say music is stupid? Why does it take courage to lie, tonto? I can't figure out whether you hate Music, hate Mathematics, or both. IAC, you seem to be living in a fantasy world. Do you have the courage to admit that both Mathematics and Music are beautiful, and that that's the reason some people enjoy both? -- Shmuel (Seymour J.) Metz, SysProg and JOAT Any unsolicited bulk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org === Subject: : Re: Are all mathematicians music lovers? > on Saturday 16 Prophet Mohammad was thought to have hated > the arts and music; he destroyed the Vedic temples that once existed at > Mecca. I don't think that has anything to do with arts or music. Correct me if I'm > wrong, but didn't these temples contain representations of God? If I am not > mistaken, representation of God in a pictural form is not allowed by Islam. Conservative Muslims today refrain from listening to music. I don't know where you hold this fact from. In fact, Muslim prayers are quite > melodic. It almost seems they are sung. On this, nindiv may have some justification. I shared an office with a very devout and scholarly Lebanese Moslem for a couple of years. He was also a great lover of western classical music. But one day he decided that if he wished to stay true to his beliefs, he should sell off his music collection as he felt it was probably prohibited. This was based on his own readings of the Quran/Koran, not the edict of some imam. This guy was not some half-educated fanatical sheep (in fact he told me that his local clerics were wrong in a number of their teachings, including the way they chanted). In fact from his research he concluded that the original prohibition probably referred to drunken revelry around the campfire. Nevertheless, he decided the edict applied to all (instrumental? non-chant?) music, no exceptions allowed. For the record, he was also VERY vocal about how un-Moslem the Moslem terrorists of his homeland were, and he could quote you chapter and verse on where they were distorting and violating scripture. Peaceful Moslems DO exist. - Randy === Subject: : Re: Are all mathematicians music lovers? > In fact from his research he concluded > that the original prohibition probably referred to drunken > revelry around the campfire. Nevertheless, he decided the > edict applied to all (instrumental? non-chant?) music, no > exceptions allowed. Well, from this, it seems it was his decision to refrain from listening to *all* music, since you're saying he thought the text probably referred only to drunken revelry. Some people decide to follow the gcd idea of the text, to be sure not to contradict it; others decide based on what they believe was the spirit and context of the writing. 2 different approaches that lead to some differences. > For the record, he was also VERY vocal about how un-Moslem > the Moslem terrorists of his homeland were, and he could > quote you chapter and verse on where they were distorting > and violating scripture. And was very right to be vocal about that. Unfortunately, most medias convey a wrong image of religions due either to bias (which is unacceptable but unfortunately very common) or lack of knowledge/understanding of the reference text. Quotes given out of context are also very frequent. > Peaceful Moslems DO exist. Of course! Sam -- Don't be afraid, I'm gonna give you the choice I never had... - Lestat in Interview with the Vampire (Ann Rice, 1976) === Subject: : Re: Armand Borel (Hans Aberg) a .8ecrit : >I found the Algebra and Commutative Algebra books useful, but I do not AFAIK, Serre was heavily involved in the writing of the Commutative Algebra book. If this is the case, any difference in quality between this and Homological Algebra probably isn't anything to do with Serre. === Subject: : Re: Armand Borel >>I found the Algebra and Commutative Algebra books useful, but I do not ... See thread Bourbaki Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: === Subject: : Re: Armand Borel dead >> It is the opposite in fact: There is a selection of facts that avoid any >> real difficulties. >Don't you think Serre's paper(s) on coherent sheaves (FAC) >were rather significant in the development of algebraic geometry? >If he was just dressing up the work of others, who were they? This is too far back in time in my interests in order to be able to comment. Experts in the field usually know such things; try one of those. I think it is time to leave this discussion. Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: === Subject: : Re: Armand Borel dead >The same ethical principles should apply to everybody. But then, there is >>the way people have *behaved* with respect to those principles. You are right on the mark. Exactly. Some behaviours do justify posthumous finger-pointing. Some don't. >>So, tell us: what crimes did Borel commit that would justify the same kind >>of finger-pointing? The question was not over that issue, but whether it was admittable to > discuss negative aspects of a deceased person. Ditto. > Then there should be the > same underlying equality principle in play as in the case of crimes > against humanity. For more, see my reply to KRamsay. In that reply (or elsewhere, for that matter), I don't see anything about Borel that could *possibly* justify such posthumous finger-pointing. As I said, it only tells us something about you, namely, that you are in serious lack of good manners and decency. Hugo === Subject: : Re: Armand Borel dead >In that reply (or elsewhere, for that matter), I don't see anything about >Borel that could *possibly* justify such posthumous finger-pointing. Well, that's your opinion. Conservative posters usually say that such matters cannot be discussed while the person is in office, being a revered scientist, nor after retiring, being a revered retired scientist, not after death, being a revered deceased scientist. So it can never be discussed by that view. In the case of Newton, such facts are only recently being brought into the light. It is just a question of moving towards a more open society, whether one takes a less or more open attitude towards bringing out the truth. The math subculture lags behind the rest of society in this and other social aspects, taking a more conservative stance. >As I said, it only tells us something about you, namely, that you are in >serious lack of good manners and decency. It sounds as though you are out to pick up a fight and nothing else. I think the bullying that is the consqeuence of not bringing out such facts is the true indecency. You have to experience it, in order to understand what it really means. Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: === Subject: : Re: Armand Borel dead > So there is a sorry combination here, a celebrity status that is allowed > to deflect much needed resources away from topics that would provide for a > more proper scientific development. Serre is, of course, a great talent, > but not a Nobel Prize level scientist or anything on that level. In that case, you'd better notify the Abel Committee immediately. No doubt they'll revoke Serre's Abel Prize when they find that such a notable as yourself thinks him unworthy. -- Wayne Brown | When your tail's in a crack, you improvise fwbrown@bellsouth.net | if you're good enough. Otherwise you give | your pelt to the trapper. e^(i*pi) = -1 -- Euler | -- John Myers Myers, Silverlock === Subject: : Re: Armand Borel dead >>Don't you think Serre's paper(s) on coherent sheaves (FAC) >>were rather significant in the development of algebraic geometry? > This is too far back in time in my interests in order to be able to > comment. Experts in the field usually know such things; try one of those. If you don't know anything about Serre's work, it makes your criticism of him even more bizarre. I imagine his work on faisceaux algebriques coherents would generally be regarded as his most significant contribution. -- Timothy Murphy e-mail: tim@birdsnest.maths.tcd.ie tel: +353-86-233 6090 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: : Re: Armand Borel dead > Have you ever met Borel? :-) I have. When I spent two semesters at the Institute in the early eighties, I ate lunch with him frequently. He was on at least one occasion a guest in my home. I was aware of his reputation for reducing people to emotional rubble, but he was always extremely kind and generous to me. I learned a lot of math from him. Possibly he just never saw me as a threat worth devastating. At the time I was only a few years out of grad school, and my father, who had never seen the point of a research career, was always gently prodding me to think of ways that a math Ph.D. could be useful in the world of commerce. One day, Borel took my father (who was visiting me at the Institute) for a long walk in the Institute woods. I don't know what got said on that walk, but I know that starting from that moment, my father believed to his core that there could be no higher calling than research in pure mathematics. He rarely read books, but he went out the next day and bought Hardy's Mathematician's Apology. A few years ago, I got on a train in France and saw (by pure coincidence) Borel sitting alone in a passenger seat. I found that despite all the lunches we'd had together, I was too shy to say hello. He *was* intimidating, but not, I think, intentionally. I cried when I saw his obituary. I wouldn't have predicted this, but his passing made me much more than ordinarily sad. I'm glad I knew him. Steven E. Landsburg www.landsburg.com/about2.html === Subject: : Re: Armand Borel dead > So there is a sorry combination here, a celebrity status that is allowed > to deflect much needed resources away from topics that would provide for a > more proper scientific development. Serre is, of course, a great talent, > but not a Nobel Prize level scientist or anything on that level. > In that case, you'd better notify the Abel Committee immediately. > No doubt they'll revoke Serre's Abel Prize when they find that such a > notable as yourself thinks him unworthy. That's... astonishing! ENS (Paris), CNRS, Princeton, Harvard; he was the youngest mathematician to be awarded the Fields Medal (1954), plus Abel's Prize, Gaston Julia Prize, Balzan Prize, Steele Prize, and many honorary degrees. But man, he's a fake!!!! === Subject: : Re: Armand Borel dead >In that reply (or elsewhere, for that matter), I don't see anything about >>Borel that could *possibly* justify such posthumous finger-pointing. Well, that's your opinion. It's also the opinion of quite a number of other people, including contributors to this thread. > Conservative posters usually say that such matters cannot be discussed > while the person is in office, being a revered scientist, nor after > retiring, being a revered retired scientist, not after death, being a > revered deceased scientist. So it can never be discussed by that view. The fact that he is revered has nothing to do with it. Assuming a human being hasn't committed crimes that are too serious to bar them from public mention, his memory deserves a minimum of respect, for a certain amount of time (ever heard about the word mourning?). > In the case of Newton, such facts are only recently being brought into the > light. Newton has died long ago. Nobody mourns his demise anymore. > It is just a question of moving towards a more open society, whether one > takes a less or more open attitude towards bringing out the truth. The > math subculture lags behind the rest of society in this and other social > aspects, taking a more conservative stance. The fact that taking a conservative stance makes one lag behind socially is your opinion. >>As I said, it only tells us something about you, namely, that you are in >>serious lack of good manners and decency. It sounds as though you are out to pick up a fight and nothing else. If I were, I would already have reacted to a far greater number of your posts in this thread than I actually have. I'm just reacting to a post I find outrageous. On the other hand, you're posting to a newsgroup, saying over and over again how the primary audience of that very newsgroup lags behind and is socially underdeveloped. Now how's *that* for picking up a fight? > I think the bullying that is the consqeuence of not bringing out such > facts is the true indecency. You have to experience it, in order to > understand what it really means. Well, if you really experience what you say, why don't you understand that you might have hurt other people's feelings in the first place. Hugo === Subject: : Re: Armand Borel dead See subject: Serre's work Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: === Subject: : Re: Armand Borel dead >> Have you ever met Borel? :-) >I have. When I spent two semesters at the Institute in the early >eighties, I ate lunch with him frequently. He was on at least one >occasion a guest in my home. I was aware of his reputation for >reducing people to emotional rubble, but he was always extremely >kind and generous to me. I learned a lot of math from him. Possibly >he just never saw me as a threat worth devastating. This is good that others that knew Borel better come forth and describe their experiences with him as counterbalance to what I have said. I just want people to know the picture that you describe as his reputation for reducing people to emotional rubble so that they do not believe that he was a kind of holy man. >A few years ago, I got on a train in France and saw (by pure coincidence) >Borel sitting alone in a passenger seat. I found that despite all >the lunches we'd had together, I was too shy to say hello. He *was* >intimidating, but not, I think, intentionally. So here we go: I just said what everybody who knew him knew. :-) Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: === Subject: : Re: Armand Borel dead See subject: Serre's work Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: === Subject: : Re: Armand Borel dead See subject: Ad Infinitum Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: === Subject: : Re: Armand Borel dead See subject: Abel Prize Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: === Subject: : Re: Armand Borel dead In response to the followng question: >> Don't you think Serre's paper(s) on coherent sheaves (FAC) >> were rather significant in the development of algebraic geometry? > This is too far back in time in my interests in order to be able to > comment. Experts in the field usually know such things; try one of those. > Serre is, of course, a great talent, > but not a Nobel Prize level scientist or anything on that level. Umm....if you're not familiar with work, then how do you know? Steven E. Landsburg www.landsburg.com/about2.html === Subject: : Re: Armand Borel dead >> Serre is, of course, a great talent, >> but not a Nobel Prize level scientist or anything on that level. >Umm....if you're not familiar with work, then how do you know? See the threads Serre's work and Abel Prize. Two things: I don't want to waste more of my time on this, whence the style of reply. And the Nobel Prize statutes require one to pinpoint specific exceptional scientific work, which is different from those say of the Fields medal and of the Abel Prize. Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: === Subject: : Re: Armand Borel dead >>In that reply (or elsewhere, for that matter), I don't see anything about >>Borel that could *possibly* justify such posthumous finger-pointing. Well, that's your opinion. It's also the opinion of quite a number of other people, including > contributors to this thread. in fact every contributor to this thread with the exception of hans aberg. *DONT FEED THE TROLL* tom === Subject: : Re: Armand Borel dead > I think it is time to leave this discussion. well bloody do it then and leave us all in peace. tom === Subject: : Re: Armand Borel dead > See subject: Ad Infinitum All messages pointed at by your See subject redirections have reached my newsserver, except for that one. The same goes for groups.google.com. So it seems that you never posted what you claim you did. What a wimpy way of running out of arguments! Even more pathetic than the I think it is time to leave in your reply to Timothy Murphy's point about the coherent sheaves paper. But such a bahaviour is not really surprising, coming from someone who doesn't hesitate to walk over a man's warm corpse to get at another man's throat. I do sympathize with your parents for an utterly failed education. Hugo === Subject: : Re: Armand Borel dead > *DONT FEED THE TROLL* I'm afraid Aberg is not even that. It's not the first time that he attacks Serre like this on sci.math (at some point, he even managed to slip past the moderator's watchful eye into sci.math.research). I had never felt inclined to react to his previous posts, but bouncing off an obituary for Borel to attack him was too indecent and untimely for me to remain silent. Hugo === Subject: : Re: Armand Borel dead > [...] so that they do not believe that he > was a kind of holy man. Nobody (either in this thread, or the NYT obituary) ever claimed he was. He was a human being, like you and me, with his good and his bad sides, like you and me, and there are certain occasions on which one should refrain from such petty judgements in public, that's all. Hugo === Subject: : Re: Armand Borel dead >I'm afraid Aberg is not even that. It's not the first time that he attacks >Serre like this on sci.math (at some point, he even managed to slip past >the moderator's watchful eye into sci.math.research). You have rather insidiously omitted the full context of that thread: This thread contained discussions about late Rota's editorial practises (of which I said nothing), and in that context, I happened to mention some other (living) mathematicians editorial practises (of which Serre was only one). These were examples of editorial practises, as that topic is of importance of mathematicians that publish. So the reason that there are strong emotional reactions does not have anything to do whether the person spoken about is deceased or not, but that some are idols considered to be immaculate and therefore negative facts about them, living or not, are inadmissible for discussion no matter how true. If a person is without that idol category, one evidently can say anything, and nobody will react. This idolatry seems also to be connected to the French school mathematicians: Say the same thing about a non-French mathematician, and nobody seems to react. Perhaps it is so that this French school in modern times is so heavily built up around arrogant prestige because it does not really have the capacity to relate the other general developments in modern science. It has perhaps become an isolated pool of enthusiasts that really need to aggressively attack any threats from outside scientific evolution in the lack of being able to follow that evolution. See more in the thread Ad Infinitum. Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: === Subject: : Re: Armand Borel dead *DONT FEED THE TROLL* >I'm afraid Aberg is not even that. It's not the first time that he attacks >>Serre like this on sci.math (at some point, he even managed to slip past >>the moderator's watchful eye into sci.math.research). You have rather insidiously omitted the full context of that thread: Ever heard about bandwith? Anybody can look up details in Google. No need to repeat them here. > See more in the thread Ad Infinitum. I.e., nowhere. Great! Hugo === Subject: : Re: Armand Borel dead >I'm afraid Aberg is not even that. It's not the first time that he attacks >Serre like this on sci.math (at some point, he even managed to slip past >the moderator's watchful eye into sci.math.research). Also, technically, I did not attack Serre, but merely described his editorial practises; see the thread Publishing. Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: === Subject: : Article Sorry for my poor english. (I have no academic affiliation nor appropriate degrees). I am looking for someone interested and knowledgeable in nonlinear science who could consider mail me: food4thought@libero.it . Many thanks. === Subject: : A simple question on distributivity Hello to everyone. I am studying category theory, and I have a question. Does exist some necessary and sufficient condition for the distributive law of product over coproduct: A x (B + C) ~= (A x B) + (A x C) (read ~= as isomorphic to) to hold over a category? (assume the category cartesian and cocartesian). It seems that there is nothing about the topic in the McLane, or more likely I'm unable to find it. Thank you very much. Pietro === Subject: : Astronomical Observations - Part 4 Comments: This message probably did not originate from the above address. It was automatically remailed by one or more anonymous mail services. You should NEVER trust ANY address on Usenet ANYWAYS: use PGP !!! Get information about complaints from the URL below X-Remailer-Contact: http://80.65.224.85/POL/ In case my abuse address is unreachable: It is because it has been flooded by , please contact X-Mail2News-Contact: http://80.65.224.85/ -----BEGIN PGP SIGNED MESSAGE----- Continued from part 3... Next we'll look at Mars, which has the longest synodic period of all the planets. We can see that Mars is the first planet beyond Earth's orbit since Mars is moving much faster through the caelestial zodiac than Jupiter or Saturn. But like the Jovian planets we can see that the elongations of Mars from the Sun reach oppositions on an observably predictable periodic basis as is true for heliacal risings, settings, squares, trines or any repeating angle of aspect to the Sun. Heliacal risings are being treated as semisextile aspect for continuity, and the nice round number thirty degrees is convenient, easy to remember and to measure by hand, since a whole hand or fist plus three closed middle fingers at arm's length together makes 15 degrees, two whole fists make about 20 degrees, depending on your physical type. The angle from outstretched thumbtip to pinky fingertip is some 25 degrees, but you must calibrate your own hands and fingers to estimate a perspective angle accurately. An easy way to do this is to stand in a rectangular or square room and see how many hands, fists, and fingers it takes to measure 90 degrees, i.e. from wall to wall. Four times a fist and 3 closed middle fingers ought to be about 90 degrees. Experiment to see what works best. Calibrating by the stars assures the greatest accuracy. No matter how close or far away an object is, ten feet or ten thousand lightyears, the angle subtended to you viewing those objects will be the same. A really sharp naked-eye astronomer can discern down to one arcminute. But I'm being conservative, so that ancient stargazers would need only resolve twenty arcminutes and estimate positions of stars and planets to plus or minus one de- gree, which for the Sun's apparent motion is about one day equals one degree. This is essential to know since adding one or more days to a predicted heliacal rising adds ~1 degree per day to the Sun's ecliptic longitude. Each consecutive heliacal rising for Mars occurs about 780 days apart. With a spectacular opposition for Mars just days away at this writing which will be August 27, at 26 Libra to the Sun 26 Scorpio. Mars will rise near above Mars, Zubenelgenubi at 20 Lib +0 & Zubeneshamali at 25 Lib +8. Add 780 days and we have January 31 2007. There's Mars at 16 Sagittarius to the Sun 16 Capricorn, is nearly impossible to see Kaus Borealis at 12 Sag -2 this close to sunrise (past astronomical twilight) and Mars may be difficult to spot here in the mountains of central Colorado. 16 Sag is 20 degrees past 26 Lib but we've witnessed Mars at opposition back on August 27th This also tells us that Mars is zipping along, so must have circuited the zodiac past 360 degrees, and is now 360 add 20 equals 380 degrees from where it was before. Simple interpolation tells us Mars takes some 739 days to complete one sidereal orbit. This is a rough figure as further observation shows. 10 times 780 is 7800. Ex- rience shows Mars has significant orbital eccentricity, and orbits quite rapidly through the caelestial zodiac. We find through experience that Mars is frequently off by a month or more from where we predicted it would be last we predicted its next heliacal rising, setting or any other repeating like-phase. Mars is at 14 Pis, and the Sun is 6 Ari on April 21, 2026. We must jump ahead to May 28, 2026, fully 37 days later, to find Mars and the Sun separated by 30 degrees sidereal longitude. As the apparent velocity of Mars is nearly as fast as the Sun's past superior conjunction it takes a few days to compensate for being just a degree off from 30 degrees. Thus to compensate for 8 degrees delta took us 37 days. Come May 28 2026, Mars is 12 Ari and the Sun is 12 Tau. Hamal at 13 Ari +10 and Sheratan at 9 Ari + 8 makes it easy to estimate Mars' position at 12 Aries. Shedir at 13 Ari +47 draws a nearly perpendicular line or arc to Hamal relative to the ecliptic making measurement easy. 26 Libra is 166 degrees from 12 Ari, meaning Mars went under eleven & a half times or 4126 degrees around the zodiac in 7837 days, making our observable average 684 days per sidereal orbit based on just two observations ten heliacal risings apart. Babylonians over centuries of observation and calibration found this to be around 687 days based on the long-term averages, which is one year, three hundred twenty-two days per sidereal orbit of Mars. These ancient astronomers-astrologers noticed that 151 sidereal orbits of Mars nearly coincided with 284 tropical years and 133 repeating synodic phases of we arrive at December 12, 2288. There's Mars at 23 Lib and the Sun 22 Sco, 29 degrees apart. Merely four days later finds Mars 26 Lib & Sun 26 Sco--right on the dot. So the ancient Babylonian sidereal-synodic multiple of Mars is off just 4 days in 284 years...very impressive. We also notice that Mars goes retrograde centered near inferior conjunction for an average of 73 days. Try it. Astrolog charts the synodic velocities of every planet. days. With every empirical observation for retrogrades and oppositions, the accuracy of this average improves. Observation proves 73 days is Mars' retrograde average. End Part 4. See Part 5 For Continuation... Daniel Joseph Min *Min's Planetary Awareness Technique (chapters 1 thru 6): http://groups.google.com/groups?selm=HFVIRNCU37838.7946990741@ Gilgamesh-frog. org *Min's Official PGP Public Key on the MIT server: http://groups.google.com/groups?selm=3XWB7QJO37766.971099537@ Gilgamesh-frog.o rg *Min's Home Page On The World Wide Web: http://groups.google.com/groups?selm=0XNKAO4L37773.8337962963@ Gilgamesh-frog. org -----BEGIN PGP SIGNATURE----- iQA/AwUBP0JnG5ljD7YrHM/nEQKEUwCfVNU4HIIAbrs73+ j1QXdkn048QOEAoNj2 R5LrlzWSRuvGr2SQSdfUypYI =Qo/4 -----END PGP SIGNATURE----- === Subject: : Astronomical Observations - Part 5 Comments: This message probably did not originate from the above address. It was automatically remailed by one or more anonymous mail services. You should NEVER trust ANY address on Usenet ANYWAYS: use PGP !!! Get information about complaints from the URL below X-Remailer-Contact: http://80.65.224.85/POL/ In case my abuse address is unreachable: It is because it has been flooded by , please contact X-Mail2News-Contact: http://80.65.224.85/ -----BEGIN PGP SIGNED MESSAGE----- Continued from part 4... Next on the list is Earth, meaning the Sun relative to Earth. Since circa ~200,000 BC, the heliacal rising of stars has consistently demonstrated to stargazers that the tropical year precesses against the stars by about five-sixths of one arcminute per year or approximately one degree per 26000 solar days, i.e. 71 years 68 days on average. The ancient Mayans were superb astronomers. They used a Haab intercalation interval so that 1508 haabs was commensurate with around 1507 tropical years (C.P. Bowditch, published 1906) since the value of one mean tropical year takes 365.2422 mean solar days, and one Haab equals exactly 365 days--you do the math. The very long-term Mayan average for the great year of pre- cession equals 5 times 13 Baktun, or 5 ages of the Sun. Interesting, since Leo is the fifth sign of the zodiac. One Baktun is 144,000 days. 13 Baktun = 1,872,000 days. Five times 1,872,000 days equals 9,360,000 days a year of precession ergo one 360-degree sidereal gyration of Earth's axis of rotation against the caelestial zodiac one zodiacal age of precession is 2,135 years 208 days. Cf. modern secular-religious estimates of ~2,150 years. The Mayan long-count is undoubtedly more accurate. The well-known date of December 21, 2012 was predicted not by modern science but by ancient Mayan astronomers. It predicted the winter solstice Sun conjunct the sacred tree or apparent intercept of the galactic & ecliptic planes at 5 Sagittarius +0...accurate to one arcminute. If we subtract 9360000 days from December 21 2012, the last conjunction of the winter solstice Sun was likely not too far from Julian Day -6903717, which the modern Gregorian calendar shows as March 1, 23615 BC, clearly way off the mark...the Gregorian calendar is erroneous for long-term prediction. The Mayan calendar is better by far but how they achieved such mastery is a mystery, unless they actually observed for many, many millennia. I believe this is how they did it, and that the Mayans and other pre-Columbian civilizations are vastly older than secular-religious archaeologists have admitted to. By watching the precession of Earth's axis, really the whole rotating Earth, long-term prediction of sidereal- synodic-tropical cycles gained accuracy over centuries and millennia of observation. The length of solar days was always the basis for counting longer periods, such as a lunar month was some 29 1/2 days, a tropical year was about 365 1/4 days, four tropical years about 1461 days etc. Each multiple was numbered by days, months & years, by the Sun the Moon & Stars also as per Genesis. Hence the multiples for the planets out to Saturn were referenced to Earth's solar days, lunar months and the Earth's tropical and sidereal years. As we've seen for Mars, Jupiter & Saturn, the sidereal motion of planets is fundamental to determining not only the position of a planet but also its sidereal year around the Sun. It is perfectly obvious that Mars, Jupiter & Saturn orbit the Sun and it is equally obvious that Mercury & Venus also orbit the Sun. Hence it follows that Earth orbits the Sun, since we can see that Mars is further away in its orbit than Earth is and Venus is closer than Earth is by its heliocentric orbit. Only the Moon sidereally appears to orbit the Earth, and the phases of the Moon show that both Earth and Moon are orbiting the Sun--in reference to the caelestial sphere. Incommensurability between Earth's tropical and sidereal years is easy to understand, yet has confounded more than a few amateur astronomers and astrologers for centuries to millennia. In tracking the synodic and sidereal motion of planets, we are referencing all positions, that of the Sun, and the planet(s) in question, to the caelestial zodiac of the stars. We are counting in solar days independently of years at first, only later by counting fractions of years in days instead of decimal places. Thus the side- real year of Earth reveals to an observer how tropical years are slightly faster than sidereal years, as year after year we see this disparity compounding enough so that we can correctly estimate the value of precession. The difference between a solar day and sidereal day on Earth is dependent on the length of a sidereal year vs. the length of a sidereal day. The faster Earth rotates sidereally, then the more solar days per sidereal year the observer will witness. The faster the Earth orbits the Sun, the fewer solar days per sidereal year we see. Since the tilt of Earth's axis circa 200,000 BC, solar days have numbered 365 and change per year with barely 50 arcseconds per year difference between sidereal and tropical year relative to the stars to wit, precession. That is why a solar day is slightly longer than a side- real day, since Earth's orbit makes the Sun rise later than distant stars which, comparatively, care not that Earth orbits the Sun with its sidereal-annual parallax having generally undiscernible effect, sidereal diurnal parallax/geocentric parallax having thousands of times less effect on the apparent positions of stars--albeit planets are affected slightly more, Moon more than any- thing else. At about a quarter million miles, the Moon can appear up to a degree off from geocentric position. End Part 5. See Part 6 For Continuation... Daniel Joseph Min *Min's Planetary Awareness Technique (chapters 1 thru 6): http://groups.google.com/groups?selm=HFVIRNCU37838.7946990741@ Gilgamesh-frog. org *Min's Official PGP Public Key on the MIT server: http://groups.google.com/groups?selm=3XWB7QJO37766.971099537@ Gilgamesh-frog.o rg *Min's Home Page On The World Wide Web: http://groups.google.com/groups?selm=0XNKAO4L37773.8337962963@ Gilgamesh-frog. org -----BEGIN PGP SIGNATURE----- iQA/AwUBP0KlSJljD7YrHM/ nEQIt5ACffH8bKDxGujY8e5gqpy2Ckz80mCsAoJGt 5OLjPwleUzpX9Z9+ZHV5zIiO =4+PY -----END PGP SIGNATURE----- === Subject: : Re: Astronomical Observations - Part 5 > Next on the list is Earth, meaning the Sun relative to > Earth. Since circa ~200,000 BC, the heliacal rising of > stars has consistently demonstrated to stargazers that > the tropical year precesses against the stars by about > five-sixths of one arcminute per year or approximately > one degree per 26000 solar days, i.e. 71 years 68 days > on average. The ancient Mayans were superb astronomers. > They used a Haab intercalation interval so that 1508 > haabs was commensurate with around 1507 tropical years > (C.P. Bowditch, published 1906) since the value of one > mean tropical year takes 365.2422 mean solar days, and > one Haab equals exactly 365 days--you do the math. The > very long-term Mayan average for the great year of pre- > cession equals 5 times 13 Baktun, or 5 ages of the Sun. > Interesting, since Leo is the fifth sign of the zodiac. How to you get each line to have the same number of characters unless the last character is a punctuation mark, in which case the line is one character longer? I do see an exception below at 'diurnal', but for such a length of prose, how do you do it? Are you possed of some hidden knowledge of numbers and cycles? > That is why a solar day is slightly longer than a side- > real day, since Earth's orbit makes the Sun rise later > than distant stars which, comparatively, care not that > Earth orbits the Sun with its sidereal-annual parallax > having generally undiscernible effect, sidereal diurnal > parallax/geocentric parallax having thousands of times > less effect on the apparent positions of stars--albeit > planets are affected slightly more, Moon more than any- > thing else. At about a quarter million miles, the Moon > can appear up to a degree off from geocentric position. ----== Posted via Newsfeed.Com - Unlimited-Uncensored-Secure Usenet News==---- http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000 Newsgroups ---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption =--- === Subject: : Re: Astronomical Observations - Part 5 haabs was commensurate with around 1507 tropical years > (C.P. Bowditch, published 1906) since the value of one > mean tropical year takes 365.2422 mean solar days, and > one Haab equals exactly 365 days--you do the math. The > very long-term Mayan average for the great year of pre- > cession equals 5 times 13 Baktun, or 5 ages of the Sun. What is truly astounding is that all of this gibberish..., er, text is that, disregarding -'s, , and .'s occuring at the end of the lines, it has _exactly_ the same length per line _without_ using additional spaces to pad the line lengths. What's the odds of that happening at random? Is this the result of obsessively painstaking editing on the part of the author? Or... is this some sort of strange encryption for an even deeper message???? Cheers - Chas === Subject: : Astronomical Observations - Parts 1, 2, 3, 4 Comments: This message probably did not originate from the above address. It was automatically remailed by one or more anonymous mail services. You should NEVER trust ANY address on Usenet ANYWAYS: use PGP !!! Get information about complaints from the URL below X-Remailer-Contact: http://80.65.224.85/POL/ In case my abuse address is unreachable: It is because it has been flooded by , please contact X-Mail2News-Contact: http://80.65.224.85/ -----BEGIN PGP SIGNED MESSAGE----- ANY MODERN ASTRONOMY program will work for this lesson. I recommend using the freeware Astrolog 5.41G with the freeware JPL-DE406 Swiss Ephemeris, Carte du Ciel 2.75 which is also freeware, and includes links to download dozens of freeware catalogues and other plugin options, or check out the SkyMap 9 demo version on my links URL: http://groups.google.com/groups?selm=IHO72C3E37766.8210763889@ Gilgamesh-frog .org This is very basic, and will show you how every planet visible to the naked eye, which includes the Sun, Moon, Mercury, Venus, Earth, Mars, Jupiter, Saturn, & Uranus, this will show you how these planets move as seen from the Earth in conspicuously repetitious and predictable patterns which are easily counted by days, months, and years between repeating sidereal and synodic multiples. This absolutely destroys any and all arguments against the ancients being perfectly able to see the motion of the planets against the night sky and counting by days, months and years to predict sidereal & synodic periods for each planet at least out to Saturn and possibly to Uranus, since it rarely can be seen with the naked eye. This is a big deal because secular academia has closed their eyes to timeless science and its reproducibility. This clearly transcends simple astronomy, but includes astrology, metaphysics, and all spiritual implications. Limit your program to what is visible to the naked eye. No guesswork & no speculation. Your astronomy software reliably emulates what we'd see when viewing the night sky in that direction, at that time from that location, conveniently, efficiently and with impressive accuracy. Of course, the view is better through a good telescope, or through the unaided, human eye, since it is assumed that ancients didn't have other means to see the stars. That's a humongous ad hoc assumption, but I'm granting modern-day atheistic science that much and I still win. Accurate positions of planets and stars is all we need for this lesson. Your favorite software will work fine. No telescope needed. We can see this all with our eyes, so reduce your software's star magnitude limit to five, and assume Uranus, Neptune and Pluto to be nonexistent (not as Gods, but to pacify the unbelieving scientist). For this lesson, we're concerned only with heliacal ri- sings of each planet separately, which depends only on sufficient angle between the planet and the Sun, so it can be spotted against background stars before sunrise. The Sun must be about 18 degrees below the horizon for full darkness and a little less for heliacal phenomena. This angle varies with each planet, and each star, and time of year, temperature, pressure, how good your eye- sight is, the geographical latitude of observation and local horizon, obstructions and circumstances of light pollution, smog, haze from forest fires, volcanos, etc. While these conditions can vary to extremes, generally, provided reasonably good seeing conditions towards the eastern horizon about an hour or so before sunrise, as you look to the east (from moderate latitudes) you can barely make out a planet that you expect to see rising heliacally on or about that date. If you miss it, then try again in a couple of days and you're bound to spot the planet you're looking for if it's Mars, Jupiter or Saturn; or plan ahead and begin looking sooner if it's Mercury whose orbit you can see is eccentric. You know that each planet has predictable orbital patterns, and although these patterns vary over the short-term, over the long-term they become more and more predictable to fractions of a degree in sidereal longitude & latitude. That's how you know that Venus is the most predictable, since Venus has the least eccentric orbit. We see this behavior of Venus through heliacal risings or settings, especially at maximum elongations inferior or superior. If getting up at four in the morning is not your style, simply open your astronomy program and set it for your geographical location and voilla! You're ready to view to heliacal risings of every planet--against the stars. In the next part we focus on Saturn's heliacal risings. Open your favorite astronomy program. As always, I use Astrolog, so all examples given refer to JPL ephemeris DE-406 with Abramov's expanded version of fixstars.ast provided by S. Moshier using the Astronomical Almanach. All data is accurate to within several milliarcseconds, which is vastly better accuracy than the plus or minus half a degree or thirty arcminutes we can achieve with an extended pinky finger at arm's length measuring one arcdegree...twice the apparent diameter of a full Moon. Three closed middle fingers spans five degrees, or the whole hand equals about ten degrees. You can calibrate simple hand measurements by memorizing bright marking stars near the ecliptic by their approximate longitude on the caelestial zodiac. The constellations and their associated myths help us to easily locate and identify stars as we become familiar with their appearances and their order in the sky. This is where Carte du Ciel or SkyMap comes in handy, since they depict the stars and planets graphically, and include millions more objects and dozens of unabridged catalogues for the astronomer. However, only Astrolog can chart the marking stars and planets by their zodiacal, constellational coordinates as used by ancient stargazers for tracking the planets. The complete list of almost 1000 stars is posted on my website, but here's an abbreviated list for convenient reference with the values rounded off to whole degrees and favoring brighter stars in the northern hemisphere. Remember the goal is not to memorize every star but is to estimate a planet's position at its heliacal rising, setting, opposition and other repeating synodic phases against the fixed background of this caelestial sphere: Name Longit. Lat. Bayer Al Pherg : 2 Ari + 5 etPsc Sheratan : 9 Ari + 8 beAri Caph : 10 Ari +51 beCas Hamal : 13 Ari +10 alAri Shedir : 13 Ari +47 alCas Cih : 19 Ari +49 gaCas Ruchbah : 23 Ari +46 deCas Segin : 0 Tau +48 epCas Algol : 1 Tau +22 bePer Alcyone : 5 Tau + 4 etTau Mirphak : 7 Tau +30 alPer Aldebaran : 15 Tau - 5 alTau Rigel : 22 Tau -31 beOri Bellatrix : 26 Tau -17 gaOri Capella : 27 Tau +23 alAur Mintaka : 28 Tau -23 deOri Alnilam : 29 Tau -25 epOri Alnitak : 0 Gem -25 zeOri Saiph : 2 Gem -33 kaOri Polaris : 4 Gem +66 alUMi Betelgeuse: 4 Gem -16 alOri Menkalinan: 5 Gem +21 beAur Alhena : 14 Gem - 7 gaGem Sirius : 19 Gem -40 alCMa Castor : 25 Gem +10 alGem Pollux : 28 Gem + 7 beGem Procyon : 1 Can -16 alCMi Asellus Au: 14 Can + 0 deCnc Kochab : 19 Can +73 beUMi Dubhe : 20 Can +50 alUMa Subra : 29 Can - 4 omiLeo Alphard : 2 Leo -22 alHya Algieba : 5 Leo + 9 ga1Leo Regulus : 5 Leo + 0 alLeo Thuban : 13 Leo +66 alDra Dhur : 17 Leo +14 deLeo Denebola : 27 Leo +12 beLeo Vindemiatr: 15 Vir +16 epVir Spica : 29 Vir - 2 alVir Arcturus : 29 Vir +31 alBoo Menkent : 18 Lib -22 thCen Zubenelgen: 20 Lib + 0 al2Lib Dschubba : 8 Sco - 2 deSco Antares : 15 Sco - 5 alSco Rastaban : 17 Sco +75 beDra : 21 Sco -12 epSco Sabik : 23 Sco + 7 etOph Rasalhague: 28 Sco +36 alOph Sargas : 1 Sag -20 thSco Gal.Center: 2 Sag - 6 SgrA* Eltanin : 3 Sag +75 gaDra Sacred Tre: 5 Sag + 0 ----- Solar Apex: 7 Sag +53 HerA* Kaus Austr: 10 Sag -11 epSgr Nunki : 18 Sag - 3 siSgr Vega : 21 Sag +62 alLyr Altair : 7 Cap +29 alAql Dabih : 9 Cap + 5 beCap Sadr : 0 Aqu +57 gaCyg Enif : 7 Aqu +22 epPeg Fomalhaut : 9 Aqu -21 alPsA Deneb : 11 Aqu +60 alCyg Markab : 29 Aqu +19 alPeg Scheat : 5 Pis +31 bePeg Algenib : 14 Pis +13 gaPeg Alpheratz : 20 Pis +26 alAnd Since we're beginning with Saturn, set restrictions in Astrolog to restrict all then uncheck only the Sun and and you'll see Saturn at opposition in 15 Gemini. This is just one pinky finger in longitude from Alhena at 14 Gemini. With the Sun in 15 Sagittarius, then Saturn adding 378 days, which is January 13, 2005. But Saturn is a little slow in getting there, reaching opposition the next day January 14, in 29 Gemini. The oppositions, which we'll skip for Jupiter and Mars, prove to us the planets Mars, Jupiter and Saturn, are orbiting the Sun beyond Earth's orbit, and these orbits are predictable, especially over long-term observations. As with Saturn, by adding 3781 days to its synodic phase, we arrive at which is May 9, 2014, again missing exactitude by only one or two days, due to Saturn's moderate eccentricity and about 2.5 degrees inclination to the ecliptic. For long-term predictions, the ancient Babylonians noticed that 9 sidereal orbits of Saturn coincided with around 256 synodic periods and 265 tropical years speaking in have January 1, 2269. Sure enough, there's Saturn near opposition in 14 Gemini directly above Alhena and just two days from true opposition January 3, 2269, showing that the Babylonians knew what they were talking about two thousand years before Christ. It's no mystery, but is readily observable, predictable and reproducible in the laboratory of the night sky, like heliacal risings. The predawn risings of stars and planets have been the carefully watched and predicted since men could mark a cave wall with a piece of coal, blood or whatever else has handy. Primitive stone observatories emerged which had much greater longevity, and showed the teamwork of prehistoric stargazers, and the importance they placed on the ephemeris of the Sun, Moon & Stars to the Earth. Naturally, the Sun is the single most important object visible in the Earth's sky. Man has watched the Sun as it rises and sets every day since humankind has walked the Earth. All life forms follow the diurnal circadian rhythm of Earth's daily rotation in one way or another. Hence the Sun formed the fundamental basis of tracking time from the beginning of every civilization that has come and gone, from primitive tribes of early hominids to more advanced human cultures, most of which are too distant in the past for their records to have survived. More recently, the Egyptians, Babylonians, Mayans, and others around the post-deluvian world are close enough in time for many of their records to be extant, mostly bits and pieces, some fairly intact, like the pyramids. In mans present time, secular-religious archaeologists prefer to believe that civilization is basically under 7000 years old worldwide, due to their historical ties to the Roman church, and continued use of the language in their laws and their sciences. This is not to blame the ancient translation of the bible, the Vulgate, but has been the politics of religion, as men serve mammon. After all the bible predicted this would happen, so it isn't surprising that the schism of religious-apostasy should continue to rule the minds of men. Yet the Moon & Stars have continued to illuminate the night sky for geological aeons and shall continue to do so for aeons. So it is that Saturn has been rising and setting helia- cally in very predictable intervals and shall continue to do so for many long ages to come. Since the initial date and time for observation of Saturn before sunrise will vary, we know the Sun needs to be some 18 degrees below the horizon to ensure visibility of any brighter star or planet from moderate latitude any time of year, weather permitting. But in fixed locations, i.e. where ancient and antediluvian population centers flourished, the heliacal risings of stars and planets were readily estimated to within a few days time and by the seasons of the year, tied directly to planting, harvesting and every single aspect of their lives. Thus astrology was the natural result of watching and predicting when the stars and planets would rise and set, by knowing where the planets are day and night. This knowledge was made by simple observation, counting days, months and years between cycles and phases. When Saturn rose heliacally, it was always about 378 days give or take a day or two since the last time it was observed to rise heliacally. With each consecutive heliacal rising of Saturn, fixed stars in the background showed that Saturn moves about 13 degrees in keeping with the Sun's progress relative to the stars some 13 days later each year--again, give or take a day or two, talking about long-term averages rounded off to integer days since the whole premise is to show that ancient stargazers could and did see that the planets clearly orbit the Sun, and that they could readily observe and recognize the sidereal and synodic orbits by watching the heliacal risings of planets and stars. The accuracy of the ancient ephemeris increased commensurate with continued calibration by observation of heliacal phenomena over the centuries and millennia of that civilization from its rise until its fall. The quality of long-lost very ancient ephemeredes is known by mans inherent ability as a man to see the night sky and to notice patterns and repetition in nature. These are perfectly natural talents that all people are born with--at least most people are. Once again, this comes down to how much credit we give prehistoric man. There are anthropologists who have recognized that early man was smarter than modern day, secular-religious science had theretofore acknowledged. Likewise the recognition that at least semi-intelligent hominids have been here many millions of years earlier than the orthodoxy used to believe albeit some still cling to their hopelessly obsolete superstitions about the antiquity of man, etc, it is clear that man and man-like sentient beings have roamed the Earth for aeons. One might reasonably argue that dolphins or whales are smart enough to notice the planets and stars rising and setting, and to count the days and years of these events. Elephants are known to remember things very well. At a minimum, we can safely say that early man was intelligent enough to count the days, months or years of observable heliacal phenomena and we see that such observable events are predictable, simply counting these events by days, months and years. ______________________________________________________ I think this is what makes modern astronomers angry at those of us who have realized that planetary motion is not nearly as mysterious as they'd like you to believe. ______________________________________________________ The Egyptians, Babylonians and Mayans showed admirable levels of sophistication in their astronomical records and their ability to predict very long-term periodical events, the great year of precession being among these, since the Earth's axis of rotation visibly gyrates one degree against the fixed stars about every 26000 solar days, which is about 71 tropical years, two months and nine days, therearound. This is according to the Mayan astronomers, whose astronomical skills were comparable to those of the Babylonians. Both left records proving that they could see the night sky, and that they could accurately count and predict periodic planetary orbits against the starry background of the caelestial sphere. As in this case, we *see* Saturn observably progresses about twelve degrees every year against the stars seen from Earth. Every twenty-nine and a half years, Saturn goes full circle against the stars, and over centuries of observation we see that Saturn circles the Sun nine times every two hundred sixty-five years--meaning that Saturn advances closer to twelve and a quarter degrees longitude per year thereby making short-term estimates of Saturn's motion a little more accurate and reliable than our round number of twelve degrees per year. Thus we may safely predict that Saturn will have moved east by closer to forty-nine degrees every four years, plus our ephemeris for Saturn has improved significantly by repeated observation and simple mathematical deduction. We'll notice Saturn's thirteen degree advance at times of entering or leaving retrograde motion and that this retrograde lasts for about one hundred thirty-eight or so days centered on inferior conjunction or opposition to the Sun. Every three hundred seventy-eight days, we see these motions repeat, when Saturn appears to stand still in the sky then begin to move backwards for some four and a half months before standing still again and returning to normal motion. Every time we see it again, about 378 days have passed and Saturn is approximately 13 sidereal degrees from where it was last time around. Carte du Ciel is especially useful for animating these apparent synodic motions against the background of the stars, since you can fine-tune increments down to days, hours and minutes, and mark the locations with finder circles to readily observe a planet's motion relative to the stars & constellation figures, and to the other planets. Although the accuracy of the ephemeris is not very reliable beyond plus or minus four thousand years, especially for the Moon, you can view distant dates to circa 20,000 years BC / AD. While tropical seasons can be way off the mark the apparent motion of a planet to the stars may not be far off the mark for say, 9000 BC. You just won't know the season, or the Moon's position at such a distant date, but other planets are probably within a couple of degrees of where they actually were. Not that this matters much, since you are simply using the present-day ephemeris to view synodic and sidereal motion of the planets that are visible and predictable. For example, most of us'll probably be up and about at should remember to walk outside for a moment and check out Saturn in 15 Gemini--just above and east of Alhena, and right below Mebsuta which marks sidereal 15 Gemini just 2 degrees above the ecliptic. Your extended thumb at arm's length spans about two arcdegrees thus you'll see that Saturn is maybe a pinky fingernail's width or so (about 2/3's of a degree) below the ecliptic at the time of observation. Since Asellus Australis (see list above) marks 14 Cancer right on the ecliptic (actually +0:04'38 but round degrees are all a stargazer needs), and bright Regulus at 5 Leo is less than half a degree above the ecliptic, you can quickly visualize the line, rather the arc of the ecliptic across the sky. Jupiter at 24 Leo and about a degree above the ecliptic should be visible in the eastern sky. Sirius at 19 Gem and 40 degrees below the ecliptic will be hard to miss in the southern sky (unless you live north of Barrow, Alaska). If you live in the southern US or similar latitude you might spot bright Canopus at 20 Gem -76 degrees barely above the south horizon. Orion should be in clear view below right of Saturn. See if you can spot Al-debaranu, the prime fiducial of the caelestial zodiac at 15Tau00 and 5 degrees below the ecliptic. As you see, when you look at a planet in the night sky the background stars help you to locate the planet's longitude and latitude, hence confirming previous predictions, and calibrating future predictions. In ancient times this was done for centuries & millennia. Let's look at Saturn heliacally. Just to be on the safe side, we'll put 30 degrees past Saturn for the predawn Sun. That ought to make it easy to spot Saturn before sunrise, whether you're watching from the old, royal Greenwich observatory at 25 meters above sea level & 00E00:00 longitude 51N28:38 latitude, or viewing atop the Great Pyramid at 31E09:00 29N58:51, or from the Sun Pyramid in Teotihuacan, Mexico ~19:44N 98:50W or from the site of ancient Babylon 44E24 32N33. Use your own default observation location, set up your favorite astronomy program to watch the sky from there. I'm using my own location here in central Colorado USA. Saturn is plainly visible at heliacal rising August 14, with Saturn 27 Gemini and the Sun 27 Cancer. We'll add the 378 days for Saturn's synodic period, to August 27, 2005, with Saturn 10 Can and Sun 9 Leo. Like before we are just a day short, so on August 28, 2005, Saturn is some 13 degrees further along in the caelestial zodiac which is 756 days, and we have September 9, 2006 which is about two days shy of Saturn 30 sidereal degrees to the Sun, thus September 11 2006 finds Saturn rising at previous observations is closer to 3781 than 3780. The date is December 21, 2014. Low and behold, Saturn's at 5 Scorpio and the Sun is 5 Sagittarius, right where we expected it to be. Remember, Saturn was at 27 Gem back Saturn is heliacally risen we see that Saturn is 5 Sco and the Sun 5 Sag. That's near 128 degrees that Saturn has progressed in ten synodic periods or ten times our round figure of 13 degrees. Again, as observations are made over longer and longer periods of time, ephemeris calibration and improvements are the inevitable result. These long-term observations of the heliacal phenomena inevitably reveal the limits as to how far the planets can appear to stray from Earth's ecliptic with the Sun, revealing each planet's orbital inclination to Earth's, and also revealing other obvious limits, such as Venus and Mercury display their orbital eccentricity when at maximum elongation, Venus very little, Mercury a whole lot more. This plainly shows the observer that Venus & Mercury are closer in heliocentric orbit than Earth is, and of course the paths of Mars, Jupiter & Saturn show that they are further away from the Sun in their helio- centric orbits than Earth is. We'll cover more on this in later parts. Jupiter is next on the list of planets. at 13 Virgo, 30 sidereal degrees from the Sun 13 Libra. just below bright Venus at 7 Virgo. Zaniah (etaVir) is between them near 10 Virgo. Remember, we are measuring the sky with our naked eye and extended hand, so round degrees, maybe down to a sixth of a degree, or ten arc- minutes, is as good of accuracy as we can achieve. You can see that the modern accuracy of JPL's ephemeris is based on observations made by large observatories, and formulated using advanced knowledge of mathematics and These values are rounded off to the nearest arcseconds of longitude and latitude, while the internal accuracy of the software is good to milliarcseconds (JPL-DE406): Aldebaran : 15Tau00'00 -5:28'00 Venus : 6Vir55'17 +1:32'47 Zaniah : 9Vir30'15 +2:35'21 Jupiter : 12Vir35'49 +1:07'05 Sun : 12Lib32'48 +0:00'00 Ancient observers would commonly use a measuring stick or metal rod notched with linear increments calibrated by the observer which he or she could comfortably hold at arm's length between both hands, ensuring a uniform perspective of sidereal measurement. But we will limit our ancient observers as having nothing but themselves to view the heavens, since that's all that they needed to clearly view the predictable motions of the planets against the fixed background of stars. Easily accurate to plus or minus one degree, simple enough so children could be taught to do this and carry on the stargazing tradition, counting the days, weeks, months, and years, planting, harvesting, worshipping by the ephemeris and its religiously-observed calendar--the religion of the stars. As each civilization developed, and became more sophisticated, they organized and specialized, so that astronomical observation, astronomy, and their logical deductions based on astronomical observations--meaning mathematics--ergo astrology, became the disciplines of specialists so that others in their community could go about their business. In ancient times, the astrologer was synonymous with the mathematician, star-logician in the most literal sense. Even in our day and age, it was only within the last few centuries that astrologer and astronomer reached a schism, since astrologers had long-since ignored the proper mathematics of astrology, and astronomers became disenchanted with the illusions yet perpetuated by today's tropicillogical astrologers and other schisms of astrology,--all who've hopelessly lost their grasp on the ancient practice of star-logic. Since this schism, astronomers have changed their ways of measuring the sky such that constellations became synonymous with unequal boundaries associated with the asterisms or some 88 familiar groups of brighter stars instead of the ancient method which divides the entire caelestial sphere into twelve equal meridians as signs with meridians of latitude from the caelestial equator. Modern astronomers began referencing positions only to Earth's terrestrial equator by its intersection on her ecliptic. Next time there's a pole shift, or crustal displacement (or both?), that'll screw up their method of measuring the sky in a heartbeat. Meanwhile Earth's slow gyration of precession continues to change modern astronomer's coordinates. For example, look at Regulus at 5 Leo near the ecliptic. In 8000 BC, Regulus was at 5 Leo. In 8000 AD, Regulus will still be at 5 Leo. The position of Regulus is easy to see and easily recalled. Only the slight, very long-term wobble of the ecliptic itself affects how we chart latitude of stars near the ecliptic, and also the longitude of stars farther away from the ecliptic. As a result, Regulus might be close to a degree from the ecliptic at some remote epoch but it's still going to mark 5 Leo for a long time to come, irrespective of precession, pole-shift or annihilation of civilization. Any survivors can point up at Regulus and confidently say Look! There's Regulus 5 Leo, and any planet passing nearby will certainly be identified by its position--relative to a recognizable fixed star, and certainly not by its RA/Dec. As for this example, on Julian Day -1200514, 1-Jan--7999 (8000 BC Gregorian) Carte du Ciel shows Regulus at 0h46m35s +4*36'06, and Carte du Ciel shows Regulus at 15h29m45s -17*53'29 on Julian Day 4643000 1-Jan-8000 (8000 AD). For a caveman marking scores on a cave wall to remember positions of planets relative to nearby stars counting days, months and years between repeating heliacal risings and other predictable synodic phases relative to the Sun, anyone can see that the positions of planets are most readily and easily tracked by their positions to visible stars, and that those stars remain fixed in their position on Earth's caelestial sphere with subtle proper motion so slow that it takes millennia even to be noticed by the best of naked-eye astronomers. Hence Orion's Belt, for example, is very close to the same position in the sky as it was when they built the Great Pyramids 10,500 BC, since the three stars of the belt have very low proper motion. So Mintaka 28 Tau, Alnilam 29 Tau, and Alnitak the Great Pyramid star at 0 Gem have illuminated the same positions on Earth's caelestial zodiac ever since. we can easily see where it is in relation to the stars before sunrise, since the stars tell us where 13 Virgo is. In this case, Zaniah at 10 Vir is nearby, so it is easy to estimate Jupiter's position to plus or minus a degree of certainty. With this simple observation, the next heliacal rising of Jupiter is easily predicted by the average period that Jupiter has been seen for ages to repeat its synodic cycles. That is December 4, 2005, but Jupiter is about five days past the 30-degree mark from the Sun. November 29, 2005 finds Jupiter 12 Libra and the Sun in 12 Sco, and Jupiter will be rising near rising Jupiter, you can be sure where 29 Virgo is. But Kappa Virgo at 10 Lib and +3 latitude--although it's a lot closer to Jupiter--may be difficult to see at 4.18 magnitude. The star called 109 Vir is a bit brighter at 3.72 magnitude and marks 13 Libra near +17 latitude. The important thing is to know which stars that you're looking at, and their approximate longitude & latitude in the zodiac. In the 395 days between heliacal rising, Jupiter will complete one sidereal orbit approximately every 12 years. Jump ahead 4000 days from October 30th late. We must go back to October 5, 2015, with Jupiter has moved from 13 Vir to 17 Leo, 26 degrees before the completion of one sidereal year for Jupiter. Estimates that Jupiter would take about 12 years to complete one sidereal orbit. Jupiter's tenth heliacal rising showed us that Jupiter moved about 334 degrees over 3992 days. We might extrapolate off this, and figure that Jupiter will make about 360 degrees in another 311 days making a rough estimate 3992 + 311 = 4303 days for a sidereal year of Jupiter based on a total of three observations. Let's look at the next rising of Jupiter 400 days from October 5, 2015, November 8 2016. Now we're about four days late, so go back to November 4, 2016, for Jupiter my location. So for eleven heliacal risings 30 degrees from the Sun, it took 4388 days, and Jupiter transited Virgo on November 4 2016. That's fully 360 degrees and 4 extra degrees that Jupiter was observed to move over the course of 4388 days and a touch more than 12 years. Simple interpolation estimates a sidereal year at 4340 days, 37 days higher than our previous estimate but is now based on four observations not just three. Further observations empirically calibrate our rough estimates. takes a little less than 12 years to complete one side- real orbit, since we are plus 4 degrees after 12 years. Repeated observation refined our estimate to 4340 days. After centuries, the ancients were able to winnow this down to some 4332 days or about 11 tropical years plus around 316 days that it takes Jupiter to orbit the Sun. It doesn't take any rocket scientist but only common sense with a little simple addition and subtraction of round degrees, days and years. The stargazer could see Jupiter go retrograde for some 121 days centered on in- ferior conjunction (opposition), and see these synodic events repeat every 400 days by the long-term averages. Ancient Babylonian astronomers were sufficiently adept to notice that 36 sidereal orbits of Jupiter was quite close to 427 tropical years and 391 synodic periods of if they knew what they were talking about. Try October 30, 2431. Just 6 days later Jupiter is 30 degrees from the Sun with Jupiter 12 Vir and the Sun 12 Vir. That's ancient synodic multiple for Jupiter is right in there. Next we'll look at Mars, which has the longest synodic period of all the planets. We can see that Mars is the first planet beyond Earth's orbit since Mars is moving much faster through the caelestial zodiac than Jupiter or Saturn. But like the Jovian planets we can see that the elongations of Mars from the Sun reach oppositions on an observably predictable periodic basis as is true for heliacal risings, settings, squares, trines or any repeating angle of aspect to the Sun. Heliacal risings are being treated as semisextile aspect for continuity, and the nice round number thirty degrees is convenient, easy to remember and to measure by hand, since a whole hand or fist plus three closed middle fingers at arm's length together makes 15 degrees, two whole fists make about 20 degrees, depending on your physical type. The angle from outstretched thumbtip to pinky fingertip is some 25 degrees, but you must calibrate your own hands and fingers to estimate a perspective angle accurately. An easy way to do this is to stand in a rectangular or square room and see how many hands, fists, and fingers it takes to measure 90 degrees, i.e. from wall to wall. Six times a fist plus 3 closed middle fingers ought to be about 90 degrees. Experiment to see what works best. Three times your outstretched thumbtip to pinky finger- tip plus three closed middle fingers equals 90 degrees. Calibrating by the stars assures the greatest accuracy. No matter how close or far away an object is, ten feet or ten thousand lightyears, the angle subtended to you viewing those objects will be the same. A really sharp naked-eye astronomer can discern down to one arcminute. But I'm being conservative, so that ancient stargazers would need only resolve twenty arcminutes and estimate positions of stars and planets to plus or minus one de- gree, which for the Sun's apparent motion is about one day equals one degree. This is essential to know since adding one or more days to a predicted heliacal rising adds ~1 degree per day to the Sun's ecliptic longitude. Each consecutive heliacal rising for Mars occurs about 780 days apart. With a spectacular opposition for Mars just days away at this writing which will be August 27, at 26 Libra to the Sun 26 Scorpio. Mars will rise near above Mars, Zubenelgenubi at 20 Lib +0 & Zubeneshamali at 25 Lib +8. Add 780 days and we have January 31 2007. There's Mars at 16 Sagittarius to the Sun 16 Capricorn, is nearly impossible to see Kaus Borealis at 12 Sag -2 this close to sunrise (past astronomical twilight) and Mars may be difficult to spot here in the mountains of central Colorado. 16 Sag is 20 degrees past 26 Lib but we've witnessed Mars at opposition back on August 27th This also tells us that Mars is zipping along, so must have circuited the zodiac past 360 degrees, and is now 360 add 20 equals 380 degrees from where it was before. Simple interpolation tells us Mars takes some 739 days to complete one sidereal orbit. This is a rough figure as further observation shows. 10 times 780 is 7800. Ex- rience shows Mars has significant orbital eccentricity, and orbits quite rapidly through the caelestial zodiac. We find through experience that Mars is frequently off by a month or more from where we predicted it would be last we predicted its next heliacal rising, setting or any other repeating like-phase. Mars is at 14 Pis, and the Sun is 6 Ari on April 21, 2026. We must jump ahead to May 28, 2026, fully 37 days later, to find Mars and the Sun separated by 30 degrees sidereal longitude. As the apparent velocity of Mars is nearly as fast as the Sun's past superior conjunction it takes a few days to compensate for being just a degree off from 30 degrees. Thus to compensate for 8 degrees delta took us 37 days. Come May 28 2026, Mars is 12 Ari and the Sun is 12 Tau. Hamal at 13 Ari +10 and Sheratan at 9 Ari + 8 makes it easy to estimate Mars' position at 12 Aries. Shedir at 13 Ari +47 draws a nearly perpendicular line or arc to Hamal relative to the ecliptic making measurement easy. 26 Libra is 166 degrees from 12 Ari, meaning Mars went under eleven & a half times or 4126 degrees around the zodiac in 7837 days, making our observable average 684 days per sidereal orbit based on just two observations ten heliacal risings apart. Babylonians over centuries of observation and calibration found this to be around 687 days based on the long-term averages, which is one year, three hundred twenty-two days per sidereal orbit of Mars. These ancient astronomers-astrologers noticed that 151 sidereal orbits of Mars nearly coincided with 284 tropical years and 133 repeating synodic phases of we arrive at December 12, 2288. There's Mars at 23 Lib and the Sun 22 Sco, 29 degrees apart. Merely four days later finds Mars 26 Lib & Sun 26 Sco--right on the dot. So the ancient Babylonian sidereal-synodic multiple of Mars is off just 4 days in 284 years...very impressive. We also notice that Mars goes retrograde centered near inferior conjunction for an average of 73 days. Try it. Astrolog charts the synodic velocities of every planet. days. With every empirical observation for retrogrades and oppositions, the accuracy of this average improves. Observation proves 73 days is Mars' retrograde average. End Parts 1, 2, 3, 4. See Part 5 For Continuation... Daniel Joseph Min *Min's Planetary Awareness Technique (chapters 1 thru 6): http://groups.google.com/groups?selm=HFVIRNCU37838.7946990741@ Gilgamesh-frog. org *Min's Official PGP Public Key on the MIT server: http://groups.google.com/groups?selm=3XWB7QJO37766.971099537@ Gilgamesh-frog.o rg *Min's Home Page On The World Wide Web: http://groups.google.com/groups?selm=0XNKAO4L37773.8337962963@ Gilgamesh-frog. org -----BEGIN PGP SIGNATURE----- iQA/AwUBP0Jq6ZljD7YrHM/ nEQK6ZwCgj4ASMKxKS5dBJ8gv7o8mmJ6x4AIAnAls toVeWbrDaKWagdCcHZvjahXm =Qkuc -----END PGP SIGNATURE----- === Subject: : Re: Astronomical Observations - Parts 1, 2, 3, 4 Gilgamesh-frog.org Sort of figures that this dickhead would be using a remailer. -- Alan Erskine alanerskine(at)optusnet.com.au John Howard doesn't speak for this Australian in the Amrosi death sentence - Jail, not death. Comments: This message did not originate from the Sender address above. It was remailed automatically by anonymizing remailer software. Please report problems or inappropriate use to the remailer administrator at . === Subject: : Re: Astronomical Observations - Parts 1, 2, 3, 4 Mail-To-News-Contact: abuse@dizum.com >Gilgamesh-frog.org >Sort of figures that this dickhead would be using a remailer. Speaking of dickheads: About A l a n E r s k i n e a l a n e r s k i n e @ o p t u s n e t . c o m . a u A l a n E r s k i n e is a well known Australian netkook, psychopath, and coward who lives in Melbourne. He usually stalks and harasses the posters of the sci.space groups, although he is known to make excursions into a wide range of other groups for the purposes of trolling and stalking and harassing the participants there. He is currently under investigation by the authorities in Australia. If you fall victim to his frequent attacks and stalking/harassment campaigns, like so many in the sci.space groups have, notify abuse@optusnet.com.au . This psycho is dangerous. Most sane participants have been forced to killfile him for his frequent violent outbursts at posters and his unrelenting attacks. Some are taking legal action against him. Here are just a few of his victims. He has been stalking and harrasing Scott and Betty Grissom relentlessly. Scott Grissom is the son of Gus Grissom, the second American in space, who died in an Apollo 1 training exercise fire, and Betty Grissom, 75, is his widow. Alan has also been waging long-time stalking and harassment campaigns against other well-known members of the space community, including Richard Katz of NASA, John Maxson, and his sons Paul and Daniel Maxson. He has even gone so far as to threaten to kill the entire Maxson family. He has been known to stalk his victims anonymously and via sock puppets, showing what a coward he is. Furthermore, he likes to challenge people to fights, telling them to come to Melbourne and look him up in the phone book, while having admitted elsewhere that he is not listed in the phone book. Coward! Alan is an enemy of freedom of speech and hates remailers, which serve a useful and necessary purpose. He has resorted to impersonating others and even himself through remailers in an effort to discredit remailers and make it look like others were attacking him through them. He regularly bombards remailer operators with hysterical complaints and threats and floods their mailboxes. Alan is currently on one of his psychotic crusades, this time to get people to killfile posts from remailers in order to prevent word of his misdeeds from getting out. Alan is a Nazi, he wants to be able to say whatever he wants about others but doesn't want anyone to mention him. He wants to express himself freely but wants to shut others up and, not only that, wants others to do as he says, read only the messages he approves of, etc. He wants to police, rule, and control newsgroups and people like the perfect little jackbooted Nazi he is. Alan is not just content to harass his victims in the newsgroups, he usually also stalks them via email. If you find yourself on the receiving end of Alan's famous psychotic emails, contact abuse@optusnet.com.au immediately, as well as the law enforcement agencies in Melbourne, Australia. This psycho is very violent. Do not attempt to reason with him yourself, many others have tried and he does not respond to reason. Let the authorities deal with him. === Subject: : Re: Astronomical Observations - Parts 1, 2, 3, 4 the point about symbolic or western astrologers is taken; they tend to use the JPL ephemeris, which has no mind of the position of the Sun on the morning horizon, relative to the zodiac; it's probably downloadable from Caltech, in any case http://www.schillerinstitute.org/ http://www.schillerinstitute.org/newspanish/InstitutoSchiller/ Ciencia/Enterr arMatematicas.html was synonymous with the mathematician, star-logician in the most literal sense. Even in our day and age, it was only within the last few centuries that astrologer and astronomer reached a schism, since astrologers had long-since ignored the proper mathematics of astrology, and astronomers became disenchanted with the illusions yet perpetuated by today's tropicillogical astrologers and other schisms of astrology,--all who've hopelessly lost their grasp on the ancient practice of star-logic. Since this schism, astronomers have changed their ways of measuring the sky such that constellations became synonymous with unequal boundaries associated with the asterisms or some 88 familiar groups of brighter stars instead of the ancient method which divides the entire caelestial sphere into twelve equal meridians as signs with meridians of latitude from the caelestial equator. > http://groups.google.com/groups?selm=0XNKAO4L37773.8337962963@ Gilgamesh-frog. org -----BEGIN PGP SIGNATURE----- > iQA/AwUBP0Jq6ZljD7YrHM/ nEQK6ZwCgj4ASMKxKS5dBJ8gv7o8mmJ6x4AIAnAls > toVeWbrDaKWagdCcHZvjahXm > =Qkuc > -----END PGP SIGNATURE----- --Dec.2000 'WAND' Chairman Paul O'Neill, reelected to Board. Newsish? http://www.rand.org/publications/randreview/issues/rr.12.00/ http://members.tripod.com/~american_almanac === Subject: : Re: Astronomical Observations - Parts 1, 2, 3, 4 > the point about symbolic or western astrologers is taken; > they tend to use the JPL ephemeris, > which has no mind of the position of the Sun > on the morning horizon, relative to the zodiac; > it's probably downloadable from Caltech, > in any case > Perhaps you meant to say: the point about symbolic or western astrologers is taken; they tend to use the JPL's ephemeris, which has no mind of the real position of the Sun on the morning horizon, relative to the zodiac; it is probably down- loadable from Caltech, in any case, the URL follows as: > http://www.schillerinstitute.org/ > http://www.schillerinstitute.org/newspanish/InstitutoSchiller/ Ciencia/Enterra rMatematicas.html > === Subject: : a very simple question Hi I have a point x, I wanna substract 4z(z is an integer) from x, so that x-4z is in [-1,3) I write this matlab codes, but it doesn't work for x< 0 case, x = rand(1,8)*20 y = x+ones(1,8); r = fix(y./4); z = x - 4*r === Subject: : Re: a very simple question > Hi I have a point x, > I wanna substract 4z(z is an integer) from x, > so that x-4z is in [-1,3) I write this matlab codes, but > it doesn't work for x< 0 case, x = rand(1,8)*20 > y = x+ones(1,8); > r = fix(y./4); > z = x - 4*r There is a newsgroup for matlab: comp.soft-sys.matlab There is a list of Frequently Asked Questions (+answers) here: http://www.mit.edu/~pwb/cssm/ David Bernier === Subject: : Re: a very simple question > Hi I have a point x, > I wanna substract 4z(z is an integer) from x, > so that x-4z is in [-1,3) I write this matlab codes, but > it doesn't work for x< 0 case, x = rand(1,8)*20 > y = x+ones(1,8); > r = fix(y./4); > z = x - 4*r I am supposing x is actually a number, and you want to find an integer z for which -1 <= x - 4 z < 3 So, why not solve for z? Adding 4z to everything gives 4z-1 <= x < 4z + 3 or 4z <= x+1 < 4(z+1). so you have z <= (x+1)/4 < z+1 Hmm. I guess that's just what you did, seeing your Matlab code. Looks to me as though you might want z = floor((x+1)/4). The fix function selects the nearest integer towards zero, whereas the floor function selects the greatest integer *less than or equal to* the value. Dale === Subject: : Re: a very simple question > Hi I have a point x, > I wanna substract 4z(z is an integer) from x, > so that x-4z is in [-1,3) I write this matlab codes, but > it doesn't work for x< 0 case, 1. You'll find comp.soft-sys.matlab better for Matlab questions. x = rand(1,8)*20 > y = x+ones(1,8); > r = fix(y./4); 2. In the case of these two calculations you don't need vector notation. You can add or multiply a vector and a scalar: y = x+1; r = fix(y/4); 3. The real problem is that you want floor(y/4), not fix(y/4). As you noticed, they behave slightly differently for negative numbers. Try fix(x) and floor(x) for different negative numbers to see how they behave. Try round(x) and ceil(x) as well. - Randy === Subject: : Re: axioms of mathematics X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Punge: Micro$oft X-Sanguinate: themvsguy@email.com X-Terminate: SPA(GIS) X-Tinguish: Mark Griffith X-Treme: C&C,DWS at 05:45 PM, bruce said: >Hi group; Since reading Simon Singh, I've been googling in vain for >information on the axioms of mathematics. There are none. There are various axiom systems for various branches of Mathematics. >I am a complete layperson, and I cannot find out what these axioms >actually are (I vaguely recall things like ab=ba, x+y=y+x, things as >elementary as that, are axiomatic). They could be axioms or they could be definitions, depending on the context. Either way, they refer to a specific branch of Mathematics, Algebra. >I repeat that I am a total layperson, who studied school mathematics >to 16 years old, more than 20 years ago! Chances are that you didn't actually learn any Mathematics in school, but rather some basic arithmetic and a little symbol manipulation. Had you gone to school 50-60 years ago then I might have expected at least a Geometry course to have some real Mathematics, but that hasn't been the case for a long time. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Any unsolicited bulk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org === Subject: : binomials over negative reals The binomial coefficients, I'll call it n choose m, have all sorts of definitions, all agreeing when both params are positive integers. It is usually defined by fiat to be 0 when m is negative. However, if you define it (er.. compute it) using factorials, that is: n! --------- (n-m)! m! and then translate to Gamma (the analytic continuation of factorials), you get something a bit different when both n and m are negative (integral or not). Since Gamma(n) has poles only at negative integers, 1/Gamma is entire, so the only problem in computing is when n is a negative integer. Does there exist a formula for the binomials whose computation does not involve such singularities? The Beta function B(n,m) (which happens to equal Gamma(n)Gamma(m)/Gamma(n+m) ) seems close with the corresponding formula Integral[u^n (1-u)^m, {u,0,1}], but again that is undefined for both n and m negative. Mitch === Subject: : Re: binomials over negative reals > The binomial coefficients, I'll call it n choose m, have all sorts of > definitions, all agreeing when both params are positive integers. It is > usually defined by fiat to be 0 when m is negative. However, if you define it (er.. compute it) using factorials, that is: n! > --------- > (n-m)! m! and then translate to Gamma (the analytic continuation of factorials), > you get something a bit different when both n and m are negative > (integral or not). Since Gamma(n) has poles only at negative integers, > 1/Gamma is entire, so the only problem in computing is when n is a > negative integer. Does there exist a formula for the binomials whose computation does not > involve such singularities? The Beta function B(n,m) (which happens to > equal Gamma(n)Gamma(m)/Gamma(n+m) ) seems close with the corresponding > formula Integral[u^n (1-u)^m, {u,0,1}], but again that is undefined for > both n and m negative. Mitch First, Gamma(n)Gamma(m)/Gamma(n+m) = integral{0 to 1} u^(n-1) (1-u)^(m-1) du where defined. (m and n replaced with (m-1) and (n-1) in beta-integral.) If the m in binomial(n,m) (ie. n choose m) is a nonnegative integer, then we can write binomial(n,m) as n*(n-1)*(n-2)*...(n-m+1)/m!, where n can be any complex number. We can write this as a polynomial: sum{k=0 to m} S(m,k) n^k (-1)^(m+k), where S(m,k) is an unsigned Stirling number of the first kind. Using this definition, then: binomial(-n,m) = (-n)*(-n-1)*(-n-2)*...(-n-m+1)/m! = (-1)^m n*(n+1)*(n+2)*...(n+m-1)/m! = (-1)^m binomial(n+m-1,m). There are many ways to calculate binomial coefficients, I am sure, but I am in too much of a hurry today to list any more, if any others would even came to mind now. :/ Thanks, Leroy Quet === Subject: : Re: binomials over negative reals >The binomial coefficients, I'll call it n choose m, have all sorts of >>definitions, all agreeing when both params are positive integers. It is >>usually defined by fiat to be 0 when m is negative. >>However, if you define it (er.. compute it) using factorials, that is: >> n! >>--------- >>(n-m)! m! >>and then translate to Gamma (the analytic continuation of factorials), >>you get something a bit different when both n and m are negative >>(integral or not). Since Gamma(n) has poles only at negative integers, >>1/Gamma is entire, so the only problem in computing is when n is a >>negative integer. >>Does there exist a formula for the binomials whose computation does not >>involve such singularities? The Beta function B(n,m) (which happens to >>equal Gamma(n)Gamma(m)/Gamma(n+m) ) seems close with the corresponding >>formula Integral[u^n (1-u)^m, {u,0,1}], but again that is undefined for >>both n and m negative. but is just fine as 1/Beta(...) > First, Gamma(n)Gamma(m)/Gamma(n+m) = > integral{0 to 1} u^(n-1) (1-u)^(m-1) du > where defined. (m and n replaced with (m-1) and (n-1) in beta-integral.) oops, yes. > If the m in binomial(n,m) (ie. n choose m) is a nonnegative integer, > then we can write binomial(n,m) as n*(n-1)*(n-2)*...(n-m+1)/m!, where n can be any complex number. ... > Using this definition, then: binomial(-n,m) = (-n)*(-n-1)*(-n-2)*...(-n-m+1)/m! = (-1)^m n*(n+1)*(n+2)*...(n+m-1)/m! = (-1)^m binomial(n+m-1,m). Yes, that's the most accepted version of binomial(n, m). it implies the accepted definition, which, if binomial(n, m) = 0 for n pos and m negative, then so is binomial(-n,m). Unfortunately, the above also depends on m being an integer. If you compute using n! Gamma(n+1) 1 --------- = ---------------------- = ------------------------ (n-m)! m! Gamma(n-m+1)Gamma(m+1) Beta(n-m+1, m+1) (n + 1) Then you get singularities in the middle of computation despite it being total. I'm looking for a nice way to compute this nonstandard binomial coeffs (meaning not defined to be zero for negative m) when negative reals in both params. I suspect there is a way to transform 1/Beta(..) into an integral directly (that is transform 1/Integral(f(n,m)) to Integral(g(n,m)), but my analysis skills are not up to it. Any hints? I've tried modifying the derivation in: http://mathworld.wolfram.com/BetaFunction.html but I was unsuccessful. > There are many ways to calculate binomial coefficients, I am sure, but > I am in too much of a hurry today to list any more, if any others > would even came to mind now. Of all the ones I've seen, they all seem to have difficulty (in computation) with negative integers in both n and m Mitch === Subject: : Bourbaki >>I found the Algebra and Commutative Algebra books useful, but I do not >AFAIK, Serre was heavily involved in the writing of the Commutative >Algebra book. If this is the case, any difference in >quality between this and Homological Algebra probably isn't anything >to do with Serre. quality and usability of whatever text at hand. That will vary with text and author, as well with the reason one is consulting the text. Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: === Subject: : Re: Calculating taxes Try posting this in sci.econ. This kind of stuff often gets discussed there. >>Let us assume someone makes $40000 a year. He pays about ~15% federal tax on >>his salary. The rest he spends on rent, food, gasoline and clothes. Now, all >>these things have taxes included in their prices - company profits, wages of >>employees at every production step. If he buys a gallon of gasoline for $2.00, >>a gallon of milk for $4.00 and pays $1000 a month rent, what percentage of >>these amounts go to the federal government? >>Or in other words, if I buy a wooden pecil for $0.10, how much do I pay to the >>federal government? The $0.10 represents all the costs and profits of a lumber >>company, a wood mill, a transportation company, wholesaler, retailer, etc. > 40% of a phone bill is taxes. > ----== Posted via Newsfeed.Com - Unlimited-Uncensored-Secure Usenet News==---- > http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000 Newsgroups > ---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption =--- > === Subject: : Re: Calculating taxes Add this to the equation: http://www.bankrate.com/brm/green/taxes/1e.asp Lurch > Let us assume someone makes $40000 a year. He pays about ~15% federal tax on > his salary. The rest he spends on rent, food, gasoline and clothes. Now, all > these things have taxes included in their prices - company profits, wages of > employees at every production step. If he buys a gallon of gasoline for $2.00, > a gallon of milk for $4.00 and pays $1000 a month rent, what percentage of > these amounts go to the federal government? > Or in other words, if I buy a wooden pecil for $0.10, how much do I pay to the > federal government? The $0.10 represents all the costs and profits of a lumber > company, a wood mill, a transportation company, wholesaler, retailer, etc. > Stan === Subject: : Re: Calculating taxes |The question is sensible, you could use input-outout tables to caluclate the |answer, OK they are not detailed enough to do it for real. I think we knew to begin with we didn't have the data handy to give a precise answer. But how would you define the question in terms of input-output tables? Keith Ramsay === Subject: : Re: Calculus is irrational? > Well, yes. By any common definition then, calculus is founded within > human rationality, as is most of math. The fact that you seem to use a > non-standard definition of rationality only reminds me of old, bad, > jokes about dogs, legs, and tails. Fie, infidel! That old bad joke is attributed to US national hero Abraham Lincoln. It was presumably fresh when he told it. - Randy === Subject: : Re: Calculus is irrational? Visiting Assistant Professor at the University of Montana. >Arturo Magidin >>Whats rational to us would be the real world. >> Again, why would this exclude unicorns under you definition? >Because my defintion is based on observation and models created on >those observations. As far as I know, no one as observed a unicorn, But it is possible to model a unicorn based on observations we make. They are basically a horse with a horn growing out of their forehead. All the particulars are observable, and we can certainly create a model of a unicorn which is complete. Your exclusion does not agree with your definition. >therefore, no one has added the observation of unicorns to our model >of nature, and therefore, unicorns do not exist in nature and they do >not exist within our rationality. Nobody has observed an eon, yet eons are part of our model of nature. Your exclusion of unicorns from your definition of real world is arbitrary. >You simply do not understand my defintion. I disagree. I think ->you<- do not understand the full implications of the definition you have given. > Please read it again, and >think carefully about it. We are much closer to agreement than you >think. Ah, mind reader, on top of everythign else? ============================================================== ======== It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) ============================================================== ======== Arturo Magidin magidin@math.berkeley.edu === Subject: : Re: Calculus is irrational? Hamish Reid > Then that's quite the non-standard definition or usage of the term > human rationality. Why not use a standard definition, or talk about > observable phenomena or something? Because rationality hasn't been properly defined yet. You said the standard defintion, the ability to reason. If this was the case, whats the difference between rational and reasonable? I'm postulating that the differece is that we have to be reasoning about what we have observed. > It's one thing to say that some parts of math -- like infinity -- lie[s] > outside of [what the rest of us call] human rationality; it's quite > another to make the trivial commonplace that large parts of math are not > rooted in observable phenomena. If we take the example of a bouncing ball and the t is the amount of time that elapes between each bounce i the folumla is something like t = 1 / (10^i). It is these types of questions I am describing as irrational because it will never hit 0, and so it will bounce infinite times (although for a finite period of time). Because we cant' observe something bouncing an infinite number of times the solution doesn't even get off to a rational start. > The apples are. But what about the numbers? Have you ever observed a > three in the wild? Or a billion people, for that matter? Sure I've observed three in the wild. 3 apples would be observing 3 in the wild. You just need to think less concretely about things. A billion? I observe billions of atoms. I represent billion with a number. A billion is very specific about the quantity it represents so we can model it accurately. Infinity is a different beast. > Your philosophy > (such as it is) seems perilously close to solipsism, and seems to rule > the vast majority of practical math out of rational (your definition) > bounds.... Indeed but the advantage is that it specifies a definitive boundry. But remember, just because its not completely within the boundries of rationality, the fact that there is a large amount of rationality too it means that it will be useful. Just not the abosolute truth. Mike Helland === Subject: : Re: Calculus is irrational? X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Punge: Micro$oft X-Sanguinate: themvsguy@email.com X-Terminate: SPA(GIS) X-Tinguish: Mark Griffith X-Treme: C&C,DWS at 03:42 PM, mhelland@techmocracy.net (Mike Helland) said: >If calculus assumes infinity to come to its answers If elephants are plankton >(for example, the >limit of a function, we sum to infinity to find an answer) No. >and because infinity is irrational (infinity being defined by p/0 No. And also no. >is it fair to say that any answer given to us by calculus is by >definition irrational as it assumes irrationality in the solution? Is it fair to say that conductors are irrational? You are very confused about what Calculus is, and also about the meaning of the word infinity. Calculus does not involve[1] infinity, and wouldn't be irrational if it did. [1] There is a concept of a function approaching plus or minus infinity, but that doesn't mean what you seem to believe that it does. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Any unsolicited bulk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org === Subject: : Re: Calculus is irrational? > Hamish Reid Then that's quite the non-standard definition or usage of the term > human rationality. Why not use a standard definition, or talk about > observable phenomena or something? Because rationality hasn't been properly defined yet. It's been quite well defined for centuries... which is one of the reasons we keep asking you to use some existing definitions rather than your own. And to keep using the *same* definitions.... >You said the > standard defintion, the ability to reason. I said *a* standard definition, and I asked why you don't use one, instead of (deliberately?) confusing things by redefining rationality to mean something no one else means by it. This thread started when you stated some very confused and non-standard definitions of infinity and calculus (e.g. that inifinity was defined as 1/0 and was therefore irrational), and you don't seem to have learned much from that point on. >If this was the case, whats > the difference between rational and reasonable? Is that meant to be a reasonable question? (I'm serious). And what does it have to do with your misuse of the word rational? Is it reasonable to keep equivocating on the various different definitions of rational and rationality the way you do? And is it reasonable to keep stating that calculus depends on infinity (and is therefore irrational) when it's clearly *not* dependent on infinity? >I'm postulating that > the differece is that we have to be reasoning about what we have > observed. Again, *why*? If you want to talk about *human experience*, do so, and if you want to talk about reasoning only from direct human experience, do so, but don't pretend that that's all that human rationality can mean. I've never observed a billion people, but I can reason quite healthily about it. Ditto with mathematical limits, or with pi (has anyone ever observed pi?) (Mmmmmmm, pi...). If you're unable to build on what you observe directly to extrapolate beyond that, well, the real world's probably not for you, no? > It's one thing to say that some parts of math -- like infinity -- lie[s] > outside of [what the rest of us call] human rationality; it's quite > another to make the trivial commonplace that large parts of math are not > rooted in observable phenomena. If we take the example of a bouncing ball and the t is the amount of > time that elapes between each bounce i the folumla is something like t > = 1 / (10^i). It is these types of questions I am describing as > irrational because it will never hit 0, and so it will bounce infinite > times (although for a finite period of time). Because we cant' observe > something bouncing an infinite number of times the solution doesn't > even get off to a rational start. You know, you just have to think less concretely about these things... all math is abstraction, and this is a classic case of that. Just like the number three is an abstraction. In any case, you seem horribly confused both about basic maths and the word rational as used by mathematicians and most of the rest of the world. There's nothing irrational here, nothing beyond our ability to reason -- almost by definition. Again, though, if you want to make the commonplace (and quite uncontroversial) observation that much such math is not grounded in our real-world experiences, then do so (after all, the math is about an ideal ball, not a real-world ball) -- but don't drag spurious uses of the word rational into it. Such math is rational by any conventional definition of the word -- very much so. The fact that it is math means that some measure of rational thought went into it (well, more or less). > The apples are. But what about the numbers? Have you ever observed a > three in the wild? Or a billion people, for that matter? Sure I've observed three in the wild. 3 apples would be observing 3 in > the wild. No, it would be observing three apples in the wild. Not the same thing at all. >You just need to think less concretely about things. And I'm tempted to say you need to think *more* concretely about things, especially existing definitions and math. It would surely help the problem you seem to have with wild and furious equivocation on the various standard and non-standard meanings of the words rational and rationality. >A > billion? I observe billions of atoms. I represent billion with a > number. Billion *is* a number. >A billion is very specific about the quantity it represents so > we can model it accurately. Infinity is a different beast. In that it doesn't represent a specific quantity, yes, In that we can model it accurately (and do so all the time), no. Pi is a very specific quantity too, which we model very accurately, but I'll bet you've never seen pi in the wild. i is also very specific, but I shudder to think what you'd do with the fact that it's an imaginary number (aren't they all?!). > Your philosophy > (such as it is) seems perilously close to solipsism, and seems to rule > the vast majority of practical math out of rational (your definition) > bounds.... Indeed but the advantage is that it specifies a definitive boundry. It may be definitive, but it's not useful, and like most solipsism, it tells us nothing much about either math or the real world. > But remember, just because its not completely within the boundries of > rationality, the fact that there is a large amount of rationality too > it means that it will be useful. Just not the abosolute truth. Could you rephrase that in English or math? I'm serious. And where does absolute truth fit into all this? It surely doesn't exist in the real world.... Hamish === Subject: : Re: Calculus is irrational? > Well, yes. By any common definition then, calculus is founded within > human rationality, as is most of math. The fact that you seem to use a > non-standard definition of rationality only reminds me of old, bad, > jokes about dogs, legs, and tails. Fie, infidel! That old bad joke is attributed to US national > hero Abraham Lincoln. Ha! Where I grew up in UnAmerica it was attributed to the ancient Greeks. > It was presumably fresh when he told it. As fresh as the observation that there's nothing new under the sun... :-) Hamish === Subject: : Re: Calculus is irrational? Hamish Reid > It's been quite well defined for centuries... which is one of the > reasons we keep asking you to use some existing definitions rather than > your own. And to keep using the *same* definitions.... But I'm saying that the defintion is lacking. Being able to reason is not all there is to being rational. I say you have to be reasoning about things that are directly observable to be rational. If you disagree, whats a more effective defintion? Also, are there any other words that already fit my defintion? If rational isn't a good word for what I'm talking about (and I happen to think it's the perfect word), then I need to find a new word, but my point will still stand that the solutions of calculus are not something observable we can reason about. >I'm postulating that > the differece is that we have to be reasoning about what we have > observed. Again, *why*? If you want to talk about *human experience*, do so, and > if you want to talk about reasoning only from direct human experience, > do so, but don't pretend that that's all that human rationality can > mean. Why not? I think it does a very good job of defining rationality. Someone asked me the question why does your rationality include unicorns? It doesn't. Because a unicorn has never been observed, it does not lie in our rationality. We can theorize about one, suggest that it might look like, but ONLY IF our defintion of rationality demands that what we reason about must be observed is the unicorn excluded from rationality. My defintion of rationality leaves out the fairy tales. Your defintion of rationality, lacking the requirement of observation, does not exclude these fairy tales. Do you disagree with my summary? If not, do you see why a more percise defintion of rationality is in order? > Because we cant' observe > something bouncing an infinite number of times the solution doesn't > even get off to a rational start. You know, you just have to think less concretely about these things... > all math is abstraction, and this is a classic case of that. Just like > the number three is an abstraction. It is an abstraction, yes. The number three is an abstraction of three things. However, whether something is an abstraction or not does not determine whether it is rational or irrational. The abstraction of 3 lies within our rationality. I observe it frequently in the real world by abstracting observed things. The abstraction that leads to infinity must include everything, including our rationality. If so, rationality cannot be contained in rationality. Incompleteness theorem and all that. > The fact that it is math means > that some measure of rational thought went into it (well, more or less). Do you understand my point that although calculus isn't completely rational I'm not saying that it is completely irrational? I totally agree that there is rationality present in calculus, that is why it isn't useless. But I'm not about to admit that its 100% rational. You agree with me that there are elements of calc that lie outside the real world. If not irrational, what is the adjective you give to something that lies outside the real world? Mike Helland === Subject: : Re: Calculus is irrational? I totally agree that there is rationality present in calculus, that is > why it isn't useless. But I'm not about to admit that its 100% > rational. > Would it be 99% rational? 74% rational? 31.4159...% rational? etc. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html His mind has been corrupted by colours, sounds and shapes. The League of Gentlemen === Subject: : Re: Classes vs. Sets <3f3a574b$15$fuzhry+tra$mr2ice@news.patriot.net> <716e06f5.0308132210.43fcdbaa@posting.google.com> <3f3bbee0$0$27001$afc38c87@> X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Punge: Micro$oft X-Sanguinate: themvsguy@email.com X-Terminate: SPA(GIS) X-Tinguish: Mark Griffith X-Treme: C&C,DWS >Yes, it was a bit harsh wasn't it? Hehe. But I don't mind. The >real question that I was getting at is how many proper classes are >there? What do you mean by that question? In what set theory. It is meaningless in, e.g., GB. The question can only be formulated in a larger system, and the answer will depend on what system you use. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Any unsolicited bulk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org === Subject: : Re: Classes vs. Sets <3f3a574b$15$fuzhry+tra$mr2ice@news.patriot.net> <716e06f5.0308132210.43fcdbaa@posting.google.com> X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Punge: Micro$oft X-Sanguinate: themvsguy@email.com X-Terminate: SPA(GIS) X-Tinguish: Mark Griffith X-Treme: C&C,DWS at 11:10 PM, w.taylor@math.canterbury.ac.nz (Bill Taylor) said: >This is a little bit harsh! It may be harsh, but it is true. It follows directly from the definition he cited. >I suspect he wanted to know, CAN you in fact gather up all the >classes (or proper classes, or whatever), into some sort of >something? And the answer is that you can't do that in any of the more common set theories. Which is not to say that you can't have a set theory with hierarchies; in fact, Russel and Whitehead did. But I don't believe that he was asking about those. >But you can make super-classes out of collections of >classes, and have greater or lesser superclasses depending on what's >been included and excluded. In what set theory? Certainly not GB or ZF. >Then (to get in ahead of the obvious next question) you can make >superduper-classes, and so on, as far as the ordinal mind of man can >go! It's not an issue of where the mind can go, but of where the axioms can go. >And the reason this is not usually done, so I'm told, is that you >can get all the same action with a lot less effort just by positing >various large cardinal axioms Then you're talking about a different set theory, and you are still faced with proper classes not being elements of other classes. By definition. The fact that you can model the original theory in the expanded theory doesn't change that. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Any unsolicited bulk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org === Subject: : Re: Classes vs. Sets |>I suspect he wanted to know, CAN you in fact gather up all the |>classes (or proper classes, or whatever), into some sort of |>something? | |And the answer is that you can't do that in any of the more common set |theories. Which is not to say that you can't have a set theory with |hierarchies; in fact, Russel and Whitehead did. But I don't believe |that he was asking about those. I think perhaps he was. But the fact that you need to go outside ZF or GB more or less tells you what the answer is. The superclass of all classes (in whatever sense it may exist) is not a set. And it's not a class. In a certain sense it's bigger. Keith Ramsay === Subject: : comparison I have to compare a double precision value with (c/d)^n where c, d and n are big positive integers. How can such a comparison be made, without actually computing c^n and d^n ? Thanks in advance, Daniel Aioanei === Subject: : Re: comparison I have to compare a double precision value with (c/d)^n where c, d > and n are big positive integers. How can such a comparison be made, > without actually computing c^n and d^n ? By taking the logarithm. x > (c/d)^n if and only if log(x) > n(log c - log d) This is true regardless of what base log you use. If you were being really clever, and had access to bit manipulations, you could do this with log base 2, which is almost trivial to calculate for binary numbers (both integer and floating point). - Randy === Subject: : correlation between 2 entities?? Hello everybody! I've got two (or more) entities, and I'd like to know how much they correlate to each other, expressed as values from 0 to 1. For this purpose I know the values of some properties of them; these values are from _N. Eg. Entity 1: A=2 C=2 G=3 Entity 2: A=1 D=1 H=2 Entity 3: A=2 C=3 G=2 Here would 1 and 3 be correlated, and 2 has not much similarity. I already found http://mathworld.wolfram.com/CorrelationCoefficient.html http://mathworld.wolfram.com/Correlation.html http://mathworld.wolfram.com/ SpearmanRankCorrelationCoefficient.html and some other but I think they don't match my problem. I got as far as interpreting the various properties as a N-dimensional space (in the example above A,C,D,G,H ... 5 dimensions), the values as the coordinates, and so I can express the similarity as the distance between any two entities (points in my space). My problems are now: To normalize this I have to check the maximum distance in every dimension - which is some work if there are lots of properties. That I can handle, I'll just cache the maximum value in each dimension and update that for new entities. Is the best way to take 1 - (distance between two entities)/maximum distance as my similariness value? or are there better values?? (perhaps it's enough to take the squared distances, so I can avoid a square root?) Furthermore, if there are many entities I'd like to avoid having to compare each to every other, which would be O(N*(N-1)) ~ O(N^2). I though that I could stop comparing one entity against others if it's near to another entity and build clusters of them for further analysis. But that's still O(N^2) in the worst case - if similar entities happen to be the last to be compared. Or I could seperate them into bins beforehand - but by which criterions?? If I snip each dimension in a few ranges I may get 3 ranges ^ (100 properties, ie. 100 dimensions) -> 3^100 bins ~ 5.2e47 I don't have enough memory for that, even if they are sparsely filled. Any suggestions?? Phil === Subject: : Re: correlation between 2 entities?? > Hello everybody! I've got two (or more) entities, and I'd like to know how much they > correlate to each other, expressed as values from 0 to 1. > For this purpose I know the values of some properties of them; these > values are from _N. In what sense do you want to measure their correlation? You've stated a value that will correspond to the correlation, but not what it should tell you. Eg. > Entity 1: A=2 C=2 G=3 > Entity 2: A=1 D=1 H=2 > Entity 3: A=2 C=3 G=2 Here would 1 and 3 be correlated, and 2 has not much similarity. I already found > http://mathworld.wolfram.com/CorrelationCoefficient.html > http://mathworld.wolfram.com/Correlation.html > http://mathworld.wolfram.com/ SpearmanRankCorrelationCoefficient.html > and some other but I think they don't match my problem. > All of these measure the correlation in ALL the data elements. You are looking for a pairwise comparison. > I got as far as interpreting the various properties as a N-dimensional > space (in the example above A,C,D,G,H ... 5 dimensions), the values as > the coordinates, and so I can express the similarity as the distance > between any two entities (points in my space). > My problems are now: To normalize this I have to check the maximum distance in every > dimension - which is some work if there are lots of properties. That I > can handle, I'll just cache the maximum value in each dimension and > update that for new entities. > Is the best way to take > 1 - (distance between two entities)/maximum distance > as my similariness value? or are there better values?? (perhaps > it's enough to take the squared distances, so I can avoid a square > root?) Furthermore, if there are many entities I'd like to avoid having to > compare each to every other, which would be O(N*(N-1)) ~ O(N^2). > I though that I could stop comparing one entity against others if it's > near to another entity and build clusters of them for further > analysis. But that's still O(N^2) in the worst case - if similar > entities happen to be the last to be compared. Or I could seperate them into bins beforehand - but by which > criterions?? If I snip each dimension in a few ranges I may get > 3 ranges ^ (100 properties, ie. 100 dimensions) -> 3^100 bins ~ > 5.2e47 > I don't have enough memory for that, even if they are sparsely filled. Any suggestions?? I would decide what you are thinking of as correlating and then formalize that definition. One way to approach this would be to have the correlation between Data A and Data B is: Correlation(A,B) = 1/(dist(A,B)^2+1) In this case, Corr=1 means A=B, and Corr -> 0 as dist(A,B) -> oo As for comparing all the data points, I don't see an obvious way to get under O(N^2). -- Will Twentyman email: wtwentyman at copper dot net electron-dot-cloud are galaxies === Subject: : Re: Coulomb barrier becomes Fusion Barrier Principle; compounding of Maxwell Equations It is obvious that when you have complex apparatus such as an electric motor that you have several of the Maxwell Equations at work. Complex EM devices have Compound Maxwell Equations at work. So that we compound the Maxwell Equations in a device. A Tokamak not only has the Gauss law at work but the other laws of Maxwell Equations. But we can generalize the entire Tokamak as Faraday's Law applied to Gauss's law of Coulomb. The compounding of Maxwell Equations in a given device or the nesting of Maxwell Equations in a given operation. Or the function or variable of one Maxwell Equation into a different Maxwell Equation. Such that as a tokamak is up and running that you have a multivariable function of Gauss's Law with Faraday's Law, or Gauss's law with Ampere's Law. In a final analysis, given any complex EM machine that it can be explained as a series of the application of the 4 Maxwell laws. Tokamaks being one of those machines. And because the Faraday and Ampere Laws are cylinders and because the Gauss Law (Coulomb) is a sphere. That Fusion-physics reduces to the generalization of sphere enclosed in cylinder and vice versa. And the limitations or barriers or maximums in mathematics applies to the generalized Maxwell Equations where 2/3 volume and surface area become in physics a upper limit for breakeven. Archimedes Plutonium, a_plutonium@hotmail.com whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: : Decidability in mathematics I've noticed an interesting lack of understanding about ring and field operations from several posters who've replied to my posts where I prove that adding pi to the ring of integers results in a field. Many of these posters have *proclaimed* that infinite operations are barred from rings and fields in particular cases, apparently lacking an understanding of in what cases they are actually barred and why. For instance, 1/2 = 1/3 + 1/3^2 + 1/3^3+... and of course 1/2 is not barred from rationals, as it is the ratio of non-zero integers, but also, the series though infinite is decidable, which just means you get a *single* number. However, a sum like 1 + -1 + 1 + -1 +... IS barred because it's undecidable. Similarly, an infinite sum like 1+1+1+... is barred because it's undecidable. However, pi^2/6 = 1 + 1/2^2 + 1/3^2 +...+1/k^2+... is not barred because it IS decidable, if you believe that pi^2 is a single number. If you don't then that's another discussion. BUT, mathematicians didn't want to include pi or pi^2/6 in the field of rationals, so even though it's decidable in the field, they exclude it with the rule that a rational must be the ratio of integers where the denominator is nonzero. Then they go to reals where they remove the rule, and call decidability, convergence. The only significant mathematical difference between reals and rationals is the rule that the field of rationals excludes elements that can't be written as the ratio of integers with a non-zero denominator. Now posters can argue about the reality, but I highlight the facts, as well as their arguing, to point out to those of you who didn't know that mathematics IS a *social* phenomena, where intriguing social fossils like the one I've shown abound. Given the subject line of this thread, I'm curious to see if any posters on the sci.math newsgroup have the depth of understanding to point out any other areas where decidability plays a key role. Also, can any of you point out how the *field* of rationals, despite being defined by integers as described above, differ from the ring of integers in terms of the *key* difference? James Harris === Subject: : Re: Decidability in mathematics > Many of these posters have *proclaimed* that infinite operations are > barred from rings and fields in particular cases, apparently lacking > an understanding of in what cases they are actually barred and why. Hum huh... who said anything is barred ? In order to do mathematics, you need rigorous definitions, that's it. We *proclaimed* the definition of a field, and you subscribed. > However, pi^2/6 = 1 + 1/2^2 + 1/3^2 +...+1/k^2+... is not barred > because it IS decidable, if you believe that pi^2 is a single number. > If you don't then that's another discussion. No. According to our definition of a field (ring, group, monoid, magma), infinite sums means NOTHING. Considering the fact that *you call* Q a *field*, you have to comply with this precise definition. If you don't, just *try to* change the definition of the algebraic structures, and see what happens. > BUT, mathematicians didn't want to include pi or pi^2/6 in the field > of rationals, so even though it's decidable in the field, they exclude > it with the rule that a rational must be the ratio of integers where > the denominator is nonzero. I'm more and more aware that your posts contain grave logical mistakes. They exclude Pi^2 because Legendre proved it is NOT a rational (using the *definition* of a rational, to which you subscribed too). If you want to include Pi^2 in the field of rationals, then you HAVE TO prove it's a rational (and the definition of a field isn't sufficient). === Subject: : Re: Decidability in mathematics However, pi^2/6 = 1 + 1/2^2 + 1/3^2 +...+1/k^2+... is not barred > because it IS decidable, if you believe that pi^2 is a single number. > If you don't then that's another discussion. BUT, mathematicians didn't want to include pi or pi^2/6 in the field > of rationals, so even though it's decidable in the field, they exclude > it with the rule that a rational must be the ratio of integers where > the denominator is nonzero. Your understanding is somewhat backwards. Having defined Q, the rationals, as the quotient field of Z, it can then be shown that pi is not a member of Q in the sense that pi cannot be expressed _using the allowed field operations_ on members of Q. One might bemoan the fact that pi isn't in Q, but that's just the way things are. That fact is _not_ the result of mathematicians deciding they didn't want to include pi any more than the fact that there can't be ten black queens at any stage in a chess game is the result of a decision of the game designers not to include that state of affairs. In both cases, it's a consequence of the rules, nothing more. Then they go to reals where they remove the rule, and call > decidability, convergence. Backwards, again. See below. > The only significant mathematical difference between reals and > rationals is the rule that the field of rationals excludes elements > that can't be written as the ratio of integers with a non-zero > denominator. You're on to something here. Fields admit something called valuations, one of the simplest of which is the ordinary absolute value, |.|, on the reals and rationals. A valuation then gives rise to a distance metric which allows one to say how far apart members of the field are. With such a metric it makes sense to talk about convergence of a sequence of elements in the field and hence about infinite series. A significant mathematical difference between the reals and rationals is that the reals are what is known as complete under |.| while the rationals aren't. In simple terms, this means that every convergent sequence of reals converges to a limit which is a real number, while the same property doesn't hold for the rationals. That's a nifty result, but it wasn't decided by fiat--it's a consequence of the rules. > Now posters can argue about the reality, but I highlight the facts, as > well as their arguing, to point out to those of you who didn't know > that mathematics IS a *social* phenomena, where intriguing social > fossils like the one I've shown abound. It's hardly a social fossil, whatever that means. It's a consequence of the mathematical rules of the game and has been known for quite a while. > Given the subject line of this thread, I'm curious to see if any > posters on the sci.math newsgroup have the depth of understanding to > point out any other areas where decidability plays a key role. Decidability is a term of art, having almost nothing to do with your use of the term. Convergence is a perfectly good term, and I'd suggest using it instead. In answer to your question, convergence plays a key role in topology, and hence in real and complex analysis. > Also, can any of you point out how the *field* of rationals, despite > being defined by integers as described above, differ from the ring of > integers in terms of the *key* difference? > The key difference is that the rationals contain multiplicative inverses of all nonzero elements; the integers don't. Surely you must know that--it's what makes a field different from a ring. Rick === Subject: : Re: Decidability in mathematics > I've noticed an interesting lack of understanding about ring and field > operations from several posters who've replied to my posts where I > prove that adding pi to the ring of integers results in a field. Many of these posters have *proclaimed* that infinite operations are > barred from rings and fields in particular cases, apparently lacking > an understanding of in what cases they are actually barred and why. > Please look up the word definition in your dictionary. Then look up the definition of ring and field in an appropriate book on algebra. For extra credit, post your findings. Now, see if you can prove from what you've learned that infinite sums are legal operations. Please post this so we can shred it to. > For instance, 1/2 = 1/3 + 1/3^2 + 1/3^3+... and of course 1/2 is not > barred from rationals, as it is the ratio of non-zero integers, but > also, the series though infinite is decidable, which just means you > get a *single* number. Now, what makes 1/2 a rational? Is it the definition of rational or the fact that the limit of the sum exists? However, a sum like 1 + -1 + 1 + -1 +... IS barred because it's > undecidable. Similarly, an infinite sum like 1+1+1+... is barred because it's > undecidable. However, pi^2/6 = 1 + 1/2^2 + 1/3^2 +...+1/k^2+... is not barred > because it IS decidable, if you believe that pi^2 is a single number. > If you don't then that's another discussion. Ok, pi^2/6 as a specific value... but I can as easily observe that all integers have specific values. Is pi^2/6 now an integer? If not, why not? BUT, mathematicians didn't want to include pi or pi^2/6 in the field > of rationals, so even though it's decidable in the field, they exclude > it with the rule that a rational must be the ratio of integers where > the denominator is nonzero. A limit existing does not mean the limit is in the same field as the summands of the limit. Then they go to reals where they remove the rule, and call > decidability, convergence. The only significant mathematical difference between reals and > rationals is the rule that the field of rationals excludes elements > that can't be written as the ratio of integers with a non-zero > denominator. Actually, there are a lot more differences than that. Minor things like the cardinality of the rationals being lower than the cardinality of the reals, the characterizations of the decimal expansions, etc. What you have cited is the difference in the definitions. Now posters can argue about the reality, but I highlight the facts, as > well as their arguing, to point out to those of you who didn't know > that mathematics IS a *social* phenomena, where intriguing social > fossils like the one I've shown abound. It's a social phenomena only in the sense that we've agreed to a common set of rules and definitions that we agree to. If someone wishes to consider a new definition, it is stated unambiguously and the results are then explored. Given the subject line of this thread, I'm curious to see if any > posters on the sci.math newsgroup have the depth of understanding to > point out any other areas where decidability plays a key role. The halting problem comes to mind. Also, can any of you point out how the *field* of rationals, despite > being defined by integers as described above, differ from the ring of > integers in terms of the *key* difference? Aside from the rationals having 0 as the only non-unit while the integers have only 1,-1 as units? There are a number of other significant differences. -- Will Twentyman email: wtwentyman at copper dot net === Subject: : Re: Decidability in mathematics I see nothing to add to what Santini and Decker have said about your *terminologie nouveau*, although it could have been shortened. what has been said about a sword and its anagram, should not be applied to one's neck (on a regular basis .-) one of the simplest of which is the ordinary absolute value, |.|, on > the reals and rationals. A valuation then gives rise to a distance > metric which allows one to say how far apart members of the field are. > With such a metric it makes sense to talk about convergence of a > sequence of elements in the field and hence about infinite series. A significant mathematical difference between the reals and rationals > is that the reals are what is known as complete under |.| while > the rationals aren't. In simple terms, this means that every > convergent sequence of reals converges to a limit which is a real > number, while the same property doesn't hold for the rationals. > That's a nifty result, but it wasn't decided by fiat--it's a > consequence of the rules. > Decidability is a term of art, having almost nothing to do with > your use of the term. Convergence is a perfectly good term, and > I'd suggest using it instead. In answer to your question, convergence > plays a key role in topology, and hence in real and complex analysis. http://buckminster.info/Ideas/03-TetGeomTheorem4-ColorProof.htm --UN HYDROGEN (sic; Methanex (TM) reformanteurs) ECONOMIE?... La Troi Phases d'Exploitation de la Protocols des Grises de Kyoto: (FOSSILISATION [McCainanites?] (TM/sic))/ BORE/GUSH/NADIR @ http://www.tarpley.net/aobook.htm. Http://www.tarpley.net/bushb.htm (content partiale, below): 17 -- L'ATTEMPTER de COUP D'ETAT, 3/30/81 23 -- Le FIN d'HISTOIRE 24 -- L'ORDEUR du MONDE NOUVEAU 25 -- THYROID STORK !?! === Subject: : Re: Decidability in mathematics > Many of these posters have *proclaimed* that infinite operations are > barred from rings and fields in particular cases, apparently lacking > an understanding of in what cases they are actually barred and why. > Hum huh... who said anything is barred ? > In order to do mathematics, you need rigorous definitions, that's it. We > *proclaimed* the definition of a field, and you subscribed. But apparently many of you, while being able to parrot, do not understand mathematics. > However, pi^2/6 = 1 + 1/2^2 + 1/3^2 +...+1/k^2+... is not barred > because it IS decidable, if you believe that pi^2 is a single number. > If you don't then that's another discussion. > No. According to our definition of a field (ring, group, monoid, magma), > infinite sums means NOTHING. > Considering the fact that *you call* Q a *field*, you have to comply with > this precise definition. > If you don't, just *try to* change the definition of the algebraic > structures, and see what happens. You're babbling, and seem emotional. Mathematics does not require such emotion. If you wish to discuss mathematics rationally, then please try to calm down. > BUT, mathematicians didn't want to include pi or pi^2/6 in the field > of rationals, so even though it's decidable in the field, they exclude > it with the rule that a rational must be the ratio of integers where > the denominator is nonzero. I'm more and more aware that your posts contain grave logical mistakes. > They exclude Pi^2 because Legendre proved it is NOT a rational (using the > *definition* of a rational, to which you subscribed too). > If you want to include Pi^2 in the field of rationals, then you HAVE TO > prove it's a rational (and the definition of a field isn't sufficient). You are correct that pi^2 is not rational, as it doesn't fit the definition. However, I'm correct in noting that there is no field of rationals, as the term is a social fossil. Rationals are defined by the definition that a rational is the ratio of two integers where the denominator is non-zero, so they are defined by the ring of integers, and that's all. James Harris === Subject: : Re: Decidability in mathematics Visiting Assistant Professor at the University of Montana. >> Many of these posters have *proclaimed* that infinite operations are >> barred from rings and fields in particular cases, apparently lacking >> an understanding of in what cases they are actually barred and why. >> Hum huh... who said anything is barred ? >> In order to do mathematics, you need rigorous definitions, that's it. We >> *proclaimed* the definition of a field, and you subscribed. >But apparently many of you, while being able to parrot, do not >understand mathematics. Many years ago I removed the irony-meter from my computer (it kept blowing up) and replaced it with a good solid hassiumy-meter. It had served me well for many years, even though I read alt.atheism. And damn it if didn't just blow up... LOL! ============================================================== ======== Why do you take so much trouble to expose such a reasoner as Mr. Smith? I answer as a deceased friend of mine used to answer on like occasions - A man's capacity is no measure of his power to do mischief. Mr. Smith has untiring energy, which does something; self-evident honesty of conviction, which does more; and a long purse, which does most of all. He has made at least ten publications, full of figures few readers can critize. A great many people are staggered to this extend, that they imagine there must be the indefinite something in the mysterious all this. They are brought to the point of suspicion that the mathematicians ought not to treat all this with such undisguised contempt, at least. -- A Budget of Paradoxes, Vol. 2 p. 129 by Augustus de Morgan ============================================================== ======== Arturo Magidin magidin@math.berkeley.edu === Subject: : Re: Decidability in mathematics ... > blowing up) and replaced it with a good solid hassiumy-meter. It had ^^^^^^^^ 'scuze my ignorance, but what's hassiumy? -- G.C. === Subject: : Re: Decidability in mathematics >> Many of these posters have *proclaimed* that infinite operations are >> barred from rings and fields in particular cases, apparently lacking >> an understanding of in what cases they are actually barred and why. >Hum huh... who said anything is barred ? >In order to do mathematics, you need rigorous definitions, that's it. We >*proclaimed* the definition of a field, and you subscribed. I'm afraid James Harris simply does not *know* the definition of a ring, field or any other mathematical structure. (JSH: Care to prove me wrong? Just give the definition of a ring - maybe looking it up in some text book helps). Peter -- Peter van Rossum, Department of Mathematics, University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands, Phone: +31-24-3652997, E-mail: petervr@sci.kun.nl === Subject: : Re: Decidability in mathematics Visiting Assistant Professor at the University of Montana. >> ... >> blowing up) and replaced it with a good solid hassiumy-meter. It had > ^^^^^^^^ >'scuze my ignorance, but what's hassiumy? Hassium is element 108, lying three spots below iron in the periodic table. Heaviest known element on the same column-group as iron. iron -> irony hassium -> hassiumy. ============================================================== ======== Why do you take so much trouble to expose such a reasoner as Mr. Smith? I answer as a deceased friend of mine used to answer on like occasions - A man's capacity is no measure of his power to do mischief. Mr. Smith has untiring energy, which does something; self-evident honesty of conviction, which does more; and a long purse, which does most of all. He has made at least ten publications, full of figures few readers can critize. A great many people are staggered to this extend, that they imagine there must be the indefinite something in the mysterious all this. They are brought to the point of suspicion that the mathematicians ought not to treat all this with such undisguised contempt, at least. -- A Budget of Paradoxes, Vol. 2 p. 129 by Augustus de Morgan ============================================================== ======== Arturo Magidin magidin@math.berkeley.edu === Subject: : Re: Decidability in mathematics >I've noticed an interesting lack of understanding about ring and field >operations from several posters who've replied to my posts where I >prove that adding pi to the ring of integers results in a field. >Many of these posters have *proclaimed* that infinite operations are >barred from rings and fields in particular cases, apparently lacking >an understanding of in what cases they are actually barred and why. >For instance, 1/2 = 1/3 + 1/3^2 + 1/3^3+... and of course 1/2 is not >barred from rationals, as it is the ratio of non-zero integers, but >also, the series though infinite is decidable, which just means you >get a *single* number. Is _that_ what decidable means? Give a reference for this please. >However, a sum like 1 + -1 + 1 + -1 +... IS barred because it's >undecidable. >Similarly, an infinite sum like 1+1+1+... is barred because it's >undecidable. >However, pi^2/6 = 1 + 1/2^2 + 1/3^2 +...+1/k^2+... is not barred >because it IS decidable, if you believe that pi^2 is a single number. >If you don't then that's another discussion. >BUT, mathematicians didn't want to include pi or pi^2/6 in the field >of rationals, so even though it's decidable in the field, they exclude >it with the rule that a rational must be the ratio of integers where >the denominator is nonzero. Uh, yes. They _defined_ a rational number in such a way that pi^2/6 is not rational. So friggin what? >Then they go to reals where they remove the rule, and call >decidability, convergence. >The only significant mathematical difference between reals and >rationals is the rule that the field of rationals excludes elements >that can't be written as the ratio of integers with a non-zero >denominator. That's the _only_ difference? So there are no significant differences between the reals and any subfield of the reals that includes at least one irrational? Huh, actually I thought there were a lot more differences than that. >Now posters can argue about the reality, but I highlight the facts, as >well as their arguing, to point out to those of you who didn't know >that mathematics IS a *social* phenomena, where intriguing social >fossils like the one I've shown abound. >Given the subject line of this thread, I'm curious to see if any >posters on the sci.math newsgroup have the depth of understanding to >point out any other areas where decidability plays a key role. _Decidability_ is important in logic. But that's _decidability_, not this thing that you're _calling_ decidability. >Also, can any of you point out how the *field* of rationals, despite >being defined by integers as described above, differ from the ring of >integers in terms of the *key* difference? >James Harris ************************ David C. Ullrich === Subject: : Re: Decidability in mathematics > ... >> blowing up) and replaced it with a good solid hassiumy-meter. It had > ^^^^^^^^ >'scuze my ignorance, but what's hassiumy? Hassium is element 108, lying three spots below iron in the periodic > table. Heaviest known element on the same column-group as iron. iron -> irony > hassium -> hassiumy. Thanks, baint in Pauling but I found it here: http://www.webelements.com/. Down my way we 'as osmiumy meters--but only for the gentry. -- G.C. === Subject: : Re: Decidability in mathematics BUT, mathematicians didn't want to include pi or pi^2/6 in the field >of rationals, so even though it's decidable in the field, they exclude >it with the rule that a rational must be the ratio of integers where >the denominator is nonzero. >>I'm more and more aware that your posts contain grave logical mistakes. >>They exclude Pi^2 because Legendre proved it is NOT a rational (using the >>*definition* of a rational, to which you subscribed too). >>If you want to include Pi^2 in the field of rationals, then you HAVE TO >>prove it's a rational (and the definition of a field isn't sufficient). > You are correct that pi^2 is not rational, as it doesn't fit the > definition. However, I'm correct in noting that there is no field of rationals, as > the term is a social fossil. Please look up definitions before saying things like this. Rationals are defined by the definition that a rational is the ratio > of two integers where the denominator is non-zero, so they are defined > by the ring of integers, and that's all. > The fact that they form a field has to do with how + and * are defined on them. A field is more than a set. It is a set along with two binary operations. You need to know how they interact before you can call something a field, not just how the set is defined. James Harris -- Will Twentyman email: wtwentyman at copper dot net === Subject: : Re: Decidability in mathematics : I've noticed an interesting lack of understanding about ring and field : operations from several posters who've replied to my posts where I : prove that adding pi to the ring of integers results in a field. Words you seem not to know the definitions of: (1) Ring (2) Field (3) Decidable Mike === Subject: : Re: Decidability in mathematics > For instance, 1/2 = 1/3 + 1/3^2 + 1/3^3+... and of course 1/2 is not > barred from rationals, as it is the ratio of non-zero integers, but > also, the series though infinite is decidable, which just means you > get a *single* number. Not decidable, convergent. > However, a sum like 1 + -1 + 1 + -1 +... IS barred because it's > undecidable. Not undecidable, nonconvergent. > Similarly, an infinite sum like 1+1+1+... is barred because it's > undecidable. Not undecidable, divergent. A rational number may be *represented* by a convergent, infinites series, but it is *defined* as the ratio of two integers. -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: : Re: Decidability in mathematics Don't you wonder how much math he actually has taken? I'd figure anyone well-versed in mathematics would know what convergent and divergent means. He might appear more credible if he knew the terminology. David Moran === Subject: : Re: Decidability in mathematics I've noticed an interesting lack of understanding about ring and field >operations So have we, a great lack in JSH's understanding of ring and field operations. === Subject: : Re: Decidability in mathematics What is much more wonderous to me is how many posters (sound as if they) feel superior over JSH just because they know the currently standard definitions of a few words, like 'ring', 'converging' and 'decidable'. Are we going back to the pedantry of mediaeval scholastics? Herman Jurjus > Don't you wonder how much math he actually has taken? I'd figure anyone > well-versed in mathematics would know what convergent and divergent means. He > might appear more credible if he knew the terminology. > David Moran === Subject: : Re: Decidability in mathematics > Are we going back to the pedantry of mediaeval scholastics? No -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html His mind has been corrupted by colours, sounds and shapes. The League of Gentlemen === Subject: : Re: Decidability in mathematics In sci.math, Michael Hochster : >: I've noticed an interesting lack of understanding about ring and field >: operations from several posters who've replied to my posts where I >: prove that adding pi to the ring of integers results in a field. Words you seem not to know the definitions of: (1) Ring > (2) Field > (3) Decidable (4) Limit. (5) Definition. After all, Q arbitrarily excludes the irrational numbers (by definition), because Q isn't closed under an infinite series, apparently. Might as well throw in i and the quaternions while we're at it. :-) Mike -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: : Re: Decidability in mathematics > Don't you wonder how much math he actually has taken? I'd figure anyone > well-versed in mathematics would know what convergent and divergent means. He > might appear more credible if he knew the terminology. David Moran I've determined that many posters don't actually know either mathematics or current terminology in mathematics, but post as if they do. And the group accepts them and their posts. It's democracy run amok, in a community that is broken, which apparently doesn't know much actual mathematics at all. Consider the following links to definitions on MathWorld: http://mathworld.wolfram.com/Ring.html http://mathworld.wolfram.com/Decidable.html http://mathworld.wolfram.com/Theory.html My assessment is that many people within the math community apparently see *people* and their level of support as the final authority, which is democracy. And they aren't even well-versed in the definitions that they claim support their positions. The math community is about style over substance, with people mostly relying on democratic processes, where if enough agree on any given point, they don't care about logic OR definitions. That is, it's a fashion show. I'm currently assessing just how broken the math community is. James Harris === Subject: : Re: Decidability in mathematics >> Rationals are defined by the definition that a rational is the ratio >> of two integers where the denominator is non-zero, so they are defined >> by the ring of integers, and that's all. > The fact that they form a field has to do with how + and * are defined > on them. > A field is more than a set. It is a set along with two binary > operations. You need to know how they interact before you can call > something a field, not just how the set is defined. It's a shame that James didn't follow Brian Chandler's advice (back in June) to read W.W. Sawyer's A Concrete Approach to Abstract Algebra. If he did, he'd have found field defined and very thoroughly explained in the first few pages of Chapter 2. (However, I realize his attention span for learning mathematics probably isn't long enough to carry him all the way to the second chapter.) -- Wayne Brown | When your tail's in a crack, you improvise fwbrown@bellsouth.net | if you're good enough. Otherwise you give | your pelt to the trapper. e^(i*pi) = -1 -- Euler | -- John Myers Myers, Silverlock === Subject: : Re: Decidability in mathematics > Don't you wonder how much math he actually has taken? I'd figure anyone > well-versed in mathematics would know what convergent and divergent means. He > might appear more credible if he knew the terminology. David Moran > I've determined that many posters don't actually know either > mathematics or current terminology in mathematics, but post as if they > do. And the group accepts them and their posts. > It's democracy run amok, in a community that is broken, which > apparently doesn't know much actual mathematics at all. > Consider the following links to definitions on MathWorld: > http://mathworld.wolfram.com/Ring.html > http://mathworld.wolfram.com/Decidable.html > http://mathworld.wolfram.com/Theory.html > My assessment is that many people within the math community apparently > see *people* and their level of support as the final authority, which > is democracy. And they aren't even well-versed in the definitions > that they claim support their positions. > The math community is about style over substance, with people mostly > relying on democratic processes, where if enough agree on any given > point, they don't care about logic OR definitions. That is, it's a > fashion show. > I'm currently assessing just how broken the math community is. > James Harris What??? Now you're posting links that refute your own arguments??? Wow! ! -- There are two things you must never attempt to prove: the unprovable -- and the obvious. By the way, your assessment has little or no value, since it does not correspond to the facts. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: : Re: Decidability in mathematics Visiting Assistant Professor at the University of Montana. >Don't you wonder how much math he actually has taken? I'd figure anyone >well-versed in mathematics would know what convergent and divergent means. He >might appear more credible if he knew the terminology. Based on his randomly occurring autobiographical posts and his description of how math is done, it seems fairly clear that James never took any mathematics course beyond the basic calculus sequence. As such, it is quite possible that he did not dwell on convergent and divergent series for too long. ============================================================== ======== It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) ============================================================== ======== Arturo Magidin magidin@math.berkeley.edu === Subject: : Re: Decidability in mathematics C. Bond skrev i melding > Don't you wonder how much math he actually has taken? I'd figure anyone > well-versed in mathematics would know what convergent and divergent means. He > might appear more credible if he knew the terminology. David Moran I've determined that many posters don't actually know either > mathematics or current terminology in mathematics, but post as if they > do. And the group accepts them and their posts. It's democracy run amok, in a community that is broken, which > apparently doesn't know much actual mathematics at all. Consider the following links to definitions on MathWorld: http://mathworld.wolfram.com/Ring.html http://mathworld.wolfram.com/Decidable.html http://mathworld.wolfram.com/Theory.html My assessment is that many people within the math community apparently > see *people* and their level of support as the final authority, which > is democracy. And they aren't even well-versed in the definitions > that they claim support their positions. The math community is about style over substance, with people mostly > relying on democratic processes, where if enough agree on any given > point, they don't care about logic OR definitions. That is, it's a > fashion show. I'm currently assessing just how broken the math community is. James Harris > What??? Now you're posting links that refute your own arguments??? Wow! ! > -- > There are two things you must never attempt to prove: the unprovable -- and the obvious. > By the way, your assessment has little or no value, since it does not correspond to the facts. > -- > Democracy: The triumph of popularity over principle. > -- > http://www.crbond.com That's because he does not understand the mathematical language. He does not understand what is written on those pages he referres to. Karl-Olav Nyberg === Subject: : Re: Decidability in mathematics >Don't you wonder how much math he actually has taken? I'd figure anyone >>well-versed in mathematics would know what convergent and divergent means. He >>might appear more credible if he knew the terminology. >>David Moran > I've determined that many posters don't actually know either > mathematics or current terminology in mathematics, but post as if they > do. And the group accepts them and their posts. It's democracy run amok, in a community that is broken, which > apparently doesn't know much actual mathematics at all. Consider the following links to definitions on MathWorld: http://mathworld.wolfram.com/Ring.html http://mathworld.wolfram.com/Decidable.html http://mathworld.wolfram.com/Theory.html Did you actually *read* these? They are exactly what we've been saying they are. My assessment is that many people within the math community apparently > see *people* and their level of support as the final authority, which > is democracy. And they aren't even well-versed in the definitions > that they claim support their positions. You mean like the ones you referenced? We understand them. The math community is about style over substance, with people mostly > relying on democratic processes, where if enough agree on any given > point, they don't care about logic OR definitions. That is, it's a > fashion show. Aren't you the one that calls excluding pi from the rationals ad hoc, even though it clearly fails to meet the definition? Aren't you the one that wants something to be a factor even though it doesn't meet the definition of factor? How is it that *we* are the ones who don't care about definitions? I'm currently assessing just how broken the math community is. > James Harris -- Will Twentyman email: wtwentyman at copper dot net === Subject: : Re: Decidability in mathematics >> Don't you wonder how much math he actually has taken? I'd figure anyone >> well-versed in mathematics would know what convergent and divergent means. He >> might appear more credible if he knew the terminology. >> David Moran >I've determined that many posters don't actually know either >mathematics or current terminology in mathematics, but post as if they >do. And the group accepts them and their posts. >It's democracy run amok, in a community that is broken, which >apparently doesn't know much actual mathematics at all. >Consider the following links to definitions on MathWorld: >http://mathworld.wolfram.com/Ring.html >http://mathworld.wolfram.com/Decidable.html >http://mathworld.wolfram.com/Theory.html This is hilarious. You've done the same thing before, so you don't get any extra credit, but it's hilarious every time you do it. Every time you do what? First say something wrong and then in defense of what you said post a link to something that you seem to think shows you're right, when in fact it shows you're simply wrong. Hint: The definition of Decidable there is this: A theory is decidable iff there is an algorithm which can determine whether or not any sentence r is a member of the theory. Which is of course no big news, we all knew that's what decidable meant. The hilarious part is that you don't seem to notice that the definition _shows_ that the word decidable does _not_ mean convergent. Hilarious every time, the sort of thing that keeps people reading: Complaining about the ignorance of everyone else, while giving abundant demonstrations of your own ignorance. (Possibly ignorance plus stupidity, if you actually think that that definition shows that decidable means convergent...) >My assessment is that many people within the math community apparently >see *people* and their level of support as the final authority, which >is democracy. And they aren't even well-versed in the definitions >that they claim support their positions. Guffaw. >The math community is about style over substance, with people mostly >relying on democratic processes, where if enough agree on any given >point, they don't care about logic OR definitions. That is, it's a >fashion show. >I'm currently assessing just how broken the math community is. >James Harris ************************ David C. Ullrich === Subject: : Re: Decidability in mathematics >[...] >What??? Now you're posting links that refute your own arguments??? Wow! ! He's done so before, maybe that was before your time. >-- >There are two things you must never attempt to prove: the unprovable -- and the obvious. >By the way, your assessment has little or no value, since it does not correspond to the facts. >-- >Democracy: The triumph of popularity over principle. ************************ David C. Ullrich === Subject: : Re: Decidability in mathematics monsieur Jurjus, monsieur Harris has linked us to the Wolframites' definitions, apparently as a form of closure (sik) as to that with which he agrees, and they happen to be close-enough for the sharing of cigars amongst all of Harris' particular audience, us few & proud; *you* are the pedant, hereinat. the question is, Will he/you ever read them & impliment?... apparently, a monetary award is all the criterion that serves, or perhaps it's Sir David's other award, not MacArthur's. (who in Hell is/was MacArthur, anyway?... shades of more British mathsology.-) > (sound as if they) feel superior over JSH just > because they know the currently standard definitions of a few > words, like 'ring', 'converging' and 'decidable'. --les ducs d'Enron! http://www.tarpley.net === Subject: : Re: Decidability in mathematics Many of these posters have *proclaimed* that infinite operations are >barred from rings and fields in particular cases, apparently lacking >an understanding of in what cases they are actually barred and why. >>Hum huh... who said anything is barred ? >>In order to do mathematics, you need rigorous definitions, that's it. We >>*proclaimed* the definition of a field, and you subscribed. > I'm afraid James Harris simply does not *know* the definition of a ring, > field or any other mathematical structure. (JSH: Care to prove me wrong? > Just give the definition of a ring - maybe looking it up in some text > book helps). Definitions? Just social fossils! Gib === Subject: : Re: Decidability in mathematics > In sci.math, Michael Hochster > : >: I've noticed an interesting lack of understanding about ring and field >>: operations from several posters who've replied to my posts where I >>: prove that adding pi to the ring of integers results in a field. >>Words you seem not to know the definitions of: >>(1) Ring >>(2) Field >>(3) Decidable > (4) Limit. > (5) Definition. (6) Proof. Gib === Subject: : Re: Decidability in mathematics > I'm currently assessing just how broken the math community is. Keep up the good work :-) Gib === Subject: : Derivative discontinuous on a dense set I would like to see an example of a differentiable function on (0,1), whose derivative is discontinous on a dense subset , say the rationals, of (0,1). Is this possible? === Subject: : Re: Derivative discontinuous on a dense set >I would like to see an example of a differentiable function on (0,1), whose >derivative is discontinous on a dense subset , say the rationals, of (0,1). >Is this possible? Yes, it's possible, in fact if (x_n) is any sequence of reals then there exists a differentiable f:R -> R such that f' is discontinuous at every x_n. Start with a differentiable function g:R -> R such that |g| <= 1 everywhere, |g'| <= 1 everywhere, g' is continuous except at the origin, and g' has oscillation equal to 1 at the origin (ie there exist s_j -> 0 and t_j -> 0 such that |g'(s_j) - g'(t_j)| tends to 1 as j -> infinity.) Let f(x) = sum 10^(-n) g(x - x_n). Seems clear that f should be differentiable, with a derivative discontinuous at every x_n, if things work out right; in fact things do work out right: First, the fact that |g'| <= 1 shows that f is differentiable (write f as the sum of the first N terms plus the tail; the sum of the first N terms is differentiable, and the tail has Lip_1 constant less than epsilon...) Note that in fact the derivative of the sum is the sum of the derivatives at every point. And f' is not continuous at x_n: This is because the sum of 10^(-k) g'(x - x_k) for k < n _is_ differentiable at x_n, the derivative of the n-th term has oscillation at least 10^(-n) at x_n, and the sum of 10^(-k) g'(x - x_k) for k > n is smaller than 10^(-n). ************************ David C. Ullrich === Subject: : Re: Derivative discontinuous on a dense set >I would like to see an example of a differentiable function on (0,1), whose >derivative is discontinous on a dense subset , say the rationals, of (0,1). >Is this possible? Yes, it's possible, in fact if (x_n) is any sequence of reals then > there exists a differentiable f:R -> R such that f' is discontinuous > at every x_n. Start with a differentiable function g:R -> R such that |g| <= 1 > everywhere, |g'| <= 1 everywhere, g' is continuous except at > the origin, and g' has oscillation equal to 1 at the origin > (ie there exist s_j -> 0 and t_j -> 0 such that |g'(s_j) - g'(t_j)| > tends to 1 as j -> infinity.) Let f(x) = sum 10^(-n) g(x - x_n). Seems clear that f should be differentiable, with a > derivative discontinuous at every x_n, if things work out > right; in fact things do work out right: First, the fact that |g'| <= 1 shows that f is differentiable > (write f as the sum of the first N terms plus the tail; > the sum of the first N terms is differentiable, and the > tail has Lip_1 constant less than epsilon...) Note that > in fact the derivative of the sum is the sum of the > derivatives at every point. And f' is not continuous at x_n: This is because the > sum of 10^(-k) g'(x - x_k) for k < n _is_ differentiable > at x_n, the derivative of the n-th term has oscillation > at least 10^(-n) at x_n, and the sum of 10^(-k) g'(x - x_k) > for k > n is smaller than 10^(-n). Here's a slightly different proof: Choose differentiable functions f_n on R with |f_n|, |f_n'| < 1/2^n on R such that f_n' is discontinuous at x_n and continuous everywhere else. Let f(x) = sum (n=1,oo) f_n(x), a uniformly convergent series. We have [f(x) - f(a)]/(x-a) = sum (n=1,oo) [f_n(x) - f_n(a)]/(x-a). By the mean value theorem and the assumed bounds on |f_n'|, the last series converges uniformly on R. So the limit of the sum is the sum of the limits and we get f'(x) = sum (n=1,oo) f_n'(x) for all x in R. The last series is uniformly convergent, and the usual argument involving uniform convergence and continuity shows f' is continuous on R {x_1,x_2, ...}. At each x_n, write f'(x) = f_n'(x) + all the rest to see that f is discontinuous at x_n. (I'm assuming the x_n's are distinct.) === Subject: : distance between 2 darts suppose you throw 2 darts onto a dart board of radius R. (center at origin) Darts land on the dart board uniformly i.e. Pr( (x,y) == (x0,y0) ) = 1/(pi*R^2) let D = sqrt( (x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2 ) What is Pr( D <= d) ? i.e. need to get the distribution function of D. I am actually interested in the distribution function of 2 points chosen uniformly inside a ball of radius R. (center at origin) but in my previous post, i stated the problem incorrectly--so if someone can help me solve this problem for 2 D I can extend it to 3 D myself. I think that I should proceed in 4 steps: 1) first get the density function (f_U) of U = x1-x2 where x1 and x2 are unform on [-R,R] AND x1^2 + y^2 <= R^2 2) then get the density function (f_V) of V = U^2 3) then get the density function (f_W) of W = V1 + V2 (where V is got from 2) 4) finally, get the density function (f_Z) of Z = sqrt( W ) thanks in advance for any tips or input (I am stuck on getting the density f_U since I don't know how to include the constraint) thanks in advance for any help === Subject: : distance between random points in a sphere it is easy for me to numerically get the distribution of the distance between 2 points picked uniformly in a sphere of radius R -- However, I want to get the closed form density. Since the points are uniformly picked the density of each point f(x,y,z) = C = 0 otherwise where C = 1/( (4/3)*pi*R^3 ) if x^2 + y^2 + z^2 <= R^2 however, when I pick 2 points according to this density and determine the distance, d, the range of the distance is 0 <= d <= 2*R However, I am stuck regarding how I can find the density of d Pr( d <= a) = Constant * int int int dx dy dz ------------/ (x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2 <= R^2 any help as to how I should proceed would be much appreciated perhaps it would be easier to use spherical coordinates in which case Pr( d <= a) = Constant * int_{theta=0 to 2*pi} int_{phi=-pi to pi} int _{r=0 to R} r*sin(theta)*dtheta*dphi*dr thanks for any help. Even an example of a distance between 2 points picked uniformly in a circle would be very helpful -- i will extend it to 3 d case. thanks === Subject: : Re: distance between random points in a sphere >it is easy for me to numerically get the >distribution of the distance between 2 points picked uniformly in a >sphere of radius R -- However, I want to get the closed >form density. For simplicity, assume R = 1; you can always scale later. Consider the two random vectors you have selected. By symmetry, the angle between them is uniformly distributed. Let R1, R2, and pi U be the random moduli and angle. By the law of cosines, D^2 = R1^2 + R2^2 - 2 R1 R2 cos(pi U). The three random variables are independent; U is uniform(0,1), and each R has density r -> 3r^2 (2r in the circle). Now use the Jacobian change-of-variable method to determine the density of D^2, or set up the integral Finish up by calculating the density of the square root. I suspect the final answer is not pretty. In fact, you may not be able to express the density without the presence of integral signs. Best regards, -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: : Re: distance between random points in a sphere > it is easy for me to numerically get the > distribution of the distance between 2 points picked uniformly in a > sphere of radius R -- However, I want to get the closed > form density. Take R = 1 (you can scale later) and let X and Y denote the points so picked. The probability that |X-Y| < d is the area of the the set of points on the unit sphere whose distance from (1,0,0) is < d, divided by 4Pi. We can calculate that as a surface of revolution. A little geometry shows that if the distance from (1,0) to another point on the unit circle is d, then the x-projection of that point is 1 - d^2/2. So P{|X-Y| < d} = (1/4Pi)*integral_[1-d^2/2,1] 2Pi*f(x)*sqrt(1 + f'(x)^2) dx, where f(x) = sqrt(1-x^2). Most everything miraculously cancels, leaving d^2/4. That's the cumulative probability function on [0,2]; its derivative is d/2, and that is your density function on [0,2]. === Subject: : Re: distance between random points in a sphere > it is easy for me to numerically get the > distribution of the distance between 2 points picked uniformly in a > sphere of radius R -- However, I want to get the closed > form density. Given two points, independently and uniformly distributed on the surface of a sphere of radius R, we may, without loss of generality, arrange our coordinate system to place the points at (-R,0,0) and (x,sqrt(R^2-x^2),0), where x is uniformly distributed on [-R,R]. Note that the uniform distribution along a diameter is valid only in 3 dimensions. The distance s is given by s^2 = (x+R)^2 + (R^2 - x^2) = 2R * (R+x) which yields x = s^2/(2R) - R. Since x is an increasing function of R, we can conclude that for each s in [0,2R], the probability of obtaining a distance less than s is the probability that a random variable, uniformly distributed on [-R,R], is less than x = s^2/(2R) - R. This latter probability is just the length of the subinterval [-R,x] divided by the length of the entire interval [-R,R], or s^2/(2R) / (2R) = s^2 / (4R^2), which produces the desired distribution. Its derivative s/(2R^2) is the density function. In particular, the expected value of s is int_0^(2R) s^2/(2R^2) = 4/3 * R. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: : Re: distance between random points in a sphere it is easy for me to numerically get the > distribution of the distance between 2 points picked uniformly in a > sphere of radius R -- However, I want to get the closed > form density. Take R = 1 (you can scale later) and let X and Y denote the points so > picked. The probability that |X-Y| < d is the area of the the set of points > on the unit sphere whose distance from (1,0,0) is < d, divided by 4Pi. We > can calculate that as a surface of revolution. A little geometry shows that > if the distance from (1,0) to another point on the unit circle is d, then > the x-projection of that point is 1 - d^2/2. So P{|X-Y| < d} = (1/4Pi)*integral_[1-d^2/2,1] 2Pi*f(x)*sqrt(1 + f'(x)^2) dx, where f(x) = sqrt(1-x^2). Most everything miraculously cancels, leaving > d^2/4. That's the cumulative probability function on [0,2]; its derivative > is d/2, and that is your density function on [0,2]. thank you for your reply, I think you misread my post, because I was interested in 2 points *IN* the sphere not *ON* it. thanks les p.s. from numerical calculations, the density looks symmetric and has a peak near 3/4 * R === Subject: : Re: distance between random points in a sphere > it is easy for me to numerically get the > distribution of the distance between 2 points picked uniformly in a > sphere of radius R -- However, I want to get the closed > form density. Given two points, independently and uniformly distributed on the surface > of a sphere of radius R, we may, without loss of generality, arrange our > coordinate system to place the points at (-R,0,0) and > (x,sqrt(R^2-x^2),0), where x is uniformly distributed on [-R,R]. Note > that the uniform distribution along a diameter is valid only in 3 > dimensions. The distance s is given by s^2 = (x+R)^2 + (R^2 - x^2) > = 2R * (R+x) which yields x = s^2/(2R) - R. Since x is an increasing function of R, we can conclude that for each s > in [0,2R], the probability of obtaining a distance less than s is the > probability that a random variable, uniformly distributed on [-R,R], is > less than x = s^2/(2R) - R. This latter probability is just the length > of the subinterval [-R,x] divided by the length of the entire interval > [-R,R], or s^2/(2R) / (2R) = s^2 / (4R^2), which produces the desired > distribution. Its derivative s/(2R^2) is the density function. In particular, the expected value of s is int_0^(2R) s^2/(2R^2) = 4/3 * R. thanks for the reply, I think you mis-read the post as what is required is the distribution of distance between 2 points picked uniformly *IN* a sphere of radius R not *ON* sphere. thanks les === Subject: : Re: distance between random points in a sphere >I think you mis-read the post as >what is required is the distribution of distance between 2 points >picked uniformly *IN* a sphere of radius R not *ON* sphere. This is a common instance of miscommunication here. Amongst mathematicians, one should be careful in the use of the words circle (a curve), disk (a region), sphere (a surface), and ball (a region). But you did clearly say >f(x,y,z) = C > = 0 otherwise >where C = 1/( (4/3)*pi*R^3 ) if x^2 + y^2 + z^2 <= R^2 My post addressed your intended problem. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: : Dodecagon algorithm maze This puzzle is based upon a couple others I have posted recently, but should be more interesting. Start with a regular dodecagon (12-gon) with its vertexes labeled as the hours on a clock's face. Draw a continuous path consisting of 12 straight line-segments where each segment starts and ends on the dodecagon's vertexes, and each being drawn using the following conditions: Each vertex is visited once and only once. (And adjacently visited vertexes MAY be adjacent vertexes on the 12-gon.) The path returns to its starting point so as to be a closed loop. Let the m_th VISITED vertex = v(m). The segments are each individually drawn by the following rules: v(1) = 12. v(2) = a prime. v(3) = v(2) +3. v(4) = v(3)/2 v(5) = such that segment(4,5) crosses 2 PREVIOUSLY drawn segments. v(6) = lowest prime > v(5). v(7) = such that segment(6,7) crosses v(7)/2 previously drawn segments. v(8) = lowest integer > v(7) and not yet visited. v(9) = (v(6) +v(8))/2. v(10) = horizontally positioned across from v(9). v(11) = GCD(v(10),v(2)) v(12) = such that segment(11,12) crosses only one previously drawn segment. And then return to v(1) = 12. And (so as to narrow the possibilities of working solutions) the TOTAL number of occurrences that the path crosses itself (crosses previously and subsequently drawn segments) is 34. (By an occurrence of the path crossing itself, I mean every time any given segment crosses any other segment. So, for example, two {exactly 2} segments crossing adds 2 crossings to the count. If 3 segments happen to cross at one point, this adds 6 crossings to the count {because each of the 3 segments crosses the 2 other segments}. Generally, n segments crossing at a single point adds n*(n-1) to the crossing-count.) (And the dodecagon's vertexes themselves, where at each 2 line-segments come together, are NOT considered to be crossings.) So, I am not sure how many solutions there are to this particular version of this puzzle. (To make certain that the solution is unique would take, for me, too much effort and/or scratchpaper and/or computer and/or motivation...) I doubt there are many, if more than 1. Hopefully, this polygonal (algorithmic) maze will inspire some of you to create your own such puzzles. Thanks, Leroy Quet === Subject: : Re: Dodecagon algorithm maze > This puzzle is based upon a couple others I have posted recently, but > should be more interesting. > Start with a regular dodecagon (12-gon) with its vertexes labeled as the > hours on a clock's face. Draw a continuous path consisting of 12 straight line-segments where each > segment starts and ends on the dodecagon's vertexes, and each being drawn using > the following conditions: Each vertex is visited once and only once. (And adjacently visited > vertexes MAY be adjacent vertexes on the 12-gon.) The path returns to its starting point so as to be a closed loop. Let the m_th VISITED vertex = v(m). The segments are each individually > drawn by the following rules: v(1) = 12. > v(2) = a prime. > v(3) = v(2) +3. > v(4) = v(3)/2 > v(5) = such that segment(4,5) crosses 2 PREVIOUSLY drawn segments. > v(6) = lowest prime > v(5). > v(7) = such that segment(6,7) crosses v(7)/2 previously drawn segments. > v(8) = lowest integer > v(7) and not yet visited. > v(9) = (v(6) +v(8))/2. > v(10) = horizontally positioned across from v(9). > v(11) = GCD(v(10),v(2)) > v(12) = such that segment(11,12) crosses only one previously drawn > segment. > And then return to v(1) = 12. And (so as to narrow the possibilities of working solutions) the TOTAL > number of occurrences that the path crosses itself (crosses previously > and subsequently drawn segments) is 34. (By an occurrence of the path crossing itself, I mean every time any > given segment crosses any other segment. So, for example, two {exactly 2} > segments crossing adds 2 crossings to the count. If 3 segments happen to > cross at one point, this adds 6 crossings to the count {because each of > the 3 segments crosses the 2 other segments}. Generally, n segments > crossing at a single point adds n*(n-1) to the crossing-count.) > (And the dodecagon's vertexes themselves, where at each 2 line-segments come > together, are NOT considered to be crossings.) So, I am not sure how many solutions there are to this particular version > of this puzzle. (To make certain that the solution is unique would take, > for me, too much effort and/or scratchpaper and/or computer and/or > motivation...) I doubt there are many, if more than 1. Hopefully, this polygonal (algorithmic) maze will inspire some of you > to create your own such puzzles. Thanks, > Leroy Quet I have a question... can we have numbers such as 8+6=2 since 8+6=14 and 14-12=2? So that the numbers loop around themselves? (...Starblade Riven Darksquall...) === Subject: : Re: Dodecagon algorithm maze > This puzzle is based upon a couple others I have posted recently, but > should be more interesting. > Start with a regular dodecagon (12-gon) with its vertexes labeled as the > hours on a clock's face. Draw a continuous path consisting of 12 straight line-segments where each > segment starts and ends on the dodecagon's vertexes, and each being drawn using > the following conditions: Each vertex is visited once and only once. (And adjacently visited > vertexes MAY be adjacent vertexes on the 12-gon.) The path returns to its starting point so as to be a closed loop. Let the m_th VISITED vertex = v(m). The segments are each individually > drawn by the following rules: v(1) = 12. > v(2) = a prime. > v(3) = v(2) +3. > v(4) = v(3)/2 > v(5) = such that segment(4,5) crosses 2 PREVIOUSLY drawn segments. > v(6) = lowest prime > v(5). > v(7) = such that segment(6,7) crosses v(7)/2 previously drawn segments. > v(8) = lowest integer > v(7) and not yet visited. > v(9) = (v(6) +v(8))/2. > v(10) = horizontally positioned across from v(9). > v(11) = GCD(v(10),v(2)) > v(12) = such that segment(11,12) crosses only one previously drawn > segment. > And then return to v(1) = 12. And (so as to narrow the possibilities of working solutions) the TOTAL > number of occurrences that the path crosses itself (crosses previously > and subsequently drawn segments) is 34. (By an occurrence of the path crossing itself, I mean every time any > given segment crosses any other segment. So, for example, two {exactly 2} > segments crossing adds 2 crossings to the count. If 3 segments happen to > cross at one point, this adds 6 crossings to the count {because each of > the 3 segments crosses the 2 other segments}. Generally, n segments > crossing at a single point adds n*(n-1) to the crossing-count.) > (And the dodecagon's vertexes themselves, where at each 2 line-segments come > together, are NOT considered to be crossings.) So, I am not sure how many solutions there are to this particular version > of this puzzle. (To make certain that the solution is unique would take, > for me, too much effort and/or scratchpaper and/or computer and/or > motivation...) I doubt there are many, if more than 1. Hopefully, this polygonal (algorithmic) maze will inspire some of you > to create your own such puzzles. Thanks, > Leroy Quet I have a question... can we have numbers such as 8+6=2 since 8+6=14 and 14-12=2? So that the numbers loop around themselves? (...Starblade Riven Darksquall...) === Subject: : Re: Dodecagon algorithm maze > This puzzle is based upon a couple others I have posted recently, but > should be more interesting. > Start with a regular dodecagon (12-gon) with its vertexes labeled as the > hours on a clock's face. Draw a continuous path consisting of 12 straight line-segments where each > segment starts and ends on the dodecagon's vertexes, and each being drawn using > the following conditions: Each vertex is visited once and only once. (And adjacently visited > vertexes MAY be adjacent vertexes on the 12-gon.) The path returns to its starting point so as to be a closed loop. Let the m_th VISITED vertex = v(m). The segments are each individually > drawn by the following rules: v(1) = 12. > v(2) = a prime. > v(3) = v(2) +3. > v(4) = v(3)/2 > v(5) = such that segment(4,5) crosses 2 PREVIOUSLY drawn segments. > v(6) = lowest prime > v(5). > v(7) = such that segment(6,7) crosses v(7)/2 previously drawn segments. > v(8) = lowest integer > v(7) and not yet visited. > v(9) = (v(6) +v(8))/2. > v(10) = horizontally positioned across from v(9). > v(11) = GCD(v(10),v(2)) > v(12) = such that segment(11,12) crosses only one previously drawn > segment. > And then return to v(1) = 12. And (so as to narrow the possibilities of working solutions) the TOTAL > number of occurrences that the path crosses itself (crosses previously > and subsequently drawn segments) is 34. (By an occurrence of the path crossing itself, I mean every time any > given segment crosses any other segment. So, for example, two {exactly 2} > segments crossing adds 2 crossings to the count. If 3 segments happen to > cross at one point, this adds 6 crossings to the count {because each of > the 3 segments crosses the 2 other segments}. Generally, n segments > crossing at a single point adds n*(n-1) to the crossing-count.) > (And the dodecagon's vertexes themselves, where at each 2 line-segments come > together, are NOT considered to be crossings.) So, I am not sure how many solutions there are to this particular version > of this puzzle. (To make certain that the solution is unique would take, > for me, too much effort and/or scratchpaper and/or computer and/or > motivation...) I doubt there are many, if more than 1. Hopefully, this polygonal (algorithmic) maze will inspire some of you > to create your own such puzzles. Thanks, > Leroy Quet Oh, yes, BTW, I found it already. ;> (...Starblade Riven Darksquall...) === Subject: : Re: Dodecagon algorithm maze > This puzzle is based upon a couple others I have posted recently, but > should be more interesting. > Start with a regular dodecagon (12-gon) with its vertexes labeled as the > hours on a clock's face. Draw a continuous path consisting of 12 straight line-segments where each > segment starts and ends on the dodecagon's vertexes, and each being drawn using > the following conditions: Each vertex is visited once and only once. (And adjacently visited > vertexes MAY be adjacent vertexes on the 12-gon.) The path returns to its starting point so as to be a closed loop. Let the m_th VISITED vertex = v(m). The segments are each individually > drawn by the following rules: v(1) = 12. > v(2) = a prime. > v(3) = v(2) +3. > v(4) = v(3)/2 > v(5) = such that segment(4,5) crosses 2 PREVIOUSLY drawn segments. > v(6) = lowest prime > v(5). > v(7) = such that segment(6,7) crosses v(7)/2 previously drawn segments. > v(8) = lowest integer > v(7) and not yet visited. > v(9) = (v(6) +v(8))/2. > v(10) = horizontally positioned across from v(9). > v(11) = GCD(v(10),v(2)) > v(12) = such that segment(11,12) crosses only one previously drawn > segment. > And then return to v(1) = 12. And (so as to narrow the possibilities of working solutions) the TOTAL > number of occurrences that the path crosses itself (crosses previously > and subsequently drawn segments) is 34. (By an occurrence of the path crossing itself, I mean every time any > given segment crosses any other segment. So, for example, two {exactly 2} > segments crossing adds 2 crossings to the count. If 3 segments happen to > cross at one point, this adds 6 crossings to the count {because each of > the 3 segments crosses the 2 other segments}. Generally, n segments > crossing at a single point adds n*(n-1) to the crossing-count.) > (And the dodecagon's vertexes themselves, where at each 2 line-segments come > together, are NOT considered to be crossings.) So, I am not sure how many solutions there are to this particular version > of this puzzle. (To make certain that the solution is unique would take, > for me, too much effort and/or scratchpaper and/or computer and/or > motivation...) I doubt there are many, if more than 1. Hopefully, this polygonal (algorithmic) maze will inspire some of you > to create your own such puzzles. Thanks, > Leroy Quet I have a question... can we have numbers such as 8+6=2 since 8+6=14 > and 14-12=2? So that the numbers loop around themselves? (...Starblade Riven Darksquall...) Actually a good question. And the answer is no, there is no use of modulo arithmetic. I realized recently that this 'maze' has only a 3 branchings (branching = more than 1 possible choices as to what the next vertex will be, based on earlier visited vertexes), and so might be actually pretty simple. So, I might as well give my solution (which might still not be unique, even if there are not many possible paths which might either work or not work), but hide it slightly below: | | V | | V | | V | | V | | V | | V | | V v(2) = lowest prime which is the sum of 2 previous primes. (where 1 is not counted as a prime) v(5) = v(2) +2. v(7) = v(5) -1. The rest is implied... Thanks, Leroy Quet === Subject: : E8 lattice question Hello I am reading SPLAG book of SLONA and CONWAY I am very confused by the lattice obtained from construction A. It says if # is an [n,k] binary code, construction A produces a lattice Lambda(#) in R^n. If the codewords in +1,-1 notation, the points of Lambda(#) consists of all vectors of the form c+4z for c in #, z in Z^n according to the book E8 consists of c+4z for c in #, z in Z^8 c is first order Reel_muller code Obviously c is integer vector. so c+4z is teger vector. However, E8 is defined as (D8+(1/2,...,1/2)) U D8 so there are some vectors like (1/2,...,1/2) is an E8 point I am very confused about this. Thank you very much! === Subject: : Re: E8 lattice question > Hello I am reading SPLAG book of SLONA and CONWAY > I am very confused by the lattice obtained from construction A. It says if # is an [n,k] binary code, construction A produces a lattice > Lambda(#) in R^n. If the codewords in +1,-1 notation, the points of > Lambda(#) consists of all vectors of the form > c+4z for c in #, z in Z^n Hmmm. Does it really? Chapter and verse please! I prefer to write my binary codes in 0,1 notation and define L(C) as {c + 2z: c in C, z in Z^n}. If your lattice is Lambda and mine is L then Lambda = (1,1, ..., 1) + 2L (i.e. it's a coset of a scaled version of L). I put lattice in quotes then since to me a lattice has gotta have 0, and I'd call yours a coset of a lattice. However, E8 is defined as > (D8+(1/2,...,1/2)) U D8 so there are some vectors like (1/2,...,1/2) is an E8 point If you define D_8 to be (in my notation) L(C) where C is the paity check code of length 8 (i.e. the set of (a_1, ..., a_8) where a_1 + ... + a_8 is even) then this works. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html His mind has been corrupted by colours, sounds and shapes. The League of Gentlemen === Subject: : Re: E8 lattice question > Hello I am reading SPLAG book of SLONA and CONWAY > I am very confused by the lattice obtained from construction A. It says if # is an [n,k] binary code, construction A produces a lattice > Lambda(#) in R^n. If the codewords in +1,-1 notation, the points of > Lambda(#) consists of all vectors of the form > c+4z for c in #, z in Z^n > Hmmm. Does it really? Chapter and verse please! It is chap 5 p137 & p139 I think the generator matrix of (D8+(1/2,...,1/2)) U D8 is G2 = [2 0 0 0 0 0 0 0; -1 1 0 0 0 0 0 0; 0 -1 1 0 0 0 0 0; 0 0 -1 1 0 0 0 0; 0 0 0 -1 1 0 0 0; 0 0 0 -1 1 0 0 0; 0 0 0 0 -1 1 0 0; 0 0 0 0 0 -1 1 0; .5 .5 .5 .5 .5 .5 .5 .5 ] The coding theory construction of E8: apply Construction A to the [8,4,4] Hamming code. SPLAG p.121 G = 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 1 1 1 0 1 0 0 0 1 0 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 1 So there must be some ways to transform G2 to G Not too sure what it is. Thanks a lot! > I prefer to write my binary codes in 0,1 notation and define > L(C) as {c + 2z: c in C, z in Z^n}. > If your lattice is Lambda and mine is L then Lambda = (1,1, ..., 1) + 2L > (i.e. it's a coset of a scaled version of L). I put lattice in quotes > then since to me a lattice has gotta have 0, and I'd call yours > a coset of a lattice. However, E8 is defined as > (D8+(1/2,...,1/2)) U D8 so there are some vectors like (1/2,...,1/2) is an E8 point > If you define D_8 to be (in my notation) L(C) where C is > the paity check code of length 8 (i.e. the set of (a_1, ..., a_8) > where a_1 + ... + a_8 is even) then this works. === Subject: : Re: E8 lattice question > Hello >> I am reading SPLAG book of SLONA and CONWAY >> I am very confused by the lattice obtained from construction A. >> It says if # is an [n,k] binary code, construction A produces a lattice >> Lambda(#) in R^n. If the codewords in +1,-1 notation, the points of >> Lambda(#) consists of all vectors of the form >> c+4z for c in #, z in Z^n >> Hmmm. Does it really? Chapter and verse please! > It is chap 5 p137 & p139 Which edition? > I think the generator matrix of (D8+(1/2,...,1/2)) U D8 > is > G2 = [2 0 0 0 0 0 0 0; > -1 1 0 0 0 0 0 0; > 0 -1 1 0 0 0 0 0; > 0 0 -1 1 0 0 0 0; > 0 0 0 -1 1 0 0 0; > 0 0 0 -1 1 0 0 0; > 0 0 0 0 -1 1 0 0; > 0 0 0 0 0 -1 1 0; > .5 .5 .5 .5 .5 .5 .5 .5 ] Aaaarrrgghh! Real Mathematicians don't write .5 when they mean 1/2 :-) > The coding theory construction of E8: > apply Construction A to the [8,4,4] Hamming code. > SPLAG p.121 > G = 2 0 0 0 0 0 0 0 > 0 2 0 0 0 0 0 0 > 0 0 2 0 0 0 0 0 > 0 0 0 2 0 0 0 0 > 1 1 1 0 1 0 0 0 > 1 0 1 1 0 1 0 0 > 1 0 0 1 1 0 1 0 > 1 0 0 0 1 1 0 1 This is a generator matrix for a lattice L' which is isometric to sqrt(2)E_8 where E_8 is as above. There is an orthogonal matrix M such that L' = sqrt(2) E_8 M. Alas I can't be bothered to find such a one for you right now, but if I find myslef at a really loose end ... :-) -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html His mind has been corrupted by colours, sounds and shapes. The League of Gentlemen === Subject: : EPR communication? Probably they are correct below. I have not yet read the paper carefully enough yet to see exactly where the orthodox rules are modified. I would be pleasantly surprised if the paper turns out to be correct. Re: QM allows non-local signaling -------------------------------------------------------------- -------------- ---- === Subject: : Re: QM allows non-local signaling -------------------------------------------------------------- -------------- ---- So has Srikanth demonstrated an observable that is an exception to this rule? I don't think so, but here I should probably defer to people more willing to dive into the details of Young interferometers... It's probably best *not* dive into the details of Young interferometers, but simply to say: Srikanth shows that if you modify the usual rules of quantum mechanics, you can get non-local signaling. It's then an experimental question to see whether Srikanth's modification of quantum mechanics is correct. If he hasn't done any experiments to support this claim, it doesn't seem very likely. === Subject: : Re: EPR communication? you're welcome, Dancing Master. > It's probably best *not* dive into the details of Young interferometers, > but simply to say: Srikanth shows that if you modify the usual rules of > quantum mechanics, you can get non-local signaling. It's then an > experimental question to see whether Srikanth's modification of quantum > mechanics is correct. If he hasn't done any experiments to support this > claim, it doesn't seem very likely. http://buckminster.info/Ideas/03-TetGeomTheorem4-ColorProof.htm --UN HYDROGEN (sic; Methanex (TM) reformanteurs) ECONOMIE?... La Troi Phases d'Exploitation de la Protocols des Grises de Kyoto: (FOSSILISATION [McCainanites?] (TM/sic))/ BORE/GUSH/NADIR @ http://www.tarpley.net/aobook.htm. Http://www.tarpley.net/bushb.htm (content partiale, below): 17 -- L'ATTEMPTER de COUP D'ETAT, 3/30/81 23 -- Le FIN d'HISTOIRE 24 -- L'ORDEUR du MONDE NOUVEAU 25 -- THYROID STORK !?! === Subject: : Re: Equidistantly distributed lattice points > It is well known that there is no equilateral triangle > with its vertices at lattice points. Extending the idea, we can define the set S of natural numbers n > satisfying the condition: > there exist n lattice points in the (n-1)-dimensional Euclidean space > such that the distances between any two of them are equal. Can we then determine all the elements of this set S? The following are the facts I have found so far: > 1. S is a multiplicatively closed set. > 2. For an odd number n, n belongs to S iff n is a perfect square. > 3. S contains all of > (a) perfect squares, > (b) multiples of 4, and > (c) even numbers without prime divisors p==-1 (mod 4) in its square-free > part. It's pretty well-known that these n are the n which are sums of 1, 2, 4 or 8 odd squares (well the last two cases are the n divisible by 4:-)). This is a nice exercise in the equivalence of rational quadratic forms (Hilbert symbols et al). There is no loss in considering points with rational coefficients. In a regular simplex the n-1 edges issuing from one vertex are vectors v_1, ..., v_{n-1} with v_i . v_i = m and v_i . v_j = m/2 for i =/= j and m is fixed. So the problem is soluble for some m if the quadratic form m sum_{i <= j} x_i x_j is equivalent to x_1^2 + ... + x_{n-1}^2 over Q. Now apply the machinery of Hilbert symbols (e.g. as in Serre's Course of Arithmetic) and the answer will fall out.... -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html His mind has been corrupted by colours, sounds and shapes. The League of Gentlemen === Subject: : expectation maximization Hi! I have to understand what is expectation maximization but I have only some additional mathematics and no statistics background. Could anyone please kindly advise me on what should I read into before I can understand this topic?? Thank you very much! Ann === Subject: : Re: expectation maximization > Hi! I have to understand what is expectation maximization but I have only > some additional mathematics and no statistics background. Could anyone > please kindly advise me on what should I read into before I can understand > this topic?? I'm guessing you need to learn about what expected value is from statistics and you need to know how to minimize functions, say, from calculus. J === Subject: : Re: expectation maximization > Hi! I have to understand what is expectation maximization but I have only > some additional mathematics and no statistics background. Could anyone > please kindly advise me on what should I read into before I can understand > this topic?? Thank you very much! Ann There is a text on this, but it is probably more advanced than you are ready for. It is cleverly called Great Expectations. === Subject: : Re: Face facts, many Americans *like* idea of killing LW 2001 > Ha. Yeah, they did sort of tinker with the details a bit. Guess that's > how it has to be when you only have 128 pages or so to work with. > Ed Howdershelt - Abintra Press > Science Fiction and Semi-Fiction > http://abintrapress.tripod.com A little off topic from this particular thread, but when did Semi-Fiction officially become a recognized genre? Is it restricted to science specifically or can the mix include a combination of any or all genre's? Just curious. -- Rhiannon rhianon@sympatico.ca All those who believe in psychokinesis raise my hand. === Subject: : Re: Face facts, many Americans *like* idea of killing > Ha. Yeah, they did sort of tinker with the details a bit. Guess that's > how it has to be when you only have 128 pages or so to work with. > Ed Howdershelt - Abintra Press > Science Fiction and Semi-Fiction > A little off topic from this particular thread, but when did Semi-Fiction > officially become a recognized genre? Is it restricted to science > specifically or can the mix include a combination of any or all genre's? > Just curious. Genre? Recognized? By definition, perhaps. Don't know if a 'Semi-Fiction' genre exists. Don't really care, either. :) I use the term 'Semi-Fiction' to describe those of my works in which only names of people and certain places have been changed to avoid lawsuits. See titles Kim, Mindy, Anne, and Field Decision. Ed Howdershelt - Abintra Press Science Fiction and Semi-Fiction http://abintrapress.tripod.com === Subject: : Re: Face facts, many Americans *like* idea of killing >> Ha. Yeah, they did sort of tinker with the details a bit. Guess >> that's how it has to be when you only have 128 pages or so to work >> with. >> Ed Howdershelt - Abintra Press >> Science Fiction and Semi-Fiction >> A little off topic from this particular thread, but when did >> Semi-Fiction officially become a recognized genre? Is it restricted >> to science specifically or can the mix include a combination of any >> or all genre's? Just curious. Genre? Recognized? By definition, perhaps. > Don't know if a 'Semi-Fiction' genre exists. > Don't really care, either. :) > I use the term 'Semi-Fiction' to describe those of my works in which > only names of people and certain places have been changed to avoid > lawsuits. See titles Kim, Mindy, Anne, and Field Decision. > Ed Howdershelt - Abintra Press > Science Fiction and Semi-Fiction > http://abintrapress.tripod.com of it? It seems that changing the names of people and places isn't always enough. I was going to jokingly mention a couple of semi-celebs in my book but now I'm not sure if it's worth the risk. http://www.publishlawyer.com/carousel4.htm 2poor === Subject: : Re: Factor Analysis/Principle Components Analysis help > Hi Tim, > what exactly it is. Principle Components is pretty simple in my opinion. > You have p random variables, build your p x p covariance matrix, grab the eigenvalues > and eigenvectors and you have your principle components. > Yes. But there are different ways to calculate them, or - in the case > of singular matrices (after removal of errorvariance) - to get them > (or approximates or...) anyway. Factor analysis, you have an error term, so I believe you have > x_1 = q_11*y_1 + ... + q_1n*y_n + e_1 > ... > x_m = q_n1*y_1 + ... q_nn*y_n + e_n where the matrix Q = (q_ii) is the loading matrix. > Yes How is the Error matrix calculated exactly? Isn't it that Iteratively, > the diagonal error matrix > is guessed until _______ (I don't know what goes in the blank). > The estimation of error variances is - in a certain interval - > arbitrary, or: infinitely many estimations are possible. You have to > decide, which seems appropriate with your data (and your model) > Some prominent estimators are: > a- they are proportional to the remainder of a multiple regression > of all other variables to one selected variable: its non-explained > variance > b- they are all equal (for instance, repeated measuring with some > physical/electronic apparatus) > c- they are similar to, or: best estimated, when starting with the > maximum correlation for each variable and then processing > iterative calculations with refinement of theses estimations. > d- or: they are known/assumed from knowledge of other replications > of the same domain of survey. > Overestimation lead to implicite negative variance of the common > factors (or complex loadings), so computational problems are > common. > I do estimations via rotations, so systematically an overestimation > cannot occur, and a) and b) can be simply solved for > a) x*E or > b) x*I > (E and I being diagonal, x any parameter estimated to be maximal with > the restraint, that a solution does not contain negative variances.) > We then have the equality S = QQ^T + E, > where S is the observed covariance matrix, and Q is the loading matrix, and > Q^T is Q transposed, > and E is the error diagonal matrix. > Yes My colleague explained it to me like this : The error matrix is computed > iteratively, and then it is subtracted from the covariance matrix, and > then principle components is performed on the remaining matrix, which > in this case would be QQ^T. > Is this how factor analysis is performed? > Yes; but a very common procedere does > - guess error matrix > - do a principal components extraction from errorfree covariance (until > the n-th principal component has eigenvalue <=1 (or negative variance) > - see what is the communality now for each variable > - derive from that new error-estimation > - repeat this, until no more change (or an math-error ;-) ) occurs. > (nobody knows, i would say, in which relation now the errors are to the > measures other than that they satisfy this iteration...) > In this case, QQ^T would be real and symmetric, so it's eigenvectors would > be orthogonal. However, the column vectors in Q would not necessarily be > orthogonal. > This statement surprises me. I always assumed, they are? > The extraction method, following Hotelling, produces othogonal components > automatically, I think? > If principal components are taken as eigenvectors of this reduced covariance matrix, > (with its eigenvalue as scaling coefficient), and usually they are taken like this, > they should be orthogonal. Well, I don't think they are necessarily orthogonal. S = QQ^T + E , so when you subtract E, you get S-E = QQ^T, and S-E is real and symmetric. So QQ^T is real and symmetric, and when you perform principal components, the eigenvectors of QQ^T are orthogonal, but those eigenvectors are not the factors, are they? I thought the factors are the column vectors of Q, which would be the loadings. The column vectors of Q don't seem like they have to be orthogonal. In fact, in factor models I have produced recently, the factors were not orthogonal. What do you think? > HTH > Gottfried Helms > -- > ------------------------------------------------------------- > Gottfried Helms Soz.P.8ad./Soz.Arb. > FB04 // FG Prevention & Rehabilitation at University > D-34109 Kassel Moenchebergstr. 19 B > -------------------------------------------------------------- -- > email: mailto:helms@hrz.uni-kassel.de > www: http://www.uni-kassel.de/~helms > ============================================================== == === Subject: : Re: Factor Analysis/Principle Components Analysis help X-ID: TJLJcrZZ8eo0do3NugdjRuE9TInoXQs8g9SmW4k4pFZ6gD+evvoD60 Well, I don't think they are necessarily orthogonal. > S = QQ^T + E , > so when you subtract E, you get S-E = QQ^T, and S-E is real and symmetric. > So QQ^T is real and symmetric, and when you perform principal components, > the eigenvectors of QQ^T are orthogonal, but those eigenvectors are not the > factors, are they? > I thought the factors are the column vectors of Q, which would be the > loadings. The column vectors of Q don't seem like they have to be orthogonal. > In fact, in factor models I have produced recently, the factors were not orthogonal. > What do you think? > A correlation matrix: r = { _ { 1.00, -0.03, 0.00, 0.04, 0.77, 0.76}, _ { -0.03, 1.00, 0.68, 0.36, 0.51, 0.59}, _ { 0.00, 0.68, 1.00, 0.31, 0.43, 0.33}, _ { 0.04, 0.36, 0.31, 1.00, 0.07, 0.13}, _ { 0.77, 0.51, 0.43, 0.07, 1.00, 0.93}, _ { 0.76, 0.59, 0.33, 0.13, 0.93, 1.00} _ } The factorloadings-Matrix, common-factors at the left in pc-position, error-loadings at the right, factors to be read vertical, variables horizontal. CE =_ { { 0.67, -0.67, 0.25, -0.15, -0.08, 0.09, 0.00, 0.00, 0.00, 0.00, 0.00}, _ { 0.70, 0.60, -0.19, 0.30, -0.03, 0.00, 0.09, 0.00, 0.00, 0.00, 0.00}, _ { 0.58, 0.60, -0.19, -0.43, -0.04, 0.00, 0.00, 0.27, 0.00, 0.00, 0.00}, _ { 0.29, 0.49, 0.74, 0.02, 0.03, 0.00, 0.00, 0.00, 0.35, 0.00, 0.00}, _ { 0.94, -0.25, -0.11, -0.05, 0.15, 0.00, 0.00, 0.00, 0.00, 0.15, 0.00}, _ { 0.95, -0.24, -0.04, 0.19, -0.06, 0.00, 0.00, 0.00, 0.00, 0.00, 0.07} } Test of orthogonality of the columns by multiplication. The left upper part (crossproduct of the first 5 factors) is a diagonal matrix, that means, the factor-loadings are pairwise orthogonal. L= CE'*CE L = _ { { 3.14, -0.00, 0.00, -0.00, 0.00, 0.06, 0.07, 0.16, 0.10, 0.14, 0.06}, _ { -0.00, 1.55, -0.00, 0.00, 0.00, -0.06, 0.06, 0.16, 0.17, -0.04, -0.02}, _ { 0.00, -0.00, 0.70, -0.00, 0.00, 0.02, -0.02, -0.05, 0.26, -0.02, -0.00}, _ { -0.00, 0.00, -0.00, 0.34, -0.00, -0.01, 0.03, -0.12, 0.01, -0.01, 0.01}, _ { 0.00, 0.00, 0.00, -0.00, 0.04, -0.01, -0.00, -0.01, 0.01, 0.02, -0.00}, _ { 0.06, -0.06, 0.02, -0.01, -0.01, 0.01, 0.00, 0.00, 0.00, 0.00, 0.00}, _ { 0.07, 0.06, -0.02, 0.03, -0.00, 0.00, 0.01, 0.00, 0.00, 0.00, 0.00}, _ { 0.16, 0.16, -0.05, -0.12, -0.01, 0.00, 0.00, 0.07, 0.00, 0.00, 0.00}, _ { 0.10, 0.17, 0.26, 0.01, 0.01, 0.00, 0.00, 0.00, 0.13, 0.00, 0.00}, _ { 0.14, -0.04, -0.02, -0.01, 0.02, 0.00, 0.00, 0.00, 0.00, 0.02, 0.00}, _ { 0.06, -0.02, -0.00, 0.01, -0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00} } R = CE*CE' ( correlation matrix) Rc = C*C' = R - E*E' ( correlation matrix with errors removed) The principal components of the reduced correlation-matrix Rc, are its rescaled eigenvectors. May be we have a misunderstanding? Please show one of your examples. Gottfried Helms r = { _ { 1.00, -0.03, 0.00, 0.04, 0.77, 0.76}, _ { -0.03, 1.00, 0.68, 0.36, 0.51, 0.59}, _ { 0.00, 0.68, 1.00, 0.31, 0.43, 0.33}, _ { 0.04, 0.36, 0.31, 1.00, 0.07, 0.13}, _ { 0.77, 0.51, 0.43, 0.07, 1.00, 0.93}, _ { 0.76, 0.59, 0.33, 0.13, 0.93, 1.00} _ } e = { _ { 0.09, 0.00, 0.00, 0.00, 0.00, 0.00}, _ { 0.00, 0.09, 0.00, 0.00, 0.00, 0.00}, _ { 0.00, 0.00, 0.27, 0.00, 0.00, 0.00}, _ { 0.00, 0.00, 0.00, 0.35, 0.00, 0.00}, _ { 0.00, 0.00, 0.00, 0.00, 0.15, 0.00}, _ { 0.00, 0.00, 0.00, 0.00, 0.00, 0.07} _ } rc = { _ { 0.99, -0.03, 0.00, 0.04, 0.77, 0.76}, _ { -0.03, 0.99, 0.68, 0.36, 0.51, 0.59}, _ { 0.00, 0.68, 0.93, 0.31, 0.43, 0.33}, _ { 0.04, 0.36, 0.31, 0.87, 0.07, 0.13}, _ { 0.77, 0.51, 0.43, 0.07, 0.98, 0.93}, _ { 0.76, 0.59, 0.33, 0.13, 0.93, 1.00} _ } === Subject: : Re: Factor Analysis/Principle Components Analysis help === Subject: : Re: Factor Analysis/Principle Components Analysis help Message-id: > Well, I am in no position to elaborate on PCA in factor analysis. I use PCA to reduce dimentionality of my input file to make stock predictions. *It does not work*, since my NNs extract all predictive information from the data anyway without prior reduction of the information burden with PCA. The reason might be that linear techniques are not good enough to model a complex nonlinear system. Another reason could be that combinations of inputs make it impossible for the networks to differentiate between strong and weak inputs. So, the name is cool, but it does not work. Stan === Subject: : Re: Factorial/Exponential Identity, Infinity > The above is riddled with error. ... What are implications of using an infinity as a number in the same > equations as finite numbers and infinitesimals? This thread started because I think that half of the sequences have > equal numbers of ones and zeros. Yet, I am unable to prove that. It's not easy giving meaning to half of the sequences. Virgil offers one interpretation and proves that under that interpretation, almost all sequences have equal numbers of ones and zeros (the fraction that don't is zero). Here's another interpretation that perhaps has already occurred in this thread: In even-length random binary sequences of length 2N, what is the asymptotic probability as N->oo that the number of 0's equals the number of 1's? Note that I am asking a question about large finite N, not infinite N. In your roundabout way you show that you're aware of the binomial distribution. This probability is quite easy to calculate. It's (2N)!/[2^(2N)*(N!)^2] > What was shown, and it was known, or at least a variation of it was an > exercise in a book, The distinction between what was shown and a variation of what was shown is important. I have no idea whether or not you have taken a valid result and made a variation of it that is no longer valid. I suspect that to be the case since your statement as written is nonsense. lim n->oo n! sqrt(n) / ( (n/2)!^2 2^n) = sqrt(2 / (Pi n )) > The limit as n->oo can't have a dependence on n in it. However, the large-n asymptotic behavior might. What was the original result that you varied? Using the Stirling approximation (I *strongly* suspect I'm repeating ground that was already covered in this thread) I get (2N)!/(N!)^2 = (approx)(1/sqrt(2*pi)) * 2^(2N+1)/sqrt(2N) So (2N)!/[2^(2N)*(N!)^2] is asymptotically approximated by sqrt(2/N)/sqrt(2*pi) = 1/sqrt(pi*N) delete the factor of sqrt(n) from the numerator on the left-hand side. I guess that's your variation. Now, this probability decreases as 1/sqrt(N). That means that the fraction becomes smaller and smaller as N increases. For N = 10^6, less than 1/3000 of the sequences have equal ones and zeros. For N = 10^12, less than 1 in 3 million have equal numbers of ones and zeros. Where do you think this trend goes as N increases without bound? Does it really surprise you that you're having trouble proving that the limit of an expression proportional to 1/sqrt(N) is 1/2? - Randy === Subject: : Re: Factorial/Exponential Identity, Infinity So what. One thing about this discussion is that where the binomial choice function is normally (n k), here we have been discussing (n n/2). The point to observe there is that instead of two variables it is considered with one variable n and an expression of n as the second variable, and the expression is considered for n in the limit. Yet, in Euler's formula for Gamma, replacing z with an expression of n yields various results depending on the expression of z in terms of n. It's valid for some expressions. Why isn't it for all? When I put a variable on both sides of a limiting equation: lim n->oo f(n) = g(n) that means lim n->oo f(n) = lim n->oo g(n) and lim n->oo f(n) / g(n) = 1 Is it a countable number of ones and zeros? I don't care. It doesn't matter. It's irrelevant. It's also useless and counterproductive. If there are half ones and half zeros in a sequence, then removing a pair of one one and one zero, repeated ad infinitum, would never change that set from having half ones and half zeros. I realize that any infinite sequence with infinite ones and infinite zeros could have a pair of each removed any number of times. An infinite sequence of the form .0101(01)... has half ones and half zeros. You can interchange any two sequence elements as many times as you want, it still has half ones and half zeros, and would never be the sequence .001001(001)... or .011011(011)..., for 1/3 and 2/3, yet it can easily be .00000000000000001111111111111111(01)..., for 1/2. The same holds true for other fractions greater than zero and less than one, there is a canonical sequence describing it and it is a different sequence than for any other fraction. What about those sequences with one or n more more or less than 1/2? They differ from the canonical sequence after canonicalization at finitely n many points. If it differs by more than finitely many points then it's a finite difference of a different canonical sequence. Why is it not possible to switch some elements of a canonical sequence and get a different canonical sequence? Here I've used the terms canonical sequence and canonicalization. What are they? What is their meaning and context? There's a quantity that defines how many of the infinite length sequences have equal numbers of ones and zeros. It's infinitesimal. When summed with the quantities of there being all the other expressions of n between 0n and 1n many zeros, and their finite differences, that is equal to one. Ross Finlayson === Subject: : Re: Factorial/Exponential Identity, Infinity > When I put a variable on both sides of a limiting equation: lim n->oo f(n) = g(n) Ross is breaking one of the customs, if not rules, of logic in using the same symbol, n, as both a bound variable and a free variable in the same statement. The custom has the desirable goal of avoiding ambiguity, which Ross seems rather to relish than avoid. As this equation has no free variable on its left hand side but does have one on its right hand side, the only sensible interpretation is that the right hand side function, denoted by g(n), must be a constant. If this is the case, and the constant is denoted by c, and the constant is not zero, then one trivially gets what Ross seems to want, namely if lim n->oo f(n) = c and c <> 0, then lim n->oo f(n)/c = 1 that means lim n->oo f(n) = lim n->oo g(n) and lim n->oo f(n) / g(n) = 1 Ross' assumption that if lim n->oo f(n) = lim n->oo g(n) then lim n->oo f(n) / g(n) = 1 is flawed without the additional assumption that neither lim n->oo f(n) nor lim n->oo g(n) is zero. Consider f(n) = 1/n^2 and g(n) = 1/n. Then lim n->oo f(n) = lim n->oo g(n) = 0 but lim f(n)/g(n) = 0, not 1 and g(n)/f(n) diverges. === Subject: : Re: Factorial/Exponential Identity, Infinity I guess canonicalization is the process of converting in place a sequence to a sequence with repeating terminating sequences and some finite beginning sequence. Can any infinite binary sequence be converted to a sequence with a repeating terminating sequence and a finite beginning sequence? There are the sequences similar to Champernowne's that can not, perhaps. .01001000100001000001... These sequences have infinitesimally many ones compared to zeros, but infinitely many ones. Thus canonicalization might result in a sequence similar to those, or to the repeating sequence. There is a function of n to approximate the density of primes in [1,n], it is similar to that. The above sequence has x many ones and sum x many zeros for a sequence length of x + sum x. The function sum x for x={1, 2, ...} is equal to (x+1)x/2. Thus the number of ones compared to sequence elements is x / (x+(x+1)(x/2)). Thus for n = (x + (x+1)(x/2)), the number of ones is x. Solving for x in terms of n: n = (x + (x+1)(x/2)) n = (x+ x^2/2 + x/2) n = (x^2/2 + 3/2 x) 2n = x^2 + 3x 2n+ (3/2)^2 = (x+3/2)^2 (2n + 9/4)^(1/2) - 3/2 = x The idea here is to say that (n x) = (n f(n)), so instead of (n n/2) / 2^n it's along the lines of (n ((2n + 9/4)^(1/2) - 3/2)) ). n! / ( Gamma(2n + 9/4)^(1/2) - 1/2) Gamma( n - ((2n + 9/4)^(1/2) - 1/2) ) 2^n ) Then, besides the sequences canonicalizable to .01001000100001... With a zero then a one then two zeros then a one then three zeros then a one then four zeros then a one, etcetera, there would be sequences with various numbers of zeros and various number of ones, where in the limit there are infinitely many more zeros than ones, with the density of the infinitely many ones being infinitesimal. For example instead of the number of contiguous zeros incrementing, it might increment by two or double. n = x + (x+1)x n = x + sum 2^x These would be sequences representing irrational numbers, and they could not be canonicalized into a rational sequence. That is different than sequences with about equal or fractional numbers of ones and zeros which are canonicalizable into sequences with repeating sequences. These are sequences with infinitely many ones, and relatively infinitely many more zeros, and vice versa. a) sequences with finite ones or zeros b) sequences with infinitely many ones and relatively infinitely many more or less zeros c) sequences with infinitely many ones and relatively finitely many more or less zeros It is not the simple case that all the irrationals are of type b, some are of type c, but all the sequences of type b are irrational. The canonical sequences of type c are rational. It's easy enough to characterize the canonical sequences with repeating terminal sequences, particularly the cases with terminating 1... or 0.... Those are basically the rationals, with the sequences terminating with 1 or 0 having finitely many zeros or ones, respectively. Type a might be considered a subset of type c, as the finite number of ones or zeros goes to an expression of n, one is 1/n, two is 2/n, etcetera, each is less than n/x for finite x, e.g n/2, n/3, n/4, 2n/5, n/5, n/6, etc., 1/n of n is 0n +1. What I'm interested in determining now is how many of the sequences of type c are irrational, and how many sequences there are of type b. There are 2^n possible sequences, I want to see if there are 2^(n-1) each of rational and irrational sequences. The problem there is determining how many sequences of type c are irrational or rational, besides determining how many sequences of type b and c there are. It is apparent that there are more than n many rational sequences: there are n many sequences with a one and infinitely many zeros, each is rational. I guess I can start with considering the canonical sequence with half ones and half zeros. .101010(10)... It's representative of a rational. So are .000111(10)... .001011(10)... .010011(10)... .100011(10)... .100101(10)... .101001(10)... .110001(10)... .110010(10)... .110100(10)... .111000(10)... .001101(10)... .011001(10)... .011010(10)... .011100(10)... .001110(10)... .010110(10)... .100110(10)... .101100(10)... .010101(10)... .101010(10)... Each there still has equal numbers of ones and zeros, they canonicalize to .(10).... For the beginning sequence of x many elements there are (x x/2) or in the above example for x=6 there are 6! / (3! 3!) = 6*5*4/3*2*1 = 20. That accounts for about x of the n many elements of the sequences, the terminating n-x many elements have obviously (n-x (n-x)/2) -1 permutations besides (10).... We can just consider sequences in terms of how many ones there are, from 1 to n many ones. There are sequences with n/2, n/3, 2n/3, n/4, 3n/4, etcetera, +-x, many ones, those are all the sequences canonicalizable into sequences with repeating endings. The sequences can be rational or irrational, their canonical form is rational. I think it is safe to say that there are relatively infinitely many more sequences of type c than type b. Yet, it is much easier to specify the relative quantities of zeros and ones for all sequences of type c than type b. In a way, that's about comparing the density in the naturals of integer multiples, parallel to type c, to squares, cubes, primes, sums, factorials, etcetera, infinite sets with infinitesimal density, type b, except sequences of type b are exclusive from sequences of type c. MathWorld notes that abundant numbers have a finite, positive density in the naturals. Then, there's the problem of separating the sequences of type c into rationals and irrationals. For the sequences S_(1/2) that canonicalize to a sequence with n/2 of n many ones, how many are irrational and thus the rest rational? We have an expression for how many sequences there are with n/2 many ones, in terms of n. What would answer that question then is how many of those are rational. The problem I confront here in that consideration is that the sequence is of infinite length. One way to consider it is to determine how many possible repeating sequences there could be, in terms of having some amount left over to be the finite beginning sequence with other than the repeating sequence yet still with half ones. I'll start with considering repeating sequences of length l 2, 4, 6, etcetera, the repeating sequence has to have a length of a multiple of 2 just as for the case of S_(1/3) it woud have to be a multiple of three. Let's say it is two: (01)... (10)... Each of those is one of the cases for l=4, 6, 8, ..., (0101)... (010101)... (01010101)... Where l=2, I will assume then that the n many elements of the sequence are divided into n/2 many subsequences. Then, of those subsequences, another variable's as many of those, a finite number, are subsequences for the beginning sequence, and the remaining subsquences each have the repeating sequence. The length of the repeating subsequence can be any multiple of two, although it should be arbitrarily less than n. Again, the idea here is to determine how many sequences end with repeating subsequences and are thus rational. How is it gone about quantifying the number of rationals compared to the number of irrationals among all the sequences with half ones and half zeros? I think it helps that I think 2^(n-1) of the sequences are rational and the other 2^(n-1) are irrational. Determining how many sequences of type c are rational would actually show that, or not. That leads me back to thinking about how the rationals and irrationals are each dense in the reals and complementary where their union is the reals. Here's a non sequitur, for integer n, n! +-1 is prime. Go back to the posts about functions for the recurrence relations. Ross === Subject: : Re: Field of rationals and pi > Euler found out that he could define pi^2 in the following nifty way: pi^2/6 = 1 + 1/4 + 1/9 + 1/16 +... Which means you can define it using members of the *field* of > rationals. However, pi is transcendant and is itself not a rational. Everything's good so far, as long as you realize that you are talking about a limit, and not an actual sum. > I'm curious about the rule mathematicians use to exclude pi^2/6 from > the field of rationals, as it itself is the result of an infinite sum > of members of that field. Is that it? Mathematicians simply exclude infinite sums from the > field of rationals? Or do they rely on the definition of a rational > as the ratio of a/b, where 'a' and 'b' are integers? It fails the definition of a/b, a and b integers, b <> 0. > Continuing in that direction, recently a leading mathematician at a > major university in the United States of America (a top 20 > university) sent me an email stating that my rule of no other integers > being units except -1 and 1 did not exclude pi if you used Z[pi]. I said it did in the following reply (Professor's name omitted): Professor ****: You assertion is easily proven false. Please consider the following. infinity. Notice that you are asserting a limit, not an infinite sum. But then you have pi^2/6 = 1 + 1/4(1 + 1/4 + 1/9 +...1/k^2) + 1/9(1+1/4 + > 1/9+...)+...1/k^2(1+k^2), which is The original expression has 1/36 one time. This has it twice. 1/4( ... +1/9+...) and 1/9(...+1/4+...). I don't see where this statement came from. pi^2/6 = 1 + pi^2/24 + pi^2/54 +...pi^2/6k^2, multiplying out and collecting to the left except for 1, 6(24)(54)...(6k^2) pi^2 [24(54)...(6k^2) - 6(54)...(6k^2) - > 6(24)...(6k^2) - ... -6(24)(54)...] = 1, which proves that you have an infinite number of units, some of which > are 6, 24, and 54, which is the result if you include pi in a ring > with integers, so my definition *does* exclude it. You cannot perform an infinite sum in a ring. Well he replied: Actually, Z[pi] has no units save 1, -1, as a consequence of > the fact that pi is transcenddental (not algebraic). Z[pi] does not > contain the numbers you are considering above, which (as I read it) > are obtained by summing infinite series. The elements of > Z[pi] are just those real numbers that can be expressed as > f(pi) where f(x) is a FINITE polynomial with integer coefficients. Do you agree with the professor, who I remind is a *leading* > mathematician? I agree with him even if he's a janitor. -- Will Twentyman email: wtwentyman at copper dot net === Subject: : Filtering images Hi all ! I've got to filter a sequence of images t extract th derivatives of the video. I can't understand wow I can construct such a filter given its ta ps (eg: [-1 -2 0 2 1]). Shall I consruct a 3D 5x5x5 matrix by replicatin g the kernel towards the two other dimensions, or put the kernel in the center of the matrix and complete with zeros, or something else ?? I couldn't found any answer on the web, whereas I found many explanation s on how to design a kernel :) Thanks a lot for answering me on my email laconicos@yahoo.fr :) Nicolas. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- Article poste via Voila News - http://www.news.voila.fr === Subject: : finding a function Hi again :) The problem I have this time is not immediately related to school but a problem I have of my own ... given that: h=7c; 2h=18c; and 3h=35c; 20c=a; 10a=l; and 3l=s is it possible to express 's' as a function of 'h' and if so what is the function and how did you arrive at that please ? ie: (i) if I want 's' to equal 700, what value 'h' do i require and why ? (ii) can I change the equalities of h ? e.g. if I say that h=9c (but nothing else changes), how will this change the function ? The reason I ask for how the function is derived is so that I can change any of the other factors and re-calculate without bothering you all again ... i.e. I want to know how to calculate the new function if I change, for example, 20c=a to 15c=a and so on I hope someone can assist me and I thank, in anticipation, all respondents regards, ivan. p.s. is this calculus or just plain algebra ? p.p.s. this is (hopefully) a way of gauging sales (s) against hours (h) used in doing the job (how many hours do I need to invest in getting a given return) === Subject: : Re: finding a function > given that: h=7c; 2h=18c; and 3h=35c; > 20c=a; > 10a=l; and > 3l=s is it possible to express 's' as a function of 'h' and if so what is the > function and how did you arrive at that please ? Well, if h=7c then 2h CAN NOT also equal 18c nor can 3h be 35c, unless both h and c are 0. If you had only one relation involving h and c, then such a task is possible. You want s so start with s and keep substituting the variable you have until you get an expression in h. s = 3l = 3(10a) = 30a = 30(20c) = 600c = ... ? > (i) if I want 's' to equal 700, what value 'h' do i require and why ? Sounds like you want h as a function of s. > (ii) can I change the equalities of h ? e.g. if I say that h=9c (but > nothing else changes), how will this change the function ? You have s = 600c, so however c related to h, just plug that in to s=600c. > The reason I ask for how the function is derived is so that I can change > any of the other factors and re-calculate without bothering you all again > ... i.e. I want to know how to calculate the new function if I change, for > example, 20c=a to 15c=a and so on Okay, so if you have c = K*h (for any coefficient you want, K) then s=600c becomes s=600K*h, so just stick that coefficient K in there (i.e. multiply it by 600) > I hope someone can assist me and I thank, in anticipation, all > respondents I expect a lot of people will repond. > p.s. is this calculus or just plain algebra ? Just plain algebra... about grade 8 or 9 level. J === Subject: : Finite element analysis of prismatic axisymmetrical shells I'm looking for an inexpensive (or free) Finite Element or Finite Strip Analysis software package for prismatic axisymmetrical shell analysis with Civil Engineering orientation. As a matter of fact, I need to design and calculate a water containment tank made of rectangular flat-wall prestressed concrete panels. If panels were curved to the tank radius, then the elastic analysis would be that of an ordinary cylindrical shell and the problem is over. However, if panels are flat -as is my current concern- the tank cross section is a polygonal, with as many sides or faces as the amount of panels used. Of course, the more panels the tank has, the more to a cylindrical shell it approximates. Now, for this prismatic axisymmetrical shell I need to perform all the elastic analysis (stresses and strains) together with inelastic time-dependent effects such as concrete creep, shrinkage and prestress losses of prestressed steel. However, it is not a mandatory feature for the software to have inelastic analysis capabilities. I performed an extensive search, but couldn't find any software designed for this specific task. Instead, I found a lot of expensive, heavy-weight, all-purpose and ultra-sophisticated Finite Element packages, but none of them fits with my current needs. I will highly appreciate if someone points me to a software package capable of assisting me in the design and analysis process explained above. Thank you, Fernando Ronci E-mail: fernandoronci@hotmail.com === Subject: : Re: Finite element analysis of prismatic axisymmetrical shells I'm looking for an inexpensive (or free) Finite Element or Finite > Strip Analysis software package for prismatic axisymmetrical shell > analysis with Civil Engineering orientation. As a matter of fact, I > need to design and calculate a water containment tank made of > rectangular flat-wall prestressed concrete panels. > If panels were curved to the tank radius, then the elastic analysis > would be that of an ordinary cylindrical shell and the problem is > over. > However, if panels are flat -as is my current concern- the tank cross > section is a polygonal, with as many sides or faces as the amount of > panels used. Of course, the more panels the tank has, the more to a > cylindrical shell it approximates. > Now, for this prismatic axisymmetrical shell I need to perform all the > elastic analysis (stresses and strains) together with inelastic > time-dependent effects such as concrete creep, shrinkage and prestress > losses of prestressed steel. > However, it is not a mandatory feature for the software to have > inelastic analysis capabilities. I performed an extensive search, but couldn't find any software > designed for this specific task. Instead, I found a lot of expensive, > heavy-weight, all-purpose and ultra-sophisticated Finite Element > packages, but none of them fits with my current needs. > I will highly appreciate if someone points me to a software package > capable of assisting me in the design and analysis process explained > above. Thank you, Fernando Ronci > E-mail: fernandoronci@hotmail.com Please post in sci.engr.civil and sci.engr.analysis. The polygonal tank is polar symmetric.( Not axisymmetric which comes by rotation about an axis). It is supposed water is contained in the polygonal prismatic/extruded column, which serves as containment and not as a support structure of an overhead wineglass/spherical tank. By having a cylindrical section instead of a polygonal, bending stresses and hence weight and cost of the concrete tank can be reduced. === Subject: : Re: Finite element analysis of prismatic axisymmetrical shells I'm looking for an inexpensive (or free) Finite Element or Finite > Strip Analysis software package for prismatic axisymmetrical shell > analysis with Civil Engineering orientation. As a matter of fact, I > need to design and calculate a water containment tank made of > rectangular flat-wall prestressed concrete panels. > If panels were curved to the tank radius, then the elastic analysis > would be that of an ordinary cylindrical shell and the problem is > over. > However, if panels are flat -as is my current concern- the tank cross > section is a polygonal, with as many sides or faces as the amount of > panels used. Of course, the more panels the tank has, the more to a > cylindrical shell it approximates. > Now, for this prismatic axisymmetrical shell I need to perform all the > elastic analysis (stresses and strains) together with inelastic > time-dependent effects such as concrete creep, shrinkage and prestress > losses of prestressed steel. > However, it is not a mandatory feature for the software to have > inelastic analysis capabilities. I performed an extensive search, but couldn't find any software > designed for this specific task. Instead, I found a lot of expensive, > heavy-weight, all-purpose and ultra-sophisticated Finite Element > packages, but none of them fits with my current needs. > I will highly appreciate if someone points me to a software package > capable of assisting me in the design and analysis process explained > above. Thank you, Fernando Ronci > E-mail: fernandoronci@hotmail.com Please post in sci.engr.civil and sci.engr.analysis. > The polygonal tank is polar symmetric.( Not axisymmetric which comes > by rotation about an axis). I searched in virtually every online bookstore (Amazon, Barnes and Noble and others) and found no single book dealing with analysis of polygonal shells. Instead, books on cylindrical axisymmetric shells and folded plates abound. > It is supposed water is contained in the > polygonal prismatic/extruded column, which serves as containment and > not as a support structure of an overhead wineglass/spherical tank. By > having a cylindrical section instead of a polygonal, bending stresses > and hence weight and cost of the concrete tank can be reduced. Yes. However, if panels were flat they could be mass-produced regardless of tank size and capacity. On the other hand, if walls were curved to the tank radius, a different wall should be cast for every tank capacity, becoming an expensive and complicated process. No matter polygonal tanks are not as convenient (in economic terms) as cylindrical ones, I think they're worth a try. Thank you. Fernando Ronci E-mail: fernandoronci@hotmail.com === Subject: : Re: Finite element analysis of prismatic axisymmetrical shells > having a cylindrical section instead of a polygonal, bending stresses and hence weight and cost of the concrete tank can be reduced. > Yes. However, if panels were flat they could be mass-produced regardless of tank size and capacity. ... No matter polygonal tanks are not as convenient (in economic terms) as cylindrical ones, I think they're worth a try. Even in floor slab designs, there is pre-shaping in funicular constructions, which reduces bending, and it is not too difficult/expensive to implement.Aim should be to make design as in-plane or membrane as possible. Choosing more faces of polygon, bending can be reduced. Please try above groups also. === Subject: : Floor(some primes *a real) divisible by n First, a specific case, then something more general: Let q = any odd integer. Then maybe, for all integers m >= some positive integer M (M is a function of q), | m | m-1 | | 1 ---- | q | |---------- | | q-1 | | | zeta(q-1) | | p(k) | |__ k=1 __| where p(k) is the k_th prime, and zeta() is the Riemann zeta function. Since the above ascii-art will most likely not appear correctly to most sci.math readers, here is the linear-mode: q^(m-1) divides floor(1/(zeta(q-1)) product{k=1 to m} p(k)^(q-1) ) So, for example, I guess, 3^(m-1) divides floor((6/pi^2) product{k=1 to m} p(k)^2 ) for all m's >= M. I THINK this is true for all positive integer m's. - And generally, Let n be any odd positive integer. Let a(p), a nonnegative integer, be such that p^a(p) is highest power of p (a prime) which divides n. Let A = sum{j=2 to oo} (p(j) -1) a(p(j)), (which is a finite sum, despite the infinity (oo) in the sum's limit). Let Z = product{j=2 to oo} zeta(p(j)-1)^a(p(j)), (a finite product). Then maybe, for all integers m >= some positive integer M (M is a function of n), | m | m-1 | | 1 ---- | n | |----- | | A | | | Z | | p(k) | |__ k=1 __| In linear mode: n^(m-1) divides floor((1/Z) product{k=1 to m} p(k)^A ) So, for example, I guess, 45^(m-1) divides floor((3240/pi^8) product{k=1 to m} p(k)^8 ) for all m's >= some M. I got these theorems, if even true, easily from the Riemann zeta product-representation and Fermat's little theorem. But I used hardly any rigor, especially in the more general theorem. If anyone cares to prove/disprove the above, I do not believe this would be too hard. Thanks, Leroy Quet === Subject: : Floor(some primes *a real) divisible by n(corrected) This is a repost, since q should be prime. (How did I miss this in proof-reading???) ---- First, a specific case, then something more general: Let q = any odd PRIME. ^^^^^ Then maybe, for all integers m >= some positive integer M (M is a function of q), | m | m-1 | | 1 ---- | q | |---------- | | q-1 | | | zeta(q-1) | | p(k) | |__ k=1 __| where p(k) is the k_th prime, and zeta() is the Riemann zeta function. Since the above ascii-art will most likely not appear correctly to most sci.math readers, here is the linear-mode: q^(m-1) divides floor(1/(zeta(q-1)) product{k=1 to m} p(k)^(q-1) ) So, for example, I guess, 3^(m-1) divides floor((6/pi^2) product{k=1 to m} p(k)^2 ) for all m's >= M. I THINK this is true for all positive integer m's. - And generally, Let n be any odd positive integer. Let a(p), a nonnegative integer, be such that p^a(p) is highest power of p (a prime) which divides n. Let A = sum{j=2 to oo} (p(j) -1) a(p(j)), (which is a finite sum, despite the infinity (oo) in the sum's limit). Let Z = product{j=2 to oo} zeta(p(j)-1)^a(p(j)), (a finite product). Then maybe, for all integers m >= some positive integer M (M is a function of n), | m | m-1 | | 1 ---- | n | |----- | | A | | | Z | | p(k) | |__ k=1 __| In linear mode: n^(m-1) divides floor((1/Z) product{k=1 to m} p(k)^A ) So, for example, I guess, 45^(m-1) divides floor((3240/pi^8) product{k=1 to m} p(k)^8 ) for all m's >= some M. I got these theorems, if even true, easily from the Riemann zeta product-representation and Fermat's little theorem. But I used hardly any rigor, especially in the more general theorem. If anyone cares to prove/disprove the above, I do not believe this would be too hard. Thanks, Leroy Quet === Subject: : Re: Floor(some primes *a real) divisible by n(corrected) > This is a repost, since q should be prime. > (How did I miss this in proof-reading???) ---- > First, a specific case, then something more general: Let q = any odd PRIME. > ^^^^^ > Then maybe, for all integers m >= some positive integer M (M is a > function of q), > | m | > m-1 | | 1 ---- | > q | |---------- | | q-1 | > | | zeta(q-1) | | p(k) | > |__ k=1 __| > where p(k) is the k_th prime, and zeta() is the Riemann zeta function. > Since the above ascii-art will most likely not appear correctly to most > sci.math readers, here is the linear-mode: q^(m-1) divides floor(1/(zeta(q-1)) product{k=1 to m} p(k)^(q-1) ) So, for example, I guess, 3^(m-1) divides floor((6/pi^2) product{k=1 to m} p(k)^2 ) for all m's >= M. I THINK this is true for all positive integer m's. - And generally, Let n be any odd positive integer. Let a(p), a nonnegative integer, be such that p^a(p) is highest power of > p (a prime) which divides n. Let A = sum{j=2 to oo} (p(j) -1) a(p(j)), (which is a finite sum, despite the infinity (oo) in the sum's limit). > Let Z = product{j=2 to oo} zeta(p(j)-1)^a(p(j)), (a finite product). > Then maybe, for all integers m >= some positive integer M (M is a > function of n), > | m | > m-1 | | 1 ---- | > n | |----- | | A | > | | Z | | p(k) | > |__ k=1 __| > The above is erroneous, unless correct by complete accident. The result SHOULD be: | m | m-B | | 1 ---- | n | |----- | | A | | | Z | | p(k) | |__ k=1 __| where B = sum{k=1 to oo} a(p(k)). (I think I read that B is referred to as omega(n).) In linear mode, the corrected: n^(m-B) divides floor((1/Z) product{k=1 to m} p(k)^A ) So, for example, I guess, 45^(m-3) divides floor((3240/pi^8) product{k=1 to m} p(k)^8 ) for all m's >= some M. > I got these theorems, if even true, easily from the Riemann zeta > product-representation and Fermat's little theorem. But I used hardly any rigor, especially in the more general theorem. If anyone cares to prove/disprove the above, I do not believe this would > be too hard. > Thanks, Leroy Quet === Subject: : Re: Focus on point of dispute, more math everyone has their areas wherein this applies, known as gambler's fallacy: never recalling one's losses. we already have a Compendium of Harris' Pithy Sayings on Mathematics and Universe, but do we have one of your Acknowledged Errors in The [On-going] Proof? no question, math is a *socialized* thing; So?... your neoligism of Uberpolyies seems to be nothing more, than what might be done with partial differential equations, not that it isn't a potentially clever device, but how could it possibly change the math having spoken?... just take the social initiaitve & ask someone -- maybe, using a different user ID, though! > Given a mathematical argument there are typically points not in > dispute, so you can delete those out until you reach the *last* point > you don't disagree with, which you should leave in for context, and > then you consider the *next* point you do disagree with, and state why > you believe it doesn't follow logically. > Remember, a math proof begins with a truth and proceeds by logical > steps to a conclusion which then must be true. > And yes readers, you *can* stick in numbers to help yourself > understand, and then you may also have the dubious pleasure of > catching posters, who have advanced math training, like from Berkeley > University's Ph.D program, lying to you. What I use is the non-polynomial factorization P(m) = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf), which in this case is P(m) = (a_1 + 7)(a_2 + 7)(a_3 + 7) = 49(2401 m^3 - 147 m^2 -144 m + 10) with a special expression I call an uber-polynomial, which again is > The reality is that these methods are something beyond what > mathematicians have used before, and with their own techniques, they > can't say much about how 49 divides off of P(m), and they can't > constrain it as a variable dependent on m. http://buckminster.info/Ideas/03-TetGeomTheorem4-ColorProof.htm --UN HYDROGEN (sic; Methanex (TM) reformanteurs) ECONOMIE?... La Troi Phases d'Exploitation de la Protocols des Grises de Kyoto: (FOSSILISATION [McCainanites?] (TM/sic))/ BORE/GUSH/NADIR @ http://www.tarpley.net/aobook.htm. Http://www.tarpley.net/bushb.htm (content partiale, below): 17 -- L'ATTEMPTER de COUP D'ETAT, 3/30/81 === Subject: : Re: Focus on point of dispute, more math consisting wholly of would-be helpers & critics, and people who otherwise have no apparent life, as your self; eh?... mea culpa, dood! > My audience is a very particular one. > * 1. The Galois theory argument has been presented before. That the polynomial in [3], -m*k*f^2 + 3*v*y^2 + y^3 is irreducible in general is easily seen by letting m = 1 > and f = 5, where it becomes y^3 + 72*y^2 - 13825, which is easily shown to be irreducible. *2. This argument does not say anything about whether e1, > e2, and e3 are coprime to f. *3. This is the key new step in the argument. *4. The same argument is implied in Harris's proof of > FLT, except there he is using objects rather than > algebraic integers. However it has not been established > whether objects are different from algebraic integers, > and a number of basic theorems for objects which are > needed for his argument have not been proven. --Dec.2000 'WAND' Chairman Paul O'Neill, reelected to Board. Newsish? http://www.rand.org/publications/randreview/issues/rr.12.00/ http://members.tripod.com/~american_almanac === Subject: : Re: Focus on point of dispute, more math > There is a much more general question. Why does anyone read these > James Harris threads at all? > I read the replies to JSH, knowing that most of the math is totally beyond me, in the hope and belief that as I study more mathemathics (which I am doing even now) that even such a casual and meaningless (to me) exposure to the terminology and presentation will benefit me. Already I have learned things like the existence of rings and fields and Galois Theory and I even know that these are to do with algebra :) I read the actual JSH posts because I have recently found some definite pleasure in watching a grasshopper wriggle after being skewered by a pin :) (and even without being able to follow the maths, I can tell by the nature of the JSH replies (or lack of) when he has been well and truly skewered) ivan. --remove the obvious to reply by email-- === Subject: : Re: Focus on point of dispute, more math > There is a much more general question. Why does anyone read these > James Harris threads at all? > I read the replies to JSH, knowing that most of the math is totally beyond > me, in the hope and belief that as I study more mathemathics (which I am > doing even now) that even such a casual and meaningless (to me) exposure to > the terminology and presentation will benefit me. Already I have learned > things like the existence of rings and fields and Galois Theory and I > even know that these are to do with algebra :) Notice the community aspect, as I've noted repeatedly. The math community is rife with people who know that pleasing the group IS considered substance. It's part of the math fashion show, where people believe belief is substance. I must admit I still find it fascinating. Let's see about tweaking things a bit more. > I read the actual JSH posts because I have recently found some definite > pleasure in watching a grasshopper wriggle after being skewered by a pin :) > (and even without being able to follow the maths, I can tell by the nature > of the JSH replies (or lack of) when he has been well and truly skewered) ivan. What's telling to me is that animosity, which is rather primitive. There's an anger response from people who go from admitting, like this poster did, a basic ignorance of mathematics, to making the claim they apparently think the group wants, which is their perception that I'm harmed in some way. So this poster who's opinion, by his own admissions in terms of mathematics is worthless, still feels that his opinion is of value, clearly because he's at least learned the true nature of the math community. My assessment is that for mathematicians the *claim* of logic and mathematical preciseness is part of the show, and not even people outside of the math community--who want in, like this fellow--actually believe it. James Harris === Subject: : Re: Focus on point of dispute, more math > The math community is rife with people who know that pleasing the > group IS considered substance. > It's part of the math fashion show, where people believe belief is > substance. > I must admit I still find it fascinating. Let's see about tweaking > things a bit more. > I read the actual JSH posts because I have recently found some definite > pleasure in watching a grasshopper wriggle after being skewered by a pin :) > (and even without being able to follow the maths, I can tell by the nature > of the JSH replies (or lack of) when he has been well and truly skewered) ivan. > What's telling to me is that animosity, which is rather primitive. > There's an anger response from people who go from admitting, like this > poster did, a basic ignorance of mathematics, to making the claim they > apparently think the group wants, which is their perception that I'm > harmed in some way. > So this poster who's opinion, by his own admissions in terms of > mathematics is worthless, still feels that his opinion is of value, > clearly because he's at least learned the true nature of the math > community. > My assessment is that for mathematicians the *claim* of logic and > mathematical preciseness is part of the show, and not even people > outside of the math community--who want in, like this fellow--actually > believe it. > James Harris You can now add failure at sociology and group psychology to your legacy of failures at logic and mathematics. (P.S. Practicing psychology without a license may be prosecutable.) -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: : Re: Four Color Graph. It is convenient to require that every vertex in Graph X has degree >=2. The polygon arrangement guarantees this. Therefore, X is > formally defined as; Well, asking about any 4-chromatic graph or any 4-chromatic-graph- > without-pendant-vertices are different things, and so would yield > different probabilities. > TRUE. > Graph X is a 4-chroma graph generated by adding (2n-6) diagonals to > an n-sided polygon! I'm assuming you aren't fixing the position of the polygon and asking if > there will be crossing edges, since that would be a geometrical problem > instead of a graph-theoretic one. Yes, the position of the polygon is fixed by labeling the vertices. No, I am just asking if the graph is planar or not planar. > But by stating that all 4-chromatic graphs (without degree-1 vertices) > are generated by starting with a cycle and then adding diagonals asserts > that all such 4-chromatic graphs are hamiltonian (i.e. that there always > exits such a cycle to start with.) Is this known? > This is a condition of the problem. > Again, what is the probability that X is planar; if X is randomly > generated? > I would like to know your a priori estimate? Say for n = 12? Once again, you need to be specific about random generation. I think > you mean randomly selected and that this is selected over all > nonisomorphic unlabeled connected simple such graphs. > My estimate for n=12? Maybe 5% ... and I guess it will decrease at n > increases. My calculation for n=12 (based on the conditions specified) is <0.008% It is noted that while the planar graph is 'restricted' to > triangulated polygons; the general 4-chroma graph is not!! So are you asking over all 4-chromatic triangulated polygons or are > you asking over the general 4-chromatic graphs? Both! > I'm also interested to know the roots of your interest in this problem > Bill J. === Subject: : Re: Fraud in Computer Science Publishing 1. Do you consider Predicate Calculus wffs to be programs? Technically, it is only source code. But by extension, we may call that a > program. 2. How do you define a program? A list of instructions that a computer can execute. Ok, then how would a computer execute a Predicate Calculus wff? You can take a specific example, such as PRIME(x)^BETW(1000,x,1005). This is pretty simple (only 2 relation references.) Could you describe how it is executed, step by step? > 3. Do you know of a better way to sepcify the largest proper factor > of a given number? A better way, maybe not; a way that's as good, yes. And it happens that this > way allow to specify things that PC does not. 1. It isn't better if the user has to input new programs to create a program. 2. Displaying a program also doesn't allow you to define non-r.e. sets or functions (used throughout the Theory of Computation.) > 4. Do you know of a way to specify it that isn't programming? Not that the computer can understand. If the computer can understand it, it's > programming. But didn't you just imply that PC wffs can be executed? > 5. Do you know of a simpler way to specify it? Depends on the task to be accomplished. The above wff. > 6. How do you think that a Mathematician would specify it? It depends. If he's doing a proof, he'll probably just say is a proper > factor. If he's working with computer, he'll use whatever language he finds > suitable for the task. Might be Maxima, Lisp, PC, or something else. What's > your point? The point is, how are sets and functions formally defined by Mathematicians? I say they use the Predicate Calculus most often - and for good reason: it is nonprocedural. It is not executed to determine what it means. formally define a set or function, as opposed to displaying a program that computes it? In general, you can't display a program to define a set or function. Some sets and functions are not computable by a program. > 7. Did you know that the state-of-the-art in Program Synthesis is to > specify the program requirement as a Predicate Calculus wff? This is meaningless. It's not because some people, however talented they are, > have decided to use a specific language, that all other languages should not > be considered. I am not saying that. I am saying that their use of PC is a reason to be careful at what you criticize. I can site people such as Zohar Manna and Richard Waldinger of Stanford U and SRI who tried for decades to perform Program Synthesis, and they used the Predicate Calculus. So, are you prepared to criticize the big boys for using the Predicate Calculus as well? (They gave up in the 1990's, blaming the inadequacy of available automated theorem provers. They were using the wrong approach, trying to prove (all A)(exists B)R(A,B) where R(input,output) defines the program.) > 9. How about the fact that a Predicate Calculus wff has no assignment, > conditional execution, loops or the possibility of not terminating? Lisp programs can have the above characteristics. the difference is, it > doesn't force you to stay within these bounds. You can't be sure whether any given program terminates. You can be sure of PC wffs, because they are not executed in the 1st place. And in your system, you do have to use general programs because the user has to enter in new programs to satisfy new requests for a program to be generated. > 10. Do you think that Predicate Calculus wffs and computer programs > are at the same level of abstraction? Depends on the language used. Check out the Shakespeare programming language. > Highly abstract :-) Actually, I wonder if the term level of abstraction is well-defined when both languages represent programs in general. It would take a long FORTRAN program to twiddle bits, and a long assembly language program to add two numbers. > several algorithms. On the other hand, some specifications cannot be > expressed with PC. Then they do have a one-to-many relationship? Each wff can be implemented by multiple different algorithm? Recall my quote: Predicate Calculus (is) an adequate logical basis for all of today's mathematics. . . . Any mathematical field can be defined and all the roofs carried out within the Predicate Calculus. - J. N. Crossley, What is Mathematical Logic?, pp. 2-3. Non-recursively enumerable sets cannot be defined by a program that lists or decides it. You cannot represent concepts from (and automate) the Theory of Computation as I have done (see details in my papers.) > 15. Do you see value in being able to determine the wff that a > particular program computes? Yes, because PC is closer to mathematical formalism than other languages. That's It! That's why I was asking what Mathematicians use. Now you see it! > 16. Do you see value in being able to determine computer programs that > implement a given predicate calculus wff? Yes; compilers for all sorts of languages are always useful :-) Especially a non-procedural language like the Predicate Calculus. > I know no system that determines programs that compute a given wff. > Sam I maintain that my system does and it is explained in detail in my two papers below, as I have given excerpts from in these messages. Charlie Volkstorf Cambridge, MA http://www.mathpreprints.com/math/Preprint/CharlieVolkstorf/ 20021008.1/1 http://www.arxiv.org/html/cs.lo/0003071