mm-269 === Subject: Re: question about group notation> What is the standard way to write a 2-genera group that is not> necessarily abelian? I'm under the impression that G= usually> stands for an 2-genera abelian group. Maybe G= where the ~> indicates some relations imposed? Please let me symmetries of a regular n-gon:(cf. Dummit and Foote, Abstract Algebra)In general, is the group of all finite products of r, s, r^(-1), ands^(-1). If no relations are explicitly given, r and s are typically someelements of a given group G and so the relations are implicit.Usually if the group being genera by a and b is abelian, this is indica bythe relation [a,b] = 1, for example: Z x Z is isomorphic to Z_n x Z_m is isomorphic to where [a,b] = a^(-1)*b^(-1)*a*b (the commutator of a and b).Without any explicit or implicit relations regarding r and s, I am not sure what means; maybe some authors assume that [r,s] = 1 is implicit in this case,in which case is Z x === standard way to write a 2-genera group that is not> necessarily abelian? I'm under the impression that G= usually> stands for an 2-genera abelian group. Maybe G= where the ~> indicates some relations imposed? Please let me Abelian group is < x,y : xyx^-1 y^-1 >.Expressions after : are the relators which reduce to e.Thus < x,y > = < x,y : xx^-1, === following equation to solve:> 1215 = 66y + (34 + 4y)x.> I know that y < 7 and x < 15. Also, y and x are integers.> What would be the general way of solving this form of equation? Also,> would the method be fast if implemen in a computer in order to> solve it for very large variables and constants?> Any ideas would be apprecia. Well, here's an inexpert idea. Write the equation as4xy + 34x + 66y = 1215and complete the oblong to get the factorized form(2x + 33)(2y + 17) = 1215 + (33 multiplied by 17) = 1776.Then think about splitting the right-hand side into suitable factors. Inthis example all integers x and y make the left-hand side odd, whereasthe right-hand side is even, so there's no solution. Ken === special form> I have the following equation to solve:> 1215 = 66y + (34 + 4y)x.> I know that y < 7 and x < 15. Also,> y and x are integers.> What would be the general way of > solving this form of equation? Also,> would the method be fast if implemen> in a computer in order to> solve it for very large variables> and constants?See the section ``Delta > 0 is Square'' inSolving the equation ax^2 + bxy + cy^2 + dx + ey + f = 0 - PDF Fileat http://hometown.aol.com/jpr2718/Either methods in that section or elsewhere in the file should cover what youask, depending on what you mean by ``this form of equation,'' and what you meanby `` very large.''As another poster no, your equation has no solutions. As written, the === Solving diophantine equation of special formFirst, either you've miswritten the problem, or else there isn't anysolution, since 1215 is odd, while 66y+(34+4y)x is even! But let'ssuppose that you meant to write 1214 instead.In general, what you want to do is complete the product on therighthand side and then use a factorization of the other side. Thus66y + (34 + 4y)x = 66y + 34x + 4xy = (17 + 2y)*(33 + 2x) - 33*17So your equation becomes (17 + 2y)*(33 + 2x) = 1214 + 561 = 1775Now take each possible factorization 1775 = A*B (there are onlyfinitely many of them) and solve the two equations 17 + 2y = A and 33 + 2x = Bfor x and y. Thus y = (A - 17)/2 and x = (B - 33)/2Since 1775 = 5*5*71, there are several possibilities for A. But if wewant x and y to be positive, then we need A > 17 and B > 33, whichcuts things down. In fact, I think that means that A = 25 and B = 71,so the solution is y = 4 and x = 19.Check: 66*4 + (34+4*4)*19 = 1214This method works quite generally to solved = a*y + b*x + c*x*y.The most time-consuming part of the algorithm is the necessity tofactor the integer that appears on the left after you complete theproduct on the right.> Hi!> I have the following equation to solve:> 1215 = 66y + (34 + 4y)x.> I know that y < 7 and x < 15. Also, y and x are integers.> What would be the general way of solving this form of equation? Also,> would the method be fast if implemen in a computer in order to> solve it for very large variables and constants?> Any === equation> In earlier releases, it can be done in the linalg package with the > undocumen function `linalg/matfunc`. See matfunc in my Maple Advisor > Database, .In Maple 7 I tried your example:with(linalg): A:= matrix([[11,3],[5,3]]); B:= matfunc(A,sqrt);and it returnedB := matfunc(A,sqrt);Am I missing some setup step? (I thought it doesn't require === matfunc is not definied in linalg as I thought.> In earlier releases, it can be done in the linalg package with the> undocumen function `linalg/matfunc`. See matfunc in my MapleAdvisor> Database, .> In Maple 7 I tried your example:> with(linalg): A:= matrix([[11,3],[5,3]]); B:= matfunc(A,sqrt);> and it returned> B := matfunc(A,sqrt);> Am I missing some setup step? (I thought it doesn't require === releases, it can be done in the linalg package with the > undocumen function `linalg/matfunc`. See matfunc in my Maple Advisor > Database, procedure! I tes it like this:G:=Matrix(6,6,[[0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0,0], [1,0,0,0,0,0],[0,0,1,1,0,0],[0,1,0,0,1,0]]);Both T:=(1-G)^(-1);(the solution of T=1+G*T)andT:=1/2*(1 + matfunc(-3-4*G,sqrt))(the solution of T=1+G+T*T)agree what the === involving cubesSeveral questions. (I include the Mensa group, becausethe first question is more relevant for them. Please,no smart-*ss remarks about geniuses. And, finallysince I had to ask these questions, I could notfigure it out myself, so I am clearly not a genius.)Many spatial IQ tests present an unfolded cube andthe tester is instruc to pick from severalperspectively drawn cubes that can/can'tbe folded from the unfolded cubeNone of these tests give any more detail aboutthe cube, or how the cube should be folded.If you imagine the unfolded cube to be yourpalm, do you fold fingers-curled manner, orthe opposite? I guess what I am trying tosay, is, do we assume the same pattern drawnon each face of the cube is the same patternon both sides of each face? It makes adifference, and it drives me insane, thatthese are never specified. (At least mysub genius mind believes it makes a difference!)Finally, I can brute force my way to finding outhow many ways to unfold a 3-d cube. Is there amore elegant way of figure that out, perhapsa method which is easily generalized toN dimensional cubes?I have observed for 3d cubes, you have a 2,3 count.If you unfold the cube, there must be 2 verticaljoining edges, and 3 horizontal joining edges.(or === involving cubes>Many spatial IQ tests present an unfolded cube and>the tester is instruc to pick from several>perspectively drawn cubes that can/can't>be folded from the unfolded cube>None of these tests give any more detail about>the cube, or how the cube should be folded.>If you imagine the unfolded cube to be your>palm, do you fold fingers-curled manner, or>the opposite? I guess what I am trying to>say, is, do we assume the same pattern drawn>on each face of the cube is the same pattern>on both sides of each face? It makes a>difference, and it drives me insane, that>these are never specified. (At least my>sub genius mind believes it makes a difference!)>Finally, I can brute force my way to finding out>how many ways to unfold a 3-d cube. Is there a>more elegant way of figure that out, perhaps>a method which is easily generalized to>N dimensional cubes?>I have observed for 3d cubes, you have a 2,3 count.>If you unfold the cube, there must be 2 vertical>joining edges, and 3 horizontal joining edges.>(or the otherway around.)basic math students in a certain group of schools. They seem hard as hell todo just in the mind, but if a cardboard model of the unfolded cube is made,then a user can fold it, and compare the folded form to the versions shown inthe drawing choices, and simply make the selection for which one fits. Curious here, if building a model to determine which drawing fits defeats thepurpose of the problems? No information with the exercise seems to === group, because>the first question is more relevant for them. Please,>no smart-*ss remarks about geniuses. And, finally>since I had to ask these questions, I could not>figure it out myself, so I am clearly not a genius.)>Many spatial IQ tests present an unfolded cube and>the tester is instruc to pick from several>perspectively drawn cubes that can/can't>be folded from the unfolded cube>None of these tests give any more detail about>the cube, or how the cube should be folded.>If you imagine the unfolded cube to be your>palm, do you fold fingers-curled manner, or>the opposite? I guess what I am trying to>say, is, do we assume the same pattern drawn>on each face of the cube is the same pattern>on both sides of each face? When you fold such objects, one side of the paper becomes the insideand the other side is the outside. Unless they present the objects tobe folded in 3 D you only see one side -- that will be the outsidesurface.> It makes a>difference, and it drives me insane, that>these are never specified. (At least my>sub genius mind believes it makes a difference!)>Finally, I can brute force my way to finding out>how many ways to unfold a 3-d cube. Is there a>more elegant way of figure that out, perhaps>a method which is easily generalized to>N dimensional cubes?>I have observed for 3d cubes, you have a 2,3 count.>If you unfold the cube, there must be 2 vertical>joining edges, and 3 horizontal joining edges.>(or the otherway around.)Don't know, I never bothered === Finally, I can brute force my way to finding out> how many ways to unfold a 3-d cube. Is there a> more elegant way of figure that out, perhaps> a method which is easily generalized to> N dimensional cubes?The candidate configurations are the hexominoes: http://mathworld.wolfram.com/Hexomino.htmlI did a web search, and found that 11 of these can be folded intoa cube, although the hexominoes themselves were not lis: http://mathforum.org/pow/sol22.htmlThis made me think that the solutions are the odd hexominoes . . .The 35 hexominoes can be divided into two parity classes based onthe numbers of black and white squares they cover on a checkerboard.For example, Odd Even B B WB W B B W B B W WThe 11 hexominoes with odd parity cover either 2 or 4 black squares,depending on how they are placed; the 24 with even parity cover 3.Because there are an odd number of odd-parity hexominoes, they cannotcombine to form any shape in the plane with equal numbers of blackand white squares, such as a rectangle.But all this is a digression. The odd-parity hexominoes are not thosethat can fold into a cube. I made this list of those that can so fold: X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X XX X X X X X X X X X X X X X X X X X X XX X X X X X X X X X X X XXOnly 4 of these are odd.It would be nice to find a way to characterize these shapes via anindicator that is simple to calculate (like the parity is).-- | Jim Ferry | Center for Simulation |+------------------------------------+ of Advanced Rockets || http://www.uiuc.edu/ph/www/jferry/ +------------------------+| jferry@[delete_this]uiuc.edu | === involving cubes>Several questions. (I include the Mensa group, because>the first question is more relevant for them. Please,>no smart-*ss remarks about geniuses. Of course not.--Mensanator2 of Clubs === Subject: Re: Spatial IQ Tests involving cubes|Finally, I can brute force my way to finding out|how many ways to unfold a 3-d cube. Is there a|more elegant way of figure that out, perhaps|a method which is easily generalized to|N dimensional cubes?Make a graph where the vertices correspond to the facetsof the cube, and where two are joined by an edge if thefacets are adjacent. That's equivalent to the complementof the graph consisting of N disjoint pairs of points eachconnec by an edge, since there's always just onefacet that isn't adjacent to a given one, the oppositefacet, and they come in N opposite pairs.Then I believe the ways of unfolding it correspond tospanning trees in this graph. It seems like there === Spatial IQ Tests involving cubes>> Finally, I can brute force my way to finding out>> how many ways to unfold a 3-d cube. Is there a>> more elegant way of figure that out, perhaps>> a method which is easily generalized to>> N dimensional cubes?> It would be nice to find a way to characterize these shapes via an> indicator that is simple to calculate (like the parity is).I found a characterization, but it's not simple.A hexomino can be folded into a cube iff it does not containone of the following 7 patterns:X X X X X X X X X X X X X X X X X X X X X X X X XX X X X X X X X X X X X XOf course, that's not much better than saying, it must be one ofthese 11 . . .-- | Jim Ferry | Center for Simulation |+------------------------------------+ of Advanced Rockets || http://www.uiuc.edu/ph/www/jferry/ +------------------------+| jferry@[delete_this]uiuc.edu | University of Illinois === of your help. My question now is how we canintegrate over a sphere using spherical coordinates when they arebadly behaved at several points, they don't constitute an atlas sounder this coordinate regime the sphere isn't a manifold, what is it,why do integrals still work. I've seen something to the effect thatpoint sets of zero-volume do not contribute to integrals, would thevolume form here be the 'area' form or the form of the space thesphere was sitting in?Pardon me if this question === manifoldsNNTP-Posting-User: of your help. My question now is how we can>integrate over a sphere using spherical coordinates when they are>badly behaved at several points, they don't constitute an atlas so>under this coordinate regime the sphere isn't a manifold, what is it,>why do integrals still work. I've seen something to the effect that>point sets of zero-volume do not contribute to integrals, would the>volume form here be the 'area' form or the form of the space the>sphere was sitting in?>Pardon me if this question ends up being innane.>>kevinThe point sets of zero volume statement is probably a good way of seeingthis. If you've never seen Lebesgue integration before, take a look atsome real analysis texts (Rudin, Royden, plenty of others). The value of afunction on a set of measure zero doesn't change the integral in Lebesgueintegration theory (and the Riemann integral equals the Lebesgue integralif the Riemann integral exists at all). In particular, if you areintegrating a function that is infinite (or undefined, or anything) onlyon a set of measure zero, it won't give you any problems.I would suggest looking up the rigorous definitions, but for your purposesit sounds like you can consider measure zero to mean having no volume(I'm assuing you are integrating over a solid sphere -- if it's just thesurface, then replace volume with === meant by D_1, D_2, Z_1, Z_2?D_i is a dihedral group. Z_i is a === Target Translation Processing Algorithm.Hi all,I have RCS data for a certain rotating target measured at twodifferentheights, signals S1 and s2, both of which are functions of frequencyand aspect angle.The two signals can be expressed in matrix form... |s1| |1 1 | |t| |n1|S = | | = | | | | + | | = Ax + n |S2| |1 e^(-i.2k.d) | . |g| |n2|Where t and g are the desired target and undesired target-groundreturns,respectively, d is the difference in targetheights,k=(2.pi)/wavelength is the wave number, and n1,2 is addednoise.Given an appropriate selection of d, the above equation represents apair of linearly-independant equations that can be solved for theunknowns t and g.Could anyone give me a hint as to how to go about actually solving theabove equation and possibly how to implement it in a software package,eg Matlab.Any help would === Ziga Habjan:> test[1] Prove Fermat's last theorem. (there exist infinitely many primes of the form 2^(2^n) + 1)[2] Prove Goldbach's conjecture. (any even number > 2 is the sum of two primes)[3] Prove that the number of real points in the line segment [0,1] (C) is equivalent to the cardinality of the set of all subsets of the natural numbers (aleph-1).:-)-- #191, ewill3@earthlink.netIt's still legal to go === Machine<10l301-l7p.ln1@ lexi2.athghost7038suus.net>:> In sci.math, Ziga Habjan> :>> test> [1] Prove Fermat's last theorem.> (there exist infinitely many primes of the form 2^(2^n) + 1)Oops. Someone already did and I transcribed the wrong thing anyway. Aargh.> [2] Prove Goldbach's conjecture.> (any even number > 2 is the sum of two primes)> [3] Prove that the number of real points in the line segment [0,1] (C)> is equivalent to the cardinality of the set of all subsets of the> natural numbers (aleph-1).>:-)-- #191, ewill3@earthlink.netIt's still legal to === Machine>><10l301-l7p.ln1@ lexi2.athghost7038suus.net>:>> In sci.math, Ziga Habjan>> :> test>> [1] Prove Fermat's last theorem.>> (there exist infinitely many primes of the form 2^(2^n) + 1)> Oops. Someone already did and I transcribed the wrong thing anyway. > Aargh.>> [2] Prove Goldbach's conjecture.>> (any even number > 2 is the sum of two primes)>> [3] Prove that the number of real points in the line segment [0,1] (C)>> is equivalent to the cardinality of the set of all subsets of the>> natural numbers (aleph-1).>>:-)In light of what you said under [1], I thought perhaps [3] wasdeliberate. But just in case it wasn't: the cardinalities of R and ofP(N) are both equal to c = 2^aleph_0. The hypothesis that c = aleph_1,the cardinality of the set of all countable ordinals, is the continuumhypothesis.-- Dave SeamanJudge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.function over a (compact, possibly simply connec, and even convex>if we restric ourselves) domain in R^2? We are aware of a test based>on the Bordered Hessian, although I don't think we can evaluate the>derivatives at every point in our domain. What would be ideal is some>sort of test we could perform on the boundary of our domain.I don't see how a test on the boundary of the domain could tell you anything about the behaviour of the function in the interior of thedomain (except for special cases such as harmonic functions where thefunction is determined by its boundary values).Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia === the constructivist's osophical> criticism of classical mathematics. But then when> they start doing mathematics, they muck things up> and make things much too complica.> As I see it, constructivists are trying to impress> the mathematicians, rather than trying to produce> something of value to those who apply mathematics.> The idea that I advocate is that probability should be> recognized as logic. That is, logic is a tool for> reasoning about the universe, and that's exactly what> probability is too. Combining that idea together with> the constructivist osophy leads to foundational> framework for mathematics which is especially useful> and intuively pleasing to those who apply mathematics.Can't wait to see it! Do you already have some material? References,examples, ...Herman === forthcoming proof|If a group of people get together and say to |themselves we have a consistent theory of the|universe, which we all agree is the truth. It|doesn't matter that the truths in our theory|have no connection to an observable reality,|because for us, the 'universe' is an abstraction|which obeys the theory, and can be presumed|to exist simply because the theory is consistent.|Furthermore, because we have this special knowledge,|we are justified in excluding and denigrating those|who question any aspect of our special knowledge,|then that group may be accep as a religious|entity, but it has no business having any authority|in the public domain.The thing is, I don't hear anybody saying this. I hear them saying thingsyou disagree with, but what you have written here makes the (common) errorof describing your opponents' views as if they were thinking in just thesame terms as you do. It would sound very weird indeed to hear othermathematicians describing mathematics or its foundations as a theory ofthe === forthcoming proof>> If a group of people get together and say to>> themselves we have a consistent theory of the>> universe, which we all agree is the truth. It>> doesn't matter that the truths in our theory>> have no connection to an observable reality,>> because for us, the 'universe' is an abstraction>> which obeys the theory, and can be presumed>> to exist simply because the theory is consistent.>> Furthermore, because we have this special knowledge,>> we are justified in excluding and denigrating those>> who question any aspect of our special knowledge,>> then that group may be accep as a religious>> entity, but it has no business having any authority>> in the public domain.There we are. Petry is advocating the suppressionof free enquiry by projecting his fanaticism onto atarget group.He should also question his notion of observable reality.Another rhetorical device for privileging his narrowworldview.-- His mind has been corrup by colousounds and === behind my great forthcoming proof> |If a group of people get together and say to> |themselves we have a consistent theory of the> |universe, which we all agree is the truth. It> |doesn't matter that the truths in our theory> |have no connection to an observable reality,> |because for us, the 'universe' is an abstraction> |which obeys the theory, and can be presumed> |to exist simply because the theory is consistent.> |Furthermore, because we have this special knowledge,> |we are justified in excluding and denigrating those> |who question any aspect of our special knowledge,> |then that group may be accep as a religious> |entity, but it has no business having any authority> |in the public domain.> The thing is, I don't hear anybody saying this. I hear them saying things> you disagree with, but what you have written here makes the (common) error> of describing your opponents' views as if they were thinking in just the> same terms as you do. It would sound very weird indeed to hear other> mathematicians describing mathematics or its foundations as a theory of> the universe.It's an illustration of the psychological concept of projection;attributing one's own neuroses to those whom one deals with.Also a straw man argument. Mathematics is an open disciplinevaluing free enquiry (although Petry opposes this) in whichall can participate. It is not special knowledgeunlike science which is not freely opento anyone due to the large capital costs, and also to the hugeontological baggage (electrons/quarks/strings whatever) whichmust be taken on to gain admission to the scientists' club.Mathematics is not a theory of the universe. It is the study ofmathematical objects. It is older and separate from physics,a socially construc doctrine which posits an underlyingreality behind the phenomenal world --- one which only physicistscan interpret correctly.I presume that Petry wants to subordinate the free enquiry of mathematicsto the commericial/governmental/military sponsoredinterests of the scientists. Luckily he has no power or influence.-- His mind has been corrup by colousounds and === behind my great forthcoming proof A8EwTYfhf*u~,Eu,tf6 $HN*MY&)u0G=N' x<%)/s=GZ_BD2Qz9m=S%4v^I+>T|'1{w70ZY=ih,=)kMY_}?{%)x0)];K~@ J6m5.EN?>ZhXh;Y V|',x(js'Jfq02joVpj|#x>> Much of what Petry says sounds to me like the things the >> late Errett Bishop used to say. Bishop was highly regarded>> by many Platonist-style mathematicians. >> But Bishop was an accomplished mathemtician.> The constructivist viewpoint is especially > appealing to students who have grown up with> computers on their minds. It appears that Robin> Chapman would like to exclude such students> from the mathematics community.On the contrary, it appears that Robin Chapman encourages someexpertise in a field before one makes bold claims that others aredoing it all wrong.To be sure, if this is what he is saying, I'm afraid that osophyof science would lose its appeal to me. I always *liked* the bitwhere I got to say what others should do. Unlike some, I was under no illusion that my pronouncements shouldhave practical effect, of course.I don't think that anyone here is claiming constructivism isn'tinteresting or worthy of study. I suppose most mathematicians objectto the claim that all mathematics ought to be constructive, but thatdoesn't mean that constructive mathematics isn't interesting oruseful. On the contrary, as computers are used in more mathematicalapplications, one expects that constructive mathematics becomes moreimportant, I think.What gets one's dander up is this nonsense about the Cantorians whohave crea a dangerous fantasy world of infinities. Before makingsuch bold claims, I suppose that one ought to have the mathematical(and osophical?) experience to support one's arguments.-- you, but just remember, I'm the guy who proved Fermat's Last Theoremin just a bit over 6 years [...] My standards are kind of high. === basic idea behind my great forthcoming proofdavid_lawrence_petry@yahoo.com says...>> What I have problems with is David Petry calling modern mathematicsdangerous and calling its adherents fanatics. >You're misinterpreting what I say. Here's the idea>that I think is wrong and dangerous, and is the>essence of fanaticism.>If a group of people get together and say to >themselves we have a consistent theory of the>universe, which we all agree is the truth. It>doesn't matter that the truths in our theory>have no connection to an observable reality,>because for us, the 'universe' is an abstraction>which obeys the theory, and can be presumed>to exist simply because the theory is consistent.>Furthermore, because we have this special knowledge,>we are justified in excluding and denigrating those>who question any aspect of our special knowledge,>then that group may be accep as a religious>entity, but it has no business having any authority>in the public domain.You say that I'm misinterpreting what you, but thenyou post something that just confirms my interpretation.Anyway, almost everything about your paragraphis mistaken. Mathematicians don't exclude anddenigrate those who question the existence orutility of infinite sets. They just shrug theircollective shoulders. Constructive mathematicsis a perfectly respectable (if unexciting) branchof mathematics.So what you're really objecting to is not the*exclusion* of constructive mathematics, becausethere is no such exclusion. You're objecting tothe *inclusion* of nonconstructive mathematics.You think it should be banned from academia. Andyet you are complaining about others excludingand denigrating *you*. That is the height ofhypocrisy, David.Your complaints about being denigra andexcluded amounts to You people are soclosed-minded and intolerant that youcompletely exclude those who would burnheretics at the stake.I would say that you have things exactly backwards,that you are the one who is claiming special knowledgeabout the proper role of mathematics, you are the onewho is excluding different points of view, you are theone === Re: The basic idea behind my great forthcoming proofHerman Jurjus says...>After i pos it, i realized it sounded strange. But, really, i think>mathematicians do not care enough about foundations.>And, yes, i still upheld that, for the working mathematician, foundation has>long reached its end point: FOL / ZF / Bourbaki, and that's it.>And even within subjects like set theory, the basics are not questioned>anymore.>Math has found a way to make progress without bothering, and i don't think>that's a good thing.It seems to me that there are two different issues here---progressin mathematical foundations, and progress in mathematical applications.Applied math doesn't really seem to depend on foundations to anyimportant extent. I think that the situation is similar to that inphysics---there are many difficult fundamental questions that areunanswered about cosmology and the ultimate building blocks of matter.But applied physics (and engineering, which I think of as applied physics)doesn't really depend on those foundational === idea behind my great forthcoming proofkramsay@aol.com () says...>The last time I asked for examples of useful nonconstructive>mathematics, someone helpfully sugges one of Bishop's>own results in functional analysis from before his switch.>Bishop-Phelps I think it was. If David Petry thinks he can show>that it's a sort of contentless baloney that could be interesting.David Petry's point seems to be that modern mathematics is toofar removed from its mission, which is to be a tool in solvingreal-world problems.But is there any reason to think that constructive mathematics*helps* in applied mathematics? I don't think it really makesmuch difference. Constructively, you can try to prove that (forexample) a differential equation has a solution, and the proofitself will provide an algorithm for computing the solution.Classically, one would break it into two different tasks:(1) Deciding whether the equation has a solution, and(2) Finding an algorithm for computing the solution. Ifthe classical mathematician manages to complete task 2,then hasn't he accomplished the same practical goal asthe constructive mathematician?The algorithms might turn out to be different, becausethe constructive mathematician will come up with an algorithmthat is provably correct *constructively*, while the classicalmathematician only needs to prove that it is correct === basic idea behind my great forthcoming proof> Much of what Petry says sounds to me like the things the> late Errett Bishop used to say. Bishop was highly regarded> by many Platonist-style mathematicians.> But Bishop was an accomplished mathematician.>> The constructivist viewpoint is especially>> appealing to students who have grown up with>> computers on their minds. It appears that Robin>> Chapman would like to exclude such students>> from the mathematics community.How it appears to Petry is of no consequence.It appears that his perceptions are rather odd.The fact is that I stress the inclusionary nature of mathematics.On the contrary it appears that Petry wishes to expel*mathematicians* from the mathematics community.> On the contrary, it appears that Robin Chapman encourages some> expertise in a field before one makes bold claims that others are> doing it all wrong.Rather I was suggesting that Bishop was taken seriouslydue to his contributions in mathematics instead of hisconstructivist polemic,> I don't think that anyone here is claiming constructivism isn't> interesting or worthy of study. I suppose most mathematicians object> to the claim that all mathematics ought to be constructive, but that> doesn't mean that constructive mathematics isn't interesting or> useful. On the contrary, as computers are used in more mathematical> applications, one expects that constructive mathematics becomes more> important, I think.You are falling into the Petrian illusion that advances in algorithmicmathematics are due to constructivist mathematiciansnot to classical mathematicians. That is not so.An instructive comparison involves the textsGalois Theory by EdwardsandA Course in Computational Algebraic Number Theory by Cohen.Both study finite field extensions of the rationals.Edwards subscribes to the constructivist dogma, for instance,refusing to admit that a polynomial over a number fieldis either reducible or irreducible until an algorithm is givenfor deciding it. Cohen's book is classical and he rejects suchencumbrances. One of these books contains a huge array of computerimplemen algorithms and much data from their output;the other contains no nontrivial computational resultsat all. Now which do you reckon is which?-- His mind has been corrup by colousounds and === behind my great forthcoming proof> everything about your paragraph> is mistaken. Mathematicians don't exclude and> denigrate those who question the existence or> utility of infinite sets. They just shrug their> collective shoulders. Constructive mathematics> is a perfectly respectable (if unexciting) branch> of mathematics.> So what you're really objecting to is not the> *exclusion* of constructive mathematics, because> there is no such exclusion. You're objecting to> the *inclusion* of nonconstructive mathematics.Looking at this whole thread, i see two things (perhaps mistakenly so) :1. David Petry is fighting against collectively shrugged shoulders,nothing else.2. The opposition he meets just shows that mathematics is indeed indesperate need of a paradigm shift.These views are too predictable and too static. No wonder nobody wants tolearn that anymore.;-)Herman === see is that it's> harder for them to produce results of value to those who> apply mathematics, than it than it is for the classical> mathematicians, precisely *because* of the restrictions> with which they (the constructivists) have saddled themselves.> I seriously don't believe they're *trying* to make things> complica. Things just *are* complica, when you ban> the simplifying conceptual tools provided by infinite sets> and a realist attitude towards them.It's also harder to prove theorems about derivatives and integrals in themodern sense,as compared to the way Leibniz and Euler did it. But is that a reason tostick with the old method?Or is === basic idea behind my great forthcoming proof <878yq8zba6.fsf@phiwumbda.localnet> A8EwTYfhf*u~,Eu,tf6 $HN*MY&)u0G=N' x<%)/s=GZ_BD2Qz9m=S%4v^I+>T|'1{w70ZY=ih,=)kMY_}?{%)x0)];K~@ J6m5.EN?>ZhXh;Y V|',x(js'Jfq02joVpj|#x[...]>> I don't think that anyone here is claiming constructivism isn't>> interesting or worthy of study. I suppose most mathematicians object>> to the claim that all mathematics ought to be constructive, but that>> doesn't mean that constructive mathematics isn't interesting or>> useful. On the contrary, as computers are used in more mathematical>> applications, one expects that constructive mathematics becomes more>> important, I think.> You are falling into the Petrian illusion that advances in algorithmic> mathematics are due to constructivist mathematicians> not to classical mathematicians. That is not so.I dele a caveat that I had written above, namely: I confess thatthe relationship between constructive and computable mathematics hasnever been terribly clear to me. I should have left that sentence in.> An instructive comparison involves the textsGalois Theory by Edwards> andA Course in Computational Algebraic Number Theory by Cohen.> Both study finite field extensions of the rationals.> Edwards subscribes to the constructivist dogma, for instance,> refusing to admit that a polynomial over a number field> is either reducible or irreducible until an algorithm is given> for deciding it. Cohen's book is classical and he rejects such> encumbrances. One of these books contains a huge array of computer> implemen algorithms and much data from their output;> the other contains no nontrivial computational results> at all. Now which do you reckon is which?But, doesn't constructive mathematics (or at least intuitionisticmathematics) arise typically in computer science semantics? This isthe connection to which I should have alluded, but it is not, I think,rela to computers [in] mathematical applications. So, mycomments above are somewhere clarification.-- Jesse HughesSurround sound is going to be increasingly important in futureoffices. -- Microsoft === basic idea behind my great forthcoming proofHerman says...>Looking at this whole thread, i see two things (perhaps mistakenly so) :>1. David Petry is fighting against collectively shrugged shoulders,>nothing else.>2. The opposition he meets just shows that mathematics is indeed in>desperate need of a paradigm shift.That seems weird to me. David says that the state of modern mathematicsis rotten. The fact that people disagree with him proves that he's right?Does the fact that you agree with him prove that he's wrong?--Daryl === great forthcoming proof> There we are. Petry is advocating the suppression> of free enquiry by projecting his fanaticism onto a> target group.Maybe. But not any more than the evolutionistsare advocating the suppression of free inquiry byproject [their] fanaticism onto the === great forthcoming proof> |If a group of people get together and say to > |themselves we have a consistent theory of the> |universe, which we all agree is the truth. [..]> The thing is, I don't hear anybody saying this. was what elici my wrong === my great forthcoming proof> It's an illustration of the psychological concept of projection;> attributing one's own neuroses to those whom one deals with.Hmmm. That's the impression I have too, on your part.> Mathematics is an open discipline> valuing free enquiry (although Petry opposes this) in which> all can participate. That's not the way I saw it when I was in grad school,and it appears to me that you are one of the most opposedto free inquiry, though you will take the name of freeinquiry to defend you own views.> I presume that Petry wants to subordinate the free enquiry of mathematics> to the commericial/governmental/military sponsored> interests of the scientists. Dang, you're on to me.> Luckily he has no power or influence.Yup, the world has been saved by good === forthcoming proof> The idea that I advocate is that probability should be> recognized as logic. That is, logic is a tool for> reasoning about the universe, and that's exactly what> probability is too. Combining that idea together with> the constructivist osophy leads to foundational> framework for mathematics which is especially useful> and intuively pleasing to those who apply mathematics.> Can't wait to see it! Do you already have some material? References,> examples, ...Sorry. I got discouraged by the reaction of themathematicians I encountered in grad school. Toomuch free inquiry, you know.I no longer have anything to do with mathematics,other than === basic idea behind my great forthcoming proof>> |If a group of people get together and say to themselves we have a>> |consistent theory of the universe, which we all agree is the truth.>> |[..]>> The thing is, I don't hear anybody saying this.wrong and dangerous comments.the truth is? Where did I say anything about theories of theuniverse? Where did I imply in the least bit that mathematics is aboutsome kind of === idea behind my great forthcoming proofdavid_lawrence_petry@yahoo.com says...>> There we are. Petry is advocating the suppression>> of free enquiry by projecting his fanaticism onto a>> target group.>Maybe. But not any more than the evolutionists>are advocating the suppression of free inquiry by>project [their] fanaticism onto the creation>scientists.You're being paranoid. Your point of view isn't beingsuppressed by the evolutionists of modern mathematics.Your point of view is just being ignored. That mighthurt your feelings, but it isn't fanaticism.--Daryl === great forthcoming proofdaryl@atc-nycorp.com (Daryl McCullough) says...>david_lawrence_petry@yahoo.com says...> There we are. Petry is advocating the suppression> of free enquiry by projecting his fanaticism onto a> target group.>>Maybe. But not any more than the evolutionists>>are advocating the suppression of free inquiry by>>project [their] fanaticism onto the creation>>scientists.>You're being paranoid. Your point of view isn't being>suppressed by the evolutionists of modern mathematics.>Your point of view is just being ignored. That might>hurt your feelings, but it isn't fanaticism.I guess I read David's post too quickly. It seems that heis considering himself the oppressor, rather than the oppressed.It's sort of the reverse of paranoia---you believe that you'reconspiring === Re: The basic idea behind my great forthcoming proof <87n0f38zl9.fsf@phiwumbda.localnet> <87fzku5vx8.fsf@phiwumbda.localnet> <874r17donm.fsf@phiwumbda.localnet> <87ptju5esn.fsf@phiwumbda.localnet> <87smop5fj5.fsf@phiwumbda.localnet> A8EwTYfhf*u~,Eu,tf6 $HN*MY&)u0G=N' x<%)/s=GZ_BD2Qz9m=S%4v^I+>T|'1{w70ZY=ih,=)kMY_}?{%)x0)];K~@ J6m5.EN?>ZhXh;Y V|',x(js'Jfq02joVpj|#x>> So your beef is with the axiom of infinity? Infinity is undeniably a useful abstraction. But as> such, it's a figure of speech. And figures of speech> don't necessarily obey the same laws of logic as do> concrete objects. > Cantor chose laws of logic to deal with infinity> based upon his religious views of how infinity ought> to behave. > There are alternate ways to deal with infinity. What> I am advocating is that we recognize computation as> the reality underlying mathematics, and assign to> infinity only those properties which help> us organize our knowledge and understanding of that> underlying reality.In another post, you write: Formal systems themselves are still a valid field of study from the point of view I advocate, so someone who held on to the belief that maybe someday Cantor's ideas will prove to be useful, could continue.It appears that you feel that no mathematics involving infinite setsof different sizes is useful in, say, scientific applications. Isthis accurate? You see no utility in the various applications ofmeasure theory?No, I'm pretty sure I still don't quite get your position.-- [N]ow for once I might actually have an audience that realizes that[my proof of Fermat's Last Theorem is correct], because you see,they'll finally know what's in it for them--cold, hard cash. -- === think you won't let yourself see is that it's>> harder for them to produce results of value to those who>> apply mathematics, than it than it is for the classical>> mathematicians, precisely *because* of the restrictions>> with which they (the constructivists) have saddled themselves.>> I seriously don't believe they're *trying* to make things>> complica. Things just *are* complica, when you ban>> the simplifying conceptual tools provided by infinite sets>> and a realist attitude towards them.> It's also harder to prove theorems about derivatives and integrals in the> modern sense,> as compared to the way Leibniz and Euler did it. But is that a reason to> stick with the old method?You raise an interesting point (did you mean Newton rather than Euler?).In my view, infinitesimals are, for *most* purposes, *still* the correctway to think about the calculus. And no, I don't require you torecite any incantations about Los's theorem in order to use them.But questions do come up where we seem to need to be able to insertthe discussion into a broader framework in order to come up with reliableanswers -- things like differentiating under the integral sign, integratinglimits of series, etc. Rigor definitely has its place, and in fact theprovision of a common agreed substructure for it is one of the signalcontributions of set theory.Unfortunately rigor came to be somewhat fetishized by certainelements of the mathematical community, who (wrongly) saw it asthe path to an (illusive) absolute certainty. There in a nutshellis the origin (or one of the origins) of the === great forthcoming proof> Herman says...>Looking at this whole thread, i see two things (perhaps mistakenly so) :>>1. David Petry is fighting against collectively shrugged shoulders,>nothing else.>2. The opposition he meets just shows that mathematics is indeed in>desperate need of a paradigm shift.> That seems weird to me. David says that the state of modern mathematics> is rotten. The fact that people disagree with him proves that he's right?> Does the fact that you agree with him prove that he's wrong?For example, if we think about the collection of all infinite sequences of0s and 1s,most mathematicians can not think about that collection, unless in thecontext of ZF.But using that framework is the same already as making strong assumptionsabout this collection(e.g., that it is a fixed, comple totality in the first place).Most mathematicians cannot think outside this framework, and the frameworkforces them into a world where many assumptions are made, and:a. they are not awareb. they have no clue how to do it otherwisePerhaps it is too hasty to say, but i would dare to say that, when it comesto the basics,the math culture is losing its ability to change viewpoint.In that sense, it is contemporary mathematics that is restricting thethoughts andresearch directions of many mathematicians, not David Petry or theintuitionists.BTW, i know that intuitionism often leads to irritation on the side ofclassical mathematicians.They think that intuitionists sort of want to replace classical math withsomething else.But did non-Euclidean geometry 'replace' Euclidean geometry?No, it just made an end to its exclusive monopoly.And that only made math richer, not poorer. === yourself see is that it's>> harder for them to produce results of value to those who>> apply mathematics, than it than it is for the classical>> mathematicians, precisely *because* of the restrictions>> with which they (the constructivists) have saddled themselves.I seriously don't believe they're *trying* to make things>> complica. Things just *are* complica, when you ban>> the simplifying conceptual tools provided by infinite sets>> and a realist attitude towards them.It's also harder to prove theorems about derivatives and integrals inthe> modern sense,> as compared to the way Leibniz and Euler did it. But is that a reason to> stick with the old method?> You raise an interesting point (did you mean Newton rather thanEuler?).> In my view, infinitesimals are, for *most* purposes, *still* the correct> way to think about the calculus.So do i. And i hope, one day, we will talk the same way about sets and ZF.BTW, i did mean Euler. In general, 18th century math is very interesting.They had not half the rigor of the 20th century, but hardly ever failed.They were never mislead into believing bizarreconsequences of their methods, on the grounds of a formal proof.They only believed those conclusions that seemed to make sense,in a down to earth way.> But questions do come up where we seem to need to be able to insert> the discussion into a broader framework in order to come up with reliable> answers -- things like differentiating under the integral sign,integrating> limits of series, etc. Rigor definitely has its place, and in fact the> provision of a common agreed substructure for it is one of the signal> contributions of set theory.> Unfortunately rigor came to be somewhat fetishized by certain> elements of the mathematical community, who (wrongly) saw it as> the path to an (illusive) absolute certainty. There in a nutshell> is the origin (or one of the origins) of the Formalist school.Complete agreement overhere.And would like to add, that also intuitionists are not entirely free fromthis same mistake.Herman === forthcoming proof> The idea that I advocate is that probability should be> recognized as logic. That is, logic is a tool for> reasoning about the universe, and that's exactly what> probability is too. Combining that idea together with> the constructivist osophy leads to foundational> framework for mathematics which is especially useful> and intuively pleasing to those who apply mathematics.> Can't wait to see it! Do you already have some material? References,> examples, ...> Sorry. I got discouraged by the reaction of the> mathematicians I encountered in grad school. Too> much free inquiry, you know.> I no longer have anything to do with mathematics,> other than occasional jabbering in this newsgroup.Can i encourage you to at least write everything down,and get it published somewhere? (Do you know www.arxiv.org?)Especially if you can focus on examples from real math, and keep theosophyonly as a part of it, this should be publishable (and there will be *some*people === discouraged by the reaction of the> mathematicians I encountered in grad school. Too> much free inquiry, you know.I no longer have anything to do with mathematics,> other than occasional jabbering in this newsgroup.> Can i encourage you to at least write everything down,> and get it published somewhere? (Do you know www.arxiv.org?)> Especially if you can focus on examples from real math, and keep the> osophy> only as a part of it, this should be publishable (and there will be *some*> people interes).O, btw:Internet is the ideal medium === rigor came to be somewhat fetishized by certain> elements of the mathematical community, who (wrongly) saw it as> the path to an (illusive) absolute certainty. There in a nutshell> is the origin (or one of the origins) of the Formalist school.The fetish in itself is older than the 20th century, i think. It at leastdates back to Plato.Only, in those days, we didn't have === You raise an interesting point (did you mean Newton rather thanEuler?).> In my view, infinitesimals are, for *most* purposes, *still* the correct> way to think about the calculus.> So do i. And i hope, one day, we will talk the same way about sets and ZF.This seems to be a rather ambiguous statement of me. Sorry.I meant: Sets and ZF might one day count as a vage, intuitive picture,while we have some other formalism that is much more down-to-earth,and that we use for === The basic idea behind my great forthcoming proofHerman Jurjus says...>For example, if we think about the collection of all infinite>sequences of 0s and 1s, most mathematicians can not think about>that collection, unless in the context of ZF.>But using that framework is the same already as making>strong assumptions about this collection (e.g., that it is>a fixed, comple totality in the first place).>Most mathematicians cannot think outside this framework,>and the framework forces them into a world where many>assumptions are made, and:>a. they are not aware>b. they have no clue how to do it otherwise>Perhaps it is too hasty to say, but i would dare to say that,>when it comes to the basics, the math culture is losing its>ability to change viewpoint.>In that sense, it is contemporary mathematics that is>restricting the thoughts and research directions of many>mathematicians, not David Petry or the intuitionists.It is certainly plausible that intuitionism can providea new way of thinking about numbers and functions thatis not possible in standard set theory. But David Petryisn't saying that mathematics is too closed-minded, he'ssaying that it is too *open-minded*, that mathematiciansthink about things (infinite sets) that shouldn't be apart of mathematics at all.I don't think that any mathematicians are saying that ZFCis the *only* way to think about real numbers. They arejust saying that it is an extremely powerful way to thinkabout them. Are there actual examples of results thatcould not be obtained using ZFC?>BTW, i know that intuitionism often leads to irritation>on the side of classical mathematicians.>They think that intuitionists sort of want to replace>classical math with something else.>But did non-Euclidean geometry 'replace' Euclidean geometry?>No, it just made an end to its exclusive monopoly.>And that only made math richer, not poorer. Right?I don't have any problemwith including intuitionist mathematics. I only have aproblem with David's wanting to *exclude* classicalmathematics.--Daryl === great forthcoming proof> wrong and dangerous comments.As long as it is logically consistent, it is === great forthcoming proof> Question:> Do you think this is rela to the 'danger' that i mentioned:> a decrease of respect and attention for the field ?Yes, certainly it is rela.In our increasingly technologically orien world,the mathematicians could be, and maybe should be,playing a leadership role. After all, mathematics isthe language of science and technology. But it is obvious to most scientists, engineers andcomputer scientists that the mathematicians are livingin another world. At many schools, the science andengineering departments teach their own mathematicscourses, for fear that the mathematicians would onlyconfuse and mislead their students.I would advise any student who wants to be a part ofthe technology revolution to avoid pure mathematics,unless he is very focused and knows === Re: The basic idea behind my great forthcoming proof> Can't wait to see it! Do you already have some material? References,> examples, ...Sorry. I got discouraged by the reaction of the> mathematicians I encountered in grad school. > I no longer have anything to do with mathematics,> other than occasional jabbering in this newsgroup.> Can i encourage you to at least write everything down,> and get it published somewhere? (Do you know www.arxiv.org?)I've thought about doing that. It would be a majorundertaking. At the moment I don't feel sufficientlymotiva to do it.The question that was on my mind in grad school wasthis: how can I get a computer (artificial intelligence)to understand mathematics, and in particular, to understand it in the same way that a person who appliesmathematics understands it?Thinking along those lines, I came up with answers thatI thought were very good answers. It occurred to me that rather than merely writing downmy ideas, I should write a program that shows a genuineunderstanding of mathematics, and then just write downa description of how the program works. The programwould then be the proof of the validity of my ideas.But that's not really how mathematics is === forthcoming proof> But David Petry> isn't saying that mathematics is too closed-minded, he's> saying that it is too *open-minded*, Clearing out the deadwood is an important task inany intellectual discipline. Deadwood has a wayof choking out new ideas and innovation.Some people think that to clear out deadwood is*closed-minded*, and leaving it in is === my great forthcoming proof>> wrong and dangerous comments.As long as it is logically consistent, it is valid>mathematics.simply isn't:>> |If a group of people get together and say to themselves we have a>> |consistent theory of the universe, which we all agree is the truth.>> |[..]>> The thing is, I don't hear anybody saying this.wrong and dangerous comments.He didn't say anything about theories of the universe, nothing abouttruth... looks to me like you're assuming he means the same bythe word mathematics as you do. Regardless of whether yourversion is the One True version, you should realize by now thatothers mean something else by the === a ManifoldX-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Punge: Micro$oftX-Sanguinate: themvsguy@email.comX-Terminate: SPA(GIS)X-Tinguish: Mark Griffith X-Treme: C&C,DWS>However, except for this text I have never (which doesn't mean too>much, never means about 25-30 texts) seen an exposition on those>manifolds that weren't both hausdorff and second countable. Is there>such an exposition. If not why don't we just redo the definition if>the only 'interesting' manifolds are those that satisfy these and not>only the more general conditions?There's inteesting and there's tractable. A manifold that is not, e.g,paracompact, may be interesting, but some of the standard tools areunavailable.-- Shmuel (Seymour J.) Metz, SysProg and JOATAny unsolici bulk E-mail will be subject to legal action. I reserve theright to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Do not replyto === line isinteresting. I really don't know anything about modular forms or theother material you mentioned. However its exciting to know that thesemanifolds are used. For me, as a future mathematical physicist itsvery intriguing to wonder at the types of conditions we impose onspace in our models of it. === of a Manifold tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Punge: Micro$oftX-Sanguinate: themvsguy@email.comX-Terminate: SPA(GIS)X-Tinguish: Mark Griffith X-Treme: C&C,DWS>For me, as a future mathematical physicist its>very intriguing to wonder at the types of conditions we impose on>space in our models of it.It appears that the Physics community is using Mathematics far moresophistica than when I was in school, including Geometries that aremore general than manifolds. It's an exciting time to be going intothe field, IMHO.-- Shmuel (Seymour J.) Metz, SysProg and JOATAny unsolici bulk E-mail will be subject to legal action. I reserve theright to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Do not replyto === Manifold> Is the leaf space of an interesting foliation on an interesting manifold> interesting? It's often a non-Haussdorf manifold.Yes, the leaf space is interesting.;-) The following comments are for theOP and others (not really Lee).Given the small number of Hausdorff 1-manifolds, many times the leaf spaceof a foliation on a 3-manifold, is a non-Hausdorff 1-manifold. Investigating these leaf spaces is a very active field of research byitself, with great implications for the corresponding theory of3-manifolds. Recently, it was shown that there are hyperbolic 3-manifolds that do nothave taut foliations. The primary tool was investigating the action ofthe fundamental group on non-Hausdorff 1-manifolds, in particularsimply-connec ones (trees). A 3-manifold with a taut foliation has anon-trivial action on such a tree, and Roberts, Sharesian, andStein showed that some fundamental groups (of hyperbolic 3-manifolds) haveonly trivial actions, thus showing there are no taut foliations on thesemanifolds. This disproves a long-standing conjecture that everyhyperbolic 3-manifold can === something disrespectful if not blasphemous)> Actually the bible says that God crea both Adam and Jesus so he> would be the father of both.The Bible specifically calls God the Father of both: the geneology inLuke 3 concludes, ...which was the son of Adam, which was the son ofGod.> I fail to see what distinction you are making between the two...The Bible itself refers to Christ as the only begotten son, so itdoes make a distinction. Luke 3 implies a distinction for Adam, aswell, who was made from the dust of the ground. Paul states in Actsthat we are all children of God, because crea (indirectly,through procreation) by God. Paul quotes a Greek poet by way ofillustration. The virgin birth is indeed trea as === paternity of Christ (was: something disrespectful if not God crea both Adam and Jesus so he>> would be the father of both.>The Bible specifically calls God the Father of both: the geneology in>Luke 3 concludes, ...which was the son of Adam, which was the son of>God.>> I fail to see what distinction you are making between the two...>The Bible itself refers to Christ as the only begotten son, so it>does make a distinction. Luke 3 implies a distinction for Adam, as>well, who was made from the dust of the ground. Paul states in Acts>that we are all children of God, because crea (indirectly,>through procreation) by God. Paul quotes a Greek poet by way of>illustration. The virgin birth is indeed trea as a special sort of>sonship in scripture.Ah -- the Bible that talks about Adam (Old Testament) -- doesn't talkabout Jesus. (New Testament). Those are two different books writtenby different writers at different times in history. The Old Testamentis basically the Jewish Bible, and the New Testament is all aboutJesus -- and obviously was written at a later time.Both works are collections of different HUMAN authors -- which may ormay not be somewhat historically correct. It is hard to tell havethey have been rearranged and transla so many different times. Theare primitive in thought, as those people writing them had onlyprimitive knowledge of the universe to use as reference.So what do you mean by the Bible? The old Testament, === people what they can't do for themselves.currently, the politic does to the people what they can't do back.I know this, 100,000 people in Townville know this and willreport to you that the government tortures adam of the biblethe politic does for the people what they can't do for themselves,as far as I am concerned, this means the same asSeperate Church and State!Let Adam decide when he wants to be filmed, stop the Truman Show.Adam is here for the future of humanity, not your personal training.I give you all this chance to do for yourselves, if the peoplecan take religion then the government will be YOUR audience.If the people can't do for themselves the seperation of churchand state then the government is not going to let go of a freenuclear blast shield that is Adams current immortality, andtheir free infinite knowledge they spy on all his life.If the people can't seperate church and state, then you areforever to live a life that the politic will do to you anythingthat you can't do back. God is on your side, be on his.phone any of 100,000 people in Townsville Australia toverify, just like the following 4 posts, that the truman showis running live there, and the truman is constantly torturedby a satelite that penetrates buildings and a team of narcisisticagenst that work around the clock to persecute Adam. MEaus.tv know The Truman Show is true_____________________________I'm from Townsville and YOU ARE the Truman!http://tinyurl.com/iky5I was in Townsville over the weekend, and I heard him.Very spooky!http://tinyurl.com/iky8>phone someone in Townsville, half of you must know someone there,>every day I go out people say THERES THE TRUMANI'm in Townsville. We're sick of you.http://tinyurl.com/iky9http://tinyurl.com/iky4You rule === correct an error of my original post first. Force depends on>Acceleration, not Speed. Inertia is dependent on Speed and Mass, IIRC>(I'm not looking thru nething to double check this), while Force>depends on Acceleration and Mass (i.e., F=m*a).>>Acceleration, I believe, IS an absolute. Therefore, my correc post>suggests you are wrong.> Well, sort of, but hitting lightspeed.True, but you do keep accelerating, if at === sciencesCut<> That's not provable but it is equally unimportant. What is important is> that sci.math is less likely as a place for discussion of your ideas.> There simply isn't anything interesting about what you are talking> about.Is that why you hide-out in sci.math Charles? Trying to === sci.math, Donald G. Shead<9vuXa.16327$Vx2.8406685@newssvr28. news.prodigy.com>:> There are several branches of science: Chemistry is one of the most> important because it pertains to living bodies and their driving forces:> Such as thirst, hunger, libido, comfort, and survival.Um...somebody's a little confused. Inorganic chemistry studiesreactants and reactions and side issues such as electropotentials.Organic chemistry studies carbon-based molecules, both aliphaticand aromatic (benzene). (It was once thought that only life couldsynthesize organic molecules -- until someone synthesized piss (actually,carbamic acid/urea) in a laboratory. )> Mechanics is the most important science for engineering and other skilled> crafts:> Mechanics is the science of bodies of matter and the forces they exert on> each other. A thorough understanding of force and its effects is> indispensible to anyone who designs various machines, roads, and bridges.I think that's more or less right, actually.> The concept of inertia as a mathematical measure of the quantity of matter> in objects,I think that's more or less wrong, actually. You are referring to massor volume here.> bodies, and masses thereof is central to mechanics, and is the _quotient of> the ratio_ of the net force [f] exer on and/or by those objects, bodies,> and masses to the forced change in motion [(vt-vi) = s/t] that they cause:> Inertia is a constant: ft/s = w/g !Brilliant. Now what can one do with this constant? Invest itin a bank?> Inertia is the measure of mass; which is expressed in units [plural]:> In the English gravitational system presently still used in the U.S.,> mass is expressed in slugs; where oneslug = one (lbf) sec/foot = 32# sec/32'!Actually 1 slug = 32 (lbm), give or take a few ounces. It'seither a numerical accident or by planning by the appropriateStandards Committee(s) long ago that the force exer by1 lbm in the Earth's gravitational field is exactly 1 lbf.> The point is: Mass is NOT a fundamental concept of measure:Define fundamental concept of measure. You may also wantto study Avogadro's Number: if you have the right number ofcarbon atoms you'll get 12 grams. Guaranteed!Sounds fairly fundamental to me. (I don't know why they used grams,in the definition of mole, though. Probably because the cgi systemwas more fashionable at the time. )> Length, Force,> and Time are! Why is NOT the kilogram defined as> one newton sec/meter?It could be.> Why is it the other way around?Lemme guess: because of a gigantic conspiracy by the world'sscientists to keep us ignorant of the true value of the slug.Yeah. Right.> Why is the newton defined as one newton = 1 kg meter/sec?> Can't anybody see that a kilogram is one newton sec/meter?How does one store a newton? At least one can conceptually count atoms.(There is a way, but it would require quite a bit ofmachinery, to generate a newton of force, given some gas,a method by which one can isothermally adjust pressures,a piston, some plumbing, a measurement ruler, and a testspring. The fact that Rydberg's Constant is involved doesnot help.)> On page 192 of Kenneth Fords BASIC PHYSICS (1968) the> gram is 'defined' as: 1 gm = 1 dyne sec/cm, and a> kilogram is 1000 times greater isn't it?> O.K. tell me again where I'm wrong.You're not. However, the problem is that a reference mass is fareasier to create, than a reference force. (The comparison oftwo masses vs. two forces or two torques I judge to be about the same,especially since the simple pan scale compares two torques anyway;however, careful design ensures that the lever arms are equal andthe weight is in fact proportional to the mass anyway, as you areso fond to point ewill3@earthlink.netIt's still legal to go === the wrong newsgroup: sci.math.O.K. Tell me again; why is that wrong? Just 'cuz you're too dumb or too> disinteres?Apparently you have cognitive problems. My previous post identified and> referenced the newsgroup sci.math, not my personal traits.Are you trying to tell me something about the newsgroup sci.math or just> about _your own_ personal ignorance?> Physics can't be done without math! If anyone else in that newsgroup thinks> so they have chosen the wrong branch of science.Physics can't be done without some symbols either, but thatdoesn't mean every physics post is appropriate for sci.math.symbolic.You are communicating in your version of the english language.That doesn't mean your posts are on-topic for english orwriting newsgroups.Do physics books get shelved under Mathematics at yourlibrary or bookstore, if you ever patronize either of thosetypes of establishments?If you are aware the answer is === sciencesCut<> Physics can't be done without some symbols either, but that> doesn't mean every physics post is appropriate for sci.math.symbolic.> You are communicating in your version of the english language.> That doesn't mean your posts are on-topic for english or> writing newsgroups.> Do physics books get shelved under Mathematics at your> library or bookstore, if you ever patronize either of those> types of establishments?> If you are aware the answer is no, why do you think that is?> - RandyWell anyway Randolf, with physics using so much mathematics, and highway andbridge designers demetricating, I believe I'll try and keep in touch withthem. I've got some pretty good feedback from some of === grateful.The two-envelope paradox', Analysis, 55 (1995), pp. 6-11.> Emanuel Rutten>>Two envelopesI looked this up on the web. Didn't find exactly the reques paper, but> dozens of write-ups. Amusing little puzzle, though I'm not sure why it> seems to have attrac so much interest. Maybe I'm missing something,> but> here's my off-the-cuff analysis.You are given two envelopes, and told that each contains money.You are also told that the amount in one envelope is twice that in the> other.You open one envelope, discover that it contained $20You are offered the chance to switch for the other unopened envelope.You reason that the other may contain $10 or $40 with equal probability,> so> that your average return for switching is $25... better than the $20 you> have, so you switch.Of course the situation is symmetrical, so that if you had chosen the> other> first the same logic would have dicta a switch as well. Indeed you> could> reason the same way without opening any envelopes at all and repealy> argue that it is better to switch, alternating forever between the two> envelopes.Obviously something is wrong with the reasoning, but it isn't immediately> clear where the problem lies.>> IMHO: The paradox lies in the deliberate use of expectation instead of> probability. P(higher amount) = 0.5; P(lower amount) = 0.5, ergo there is> no preference. Why is expectation not appropriate in this puzzle? Because> these are not repea experiments. The expectation of $25 would come in> over time because $10 and $40 would appear roughly evenly. Think of it this> way. Let's say there is a lottery where tickets cost $1. P(winning any> prize) = 0.05 and an expectation of $0.60 (due to a HUGE top prize). Over> time you would win 60% of what you spent - but buying ONE ticket only, you> have a 95% chance of winning NOTHING.> Again - just my opinionOK, assuming you are always given the option to swap....you get the $20, so you know the other envelopes are either$10 or $40.for starters I would SWAP. Unless I was starving for a sandwich that day,$40 - $20 is more than $20 - 10. You win $20 or lose $10, its a 2 to 1payout for an even money bet.put it this way, if you could take a million times the amount, but riskgetting only 1 millionth of the amount, gee, 50% chance of winning lotto I thinkI'd take it for a $19.99 ticket. But you know if the envelope you have has$20,000,000 you don't swap.Its not symmetrical, given an envelope and told the OTHER envelop has$20, you keep the mystery envelope. The 2 to 1 payout for === envelope paradox>For n >= 1, with probability 1/2^n, I place 3^(n-1) and 3^n dollars in >the two envelopes, and you know that I have followed this rule. Now you>play the game as before.>>You open one envelope and observe see x = 3^m dollars. Of course, if x=1, then>you definitely swap.>>Otherwise, you reason, that the n that I used must have been m-1 or m.>Since it is twice as likely to have been m-1 as m, the other envelope>contains x/3 with probability 2/3, and 3x with probability 1/3.>So your expec gain from swapping is 2x/9 + x = 11x/9, and you should >always swap.> That's a good one.> The first thing to note is that before the game your expec win is> infinite, whether you swap or not. So there's no paradox there.> As a strategy for a sequence of games, swapping just changes your> expec gain from infinite to infinite*11/9.No. Infinite expectations are irrelevant since there exist paradoxicalcases that do not include infinite expectations.Lets construct such a case.Let X denote the smallest of the two values in both envelopes. Thereexist a probability distribution for X such that the mean expecgain on swapping is finite (see M. Clark and N. Shackel, Mind Vol.109, 2000).Now pick this distribution for X and use the following game setup toavoid infinite expectations:* You get nothing unless you swap,* If you swap you will get the amount gained or must pay the amountlost.In this case the paradox emerges without using infinite expectations.> To make the expectation be finite, you need to do something like> consider only cases where the first envelope contains less than K> dollars. But then it seems less paradoxical: given the distribution> and the fact that the first envelope contains (say) less than 10> dollait's not so surprising that the expec value of the === touch with this. > I hope it will be helpful to you.> The identity sign is * for T and + for R.OK - that is no problem.One problem is that in the standard math we have UNARY and BINARY + aswell as UNARY and BINARY -Now we will require UNARY and BINARY *That implies a NEW function - NEITHER an addition nor a subtraction.The A*B function.I can imagine your bank manager writing to you DEAR MR. GOLDEN, YOURACCOUNT IS NEITHER IN CREDIT (+) NOR IN DEBIT (-), BUT IN STAR (*) -HOW DO YOU PROPOSE TO REMEDY THE SITUATION?In addition, in any new commutativity &c. For example, the Boolean AND and OR,together with NAND, NOR, XOR and XNOR are ALL commutative.In conventional maths, + and * are commutative whilst - and / are not.We need a lot of precise definitions of this system before we can seewhether it is of use.Regarding Quarks &c. - I am just looking loosely for places where aweird math might === : T spaceI believe that all of these are already inherent in the signmechanics.Subtraction is still summation, just of a negative sign.i.e. in R: 4 - 5 = (+4) + (-5) = -1.In three-signed, star (*) is the general summation operator.The plus operator in R and the star operator in Y both preserve thesign of the values they are applied to. There is no actual thirdoperator. It is just sign modification.Suppose y1, y2 in Y. y1 * y2 means the sum of y1 and y2. y1 - y2 = y1 * (-y2) which means the sum of these values with y2'ssign modified.Suppose y1 = - 2 + 3 and y2 = + 2 * 4. y1 * y2 = - 2 + 3 + 2 * 4 = - 2 + 5 * 4 = + 3 * 2.Because -(y2) = -( + 2) -( * 4 ) = * 2 - 4: y1 - y2 = - 2 + 3 * 2 - 4 = - 6 + 3 * 2 = - 4 + 1.And y1 + y2 = - 2 + 3 - 2 + 4 = - 4 + 7.It believe it is best just to use the term summation and let the signoperators do their thing. It's all consistent and the concern overunary and binary goes away.> One problem is that in the standard math we have UNARY and BINARY + as> well as UNARY and BINARY -> Now we will require UNARY and BINARY *> That implies a NEW function - NEITHER an addition nor a subtraction.> The A*B function.> I can imagine your bank manager writing to you DEAR MR. GOLDEN, YOUR> ACCOUNT IS NEITHER IN CREDIT (+) NOR IN DEBIT (-), BUT IN STAR (*) -> HOW DO YOU PROPOSE TO REMEDY THE SITUATION?> In addition, in any new math every function has to the Boolean AND and OR,> together with NAND, NOR, XOR and XNOR are ALL commutative.> In conventional maths, + and * are commutative whilst - and / are not.But they have simple conversions to the straight product and sum:for a, b in R a / b = a(1/b) = (1/b)a. a - b = a + (-b) = (-b) + a.This method is applicable in three-signed as well.The reciprocal value should be *1.> We need a lot of precise definitions of this system before we can see> whether it is of use.> Regarding Quarks &c. - I am just looking loosely for places where a> weird math might fit.> Charles las WehnerMaybe one way to convince you is to complicate sign modification bynesting more signs:In R: -(+(-(-1) = -1In Y: -(+(-(-1) = +1Working in R you count around the number of poles you pass and thelast one you come to is the sign of the number. The same is true in Y.In effect the sign operator should be viewed as a discrete === missed my post yesterday so I've cut and paste it as follows: You seem to be quite a dedica amateur mathematician. Good for you. Iwas wondering what you do for a living? Also, Nora Baron has asked that you refute her objections to your proofin her is in your interest to do so? If you're right about you proofof === Dumbass, JSH, and FLTHarris ALMOST makes me miss Archimedes Plutonium.> To cracksmokin20003@yahoo.com ()> Stop forgering James!> That's no way of stopping him. And we shouldn't! He is funny some times. skrev i melding> === monkeys typing Shakespeare> Working on averages...> A page is 20 lines by 10 words = 200 words.> 1000 pages is 200,000 words> 200,000 words by 5 characters = 1 million chracters.> Probablity a monkey that typed 1 million characters came up with Shakespeare> is 30^1,000,000 (26 chars plus punctuation).to 1....A million monkeys with a million typewriters however,Average lifespan of monkey = 50 years,Average characters per day = 1 per second * 10 hoursCharacters typed = 36000 * 365 * 50 = 657 million.So what is the longest specific sequence you can expect to be typed?sequence length = number of sequences * prob. of sequence being corrects = (657000000/s) * (1/30)^ss/5 = number of continuous words of Shakespeare a million monkeys will typemake that s/8, Shakespeares vocab was 20,000 === there are all sorts of possible outputs> of the monkeys that don't contain the works of Shakespeare. Maybe afterTo be or not to be, that is the they always type banana. Cardinality> is not particularly relevant here.I don't think that the monkeys _always_ type banana after To be or not tobe, that is the.There is at least one report of a monkey typing gezorkenplatt at thatpoint.http://aroundcny.com/technofile/texts/compupoet85. htmlHowever, there seems to be some doubt as to the exact spelling of that word.I have come across: gazornenplat Gazornumplat gerZonoNPplat gezorkenplatt gezornenbltzx gezornenplatz grrdnm zsplkt g'zorn'nplatt kazornenplat mzyxrplx qqhop90e;f[ewfweqfI think that the person who used mzyxrplx was getting mixed up with MrMxyzptlk, an enemy of === Towards a proof of infinite monkeys typing Shakespeare> Now before you look at the title and think I'm some kind of anti-evolution> nut, don't. I got into a discussion on the forums at Ars Technica> ( http://www.arstechnica.com , discussion link ishttp://arstechnica.infopop.net/OpenTopic/page?a=tpc&s= 50009562&f=34709834&m=9270966775> -- I'm CanSpice in that thread and the person on the other side of the> argument is Aylutar) about trying to prove the old adage that if you had> an infinite number of monkeys typing on an infinite number of typewriters> for an infinite amount of time, you would produce the works of> Shakespeare.> I initially tackled it from a probability point-of-view, showing that the> probability that you do _not_ have the works of Shakespeare as you take> time to go to infinity goes to zero. Thus the probability that you do have> the works of Shakespeare in infinite time is one. That's fairly easy to> show.> Then the other side presen an infinitely long string that did not have> the works of Shakespeare in it: aaaXaaaXaaaXaaaX.... where X is one of> the choosable characters (i.e. the alphabet, space, and punctuation). It's> still random in some sense of the word, so it should be one of the> strings typed by one of the monkeys.> Then I proved that the probability of getting this string is zero.> Then Aylutar broke out set theory. He first sta that there are a> countable number of monkeys (aleph-0 of them) and they would produce a> countable number of strings (aleph-0 of them). Switching to numbers (which> is fine because we're merely reducing the number of possible things to> choose from -- instead of 30-odd things we have 10), he then said that the> domain of possible outputs of the monkeys is (0,1]. Since the number of> numbers in (0,1] is larger than the number of strings the monkeys can> produce, it's perfectly valid to say that 0.1001001001001001... is a> possible output of the monkeys, so since he's provided a counter-example> that does not have the works of Shakespeare in it (let's say Hamlet is> actually 1234978656235634), you cannot be guaranteed that the monkeys> will produce Shakespeare.> To which I got confused because I don't know set theory, but I have a> sneaking suspicion that you can't just jump into real numbers the way he> did by bringing in (0,1] like that (and, for that matter, I wonder why he> didn't choose [0,1], since 0 should be a possible output too, according to> him...).> I tried responding with something along those lines. The monkeys are> outputting integenot real numbeso you can't say that the domain of> all possible outputs is the real numbeit's integers.> To which he said that we'll just put a decimal point in front of whatever> number the monkey types and hey presto, we've got a real number.> To which I believe he's doing a one-to-one mapping from the integers to> the reals, showing that there are just as many integers as there are reals> (which is inherently false).> So, my question is, who's right? And further, why? And further further,> why is the other person wrong? Where's the flaw in their reasoning?> If you want to read the thread in question, it only really heats up after> the third page or so.I think the situation might be clearer if we change the initial conditionsslightly: We have an infinite number of monkeys but only a finite time.Let us make the following suppositions:# The monkeys type at one character per minute, at the chime of a bell.# We have a string of characters available which is the definitive Works ofShakespeare.# This definitive string is exactly 1 million characters long.# Each monkey chooses its next character at random from a fixed set of 100characters. Each character being chosen with a probability of 0.01On the first chime 99% of the monkeys press the wrong key. We ignore themfrom now on. But there are still an infinite number of monkeys who havepressed the right key.On the second chime 99% of the monkeys who had got the first character rightpress the wrong key. We ignore them from now on. But there are still aninfinite number of monkeys who are still following the definitive text.On the third chime 99% of the monkeys who had got the first two charactersright press the wrong key. We ignore them from now on. But there are stillan infinite number of monkeys who are still following the definitive text....On the millionth chime (after about two years of typing) 99% of the monkeyswho had got the first 999,999 characters right press the wrong key. Butthere are still an infinite number of monkeys who have now typed out thecomplete works of Shakespeare.-- Clive === of infinite monkeys typing Shakespeare>>Then Aylutar broke out set theory. He first sta that there are a >>countable number of monkeys (aleph-0 of them) and they would produce a >>countable number of strings (aleph-0 of them). Switching to numbers (which >>is fine because we're merely reducing the number of possible things to >>choose from -- instead of 30-odd things we have 10), he then said that the >>domain of possible outputs of the monkeys is (0,1]. Since the number of >>numbers in (0,1] is larger than the number of strings the monkeys can >>produce, it's perfectly valid to say that 0.1001001001001001... is a >>possible output of the monkeys, so since he's provided a counter-example >>that does not have the works of Shakespeare in it (let's say Hamlet is >>actually 1234978656235634), you cannot be guaranteed that the monkeys >>will produce Shakespeare.>> This is a red herring. Of course there are all sorts of possible outputs>> of the monkeys that don't contain the works of Shakespeare. Maybe after To be or not to be, that is the they always type banana. Cardinality >> is not particularly relevant here.>Yes, but how do you definitively prove that? Since, as you say, there are >all sorts of possible outputs that don't contain the works of Shakespeare, >how can you be guaranteed that they won't type one of those outputs? It >completely clashes with showing the probability that they won't type >Shakespeare goes to zero as time goes to infinity, but how do you resolve >the conflict?There is no conflict, because proability = 0 does not imply impossible, and probability = 1 does not imply certain.Here's an informal proof that if you choose a string of infinitely many letters 'kugiufwnv...' at random thenthe probability that the word 'cat' appears is 1: Weprove this in steps:1. The probability that the letter 'c' appears at least once is 1:Pf: This is the same as saying the probability that the letter'c' never appears is 0. Think about what has to happenif the letter 'c' never appears The first letter has to beother than 'c', which happens with probability 25/26.The second letter also has to be other than 'c'; theprobability that neither of the first two letters is 'c'is (25/26)^2, that is, 25/26 squared.Now the probability that _none_ of the first N lettersis 'c' is (25/26)^N. This probability tends to 0 as N tends to infinity, so the probability that none ofthe infinitely many letters is 'c' is 0. QED(Step 1).2. The probability that the letter 'c' appears infinitelymany times is 1.Pf: The probability that it appears at least once is1. Now after it appears once repeat the sameargument: The probability of an occurence of 'c'after the first one is also 1, so the probabilitythat there are at least two 'c's is 1. And theprobability that there are at least 3 'c's is also1, for the same reason... QED.3. The probability that the string 'ca' appearsat least once is 1.Pf: We know that with probability 1, there areinfinitely many 'c's. What has to happen forthere to be _no_ 'ca's? The letter after thefirst 'c' has to be other than 'a'; this happenswith probability 25/26. The probability thatneither the first nor the second 'c' is followedby 'a' is (25/26)^2; the probability that noneof the first N 'c's is followed by 'a' is (25/26)^N.Since this tends to 0, the probability thatnone of the infinitely many 'c's is followed by'a' is 0. QED.4. The probability that there are infinitely many'ca's is 1.Pf: See the proof for Step 2.5. The probability that there is at least one 'cat'is 1.Pf: === Re: Towards a proof of infinite monkeys typing Shakespeare> 5. The probability that there is at least one 'cat'> is 1.> Pf: See the proof of Step 5.In the index to Rotman's Theory of Groups we find Navel, Maurice, seePippik, Moishe and Pippik, === infinite monkeys typing Shakespeare> This is a red herring. Is that some kind of Girdle Sentence, or something? (I like thescales > a lot, in any case--very iconic.)Lee Rudolph, wondering how long it would take an infinite number ofred === proof of infinite monkeys typing Shakespeare>> 5. The probability that there is at least one 'cat'>> is 1.>> Pf: See the proof of Step 5.Just checking to see if you were paying attention...>In the index to Rotman's Theory of Groups we find Navel, Maurice, see>Pippik, Moishe and Pippik, Moishe, see === a proof of infinite monkeys typing Shakespeare> sequence length = number of sequences * prob. of sequence being correct> s = (657000000/s) * (1/30)^sGuess I'll work it out then,,make it s = 1,000,000,000 / s * 0.1^stry s = 1010 = 100,000,000 * 0.0000000001pretty close, a million monkeys on a million === Re: Towards a proof of infinite monkeys typing ShakespeareOriginator: hack@watson.ibm.com (hack)> ... For example, a sufficient assumption is that there is epsilon > 0 >such that, for any letter x, at all times the conditional probability of >typing x next, given what has been typed already, is greater than epsilon.Note that the order of quantifiers is essential. It would not be sufficientthat the probability of typing x be non-zero given what has been typed already.(Shades of uniform vs not, in terms of === does not imply certaintywhere he states that a probability of 0 does not imply impossibleand a probability of 1 does not imply certainty.Please help the slow pokes!the limit of p as the number of tries goes to infinity, p approaches1 (or zero).But, does that mean that the limit of p as it approaches asymptoticallyto 1 (or 0) is equal to 1 (or 0)? I just thought it is, but onlyby convention? and that there is just a smidgen of difference?Please clarify.Now, onto a rela subject. In the book store, I read the back coverof a book entitled Non-standard Analysis by Abraham Robinson. I recalla review stating that the book contains really cool stuff involving infinitesimals (cool stuff is not the exact wording). Well the backof the book states that Robinson's work remains controversial even tothis day. Well, why would it be controversial? I thought controversywas for the inexact === Re: Ullrich says probability 1 does not imply certainty> where he states that a probability of 0 does not imply impossible> and a probability of 1 does not imply certainty.> Please help the slow pokes!> the limit of p as the number of tries goes to infinity, p approaches> 1 (or zero).> But, does that mean that the limit of p as it approaches asymptotically> to 1 (or 0) is equal to 1 (or 0)? I just thought it is, but only> by convention? and that there is just a smidgen of difference?> Please clarify.Well its probability its not what WILL happen.A coin tossed an infinite number of times has no reason to not go :H H H H H H H H H H H H H H H H H Heven though the probability of atleast 1 tail is 1, it is not certain.i.e. an outcome === not imply certainty> Now, onto a rela subject. In the book store, I read the back cover> of a book entitled Non-standard Analysis by Abraham Robinson. I recall> a review stating that the book contains really cool stuff involving > infinitesimals (cool stuff is not the exact wording). Well the back> of the book states that Robinson's work remains controversial even to> this day. Well, why would it be controversial? I thought controversy> was for the inexact sciences where mathematical proof is out of reach?Non-standard analysis itself is not subject to controversy. However, various claims about it's naturality as a formalisation of the intuitive concepts of analysis is. This is basicly a osophical (and partly pedagogical) controversy.-- Aatu Koskensilta (aatu.koskensilta@xortec.fi)Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus === does not imply certainty>where he states that a probability of 0 does not imply impossible>and a probability of 1 does not imply certainty.I would say logical necessity rather than certainty. If I flip a fair coin infinitely many times, it is not logically necessarythat it will come up tails at least once: the sequence of outcomes(H, H, H, ...) is just as possible as any other particular sequence,and I know _some_ sequence will occur.But I am certain that it will happen (the probability is 1).>the limit of p as the number of tries goes to infinity, p approaches>1 (or zero).a sequence of trials. In the infinite sequence, the probability is 1. This particular type of event happens if and only if it happens in some finite number of trials, and the probability p(N) that it happens at least once in N trials approaches 1 as N -> infinity. On the other hand, there are other events having to do with an infinite sequence of trials where you can't just look at a finite number of trials and decide that the event occurs. So not every event of probability 1 arises in this way.>But, does that mean that the limit of p as it approaches asymptotically>to 1 (or 0) is equal to 1 (or 0)? I just thought it is, but only>by convention? and that there is just a smidgen of difference?I don't understand what you're asking about. Whether the limit of p as p -> 1 is 1? Of course it is! That's one of the easiest facts to proveabout limits, but what does it have to do with the question of whetherthe limit of p(n) as n -> infinity is 1?Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia === probability 1 does not imply certainty> where he states that a probability of 0 does not imply impossible> and a probability of 1 does not imply certainty.> Please help the slow pokes!The technical term is almost certain for an event ofprobability 1. This emphasizes the difference between P=1and certainty.> the limit of p as the number of tries goes to infinity, p approaches> 1 (or zero).> But, does that mean that the limit of p as it approaches asymptotically> to 1 (or 0) is equal to 1 (or 0)? I just thought it is, but only> by convention? and that there is just a smidgen of difference?Well, you have to keep in mind the difference between finiteand infinite, first of all. What a limit of 1 means here is thatfor all finite sequences of monkey-typewriter outputs thereis indeed a smidgen of difference from 1. This smidgen isfinite. If you probability that it contains Hamlet is closeto 1, but still not equal to 1. And the longer you go, thesmaller that difference gets.The LIMIT is exactly 1, no difference at all. But the limit isan abstract, another way of describing the trend as the finiteexperiments get longer and longer. The limit is 1 meansthe finite difference of P from 1 for finite experimentscan get as small as you like. All realizable experiments involvingmonkeys and typewriters will involve events at finite timeswith finite numbers of monkeys and finite length strings.That's all a separate issue from why P=1 (exactly) doesn'tmean certain. Very informally, if you have an infinite numberof equally probable events (such as possible monkey outputs)then the probability of any given one is 0. Thus the probabilitythat it doesn't happen is 1. Yet obviously any given stringis possible.Formally, it has to do with the measure-theory foundations ofmodern probability theory. A finite subset of an infinitecollection of points has measure 0 and thus usuallyprobability 0 of occurring. Yet it can still be part ofthe set of possible events.Again informally, the terminology almost certain suggeststhat for all practical purposes you can consider it certaineven though technically it's not. If you bet on an a.c. event,the probability of === 1 does not imply certainty>where he states that a probability of 0 does not imply impossible>and a probability of 1 does not imply certainty.That's correct. Did you have a question about that?Suppose that you flip a coin infinitely many times. What isthe probability that you get all heads? Zero. Does that meanthat it's impossible to get all heads? No, you get _some_sequence, and any given sequence is just as likely as anyother.>Please help the slow pokes!>the limit of p as the number of tries goes to infinity, p approaches>1 (or zero).Yes. If, for example, p(n) is the probability that there is no 'c'among the first n letters in that infinite random string, I gavea proof that p(n) tends to 1 as n tends to infinity; it follows thatthe probability that there is a 'c' somewhere in the (infinite)string is 1.>But, does that mean that the limit of p as it approaches asymptotically>to 1 (or 0) is equal to 1 (or 0)? I just thought it is, but only>by convention? and that there is just a smidgen of difference?>Please clarify.I have no idea what the question is, sorry.In case it helps, all of the following expressions mean exactlythe same thing:p(n) tends to 1 as n tends to infinityp(n) -> 1 as n -> infinitythe limit of p(n) as n tends to infinity is equal to 1the limit of p(n) as n tends to infinity is equal to 1,exactly, without even a smidgeon of a differenceyou can make p(n) as close as you like to 1, merelyby taking n large enoughfor any number e > 0 there exists N such that |p(n) - 1| < efor all n greater than Nlim_{n->infinity} p(n) = 1.>Now, onto a rela subject. In the book store, I read the back cover>of a book entitled Non-standard Analysis by Abraham Robinson. I recall>a review stating that the book contains really cool stuff involving >infinitesimals (cool stuff is not the exact wording). Well the back>of the book states that Robinson's work remains controversial even to>this day. Well, why would it be controversial? I thought controversy>was for the inexact sciences where mathematical proof is out of reach?It's a totally unrela subject. To answer your question, there'snothing controversial about the correctness of the mathematics,the controversy (if there is one) is over the question of whetherit's the best way to do === probability 1 does not imply certainty> Suppose that you flip a coin infinitely many times. What is> the probability that you get all heads? Zero. Does that mean> that it's impossible to get all heads? No,...Yes.> you get _some_sequence, and any given sequence is just> as likely as any other.Although, a priori, any_finite_sequence of a given length is as likelyas any other, I claim: When flipping a fair coin, an infinite sequenceof heads is impossible, theoretically. Such a sequence, violates theaxioms of probability. If an event has non-zero probability it mustoccur in infinitely-many trials with probability 1. Gran, though,there may be arbitrarily long sequences, beginning HHH...HT...As you write:If, for example, p(n) is the probability that there is no 'c'> among the first n letters in that infinite random string, I gave> a proof that p(n) tends to 1 as n tends to infinity; it follows that> the probability that there is a 'c' somewhere in the (infinite)> === not imply certainty>> Suppose that you flip a coin infinitely many times. What is>> the probability that you get all heads? Zero. Does that mean>> that it's impossible to get all heads? No,...>Yes.>> you get _some_sequence, and any given sequence is just>> as likely as any other.>Although, a priori, any_finite_sequence of a given length is as likely>as any other, I claim: When flipping a fair coin, an infinite sequence>of heads is impossible, theoretically. You can claim that if you want. It's not so.>Such a sequence, violates the>axioms of probability. Exactly which axiom is that?>If an event has non-zero probability it must>occur in infinitely-many trials with probability 1. That's not an axiom in the probability theory I know.It's a theorem (or it would be if we added the assumptionsthat have been left unsta through all this.) But itdoes not imply that an infinite sequence of heads isimpossible, just that it has probability zero.Here's an example that may or may not helpclarify why probability 0 does not imply impossible:Chose a real number x between 0 and 1, at random.What's the probability that x = 1/2? That probabilitymust be zero, because it's the same as the probabilityyou get any other value, and there are infinitely manypossible values. But x = 1/2 is _possible_, for the samereason: x has _some_ value, and they _all_ haveprobability zero.(Note that actually choosing that number between 0 and1 is exactly equivalent to choosing a sequence of headsand tails - think about binary expansions. The reason Ichanged it to a real between 0 and 1 was hoping itmight be more clear in that setting why every outcomeis as likely as every other outcome.)>Gran, though,>there may be arbitrarily long sequences, beginning HHH...HT...>As you write:>If, for example, p(n) is the probability that there is no 'c'>> among the first n letters in that infinite random string, I gave>> a proof that p(n) tends to 1 as n tends to infinity; it follows that>> the probability that there is a 'c' somewhere in the === Re: Ullrich says probability 1 does not imply certainty> where he states that a probability of 0 does not imply impossible> and a probability of 1 does not imply certainty.> Please help the slow pokes!> the limit of p as the number of tries goes to infinity, p approaches> 1 (or zero).If the set of possible outcomes to a probabilistic experiment is finite, then an event with 0 probability is impossible, but when the set of possible outcomes is suitably infinite, as in the set of reals betwen 0 and 1, it may happen that every single outcome event has zero probability.Such situations are more theoretical than actual, since there is no way physically of actually having an infinite set of outcomes to any practical probabilistic === imply certainty>> Suppose that you flip a coin infinitely many times. What is>> the probability that you get all heads? Zero. Does that mean>> that it's impossible to get all heads? No,...>Yes.>> you get _some_sequence, and any given sequence is just>> as likely as any other.>Although, a priori, any_finite_sequence of a given length is as likely>as any other, I claim: When flipping a fair coin, an infinite sequence>of heads is impossible, theoretically.> You can claim that if you want. It's not so.>Such a sequence violates the axioms of probability.> Exactly which axiom is that?>If an outcome has non-zero probability it must>occur in infinitely-many trials with probability 1. > That's not an axiom in the probability theory I know.I am saying it would lead to a contradiction from the axioms...> It's a theorem (or it would be if we added the assumptions> that have been left unsta through all this.) But it> does not imply that an infinite sequence of heads is> impossible, just that it has probability zero.> Here's an example that may or may not help> clarify why probability 0 does not imply impossible:I was not disputing that. But now that you mention it....> Choose a real number x between 0 and 1, at random.This is not a well-defined experiment.> What's the probability that x = 1/2? That probability> must be zero, because it's the same as the probability> you get any other value, and there are infinitely many> possible values. But x = 1/2 is _possible_, for the same> reason: x has _some_ value, and they _all_ have> probability zero.Any set of mesure zero has probability zero.But what is one to do? Maybe generate a ternarydecimal expansion by some random process andcheck if the genera number is in the Cantor set?!We can find out in finitely many steps if a numberis NOT, say, 1/2 but we can never be sure if such anumber is 1/2.> (Note that actually choosing that number between 0 and> 1 is exactly equivalent to choosing a sequence of heads> and tails - think about binary expansions. The reason I> changed it to a real between 0 and 1 was hoping it> might be more clear in that setting why every outcome> is as likely as every other outcome.)>Gran, though,>there may be arbitrarily long sequences, beginning HHH...HT...>As you write:>>If, for example, p(n) is the probability that there is no 'c'>> among the first n letters in that infinite random string, I gave>> a proof that p(n) tends to 1 as n tends to infinity; it follows that>> the probability that there is a 'c' somewhere in the === probability 1 does not imply certainty: where he states that a probability of 0 does not imply impossible: and a probability of 1 does not imply certainty.Maybe it's easier to think about subsets of [0,1].Some subsets of the unit interval have length 0 without beingempty. These correspond to possible but probability 0.Some subsets have length 1 without covering the whole interval,e.g. (0,1). A point picked from [0,1] has probability1 of being in (0,1) (the quotient of their lengths), butit's possible that you could pick an === not imply certaintyX-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Punge: Micro$oftX-Sanguinate: themvsguy@email.comX-Terminate: SPA(GIS)X-Tinguish: Mark Griffith X-Treme: C&C,DWS at 11:39 PM, buffcoder@hotmail.com (Buffy The Cache Coder) said:>where he states that a probability of 0 does not imply impossible and>a probability of 1 does not imply certainty.Of course. Pick an integer at random. What is the probability ofpicking any particular integer? Unless the odds asre skewed, it'szero.>Now, onto a rela subject. In the book store, I read the back>cover of a book entitled Non-standard Analysis by Abraham Robinson. It's a classic.>Well, why would it be controversial? The book itself shouldn't be controversial to anyone doing ModelTheory. Using NSA in introductory courses is controversial becausethere is no agreement as to whether the students would understand whatwas going on.>I thought controversy>was for the inexact sciences where mathematical proof is out of>reach?The controversies in Mathematics are over foundational issues: whataxiom systems are reasonable and what proof methods are reasonable.Google for intuitionist as an example.-- Shmuel (Seymour J.) Metz, SysProg and JOATAny unsolici bulk E-mail will be subject to legal action. I reserve theright to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Do not replyto === probability 1 does not imply certainty> at 11:39 PM, buffcoder@hotmail.com (Buffy The Cache Coder) said:>>where he states that a probability of 0 does not imply impossible and>>a probability of 1 does not imply certainty.> Of course. Pick an integer at random. What is the probability of> picking any particular integer? Unless the odds asre skewed, it's> zero.The probability can't be zero for each integer, because probabilitymeasures are required to have two properties that make them incompatiblewith your claim: (a) probability measures are countably additive, and(b) the probability of the entire space must be 1.There is no such thing as a uniform probability distribution on theintegers.-- Dave SeamanJudge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Suppose that you flip a coin infinitely many times. What is> the probability that you get all heads? Zero. Does that mean> that it's impossible to get all heads? No,...>>Yes.> you get _some_sequence, and any given sequence is just> as likely as any other.>>Although, a priori, any_finite_sequence of a given length is as likely>>as any other, I claim: When flipping a fair coin, an infinite sequence>>of heads is impossible, theoretically.>> You can claim that if you want. It's not so.>>Such a sequence violates the axioms of probability.>> Exactly which axiom is that?>>If an outcome has non-zero probability it must>>occur in infinitely-many trials with probability 1.> That's not an axiom in the probability theory I know.>I am saying it would lead to a contradiction from the axioms...Exactly how does it lead to a contradiction?>> It's a theorem (or it would be if we added the assumptions>> that have been left unsta through all this.) But it>> does not imply that an infinite sequence of heads is>> impossible, just that it has probability zero.>> Here's an example that may or may not help>> clarify why probability 0 does not imply impossible:>I was not disputing that. But now that you mention it....>> Choose a real number x between 0 and 1, at random.>This is not a well-defined experiment.Of course not - the entire discussion has been informal.You can't actually flip a coin infinitely many times, either.Here's the correct way to say Choose a real number cbetween 0 and 1 at random:Let X be a random variable uniformly distribu on [0,1].>> What's the probability that x = 1/2? That probability>> must be zero, because it's the same as the probability>> you get any other value, and there are infinitely many>> possible values. But x = 1/2 is _possible_, for the same>> reason: x has _some_ value, and they _all_ have>> probability zero.>Any set of mesure zero has probability zero.>But what is one to do? Maybe generate a ternary>decimal expansion by some random process and>check if the genera number is in the Cantor set?!>We can find out in finitely many steps if a number>is NOT, say, 1/2 but we can never be sure if such a>number is 1/2.>> (Note that actually choosing that number between 0 and>> 1 is exactly equivalent to choosing a sequence of heads>> and tails - think about binary expansions. The reason I>> changed it to a real between 0 and 1 was hoping it>> might be more clear in that setting why every outcome>> is as likely as every other outcome.)>>Gran, though,>>there may be arbitrarily long sequences, beginning HHH...HT...>>As you write:>>If, for example, p(n) is the probability that there is no 'c'> among the first n letters in that infinite random string, I gave> a proof that p(n) tends to 1 as n tends to infinity; it follows that> the probability that there is a 'c' somewhere in the (infinite)> string is === probability 1 does not imply certainty>> where he states that a probability of 0 does not imply impossible>> and a probability of 1 does not imply certainty.>> Please help the slow pokes!>> the limit of p as the number of tries goes to infinity, p approaches>> 1 (or zero).>If the set of possible outcomes to a probabilistic experiment is >finite, then an event with 0 probability is impossible, but when the >set of possible outcomes is suitably infinite, as in the set of >reals betwen 0 and 1, it may happen that every single outcome event >has zero probability.It's not quite that simple - what you say is true if all the outcomesare equally likely. Here's an experiment: Toss a coin infinitelymany times, and set X = 0 if you get HHH...., X = 1 otherwise.There are only two possible outcomes, but P(X=0) = 0.>Such situations are more theoretical than actual, since there is no >way physically of actually having an infinite set of outcomes to any >practical probabilistic experiment.True. But it does not follow that such things are of no practicalimportance! Experiments involving infinitely many trials areuseful in understanding real experiments with a large finitenumber of === probability 1 does not imply certaintyX-mimeole: Produced By Microsoft MimeOLE V6.00.2727.1300>I claim: When flipping a fair coin, an infinite sequence> of heads is impossible, theoretically.> You can claim that if you want. It's not so.> Such a sequence violates the axioms of probability.> Exactly which axiom is that?>> If an outcome ( of a flip) if has non-zero probability it must>occur in infinitely-many trials with probability 1.> That's not an axiom in the probability theory I know.>I am saying it would lead to a contradiction from the axioms...> Exactly how does it lead to a contradictionIf you specify a number, say zero, then its decimalexpansion is what it is.However, if you reverse the process and claim tobe able to generate binary expansions of all numbersbetween zero and 1, by flipping a fair coin infinitelymany times, then you will fail to generate even_one_rationalnumber. In this way, one gets that the probability of pickingrational is zero and sees that the set === probability 1 does not imply certaintyX-mimeole: Produced By Microsoft MimeOLE V6.00.2727.1300> Experiments involving infinitely many trials are> useful in understanding real experiments with> a large finite number of trials.Maybe but surely not always.There are limits. And don'tprentend to be a === imply certainty> Now, onto a rela subject. In the book store, I read the back cover> of a book entitled Non-standard Analysis by Abraham Robinson. I recall> a review stating that the book contains really cool stuff involving > infinitesimals (cool stuff is not the exact wording). Well the back> of the book states that Robinson's work remains controversial even to> this day. Well, why would it be controversial? I thought controversy> was for the inexact sciences where mathematical proof is out of reach?> Non-standard analysis itself is not subject to controversy. However, > various claims about it's naturality as a formalisation of the intuitive > concepts of analysis is. This is basicly a osophical (and partly > pedagogical) controversy.Non-standard analysis has a funny situation. As no, from atechnical point of view it is absolutely correct and rigorous. Moreover, most analysts use it informally all the time. Then whenthey are satisfied with their results, they recast them inepsilon-delta form. Robinson showed that this was unnecessary, butthey do it anyway. For one thing, the construction of non-standardnumbers is considered funny-business, involving logic, ultrafiltersand that sort of thing most analysts don't feel comfortable with.It reminds me a little about the introduction of homology groupsreplacing Betti numbers and torsion numbers in topology. This wasfirst done in print by Emmy Noether. A decade or so ago someone askedLeopold Vietoris (who died at age 109 a year or so ago) about this andhe said that every knew there were homology groups, but it was notthought appropriate to write papers using them. Once Noether brokethe taboo, everyone did it. Will this happen here? I don't know. But the controversy is primarily pedagogical. Should we teachcalculus using them when their development is rather sophistica? Well Dedekind cuts are fairly sophistica too and we don't worryabout omitting any discussion of the real line from elementarycalculus.Consider the following two statements:For every epsilon > 0, there is a delta such that |y - x| < deltaimplies|f(y) - f(x)| < epsilon.For every infinitesimal delta, f(x+delta) - f(x) is infinitesimal. An AE=> statment has been reduced to a simple A. All the rest of thecomplication has been hidden in the === probability 1 does not imply certainty> where he states that a probability of 0 does not imply impossible> and a probability of 1 does not imply certainty.> Please help the slow pokes!The issue involved belongs to what they call measure theory. Here's(hopefully) enough to get your intuitions working the right way.Measure may be defined for sets of points in the plane. I'll just dothe easiest.The measure of a rectangle is its area (length * width).Fact: A line segment has measure 0Proof: Let the segment have length l. Then the segment is a rectangleof length l and width 0, and as such, has measure 1*0 = 0Fact: A point has measure 0Proof: A slight adaptation of the previous proof.The relevance of this is that probability is just a ratio between 2quantities: the area of everything, and the sub-area that you'reinteres in.Take a dart board (somewhat idealized). The ratio between the 13area and the whole dart board is approximately 1/20 (hell if I knowhow much space the bullseye takes :). In probability-speak, theprobability of hitting the 13 with randomly-aimed darts is 1/20.(There is stuff about in the long run involved in some the technicaldetails here, but I don't think it's particularly relevant or helpfulto say anything about it.)Q: What's the probability of hitting some point exactly on the outerborder of the dart board? Well, what's the *area* of that set ofpoints that we're interes in? Of course a circumfrence is a(curved) line segment, one-dimensional, and thus has 0 area (asabove). So the probability of hitting the exact border of the dartboard with a randomly-aimed dart is 0/20 = 0.Note that it isn't *impossible* (in just about any reasonable sense)that a dart (idealized) hit the border - there's jillions ofborder-points just waiting to be hit. This same line of thought couldbe applied to any line segment drawn within the dart board.Most boundaries that you can easily think of have measure 0. But thereare boundaries that actually have non-zero measure. I'll let otherpeople fill in various technical details about boundaries, areas,inner/outer limits of this-and-that - but the above is the basic ideabehind why probability 0 doesn't mean impossible.BTW, there's nothing special about 2-dimensional stuff (or even beingdimensional at all). For example, let the experiment be to pick realnumbers out of [0,10]. A piece of the real line is 1-dimensional, sowe'll use the 1-d analogue of rectangles: intervals. The area ormeasure of an interval [a,b] is its length, b-a. Then the first ofthe above Facts, in this context, becomes:Fact: A single number has measure 0Proof: A single number r is an interval, namely [r,r]. The measure ofthe interval is r-r=0The boundary of an interval [a,b] is the pair of points === does not imply certainty>> Now, onto a rela subject. In the book store, I read the back cover>> of a book entitled Non-standard Analysis by Abraham Robinson. I recall>> a review stating that the book contains really cool stuff involving >> infinitesimals (cool stuff is not the exact wording). Well the back>> of the book states that Robinson's work remains controversial even to>> this day. Well, why would it be controversial? I thought controversy>> was for the inexact sciences where mathematical proof is out of reach?>> Non-standard analysis itself is not subject to controversy. However, >> various claims about it's naturality as a formalisation of the intuitive >> concepts of analysis is. This is basicly a osophical (and partly >> pedagogical) controversy.>Non-standard analysis has a funny situation. As no, from a>technical point of view it is absolutely correct and rigorous. >Moreover, most analysts use it informally all the time. Then when>they are satisfied with their results, they recast them in>epsilon-delta form. They do? Any evidence for the idea that this is how most analystswork?>Robinson showed that this was unnecessary, but>they do it anyway. For one thing, the construction of non-standard>numbers is considered funny-business, involving logic, ultrafilters>and that sort of thing most analysts don't feel comfortable with.>It reminds me a little about the introduction of homology groups>replacing Betti numbers and torsion numbers in topology. This was>first done in print by Emmy Noether. A decade or so ago someone asked>Leopold Vietoris (who died at age 109 a year or so ago) about this and>he said that every knew there were homology groups, but it was not>thought appropriate to write papers using them. Once Noether broke>the taboo, everyone did it. Will this happen here? I don't know. >But the controversy is primarily pedagogical. Should we teach>calculus using them when their development is rather sophistica? >Well Dedekind cuts are fairly sophistica too and we don't worry>about omitting any discussion of the real line from elementary>calculus.>Consider the following two statements:>For every epsilon > 0, there is a delta such that |y - x| < delta>implies>|f(y) - f(x)| < epsilon.>For every infinitesimal delta, f(x+delta) - f(x) is infinitesimal. >An AE=> statment has been reduced to a simple A. All the rest of the>complication has been hidden in the construction of the === says probability 1 does not imply certainty>>I claim: When flipping a fair coin, an infinite sequence>> of heads is impossible, theoretically.>> You can claim that if you want. It's not so.>> Such a sequence violates the axioms of probability.>> Exactly which axiom is that?> If an outcome ( of a flip) if has non-zero probability it must>>occur in infinitely-many trials with probability 1.>> That's not an axiom in the probability theory I know.>>I am saying it would lead to a contradiction from the axioms...>> Exactly how does it lead to a contradiction>If you specify a number, say zero, then its decimal>expansion is what it is.>However, if you reverse the process and claim to>be able to generate binary expansions of all numbers>between zero and 1, by flipping a fair coin infinitely>many times, then you will fail to generate even_one_rational>number. In this way, one gets that the probability of picking>rational is zero and sees that the set of rationals has measure>zero.Huh? The key point here is your statement then you will fail to generate even_one_rational number - that's just a slightgeneralization of what I asked for a proof of, namely thatHHH... would lead to a contradiction from the axioms.But you don't give _any_ proof here whatever! Yousimply _assert_ this.What is the _proof_ of a contradiction from the axioms?(For that matter === Ullrich says probability 1 does not imply certainty>> Experiments involving infinitely many trials are>> useful in understanding real experiments with>> a large finite number of trials.>Maybe but surely not always.>There are limits. If various other people had written that I'd assume itwas a pun... never mind.>And don't>prentend to be a === says probability 1 does not imply certainty>If you specify a number, say zero, then its decimal>expansion is what it is.>>However, if you reverse the process and claim to>be able to generate binary expansions of all numbers>between zero and 1, by flipping a fair coin infinitely>many times, then you will fail to generate even_one_rational>number. In this way, one gets that the probability of picking>rational is zero and sees that the set of rationals has measure>zero.> Huh? The key point here is your statement then you will fail> to generate even_one_rational number - that's just a slight> generalization of what I asked for a proof of, namely that> HHH... would lead to a contradiction from the axioms.> But you don't give _any_ proof here whatever! You> simply _assert_ this.I think you mean to write whatsoever.> What is the _proof_ of a contradiction from the axioms?> (For that matter what _are_ the axioms?)You name the axioms.You give a refutation from the axioms, if you want.To me, its obvious that an infinite sequence of randomcoin flips cannot be === imply certainty> Although, a priori, any_finite_sequence of a given length is as likely> as any other, I claim: When flipping a fair coin, an infinite sequence> of heads is impossible, theoretically.Why is it any more impossible than any other specific infinite sequence?The probability of any given finite sequence of length N is 2^(-N). As N goesto infinity, this probability goes to zero, no matter what the sequence is. Aninfinite sequence of heads is no more or less likely than any other sequence.They all have zero probability in the limit. Obviously, if you could conductthe experiment, you'd get a sequence, which demonstrates that probability zerodoes not mean impossible. This is no different in principle from choosing arandom real number from a uniform distribution on [0,1].> Such a sequence, violates the axioms of probability.Which axiom does it violate?> If an event has non-zero probability it must occur in> infinitely-many trials with probability 1.So what? All this says is that the collection of sequences in which H doesnot occur infinitely often has probability zero. As above, this does not implyimpossible. *Every* infinite sequence genera in this way === does not imply certainty>> Suppose that you flip a coin infinitely many times. What is>> the probability that you get all heads? Zero. Does that mean>> that it's impossible to get all heads? No,...>Yes.>> you get _some_sequence, and any given sequence is just>> as likely as any other.>Although, a priori, any_finite_sequence of a given length is as likely>as any other, I claim: When flipping a fair coin, an infinite sequence>of heads is impossible, theoretically. Such a sequence, violates the>axioms of probability.EVERY sequence has zero probability. Do your infinite flip. Supposethe result is H T T H H H T H T T T H H ....Now calculate the probability that an infinite sequence of flips willjust happen to come up H T T H H H T H T T T H H followed by theparticular sequence you got in your experiment. The probability ofjust that much is 0.5^13, or about one in 8000. Flip 10 more times andit's closer to 1:8 million of getting the result that you observed.Did a miracle occur if this was your first 23 flips? === imply certainty> Experiments involving infinitely many trials are> useful in understanding real experiments with> a large finite number of trials.>>Maybe but surely not always.>>There are limits.>> If various other people had written that I'd assume it>> was a pun... never mind.>Good one; don't you think?!>>And don't prentend to be a istine.>> Huh???>Let's not pretend the study of infinite sequences>can directly be applied mathematics.The study of infinite sequences is important in a lot of appliedmathematics. No pretending necessary. - === certainty> What is the _proof_ of a contradiction from the axioms?> (For that matter what _are_ the axioms?)> You name the axioms.> When flipping a fair coin, an infinite sequence> of heads is impossible, theoretically. Such a sequence, violates the> axioms of probability.What theory does theoretically refer to? Which axioms does it violate? What is the mathematical definition of impossible? Pray === Unbelievable!http://mr-31238.mr.valuehost.co.uk/assets/Flash/ === worldIndeed it is =)Neocomplex> === Subject: Re: Unbelievable!> http://mr-31238.mr.valuehost.co.uk/assets/Flash/ psychic.swfWell... With 40, I decided to do: 40 + 4+0 = 44, === Re: Unbelievable!> http://mr-31238.mr.valuehost.co.uk/assets/Flash/psychic.swf> Well... With 40, I decided to do: 40 + 4+0 = 44, and the symbol> for 44 didn't appear...You need to substract 4+0 from === http://mr-31238.mr.valuehost.co.uk/assets/Flash/psychic.swf === field and F is the minimal Borel field containing it. H isthe collection of all sets in F which are well-approxima by G. Howcan one show that the union of any 2 sets in H is also === union proof>Say G is a field and F is the minimal Borel field containing it. H is>the collection of all sets in F which are well-approxima by G. How>can one show that the union of any 2 sets in H is also well>approxima by G.What does well-approxima === Well-approxima union proof>>Say G is a field and F is the minimal Borel field containing it. H is>>the collection of all sets in F which are well-approxima by G. How>>can one show that the union of any 2 sets in H is also well>>approxima by G.>What does well-approxima by G mean?This was terminology I made up in my reply to an earlier question ofAgapito on the rather garbled Subjectterminology. To make matters worse, Agapito isn't mentioning themeasures u and q, and outside the context of measures there's noway to make sense of this notion at all. What I said was| Say that a set A in F | is well-approxima by G if for any epsilon > 0 there is B in G with| u((B A) union (A B)) < epsilon and the same for q. Hint: Given A_1 and A_2 that are well-approxima by G, take B_1 andB_2 as in the definition but using epsilon/2.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia === union proof>Say G is a field and F is the minimal Borel field containing it. H is>the collection of all sets in F which are well-approxima by G. How>can one show that the union of any 2 sets in H is also well>approxima by G.>>What does well-approxima by G mean?>This was terminology I made up in my reply to an earlier question of>Agapito on the rather garbled Subject>terminology. To make matters worse, Agapito isn't mentioning the>measures u and q, and outside the context of measures there's no>way to make sense of this notion at all. Not that it's possible to figure out definitions from how the wordssound, but it _sounded_ like there must be a measure aroundsomewhere.>What I said was>| Say that a set A in F >| is well-approxima by G if for any epsilon > 0 there is B in G with>| u((B A) union (A B)) < epsilon and the same for q. >Hint: Given A_1 and A_2 that are well-approxima by G, take B_1 and>B_2 as in the definition but using epsilon/2.And then apply a little set theory...>Robert Israel israel@math.ubc.ca>Department of Mathematics http://www.math.ubc.ca/~israel >University of British Columbia >Vancouver, BC, Canada V6T === union proof>Say G is a field and F is the minimal Borel field containing it. H is>the collection of all sets in F which are well-approxima by G. How>can one show that the union of any 2 sets in H is also well>approxima by G.> What does well-approxima by G mean?Sorry I didn't spell this out. Let u(X) be a measure on F. Set A iswell approxima by G if for each epsilon>0, there exists a set B inG such that u(AB) + === simplest way to find unitary matrix U to transform A to upper problem:If I have a square matrix A, how to find a unitary matrix U such thatU^(-1)*A*U=T where T is an upper triangular matrix...?There are several methods to do this, Strang's text book gives one,which I think is quite cubersome. What is the simplest way to do this?If A is real symmetric, or complex Hermitian, then there is no problemto find such U, since the orthonormal eigenvectors are adequate.But what if A is [1 2 0; 0 1 0; -1 2 2]It has eigenvalue 1(with eigenvector [1, 0, 1]'), eigenvalue 2(witheigenvector [0 0 1]'), ... the orthonormal eigenvectors are notenough. What to do with these two === unitary matrix U to transform A to upper triangular matrix T?>If I have a square matrix A, how to find a unitary matrix U such that>U^(-1)*A*U=T where T is an upper triangular matrix...?>There are several methods to do this, Strang's text book gives one,>which I think is quite cubersome. What is the simplest way to do this?It's called reduction to Schur form. I don't think there's really asimpler way than Strang's.>But what if A is [1 2 0; > 0 1 0; > -1 2 2]>It has eigenvalue 1(with eigenvector [1, 0, 1]'), eigenvalue 2(with>eigenvector [0 0 1]'), ... the orthonormal eigenvectors are not>enough. What to do with these two eigenvectors?OK, first eigenvalue 1 and normalized eigenvector [ 1/sqrt(2) ] [ 0 ][ 1/sqrt(2) ]Use Gram-Schmidt, say, to complete this to a unitary matrix, e.g. [ 1/sqrt(2) 1/sqrt(2) 0 ]U1 = [ 0 0 1 ] [ 1/sqrt(2) -1/sqrt(2) 0 ] [ 1 -1 2 sqrt(2) ]Now A1 = U1^(-1) A U1 = [ 0 2 0 ] [ 0 0 1 ]Well, that was too easy because it's already upper triangular.But if that 0 in the second column wasn't there, what you'd donext would be to take an eigenvector of the lower right 2 x 2submatrix, giving you U2 of the form[ 1 0 0 ][ 0 x y ][ 0 z w ] where [x,z]' is that eigenvector, and A2 = U2^(-1) A1 U2 wouldbe upper triangular.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia === simplest way to find unitary matrix U to transform A to upper triangular matrix T?>But what if A is [1 2 0;> 0 1 0;> -1 2 2]>It has eigenvalue 1(with eigenvector [1, 0, 1]'), eigenvalue 2(with>eigenvector [0 0 1]'), ... the orthonormal eigenvectors are not>enough. What to do with these two eigenvectors?> OK, first eigenvalue 1 and normalized eigenvector> [ 1/sqrt(2) ]> [ 0 ]> [ 1/sqrt(2) ]> Use Gram-Schmidt, say, to complete this to a unitary matrix, e.g.> [ 1/sqrt(2) 1/sqrt(2) 0 ]> U1 = [ 0 0 1 ]> [ 1/sqrt(2) -1/sqrt(2) 0 ]> [ 1 -1 2 sqrt(2) ]> Now A1 = U1^(-1) A U1 = [ 0 2 0 ]> [ 0 0 1 ]Yeah, the professor also did this, he showed that instead Strang's step bystep method, he can do this by only one step: first write down [1 0 1]'unified to [1/sqrt(2), 0 , 1/sqrt(2)]'... then by G-S method, complete it tobe a unitary matrix, exactly as what you did.But I tried this on other matrix, it does not work. The professor failed toexplain why only one step(seems like sole guessing to us...) but thie ONEstep does not work elsewhere.> Well, that was too easy because it's already upper triangular.> But if that 0 in the second column wasn't there, what you'd do> next would be to take an eigenvector of the lower right 2 x 2> submatrix, giving you U2 of the form> [ 1 0 0 ]> [ 0 x y ]> [ 0 z w ]> where [x,z]' is that eigenvector, and A2 = U2^(-1) A1 U2 would> be upper triangular.So this example is too special, that's why it can be done in one step,instead of Strang's three steps? You said: But if that 0 in the secondcolumn wasn't there, ...Which zero? U1?> [ 1/sqrt(2) 1/sqrt(2) 0 ]> U1 = [ 0 0 1 ]> [ 1/sqrt(2) -1/sqrt(2) 0 ]But my professor gives U1= [ 1/sqrt(2) 0 1/sqrt(2)] [ 0 1 0 ] [ 1/sqrt(2) 0 -1/sqrt(2)]Which does not have ZERO in the second column, but it did the same thing...So can you explain more clearly when can I use the simplified GUESSING,when I need to use Strang's === darts than who?I played darts with four friends. Their indexes are i=1, ... ,4. Eachone of us threw his dart a different number of times N(i), i=1, ...,4. We could either reach a target or not, each time. Each one of usreached the target a different number of times m(i), i=1, ..., 4. Iwant to know which one of us is better, which one is the secondbetter, etc? I imagine that the answer is not just ordering theseamounts: m(i)/N(i).Is there a simple way to solve this? There might be several errorsinvolved in the measurement process. If some N(i) are too small themeasurements may not be significant. Should each one of us throw atleast a given nummer of times T for the comparison to be possible? Andhow does the error in the comparisons behave with regard to T, and howis T rela to the number of friends? (And if no T is needed, whatshould I do?)I'm rather frustra by this simple problem. Searching the net, (andif I understood well), I found that ANOVA might be able to tell me ifa comparison is possible, but not the results of such a comparison.And I don't know what else to === indexes are i=1, ... ,4. Each> one of us threw his dart a different number of times N(i), i=1, ...,> 4. We could either reach a target or not, each time. Each one of us> reached the target a different number of times m(i), i=1, ..., 4. I> want to know which one of us is better, which one is the second> better, etc? I imagine that the answer is not just ordering these> amounts: m(i)/N(i).Why not? Isn't that similar to how batting averages are compu?> Is there a simple way to solve this? There might be several errors> involved in the measurement process. If some N(i) are too small the> measurements may not be significant. Should each one of us throw at> least a given nummer of times T for the comparison to be possible? And> how does the error in the comparisons behave with regard to T, and how> is T rela to the number of friends? (And if no T is needed, what> should I do?)Yes N(i) shouldn't be small.T can be any number you agree === played darts with four friends. Their indexes are i=1, ... ,4. EachI presume that should be three friends?> one of us threw his dart a different number of times N(i), i=1, ...,> 4. We could either reach a target or not, each time. Each one of us> reached the target a different number of times m(i), i=1, ..., 4. I> want to know which one of us is better, which one is the second> better, etc? I imagine that the answer is not just ordering these> amounts: m(i)/N(i).> Is there a simple way to solve this? There might be several errors> involved in the measurement process. If some N(i) are too small the> measurements may not be significant. Should each one of us throw at> least a given nummer of times T for the comparison to be possible? And> how does the error in the comparisons behave with regard to T, and how> is T rela to the number of friends? (And if no T is needed, what> should I do?)> I'm rather frustra by this simple problem. Searching the net, (and> if I understood well), I found that ANOVA might be able to tell me if> a comparison is possible, but not the results of such a comparison.Search for multiple comparisons or Tukey or even Bonferroni.This will allow you to do the comparisons you want. (Or tell youthat the difference is too small to tell === who?> I played darts with four friends. Their indexes are i=1, ... ,4. Each> one of us threw his dart a different number of times N(i), i=1, ...,> 4. We could either reach a target or not, each time. Each one of us> reached the target a different number of times m(i), i=1, ..., 4. I> want to know which one of us is better, which one is the second> better, etc? I imagine that the answer is not just ordering these> amounts: m(i)/N(i).Well, it is, sort of. In the language of statistics, whatyou want to compare is the (unknown) probability of hitting the target for each person, and what you have is an estimatorm(i)/N(i) for each person.A statistician will tell you that m(i)/N(i) is an unbiasedestimator: Roughly, it's as likely to be high as to be low.More precisely, the expectation value is the value p(i) youare looking for.> Is there a simple way to solve this? There might be several errors> involved in the measurement process. If some N(i) are too small the> measurements may not be significant. Should each one of us throw at> least a given nummer of times T for the comparison to be possible? And> how does the error in the comparisons behave with regard to T, and how> is T rela to the number of friends? (And if no T is needed, what> should I do?)What I think you're trying to do is to put confidence limits on each measurement then decide which differences aresignificant. For instance:Player 1: 0.6 +- 0.2Player 2: 0.55 +- 0.1I think your intuition would tell you those aren't significantlydifferent. The exact test you want you'll find under hypothesistesting, difference between two means. It can be hard toread such stuff if you're not used to the language ofstatistics. I don't claim to be a statistician, just a guywho's had to read and use a lot of that stuff.Here's just enough probability theory to give you an informalapproach to your problem. 1. m/N is random variable. For a player with a givenp of success, m/N is not going to be the same fromgame to game, but it will be drawn from a specificdistribution.2. Specifically, m is a binomial random variable. Theprobability of scoring exactly m hits in N throws isC(N,m)*p^m*(1-p)^(N-m) where C(N,m) is the combinationfunction (number of ways of choosing m things from amongN objects) = N!/[m!(N-m)!] and ^ means raised to the power of.3. The mean of the binomial distribution is Np, so themean of m/N is p.4. The variance of the binomial distribution is Np(1-p).That means that the variance of m/N is p(1-p)/N or thatthe standard deviation is sqrt(p(1-p))/sqrt(N).So if you want to write the +- error bars for eachmeasurement, use your estimate of p = m/N and calculatesqrt(p(1-p))/sqrt(N). Multiply it by 2 and you've got2-sigma limits which I believe 95% confidence limits.All conclusions from such stuff are probabilistic.If two scores with their error bars overlap, you can'tsay with X% confidence (95% for my choice) that they'resignificantly different. - === played darts with four friends. Their indexes are i=1, ... ,4. Each> one of us threw his dart a different number of times N(i), i=1, ...,> 4. We could either reach a target or not, each time. Each one of us> reached the target a different number of times m(i), i=1, ..., 4. I> want to know which one of us is better, which one is the second> better, etc? I imagine that the answer is not just ordering these> amounts: m(i)/N(i).I tend to agree with you. If I miss the target by 1 inch and you miss it bytwo inches consistantly - I would be considered the better dart playeralthough both of our m(i)s are 0. The real question would be if you aretwice as bad as I am (i.e. linear).One approach I would use is to use a Gaussian (standard) distribution withthe mean at the bullseye. The difficulty would be in determining thevarience. I would tend towards having the outside of the double ring of theboard be at the 3rd standard deviation.Let G(x) be the Gaussian formula centered at x=0 with the third SD at(x=distance determined above). For each throw distance d from the bullseyeweigh it === than who?> I played darts with four friends. Their indexes are only three friends,unless I am a friend of mine.I found this interesting free software:http://www.mrs.umn.edu/~sungurea/statlets/free/ WebStat.html--Analise->Sample Size Determination -> Comparison of ProportionsApparently it tells you how big the sample size has to be for thecomparisons to be done with a given error. (It compares only twoestimators.)Before your suggestions, I was able to find some multiplecomparisons methods, like the ones one of you mentioned. They seemquite advanced for me, but I will look for more info using the newsuggestions.Multiple Comparisonshttp://www.itl.nist.gov/div898/handbook/prc/ section4/prc474.htmhttp://www.itl.nist.gov/div898/handbook/prc /section4/prc47.htmhttp://www.itl.nist.gov/div898/handbook/prc work to do understandingthis stuff.There is a gaussian probability associa to throwing darts, I readabout it once. (But in this case, I wan to consider the event hitthe target only, so the variable would be binomial, I guess.)I will also consider the confidence intervals solutions, but it may bethe case that there are troubles with it. (Some texts I found seem toimply that to compare my friends and myself, a pair at a time, is nota good solution. Some references said this is done in scientificpapeand it's a mistake sometimes.) Until now, I === 5887x4 - 44573x3 + 219501x2 + 3219348x + 8511761>7-cyclic Galois groups. Well, one of the 35th degree resolvents as >determined by my home-brewed software is (if you'd really like to know)>x35 - 1044x33 + 841x32 + 497031x31 - 731670x30 - 144163379x29 + 81619783x28 >+ 28845748304x27 - 63809477258x26 - 4294050065448x25 + 9652013429526x24 >+ 501700755980845x23 - 1068836835858258x22 - 47728538812128418x21>+ 95317589771012373x20 + 3772588089240471043x19 - 7514288912209449175x18>- 249346071676979434656x17 + 549166864851988901349x16>+ 13734829103580245950476x15 - 36370488098612855957537x14 >- 622461008897883870980409x13 + 2048419667363957805659845x12 >+ 22632693775044165407845938x11 - 91647017502583306794280776x10 >- 635475108592085078471165068x9 + 3073318036732526406455843305x8 >+ 13077954305533155702758092173x7 - 73251959811253883248449646829x6 >- 183012946486207601497540283941x5 + 1159739627804541364505229077322x4 >+ 1535341586989039339304749450037x3 - 10846819133957260138036730698342x2 >- 5778715680179941767528762943506x + 45149231556380408446556072691041>My implementation of Davidson's EIFA (EIFA = Extremely Inefficient >Factoring Algorithm) produces one 7th degree polynomial factor which is the >subject line of this post:>p = x7 + 29x6 + 29x5 - 5887x4 - 44573x3 + 219501x2 + 3219348x + 8511761 Maple finds your degree-35 polynomial to be irreducible.But if we add 200000000*x^28, changing the coefficient to281619783, then it finds your degree-7 factor and anirreducible factor of degree 28.>Second question: what IS the Galois group of the septic? Or of the original >octic? The septic has cyclic galois group, order 7.You can express the roots of the septic in terms of the primitive 29-th roots of unity(acually as polynomials in w29 + w29^12 + w29^17 + w29^28where w29^29 = 1 and w29 <> 1, or in terms of cos(2*Pi/29)).The original octic has galois group of order 56.>I haven't yet obtained software to compute this, but am looking into the >matter. Does Maple do this? Should I just bite the bullet and purchase it? If r8 is a root of the original octic, and w29is a primitive 29-th root of unity, then Maple (after a few hours)gives eight linear factors over Q(r8, w29)(one being x - r8). One of the other seven roots is below.To get the rest, replace w29 by a power of w29.with(numtheory):f := x^8 - x^7 + 29*x^2 + 29;alias(r8 = RootOf(f, x));w29poly := cyclotomic(29, u);alias(w29 = RootOf(w29poly, u));r8new := ( 483981-139229*w29^19-139229*w29^10+279908*w29^22+102689*w29^21 +102689*w29^20+271005*w29^23+279908*w29^26+1286295*w29^27+ 1286295*w29^24+163038*w29^18+163038*w29^16+271005*w29^15+ 271005*w29^14+163038*w29^13+163038*w29^11+102689*w29^9+102689* w29^8+279908*w29^7+271005*w29^6+1286295*w29^5-139229*w29^4+ 279908*w29^3+1286295*w29^2-139229*w29^25+ (19759+41930*w29^19+41930*w29^10+3406*w29^22+57268*w29^21+ 57268*w29^20+25471*w29^23+3406*w29^26-50868*w29^27-50868*w29^ 24+61106*w29^18+61106*w29^16+25471*w29^15+25471*w29^14+61106* w29^13+61106*w29^11+57268*w29^9+57268*w29^8+3406*w29^7+25471* w29^6-50868*w29^5+41930*w29^4+3406*w29^3-50868*w29^2+41930*w29 ^25)*r8^4+(41863+44452*w29^19+44452*w29^10+46831*w29^22-27344* w29^21-27344*w29^20+53182*w29^23+46831*w29^26+118023*w29^27+ 118023*w29^24+57897*w29^18+57897*w29^16+53182*w29^15+53182*w29 ^14+57897*w29^13+57897*w29^11-27344*w29^9-27344*w29^8+46831* w29^7+53182*w29^6+118023*w29^5+44452*w29^4+46831*w29^3+118023* w29^2+44452*w29^25)*r8^3+(26445-70073*w29^19-70073*w29^10- 22930*w29^22+64254*w29^21+64254*w29^20-1859*w29^23-22930*w29^ 26+177365*w29^27+177365*w29^24+38358*w29^18+38358*w29^16-1859* w29^15-1859*w29^14+38358*w29^13+38358*w29^11+64254*w29^9+64254 *w29^8-22930*w29^7-1859*w29^6+177365*w29^5-70073*w29^4-22930* w29^3+177365*w29^2-70073*w29^25)*r8^2+(303731+149318*w29^19+ 149318*w29^10+903854*w29^22+683040*w29^21+683040*w29^20+808341 *w29^23+903854*w29^26+425162*w29^27+425162*w29^24+580563*w29^ 18+580563*w29^16+808341*w29^15+808341*w29^14+580563*w29^13+ 580563*w29^11+683040*w29^9+683040*w29^8+903854*w29^7+808341* w29^6+425162*w29^5+149318*w29^4+903854*w29^3+425162*w29^2+ 149318*w29^25)*r8+(18746+1028*w29^19+1028*w29^10+32149*w29^22+ 20968*w29^21+20968*w29^20+33299*w29^23+32149*w29^26+18448*w29^ 27+18448*w29^24+25330*w29^18+25330*w29^16+33299*w29^15+33299* w29^14+25330*w29^13+25330*w29^11+20968*w29^9+20968*w29^8+32149 *w29^7+33299*w29^6+18448*w29^5+1028*w29^4+32149*w29^3+18448* w29^2+1028*w29^25)*r8^7+(-5792-6935*w29^19-6935*w29^10-19537* w29^22-15527*w29^21-15527*w29^20-21658*w29^23-19537*w29^26+ 41219*w29^27+41219*w29^24-18106*w29^18-18106*w29^16-21658*w29^ 15-21658*w29^14-18106*w29^13-18106*w29^11-15527*w29^9-15527* w29^8-19537*w29^7-21658*w29^6+41219*w29^5-6935*w29^4-19537*w29 ^3+41219*w29^2-6935*w29^25)*r8^6+(-54809-43894*w29^19-43894* w29^10-56228*w29^22-49365*w29^21-49365*w29^20-73611*w29^23- 56228*w29^26-102659*w29^27-102659*w29^24-57906*w29^18-57906* w29^16-73611*w29^15-73611*w29^14-57906*w29^13-57906*w29^11- 49365*w29^9-49365*w29^8-56228*w29^7-73611*w29^6-102659*w29^5- 43894*w29^4-56228*w29^3-102659*w29^2-43894*w29^25)*r8^5 )/(29 * 49109);evala(subs(x = r8new, f)); # Zero-- Spammers: I don't want a small digital camera to post photos of a large, lowweight, penis on a re-financed Nigerian domain site. Peter-Lawrence.Montgomery@cwi.nl Home: San Rafael, California === was struck by the title of your post. You can't foolme by dropping a hyphen.My coterie of omnicopulatant dullards was supposed to bekept secret until they could form their own fraternity.(Still waiting for the results of an experimental === ing idiotshttp://tinyurl.com/j8lf> I was struck by the title of your post. You can't fool> me by dropping a hyphen.> My coterie of omnicopulatant dullards was supposed to be> kept secret until they could form their own fraternity.> (Still waiting for the results of an experimental cross-cloning> of === gravityHal just replied (another e-mail program) that eq. 39 of his Foundations of Physics paper has the metric he alludes to below.I do not have that paper ready at hand, but some immediate remarks are obvious.OK just inIts online right? Whats the URL?It's on our website www.earthtech.orgAlso attached here as a pdf file.Proceeding below without looking at Hal's eq. 39 - will look at it later to compare.First Einstein's theory in the non-rotating SSS charged vacuum metric. The only difference is that(1 - rs/r) is replaced by (1 - rs/r + (rq/r)^2)In the usual phenomenological theoryrs = 2G*m/c^2rq^2 = G*e^2/c^4because e^2 and G*M^2 have the same dimensions.Note that's spacetime stiffness factor G*/c^4 multiplying e^2.In the current theory G* >> G(Newton) at the micro-scale L* = Lp^2/3(c/Ho)^1/3 ~ 1 fermi where Ho is the cosmological Hubble parameter and Lp^2 is the quantum gravity exchange rate between IT geometrodynamics and BIT in the holographic universe.In Newtonian terms the effective potential is-G*m^2/r + e^2/rTo balance the attractive gravity against the self charge repulsion requires the Blackett equation (actually observed in astronomy e.g. Sirag and also Wesson papers)G*m^2 = e^2In my new exotic vacuum theory the source geon rest mass is induced by zero point energy dark matter core according to the formulam = e^2|/zpf|^1/2ThereforeG*e^4|/zpf| = e^2G* = (|/zpf|e^2)^-1Note again a characteristic non-perturbative BCS superconducting singular limit (Sir Michael Berry) non-analytic dependence of the zero point energy induced Sakharov emergent gravity on the vanishing the electromagnetic coupling and the dark matter exotic vacuum core.In Einstein's theory, thenThe effective image size lscattering of the spatially-extended electron Bohm hidden variable (Wheeler's IT) in the micro-quantum Heisenberg scattering microscope wherer ~ h/pp ~ momentum transfer in the scattering imaging processIt is obvious intuitively (check algebra later)rs ~ 2G*m/c^2 = 2|/zpf|^-1/2rq^2 ~ |/zpf|^-1a more precise analysis later will have the dimensionless self-energy factor in eq. (1.8) ofhttp://qedcorp.com/APS/Vigier4.pdfTherefore in Einstein's theory what we have islscattering = [1 - 2(|/zpf|^-1/2p/h) + (|/zpf|^-1/2p/h)^2]^1/2(e/mc^2)where e/mc^2 = |/zpf|^-1/2 ~ 1 fermiz = |/zpf|^-1/2p/hThis toy model event horizon polynomial is1 - 2z + z^2 = 0z = (2 +-[4 - 4]^1/2)/2 = 1Puthoff has not escaped the problem since his model giveslscattering = e^(2z + z^2)/2(e^2/mc)^2 ---> infinity as p ---> infinity!There is the issue of the relation between r(isotropic) and r(curvature.)I think Ibison showedr(curvature) = r(isotropic)K(e(isotropic)^1/2That must be looked at more closely, but it will not change the basic result since the exponential factor will always be there.Puthoff's theory without an event horizon predicts the electron shouldgrow larger as the imaging probe scale gets smaller. The opposite isobserved.Ho hum. Nonsense as usual. The PV approach to the Reissner-Nordstrom Hal PuthoffMaybe so. Where is your proof? Show us the math. What is your PV metric explicitly with charge? You have no rotation as yet? Is that true? Where is your paper with the metric - not just your old paper with the action with the EM field. What is the metric form you use?Will === zero point energy induced gravitylscattering = e^(2z + z^2)/2(e^2/mc)^2should belscattering = e^(2z + z^2)/2(e^2/mc^2)